Computational Fluid and Solid Mechanics

COMPUTATIONAL FLUID AND SOLID MECHANICS

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COMPUTATIONAL FLUID AND SOLID MECHANICS Proceedings First MIT Conference on Computational Fluid and Solid Mechanics June 12-15,2001

Editor: K.J. Bathe Massachusetts Institute of Technology, Cambridge, MA, USA

VOLUME 1

2001 ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

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Preface

Mathematical modeling and numerical solution is today firmly established in science and engineering. Research conducted in almost all branches of scientific investigations and the design of systems in practically all disciplines of engineering can not be pursued effectively without, frequently, intensive analysis based on numerical computations. The world we live in has been classified by the human mind, for descriptive and analysis purposes, to consist of fluids and solids, continua and molecules; and the analyses of fluids and solids at the continuum and molecular scales have traditionally been pursued separately. Fundamentally, however, there are only molecules and particles for any material that interact on the microscopic and macroscopic scales. Therefore, to unify the analysis of physical systems and to reach a deeper understanding of the behavior of nature in scientific investigations, and of the behavior of designs in engineering endeavors, a new level of analysis is necessary. This new level of mathematical modeling and numerical solution does not merely involve the analysis of a single medium but must encompass the solution of multi-physics problems involving fluids, solids, and their interactions, involving multi-scale phenomena from the molecular to the macroscopic scales, and must include uncertainties in the given data and the solution results. Nature does not distinguish between fluids and solids and does not ever exactly repeat itself. This new level of analysis must also include, in engineering, the effective optimization of systems, and the modeling and analysis of complete life spans of engineering products, from design to fabrication, to possibly multiple repairs, to end of service. The objective of the M.I.T. Conferences ^ on Computational Fluid and Solid Mechanics is to bring together researchers and practitioners of mathematical modeling and numerical solution in order to focus on the current state of analysis of fluids, soUds, and multi-physics phenomena and

^ A series of Conferences is planned.

to lead towards the new level of mathematical modeling and numerical solution that we envisage. However, there is also a most valuable related objective indeed a "mission" - for the M.I.T. Conferences. When contemplating the future and carving a vision thereof, two needs stand clearly out. The first is the need to foster young researchers in computational mechanics, because they will revitaUze the field with new ideas and increased energy. The second need is to bring Industry and Academia together for a greater synthesis of efforts in research and developments. This mission expressed in 'To bring together Industry and Academia and To nurture the next generation in computational mechanics'' is of great importance in order to reach, already in the near future, the new level of mathematical modeling and numerical solution, and in order to provide an exciting research environment for the next generation in computational mechanics. We are very grateful for the support of the sponsors of the Conference, for providing the financial and intellectual support to attract speakers and bring together Industry and Academia. In the spirit of helping young researchers, fellowships have been awarded to about one hundred young researchers for travel, lodging and Conference expenses, and in addition. Conference fees have been waived for all students. The papers presented at the Conference and published in this book represent, in various areas, the state-of-the-art in the field. The papers have been largely attracted by the session organizers. We are very grateful for their efforts. Finally, we would like to thank Jean-Frangois Hiller, a student at M.I.T, for his help with the Conference, and also Elsevier Science, in particular James Milne, for the efforts and help provided to publish this book in excellent format and in due time for the Conference. K.J. BATHE, M.I.T.

Session Organizers

We would like to thank the Session Organizers for their help with the Conference. G. Astfalk, Hewlett-Packard Company, U.S.A. N. Bellomo, Politecnico di Torino, Italy Z. Bittnar, Prague Technical University, Czech Republic D. Boffi, University of Pavia, Italy S. Borgersen, SciMed, U.S.A. M. Borri, Politecnico di Milano, Italy M.A. Bradford, University of New South Wales, Australia M.L. Bucalem, University of Sao Paulo, Brazil J. Bull, The University of Newcastle upon Tyne, U.K. S.W. Chae, Korea University, South Korea D. Chapelle, INRIA, France C.N. Chen, National Cheng Kung University, Taiwan G. Cheng, Dalian University of Technology, PR. China H.Y. Choi, Hong-Ik University, South Korea K. Christensen, Hewlett-Packard Company, U.S.A. M.A. Christon, Sandia National Laboratories, U.S.A. R. Cosner, The Boeing Company, U.S.A. S. De, Massachusetts Institute of Technology., U.S.A. Y.C. Deng, General Motors, U.S.A. R.A. Dietrich, GKSS Forschungszentrum, Germany J. Dolbow, Duke University, U.S.A. E.H. Dowell, Duke University, U.S.A. R. Dreisbach, The Boeing Company, U.S.A. E.N. Dvorkin, SIDERCA, Argentina N. El-Abbasi, Massachusetts Institute of Technology, U.S.A. C. Felippa, University of Colorado, Boulder, U.S.A. D. Ferguson, The Boeing Company, U.S.A. D. M. Frangopol, University of Colorado, Boulder, U.S.A. L. Gastaldi, University of Pavia, Italy P. Gaudenzi, University of Rome, Italy A. Ghoniem, Massachusetts Institute of Technology, U.S.A. R. Glowinski, University of Houston, U.S.A. P. Gresho, Lawrence Livermore National Laboratory, U.S.A. N. Hadjiconstantinou, Massachusetts Institute of Technology, U.S.A. M. Hafez, University of California, Davis, U.S.A. K. Hall, Duke University, U.S.A. 0. Hassan, University of Wales, U.K. A. Ibrahimbegovic, ENS-Cachan, France S. Idelsohn, INTEC, Argentina A. Jameson, Stanford University, U.S.A. 1. Janajreh, Michelin, U.S.A.

R.D. Kamm, Massachusetts Institute of Technology, U.S.A. S. Key, Sandia National Laboratories, U.S.A. W. Kirchhoff, Department of Energy, U.S.A. W.B. Kratzig, Ruhr-Universitat Bochum, Germany A. Krimotat, SC Solutions, Inc., U.S.A. C.S. Krishnamoorthy, Indian Institute of Technology, Madras, India (deceased) Y. Kuznetsov, University of Houston, U.S.A. L. Martinelli, Princeton University, U.S.A. H. Matthies, Technical University of Braunschweig, Germany S.A. Meguid, University of Toronto, Canada K. Meintjes, General Motors, U.S.A. C. Meyer, Columbia University, U.S.A. R. Ohayon, CNAM, France M. Papadrakakis, National Technical University of Athens, Greece K.C. Park, University of Colorado, Boulder, U.S.A. J. Periaux, Dassault Aviation, France O. Pironneau, Universite Pierre et Marie Curie, France E. Rank, Technical University of Munich, Germany A. Rezgui, Michelin, France C.Y Sa, General Motors, U.S.A. G. Schueller, University of Innsbruck, Austria T. Siegmund, Purdue University, U.S.A. J. Sladek, Slovak Academy of Sciences, Slovak Republic S. Sloan, University of Newcastle, Australia G. Steven, University of Sydney, Australia R. Sun, DaimlerChrysler, U.S.A. S. Sutton, Lawrence Livermore National Laboratory, U.S.A. B. Szabo, Washington University, St. Louis, U.S.A. J. Tedesco, University of Florida, U.S.A. T. Tezduyar, Rice University, U.S.A. B.H.V. Topping, Heriot-Watt University, U.K. F.J. Ulm, Massachusetts Institute of Technology, U.S.A. J.M. Vacherand, Michelin, France L. Wang, University of Hong Kong, Hong Kong X. Wang, Polytechnic University of New York, U.S.A. N. Weatherill, University of Wales, U.K. J. White, Massachusetts Institute of Technology, U.S.A. P. Wriggers, University of Hannover, Germany S. Xu, General Motors, U.S.A. T. Zohdi, University of Hannover, Germany

Fellowship Awardees

M. Al-Dojayli, University of Toronto, Canada B.N. Alemdar, Georgia Institute of Technology, U.S.A. M.A. Alves, Universidade do Porto, Portugal R. Angst, Technical University of Berlin, Germany D. Antoniak, Wroclaw University of Technology, Poland S. J. Antony, University of Surrey, U.K. A. Badeau, West Virginia University, U.S.A. W. Bao, The National University of Singapore, Singapore M. Bathe, Massachusetts Institute of Technology, U.S.A. A.C. Bauer, University of New York, Buffalo, U.S.A. C. Bisagni, Politecnico di Milano, Italy S. Butkewitsch, Federal University of Uberlandia, Brazil S. Cen, Tsinghua University, China G. Chaidron, CNAM, France M. Council, Chalmers University of Technology, Sweden A. Czekanski, University of Toronto, Canada C. E. Dalhuysen, Council for Scientific and Industrial Research, South Africa D. Dall'Acqua, Noetic Engineering Inc., Canada S. De, Massachusetts Institute of Technology, U.S.A. D. Demarco, SIDERCA, Argentina J. Dolbow, Duke University, U.S.A. J.E. Drews, Technische Universitat Braunschweig, Germany J.L. Drury, University of Michigan, U.S.A. C.A. Duarte, Altair Engineering, U.S.A. F. Dufour, CSIRO Exploration and Mining, Australia A. Ferent, INRIA, France M.A. Fernandez, INRIA, France Y. Fragakis, National Technical University of Athens, Greece A. Frangi, PoUtecnico di Milano, Italy T. Fujisawa, University of Tokyo, Japan J.R. Fernandez Garcia, Universidade de Santiago de Compostela, Spain J.F. Gerbeau, INRIA, France M. Gliick, Friedrich-Alexander University, Erlangen, Germany C. Gonzalez, Politecnica de Madrid, Spain K. Goto, University of Tokyo, Japan S. Govender, University of Natal, South Africa T. Gratsch, University of Kassel, Germany B. Gu, Massachusetts Institute of Technology, U.S.A. Y. T. Gu, National University of Singapore, Singapore S. Gupta, Indian Institute of Science, Bangalore, India M. Handrik, University of Zilina, Slovakia

L. Haubelt, Rice University, U.S.A. V. Havu, Helsinki University of Technology, Finland N. Impollonia, University of Messina, Italy R. lozzi. University of Rome, "La Sapienza", Italy H. Karaouni, Ecole Polytechnique, France R. Keck, University of Kaiserslautern, Germany C.W. Keierleber, University of Nebraska, Lincoln, U.S.A. K. Kolanek, Polish Academy of Sciences, Poland L. Ktibler, University of Erlangen-Niimberg, Erlangen, Germany D. Kuzmin, University of Dortmund, Germany N.D. Lagaros, National Technical University of Athens, Greece R. Garcia Lage, Instituto de Engenharia Mecanica, Portugal P.D. Ledger, University of Swansea, Wales, U.K. J. Li, Courant Institute, New York, U.S.A. J. Li, Massachusetts Institute of Technology, U.S.A. G. Limbert, University of Southampton, U.K. K. Liu, Polytechnic University of New York, U.S.A. M.B. Liu, National University of Singapore, Singapore J. Long, University of New York, Buffalo, U.S.A. I. Lubowiecka, Technical University of Gdansk, Poland A.A. Mailybaev, Moscow State Lomonosov University, Russia M. Malinen, Helsinki University of Technology, Finland E.A. Malsch, Columbia University, U.S.A. Y. Marzouk, Massachusetts Institute of Technology, U.S.A. M. Meyer, Technische Universitat Braunschweig, Germany B. Miller, Rzeszow University of Technology, Poland D.P. Mok, University of Stuttgart, Germany G. Morgenthal, University of Cambridge, U.K. M. Moubachir, Laboratoire Central des Fonts et Chaussees, France S.K. Nadarajah, Stanford University, U.S.A. J. Nemecek, Czech Technical University, Prague, Czech Republic T.S. Ng, Imperial College, U.K. N. Nuno, Universita di Parma, Italy M. Palacz, Polish Academy of Sciences, Poland H. Pan, Nanyang Technological University, Singapore G. Pedro, University of Victoria, Canada X. Peng, Northwestern University, U.S.A. R.C. Penmetsa, Wright State University, U.S.A. R. Premkumar, Indian Institute of Technology, Madras, India

Fellowship Awardees C. Prud'homme, Massachusetts Institute of Technology, U.S.A. K. Roe, Purdue University, U.S.A. S. Rugonyi, Massachusetts Institute of Technology, U.S.A. M.L. Munoz Ruiz, Universidad de Malaga, Spain N. Ruse, University of Stuttgart, Germany S. Sarkar, Indian Institute of Science, Bangalore, India C.A. Schenk, University of Innsbruck, Austria S. Shankaran, Stanford University, U.S.A. D. Slinchenko, University of Natal, South Africa D.O. Snyder, Utah State University, U.S.A. K.A. S0rensen, University of Swansea, Wales, U.K. A. Takahashi, University of Tokyo, Japan S. Ubal, Universidad Nacional del Litoral, Argentina

U.V. Unnithan, Indian Institute of Technology, Chennai, India F. Valentin, National Laboratory of Brazil for Scientific Computing, Brazil R. Vodicka, Technical University of Kosice, Slovakia V.M. Wasekar, University of Cincinnati, U.S.A. S. Wijesinghe, Massachusetts Institute of Technology, U.S.A. M.W. Wilson, Georgia Institute of Technology, U.S.A. W. Witkowski, Technical University of Gdansk, Poland A.M. Yommi, Universidad Nacional del Litoral, Santa Fe, Argentina Y. Zhang, Dalian University of Technology, China K. Zhao, General Motors Corp., U.S.A.

Sponsors

The following organizations are gratefully acknowledged for their generous sponsorship of the Conference:

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Contents Volume 1

Preface

•

v

Session Organizers

vi

Fellowship Awardees

vii

Sponsors

ix

Plenary Papers Alum, N., Ye, W., Ramaswamy, D., Wang, X., White, J., Efficient simulation techniques for complicated micromachined devices

2

Brezzi, R, Subgrid scales, augmented problems, and stabilizations

8

Dreisbach, R.L., Cosner, R.R., Trends in the design analysis of aerospace vehicles

11

Ingham, T.J., Issues in the seismic analysis of bridges

16

Lions, J.L., Virtual control algorithms

20

Makinouchi, A., Teodosiu, C, Numerical methods for prediction and evaluation of geometrical defects in sheet metal forming

21

McQueen, DM., Peskin, C.S., Zhu, L., The Immersed Boundary Method for incompressible fluid-structure interaction

26

Ottolini, R.M., Rohde, S.M., GMs journey to math: the virtual vehicle

31

Solids & Structures Antony, SJ., Ghadiri, M., Shear resistance of granular media containing large inclusions: DEM simulations

36

Araya, R., Le Tallec, R, Hierarchical a posteriori error estimates for heterogeneous incompressible elasticity

39

Augusti, G., Mariano, P.M., Stazi, F.L., Localization phenomena in randomly microcracked bodies

43

Austrell, P.-E., Olsson, A.K., Jonsson, M., A method to analyse the nonlinear dynamic behaviour of rubber components using standard FE codes

47

Ba§ar, Y., Hanskotter, U., Kintzel, O., Schwab, C, Simulation of large deformations in shell structures by the p-version of the finite element method

50

Bardenhagen, S.G., Byutner, O., Bedrov, D., Smith, G.D., Simulation of frictional contact in three-dimensions using the Material Point Method

54

xii

Contents Volume 1

Bauchau, O.A., Bottasso, C.L., On the modeling of shells in multibody dynamics

58

Bay lot, J.T., Papados, P.P., Fragment impact pattern effect on momentum transferred to concrete targets

61

Becache, E., Joly, P., Scarella, G., A fictitious domain method for unilateral contact problems in non-destructive testing

65

Belforte, G., Franco, W., Sorli, M., Time-frequency pneumatic transmission line analysis

68

Bohm, R, Duda, A., Wille, R., On some relevant technical aspects of tire modelling in general

72

Borri, M., Bottasso, C.L., Trainelli, L, An index reduction method in non-holonomic system dynamics

74

Boucard, PA., Application of the LATIN method to the calculation of response surfaces

78

Brunet, M., Morestin, R, Walter, H., A unified failure approach for sheet-metals formability analysis

82

Bull, J. W., Underground explosions: their effect on runway fatigue life and how to mitigate their effects

85

Cacciola, P., Impollonia, N., Muscolino, G., Stochastic seismic analysis of R-FBI isolation system

88

Carter, J.P, Wang, C.X., Geometric softening in geotechnical problems

91

Cen, S., Long, Y., Yao, Z., A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates

95

Chakraborty, S., Brown, D.A., Simulating static and dynamic lateral load testing of bridge foundations using nonlinear finite element models . .

99

Chapelle, D., Rerent, A., Asymptotic analysis of the coupled model shells-3D solids

104

Chapelle, D., Oliveira, D.L., Bucalem, M.L., Some experiments with the MITC9 element for Naghdis shell model

107

Chen, X., Hisada, T, Frictional contact analysis of articular surfaces

HI

Choi, H.Y., Lee, S.H., Lee, LH., Haug, E., Finite element modeling of human head-neck complex for crashworthiness simulation

114

Chun, B.K., Jinn, J.T., Lee, J.K., A constitutive model associated with permanent softening under multiple bend-unbending cycles in sheet metal forming and springback analysis

120

Crouch, R.S., Remandez-Vega, J., Non-linear wave propagation in softening media through use of the scaled boundary finite element method . . . .

125

Czekanski, A., Meguid, S.A., Time integration for dynamic contact problems: generalized-of scheme

128

Dai, L., Semi-analytical solution to a mechanical system with friction

132

Davi, G., Milazzo, A., A novel displacement variational boundary formulation

134

David, S.A., Rosdrio, J.M., Investigation about nonlinearities in a robot with elastic members

137

Contents Volume 1

xiii

De, S., Kim, /., Srinivasan, M.A., Virtual surgery simulation using a collocation-based method of finite spheres

140

Deeks,AJ.,WollJ.R, Efficient analysis of stress singularities using the scaled boundary finite-element method

142

Djoudi, M.S., Bahai, K, Relocation of natural frequencies using physical parameter modifications

146

Duddeck, F.M.E., Fourier transformed boundary integral equations for transient problems of elasticity and thermo-elasticity

150

Dufour, E, Moresi, L., Muhlhaus, H., A fluid-like formulation for viscoelastic geological modeling stabilized for the elastic limit

153

Dvorkin, E.N., Demarco, D., An Eulerian formulation for modehng stationary finite strain elasto-plastic metal forming processes

156

Dvorkin, E.N., Toscano, R.G., Effects of internal/external pressure on the global buckling of pipelines

159

El-AbbasU N., Bathe, K.J., On a new segment-to-segment contact algorithm

165

El-Abbasi, N., Meguid, S.A., Modehng 2D contact surfaces using cubic splines

168

Eelippa, C.A., Optimal triangular membrane elements with drilling freedoms

171

FemdndeZ'Garcia, J.R., Sofonea, M., Viaho, J.M., Numerical analysis of a sliding viscoelastic contact problem with wear

173

Frangi, A., Novati, G., Springhetti, R., Rovizzi, M., Numerical fracture mechanics in 3D by the symmetric boundary element method

177

Galbraith, P.C., Thomas, D.N., Finn, M.J., Spring back of automotive assembhes

180

Gambarotta, L., Massabd, R., Morbiducci, R., Constitutive and finite element modehng of human scalp skin for the simulation of cutaneous surgical procedures

184

Gebbeken, N., Greulich, S., Pietzsch, A., Landmann, F, Material modelling in the dynamic regime: a discussion

186

Gendron, G., Fortin, M., Goulet, R, Error estimation and edge-based mesh adaptation for solid mechanics problems

192

Gharaibeh, E.S., McCartney, J.S., Erangopol, D.M., Reliability-based importance assessment of structural members

198

Ghiocel, D.M., Mao, H., ProbabiUstic life prediction for mechanical components including HCF/LCF/creep interactions

201

Giner, E., Fuenmayor, J., Besa, A., Tur, M., A discretization error estimator associated with the energy domain integral method in linear elastic fracture mechanics

206

Gonzalez, C, Llorca, J., Micromechanical analysis of two-phase materials including plasticity and damage

211

Goto, K., Yagawa, G, Miyamura, T, Accurate analysis of shell structures by a virtually meshless method

214

Guilkey, J.E., Weiss, J.A., An implicit time integration strategy for use with the material point method

216

Gupta, S., Manohar, C.S., Computation of reliabihty of stochastic structural dynamic systems using stochastic FEM and adaptive importance sampling with non-Gaussian sampling functions

220

xiv

Contents Volume 1

Guz, LA., Soutis, C., Accuracy of analytical approaches to compressive fracture of layered solids under large deformations

224

Hadjesfandiari, A.R., Dargush, G.F., Computational elasticity based on boundary eigensolutions

227

Haldar, A., Lee, 5.K, Huh, / , Stochastic response of nonlinear structures

232

Han, S., Xiao, M., A continuum mechanics based model for simulation of radiation wave from a crack

235

Handrik, M., Kompis, V., Novak, P., Large strain, large rotation boundary integral multi-domain formulation using the Trefftz polynomial functions . .

238

Hamau, M., Schweizerhof, K., About linear and quadratic 'Solid-Shell elements at large deformations

240

Hartmann, U., Kruggel, R, Hierl, T., Lonsdale, G., Kloppel, R., Skull mechanic simulations with the prototype SimBio environment

243

Havu,V,Hakula,H, An analysis of a bilinear reduced strain element in the case of an elliptic shell in a membrane dominated state of deformation

247

Ibrahimbegovic, A., Recent developments in nonlinear analysis of shell problem and its finite element solution

251

Ingham, T.J., Modeling of friction pendulum bearings for the seismic analysis of bridges

255

lozzi, R., Gaudenzi, P., MITC finite elements for adaptive laminated composite shells

259

Janajreh, L, Rezgui, A., Estenne, V., Tire tread pattern analysis for ultimate performance of hydroplaning

264

Kanapady, R., Tamma, K.K., Design and framework of reduced instruction set codes for scalable computations for nonlinear structural dynamics

268

Kang,M.-S.,Youn,S,-K., Dof splitting p-adaptive meshless method

272

Kapinski, S., Modelling of friction in metal-forming processes

276

Kashtalyan, M., Soutis, C., Modelling of intra- and interlaminar fracture in composite laminates loaded in tension

279

Kawka, M., Bathe, K.J., Implicit integration for the solution of metal forming processes

283

Kim, H.S., Tim, HJ., Kim, C.B., Computation of stress time history using FEM and flexible multibody dynamics

287

Kong, J.S., Akgul, K, Frangopol, DM., Xi, Y., Probabilistic models for predicting the failure time of deteriorating structural systems

290

Koteras, J.R., Gullemd, A.S., Porter, V.L., Scherzinger, W.M., Brown, K.H., PRESTO: impact dynamics with scalable contact using the SIERRA framework

294

Kratzig,W.B.,Jun,D., Layered higher order concepts for D-adaptivity in shell theory

297

Krishnamoorthy, C.S.,Annamalai, V, Vmu Unnithan, U., Superelement based adaptive finite element analysis for linear and nonlinear continua under distributed computing environment

302

KUbler, L, Eberhard, P., Multibody system/finite element contact simulation with an energy-based switching criterion

306

xv

Contents Volume 1 Laukkanen, A., Consistency of damage mechanics modeling of ductile material failure in reference to attribute transferability . . .

310

LeBeau, K.H., Wadia-Fascetti, SJ., A model of deteriorating bridge structures

314

Leitdo, VM.A., Analysis of 2-D elastostatic problems using radial basis functions

317

Limbert, G., Taylor, M , An explicit three-dimensional finite element model of an incompressible transversely isotropic hyperelastic material: application to the study of the human anterior cruciate ligament

319

Liu, G.R., Liu, M.B., Lam, K.Y., Zong, Z., Simulation of the explosive detonation process using SPH methodology

323

Liu, G.R., Tu, Z.H., MFree2D®: an adaptive stress analysis package based on mesh-free technology

327

Lovadina, C, Energy estimates for linear elastic shells

330

Lubowiecka, L, Chroscielewski, J., On the finite element analysis of flexible shell structures undergoing large overall motion

332

Luo, A.C.J., A numerical investigation of chaotic motions in the stochastic layer of a parametrically excited, buckled beam . .

336

Lyamin, A.V., Sloan, S.W., Limit analysis using finite elements and nonlinear programming

338

Malinen, M., Pitkdranta, J., On degenerated shell finite elements and classical shell models

. ••

342

Martikainen, J., Mdkinen, R.A.E., Rossi, T, Toivanen, J., A fictitious domain method for linear elasticity problems

346

Massin, R, Al Mikdad, M., Thick shell elements with large displacements and rotations

351

Mathisen, K.M., Tiller, L, Okstad, K.M., Adaptive ultimate load analysis of shell structures

355

Matsumoto, T, Tanaka, M., Okayama, S., Boundary stress calculation for two-dimensional thermoelastic problems using displacement gradient boundary integral identity

359

Mitchell, J.A., Gullerud, A.S., Scherzinger, W.M., Koteras, R., Porter, V.L., Adagio: non-hnear quasi-static structural response using the SIERRA framework

361

Toukourou, M.M., Gakwaya, A., Yazdani, A., An object-oriented finite element implementation of large deformation frictional contact problems and applications

365

Nemecek, J., Patzdk, B., Bittnar, Z., Parallel simulation of reinforced concrete column on a PC cluster

369

Noguchi, H., Kawashima, T, Application of ALE-EFGM to analysis of membrane with sliding cable

372

Nuno, N., Avanzolini, G., Modeling residual stresses at the stem-cement interface of an idealized cemented hip stem

374

Obrecht, H., Briinig, M., Berger, S., Ricci, S., Nonlocal numerical modelling of the deformation and failure behavior of hydrostatic-stress-dependent ductile metals

378

Olson, L, Throne, R., Estimation of tool/chip interface temperatures for on-line tool monitoring: an inverse problem approach

381

xvi

Contents Volume 1

Pacoste, C, Eriksson, A., Instability problems in shell structures: some computational aspects

385

Palacz, M, Krawczuk, M , Genetic algorithm for crack detection in beams

389

Papadrakakis, M., Fragakis, K, A geometric-algebraic method for semi-definite problems in structural mechanics

393

PatzdK B., RypU D., Bittnar, Z , Parallel algorithm for explicit dynamics with support for nonlocal constitutive models

396

Pawlikowski, M., Skalski, K., Bossak, M , Piszczatowski, S,, Rheological effects and bone remodelling phenomenon in the hip joint implantation

399

PeiLu,X., Computational synthesis on vehicle rollover protection

403

Peng,X., Cao,J., Sensitivity study on material characterization of textile composites

406

Penmetsa, R.C., Grandhi, R.V, Uncertainty analysis of large-scale structures using high fidelity models

410

Perez-Gavildn, J.J., Aliabadi, M.H., A note on symmetric Galerkin BEM for multi-connected bodies

413

Pradhan, S.C., Lam, K.Y., Ng,TY., Reddy, J.N., Vibration suppression of laminated composite plates using magnetostrictive inserts

416

Pradlwarter, H.J., Schueller, G.I., PDFs of the stochastic non-linear response of MDOF-systems by local statistical linearization

420

Proppe, C, Schueller, G.L, Effects of uncertainties on lifetime prediction of aircraft components

425

Randolph, M.F., Computational and physical modelling of penetration resistance

429

Rank, E., Duster, A., h- versus p-version finite element analysis for J2 flow theory

431

Roe, K., Siegmund, T, Simulation of interface fatigue crack growth via a fracture process zone model

435

Rosson, B.T, Keierleber, CM, Improved direct time integration method for impact analysis

438

Rucker, M., Rank, E., The /7-version PEA: high performance with and without parallelization

441

Ruiz, G., Pandolfi, A., Ortiz, M., Finite-element simulation of complex dynamic fracture processes in concrete

445

Sdez, A., Dominguez, J., General traction BE formulation and implementation for 2-D anisotropic media

449

Sanchez-Hubert, J., Boundary and internal layers in thin elastic shells

452

Sanchez Palencia, E., General properties of thin shell solutions, propagation of singularities and their numerical incidence

454

Savoia, M., Reliability analysis of structures against buckling according to fuzzy number theory

456

Scheider, I., Simulation of cup-cone fracture in round bars using the cohesive zone model

460

Schenk, C.A., Bergman, L.A., Response of a continuous system with stochastically varying surface roughness to a moving load

463

Contents Volume 1

xvii

Schroder, J., Miehe, C, Elastic stability problems in micro-macro transitions

468

Semedo Gargdo, J.E., Mota Soares, CM., Mota Soares, C.A., Reddy, J.N., Modeling of adaptive composite structures using a layerwise theory

471

Sladek, /., Sladek, V, Van Keer, R., The local boundary integral equation and its meshless implementation for elastodynamic problems

473

Slinchenko, D., Verijenko, VE., Structural analysis of composite lattice structures on the basis of smearing stiffness

475

Soric, J., Tonkovic, Z., Computer techniques for simulation of nonisothermal elastoplastic shell responses

478

Stander, N., The successive response surface method applied to sheet-metal forming

481

Szabo, BA.,Actis, R.L, Hierarchic modeling strategies for the control of the errors of idealization in FEA

486

Tahar, B., Crouch, R.S., Techniques to ensure convergence of the closest point projection method in pressure dependent elasto-plasticity models

490

Takahashi, A., Yagawa, G., Molecular dynamics calculation of 2 billion atoms on massively parallel processors

496

Tedesco, J.W., Bloomquist, D., Latta, T.E., Impact stresses in A-Jacks concrete armor units

499

Thompson, L.L., Thangavelu, S.R., A stabilized MITC finite element for accurate wave response in Reissner-Mindlin plates

502

Tijssens, M.G.A., van der Giessen, E., Sluys, L.J., Modeling quasi-static fracture of heterogeneous materials with the cohesive surface methodology

509

Tsukrov, I., Novak, J., Application of numerical conformal mapping to micromechanical modeling of elastic solids with holes of irregular shapes

513

Tyler-Street, M., Francis, N., Davis, R., Kapp, J., Impact simulation of structural adhesive joints

517

Vermeer, P.A., Ruse, N., On the stability of the tunnel excavation front

521

Verruijt, A., Numerical aspects of analytical solutions of elastodynamic problems

524

Vidrascu, M., Delingette, H., Ayache, N., Finite element modeling for surgery simulation

527

Vlachoutsis, S., Clinckemaillie, J., Distributed memory parallel computing for crash and stamp simulations

530

Vodicka, R., The first-kind and the second-kind boundary integral equation systems for some kinds of contact problems with friction

533

Wagner, W., Klinkel, S., Gruttmann, E, On the computation of finite strain plasticity problems with a 3D-shell element

536

Wang, J.G., Liu, G.R., Radial point interpolation method for no-yielding surface models

538

Wang, X., Bathe, K.J., Walczak, J., A stress integration algorithm for /s-dependent elasto-plasticity models

542

Whittle, AJ., Hsieh, Y.M., Pinto, E, Chatzigiannelis, ¥., Numerical and analytical modeling of ground deformations due to shallow tunneling in soft soils

546

xviii

Contents Volume 1

Witkowski, W, Lubowiecka, /., Identification of chaotic responses in a stable Duffing system by artificial neural network

550

Yang, C., Soh, A. -K., Special membrane elements with internal defects

554

Zarka, 7., Kamouni, //., Fatigue analysis during one-parametered loadings

559

Zdunek, A., Non-linear stability analysis of stiffened shells using solid elements and the p-version FE-method

562

Zhang, K, Lin, J., Random vibration of structures under multi-support seismic excitations

566

Zhao, K., On simulation of a forming process to minimize springback

568

Zhou, X., Tamma, K.K., Sha, D., Linear multi-step and optimal dissipative single-step algorithms for structural dynamics

571

Zhu, P., Abe, M, Fujino, K, A 3D contact-friction model for pounding at bridges during earthquakes

575

Zohdi, T.L, Wriggers, P., Computational testing of microheterogeneous materials

579

Optimization & Design Al-Dojayli, M., Meguid, S.A., Shape optimization of frictional contact problems using genetic algorithm

584

Bartoli, G., Borri, C, Facchini, L, Paiar, F, Simulation of non-gaussian wind pressures and estimation of design loads

588

Bisagni, C, Optimization of helicopter subfloor components under crashworthiness requirements

591

Bull,J.W., Some results from the Self-Designing Structures research programme

595

Butkewitsch, S., On the use of 'meta-models to account for multidisciplinarity and uncertainty in design analysis and optimization

599

Cardona, A., Design of cams using a general purpose mechanism analysis program

603

Cheng, G., Guo, X., On singular topologies and related optimization algorithm

606

Connell, M., Tullberg, O., Kettil, P, Wiberg, N.-E., Interactive design and investigation of physical bridges using virtual models

608

Consolazio, G.R., Chung, J.H., Gurley, K.R., Design of an inertial safety barrier using explicit finite element simulation

612

DalVAcqua, D., Lipsett, A.W., Faulkner, M.G, Kaiser, T.M.Y, An efficient thermomechanical modeling strategy for progressing cavity pumps and positive displacement motors

616

Doxsee Jr, L.E., Using Pro/MECHANICA for non-linear problems in engineering design

620

Dreisbach, R.L, Peak, R.S., Enhancing engineering design and analysis interoperability. Part 3: Steps toward multi-functional optimization . .

624

Ghiocel, DM., Stochastic process/field models for turbomachinery applications

628

Contents Volume 1

xix

Gu, Z, Zhao, G., Chen, Z, Optimum design and sensitivity analysis of piezoelectric trusses

633

Hagiwara, L, Shi, Q.Z., Vehicle crashworthiness design using a most probable optimal design method

637

Harte, R., Montag, U., Computer simulations and crack-damage evaluation for the durability design of the world-largest cooling tower shell at Niederaussem power station

641

Hartmann, D., Baitsch, M., Weber, H., Structural optimization in consideration of stochastic phenomena - a new wave in engineering

645

Hollowell, W.T., Summers, S.M., NHTSAs supporting role in the partnership for a new generation of vehicles

649

Ivdnyi, P., Topping, B.H.V., Muylle, J., Towards a CAD design of cable-membrane structures on parallel platforms

652

James, R.J., Zhang, L, Schaaf, DM., Wemcke, G.A., The effect of hydrodynamic loading on the structural reliability of culvert valves in lock systems

655

Kolanek, K., Stocki, R., Jendo, S., Kleiber, M., An efficiency of numerical algorithms for discrete reliability-based structural optimization

660

Krishnamoorthy, C.S., Genetic algorithms and high performance computing for engineering design optimization

663

Launis, S.S., Keskinen, E.K., Cotsaftis, M., Dynamics of wearing contact in groundwood manufacturing system

668

Liu, S., Lian, Z , Zheng, X, Design optimization of materials with microstructure

672

Liu, C, Wang, T.-L., Shahawy, M., Load lateral distribution for multigirder bridges

676

Maleki, S., Effects of diaphragms on seismic response of skewed bridges

681

Matsuho, A.S., Frangopol, D.M., Applications of artificial-life techniques to reliability engineering

685

Maute, K., Nikbay, M., Farhat, C, HPC for the optimization of aeroelastic systems

688

Miller, B., Ziemiahski, L., Updating of a plane frame using neural networks

692

Ogawa, Y., Ochiai, T, Kawahara, M., Shape optimization problem based on optimal control theory by using speed method

696

Papadrakakis, M., Lagaros, N.D., Reliability based optimization using neural networks

698

Papadrakakis, M., Lagaros, N.D., Fragakis, Y., Parallel computational strategies for structural optimization

701

Peak, R.S., Wilson, MM, Enhancing engineering design and analysis interoperability. Part 2: A high diversity example

704

Peri, D., Campana, E.F, Di Mascio, A., Development of CFD-based design optimization architecture

708

Peterson, DM., The functional virtual prototype: an innovation framework for a zero prototype design process

711

Prasad Varma Thampan, C.K., Krishnamoorthy, C.S., An HPC model for GA methodologies applied to reliability-based structural optimization

714

XX

Contents Volume 1

Rovas, D.V, Leurent, T, Prud'homme, C , Patera, A.T., Reduced-basis output bound methods for heat transfer problems

718

Schramm, U., Multi-discipUnary optimization for NVH and crashworthiness

721

Sedaghati, R., Tabarrok, B., Suleman, A., Optimum design of frame structures undergoing large deflections against system instability

725

Senecal, PK., Reitz, R.D., CFD modeling applied to internal combustion engine optimization and design

729

Shan, C, Difficulties and characteristics of structural topology optimization

733

Shankaran, 5., Jameson, A., Analysis and design of two-dimensional sails

737

Sheikh, S.R., Sun, M., Hamdani, H., Existence of a lift plateau for airfoils pitching at rapid pitching rates

739

Stander, N., Burger, M., Shape optimization for crashworthiness featuring adaptive mesh topology

743

Steven, G.P, Proos, K., Xie, Y.M., Multi-criteria evolutionary structural optimization involving inertia

747

Wilson, MM, Peak, R.S., Fulton, R.E., Enhancing engineering design and analysis interoperability. Part 1: Constrained objects

750

Wolfe, R.W,Heninger,R., Retrofit design and strategy of the San Francisco-Oakland Bay Bridge continuous truss spans support towers based on ADINA

755

Wu, J., Zhang, R.R., Radons, S., Vibration transmissibility of printed circuit boards by calibrated PEA modeling

758

Plenary Papers

Efficient simulation techniques for complicated micromachined devices N. Alu^u^ W. Ye^ D. Ramaswamy^ X. Wang^ J. White'='* ^ Department of General Engineering, University of Illinois, Urbana, IL 61801-2996, USA ^Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA ^ Department of Electrical Engineering and Computer Science, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract In this short paper, we briefly describe techniques currently used for simulating micromachined devices. We first survey the fast 3-D solvers that make possible fluid and field analysis of entire micromachined devices and then describe efficient techniques for coupled-domain simulation. We describe the matrix-implicit multilevel-Newton method for coupling solvers which use different techniques, and we describe a mixed-regime approach to improve the individual solver's efficiencies. Several micromachined device examples are used to demonstrate these recently developed methods. Keywords: M E M S ; Fast Stokes; CAD; Pre-corrected FFT; Simulation; Mixed regime

1. Introduction In this short paper, we briefly describe techniques currently used for simulating micromachined devices. We first survey the recently developed fast 3-D solvers that make possible the fluid and field analysis of entire micromachined devices. Then, we discuss the recently developed techniques for efficient coupled domain and mixed regime analysis, as they have made it possible to efficiently simulate devices whose operation involves several physical domains. In each section, we present computational results on real micromachined devices both to make clear the problem scale and to demonstrate the efficiency of these new techniques.

2. Fast 3-D solvers The exterior fluid and electrostatic force on a surfacemicromachined device can, in principle, be computed using finite-difference or finite-element methods. Such methods are becoming less popular, primarily due to the development of fast 3-D solvers which are much more efficient in this setting. In particular, for surface-micromachined

devices: (1) exterior forces need only be evaluated on poly silicon surfaces, (2) the geometries are innately 3-D and extremely complicated, (3) the exterior fields usually satisfy linear space-invariant partial differential equations. Since forces are not needed in the volume of the exterior, only on the surface, the exterior volume-filling grid for finite-element and finite difference methods seems inefficient. In addition, the geometrically complicated nature of micro-machined devices makes generating such an exterior volume grid difficult. The electrostatic problem is linear and space invariant, and so the Laplace's equation that describes the exterior electrostatics can be replaced with an integral equation which relates the surface potentials to the surface normal electric fields. In many cases, the fluid forces are reasonably well described by the linear Stoke's equation, and so an integral formulation involving only surface quantities can be used to determine fluid traction forces. The electrostatic potential and the fluid velocity, assuming Stoke's flow, both satisfy an integral equation over the poly silicon surface given by Green's theorem: u(x)

* Corresponding author. E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

/

G{x,x)—^

9n

\

^ 9n

M(x)dfl,

(1)

N. Alum et al. /First MIT Conference on Computational Fluid and Solid Mechanics where u is either the electrostatic potential or the fluid velocity, ;c is a point on the surface, and d/dn is the derivative in the direction normal to the polysilicon surface. Discretization of the above integral equation leads to a dense system of equations which becomes prohibitively expensive to form and solve for complicated problems. To see this, consider the electrostatics problem of determining the surface charge given the potential on conductors. A simple discretization for the electrostatics problem is to divide the polysilicon surfaces into n flat panels over which the charge density is assumed constant. A system of equations for the panel charges is then derived by insisting that the correct potential be generated at a set of n test, or collocation, points. The discretized system is then Pq = ^

(2)

where q is the n-length vector of panel charges, ^ is the w-length vector of known collocation point potentials. Since the Green's function for electrostatics is the reciprocal of the separation distance between x and x\

'••' = f

panel.

4n.!.

' ^'

X^

(3>

where xt is the iih collocation point. Since the integral in (3) is nonzero for every panel-collocation-point pair, every entry in P is nonzero. If direct factorization is used to solve (2), then the memory required to store the dense matrix will grow like n^ and the matrix solve time will increase like n^. If instead, a preconditioned Krylov-subspace method like GMRES [1] is used to solve (2), then it is possible to reduce the solve time to order n^, but the memory requirement will not decrease. In order to develop algorithms that use memory and time that grows more slowly with problem size, it is essential not to form the matrix explicitly. Instead, one can exploit the fact that Krylov-subspace methods for solving systems of equations only require matrix-vector products and not an explicit representation of the matrix. For example, note that for P in (2), computing Pq is equivalent to computing n potentials due to n charged panels and this can be accomplished approximately in nearly order n operations [2,3]. To see how to perform such a reduction in cost, consider Fig. 1. The short-range interaction between close-by panels must be computed directly, but the interaction between the cluster of panels and distant panels can be approximated. In particular, as Fig. 1 shows, the distant interaction can be computed by summing the clustered panel charges into a single multipole expansion (denoted by M in the figure), and then the multipole expansion can be used to evaluate distant potentials. Several researchers simultaneously observed the powerful combination of integral equation approaches, Krylovsubspace matrix solution algorithms, and fast matrix-vector

Short-range stiiiimed direct!J

Fig. 1. A cluster of collocation points separated from a cluster of panels. products [4,5]. Perhaps the first practical use of such methods combined the fast multipole algorithms for charged particle computations with the above simple discretization scheme to compute 3-D capacitance and electrostatic forces [6]. Higher-order elements and improved efficiency for higher accuracy have been the recent developments [8,10]. The many different physical domains involved in micromachined devices has focussed attention on fast techniques which are Green's function independent, such as the precorrect-FFT schemes [3,9]. 2.1. Example fluid simulation As an example of using a fast solver, consider determining the quality factor of a comb-drive resonator packaged in air. To compute the quality factor, it is necessary to determine the drag force on the comb. The small spatial scale of micromachined combs implies that flow in these devices typically have very low Reynolds numbers, and therefore convection can often be ignored. In addition, fluid compression can be ignored for devices which use lateral actuation, like many of the comb-drive based structures fabricated using micromachining. The result of these two simpUfications is that fluid damping forces on laterally actuated microdevices can be accurately analyzed by solving the incompressible Stokes equation, rather than by solving the compressible Navier-Stokes equation. That the fluid can be treated as Stokes flow, and that the quantity of interest is the surface traction force, makes it possible to use a surface integral formulation to compute comb drag [11]. Then, the methods described above can be used to rapidly solve a discretization of the integral equation [12,13]. In Fig. 2, the discretization of a comb is shown. Notice that only the surface is discretized, yet still the number of unknowns in the system exceeds 50,000. An accelerated Stoke's flow solver completed the simulation in under 20 min, direct methods would have taken weeks and required over 16 gigabytes of memory. The simulated traction force in the motion direction is shown in Fig. 3. Note the surprisingly high contribution to the force from the structure sides. It should be noted that the quality factor computed from the numerical drag force analysis matched measure quahty factor for this structure to better than 10% [14].

N. Alum et al. /First MIT Conference on Computational Fluid and Solid Mechanics

2.5

2.5

Fig. 2. A discretized comb drive resonator over a substrate.

R

Fx

-2351.96 -4937.22 -7522.49 •-10107.7 -12693 ^ -15278.3 17863.5 20448.8 23034.1 I—I -25619.3 28204.6 -30789.8 -33375.1 -35960.4 -38545.6

' ~ ^

1

E-05

0.00015

5E-05

0.0001

Fig. 3. Drag force distribution on the resonator, bottom (substrate-side) view. 3. Coupled-domain mixed-regime simulation Self-consistent electromechanical analysis of micromachined polysilicon devices typically involves determining mechanical displacements which balance elastic forces in the polysilicon with electrostatic pressure forces on polysilicon surface. The technique of choice for determining elastic forces in the polysilicon is to use finite-element methods

to generate a nonlinear system equations of the form Fiu)-

P{u,q)=0

(4)

where w is a vector of finite-element node displacements, F relates node displacements to stresses, and P is the force produced by the vector representing the discretized surface charge q. Note that as the structure deforms, the pressure changes direction, so P is also a function of u. One can

N. Aluru et al. /First MIT Conference on Computational Fluid and Solid Mechanics view this mechanical analysis as a 'black box' which takes an input, q, and produces an output u as in HMiq)

(5)

In order to determine the charge density on the polysilicon surface due to a set of appHed voltages, one can use a fast solver, as described above. One can view the electrostatic analysis as a 'black box' which takes, as input, geometric displacements, w, and produces, as output, a vector of discretized surface charges, ^, as in q=

200 h

HE{U)

150

100

(6)

Self-consistent analysis is then to find a u and q which satisfies both (5) and (6). 3.1. Multilevel-Newton -50 h A simple relaxation approach to determining a self-consistent solution to (5) and (6) is to successively use (5) to update displacements and then using (6) to update charge. Applying (5) implies solving the nonlinear equation, (4), typically using Newton's method [15]. Although the relaxation method is simple, it often does not converge. Instead, one can apply Newton's method to the system of equations

q u

HE(U)

HM{q)_

=

0

(7)

0

in which case the updates to charge and displacement are given by solving

/ L

^q

dHE\ _ du

Aq

I

Au

(8) HAA

The above method is referred to as a multi-level Newton method [16,17], because forming the right-hand side in (8) involves using an inner Newton's method to apply HM. In order to solve (8), one can apply a Krylov-subspace iterative method such as GMRES. The important aspect of GMRES is that an explicit representation of the matrix is not required, only the ability to perform matrix-vector products. As is clear from examining (8), to compute these products one need only compute (dHM/dq)Aq and (dHE/du)Au. These products can be approximated by finite differences as in ^HM ^ dq

^ Huiq+aAq)

a

Huiq)

-50

0

50

Fig. 4. Comb drive accelerometer. tion. Computing Huiq + oid\) means using an inner loop Newton method to solve (4), which is expensive, though improvements can be made [19]. An important advantage of matrix-free multilevel-Newton methods is that it is not necessary to modify either the mechanical or electrostatic analysis programs. 3.2. Mixed regime simulation

\-HE U —

-100

(9)

where is a very small number. Therefore, this matrix-free multilevel-Newton method [18] can treat the individual solvers as black boxes. The black box solvers are called once in the outer Newton loop to compute the right hand side in (8) and then called once per each GMRES itera-

In many micromachined devices, such as the mechanical structure in Fig. 4, much of the structure acts as a rigid body. Therefore, many finite-element degrees of freedom can be eliminated and replaced with a rigid body with only 6 degrees of freedom i/rigid = {^, 0. V^, ^R^ jR, zR). The u in (4) is then ^elastic U Mrigid. The rigid/elastic mechanical solver greatly reduces the size of the stiffness matrix with the bulk shrinking to a dense 6 x 6 block (see Fig. 5). The surface of the rigid body still has to be discretized finely to properly resolve the electrostatic forces. The rigid/elastic interface should be intruded into the rigid block for a small area around the tether-block mass interface in order to avoid sharp singularities in stress across the tether-block interface. 3.3. Tilting mirror example A coupled domain mixed regime solver was tested against the experimental data of a scanning mirror (see Figs. 6 and 7) [20] with 12 x 50 x 1.1 |xm SiN hinges (Young's Modulus = 243.2 MPa, Poisson's Ratio = 0.28)

N. Alum et al. /First MIT Conference on Computational Fluid and Solid Mechanics Rigid/elastic ; fully elastic (8x10x2 block 2x2x3 hinges)

12

Ov 500

251

.22 37.5 +v

37.5 -V

All dim in microns Fig. 7. Cross-section of scanning mirror. o experiment ; - simulation (30x30x3 block 3x4x3 hinges)

5

10 15 differential voltage in volts

20

Fig. 5. Elastic/rigid matrix reduction. and 500 x 600 x 25 [xm SiN on Si central plate kept at 0 v. The ground electrodes are kept at 37.5 ± v volts. The plot (Fig. 8) shows a close match of the simulation in the linear regime and convergence failure corresponding to pullin is obtained at 12.13 v as opposed to 13.4 v of the experimental data. On an average each load step took 80 min (Digital Alpha 433 MHz). For a coarse mesh the elastic/rigid simulation is compared with the fully elastic simulation (Fig. 5) to show a very close match. The CPU time for 10 load steps for the fully elastic case was 16.8 h as opposed to 58 min for the rigid/elastic case.

2

4 6 8 10 12 Differential voltage in v for scanning mirror

Fig. 8. Mirror tilt with differential voltage v. for coupled-domain analysis, and mixed-regime techniques. It is now possible to simulate the coupled-domain behavior of an entire micromachined design in under an hour on a workstation rather than days or weeks on a supercomputer. The next step is to use these tools to automatically generate macromodels of micromachined devices, and make possible accurate simulation of systems which use micromachined devices.

4. Conclusions Simulation of entire microdevices is becoming more routine in engineering design thanks to a combination of fast integral equation solvers, multilevel-Newton methods

0

"^

-200

Fig. 6. Scanning mirror (coarse mesh).

A^. Aluru et al. /First MIT Conference on Computational Fluid and Solid Mechanics Acknowledgements The authors would like to thank the many students who have developed codes described above including Keith Nabors, Joel Phillips, and Joe Kanapka. This work was supported by the DARPA composite CAD, microfluidics and muri programs, as well as grants from the Semiconductor Research Corporation and the National Science Foundation.

[11]

[12]

[13] References [1] Youcef Saad, Schultz MH. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Statist Comput 1986;7(3): 105-126. [2] Barnes J, Hut P. A hierarchical 0{N\ogN) force-calculation algorithm. Nature 1986;324:446-449. [3] Hockney RW, Eastwood JW. Computer simulation using particles. New York: Adam Hilger, 1988. [4] Rokhlin V. Rapid solution of integral equation of classical potential theory J Comput Phys 1985;60:187-207. [5] Hackbusch W, Nowak ZP. On the fast matrix multiplication in the boundary element method by panel clustering, Numer Math 1989;54:463-491. [6] Nabors K, White J. Fastcap: a multipole accelerated 3-D capacitance extraction program. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, November 1991;10:1447-1459. [7] Nabors K, Korsmeyer FT, Leighton FT, White J. Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional first-kind integral equations of potential theory. SIAM J Sci Statist Comput 1994;15(3):713-735. [8] Bachtold M, Korvink JO, Bakes H. The Adaptive, Multipole-Accelerated BEM for the Computation of Electrostatic Forces, Proc. CAD for MEMS, Zurich, 1997, pp. 14. [9] Phillips JR, White JK. A precorrected-FFT method for electro-static analysis of complicated 3-D structures. IEEE Trans, on Computer-Aided Design, October 1997; 16(10): 1059-1072. [10] Greengard L, RokhUn V. A new version of the fast multi-

[14]

[15] [16]

[17]

[18]

[19]

[20]

pole method for the Laplace equation in three dimensions. Acta Numer 1997, pp. 229-269. Pozrikidis C. Boundary integral and singularity methods for linearized viscous flow, Cambridge University Press, Cambridge, 1992. Aluru NR, White J. A fast integral equation technique for analysis of micro flow sensors based on drag force calculations. International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, April 1998, pp. 283-286. Ye W, Kanapka J, Wang X, White J. Efficiency and accuracy improvements for FastStokes, a precorrected-FFT accelerated 3-D Stokes Solver. International Conference on ModeHng and Simulation of Microsystems, Semiconductors, Sensors and Actuators, San Juan, April 1999. Ye W, Wang X, Hemmert W, Freeman DM, White J. Viscous drag on a lateral micro-resonator: fast 3-D fluid simulation and measured data. IEEE Solid-State Sensor and Actuator Workshop, Hilton-Head Island, SC, June 1999. Bathe KJ. Finite Element Procedures, Prentice-Hall, Englewood Chffs, NJ, 1996. Rabbat NB, Sangiovanni-VincenteUi A, Hsieh HY. A Multilevel-Newton algorithm with macromodeling and latency for the analysis of large scale nonlinear circuits in the time domain. IEEE Trans, on Circuits and Systems, CAS-26(9):733-741, Sept. 1979. Brown PN, Saad Y Hybrid Krylov Methods for Nonlinear Systems of Equations, SIAM J Sci Statist Comput 1990;11: 450-481. Aluru NR, White J. A coupled numerical technique for selfconsistent analysis of micro-electro-mechanical systems, microelectromechanical systems (MEMS). ASME Dynamic Systems and Control (DSC) Series, New York 1996;59: 275-280. Ramaswamy D, Aluru N, White J. Fast coupled-domain, mixed-regime electromechanical simulation. Proc. International Conference on Solid-State Sensors and Actuators (Transducers '99), Sendai Japan, June, 1999, pp. 314-317. Dickensheets DL, Kino GS. Silicon - Micromachined Scanning Confocal Optical Microscope. J Microelectromech Syst Vol. 7, No. 1, March 1998.

Subgrid scales, augmented problems, and stabilizations Franco Brezzi * Dipartimento di Matematica and I.A.N.-C.N.R., Via Ferrata 1 27100 Pavia, Italy

Abstract We present an overview of some recent approaches to deal with instabiUties of numerical schemes and/or subgrid phenomena. The basic idea is that of enlarging (as much as one can) the finite element space, then to do an element-by-element preprocessing, and finally solve a problem with the same number of unknowns as the one we started with, but having better numerical properties. Keywords: Residual free bubble; Stabilization

1. Introduction

diameter of Q) is much smaller than |c| in a non-negligible part of the domain. The variational formulation of (1.1) is

In a number of applications, subgrid scales cannot be neglected. Sometimes, they are just a spurious by-product of a discretized scheme that lacks the necessary stability properties. In other cases, they are related to physical phenomena that actually take place on a very small scale, but still have an important effect on the solution. In recent times, it was discovered that some mathematical tricks to deal with these problems can help in both situations. One of these tricks is based on the so-called Residual Free Bubbles (RFB). In what follows, we are going to discuss its application, by considering two typical examples, one for each category: the case of advection diffusion problems and the case of composite materials. For dealing with these problems, in a typical mathematical fashion, we shall choose very simple toy problems that will, however, still retain some of the basic difficulties of their bigger industrial counterparts. In particular, we consider: 1: Advection-dominated scalar equations: find umV:= H^(Q) such that Lu:= -sAu-{-c-S/u

= f in ^ ,

w = 0 on dQ. (1.1)

Here Q is, say, a convex polygon, c a given vector-valued smooth function (convective term), / a given smooth forcing term, and s a positive scalar (diffusion coefficient). Clearly, x = (xi,X2). The numerical approximation of the problem becomes nontrivial when the product of s times a characteristic length of the problem (for instance, the * E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

find u e V such that C(u, v) := I eVu -Vvdx -\- /

-i"

/•

C-VUV&K

(1.2)

doc Vi; € V.

2: Linear elliptic problems with composite materials: find M in V := H^(^) such that: Lu := - V . (a{x)Vu) = / in ^ ,

M = 0 on dQ.

(1.3)

As before, Q is, say, a convex polygon, and / a given smooth forcing term. The (given) scalar function a{x) is assumed to be greater than a given positive constant ao in the whole domain Q, and represents, somehow, the characteristics of a composite material. The numerical approximation of (1.3) becomes nontrivial when a has a fine structure, exhibiting sharp changes on a scale that is much smaller than the diameter of ^ . The variational formulation of (1.3) is find M e V such that £(M, V) := / a(jc)Vw • Vvdx

-I

fvdx

VUG V

(1.4)

The first example corresponds to problems where an unsuited numerical scheme can generate spurious oscillations in the numerical solution, which are not present in the exact solution (that in general, will just exhibit a boundary layer

F. Brezzi/First MIT Conference on Computational Fluid and Solid Mechanics near the part of the boundary where c • n > 0, where n is the outward unit vector normal to 9 ^ . On the contrary, the second example corresponds to problems where a fine structure is already present, all over the domain, and needs to be captured by the numerical scheme, at an affordable cost. In the sequel, we are going to give the basic idea of a general strategy that can prove useful, possibly in different ways, for both types of problems.

V e Bh(K) and obtain, from (2.4) that the restriction wf of UB to K is the unique solution of the following local bubble equation:

2. The residual free bubbles approach

C{SK{g),v) = {g,v)

We notice, to start with, that the two problems presented in the Section 1 have variational formulations sharing the same structure:

and write the solution i/f of (2.5) as wf = SK^/ - Luh). We are now ready to go back to (2.4), take v = Vh, and substitute in UA = Uh + UB its expression as given by (2.5) and (2.6) to obtain

I find u ^V such that I C{u, v) = (/, i;)

(2.1)

Vi; e V,

where, in both cases, V := HQ(Q) and, from now on, ( , ) denotes the inner product in L^(^). The difference is just in the type of biUnear form C(u,v) to be used for each problem. Fixing our ideas on either one of the abstract formulations (2.1), we assume now that we are given a decomposition 7^ of ^ into triangles, with the usual nondegeneracy requirements. For the sake of simplicity we assume that we start with finite element spaces Vh made of piecewise linear continuous functions vanishing on 9^. We also play the game that the dimension of Vh is the biggest one we are ready to afford, in the end, when we solve the final system of linear equations. However, we are ready to afford some extra work, as a pre-processor before building the stiffness matrix, provided that such work could be done in parallel, and in particular element-by-element. Under these assumptions (that is given these rules) we can proceed as follows. We start by considering the space of bubbles Bn-TlKBhiK),

Bh(K):=H^(K)

V^ € 7^. (2.2)

We consider now the augmented space (2.3)

VA:=VheBh, and the corresponding augmented problem Ifindu e VA such that C(UA, VA)

= (/, VA)

^VA

e VA-

(2.4)

Notice that (2.4) is infinite dimensional, and therefore unsolvable. Still we can consider it, for the moment, at the level of an abstract speculation. We then notice that, according to (2.3), we can split UA as UA = UU + UB. In its turn, UB will be a sum of local bubble functions wf, that is: UB = J2K "f • Therefore, in each K e % ^Q can take

find UB ^ Bh(K) such that C(u^s, V) = -C(UH, V) + (/, V) Wv e Bh(K).

(2.5)

Equation (2.5), if solvable, would allow to express each wf in terms of Uh. At the formal level, we can introduce the solution operator SK, that associates to every function g (for instance in L^(K)) the solution SK(g) e H^{K) of (2.6)

yveH^(K)

C{uh, Vh) - Y^C{SK{Luh), Vh) = (/,^/.)-X!>^(<5i^(/).^A)

^Vh^Vh.

(2.7)

This is the linear system that, in the end, we are going to solve. It can be seen (see e.g. [2-4,6,7]) that, for the first example, this corresponds to classical stabilized methods like SUPG (see e.g. [8,9]). For the second example, this would correspond to a two-level method of the type of the ones studied, for instance, in [13,14]. Clearly, the major difficulty is in the actual solution of the local problems (2.5) that, in principle, present difficulties that look similar to solving the original problems. However, looking at (2.7), we notice that, in practice, we have to evaluate only terms of the type C(SK(g), Vh) that, in turn, can be written as (SKig), L*Vh), where L* is the adjoint operator of L. In our two examples, we have L*v = —sAu — c - Vu for the first one, and L* = L for the second one (where L is self-adjoint). An important observation is now that, considering for instance the first example, L*Vh will be constant in each element. Hence, only the mean value of SK(g) is needed. This implies that a rough approximate solution of (2.5) could still be acceptable. This will not be the case for our second example, where SKig) will be integrated against a term depending on a(x). This term, however, will have a very definite structure, that we might think of to exploit. It is also possible to check that, in order to compute the terms depending on SK appearing in (2.7), it is sufficient to compute the quantities Sl:j:=(SK(vi),L''vi) Fr.= L%SK(f),vi)

and V/,7

WKeTh,

(2.8)

where the v^ are the usual nodal basis for Vh. Clearly the terms appearing in (2.8) have to be computed in some approximate way, see for instance [5,7,10]. However, the implementation could also follow a path that is apparently quite different. Indeed, to every basis

10

F. Brezzi / First MIT Conference on Computational Fluid and Solid Mechanics

function v'^ G VH we can associate two other functions wi and If* that, in each K, are solutions of the problems Lwi =Q

mK

Wi = v[

on dK,

(2.9)

[5]

and L*K;* = 0

in

^

ondK.

(2.10)

[6]

Clearly wt = w* whenever L is selfadjoint. It can be checked that the nodal values of the solution M^ of (2.4) coincide with the nodal values of the solution of the problem: find Wh, linear combination of the wj/s, such that

[7]

C(wh, w*) = (/, O

[8]

V/ = 1 , . . . , dim(V,).

(2.11)

On the other hand, the computation of the solution in the form (2.11) requires essentially the same amount of work as the computation in the form (2.7). It is also interesting to notice that, for the first example, this corresponds to the use of suitable basis functions (adapted to the operator) in the Petrov-Galerkin formulation, as discussed, for instance, in [15]. For the second example, (2.11) is actually the original formulation of [13]. For applications of these concepts to different problems see for instance [1,10-12].

References [ 1 ] Arbogast T. Numerical subgrid upscaling of two-phase flow in porous media. In: Chen Z, Ewing RE, Shi Z-C (Eds), Mulfiphase Flows and Transport in Porous Media: State of the Art. Lecture Notes in Physics. Berlin: Springer, 2000. [2] Brezzi F, Bristeau M-O, Franca LP, Mallet M, Roge G. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput Methods Appl Mech Eng 1992;96:117-129. [3] Brezzi F, Franca LP, Hughes TJR, Russo A. b = f g. Comput Methods Appl Mech Eng 1997;145:329-339. [4] Brezzi F, Hughes TJR, Marini LD, Russo A, Suli E. A

[9]

[10]

[11]

[12]

[13]

[14]

[15]

priori error analysis of a finite element method with residual-free bubbles for advecfion-dominated equations. SIAM JNumer Anal 1999;36:1933-1948. Brezzi F, Marini D, Russo A. Applications of pseudo residual-free bubbles to the stabilization of convection-diffusion problems, Comput Methods Appl Mech Eng 1998; 166:5163. Brezzi F, Marini D, SUli E. Residual-free bubbles for advection-diffusion problems: the general error analysis. Numer Math, to appear. Brezzi F, Russo A. Choosing bubbles for advection-diffusion problems. Math Mod Methods Appl Sci 1994;4:571587. Brooks AN, Hughes TJR. Streamline Upwind/PetrovGalerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng. 1982;32:199259. Franca LP, Frey SL, Hughes TJR. Stabilized finite element methods: I. Applications to advective-diffusive model, Comput Methods Appl Mech Eng 1992;95:253-276. Franca LP, Macedo AP. A two-level finite element method and its application to the Helmholtz equation. Int J Numer Methods Eng 1998,43:23-32. Franca LP, Russo A. Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Appl Math Lett 1996;9:83-88. Franca LP, Russo A. Approximation of the Stokes problem by residual-free macro bubbles. East-West J Numer Math 1996;4:265-278. Hou TY, Wu XH. A multiscale finite element method for elliptic problems in composite materials and porous media. J Comput Phys 1997;134:169-189. Hou TY, Wu XH, Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math Comp 1999;68:913-943. Mitchell AR, Griffiths DF. Generalised Galerkin methods for second order equations with significant first derivative terms. In: Proc. Bienn. Conf., Dundee, 1977, Lect Notes Math 1978;630:90-104.

11

Trends in the design analysis of aerospace vehicles Rodney L. Dreisbach^*, RaymondR. Cosner^ ^ Senior Technical Fellow, The Boeing Company, Computational Structures Technology, Mail Code 67-MW, 535 Garden Avenue North, Renton, WA 98055, USA Senior Technical Fellow, The Boeing Company, Computational Fluid Dynamics, Mail Code SI06-7126, P.O. Box 516, St. Louis, MO 61366, USA

Abstract Evolution of the airframe design analysis process during the past seven (7) decades is summarized from engineering technology, computing and process viewpoints. On-going trends are presented, using examples of typical structural and aerodynamic applications, especially that of the finite element method and the computing architecture that supports these tools. Current thrusts and overall integration strategies for product simulation integration (PSI) in Boeing are highlighted relative to the objectives of reducing costs and cycle time in the design, analysis, manufacturing and support of conmiercial airplanes. Finally, opportunities for advancing certain engineering, information, and computing technologies are enumerated, by identifying selected problem areas being addressed by today's industries. Keywords: Computational structures technology (CST); Aeroelastic analysis; Computational fluid dynamics (CFD)

1. Background A high-level overview is presented of how the designanalysis process for airframe vehicles has evolved from 1930 to the present time. Beginning with a 'real' single design office that relied on drawing boards, this process changed dramatically during the 1960s when computers were introduced into the technical workplace. Specific engineering technologies, however, were advanced by independent organizations. As we moved into the 1970s, faster and larger computers were best, but specialized engineering applications and data had to be interfaced from one computer code to another. As the 21st century is entered, the primary objective is to perform product lifecycle simulation with a design office that is virtually collocated using geographically distributed, collaborative computing. 2. The aeroelastic design process The aeroelastic design process for aerospace vehicles, as shown in Fig. 1, is iterative because of the complex * Corresponding author. Tel.: +1 (425) 234-3407; Fax: +1 (425) 234-8539; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

operating environments in which the vehicles must perform, the tightly coupled system integration of different disciplines, and because of the complex structural arrangements required within the vehicles. To obviate exhaustive static and dynamic physical laboratory and flights tests for optimally designing the aerodynamic vehicle and sizing the various structural components for all flight regimes, extensive use of analytical and computational methods are currently used during the design, development and certification of flight vehicles. These methods rely heavily on the well-established FEA (finite element analysis) and finite volume techniques initially developed for industrial applications during the late 1950s.

3. The finite element method for structural analysis Whereas the FEA method was used only for structural verification purposes during the 1960s, it is currently used in the design development of all primary aerospace structure beginning with the configuration development phase, through certification activities and customer support. The mainframe computing capabilities during the 1960s limited the maximum size of the mathematical system of equations to less than 6000. This constraint provided the impetus

12

R.L. Dreisbach, R.R. Cosner /First MIT Conference on Computational Fluid and Solid Mechanics

Sttffness

Balaiioed toads

Presses

Fig. 1. The iterative aeroelastic design process for aerospace vehicles.

to develop the substructured FEA analysis method that is published widely and is provided as an optional solution technique in many vendor-supplied codes. Today's use of the FEA method is extremely diverse, where its use for structural analysis spans static, dynamic and weight computations, cross-functional interactions between aeroelastics, flutter, propulsion and acoustics, linear and nonlinear geometry and material characteristics, including tool design and manufacturing process improvements. FEA models of total transport airplanes typically represent fairly detailed structural arrangements when the analysis objectives are to predict internal loads and stresses in the airframes. As a result, typical airplane FEA model sizes have exceeded 300,000 degrees of freedom (equations). This is quite a contrast to being limited to 6000 equations during the 1960s. Another recent trend is use of the FEA method much earlier in the product definition process. That is, the FEA tools have become very easy to use by designers interested in early-looks at how their structural design will perform in its operating environment. This change from the previously used, sequential 'design-then-analyze' process allows early computer-based analyses performed by designers to be shared with the analysts. Front-loading the design process by having designers perform rudimentary analyses is a step toward true con-

current engineering. This approach has resulted in early FEA models of complex single parts that exceed 300,000 degrees of freedom! However, with the on-going revolutionary advancements in computing power, solution of this type of large problem for a single load case can be performed in less than 30 minutes! Furthermore, shape optimization of structural parts using design-geometry parameters having automated associativity can be performed just as easily. These techniques have allowed flow times for selected design/analysis processes to be reduced by orders of magnitude!

4. Aerodynamic analysis characteristics The general trends in aerodynamic analysis are the same as previously discussed for structural analysis. With ever-increasing computing power and more capable tools, there is a clear desire for steadily improving the geometric fidelity of the CFD models, and for increasing the sophistication and detailed resolution of the fluid physics models (e.g., turbulence models). These factors lead to larger computational grids, more solution variables per grid point, and more stringent convergence criteria to attain high accuracy. Today, for 3-D multi-block analyses on structured grids, computational grids of 5,000,000 points are very common.

R.L Dreisbach, R.R. Cosner /First MIT Conference on Computational Fluid and Solid Mechanics

A#m@lastle Finite Elmmtt Analysis

13

DstaiM Analysis - Si Jug - Analysis

^ r ^ s Ar^lyas Ftepc^itwy

liliiliBiiiii ^ructurari T ^ Data Ftediwtbn

Automated Airplano SIsdng

h-

J

FJI

Flight Test IMM F M I ^ I I C M I

"'^

f%oduet Inforrtmtlon R0trl#val

^

CuMomw Airline Ef^ii^re ManyfoetUiing

Fig. 2. Product simulation integration (PSI) technology and data relationships for aerospace vehicle design.

and grids of up to 20,000,000 points are fairly common. In perfect gas analyses, there are five to seven solution variables per point. For more complex problems involving chemical reactions, there can be several dozens of solution variables per point. For the larger problems, computing times of a few hundred CPU-hours are fairly common. Parallel computing has been widely adopted for these types of analyses. Typical solution files can range from a few hundred megabytes up to nearly a gigabyte today, with the problem sizes steadily increasing.

5. Computing architecture The trend in computing hardware architecture for aerospace vehicle design and analysis processes have been moving away from mainframe computing campuses to that of client-server distributed networked configurations. Current trends are away from using multiple computers in support of different technological functions, to that of using a single computing workstation user-interface to perform all of the necessary computing functions. Another significant trend in computing is tighter vertical integration of functionalities within single computing systems. This allows data to be re-used and shared by multiple technologies, where data translators are passe and commonality in the man-machine interfaces is unified.

6. The PSI (product simulation integration) project A strategic initiative at BCAG, known as the PSI for Structures project, is underway to reduce costs and cycle time in the design, analysis, and support of commercial airplanes. The 'Product' is the airplanes we design and build, and the services we provide to customers for their airplane operations. 'Simulation' is the analytical and test processes performed to predict in-service behavior of the airplane structure in support of design requirements and objectives. 'Integration' is the close binding of our design, analysis, manufacturing and support processes with the associated product information, as it supports reduced costs and cycle time. The overall technologies and data relationships associated with the PSI project are shown in Fig. 2. The primary objectives of PSI are: (1) establish and enhance preferred engineering and business processes; (2) improve the suite of engineering methods and tools, and migrate legacy applications and data; and (3) integrate structural analysis and test with product definition information and manufacturing to reduce cycle time and costs. Fundamental to the success of the PSI project in meeting its goals are establishing standard processes, associating lifecycle information to the product definition data for easy, reliable, and consistent retrieval, and adopting industry standards for sharing of these data to facilitate long-term data access.

14

R.L. Dreisbach, R.R. Cosner /First MIT Conference on Computational Fluid and Solid Mechanics

6.1. Standard processes and computing systems Standard processes reduce variability in the way we design, analyze, and support our airplane products, thus lowering training, computing, process support, and sustaining costs. Standard computing systems reduce training due to a common look and feel of the system, as well as provide easy access to multiple computing operating systems and environments, where required.

•

•

• 6.2. Tie to digital product definition By linking analyses to the product definition data, records that substantiate the design decisions, strength, durability, damage tolerance analyses, and service history of the airplane parts and assemblies are made available for derivative airplane design and analysis, as well as sustaining. To be successful, these data must be available for the life of the airplane products. Current efforts are underway to extend the definition of SSPD (single source product definition) to include analysis and test data that may not necessarily be physically linked, but at a minimum will be logically linked.

•

•

• 6.3. Data exchange standards • Evolving computing software and hardware systems have made the task of information retrieval increasingly difficult with time. Out best opportunity to preserve the data we generate today and minimize regeneration tomorrow is through the adoption of standards for information exchange. Then, in principal, we can unplug the old analysis or information management tool and plug in a new one without extensive conversion and disruption to the engineers and customers.

7. Opportunities for advancement In developing future aerospace vehicles during the 21st century, challenges abound for more innovative technologies and products than ever before. These needs are being driven by increased demands for efficiency, safety and multi-functional operational requirements placed on future aerospace systems [1,2]. Opportunities that currendy exist for advancing numerous areas of computational mechanics to virtually simulate, in a realistic manner, the lifecycle of an aerospace vehicle before physical prototyping, are noted below. • Current design/analysis tools are mostly stand-alone; most tools operate in a local environment, with little integration. • An integrated, comprehensive computing architecture for a global design/analysis system does not exist. • Free exchange of accurate product definition informa-

•

•

tion is difficult; proprietary data representations are used; need standards for data modeling and information sharing. Product data redundancy is prevalent; many different data models are created by translations to specific technology application codes. Focus has been on optimizing the mathematical models and not the product itself (e.g., strength optimization of structural gages vs. shape vs. topology vs. topography). Increased demands on the operational requirements of products have provoked interactions between multiple technology domains; focus has been on a federated data environment, but an integrated data environment is preferred; need fully coupled solution techniques (e.g., combustion simulation on structural response). Design constraints with different fidelities across multidisciplines are different; need smart techniques for product definition information representation, mapping and integration in support of the continuous design evolution process. Simulation of lifecycle systems using a common, single-source, product information management system is essentially non-existent. Costing tools and methods in support of product design are inadequate. Transfer of new technologies into practice takes many years; need stronger university/industry internships and innovative facilitated educational (advisory) techniques for 'just-in-time' learning. Solutions to multi-physics problems are overly compromised by expansive assumptions (decoupling of analysis fields such as combustion simulation from structural response simulation). The current throughput of computational mechanics solutions is marginally acceptable for single-disciplined engineering problems; need concurrent engineering solutions of multi-physics-based problems based on knowledge sharing.

8. Summary Incredible advances have been made in multiple areas of computational mechanics technologies and in process implementations within industry for developing new aerospace vehicles during the past seven (7) decades. However, more advanced computational engineering techniques for performing design-analysis-optimization-synthesis activities concurrently, in satisfying the multi-functional operational specifications of an aerospace vehicle, are needed to attain higher levels of product functional prototyping in a virtual environment. Major advances are required in numerous areas of computational mechanics to virtually simulate, in a realistic manner, the lifecycle of an aerospace vehicle before physical prototyping.

R.L. Dreisbach, R.R. Cosner /First MIT Conference on Computational Fluid and Solid Mechanics References [1] Dreisbach RL, Peak RS. Enhancing engineering design and analysis interoperability. Part 3: Steps toward multi-functional optimization. In: First MIT Conference on Compu-

15

tational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [2] Noor AK, Venneri SL, Paul DB, Hopkins MA. Structures technology for future aerospace systems. Comput Struct 2000;74(5):507-519.

16

Issues in the seismic analysis of bridges Tim J. Ingham*'^ T.Y. Lin International, 825 Battery Street, San Francisco, CA 94111, USA

Abstract The use of local and global models and the trade-offs between simple and detailed models are discussed in the context of the seismic analysis of bridges; examples are presented from various projects. The management of time history analysis using a database is also presented. Keywords: Seismic analysis; Bridge; Database

1. Introduction Three issues related to the complexity of modeling bridges for seismic analysis are discussed in this paper: the use of local and global models, the trade-offs between simple and detailed models, and the management of analysis using a database.

2. Local and global modeling It is often impractical to include every detail of a large bridge in a comprehensive 'global' seismic analysis. A global model is a complete model of a bridge, from abutment to abutment, including the foundations, piers, and superstructure. The size of this model is limited by the demanding requirements of a time history analysis, which may include 2000-3000 time steps. A commonly used strategy to deal with this issue is to conduct detailed 'local' analyses to supplement the 'global' analysis. This issue is illustrated by the analysis of the towers of the Golden Gate Bridge, made for the seismic retrofit of that bridge [5,7]. The global model of the bridge is shown on the left in Fig. 1; the figure only shows a portion of the model, near one of the towers. The modeling of the base of the tower is the minimum able to capture the important nonlinear response of the tower. This includes yielding of the extreme fibers of the base, which is modeled by *Tel.: +1 (415) 291-3781; Fax: + \ (415) 433-0807; E-mail: [email protected] 1 Ph.D., S.B. Associate. © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

finite elements with an elastoplastic material, and rocking of the base, which is modeled by gap elements supporting the finite element mesh. The global model is used to compute the total response of the bridge, including the overall demands on the tower. The local model of the tower base, used for detailed evaluation and the design of retrofit measures, is shown on the right in Fig. 1. This shell element model, with nonlinear geometry and an elastoplastic material, was used to analyze the stability of the plates making up the individual cells of the tower. The local buckling predicted by the analysis will be prevented by installation of stiffeners inside the cells.

3. Simple versus direct modeling Another issue related to the level of detail used in a global model is the use of 'simple' models versus 'direct' models. A simple model is one that is easily constructed and understood, with a minimum number of parameters. An example of a simple model is the model of the pile foundation supporting a typical pier of the new East Bay Bridge, shown on the left in Fig. 2. This model uses a 6 x 6 stiffness matrix to represent each pile below the mudline; the pile behavior is simple and readily understood. A more 'direct' model of the same foundation is shown on the right in Fig. 2; only the piles are shown. Each pile is modeled with a beam element and the surrounding soil is modeled with nonlinear p-y springs (perpendicular to each pile) and nonlinear t-z springs (parallel to each pile). The relationship between the two models is that the simple model is a linearization of the direct one.

T.J, Ingham /First MIT Conference on Computational Fluid and Solid Mechanics

17

Fig. 1. Local and global models for analysis of the Golden Gate Bridge.

Superstructure

Pile above mudline Foundation

Fig. 2. Simple and direct models for analysis of the East Bay Bridge. The simple model has the virtue of running more quickly, and the volume of results to be handled is less, but underlying its simplicity is the linearization of the behavior. This depends on the level of deformation of the pile, so the analyst must constantly verify that the linearization is compatible with the results obtained. With the direct model the computer solves the equations of motion of the pile and soil at each time step, and the initial work of linearization and the tedious job of constantly checking it are eliminated. The virtue of the direct model is that the assumptions (about soil behavior, in the case of the pile) are at a more fundamental level and more easily appreciated, hence the term 'direct'. If automated methods are used to generate the more detailed model and to process the results (see below) the direct model only has the disadvantage of needing more CPU time for its solution. In any case, the level of detail to in-

clude in a model is the analyst's choice, balancing the effort required to generate the model and to process the results against the clarity of the assumptions involved. Other examples of this trade-off may be found in references [2-4].

4. Data management The time history analysis of a large bridge, like the replacement spans of the East Bay Bridge [6], produces a large volume of data. The management of this data is an important issue in the design process. For instance, using ADINA [1], the analysis of the model shown in Fig. 3 for 60 s of an earthquake (3000 time steps at 0.02 s) produces a result file-the porthole file-that is over three gigabytes in size. This file must be searched for the critical

T.J. Ingham/First MIT Conference on Computational Fluid and Solid Mechanics

t^^ h^

K k Fig. 3. East Bay Bridge, model of main span.

combinations of axial force, shear, and moment for each member and the results summarized for easy interpretation and design. Also, the maintenance of the model to reflect design changes is a significant problem. On the East Bay Bridge project, both of these issues were addressed by using a database to store structure and model data, to generate input files for analysis, and to summarize analysis results for design. This approach is shown schematically in Fig. 4, which shows the different files involved, and in Fig. 5, which shows the process of analysis and interpretation of results. As shown in Fig. 4, the process is managed by a compiled Microsoft Access database that contains the forms used to

Access

Database; Forms

Code Reports

,rnde File | '•

Fig. 4. East Bay Bridge analysis, file structure. Access Database; .nde File Forns Code Reports

/ Model and / / Result Data /

f

^.rf

— J A D I M A >—

, Po'^trcle ~^^ File

Fig. 5. East Bay Bridge analysis, process.

Reports Eesigr Process

1

define the structure, the code needed to generate models — and input files — for analysis, and the reports used to present results. The ground motions and the data defining the structure are kept in separate Access data files. The structure file contains the data describing the complete structure; using the database system an analyst may choose to analyze the complete structure or just a part of it — e.g., the main span, or a single pier. For the chosen portion of the structure the elements and nodal connectivity describing the resulting model are written to the model data file at the same time that the ADINA input files are produced. The ADINA program stores results in a 'porthole' file; which is a binary file with a complex structure. A 'porthole reader' program is used to scan this file and transfer the maximum and minimum forces for each member to the result data file. This program may be contained in a dynamic link library integrated with the compiled database or it may be a standalone program [8]. Finally, combining the model and result data, the compiled database produces the reports needed for design, and passes the data onto specialty design programs. A typical form in the database system is shown in Fig. 6. This particular form is used to describe the layout of the piles at each pier, and to specify the pile type. The advantages of the database are several. The structure data file provides a central location for the storage of design data. Analysts working on different parts of the structure can generate models from this single file and they have ready access to common data — e.g. standard pile types. Automating and standardizing the production of input files eliminates tedious work and minimizes errors. And, the database is ideally suited to summarizing the results for a large number of members and for several ground motions.

TJ. Ingham /First MIT Conference on Computational Fluid and Solid Mechanics

_ W» W^ Im

Vi^^ ffietmn. i^eot^

19

kin ^ B

Deflection Damping Factor

o.i v The ctef)K:t«ai and dampBig facfty ^-e onKf used "]> fix piie ir»¥«d^ce matriK aid hytrid models aid 0.015915 X for pfe cap Irrf)edarc8 matrix mocteis

Re Lajfout Sdtrfamr ^rMod^io yge Cap" to tem aph cap mpedegxem^k^

u u

Fig. 6. Access database for model generation, pile modeling and layout form. 5. Conclusions The seismic analysis of large bridges presents many choices regarding the level of detail to include in a global model and the analysis of critical components. The use of automated methods for data storage, model generation, and the manipulation of results is an important factor in the complexity of the models that can be practicably handled.

[4]

[5]

[6] References [1] ADINA Theory and Modeling Guide. ADINA R&D, Cambridge, MA, 1999. [2] Baker G, Ingham T, Heathcote D. Seismic retrofit of Vincent Thomas suspension bridge. Transportation Research Record No. 1624. Transportation Research Board, 1998. [3] Ingham TJ. ModeUng of friction pendulum bearings for

[7]

[8]

the seismic analysis of bridges. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. Ingham TJ, Rodriguez S, Donikian R, Chan J. Seismic analysis of bridges with pile foundations. Comput Struct 1999;72:49-62. Ingham TJ, Rodriguez S, Nader M. Seismic modeling and analysis of the Golden Gate Bridge. Proceedings of the Structural Engineers World Congress, San Francisco, CA, 1998. Nader M, Manzanarez R, Ingham T, Baker G. Seismic Design Strategy for the New San Francisco Oakland Bay Bridge Suspension Span. Proceedings of the 16th International Bridge Conference, Pittsburgh, PA, 1999. Rodriguez S, Ingham TJ. Seismic Protective Systems for the Stiffening Trusses of the Golden Gate Bridge. Proceedings of the National Seismic Conference on Bridges and Highways, San Diego, CA, 1995. SC-Porthole7 Program. SC Solutions, Santa Clara, CA.

20

Virtual control algorithms J.L. Lions * Institut de France, 23 quai de Conti, 75006 Paris, France

Abstract Some recent advances in the development of virtual control algorithms for the approximate solution of boundary value problems are presented. Keywords: Virtual control algorithms; Controllability; Domain decomposition; Heterogeneous decomposition

Let us consider an equation

A(u) = f

(1)

in a domain ^ c R'^, where A is an elliptic operator (linear or not, scalar or vectorial), and where u is subject to boundary conditions, not specified here. We embed the problem in a family of relaxed problems By = g + k

(2)

in a domain Q (which can coincide with Q, or not), where B is an elliptic operator, related to A but 'simpler' than A, where y is subject to adequate boundary conditions on 9 ^ . In (2) the RHS contains two terms. The function g is constructed depending on / and the function X (scalar or vectorial) is a virtual control. It is to be chosen in such a way that y allows to recover the solution u of (1), exacdy (resp. approximately). In control theory terminology, it corresponds to exact (resp. approximate) controllability. This type of idea, of course made precise, allows a lot of flexibility in the construction of algorithms for the approximation of the solution of (1), the so-called virtual control algorithms. The idea was introduced in a note by JL Lions and O Pironneau [1] and since then it has been applied to a number of situations. The lecture will try to present the main ideas of the following ones. (1) Domain decomposition methods: see [1] above and [2]. (2) Decomposition of operators: [3]. (3) Decomposition of energy spaces: [4]. *Tel.: +33 (1) 4427-1708; Fax: +33 (1) 4427-1704; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(4) Heterogeneous decompositions: follows a paper by Gervasio et al. [5], to appear in Numerische Mathematik. (5) High precision with low order finite elements: [6], to appear. (6) Time decomposition: [7], [8]. Cf. also a paper in preparation with Y. Maday. (7) Towards meshless methods: paper in preparation.

References [1] Lions JL, Pironneau O. Algorithmes paralleles pour la solution de problemes aux limites. C.R.A.S. Paris 1998;327(I):947-952. [2] Lions JL, Pironneau O. Domain decomposition methods for CAD. C.R.A.S. Paris 1999;328(I):73-80. [3] Lions JL, Pironneau O. Virtual control, replicas and decomposition of operators. C.R.A.S. Paris 2000;330(I):47-54. [4] Glowinski R, Lions JL, Pironneau O. Decomposition of energy spaces and applications. C.R.A.S. Paris 1999;329(I):445-452. [5] Gervasio P, Lions JL, Quarteroni A. Heterogeneous coupling by virtual control methods. Numer Math, to appear. [6] Lions JL, Pironneau O. to appear. [7] Lions JL. Virtual and effective control for distributed systems and the decomposition of everything. J Anal Math, Hebrew Univ. of Jerusalem 2000;80:257-297. [8] Lions JL. Remarks on the control of everything. Eccomass, Barcelona, September 2000.

21

Numerical methods for prediction and evaluation of geometrical defects in sheet metal forming A. Makinouchi^'*, C. Teodosiu^

^ The Institute of Physical and Chemical Research — RIKEN, Materials Fabrication Laboratory, 2-1 Hirosawa, Wako 351-0199, Japa ^ LPMTM — CNRS, University Paris Nord, Villetaneuse , France

Abstract This paper presents a short overview of the state-of-the-art prediction and evaluation of geometrical defects in sheet metal forming, focusing on recent advances in the finite element (FE) simulation, on the benchmark tests organized to obtain reference experimental data for appraising ability of simulation codes, and on the attempt to define numerical measures for quantitatively evaluating various geometrical defects. Keywords: Sheet metal forming; Geometrical defects; Springback; Benchmark test

1. Introduction Sheet forming simulation is becoming a key technology for automotive manufacturers, sheet metal parts producers and stamping tool makers, aiming at predicting forming defects by using finite element software, in order to replace the actual tryout of stamping dies by a computer tryout. The main types of defects occurring in sheet metal forming are tearing, surface deflection, wrinkling, and springback (see Fig. 1). The last three types are also called geometrical defects. Among the three geometrical defects springback is a very sensitive forming defect, as the cumulative geometrical inaccuracy of the stamped parts may lead to serious trouble during assembling of various parts. Moreover, this difficulty tends to increase with the recent use of aluminum alloys and high-strength steels by the car manufacturers. Fig. 2 illustrates the main types of geometrical defects produced by springback (edited by Yoshida [1]).

2. Requirement from industries In 1998, the authors visited automotive industries and sheet steel suppliers in Europe, Japan and the United States, * Corresponding author. Tel.: +81 (48) 467-9314; Fax: +81 (48) 462-4657; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to discuss with engineers and researchers working at die shops and in sheet stamping sections. The reason of this visit was to prepare our keynote paper on the advance in FEM simulation and its related technologies in sheet metal forming for the CIRP Annual Meeting [2]. The visited companies were Daimler Benz, Renault Automobiles, Volvo Car Corporation and SOLLAC in Europe, Mazda, Nissan Motor, Toyota Motor and Nippon Steel in Japan, and Ford Motor, Chrysler Corporation, US Steel and National Steel in the United States. A large number of international conferences have been devoted to the sheet metal forming simulation, and an extensive literature has been published on this topic throughout the last two decades. However, the information obtained from these sources was not considered sufficient to address the above issues, because the very trend of sheet forming simulation had undergone significant changes during the last ten years. Indeed, most engineers working in automakers and sheet suppliers are software users, and their opinion does rarely appear in publications. Therefore, the authors considered that a direct contact with the technical staff involved in sheet metal forming simulations was a highly necessary prerequisite for learning the actual evaluation of the software used for industrial applications. A quite interesting bulk of information has been obtained in this way. Although a wide variety of FE codes are employed in the industries, these codes may be divided into five categories based on the formulation and solution

22

A. Makinouchi, C. Teodosiu /First MIT Conference on Computational Fluid and Solid Mechanics

Tearing

Surface deflection

Springback

Wrinkling

Fig. 1. Main types of defects encountered in sheet metal forming. Table 1 Assessment of FE codes by industrial researchers and engineers for each category classified by formulation and solution strategy Solution strategy Formulation

Dynamic explicit Incremental method

Static explicit

FE codes

LS-DYNA3D PAM-STAMP OPTRIS

ITAS3D

Company employing codes

All the companies

Nissan Nippon Steel

Defects predicted: wrinkling thickness/tearing surface defects geometrical defects after springback

Static implicit Large step method

One step method

MTLFRM

AUTO FORM

SIMEX ISOPUNCH A F ONE STEP FAST FORM3D

Ford

Benz Volvo Ford Chrysler Nissan Sollac

Renault Benz Volvo Sollac National Steel

A, X o, A A, X

X

X

A, X

X

: satisfactorily predicted; A = possible to simulate but poor results; x = impossible to simulate. strategy used. The assessment of the codes by industrial researchers and engineers is summarized for each category in Table 1. Inspection of this table reveals that the tearing and wrinkling are rather satisfactorily predicted, while pre-

diction of the springback is very poor, while the surface deflection is not simulated. Most of the engineers strongly emphasized the importance of an accurate springback prediction.

A. Makinouchi, C. Teodosiu /First MIT Conference on Computational Fluid and Solid Mechanics Rail

Springback angle

23

Panel

Side wall curl

Twisting

Warping

Shape fixing defect at punch bottom

Fig. 2. Geometrical defects produced by springback.

3. FE approach to simulate geometrical defects We shall recall here briefly some of the merits and drawbacks of three main types of FE approaches employed in the simulation of sheet metal forming, namely the dynamic explicit, the static implicit, and the static explicit codes. The dynamic-explicit codes are very robust and efficient for large-scale problems. The central difference expUcit scheme is used to integrate the equations of motion, whereas the non-equilibrated forces are transformed into inertial forces at each step. Lumped mass matrices are used, and hence no system of equations has to be solved. In spite of its success for industrial applications, dynamic explicit codes have also some intrinsic drawbacks. Thus, in order to reduce the number of steps necessary to simulate the almost quasi-static deformation processes, several numerical artifacts have to be employed, e.g. the increase

of the mass density and of the punch velocity by at least one order of magnitude and the introduction of artificial damping in order to limit the inertial effects. Moreover, the results obtained when simulating the springback depend on the type and dimensions of the finite elements and even of the number of integration points [3]. Thus, the simulation of forming defects requires a considerable experience on the user side for adequately designing the finite element mesh and choosing the scaling parameters for mass, velocity and damping (see, e.g. [4]). The static-implicit approach may seem ideally suited for metal forming problems, since the equilibrium equations are solved iteratively, thus ensuring that the equilibrium conditions are fulfilled at every step. However, in practice, complex nonlinear problems involving many contacts, may result in slow, or even lack of convergence. In the static-explicit approach, the rate forms of the

24

A. Makinouchi, C. Teodosiu /First MIT Conference on Computational Fluid and Solid Mechanics

kinematic, constitutive and equilibrium equations are integrated by a simple forward Euler scheme, involving no iterations (see, e.g. [5]). This implies that equilibrium equations are satisfied only in rate form, and thus the obtained solution can gradually drift away from the true one. In order to reduce the errors involved by linearizing the incremental analysis, a relatively large number of small incremental steps have to be used. The main advantage of this approach is its robustness, since it requires no iterative processes. Furthermore, by the very existence of intrinsic deviations from perfect equilibrium, the static-explicit algorithm is able to simulate defects arising from local instabilities, like wrinkling (see, e.g. [6]), while the static implicit codes are hardly able to treat such situations, unless such instabilities are allowed for by special numerical techniques, which require a considerable computational effort.

4. Benchmark tests to evaluate ability of FE codes for prediction of geometrical defects At several international conferences, like the VDI International Conference held at Zurich, Switzerland in 1991 [7], NUMISHEET'93 at Isehara, Japan in 1993 [8], NUMISHEET'96 at Dearborn, USA in 1996 [9], and NUMISHEET'99 at Besangon, France in 1999 [10], benchmark tests were organized in order to appraise the capability of FE codes to predict forming defects. The experimental benchmark tests have been concurrently performed by several teams over the world, in order to obtain reference data. However, most of the benchmark experimental results obtained by different participants disagreed greatly with each other and thus provided rather poor reference data for evaluating the codes. It is eventually possible to find out a posteriori the reasons for this scattering of experimental data. However, because the benchmark results are evaluated by the conference organizing committee, which dissolves after the event, it has been practically impossible to further analyze the discrepancies noticed during the conference. For the purpose of solving this problem, a three-year international research project named Digital Die Design System (3DS) started its activity in 2(XX), under the framework of the international collaborative program. Intelligent Manufacturing System (IMS). Fourteen industrial partners and seven academic and research institutes participate to the project from Canada, European Union and Japan, the present authors being deeply involved with the technical management of this project. The obtaining of reliable experimental data, with a controlled and minimized scatter, is one of main targets of the project. Such carefully performed and comprehensively documented experimental tests are expected to become a worldwide recognized database for the validation of numerical methods and codes dealing with the simulation of sheet metal forming processes.

5. Numerical representation of geometrical defects Assuming that a powerful FE code could accurately predict all geometrical defects illustrated in Fig. 2, this will be still not enough for the present requirements of the stamping industry. Indeed, the final goal of simulations is to quantitatively evaluate the geometry of stamped parts and, on this basis, to find the optimized die shapes that are able to produce parts of the exactly designed shape. To meet such requirements, it is essential to have clear definitions of forming defects and of the intrinsic values used to evaluate each geometrical defect. This problem is also a major concern in the 3DS Project. The surface of each defect model possesses some global features, which describe the overall distortions, such as the surface being 'bent' or 'twisted', and local features, which describe local distortions and their locations. There are many ways of defining such measures. One of the most promising way is to describe the local intrinsic character of the surface by the Gaussian curvature, and to represent the global features by the aggregate normal vectors to the surface [11].

6. Conclusions A short overview of recent activity in numerical methods to predict and evaluate geometrical defects in sheet metal forming is presented. Although FE codes were introduced into many industries, further intensive research effort is necessary to approach to the final goal: designing the optimum tool geometry directly by simulation.

References [1] Yoshida K (Ed). Handbook of Ease or Difficulty in Press Forming, Tokyo, 1987. (English translation, Ann Arbor, MI: National Center for Manufacturing Science, Inc., 1993.) [2] Makinouchi A, Teodosiu C, Nakagawa T. Advances in FEM simulation and its related technologies in sheet metal forming. Ann CIRP 1998;47(2):641-649. [3] Mattiasson K, Thilderkvist P, Strange A, Samuelsson A. Simulation of springback in sheet metal forming. In: Shen S, Dawson PR (Eds), Proc. NUMIFORM'95. Rotterdam: Balkema, 1995, pp. 115-124. [4] Lee SW, Yang DY. An assessment of numerical parameters influencing springback in explicit finite element analysis of sheet metal forming processes. J. Mater Process Technol 1998:80-81:60-67. [5] Kawka M, Makinouchi A. Shell-element formulation in the static explicit FEM code for the simulation of sheet stamping. J Mater Process Technol 1995;50: 105-115. [6] Kawka M, Olejnik L, Rosochowski A, Sunaga H, Makinouchi A. Modeling wrinkling phenomena in sheet metal forming. Proceedings of AEPA'98, 1998. [7] Proceedings of VDI International Conference. FE Simula-

A. Makinouchi, C. Teodosiu/First MIT Conference on Computational Fluid and Solid Mechanics tion of 3-D Sheet Metal Forming Processes in Automotive Industry, Zurich, Switzerland, 1991. [8] Proceedings of NUMISHEET'93, Isehara, Japan, 1993. [9] Proceedings of NUMISHEET'96, Dearborn, USA, 1996.

25

[10] Proceedings of NUMISHEET'99, Besan9on, France, 1999. [11] Kase K, Makinouchi A, Nakagawa T, Suzuki H, Kimura F. Shape error evaluation method of free-form surfaces. Comput-Aided Design 1999;31(8):495-505.

26

The Immersed Boundary Method for incompressible fluid-structure interaction David M. McQueen, Charles S. Peskin *, Luoding Zhu Coumnt Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA

Abstract In this paper the Immersed Boundary Method is presented, with some recent developments. The method is used to analyze fluid-structure interaction problems. Different aspects of the method are illustrated by applying it to blood flow in the heart and a flapping filament (flag-in-wind) problem. Keywords: Immersed Boundary Method; Fluid-structure interaction; Cardiac fluid dynamics; Flapping filament; Flag in wind; Computational fluid dynamics; Incompressible elasticity; Heart valves

1. Introduction In the study of fluid-structure interaction, it is useful to think of the structure as a part of the fluid where additional forces are applied, and where additional mass may be localized. In this paper, we consider the case of a viscous incompressible fluid that interacts with an immersed structure that is made of an incompressible viscoelastic material. To keep things as simple as possible, we assume that the viscosity is Newtonian and uniform throughout the system. This restriction can certainly be removed, but we shall not address that complication here. The mass density of the ambient fluid is also assumed to be uniform, but the structure is allowed to have a nonuniform mass density which may be greater or lower than that of the fluid. Instead of separating the system into its two components coupled by boundary conditions, as is conventionally done, we use the incompressible Navier-Stokes equations, with a nonuniform mass density and an applied elastic force density, to describe the coupled motion of the hydroelastic system in a unified way. In order to do this, however, we need to supplement the Navier-Stokes equations by a Lagrangian description of the elastic material, from which the elastic force density and the nonuniform mass density that appear in the Navier-Stokes equations may be calculated. Moreover, we need a mathematical apparatus to translate in either direction between Lagrangian quantities * Corresponding author. Tel.: +1 (212) 998-3126; Fax: -Hi (212) 995-4121; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

and the corresponding Eulerian quantities. This apparatus is conveniently provided by the Dirac delta function. The equations of motion that result from this point of view directly motivate a numerical method known as the "Immersed Boundary Method" [1-5]. This name emphasizes an important feature of the method: that it can handle not only immersed elastic structures that displace a finite volume, but also immersed elastic boundaries like heart valve leaflets (for which the method was originally designed), insect wings, sails, and parachutes, all of which may be idealized as surfaces which, despite having zero volume, nevertheless apply finite forces to the fluid in which they are immersed. Clearly, the Dirac delta function is particularly well suited to this situation.

2. Equations of motion As described in Section 1, we use an Eulerian description of the system as a whole (fluid -h structure) supplemented by a Lagrangian description of the structure. The independent variables of the Eulerian description are the Cartesian coordinates x and the time t, and the independent variables of the Lagrangian description are curvilinear material coordinates q,r,s and again the time t. The Eulerian description of the system as a whole involves the velocity field w(jc, r), the hydrostatic pressure field p(x,t), th^ mass density p{x, t) and the Eulerian elastic force density/(jc, 0The Lagrangian description of the immersed elastic material involves its configuration X{q,r,s,t), its Lagrangian

D.M. McQueen et al. /First MIT Conference on Computational Fluid and Solid Mechanics elastic force density F{q, r,s,t), and its Lagrangian additional mass density M(q,r,s), the integral of which over any chunk of the material gives the mass of that chunk minus the mass of the fluid displaced. Since both the mass and volume of any such chunk of the immersed elastic material are conserved, M is independent of time. Note that M = 0 in the case of a neutrally buoyant structure, and that M will be negative at any material point for which the mass density of the immersed elastic material is less than that of the ambient fluid. To complete the Lagrangian description of the elastic material, we need to specify the elastic potential energy functional, E[X], which is used in the calculation of the elastic forces from the configuration X(, , ,t) at any given time. The mass density po of the ambient fluid and the viscosity /x of the system as a whole are constant parameters. With this notation, our equations of motion read as follows: p{x,t) (-^JrU'Vu\+Vp

= ixV^u +f{x, t)

W u=0 fix, t)=

(1) (2)

F{q, r, s, t) 8 (x - X(q, r, s, t)) dq dr ds

(3)

p{x, t) = po-\- / M(q, r, s) 8 (x - X(q, r, s, t)) dq dr ds (4)

Tt

(q, r, s,t) =u {X(q, r, s, t), t)

= I u(x, t)8(x-X(q,r,s,t)) F =

dE

"dx

Note that Eq. (1) also involves the non-uniform mass density p{x, t). Since the fluid and the structure are both incompressible, it must be the case that p{x, t) at any given material point is independent of time, i.e., that Dp/Dr = 0, where D/Dr is the material derivative: 9/9f -I- a • V. This constraint is implicit in Eqs. (4) and (5); it does not have to be imposed separately. Eqs. (3) and (4) provide conversions from the Lagrangian force and mass densities F{q, r,s,t) and M(q, r, s) to the corresponding Eulerian force and mass densities,/(x, t) and p(x, t), respectively. The relationship between corresponding densities is not that their values are the same at corresponding points, but rather that their integrals over corresponding regions are equal. One can confirm that this is satisfied in our case by integrating Eq. (3) or Eq. (4) over some arbitrary region of space, changing the order of the integrals on the right-hand side, and noting that the integral of the Dirac delta function yields 1 or 0 depending on whether or not the domain of integration includes the point x = X(q, r,s,t). It is important to note that Eqs. (3) and (4) still make sense in the special case that the immersed elastic structure takes the form of a surface instead of displacing any volume. In the case of such a structure (like a sail or parachute canopy), we need only drop one of the three Lagrangian coordinates q,r,s so that Eqs. (3) and (4) become

fix, t)

-I

F(r, s, t)8(x-

I

p(x,t) = Po + dx

(5)

(6)

These equations (without the viscous term) can be formally derived from the principle of least action, see [6] for details. Here we just give an informal discussion of their meaning. Eqs. (1) and (2) are the famihar Navier-Stokes equations of a viscous incompressible fluid, with a variable mass density p(x,t) and an applied force density/(jc, r). Although it may be unconventional to use these equations in the case of an elastic material, one should recall that in the derivation of the incompressible Navier-Stokes equations the only ingredients are Newton's laws of motion, incompressibility, and a particular form of the stress tensor. It follows that the incompressible Navier-Stokes equations are applicable to any incompressible material, provided that appropriate allowance is made for the particular stress-tensor of the material, which may, of course, be different from that of a fluid. Here, the applied force density/(jc, t), the divergence of the elastic stress tensor, plays that role.

27

X(r, s, t)) dr ds

M(r,s)S{x-X(r,s,t))drds

(7)

(8)

In each of these equations, the Dirac delta function is still three-dimensional, but there are only two integrations to perform so the result is singular like a one-dimensional delta function. Again, the integral off(x, t) or p(x, t) over any finite three-dimensional region gives a finite result. Eq. (5) states that the velocity of any material point of the structure may be found by evaluating the Eulerian velocity field u{x,t) at the current location of that material point. This is essentially the definition of the Eulerian velocity field, but it also enforces the no-slip condition at the interface between the fluid and the structure, since we require that u be continuous. The second form of Eq. (5), in which the Dirac delta function appears, shows that the conversion from Eulerian to Lagrangian velocity can be expressed in a manner that resembles the conversions from Lagrangian to Eulerian force and mass densities, Eqs. (3) and (4). All of these conversions involve integral operators in which the Dirac delta function appears as a kernel. In Eq. (5), however, the integral is over the fixed Cartesian coordinates x, whereas in Eqs. (3) and (4) the integrals are over the moving curvilinear material coordinates q,r, s. Eq. (6) is shorthand for the statement that F is minus the Frechet derivative of E. That is, d^" = - / F • dZd^ dr d^.

28

D.M. McQueen et al. /First MIT Conference on Computational Fluid and Solid Mechanics

for any perturbation dX, up to terms of higher order in 6X. This is essentially the principle of virtual work.

3. Numerical method The Immersed Boundary Method is obtained by discretization of the above equations of motion. For details in the uniform density case, see [2-5]. The case of non-uniform mass density is similar, except that the Navier-Stokes solver involves the solution of difference equations with non-constant coefficients at each time step. Thus, Fourier transform methods are no longer applicable, and some iterative method such as multigrid must be used. An example of such a computation can be found in [7], and we report on another such example here.

4. Results In this section, we present results of two different immersed boundary computations, illustrating different aspects of the method. The first is a computer simulation of the heart. It involves all aspects of the mathematical formulation mentioned above except that the density of the system is considered uniform. In particular, heart muscle is modeled as an anisotropic, incompressible, elastic material that is neutrally buoyant in blood, and the heart valve leaflets are modeled as massless fiber-reinforced elastic membranes. The elastic parameters of the heart muscle are time-dependent, which is what makes it possible for the model heart to beat. The second computation presented here is a simulation of a laboratory experiment involving a flexible filament suspended in a flowing soap film with the upstream end of the filament held fixed. Because the fluid is in the form of a soap film, the whole problem is inherently two-dimensional, and the immersed boundary (the flexible filament) is one-dimensional. Filament mass, we have found, is an essential feature of the problem. Therefore, this computation illustrates those aspects of the Immersed Boundary Method that are concerned with non-uniform density. The heart model [2,8] is shown in Figs. 1-3. It is made entirely of elastic and contractile fibers immersed in viscous incompressible fluid. The model includes the four cardiac chambers and all four valves; it also includes the great vessels to which the heart is connected. These great vessels of the model have blind ends but are equipped with sources and sinks that provide appropriate loads for the model heart. An external source/sink allows for changes in cardiac volume and also provides a convenient reference pressure. The specific form of the Immersed Boundary Method used for these computations is described in [5], see also [4]. Parameters, including the Reynolds number, are those of the human heart.

Fig. 1. Cutaway view of the three-dimensional heart model during ventricular filling. The heart is viewed from the front, so the left ventricle is on the right side of the figure and the right ventricle is on the left. Structures that appear above the ventricles are (from left to right in the figure) the main pulmonary artery (with closed pulmonic valve), the ascending aorta (with closed aortic valve), and the left atrium (with open mitral valve). Two pulmonary veins are visible behind and connecting to the left atrium. Fluidflowis shown in terms of streaklines: dots mark the current positions of blood particles, and tails attached to these dots show the trajectories of these particles over the recent past. Note the prominent vortex that was shed from the anterior leaflet of the mitral valve and has migrated down towards the apex of the left ventricle. Figs. 1 and 2 show cutaway views of the heart in diastole from different perspectives. In Fig. 1 the clipping plane cuts through the mitral valve, the aortic valve, and the apex of the heart. Note the prominent vortex that was shed primarily from the anterior leaflet of the mitral valve and has then been convected towards the apex of the heart by the jet of left ventricular filling. In Fig. 2 the model heart has been turned so that the right ventricle faces the viewer. A large swirling vortex with an interesting 3D structure fills the relaxing right ventricular chamber. Fig. 3 shows the flow pattern of blood on the left side of the heart during ejection. Note the closed mitral valve, supported by papillary muscles and chordae tendineae, that prevents backflow into the left atrium, and the open aortic valve that allows the left ventricle to eject blood into the aorta.

DM. McQueen et al. /First MIT Conference on Computational Fluid and Solid Mechanics

29

Fig. 3. The computed flow pattern of left ventricular ejection. Note the tension in the closed mitral valve and the jet of blood entering the ascending aorta through the open aortic valve. Fig. 2. Transparent view of the predicted flow pattern of right ventricularfilUng.The heart model has been turned so that the free wall of the right ventricle is in front. At the upper left in the figure, the superior vena cava and inferior vena cava join to form the right atrium. The open tricuspid valve is visible at the atrioventricular junction. Other structures seen above the ventricle are (from left to right in the figure) the ascending aorta and the main pulmonary artery. Note the flow pattern of the prominent vortex that seems to fill the entire right ventricle. There is a hint of 3D structure in the way that the flow comes down through the tricuspid valve in the foreground but swirls around the vortex core into the background behind that inflow jet. It is our hope that this model will prove useful as a computer test chamber for the design of prosthetic cardiac valves. (For early studies of this kind in a two-dimensional left heart model, see [9-11].) Computer simulation of a flapping filament in a flowing soap film is shown in Fig. 4. The filament, a flexible thread, is anchored at its upper end in a soap film which flows downwards under the influence of gravity, constrained by two vertical wires at the edges of the film. Air resistance flattens the velocity profile of the flowing soap film. This simulation is based on an experiment performed in the Courant Institute WetLab by Jun Zhang [12]. Zhang's key discovery is that under a range of conditions the filament exhibits bistable behavior. Its two stable states are: (1) a steady state in which the filament points straight downstream; or (2) a sustained oscillation in which the filament

flaps like a flag in the wind and alternately sheds vortices of opposite sign creating a wake that resembles the Karman vortex street behind a cylinder. Either state is stable against small perturbations (hence the term 'bistable') but can be converted to the other state by a sufficiently large perturbation. Our principal finding is that the flapping state requires filament mass. With a massless filament, the steady state in which the filament points straight downstream is globally stable. Fig. 4 shows a simulation in which the filament mass per unit length is twice that of the experimental filament (saturated with water), the extra mass being explained by a bulge in the soap film that forms around the thread as a consequence of surface tension, thus raising the effective filament mass. Although the Reynolds number of the computation (Re = 210) is lower than that of the laboratory experiment by two orders of magnitude, the results of the simulation are in good agreement with those of the experiment, including the observed flapping frequency of about 50 Hz.

5. Conclusions The Immersed Boundary Method is a practical way to simulate fluid-structure interaction in the incompressible case. It can handle immersed elastic structures which displace finite volumes (like muscle), and also immersed

30

D.M. McQueen et al. /First MIT Conference on Computational Fluid and Solid Mechanics Center under an allocation of resources MCA93S004P from the National Resource Allocation Committee.

\

Fig. 4. Computer simulation of a flapping filament in a flowing soap film. Selected time step from a simulation showing sustained oscillation at about 50 Hz. Two different visualization techniques are used. The left panel of the figure shows the instantaneous positions of fluid markers created in bursts along the upper (inflow) boundary, as in a hydrogen bubble flow visualization. The right panel of the figure shows the corresponding vorticity contours. In both panels flow is from top to bottom (driven by gravity, working against air resistance) at an inflow velocity of 280 cm/s. The filament length is 3 cm, and the width of the channel is 8.5 cm. The Reynolds number of the computation (based on inflow velocity and filament length) is Re = 210. The flapping filament sheds vortices of alternate sign which then form the sinuous wake seen in the figures. elastic membranes (like sails, parachutes, and heart valve leaflets). Recent developments have extended the range of Reynolds numbers that the method can handle (up to and including that of the human heart), and have also made possible the simulation of immersed elastic structures which are not neutrally buoyant in the ambient fluid.

Acknowledgements The authors are indebted to the National Science Foundation (USA) for support of this work under KDI research grant DMS-9980069. Computation was performed in part on the Cray T-90 computer at the San Diego Supercomputer

References [1] Peskin CS. Flow Patterns Around Heart Valves: A Digital Computer Method for Solving the Equations of Motion. Ph.D. Thesis, Albert Einstein College of Medicine, July, 1972, 211 pp. (available at http://www.umi.com/hp/ Products/DisExpress.html, order number: 7230378) [2] Peskin CS, McQueen DM. Fluid dynamics of the heart and its valves. In: Othmer HG, Adler FR, Lewis MA, Dallon JC (Eds), Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology. Englewood Cliffs NJ: Prentice-HaU, 1996, pp. 309-337. [3] McQueen DM, Peskin CS. Shared-memory parallel vector implementation of the immersed boundary method for the computation of blood flow in the beating mammalian heart. J Supercomput 1997;ll(3):213-236. [4] Lai M-C, Peskin CS. An immersed boundary method with formal second order accuracy and reduced numerical viscosity. J Comput Phys 2000;160:705-719. [5] McQueen DM, Peskin CS. Heart Simulation by an Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity. ICTAM 2000 Proceedings, New York: Kluwer (in press), [6] Peskin CS, McQueen DM. Computational biofluid dynamics. Contemp Math 1993;141:161-186. [7] Fogelson AL, Zhu J. Implementation of a variable-density Immersed Boundary Method. Unpublished, http:/www. math.utah.edu/~fogelson. [8] McQueen DM, Peskin CS. A three-dimensional computer model of the human heart for studying cardiac fluid dynamics. Comput Graph 2000;34:56-60. [9] McQueen DM, Peskin CS. Computer-assisted design of pivoting-disc prosthetic mitral valves. J Thorac Cardiovasc Surg 1983;86:126-135. [10] McQueen DM, Peskin CS. Computer-assisted design of butterfly bileaflet valves for the mitral position. Scand J Thor Cardiovasc Surg 1985;19:139-148. [11] McQueen DM, Peskin CS. Curved Butterfly Bileaflet Prosthetic Cardiac Valve. US Patent Number 5,026,391; June 25, 1991. [12] Zhang J, Childress S, Libchaber A, Shelley M. Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 2000;408:835.

31

GM's journey to math: the virtual vehicle Robert M. Ottolini *, Steve M. Rohde General Motors Corporation, 6440 East 12 Mile Road, Warren, MI 48090-9000, USA

Abstract A recent study sponsored by the United States Government^ concluded that enterprise-wide "... modehng and simulation are emerging as key technologies to support manufacturing in the 21st century, and no other technology offers more potential than modeling and simulation for improving products, perfecting processes, reducing design-to-manufacturing cycle time, and reducing product realization costs..." General Motors has understood this potential for many years and has developed a math-based strategy to implement it. That strategy, termed 'Math-Based Synthesis Driven Vehicle Development Process', spans all facets of the vehicle creation process including the use of mathematical models to: optimally position products in the marketplace; translate the customers' voice into product functional characteristics; and synthesize robust physical reahzations, i.e., vehicle designs to meet both the physical and functional requirements, as well as producibility requirements. This involves the utilization of a multitude of different types of mathematical models and computer-based methods at different levels of detail. The logical integration of these models into the GM Vehicle Development Process (VDP), together with the exphcit definition of discrete virtual 'build events', yields the 'Virtual Vehicle,' a key component of the GM strategy and the subject of this paper. More specifically, the virtual vehicle is defined, and examples of its use and associated benefits throughout the VDP are shown. In addition, to effectively create virtual vehicles consistent with the timing requirements of a 'fast' VDP, a 'virtual environment' including a superior IT infrastructure is required. The approach presented leads to shorter product development cycles at reduced cost, fewer prototype hardware builds, and improved quality product for the customer. Keywords: Math-based; Virtual vehicle; Synthesis/analysis; Computer-aided engineering; Systems engineering; Vehicle development process

1. Introduction A major driver in the automotive industry today is the competitive pressure to shorten the product development cycle, and to provide superior functionality and quality to the customer at affordable prices. Vehicle development times have decreased from 60 months to less than 18, and will continue to decrease. Safety, environmental friendUness, and energy efficiency are additional paramount customer requirements. To help achieve these goals computers and electronics, new materials, and other technologies have been integrated into vehicles, increasing the complexity and required degree of integration of vehicle subsystems. Indeed, modem automotive vehicles are at an integration * Corresponding author. E-mail: [email protected] ^ Integrated Manufacturing Technology Roadmapping Project: Modeling and Simulation, http://imtr.ornl.gov © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

level associated with 'mechatronics'. To design such vehicles requires a considerably more sophisticated approach than that which had been used in the past, particularly to do so in a timely fashion. Fortunately, in tandem with the increasing sophistication and complexity of automotive vehicles, there has been a very rapid growth in the ability to design and engineer vehicles using computer-based methods, i.e., simulation technology. Historically, mathematical models were used to 'troubleshoot' designs, i.e., a posteriori. More recently, mathematical models have been used extensively to create and evaluate product designs via CAD and CAE tools. In General Motors a process-driven approach based on a systems engineering paradigm using mathematical models has been developed to define, design, and engineer vehicles. That process, termed 'Math-Based, Synthesis Driven', differs from traditional simulation-based design in the use of math-based synthesis. With this approach, math-models

32

R.M. Ottolini, S.M. Rohde/First MIT Conference on Computational Fluid and Solid Mechanics

are used in all phases of the vehicle development process: from quantifying the needs of the customer to validating the product using detailed, computationally intensive simulations. The logically configured mathematical representations of the vehicle as it evolves through the VDP are termed 'virtual vehicles'. These virtual vehicles are used to ensure that the vehicle will meet its specific requirement set.

2. GM's Math-Based Synthesis Driven Vehicle Development Process GM's Math-Based Synthesis Driven Vehicle Development Process is based upon systems engineering, which may be simply defined as an orderly process for the design of man-made systems to satisfy operational needs. It requires the explicit determination of functional requirements from the operational needs, and then using well-defined procedures, to translate those requirements into a physical realization that meets the needs in an optimal manner. During this translation, interactions among components are treated explicitly to ensure compatibility of all functional, physical, and program interfaces. Analytical models typically are used extensively in the systems engineering process. General Motors, beginning in the mid-eighties, has converged on a systems engineering-based vehicle engineering process as depicted by the icon in Fig. 1. The process is driven by the customers' wants and needs from the top, left-hand comer of the trapezoid. The left leg represents re-

quirements engineering. It includes developing and allocating the requirements for the vehicle, and for the manufacturing and assembly processes, to build the vehicle. These requirements 'flow' directly from, and are thus traceable to the customers' wants and needs. The flow is from the customer to the vehicle, then to the subsystems, and then to the components. On the bottom of the trapezoid we show the detailed design of the individual parts and components, which are assembled and developed to form the vehicle as shown on the right leg of the trapezoid. In the middle of the trapezoid is the validation process which includes both validation of the requirements and of the design to meet those requirements. Validation of the design is done at the component, subsystem, and vehicle levels in that order. Synthesis and analysis are key to the effective implementation of systems engineering-based vehicle development. Synthesis is a process for designing a system in which multiple and competing requirements are balanced and allocated to the subsystems and components through a systematic analytical process. Thus, synthesis forms the basis for requirements engineering and for design — a synthesis process by definition. Analysis, on the other hand, is the use of mathematical models to assess the performance of a given system, or to better understand its behavior. Analysis is used for validation at the component, subsystem, and vehicle levels as well as hardware development, e.g., debugging/tuning. More simply put, synthesis is creation driven by requirements whereas analysis is evaluation to those requirements.

CUSTOMER 4. Perform Analytical Validation to Minimize the Use of Hardware Based Techniques

DEVELOP & ALLOCATE REQUIREMENTS

ASSEMBLE & DEVELOP SYSTEM 3. Use S/A & Simulation to Support the Development of Components, Subsystems, Vehicles, and Manufacturing & Assembly ^ Processes

1. Support the Requirements Definition Process*: • Define Specific Requirements • Allocate & Balance Functional Requirements • Optimize & Integrate Requirements i * At vehicle, subsystem, & component levels

iiNlMlliiiii

MANUFACTURING

PROCESS

2. Perform Synthesis and Analysis to Achieve Optimal Balanced Vehicle, Subsystem and Component Designs That Meet Customer Requirements

Fig. 1. Systems engineering-based vehicle development process showing the role of synthesis and analysis.

R.M. Ottolini, S.M. Rohde/First MIT Conference on Computational Fluid and Solid Mechanics

33

MATH BASED VISION

24-36+ Month VDP

<18 Month VDP

18-36 Month VDP

- Consecutive, Hardware Based Learning - Math Basic for Analysis ('Check & Fix')

- Concurrent, Compressed Hardware Based Learning - Math Basis for Synthesis (Virtual Builds) and Analysis

- Continuous, Synthesis Based Learning - Hardware to Confirm Synthesis and Validate Integrated Vehicle

Fig. 2. GM's math-strategy. A key point here is the use of math models with an appropriate level of detail. For example, early in the development process, conceptual models based upon regressions or algebraic equations are often used. Conversely, for validation, finite-element models containing hundreds of thousands of degrees of freedom are often used. Having introduced the concepts of a systems engineering-based vehicle development process that is implemented through the use of synthesis and analysis, GM's mathbased strategy can be simply stated as moving from a hardware-driven, analysis-supported VDP to a synthesisdriven, hardware-supported VDP as shown in Fig. 2. This simple statement has a profound influence on how vehicles are engineered. It involves moving from a 'bottoms up' VDP paradigm in which hardware is built and tested to determine what was 'done wrong' to a 'top down' VDP

Styling

paradigm in which hardware is built to confirm the math modeHng. Viewing the VDP as a 'learning process', the learnings transit from a sequential set of hardware-based learnings to a more overlapping set using math-based synthesis and analysis; and, ultimately, to a continuous learning process via the mathematical representations as also shown in Fig. 2. Of course, some prototype hardware will still be built to correlate and confirm the math models and to ensure that the customers' needs are indeed met.

3. The virtual vehicle Given the process described above, the virtual vehicle concept can be viewed as the extensive use of coordinated math models to guide decisions regarding the definition.

Design CAD

Form

&Fit

Virtual Velilcle

Marketing Pmc&ss^^.

\ Correiation/ I Feedback

Manufacturing/ ] Assembly ]

Fig. 3. The virtual vehicle concept.

PhysmMGbMrmati&m:

Lab/PG ]•

CAT::

:^

R.M. Ottolini, S.M. Rohde/First MIT Conference on Computational Fluid and Solid Mechanics

34

1

V S A S CAPABILITY Energy Management Powertrain

1

Aerodynamics

4^ 0

Crash/Safety

1

Harmony & Optimization & Human Factors Robust Engineering j | : , i;—~r fN ^ O OI^S ^HBBBftl^ structures .^^•^^H

I 1 1 | |

^^^HV^^HD Noise & H ^ ^ L ^ S n B ^ P - - - ^ ^ _ Vibration

1 1

(^^^HUHH^B^B^SBM

• 1 ^ 9 ^ ^ ||fljW||K^jKi ^Bl^^\

Manufacturing CAE & Dimensional integration ^^ ^

\

1 1

QRD Bectricai/ \ Bectronlcs

1 1

Control Systems \

6VSAS Process & Integration/ADV, CAE Infrastructure

o

1 PaP^ 1 •sSUassr 1

Vehicle Dynamics

-^^^^I^^^^^^^^^HH^PT

t

|

t

I

I

VeWde Synthesis. Analysis, and Simulatiai j

=W»^

8.21430 1

Fig. 4. Scope of applications used in GM. form, fit, function, manufacturing, assembly, marketing, and sales and service of a vehicle through the VDP. Fig. 3 shows this conceptually. To implement the virtual vehicle concept requires the integration of math capabilities that span the vehicle's functionality as shown schematically in Fig. 4. GM has been developing this capability since the 1960s, but only in the recent past has the computing infrastructure and application software been at the point of making 'virtual test labs and proving grounds' a reality. Fig. 5 shows GM's recent rapid growth in high performance computing

^ 350

i 300 i . 250 I 200 I 150 •

100

•5

50 1980

1985

1990

1996

2000

2005

Year

Fig. 6. Growth in the number of finite elements in a typical crash worthiness simulafion. to support the virtual vehicle. The growth rate is almost 100% per year, significantly outpacing Moore's law. That growth is mandatory and enables the development of higher fidelity math models in areas such as structures and CFD as shown in Fig. 6.

4. Closure

1995

1996

1997

1998

1999 2000 Y T D Jan 2001 Jul 2001

Fig. 5. GMNA high performance computing growth since 1995.

In this paper we have attempted to give the reader a flavor for the benefits and the potential of using modem computational methods in a systems engineering process-driven framework to define and engineer automotive vehicles. The concepts of math-based synthesis and analysis were introduced. Examples drawn from actual product development were presented to illustrate the approach.

Solids & Structures

36

Shear resistance of granular media containing large inclusions: DEM simulations S.J. Antony*, M.Ghadiri Department of Chemical and Process Engineering, University of Surrey, Guildford, GU2 7XH, UK

Abstract In this paper we present the effect of size ratio on the shear resistance of dense granular media containing large inclusions. We also present the microscopic evolution of contact orientations in terms of fabric anisotropy tensor. We present how the structural orientations of the contacts are influenced by the size ratio of large inclusions. It has been shown that, as the size of submerged particle in the periodic granular cell increases, the overall shear resistance of the granular system decreases. This could be attributed to the weak fabric anisotropy of the system develops for an increase in size of large inclusion. These findings help us to understand the fundamental flow characteristics of granular media under slow shear regime. Keywords: Granular material; Shear resistance; Slow shear flow; Fabric anisotropy; Size effects; Discrete element analysis; Particle interactions

1. Introduction Granular materials are an important part of several engineering and industrial processes. The properties of the constituent particles strongly influence the deformation characteristics of the particulate medium. An estimate of the shear resistance of the particulate medium is of great importance to facilitate better process control. It is often necessary to specify the mechanical conditions required for such an operation. To control the behaviour of granular materials needs understanding of the physical processes that control the behaviour and interactions of their constituent particles. This has been facilitated greatly by the rapid growth of computer power, which has enabled an insight to be gained of the complex and often mysterious behaviour of granular materials using numerical simulations. Studies on the influence of inclusion on the behaviour particulate medium has been of recent interest. For example, the vibration induced size segregation problem, also known as 'brazil-nut effect' has been the subject of several investigations [4]. When a container having larger particles embedded in smaller granular particles is vibrated, for ex* Corresponding author. Tel.: +44 (1483) 789-477; Fax: -f-44 (1483) 876-581; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ample vertically, the bigger particles tend to move toward the top of the container. Conflicting reasons have been attached to this phenomena. Nevertheless, two-dimensional studies have indicated that [5] there exists a threshold size ratio (diameter ratio of large particle to the surrounding monodispersed particle), above which, the movement of larger particle increases. For smaller size ratio (less than about 3), no ascent of the large particle (intruder) was observed; for size ratio 5.3, the intruder undergoes an intermittent ascent; and for size ratio greater than ca. 10, the intruder ascents continuously and hence the fluidity (movement) of the large particles becomes higher. Recent studies on the size effects in compacted beds based on experiments and micromechanical modelling have shown interesting characteristics. Bonnenfant et al. [6] have studied the effect of presence of hard inclusions (glass) in a polymethylmethacrylate (PMMA) matrix on compaction in a triaxial cell. Their experimental and analytical studies have shown no influence of large inclusions on the global stiffness for the size ratio of the inclusion equal to 2 as considered by them. In this paper, we carry out three-dimensional simulations in a periodic cell for the shear resistance of granular media containing large inclusions (Fig. 1) using DEA. The interaction between contiguous particles are modelled as a

S.J. Antony, M. Ghadiri /First MIT Conference on Computational Fluid and Solid Mechanics I

T^ . I D<4

M\ Size ratio=D/d

Fig. 1. Schematic diagram (front/top view) showing the periodic cell with inclusion.

dynamic process. This allows us to get an insight into what happens inside the granular media during shearing. Vital information on the influence of inclusion on the macroscopic shear resistance and internal contact orientations of the granular assembly are obtained during shearing.

2. Simulations The simulations were carried out using Discrete Element Method (DEM), which was originally developed by Cundal and Strack [1]. The interactions between contiguous particles are modelled as a dynamic process and the time evolution of the particles is advanced using an explicit finite difference scheme. The interactions between the neighbouring particles are modelled by algorithms based on theoretical contact mechanics provided by Thornton and Yin [2] and Thornton [3]. For detailed information about the numerical methodology, the readers could refer to Cundall and Strack [1]. The simulations are performed in a periodic cell in which a large size particle (submerged particle) is created at the centre of the cell and surrounded by monodispersed spherical particles (generated randomly). The boundaries of the periodic cell from the centre of the cell were at a distance of more than ca. 4 times radius of the submerged particle. The following periodic systems were considered: (i) For comparison, an entirely monodispersed system of particles, (ii) System with a large inclusion, otherwise all other particles in the periodic cell are monodispersed. Different values of size ratio (ratio of the diameter of submerged particle to that of surrounding particles) were considered, viz., 5 and 10. The random assembly created were isotropically compressed to a stress level of 100 kPa. All the samples considered here were having elastic properties corresponding to

37

'hard' particles (Young's modulus E = 10 GPa Poisson's ratio V = 0.3, coefficient of interparticle friction fi = 0.3, and interface energy F = 0.6 J m~^). After the particles were initially generated, a servo-control algorithm was used to isotropically compress until a mean stress p = 100 kPa was achieved. At the end of the isotropic compression, the microstmcture of the samples was isotropic. At this stage, the solid fraction and mechanical coordination number (average number of load bearing contacts) of the samples considered in this study were 0.650 ± 0.017 and 5.83 ± 0.26, respectively. For shearing, a strain rate of 10~^ s~^ was employed in the simulations. The samples were subjected to the axi-symmetric compression test (ai > a2 = 0-3). During shearing, the mean stress p = (ai -\- a2 -\- 0-3)73 was maintained constant at 100 kPa using the servo-control algorithm.

3. Results and discussion Fig. 2 shows the variation of macroscopic shear resistance of the granular systems during shearing (deviator strain = Si —S3). The shear resistance has been presented in terms of the shear stress ratio q/p, defined as the ratio of deviator stress q (= ai — as) to the mean stress p. For an increase in size ratio, the granular system tends to develop maximum shear resistance at an early stage of shearing. It is shown that the mobilised shear resistance of the granular system (at steady state) reduces for an increase in the size of the submerged particle. Earlier numerical investigations on the quasi-static behaviour of granular systems have revealed [7-9] new insights into the physics of granular media. For a granular system undergoing slow shearing, the shear strength of the system depends on the ability of the system to build strongly anisotropic fabric network of contacts carrying greater than average (strong) normal force. The fabric anisotropy in the granular assembly is defined by the distribution of contact orientations, defined

0.10 deviator strain

Fig. 2. Variation of shear stress ratio during shearing.

0.20

S.J. Antony, M. Ghadiri /First MIT Conference on Computational Fluid and Solid Mechanics

38

0.00 0.00

0.10 deviator strain

0.15

submerged particle increases. The weak contacts for all the systems are nearly isotropic at all stages of shearing. It may be recalled that (Fig. 2), the granular system developed less shear resistance for an increase in the size of inclusion and this could be attributed to the fact that the system is unable to build up strong anisotropic fabric net work as the size of inclusion increases. However, this trend could change if the size of the periodic cell reduces (boundaries are at a distance of more than ca. 4 times radius of the inclusion in this study) and this is yet to be investigated.

0.20

(a)

0.75

Acknowledgements

Mono-dispersed (strong contacts) - Size ratio - 5 (strong contacts) "Size ratio -10 (strong contacts) Mono-dispersed (weak contacts) Size ratio - 5 (weak contacts) Size ratio -10 (weak contacts)

0.65

a 0.55 o

This work has been supported by EPSRC and ICI Strategic Technology Group Technology Ltd., Wilton, U.K (Grant No. GR/M33907).

References

-0.051^ 0.00

0.05

0.10 deviator strain

0.15

0.20

(b) Fig. 3. Variation of fabric anisotropy during shearing, (a) Entire system, (b) due to strong contacts only.

by a 'fabric tensor' 0,y, suggested by Satake [10] as 1

^

(1)

where M is the number of contacts in the representative volume element and rij define the components of the unit normal vector at a contact between two particles. The variation of deviator fabric (0i — (ps) of the entire assembly is presented in Fig. 3(a) while in Fig. 3(b), the deviator fabric of contacts carrying strong and weak force are bifurcated. It may be observed that there is a strong anisotropic structure for contacts carrying strong forces within the overall system. However, the granular system develops a less anisotropic fabric structure of strong forces as the size of

[1] Cundal PA, Strack ODL. A discrete numerical model for granular assemblies. Geotechnique 1979;29:47-65. [2] Thornton C, Yin KK. Impact of elastic spheres with and without adhesion. Powder Technol 1991;65:153-166 [3] Thornton C. Coefficient of restitution for collinear collisions of elastic-perfectly plastic spheres. J Appl Mech 1997;64:383-386 [4] Huntley JM. Fluidization, segregation and stress propagation in granular materials. Philos Trans R Soc Lond A 1998;356:2569-2590. [5] Duran J, Mazozi T, Clement E, Rajchenbach J. Size segregation in a two dimensional sample: Convection and arching effects. Phys Rev E 1994;50(6):5138-5141. [6] Bonnenfant D, Mazerolle F, Suquet P. Compaction of powders containing hard inclusions: experiments and micromechanical modelling. Mech Mater 1998;29:93-109. [7] Antony SJ. Evolution of force distribution in three dimensional granular media. Phys Rev E 2001; 011302. [8] Thornton C, Antony SJ. Quasi-static deformation of particulate media. Philos Trans R Soc Lond A 1998;356:27632782. [9] Thornton C, Antony SJ. Quasi-static deformation of a soft particulate system. Powder Technol 2000;109:179-191. [10] Satake M. In: Vermeer PA, Luger HJ (Eds), Deformation and Failure of Granular Materials. Rotterdam: Balkema, 1982, pp. 63-68.

39

Hierarchical a posteriori error estimates for heterogeneous incompressible elasticity R. Araya%P.LeTallec'''* ^ INRIA, France, and University of Conception, Chili ^ Universite de Paris-Dauphine, and Ecole Poly technique, 91 128 Palaiseau Cedex, France

Abstract In this work we present a recent a posteriori parameter free error estimate of hierarchical type that we apply to the finite element solution of elasticity problems involving heterogeneous and piecewise incompressible materials. This estimate is proved to be optimal, independently of the material heterogeneities or Poisson ratio. Insight on the industrial relevance, numerical implementation and various numerical examples will also be presented. Keywords: Error estimates; Reliability; Heterogeneity; Elasticity

1. Introduction Recent accidents have clearly demonstrated that reliable a posteriori error estimates and mesh adaption techniques were imperatively needed at an industrial level when computing large scale structures. From the theoretical point of view, this problem can apparently be solved either by using consistent residual estimates or by solving local auxiliary equilibrium problem at the element level. When used on real industrial problems, such as in tire industry, these theoretical strategies are faced with two main difficulties: • the constitutive materials are complex, anisotropic and strongly heterogeneous, • most engineering codes use second order or higher order elements, for which the theoretical tools are harder to implement and to derive. To overcome these difficulties, we have developed a parameter free optimal a posteriori error estimator. For this purpose, we have extended the general theory developed by Bank and Weiser (cf. [3]), and have proposed different choices of local spaces. The resulting estimate uses easy to compute element and interface residuals, and inverts them locally by solving local tangent elasticity problems. By adding a weighted estimate of the error on the pressure term, it can be extended to handle the case of incompress* Corresponding author. Tel.: +33 1 69 33 40 02; Fax: +33 1 69 33 30 31; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ible materials. It is completely parameter free, and as such is easy to implement within an industrial code. On the theoretical side, this estimate can be proved to be correctly adimensionalized with respect to the physical data, and to be uniformly valid with respect to material heterogeneities, and bulk modulus, even at the limit of incompressible or almost incompressible materials.

2. The continuous problem Let ^ be a bounded domain of R^ occupied by an elastic body. The body is supposed to be fixed on a part TB of its boundary F := 9^, with meas{Tj)) ^ 0, and subjected to applied loads on its remaining part F^^. In this framework, we consider the following elasticity problem

I —diva = f (P)

I [

in ^,

u = 0 onFz), a • n = g on Fjv,

where f e L^(^)^ and g e l?{Tj^)^ are the given external forces and a is the stress tensor. For compressible materials, this tensor satisfies the constitutive law
A(x)e(u),

with A(x) the elasticity tensor of the constitutive material and Sij(u) := \(uij -\-Ujj) the components of the linearized strain tensor e(u) associated to u.

40

R. Araya, P. Le Tallec/First MIT Conference on Computational Fluid and Solid Mechanics

For incompressible or almost incompressible materials with locally very large Lame coefficients X, the stress tensor involves in addition a pressure like term, and can be split into

that this finer space can be easily split into small local finite element subspaces, meaning that there exist M subspaces H, of W/, such that

W, =HO

a = (aij) = A(x)e{vL) + pX, -p = div u on Qjnc,

+ X]H„

(5)

A

where A{x) denotes a bounded compressible like elasticity tensor, and where the very large value of volume change stiffness is transformed into a very large value of the coefficient X{x). Nonlinear elastic laws can be handled in a similar way, and in particular, such formulations characterize the pressure in terms of the stored energy gradient by W-\p)

= d e t ( / + V u ) - 1.

For the compressible case, the standard weak formulation of problem (P) is then: Find u € H such that a{u,Y) = (F,\)

,VVGH,

(1)

where H

: = { v € H n ^ ) ^ I v^OonFz)},

aiu,\)

:= f^A{x)e(u)

with Ho = H;,. Associated with each subspace H/ there is a projection operator P/ : H -> H, given by the solution of the local, easy to solve, compressible equilibrium problem fl(P/V, w/) = fl(v, w/),

"iy/i G H/, P/V G H/.

Similarly, we can introduce the global projection operator Pw : H -^ W;, defined by fl(PwV, Zh) = «(V, Zh)

VZ;, G W;,, P^^V G W/,.

Under this notation, we obviously have W/j = P^u. In a compressible framework, the error e = u — u/j in the finite element approximation of the displacement solution of (1) by elements of H/, satisfies the variational equation (6)

VVGH,

a(e,\) = {Rh,y),

where the residual} Rh is the element of H' given by the abstract form

: e(v),

(F,v> : = ^ f . v + / ^ ^ g . v .

(P,,v>:=(F,v)-«(u,,v),

VVGH.

(7)

Similarly, partially incompressible materials are governed by the mixed formulation Find u G H and p in P such that

Using the Green's theorem, the residual Rh is given as a sum of local components

a(u, v) + / pdiw\=

(2)

(P,,v>= ^ ( P r , v ) o T +

(3)

where the local element and face components Rj and RE are given by

/

(divu

A

p)q =0,

(F, v>,

Vv G H ,

"iq e P.

We finally define the (bounded) elastic energy norm by: := a{y, v)

/

A(x)ei\)

RT = (div
Yl

(^^'^)o,F

T eTh

(8)

(9)

and : e{\)

Vv G H.

(4)

fO RF =

g-cTh

if n

[a,-n]

3. Hierarchical intrinsic error estimator 3.1. Abstract construction and fundamental example Let us approximate this basic elasticity problem by an initial finite element space H/, c H, with finite element solution Uh. In order to estimate the quality of the numerical solution obtained on this finite element mesh, we assume that we can construct a finer finite element space W/, such that H/j c W/j C H and we denote by W/, the corresponding unique solution of problem (1) in this finer space W/,. This solution is not to be computed explicitly. We only assume

FeShrWo.

if F G^'/JOFA^,

(10)

if F G ^ , - F .

With the above definitions our hierarchical a posteriori error estimate rjn in the compressible case is simply defined by the local additive decomposition ( M

^/^ = r^«(/'/e, P/e)

.1/2

,

(11)

where each local displacement P/C is solution of the local equilibrium subproblem: Find e, G H, such that a{ei,\i) = {Rh,yi),

Vv/ G H / .

(12)

R. Araya, P. Le Tallec/First MIT Conference on Computational Fluid and Solid Mechanics This error estimate can be formally justified by the abstract calculation outlined below: {Res, vf e ^^= A ^Res j = sup -

41

where Ej is the restriction of Young's modulus to the element T, and FT

UGH

\2

{Res, vY

' sup

Then, for a proper1/2choice of subspaces Hi, we have

(calculation of dual energy norm)

= a{P^^v, Pwv) ^ Y^aiPrVR,

(saturation lemma)

PTVR)

(partition lemma)

e

^ ^a{A:^^VjRes,

A-^I'j^Res)

{Rh,hF) _ fl(b^,b/.)i/2 ~

T

+ ^fl(A;i/;/?^^,

A-'llRes)

{F,hF)-a(Uh,hF) ^(b/.,b/.)i/2

_

e

HBT = span{bjRT],

T eTh,

(13)

HB, = span{b*j,RF},

F e EH,

(14)

L^'^F-f^(r(Uh):e{hF) fl(bF,b^)i/2

Instead of detailing the above proof, we restrict our attention to the choice of the finite dimensional subspaces H/. The idea is to choose these subspaces as local as possible and thus to compute the projections P/e in a cheap way. As a basic example we will use local face (one per face F e Sh) and element (one per finite element T) subspaces given by

v-^ f ITI (f + diva/,) • bF - f^j. C/, • n • b^ 1

^ ^ [ ~ ^ |

E +

a(hF,hFy/^

3.2. A particular case In the particular case where the local subspaces Hj are one dimensional (H/ = span{\i}, i = 1,... , M), it is easy to prove that we have simply

=E

1/2

a{\i,\i)

We can then show the relationship between the weighted residual error estimate proposed by Araya and Le Tallec [2] and the above hierarchical error estimate. Lemma 3.1. For isotropic heterogeneous materials, let rinj be the error estimator given by

j"^

«(bF,bF)i/2

e(hF) o,Ti

V^

Wh ' n] o,F b/7 o,F E\ e(bF) o,ri + V ^2 ei^^p) 0J2

V^l + F2

with weights automatically obtained from the local elasticity tensor at considered point.

/^[g, -nl-b^

f+divcT/, 0,7- b/7 0^7.

11/2

(

rjH

J

a(hF,hFy/^ f/^^.(f+div(T,).bF]

where b^ = A1A2A3A4 and b*p- = A1A2A3 are the usual element and face polynomial bubble functions with support strictly included in the corresponding element or face, respectively. Thus, our error estimate Y)H can be written as the following weighted sum of element and face residuals

^H

(16)

Z ! ^^Rj I

Proof outline. We define the subspace W/^ by W/^ = Hh + E ^ i H/ where the one dimensional subspaces H/ are spanned by the functions b^ = l];=i ^F^y with F e Sh, yielding

T

+ 22, ^(^eVR, PCVR)

nn

Wh-n]\\ \\Q,F'

3.3. Incompressible extension Up to now, our hierarchical error estimate has only been introduced for compressible materials. The introduction of locally very large values of Lame coefficients X changes the definition of the local (bounded) elasticity tensor A, and motivates the introduction of an auxiliary pressure space to approximate the extra stress term p = A div M. This constitutive law is no longer exactly satisfied at the finite element level, and therefore, the full a posteriori error estimate must add an energy error term associated to this volume change, and given by II^PII'

= Yl f\\A\\{det(I^VuH) - 1 - W-/{C,pH)y. T

^

The total error estimate is then given by

= J2 ''(Pre, Pre) + Y, «(^^^' P^e) + \\ep ||'. TeTh

F^^h

(17)

42

The above formula can be proved to be scale invariant, and to be asymptotically correct for heterogeneous isotropic materials independently of coefficients jumps, and of the local values of the Poisson coefficients. In other words, we can prove that there exists constants independent of mesh parameters h and of elasticity constants A and X such that •Uh

\ln
as it would be the case by using more sophisticated balanced residuals as advocated by Ainsworth and Oden [1], or Ladeveze and Pelle [4].

Acknowledgements The work of RA was partially supported by FIRTECH Calcul Scientifique.

nr < C_\ \\u - uA References

hi. T'CWT

^ 1/2

(18) EG£(T)nSM

^

Experimental results to be presented at the conference indicate the relevance of the proposed estimate. It turns out to be both practical and theoretically sound. It can be implemented in a fully automatic and local way in any industrial finite element code. Its present limitations are twofold: On one hand, the theory cannot handle strong anisotropic effects. On the other hand, the practical calculations do not permit a locally accurate stress reconstruction,

[1] Ainsworth M, Oden JT. A posteriori error estimation in finite element analysis. Comp Methods Appl Mech Eng 1997;142:1-88. [2] Araya R, Le Tallec R Adaptive finite element analysis for strongly heterogeneous elasticity. Rev Eur Elem Finis 1998;7:635-655. [3] Bank RE, Weiser A. Some aposteriori error estimators for elliptical partial differential equations. Math Comput 1985;44:283-301. [4] Ladevze P, Pelle JP, Rougeot PH. Error estimation and mesh optimization for classical finite elements. Eng Comput 1991;8:69-80.

43

Localization phenomena in randomly microcracked bodies G. Augusti, P.M. Mariano*, F.L. Stazi Dipartimento di Ingegneria Strutturale e Geotecnica, Universitd di Roma 'La Sapienza', via Eudossiana 18, 00184 Rome, Italy

Abstract Stochastic finite elements are obtained within the setting of multifield theories of soHds for randomly microcracked bodies. Strain localization effects appear even if the constitutive relations are linear and microcracks are elastic and do not grow. Keywords: Strain localization; Microcracks; Multifield theories; Random finite elements

1. Introduction To analyze a microcracked body as a continuum, the relevant region B of the three dimensional Euclidean space can be described by two fields: the placement field x and an order parameter field d [1]. Both fields are considered as observable variables, thus balanced interactions are associated to them and their gradients. Interactions pertaining to the field d and its gradient Vd (called sub structural) provide extra power and satisfy appropriate balances besides Cauchy's. Substructural interactions are represented through a tensor S (micro-stress) and a vector z (selfforce). The problem of finding coupled constitutive relations for the stress measures has been tackled in [2,3]: they were obtained from a discrete model by means of an identification procedure based on power equivalence with the continuum model. The discrete model is made by two lattices connected each other by elastic links: the former (macrolattice) describes the body at the molecular level and is constituted by rigid spheres connected by elastic links, while the latter (mesolattice) represents the mesolevel of the microcrack distribution and is made by empty shells connected by elastic links. In the present paper the attention is focused on the case in which the distribution of microcracks is stochastic within the body. This may be accounted for by considering random the number of microcracks and their position [4,5] in the discrete model or by introducing some random stiffness in the links connecting the two lattices and determining the * Corresponding author. Tel.: +39 06 4458-5276; Fax: +39 06 488-4852; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

degree of coupling. Finite elements already obtained in the deterministic case in [6] may show random features.

2. Continuum model Let B be the regular connected region of the threedimensional Euclidean space 8^ occupied by the body in its reference configuration. Points in B are labeled by X. We consider B as free of discontinuities due to microcracks but define on it two fields: x(-) and d(').ln particular, d(-) accounts for the presence of microcracks. At each X, x(X) is the actual placement of the point X, then u = x — X is the displacement field and d its perturbation induced by microcracks so that (u -f d)(X) is the overall displacement. Motions are indicated by x(X, t) and d(X, t), with t the time, and velocity fields by x and d. In particular, if q(0 is the characteristic vector of a time parameterized family of rotations, velocity fields are called 'rigid' if [1,3] : c(0 + q(0 X (x - xo),

d = d X q(t)

(1)

with c(0 representing translational velocity and XQ some fixed point. The order parameter d is not affected by translational velocity because it is a relative displacement. If we indicate with the term part any regular connected subset of B and indicate it with B*, then the overall power V(B*) developed by interactions on B* is given by V{B*) = / (Tn • 11 + <Sn . J) - MX . F + z . J + SVd) 35*

I

(2)

where n is the outward unit normal to dB*,F = Vx, T the

44

G. Augusti et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Stress tensor. Body forces are here neglected for simplicity. The balance of 'forces' is assured by assuming that V(B*) vanishes for any choice of the velocity fields and of B*. This implies Div T = 0,

DiwS-z

= 0 inB

VR

(3)

SkwCTF^ + z 0 J + (VdfS)

(4)

3. Discrete model, constitutive equations and finite element The topology of the discrete model has been described in the introduction; we assume additionally that the discrete model is periodic and focus our attention on its characteristic cell (RVE). Measures of deformation in the discrete model are d^, d^ - d^, u^ - d^, u^ - u^ where d^ is the displacement between the margins of the shell at h and u^ the displacement of the sphere at a. There are links between h and k, a and h, a and b: they carry only axial forces. In the following, t, represents the force exerted by the /th link in the macrolattice, ZQ the force due to the relative displacement d^ of the margins of the hih shell, Zj the force in the 7 th link of the microlattice, Z/ the force in the /th interlattice link. The identification procedure of the constitutive equations in the continuum model goes as follows: first one equalizes the power developed in the RVE with the density of the internal power in the continuum, then deformation measures in the RVE are expressed in terms of the measures in the continuum [2,3]. In particular, we choose a point x in the RVE such that

d^ =

dix)-\-Vd(x)(h-x),

(5)

d^-d^ = Vd(x)(h~k)

(8)

^zS(8)(h-x)

VR

The balance of 'torques' is assured by requiring that the internal power /^^ (T • F + z • ^ + 5 • V j ) vanishes for any choice of 'rigid' velocity fields and of B*: it follows that where Skw() extracts the symmetric part of its argument. Cauchy stress TF^ is not symmetric in this treatment unless the microcrack distribution is such that the second order tensor z ^ d ~\- {Wd)^S be symmetric. It is just the analytical structure implied by equation (3) (a partial differential equation) and the constitutive dependence on Vd that assure the possibility of obtaining localization phenomena.

•E^s

+ £]z,0(h-k)- ^z/0(h-x)j /=i

j=\

When appropriate constitutive equations are chosen for the interactions in the lattices, tensors T, S and vector z can be expressed in terms of Vu, J, Vd. In the simplest case one may write T = AVu + A'VJ,

z = Cd,

<S = A^Vu + GVJ (10)

where k, h!,Q are fourth order tensors, C is second order and all of them have major symmetries (see [3] for explicit expressions). Finite elements can be built up by selecting any regular tessellation less of B and indicating nodal displacements for each element B^ e tess with u^ and d^. The element displacements u^ and d^ are related to u^ and d^ by matrices of shape functions: u^ = O^u^ d^ = OJ J^ Different discretized problems can be obtained: the simplest one is given by K

j{VlYAV<^l Be

|(VcD^)VvcI>^, Be

|(VO^,)VvcD^ I ^ f CO^ + (VO^f GVd)^ /

^f.t

9 Be

/

(11) O^r

where t and r are boundary data. The extended stiffness matrix K in (11) depends on the number M of the shells in the RVE. If this number is a random variable we can expand K(M) by Taylor expansion as follows: K = K + dMK\^dM + i9^^K|^dMdM . . .

u ^ - u ^ ^ Vu(x)(a-b), u^-d^

= Vu(x)(a - x) + VJ(x)(h - x)

(6)

At the end of calculations one obtains the measures of interaction in the continuum in terms of the forces in the links of the RVE [3]: 1

L

^

LN

t,- 0 (a - b) + ^ z/ (g) (a - x)

(7)

(9)

/

(12)

where the superposed line indicates mean value and dM the first variation of M about M. However, the choice of an expansion about the mean value of M need be matched by some rule establishing the topology of links between shells: many lattices correspond to any given number of shells. M could be also considered as a random field: in each cell we could have random geometry of the microlattice and hypothesis of lattice periodicity would result weakened. This

G. Augusti et al. /First MIT Conference on Computational Fluid and Solid Mechanics

45

MSPLACEMENT1 3.47E-03 2,08E^2

CMSPLACEMENT2

S

-2.S2E-03 2.82E-03

; "til II i*»fcJvlS!ll*lk«*»"''-"--»-- -•-

:::::: :::rf :ti*:t:H._-

-

c

nil—

"""•

d

Fig. 1. In-plane overall displacements of a membrane in tension, shown in a normalized intensity color scale (a, b) and by the magnitude of the nodal overall displacement vector (c, d). (a, c) Horizontal component (Displacement 1); (b, d): vertical component (Displacement 2). procedure could introduce great difficulties in developing calculation. Alternatively, to account for the influence of randomly distributed microcracks on the gross mechanical behavior of the body, one could fix a microlattice and assume that interlattice links, which govern the degree of coupling in the mechanical problem, have all the same random stiffness H. Under these assumptions, expansion (12) becomes:

K = K + anKlj^de + |a2jjK|j|dMde...

Fig. 1): interaction measures T, z, S associated to K need be considered as averaged quantities on some class of admissible geometries for the microlattice. By indicating with ()graph the average on some class of admissible graphs, we may interpret T, z, S as 1

^ t , - 0 (a - b) + ^ z / (8) (a - x ) |

VR

(14)

graph

(13)

Even M could be considered a random field and procedures in [7] appHed. Fig. 1 (taken from [3]) is an example of in-plane displacements calculated for a square membrane of stiffness K, fixed on the left-hand side and loaded by a concentrated force in the middle of the right-hand side. The constitutive equations have been derived considering a discrete model with square symmetry (4 spheres, 4 shells). Fig. la,c show, by two different representations, the horizontal component (Displacement 1) of the overall displacement while Fig. lb,d show the vertical component (Displacement 2). If the relevant quantities are random in the sense explained above, K can be taken as the average value K (see

E^

VR

s=

(15) graph

^zS(8)(h-x)

+ ^ z , (8) (h - k) - J2zi (8) (h - X) \ ^•=1

^=1

(16)

/graph

Of course, in the sense of (13) ( )graph must be interpreted as the average with respect to H at a fixed graph.

46

G. Augusti et al. /First MIT Conference on Computational Fluid and Solid Mechanics

References [1] Mariano PM. Some remarks on the variational description of microcracked bodies. Int J Non-linear Mech 1999;34:633642. [2] Mariano PM, Trovalusci P. Constitutive relations for elastic microcracked bodies: from a lattice model to a multifield continuum description. Int J Damage Mech 1999;8:153-173. [3] Mariano PM, Stazi FL. Strain localization in elastic microcracked bodies. Comput Methods Appl Mech Eng, in print.

[4] Klain DA, Rota GC. Introduction to Geometric Probability. Lezioni Lincee. Cambridge: Cambridge University Press, 1977. [5] Kolchin VF. Random Graphs. Cambridge: Cambridge University Press, 1999. [6] Mariano PM, Augusti G, Stazi FL. Finite element simulations of strain localization induced by microcracks. Mech Mater, in print. [7] Liu WK, Belytschko T, Mani A. Random field finite elements. Int J Numer Methods Eng 1986;23:1831-1845.

47

A method to analyse the nonhnear dynamic behaviour of rubber components using standard FE codes Per-Erik Austrell *, Anders K. Olsson, Martin Jonsson Department of Mechanics and Materials, Division of Structural Mechanics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden

Abstract For filled elastomers damping is caused by two different mechanisms at material level, resulting in viscous (rate dependent) and frictional (amplitude dependent) damping respectively [1,3]- In the one-dimensional case this can be modelled with a rheological model consisting of a viscoelastic component coupled in parallel with an elastoplastic component according to Fig. 1. Constitutive models for rubber used in standard FE codes are usually either hyperelastic or viscoelastic. Elastoplastic models, needed to model the frictional damping, are also normally supplied in order to model the plastic behaviour of highly stressed metal. The aim of this work is to propose an FE procedure that is able to represent the dynamic behaviour of rubber materials including both rate and amplitude dependence as well as nonlinear elastic behaviour. The overlay method offers a method to obtain such a model using only the already implemented constitutive models in standard FE codes. The result is an FE model corresponding to the one-dimensional generalized rheological viscoplastic model discussed in Section 1. Keywords: Filled rubber; Viscoelastic; Elastoplastic; Damping; Finite element method; Amplitude dependence

1. Introduction Carbon black filled rubber consists of long polymer chains and a structure of microscopical carbon particles connected by weak crosslinks. Reorganization of the rubber network during periodic loading results in a viscous type of damping. The frictional damping is attributed to the filler structure and the breaking and reforming of the structure which take place during loading and unloading. The stresses obtained in a filled rubber material can thus be divided into a dominant elastic part, but also a viscous and a frictional part. Combining the viscoelastic and the elastoplastic models in parallel yields a material model which sums the elastic, viscous and frictional stresses. A simple five-parameter model of this viscoplastic type is shown in Fig. 1. The model simulates the rate and amplitude dependence in a physically correct manner. Filled rubber materials subjected to harmonic loading show combined frequency and amplitude dependence of * Corresponding author. Tel: +46 (46) 222-4798; Fax: +46 (46) 222-4420; E-mail: [email protected]; URL: http://byggmek.lth.se/ © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

the dynamic modulus and phase angle. The behaviour of the material model in Fig. 1 in harmonic loading is illustrated in Fig. 2. The phase angle is a measure of the damping and thus also a measure of the hysteresis, i.e., a large phase angle yields a large difference between the loading and unloading curves. Values of the modulus and phase angle, for which the amplitude and frequency results in a power output which exceeds a certain limit have been excluded from the figure. The one-dimensional model shown in Fig. 1 can be generalized by adding more viscous and frictional elements in parallel. The model can then be given a quantitative better fit to experimental data. In Section 2 this model is

Fig. 1. Mechanical analogy illustrating a simple viscoplastic material model resulting in a frequency and amplitude dependent dynamic modulus and damping.

48

RE. Austrell et al. /First MIT Conference on Computational Fluid and Solid Mechanics Dynamic shear modulus: Gdyn(A,f)

Equivalent phase angle: Arg(A,f)

100

100 Shear strain amplitude

Shear strain amplitude

Fig. 2. One-dimensional viscoplastic material model. Amplitude and frequency dependence of the dynamic modulus and phase angle.

generalized into three dimensions for the purpose of finite element calculations. 2. The overlay method According to the one-dimensional viscoplastic model shown in Fig. 1, the total stress is obtained by adding the elastic stress, the viscous, and the plastic stress. A direct generalization for a three dimensional stress state would be to add the elastic, plastic and viscous stress tensors. The total stress tensor a is then given by a ^a' +G'P + G'

(1)

where the different stress tensors are obtained from a hyperelastic, a elastoplastic and a viscoelastic material model. The hyperelastic contribution is in this paper according to a model by Yeoh [5]. The elastoplastic part of the stress tensor is given by a summation M

(2)

where the terms are obtained from a non-hardening plasticity model, according to von Mises, implemented for large strains. The model used in this paper uses three terms in the summation above. The viscoelastic stress contribution is also given by a summation according to N

E<^

(3)

k=\

where the terms are obtained from a visco-hyperelastic model, suitable for large strains. 2.1. Implementation of the overlay method An easy way to obtain a model according to Section 2 using standard FE codes, without having to program a new constitutive model, is to use an overlay of FE meshes. The basic approach using the overlay method, is to create one hyperelastic, one viscoelastic and one elastoplastic FE model, all with identical element meshes. Assembling the nodes of these models according to Fig. 3, yields a finite element model that corresponds to the five-parameter model discussed earlier. In order to create a model corresponding Hyperelastic FE-model

Rheological model

Viscoelastic FE-model

^-^r

Elastoplastic FE-model

FE-model containing: -Non-linear elasticity -Frequency dependence -Amplitude dependence Fig. 3. Basic idea of the overlay method. The different basic FE models are assembled into one model containing both frequency and amplitude dependent properties as well as non-linear elastic characteristics.

P.-E. Austrell et al /First MIT Conference on Computational Fluid and Solid Mechanics

49

5r

0

2 4 6 Displacement [mm]

Fig. 4. Amplitude dependent dynamic stiffness. Analysis of the cylindric component submitted to a radial cyclic load. to the generalized viscoplastic rheological model, a suitable number of viscoelastic or elastoplastic FE models are added in parallel. Preliminary investigations indicate that the material parameters needed for the finite elements models can simply be copied from the one-dimensional model which has been fitted to experimental data. The reason why the one-dimensional rheological model seems to be easily generalized into three-dimensions has not been thoroughly investigated. However, one reasonable explanation for this behaviour is that the isotropic and incompressible characteristics of rubber provides a constraint that reduces the degrees of freedom in the three-dimensional model.

3. Cylindric rubber bushing A cylindric component according to Fig. 4 has been studied in [2]. The component is submitted to large amplitudes at low frequencies. The very slow load rate makes it possible to neglect the viscous contribution. Hence, the material model used in this paper contains only the hyperelastic and the elastoplastic stress contributions. The component has been submitted to a variety of different load cases. Only the radial load case is presented in this abstract. Fig. 4 shows the cylindric component submitted to a radial loading. The load case is displacement controlled

and cyclic, with gradually increasing amplitude. The graph shows the relation between the radial force F, obtained from the finite element analysis, and the radial displacement. The graph also shows the influence of the nonlinear elastic stress contribution on the hysteretic response. The sharp corners of the hysteretic response is characteristic for the behaviour of highly filled rubber materials. If only the viscous damping was modelled the shape of the hysteretic response would be almost elliptic. References [1] Austrell PE. Modeling of elasticity and damping for filled elastomers. Lund University, Lund Institute of Technology, Division of Structural Mechanics, Sweden, 1997, Report TVSM-1009 [2] Austrell PE, Jonsson M. Analys av nagra axialsymmetriska gummikomponenter. Lund University, Lund Institute of Technology, Division of Structural Mechanics, Sweden 1999, Report TVSM-99/7129-SE (1-42) [3] Kaliske M, Rothert H. Constitutive approach to rate independent properties of filled elastomers. Int J Solids Struct 1998;35(17):2057-2071. [4] Simo JC. On a fully three-dimensionalfinite-strainviscoelastic damage model: formulation and computational aspects. Comput Methods Appl Mech Eng 1987;60:153-173. [5] Yeoh OH. Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chem Technol 1993;66:754-772.

50

Simulation of large deformations in shell structures by the p-version of the finite element method Y. Ba§a^^ U. Hanskotter% O. KintzeP*, Ch. Schwab'' " Institute for Structural Mechanics, Ruhr-University Bochum, Bochum, Germany ^ Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zurich, Switzerland

Abstract For the analysis of shell structures with large strains finite elements are developed ensuring an optional selection of the kinematic assumption, interpolation polynomials and particularly hyperelastic constitutive models. The essential idea of the development is to construct all the partial derivatives of the finite element procedure, e.g. with respect to the strains via the analytical tool of MATHEMATICA. The actual shell configuration is described by non-hierarchical as well as hierarchical higher-order polynomials. The reference configuration is considered exactly by means of algorithms applicable to various practically relevant cases e.g. geometry intersections. The use of higher-order interpolation polynomials substantially reduces the well-known locking phenomena connected with lower-order finite element formulations. The possibility to compute an entire shell which may be additionally bounded by arbitrary curves turns out to be a further significant advantage. The effectivity of the formulation particularly concerning the application of the hierarchical p-extension will be demonstrated by adequate numerical examples. Keywords: Finite shell elements; Hierarchical and non-hierarchical interpolations; Large strains; Finite rotations; Hyperelasticity

1. Shell equations Upper case letters denote geometrical elements of the reference state and lower case ones their counterparts in the actual configuration. Latin indices represent the numbers 1, 2, 3 and the greek ones the numbers 1, 2. The notations used are essentially adopted from [3]. We consider the reference state of a finite element continuum with a variable thickness H measured in the direction of the unit normal vector N of the midsurface. Let X = Xi^"",^^) be the position vector of an arbitrary point, where f' (/ = 1,2,3) are curvilinear coordinates selected such that the values ?' = dzl determine the curved boundaries of the finite element. The vector X is described by the following linear expression in thickness coordinate

with 0 1

;' = 0,1

* Corresponding author. E-mail: o.kintzel @ sd.ruhr-uni-bochum.de © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(1)

Xfl),

(2)

where X is the position vector of the midsurface. For single 1

0

shells the director X is selected as X = y N, while in the case of geometry intersections it can be advantageously determined according to (2) in terms of the position vectors XT and X^ of the top and bottom faces of the finite element continuum. The position vector x = xi^"^,^^) of the deformed configuration (actual state) is approximated by

X = X (X, ?^) = x ( r ) + ? ' x(r) + (?')' x(r), (3)

p = 0,1,2 X = X(X,t') = X ( r ) + §'X(r),

X = i (Xr

X = - (Xr + Xs),

with the 2D quantities x presenting the primary unknowns of the finite element procedure. The above quadratic polynomial provides the consideration of transversal strains de1

2

pending on x and x. Note that, within the present concept, the polynomial (3) may be enriched by further higher-order

Y. Ba§ar et al. /First MIT Conference on Computational Fluid and Solid Mechanics

terms in ^^. However, if more accuracy is required, the combined application of (3) with the multilayer concept [4] has been proved to be decisively more efficient. Starting from (1) and (3) the geometrical elements of the reference and actual states can be derived by the standard procedure [3], e.g. the covariant base vectors G/ = X,/ and g/ = x,/, whereas in other cases the partial differentiation ( ),/ with respect to §' is to be carried out through MATHEMATICA.

As deformation (strain) measures the right CauchyGreen tensor C and the Green-Lagrange strain tensor E are utilized subjected to the following kinematic constraints: E := Eij G' (g) G^' = i (C - G) (g/-g;-G,-Gy)G^'0G^

(4)

Any hyperelastic material can be modelled by a strain energy function W = W(Eij, G'J, a^) depending on the covariant components of E, the contravariant metric tensor components G'^ and a number of material parameters a^. An example is the Mooney-Rivlin model W = Ci(/c - 3) + €2(1 Ic — 3) given in terms of the invariants /c, / / c of C and including two material constants Ci,C2. This model is appHcable to incompressible rubber-like materials. Once a special function is selected for W the associated stress tensor can be obtained by partial differentiation with respect to E as S = W,E, which is again to be formed

51

the classical shape functions is their direct applicability for the interpolation. But, if higher-order approximation is required by adaptive strategies, they have to be completely reconstructed. The hierarchical shape functions do not use solely nodal values. They are built in this contribution by means of the Legendre polynomials. The well-known orthogonality property of these polynomials provides the significant advantage that, with increasing of the polynomial order, the new shape functions are obtainable from the foregoing ones simply through an additional extension, which saves considerably the computational efforts. This explains clearly the significance of the hierarchical shape functions for the application of adaptive hierarchical p-extension [5]. In both cited cases the 2D shape functions ^^Ar'"«(§«) = ^N'^(^^)^N''(^^) are constructed starting from the ID ones ^N"^, ^N"". Thus, the interpolation of the unknown parameter x' can be presented in a unified form by Pi

X

J:'<

(7)

Ij^m

where the indices mn are used to define a sum over the chosen 2D shape functions ^^N"^^. Note that in the case of a hierarchical approach the x^^ are not only nodal values. The construction of the shape functions are described in [2, 6]. In the latter work explanations are also given concerning the numerical implementation.

through MATHEMATICA.

For the finite element procedure the nonlinear shell equations are to be linearized. The linearization of the variation of the strain energy function 8W = 8W(\) and the kinematic constraint (4) with respect to the column vector v^ = [x x x] at the state x delivers: L8W = AS: 8E-\-S:

A8E-hS:8E

(5)

A8E = -{A 8gi • g,- + Ag,- • 8gj + 8gi • Agj +

gi-A8gj)G'^GJ.

(6)

3. Reference shell geometry A crucial point in developing higher-order shell models is the consideration of the reference configuration with an adequate accuracy. The first problem to be solved in this context is the definition of a finite element volume being convenient for the requirements of the given shell structure. This aspect will be enlighted here by two practically relevant cases. For more on this, especially concerning problems with geometry intersections we refer to [1].

The symbol A used above defines an operation to be performed similar to the variation 8. Both operations 8 and A imply partial differentiations with respect to the kinematic unknowns v to be built systematically through MATHEMATICA.

2. Finite element formulation For more flexibility the primary kinematic parameters p p . X = xJ ij (p = 0,1,2) involved in (3) are interpolated by non-hierarchical (classical) as well as hierarchical shape functions. The classical shape functions are closely related to the nodal points and are constructed here by means of Lagrange and Serendipity polynomials. The advantage of

Fig. 1. Discretization.

52

Y. Ba§ar et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Geometry and material data Geometry: A = 1.00 B= 1.00 Z7 = 0.00, 0.25, 0.50, 0.75, 1.00 d = 0.01 Material data: E= 1.00-10^ V = 0.20

We first assume that the shell is determined by its thickness H and its midsurface AQ described through the equation X = X ( 0 " ) in terms of arbitrary parameters @". To define a finite element area we first select four arbitrary nodal points A' ( ^ = 1 to 4) with the coordinates 0^^^ on AQ. If the transformation

4

4

?')(i + r)e^ (8)

= E^^(?")®5^ 0

0« = 0« [t (?2)]

0

X = X [ 0 " (^^)] = X ( ^ ^ ) , then a finite element area AAQ on AQ is determined whose boundaries are described by the discrete values ^'^ = ± 1 . Now, we suppose that one of the boundaries of the finite element area A A Q , e.g. the boundary passing through the nodes 2 and 4, has to correspond to a given curve C given by the relations 0 " = 0 J (t). In this case the first step is to replace the parameter t = r (^^) by the dimensionless coordinate ^^ e ( - 1 , 1). By using the corresponding result

0 ^ (?^) the transformation (8) is then

replaced by 0« = AT, 0« + TV^ 0« + - ( l - f ^1) 0« (^2)

1

vectors X and X so that the problem is reduced to the one discussed above. 0

1

The vectors X and X entering in (1) can be considered in an exact form (classical formulation) or alternatively 0

approximated in the same form as their counterparts x and X in the actual state (isoparametric approach). To save computation efforts the classical formulation is used in the present development. A well-known failure of this approach is that the rigid body motion criterion is not satisfied exactly. But it has been proved that this is only

-0.05

-0.15

-0.20

-0.25

200

400

(9)

In some cases it may be suitable to determine the finite element volume through the bottom Ag and top faces AT. Then, the finite element faces on A^ and AT can be determined by the same procedure as described above. The consideration of the corresponding results in (1) and (2) finally defines the finite element volume, more strictly, the 0

with ^" G (—1, 1) is considered in the midsurface equation 0

Fig. 2. Boundary conditions.

600

Fig. 3. Vertical displacement np^, of point P.

800 Degrees of Freedom

Y. Ba§ar et al. /First MIT Conference on Computational Fluid and Solid Mechanics

53

100.0 r ^ f e r — = = = r : # ^ ^ ^

10.0 ^

1000 Degrees of Freedom

Fig. 4. Relative error e of the total potential energy jr. a minor weakness having no particular influence on the numerical response. 4. Example The example is a thin plate under constant dead load P = 1.00, which has been applied on the face of the plate, acting in negative Xa-direction (Fig. 1). It is discretized with two elements, which will be distorted with a varying factor b: 0 means no distortion and 1 means distortion to a triangle. The goal of this experiment is to demonstrate the insensitivity of high-order FE-discretization to element distortion and irregular element shapes. Linear elastic material properties are taken into account (Table 1) and furthermore all boundaries in horizontal direction are fixed (Fig. 2). The convergence behaviour of the vertical displacement can be seen in Fig. 3. Each displacement curve mirrors a different mesh distortion and each point of the curve characterises a polynomial order. From polynomial order 4 all curves converge to the exact solution. In Fig. 4 the

asymptotic convergence of the relative error of the total potential energy in double logarithmic scale is to be seen.

References [1] Ba§ar Y, Hanskotter U, Omurtag MH, Schwab Ch. On the exact geometry description in the p-finite element formulation for hyperelastic shells withfiniterotations. 2001, in prep. [2] Ba§ar Y, Hanskotter U, Schwab Ch. A general high-order finite element formulation for shells at large strains and finite rotations. 2001, in prep. [3] Ba§ar Y, Weichert D. Nonlinear Continuum Mechanics of SoHds. Berlin: Springer, 2000. [4] Ba§ar Y, Ding Y Interlaminar stress analysis of composites. Layer-wise shell finite elements including transverse strains. Composites Eng 1995;5(5):485-499. [51 Szabo BA, Actis R, Schwab Ch. Hierarchic models for laminated plates and shells. Comput Methods Appl Mech Eng 1999;172:79-107. [6] Szabo B, Babuska I. Finite Element Analysis. New York: Wiley, 1991.

54

Simulation of frictional contact in three-dimensions using the Material Point Method S.G. Bardenhagen'''*, O. Byutne^^ D. Bedrov'', G.D. Smith'' " University' of Utah, Mechanical Engineering Department, Salt Lake City, UT, USA ^ University of Utah, Material and Engineering Science Department, Salt Lake City, UT, USA

Abstract An algorithm for applying frictional contact conditions in three-dimensions using the Material Point Method is described. The algorithm is computationally efficient, robust, and avoids the use of an interface stiffness parameter. Performance is assessed via a simple test problem involving large material deformations. Preliminary results on the dynamic compaction of granular material are presented. Keywords: Frictional contact; Arbitrary Lagrangian/Eulerian; Material point method; Finite deformations; Dynamic compaction; Granular material

1. Introduction During the performance of engineering systems, the majority of the loading applied to components is by contact with other components. Component contact loading can often be idealized in simulations of systems operating under design conditions, where these interactions are tightly controlled (e.g. by using bearings, lubrication or joints). However, the performance of a system outside of normal operating conditions may be equally important. Under severe loading, large deformation or failure of one component may result in unanticipated contact with other components. The classic example is a car crash. Contact and impact have received substantial attention over the past several decades, as witnessed by a review of the subject by Zhong [1], which lists nearly 500 papers. The majority of these papers describe numerical modeling approaches and/or applications using the finite element method. The problem is a very difficult one, as contact must be sensed, surface normals constructed, and interaction forces imposed to prevent interpenetration without making the system of equations to be solved ill-conditioned. Here we briefly describe an alternate approach using an arbitrary Lagrangian/Eulerian (ALE) particle-incell numerical technique for solid mechanics, the Material Point Method (MPM) [2]. * Corresponding author. Tel: +1 (801) 587-9819; Fax: +1 (801) 585-9826; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The numerical simulations described here are performed using the University of Utah's Uintah Computational Framework (UCF) in support of large scale, massively parallel computations, under the Center for the Simulation of Accidental Fires and Explosions (C-SAFE). The simulation scenario ultimately of interest to the Center is the thermo/mechanical/chemical response of a container filled with plastic bonded explosives (PBXs) in a fire. Algorithms designed to handle contact between container and explosive during initial heating, chemical decomposition, and fragmentation are needed. An MPM code has been implemented in the UCF to model the mechanical response of the container and explosives, and will ultimately couple with a fluid dynamics (fire) code. Simulation of the compaction of granular material provides a test of the mechanics code and an opportunity to model the response of a complex system only practical using large scale numerical simulations.

2. Approach Inherent in the MPM algorithm is a no slip contact condition between adjacent materials [3]. In addition, the MPM algorithm provides a convenient framework for applying more general contact conditions, including frictional contact and debonding. This framework avoids the use of an interface stiffness parameter (as for a penalty formulation), which can be difficult to select. The algorithm takes advantage of the overlying Eulerian grid to define

S.G. Bardenhagen et al /First MIT Conference on Computational Fluid and Solid Mechanics 120

^

en

^

80

-

60

-

40

-

'

Rigid Splnere Total Energy J e l l - 0 Sphere Total Energy J e l l - 0 Sphere Kinetic Energy J e l l - 0 Sphere Strain Energy Rubber Sphere Total Energy Rubber Sphere Kinetic Energy Rubber Sphere Strain Energy

/y/

-

///

55

/V // // // // — // // // / //

' -

/ /

-

y^!^^^y

-

^^-•r--

0.5

1.0 1.5 Time (sec)

2.0

Fig. 1. Specific energies for rigid and deformable sphere on inclined plane test problems. interfaces, calculate frictional forces and surface normals, and apply Coulomb frictional contact conditions [4]. The algorithm is computationally efficient, the cost is linear in the number of contacting materials. A separate contact detection step is unnecessary, and a solution is achieved with one sweep through the computational mesh. The algorithm reduces to the no slip condition inherent in the MPM algorithm when interfaces stick. These qualities make calculations involving large numbers of contacting materials tractable. However, extensive testing revealed a shortcoming. The formulation was found to violate the explicit stability condition on rare occasions when material point registration on the overlying computational grid met specific conditions. An addition to the algorithm was made to check for violation of the stability criterion and rescale the contact impulse as necessary. The modified algorithm retains the efficient qualities of the original, plus greatly increased robustness. One of the test problems investigated was that of a sphere on an inclined plane under gravity, initially at rest. For rigid bodies an analytic solution exists corresponding to rolling without slipping. For the (elastic) deformable cases the computational cell size is Z)/8, where D is the sphere's initial diameter. Eight material points per cell are used. Energies developed during rolling are plotted for the rigid sphere and two deformable sphere simulations in Fig. 1. For the rigid case, the total energy is equal to the kinetic energy, and is plotted with a thick black line. For the deformable cases, the total energy is the sum of the sphere's kinetic and strain energies. The first case is shown in gray and corresponds to both sphere and plane having stiffnesses approximately that of natural rubber. The strain energy reflects the regular occurrence of mild collisions, the contact algorithm results in the sphere skipping slightly.

Fig. 2. Initial and deformed configurations for the 'Jell-0' sphere, depicted by plotting material points. Simulation times for the deformed configurations are indicated by dotted lines in Fig. 1.

The majority of the energy is kinetic for the rubber case. Although natural rubber is fairly soft, deformability plays a small role and a reasonable resolution of the geometry results in a total energy very similar to the rigid case. Note that the rubber case has larger total energy in part because it free falls for a fraction of a cell length before contacting the plane. There is also some error accumulated during the simulation, which can be reduced by decreasing the explicit time step. To demonstrate the algorithm's ability to easily handle large deformations and the corresponding variation in the contact area, the sphere's elastic properties were reduced by a factor of 1000, resulting in a material approximating 'Jell-0'. The Jell-0 case is plotted in black, and inspection reveals the trade off between kinetic and strain energy as the sphere slumps down the plane. Snapshots displaying the initial configuration and deformed configurations at approximately 0.8-s intervals (indicated with dotted vertical lines in Fig. 1) are shown in Fig. 2. For this case, there is variation in interface velocity over the contact area resulting in some sliding during rolling, and corresponding dissipation of kinetic energy.

3. Application The behavior of granular material has received a fair amount of attention within the scientific community recently [5]. Because of the rich behavior granular material has been found to exhibit, and the ability to collect data

56

S.G. Bardenhagen et al /First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 3. Initial configuration consisting of 1000 spheres in a 1-mm cube. The grains are shaded differently only to distinguish one from another.

both on the scale of the individual grains and en masse, there is a large database available for validation. It has been found that a prevalent load carrying mechanism in granular material is provided by a small subset of the grains forming highly loaded connected paths of contacting grains, or 'force chains'. Dry granular material provides a relatively simple starting point and data for validation, but requires accurate modeling of many contacting grains. The simulation of the dynamic compaction of dense granular material further requires accurate modeling of grain deformation. These capabilities are precisely the strengths of the current state of the MPM code. A tool to generate dense packings of spheres with given size distribution using Monte-Carlo techniques was developed and used to create the initial configuration depicted in Fig. 3. There are 1000 spheres in a 1 mm cube with an 80% packing fraction. The size distribution is representative of that for the energetic grains in the PBX ultimately of interest within C-SAFE. The grains are modeled using a compressible Neo-Hookean plasticity formulation [6] with elastic material properties determined by molecular dynamics simulations, Sewell et al. [7]. The stress wave structure from a preliminary calculation with 10^ cells and 6 x 10^ material points is shown in Fig. 4. This resolution provides for five cells across the diameter of the smallest grains. The packing has been impacted from above by a piston with velocity 100 m / s . Only stressed grains are shown, displaying the non-uniform structure of the stress wave 0.12 JJLS after impact. Stress propagates most quickly through the large grains. It propagates more slowly through the smaller grains because grains must be brought into contact, and a meandering path must be traversed to reach a given depth. A closer look at an interior slice in the inset of Fig. 4

Fig. 4. Depiction of a stress wave propagating through the granular bed. Only stressed grains are shown, with maximum stresses in white. Two large grains are prominent in approximately opposite comers. Stress propagates more slowly in areas rich in smaller grains, as also seen in the diagonal slice inset.

(with the stress rescaled to emphasize stress paths) indicates the development of force chains among the smaller grains. The large grain in the middle of the slice carries large stresses, as do chains of small grains on either side. The sample size is too small to determine the effects on the stress wave structure of the interplay between large and small grains. Much larger simulations will be performed to provide a better representation of the measured grain size distribution, and to determine the sample size required for statistics representative of an essentially infinite number of grains (i.e. statistical information representative of the continuum scale).

4. Conclusions MPM is found to provide a convenient environment for the implementation of frictional contact. Preliminary results on granular compaction are encouraging, with simulations indicating preferential load paths developing during dynamic compaction. Ultimately of interest is a fundamental understanding the load carrying mechanisms and connections with continuum constitutive models via state statistics at the microscale. The inhomogeneous stress state resulting from the development of force chains may play a role in energy localization by promoting frictional sliding, plastic deformation and/or fracture. Work to incorporate fracture in these simulations is ongoing. The longer term objective is to include an interstitial material and simulate initiation mechanisms in PBXs and

S.G. Bardenhagen et al. /First MIT Conference on Computational Fluid and Solid Mechanics support the development of constitutive models. Composed of >90% by volume energetic grains in a weak matrix, there is evidence that force chains occur in PBXs as well [8]. It is generally agreed upon that non-shock initiation of PBXs is due to energy localization at the microscale and the development of 'hot spots'. Experimental information, already difficult and expensive to obtain for bulk energetic materials, is decidedly more difficult to obtain on the microscale. A fundamental understanding of the mechanisms of initiation in energetic materials will likely yield only to multi-disciplinary expertise and a closely coupled combination of numerical simulation and experimental validation.

Acknowledgements This work was supported by the U.S. Department of Energy through the Center for the Simulation of Accidental Fires and Explosions, under Grant W-7405-ENG-48.

References [1] Zhong Z-H, Mackerie J. Contact-impact problems: a review with bibhography. Appl Mechan Rev 1994;47(2):55-76.

57

[2] Sulsky D, Zhou S-J, Schreyer HL. Application of a particle-in-cell method to solid mechanics. Comput Phys Commun 1995;87:236-252. [3] Sulsky D, Schreyer HL. The particle-in-cell method as a natural impact algorithm. Adv Comput Methods Mater Model 1993;268:219-229. [4] Bardenhagen SG, Brackbill JU, Sulsky D. The material point method for granular materials. Comput Methods Appl Mechan Eng 2000;187:529-541. [5] Herrmann HJ, Hovi J-P, Luding S. Physics of Granular Media. Dordrecht: Kluwer Academic, 1998. [6] Simo JC, Hughes TJR. Computational inelasticity. New York: Springer, 1998. [7] Sewell TD, Menikoff R, Bedrov D, Smith GD, Ayyagari C. Elastic coefficients and sound speeds for HMX polymorphs from molecular dynamics simulations. J Appl Phys, submitted. [8] Foster JC Jr, Glenn G, Gunger M. Meso-scale origins of the low pressure equation of state and high rate mechanical properties of plastic bonded explosives. In: Furnish MD, Chabildas LD, Hixson RS (Eds), Shock Compression of Condensed Matter-1999. Woodbury: AIP Press, 2000.

58

On the modeling of shells in multibody dynamics Olivier A. Bauchau^*, Carlo L. Bottasso^ " Georgia Institute of Technology, School of Aerospace Engineering, Atlanta, GA 30332-0150 USA ^ Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milan, Italy

Abstract Energy preserving/decaying schemes are presented for the simulation of the nonlinear multibody systems involving shell components. The proposed schemes are designed to meet four specific requirements: unconditional nonlinear stability of the scheme, a rigorous treatment of both geometric and material nonlinearities, exact satisfaction of the constraints, and the presence of high frequency numerical dissipation. The kinematic nonlinearities associated with arbitrarily large displacements and rotations of shells are treated in a rigorous manner, and the material nonlinearities can be handled when the constitutive laws stem from the existence of a strain energy density function. Keywords: Shell analysis; Multibody dynamics; Energy preserving schemes

1. Introduction and motivation This work is concerned with the numerical simulation of geometrically exact shell models within the context of multibody system dynamics. While the partial differential equations that govern shell problems are well known, their numerical treatment is still the subject of active research. Indeed, numerical analysis tools for partial differential equations have significantly changed in recent years. In the past, general purpose discretization methods were developed, with emphasis on robustness, performance, and accuracy. These methods aimed at solving vast classes of problems such as ordinary differential equations, differential/algebraic equations, or hyperbolic conservation laws. This approach is now changing. Indeed, the differential equations that govern many problems in mathematical physics possess qualitative and structural characteristics that can be determined by studying their geometry. Classical examples of such characteristics are the invariants associated with Hamiltonian systems, the symplectic structure of the governing equations, or symmetries and attractors. There is increasing evidence that numerical methods that correctly recover the qualitative features of the underlying differential equations are often endowed with superior com* Corresponding author. E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

putational performance, greater robustness and improved accuracy. This new paradigm has resulted in the development of a new mathematical discipline, called geometric integration: a bridge that links the work of pure, applied and computational mathematicians. Simo and his co-workers were among the first to develop special integration procedures for nonlinear structural dynamics. They analyzed the problem of the dynamics of nonlinear elasto-dynamics [1], geometrically exact shells and beams [2]. In all cases, the idea was to design algorithms that ensure the discrete preservation of the total mechanical energy of the system, therefore obtaining unconditionally stable schemes in the nonlinear regime. However, increasing evidence points toward the fact that geometric integration is not sufficient, per se, to obtain robust integration schemes. While these schemes perform well for problem with a small number of degrees of freedom featuring a "smooth" dynamic response, they tend to be quite unsatisfactory when applied to the complex simulations encountered in many engineering applications [3]. In fact, the predicted time histories of internal forces and velocities can present a significant high frequency content. Furthermore, the presence of these high frequency oscillations hinders the convergence process for the solution of the nonlinear equations of motion. These oscillations are particularly violent in multibody dynamics simulations because these systems are rather stiff due to the presence of numerous algebraic constraints, while the nonlinearities of

O.A. Bauchau, C. L Bottasso /First MIT Conference on Computational Fluid and Solid Mechanics

Shell (0.0,0) (0.0.0)

Revolute joints

Crank Fig. 1. Schematic of tlie snap-through problem of a cyhndrical shell activated by a crank and link mechanism.

the system provide a mechanism to transfer energy from the low to the high frequency modes. Consequently, the presence of high frequency numerical dissipation appears

t= 0.086 sec

t=0.265sec

to be an indispensable feature of robust time integrators for multibody systems. This paper focuses on the development a geometric integrator for shell structures that preserves important qualitative features of the underlying equations, and is equipped with high frequency numerical dissipation. In order to achieve these goals, the specific features of the equations governing nonlinear flexible multibody systems with shells are reviewed. First, the governing equations are characterized by linear and rotational tensorial fields describing kinematic (displacements, velocities) and co-kinematic (forces, momenta) quantities. Second, the equations are nonlinear because of large displacements and finite rotations (geometric nonlinearities), and possibly because of nonlinear constitutive laws (material nonlinearities). Third, the presence of joints imposes different types of kinematic constraints between the various bodies of the system. In this work, the Lagrange multipliers technique is used to enforce the constraints, giving the governing equations a differential/algebraic nature. Fourth, the equations of motion imply the preservation of a number of dynamic invariants, in particular the total mechanical energy, and the total linear and angular momenta. The proposed geometric integration procedure is designed to satisfy specific requirements. First, a discretiza-

t=0.156sec

/

59

t= 0.291 sec

I

Fig. 2. System configurations at various time instants during the simulation.

60

OA. Bauchau, C. L. Bottasso /First MIT Conference on Computational Fluid and Solid Mechanics

tion process is developed that preserves the total mechanical energy of the system at the discrete solution level. This process is independent of the spatial discretization procedure that is left arbitrary. In the present implementation, the finite element method is used, and the mixed interpolation of tensorial components [4] is implemented to avoid the shear locking problem. Next, the reaction forces associated with the holonomic and non-holonomic constraints imposed on the system are discretized in a manner that guarantees the satisfaction of the nonlinear constraint manifold, i.e. the constraint condition will not drift. At the same time, the discretization implies the vanishing of the work performed by the forces of constraint at the discrete solution level. Consequently, the discrete energy conservation laws proved for the flexible members of the system are not upset by the introduction of the constraints. The resulting Energy Preserving (EP) scheme is a geometric integrator for multibody systems with shells that provides nonlinear unconditional stability. Using a simple procedure [5,6] based on the EP scheme, it is possible to derive a new discretization that implies a discrete energy decay statement. In the resulting Energy Decaying (ED) scheme, the system no longer evolves on the constant energy level set, but is allowed to drift away from it in a controlled manner. The discretization process for the forces of constraint is left unchanged: the work they perform vanishes exactly, while the system evolves on the constraint manifold without drifts. ED schemes satisfy all the requirements set forth earlier.

2. Snap-through of a cylindrical shell A crank and link mechanism is used to drive a cylindrical shell through an unstable, snap-through configuration. The system geometry is depicted in Fig. 1. The shell consists of a 60° sector of a cylinder of height h = 2.5 m.

radius r = 5 m and thickness t = 0.1 m. Material properties are: Young's modulus E = 210 GPa, Poisson ratio y = 0.25 and density p = 10"* kg/m^. The two straight edges of the shell are simply supported, and one of the curved edges is free. The last edge is connected at its midpoint to a link by means of a revolute joint. Furthermore, its displacement along x and its rotations about the y and z directions are constrained to zero. The crank length is Lc = 1.5 m and its axis of rotation is located 5 m below the connection point with the shell. The crank is modeled as a rigid body, while the link is represented by a beam of rectangular cross section of side s = 0.2 m, with the same material properties as the shell. The two elements are connected by a revolute joint. The crank rotates at constant angular velocity ^ = 0.1 rad/s for half a revolution, and stops at time ^ = TT 10~^ s, while the simulation is continued until t = 0.4 s. Fig. 2 shows the response of the shell.

References [1] Simo JC, Tamow N. The discrete energy-momentum method conserving algorithms for nonlinear dynamics. ZAMP 1992;43:757-792. [2] Simo JC, Tamow N. A new energy and momentum conserving algorithm for the nonlinear dynamics of shells. Int J Numer Methods Eng 1994;37:2527-2549. [3] Bauchau OA, Damilano G, Theron NJ. Numerical integration of nonlinear elasfic mulfi-body systems. Int J Numer Methods Eng 1995;38:2727-2751. [4] Bucalem ML, Bathe KJ. Higher-order mite general shell elements. Int J Numer Methods Eng 1993;36:3729-3754. [5] Bauchau OA. Computational schemes forflexible,nonlinear mulfi-body systems. Multibody Syst Dyn 1998;2:169-225. [6] Bauchau OA, Bottasso CL. On the design of energy preserving and decaying schemes forflexible,nonlinear multi-body systems. Comput Methods Appl Mech Eng 1999;169:61-79.

61

Fragment impact pattern effect on momentum transferred to concrete targets J.T. Baylot^'*, P.P. Papados'' ^ U.S. Army Engineer Research and Development Center, Vicksburg, MS 39180, USA ^ U.S. Army Research Laboratory, Adelphi, MD 20783, USA

Abstract Impulses resulting from metal fragment impacts on concrete targets are needed to predict the structural response of those targets. Recent experiments indicate that the momentum transferred to the target exceeds the momentum of the fragments impacting the slab. These experiments indicate that the amount of the excess impulse is a function of the pattern of impact of the fragments on the slab. Finite- element (FE) analyses have been used successfully to predict damage to concrete targets from multiple fragment impacts. In this paper, these same analysis techniques are used to investigate the effect of fragment impact pattern on momentum transferred to the concrete target. Keywords: Reinforced concrete; Dynamic loads; Impact loads; Momentum transfer; Finite-element analyses

1. Introduction In this paper, excess impulse is defined as the percent difference between the impulse applied to the slab and the total momentum of all fragments striking the slab. The momentum transferred to reinforced concrete slabs by steel fragments impacting the slabs has been studied in two recent series of experiments, Dallriva [1]. In the first set of experiments, a single steel fragment was fired at a reinforced-concrete target. On average, the excess impulse was approximately 70%. The concrete on the front face of the slab near the impact point was ejected towards the direction of the fragment launcher at a reasonably high velocity. The excess impulse has been attributed to the ejection of this concrete. Thirty similar experiments were conducted for multiple fragment impacts. In these experiments, 24 fragments were fired at the concrete target. The excess impulse ranged from 24 to 54%, indicating a loss in effectiveness as compared to a single fragment impact. The experiments indicated that the reinforcing steel does not have a large effect on the momentum transferred to the slab.

* Corresponding author. Tel.: +1 (601) 634-2137; Fax: +1 (601) 634-2309; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Damage to the concrete targets in some of these experiments has been successfully predicted using FE models and applying the loads as a pressure-time history on the surface of the slab, Papados [2]. Papados used the large-deformation, explicit-dynamic FE code, ParaDyn [3], which is the scalable version of the code, DYNA3D-LLNL [4]. Details of the constitutive model are discussed in [2]. The constitutive model is a three-invariant, three-failure surface model as suggested by Willam and Warnke [5]. The surfaces represent the yield, maximum, and residual capacity of the concrete. Failure in tension is based on fracture energy. Once the material has reached the residual surface, it cannot support tension and cannot support shear in the absence of pressure.

2. Analyses performed Finite-element analyses were performed to assess the effect of the fragment impact pattern on the impulse transferred from 150-grain steel fragments impacting a 9-in.-thick concrete target at about 4,200 fps. Analyses were performed for one, two, and three fragments. Maximum concrete ejecta velocity and damaged surface area were used as a measure of excess impulse applied to the target.

62

J.T. Baylot, P.P. Papados / First MIT Conference on Computational Fluid and Solid Mechanics

A 30- by 30- by 9-in. slab was discretized into 518,400 1/4-in. cube constant stress continuum elements. The constitutive model and rate effects curves used were identical to those used by Papados. Reinforcing steel was not modeled. The fragment impacts were represented using pressure boundary conditions that preserve the momentum of the fragment. The pulse duration was selected to match the expected fragment penetration. The fragment size is consistent with the surface area of four of the 1/4-in. cube elements. The peak pressure and pulse duration were 224,000 psi and 0.1 ms, respectively. A rise time of l/4th of the pulse duration gives a good estimate of the damage for the multiple fragment experiments, and was used for these simulations.

18,000 15.000 12.000 (0

9.000 6,000 3,000

o

1

0 -3,000

/ /" u

Dlst. from impact, In. i 0 0 25 0.5 0.75 1

rz<

-6,000 \ ^"' j^

-9.000 -12,000

0

0.04

0.08

0.12

1

0.16

0.2

Time, ms Fig. 2. Velocity histories near impact point.

3. Single fragment impact A simulation for a single fragment impacting the center of the target was performed (4 elements are loaded). The impact location (displacements scaled by 5) at the end of the fragment pulse duration is shown in Fig. 1. This figure shows that the FE model initially captures the ejection of the concrete off of the front surface of the model. Positive displacements (shown as white) are displacements in the direction of the initial fragment velocity. Velocity histories of the nodes near the fragment impact are shown as Fig. 2. The nodes at 0 and 0.25 in. were loaded by the pressure boundary condition representing the fragment impact. These nodes initially moved in the direction of the fragment velocity, but reversed at a high velocity at the end of the pulse duration. The nodes not loaded by the fragment impact immediately moved opposite to the fragment velocity. Later in time, all of the nodes reversed direction and moved away from the fragment source. Because of this behavior, the concrete did not actually eject in the simulation, and the momentum initially gained was lost.

Fig. 1. Displacements near impact point.

Analysis results were examined to determine the source of the force that caused these 'ejected' nodes to be pulled back into the remainder of the slab. At 0.11 ms, the node at the fragment impact location accelerated with the velocity very quickly becoming positive. The externally applied forces were zero after 0.1 ms. Each of the elements connected to this node has failed and all stress histories for these elements remained at zero. The only other forces that could have been acting on this node are the hourglass control and the bulk viscosity. Since the bulk viscosity forces are only active in compression, the forces must have been due to hourglass control. Attempts were made to adjust the hourglass control and to delete the failed elements on the exposed surface. Neither of these efforts was successful in overcoming the problem of the reversal of direction of the failed concrete. The mass of the element was maintained upon element deletion, and apparently so were the hourglass forces. Since the recovery of the ejecta material could not be prevented, the ejecta velocities before the recovery were used to evaluate the excess impulse. Velocity histories of nodes totally surrounded by failed elements were examined to determine, the mass and average velocity of concrete that would be ejected. The area of failed elements on the surface of the slab is shown in Fig. 3. The maximum and average magnitudes of the ejecta velocity were 11,456 ips and 1,457 ips, respectively. Average velocities and damage volumes for the next two layers were also computed. Based on the first three layers, the excess impulse applied to the target would be about 41%. Since 75% of the simulated excess impulse was due to the ejection of the first layer of elements, the first-layer impulse was selected as a measure of excess impulse applied to the target.

J.T. Bay lot, P.P. Papados / First MIT Conference on Computational Fluid and Solid Mechanics

a) Single

b) Two, 0.5-m. apart

d) Two, 3.0-in. apart

e) Tliree

63

c) Two, LS-in. apart

Fig. 3. Front surface damage.

4. Multiple fragment impact Simulations were performed for the four multiple fragment impact cases listed in Table 1. The maximum and average magnitudes of the ejecta velocity did not vary significantly in the five simulations performed. Therefore, the Table 1 Multiple fragment simulations Simulation ^

Fragment no.

x^ (in.)

y' (in.)

Damaged area (in.^)

Excess impulse

16.25

70

(%)

Single

1

0.0

0.0

0.5-in.

1 2

0.0 0.0

-0.25 0.25

32.5

70

1.5-in.

1 2

0.0 0.0

-0.75 0.75

26.75

40

3.0-in.

1 2

0.0 0.0

-1.5 1.5

29.75

55

Three

1 2 3

0.25 -0.25 0.0

-0.25 0.25 -0.25

42.19

47

^ Fragment spacing is listed for two fragment impacts. ^ The origin of the coordinate system is at the center of the slab. Horizontal and vertical coordinates are represented by x and y, respectively. Up and to the right are positive.

total area of surface damage is a relative measure of excess impulse. Exposed surface concrete damages for the four multiplefragment simulations are compared with the single fragment analysis in Fig. 3. Excess impulses were estimated for the multiple fragment runs by dividing the damaged area for that run by the damaged area in the single fragment run and by the number of fragments. That number was multiplied by 1.7 in order to adjust the excess impulse to match the single fragment experiments. As seen in Fig. 3 and in Table 1, the damaged area increases by a factor of two when two fragments are placed very close together. This results in an excess impulse equivalent to the single fragment result. The damaged area further increases with the addition of a third fragment close to these two. In this case, however, the increase in area does not offset the addition of the third fragment, and the excess impulse is reduced to 47%. When the two fragments are moved further apart, the damaged area is greater than for the single fragment, but less than for two fragments hitting close to each other. This results as an excess impulse of 40%. The damaged area then grows as the fragments are moved further apart. At a spacing of 3 in., the excess impulse grows to 55%. The limit on the growth of damaged area of twice the single fragment area would be reached in the case when the two fragments are so far apart that their areas of influence would not overlap.

64

J.T. Bay lot, P.P. Papados / First MIT Conference on Computational Fluid and Solid Mechanics

5. Conclusions

References

The analyses initially captured the front-face ejection of concrete that leads to the excess impulse applied to concrete targets by steel fragment impacts. The excess momentum could not be maintained because the ejected elements could not be effectively removed from the simulation. The analyses did indicate the importance of fragment impact pattern on the impulse applied to the target. The addition of an option allowing the user to effectively delete a failed element, and the associated mass would allow the problem to be computed more accurately.

[1] Personal Communication with Mr. Frank D. Dallriva, U.S. Army Engineer Research and Development Center, Vicksburg, MS, on Aug 16, 2000. [2] Papados PP. A reinforced concrete structure under impact: response to high rate loads. In Jones N, Brebbia CA (Eds), Structures under Shock and Impact Loads VI. Wessex Institute of Technology: WIT Press, 2000. [3] Hoover CO, DeGroot AJ, Pocassini RJ. Paradyn: DYNA3D for massively parallel computers. Lawrence Livermore National Laboratory, UCRL 53838-94, 1995. [4] Whirley RG, Engelmann BE. DYNA3D — a nonlinear explicit, three-dimensional finite element code for solid and structural mechanics — users manual. Lawrence Livermore National Laboratory, UCRL-MA-107254, rev. 1, 1993. [5] Chen WF. Plasticity in Reinforced Concrete. New York: McGraw Hill, 1982.

Acknowledgements This research was conducted at the U.S. Army Engineer Research and Development Center. The authors gratefully acknowledge permission from the Chief of Engineers to present and publish this paper.

65

A fictitious domain method for unilateral contact problems in non-destructive testing E. Becache*, P. Joly, G. Scarella INRIA, Domaine de Voluceau-Rocquencourt, BP 105, F-78153 Le Chesnay Cedex, France

Abstract In this work, we present a numerical method for solving the diffraction of transient elastic waves by cracks of arbitrary shapes in complex media, with Signorini's boundary conditions on the crack. We use a fictitious domain method based on a mixed displacement-stress formulation for elastodynamics. We propose an off-centered time discretisation scheme for enforcing the stability. Keywords: Elastodynamics; Unilateral contact; Fictitious domain method; Non-destructive testing; Crack

1. Introduction In this paper, we are interested in solving the diffraction of transient elastic waves by cracks of arbitrary shapes in complex media, with Signorini's boundary conditions on the crack. This is the continuation of a previous work [1] done on the linear problem, that is when the boundary condition on the crack is a free surface boundary condition. To get an efficient method, we want to use regular meshes and at the same time respect the geometry of the crack. This is possible thanks to the fictitious domain method, which takes into account the boundary condition via a Lagrange multiplier defined on the crack, which can be interpreted as the jump of the displacement through the crack. This allows to work with a uniform mesh in the whole domain and an independent mesh on the crack. In order to consider the unilateral contact boundary condition as a constraint, we are led to use the mixed displacement-stress formulation for elastodynamics. We will present a fictitious domain formulation of this problem in which the boundary conditions are taken into account by a variational inequality for the Lagrange multiplier. For the space discretisation of this problem, we propose to use the mixed finite element using spaces of symmetric tensors for the stress [1]. This choice was shown to allow the obtention of an explicit time discretization scheme (mass-lumping) in the linear case. In the non-linear case, we cannot use a centered difference scheme for the time discretisation which would lead to an * Corresponding author. E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

unconditionally unstable scheme. That is why we propose an off-centered scheme that we show to be stable. This scheme is explicit in the volume unknowns (displacement and stress), but impHcit in the Lagrange multiplier: one has to solve an optimisation problem with bound constraints at each time step.

2. Presentation of the dynamic unilateral contact problem We want to solve the displacement-stress formulation of elastodynamics in a domain ^ = C \ F G R^ (see Fig. 1): p—-—diver ^ dt^ Aa

=/" ^ =s(u)

i n ^ x ] 0 , r[, (1) in

Qx]0,Tl

Fig. 1. The geometry of the problem.

66

E. Becache et al /First MIT Conference on Computational Fluid and Solid Mechanics

where u is the displacement field, a the stress tensor, and s{u) the strain tensor defined as £,;(«) == (9/Wy + djUi)/2. We add to (1) Dirichlet boundary conditions on the exterior boundary: u = 0 on dCx]0,T[ \ Signorini boundary conditions without friction on the crack F, [3]: [UN]T

>0

on r ,

O-yv

< 0

on r ,

a^[w/v]r = 0

on r ,

[an]r

=0

on r ,

GT

=0

on r .

(2)

4. Discretisation 4.1. Semi-discretisation

in space

We now introduce some finite dimensional spaces Xh C X, Mh C M, QH C Q of dimensions, respectively, Nx, NM, Ng. We define, respectively, (LM)H = GH (^ L^ and {LT)H = {{^T)H ^ (^r)// ^ GH) where t is the unit tangent vector to F. The semi-discretisation in space can then be written in a matricial form as: finding ll^ X R^G such that (f/, E, AA.,Ar) € Mai:-\-D*U

where, if n denotes the unit normal to F, we set: MA^ = M •«, a^ — on • n, oj — an — a^n and prescribed initial data that we will systematically omit.

-\-

B*AT

+ B*AA.

BTI:

3. A fictitious domain formulation

{BNH,

The fictitious domain method consists in extending the two unknowns (w, a ) in the whole domain C and introducing Lagrange multipliers to take into account Signorini's boundary conditions (2). Setting Q = H^^iV), we introduce

M X

={uelLHC)Y}, ={Te

[LHC)]\divT

= r,,},

e [L\C)f/Tij

A-r)

) - d{G, v)

Z?r(o-, Mr) [Z?yv(a,/XA^ --^N)

= 0

Vr € X,

= iL v) Wv e M,

(3)

= 1 Aa : T dx,

d(T, w)

-I

= 0,

(iii)

< 0,

V/XAT e E+^.

(4)

(iv)

In practice, and this is the interesting point in the fictitious domain method, we introduce two meshes: the volumic unknowns U and E are defined on a regular grid, Th made of squares Kj of size h while the surfacic unknowns A/v and A^ are computed on a nonuniform mesh on V, TH made of segments Sj of size / / / , H = sxxpjHj (see Fig. 2).

fiA.E)

(5)

Choice of the finite elements. We intend to use the same discretisation than for the linear problem (see [1]). For the lowest order element, this choice corresponds to: XH = {cJh e X/WK

e %,

(JHIK e {QdK))

Mh = {vh e M/WK

e %,

VHIK €

}

{Qo{K)f}

= 0

^f^T

eLr,

GH = {M// e G/^S e TH, ^H\S e

< 0

VjUyv

e

Its main interest is that it leads to block diagonal mass matrices (even diagonal for My) so that My and M^ are very easy to invert.

LN-

with fl(a, r )

(ii)

where Y\ is the orthogonal projection on R^^.

The fictitious domain formulation consists in finding (a, u, Ayv, XT) : ]0, T[-> X x M x L/^ x Lj

>^N)

= F,

Ayv = n(Ayv +

LT = [/foo^^(F)]2 =. {^r e S V ^ r n = O]

+ /?yv(r

— Ayv)

(i)

Remark 1. The inequality (4)-(iv) can be reinterpreted as

LN = ^ J o + ( r ) = {fiM ^ G/l^N > 0 a.e. on F j ,

a{a, r) -|- d(T, u) + ^/-(T,

P^M

= 0,

^ ... 11

w ' div r djc,

14 -

^

To Z?Ar(T,/x/v) = {TN, lJ^N)g'.g-

The Lagrange multipliers can be interpreted SLS X^ = [MTV] and Xj = [uj], with uj = u — u^n.

Pi(S)}

iT "

4^

>--

J f^ i2 yC-

^1

/f

y

*

^T ~ '

— [1 - -

-H

Fig. 2. The two meshes.

E. Becache et al. /First MIT Conference on Computational Fluid and Solid Mechanics

This quantity is an energy under the CFL condition (7), and one has the identity

4.2. The fully discretised scheme It would be tempting to discretize (4) using centered finite difference operators, for instance: A n+l

67

E;^+I - E \

=

At

A n-1

I

which shows that ^"+^ < E^ thanks to 6-(iv). which would give an explicit scheme. However, one can show that this choice leads to an unconditionally unstable scheme! That is why we propose the following off-centered scheme: M,E"-f Z)*f/" + 5*A"^ + 5;^A^

=0,

(i)

M„

= F\

(ii)

- 0,

(iii)

A

—

DTP

A^2

r = n (A-/+ Bj

IT + S^+^ ^

(6)

(iv)

Note that if the mass matrices are block diagonal, this scheme is only impHcit in A^^ and is explicit in the other unknowns. We can show a stabiUty result: Theorem 1. Scheme (6) is stable under the usual CFL stability condition - - D*D < 1, with D*D =sup^^ (7) 4 ~ E (M.E,!:) For proving this result, we show the decay of an energy. The precise result is the following. We set yn+l/2

_

Ijn+l _

jjn

At £^1+' = ^((M^s^+i, E"+^) + (M,y'^+3/2^ y"+i/2))_

(8)

5. Numerical aspects The implementation of the method amounts to combine an explicit scheme for the unknowns U and S with an optimisation problem (quadratic functional with bounds constraints) to be solved at each time step for the unknown A. The algorithm has been tested in ID (comparison with analytic solutions). We are currently developing a 2D code, the optimisation procedure being handled by an algorithm combining the active set method with gradient projection method [2]. Numerical results will be presented at the conference.

References [1] Becache E, Joly P, Tsogka C. Fictitious domains, mixed finite elements and perfectiy matched layers for 2d elastic wave propagation. J Comp Acous (Tech. Report INRIA 3889, 2000), to appear. [2] Nocedal J, Wright SJ. Numerical Optimization. Springer, 1999. [3] Willis JR, Smyshlyaev VR Effective relations for nonlinear dynamics of cracked solids. J Mech Phys SoUds 1996;44(l):49-75.

68

Time-frequency pneumatic transmission line analysis G. Belforte, W. Franco*, M. Sorli Department of Mechanics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Abstract This paper presents a theoretical and experimental method for analyzing pneumatic transmission lines in both the time and frequency domains. The test bench developed for this purpose is described together with the experimental analysis method. The theoretical analysis models implemented in the Matlab-Simulink environment are then briefly illustrated. The paper concludes with a comparison of some of the theoretical and experimental results obtained in the investigation. Keywords: Pneumatic transmission line; Pneumatic transient; Pneumatic servosystem; Impedance method; Characteristics method; Fluid borne noise

1. Introduction

2. Experimental set-up

Transmission line dynamic behavior affects the dynamic performance and noise of pneumatic servosystems. A line having a length of 1 m, for example, introduces a delay of several ms [1]. In addition, the flow and pressure pulses generated by the compressor propagate towards the user through the lines, generating noise [2]. The dynamic performance of transmission lines must thus be considered in designing a pneumatic servosystem. Experimental studies in this field, though indispensable, are time-consuming and must be backed up by a preliminary theoretical analysis. In particular, they call for easily used computer codes capable of predicting the dynamic behavior of a line in both the time and frequency domains on the basis of the line's geometry and mechanical properties and of the properties of the air. In addition, the models' parameters must be readily identifiable. This paper describes a theoretical and experimental method for dynamic analysis of pneumatic lines. A test bench developed for this purpose is illustrated. The use of the characteristics method and the impedance method in theoretical analyses of pneumatic lines is then discussed. Finally, a number of theoretical and experimental results are presented, compared and discussed.

Fig. 1 shows a photograph of the test bench developed for investigating the dynamic behavior of pneumatic lines in the time and frequency domains [3]. The bench can accommodate lines of different geometry and material, which may feature pressure pulse-reducing devices such as accumulators, T filters, and Helmholtz resonators. Two resistive transducers (TRl) and (TR2) (ENTRAN EPNMIO, F S . 10 bar, Unearity 0 . 1 % FS.) for measuring mean pressure and two piezoelectric transducers (TPl) and

* Corresponding author. Tel.: +39 (Oil) 5646939; Fax: +39 (Oil) 5646999; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics KJ. Bathe (Editor)

^-^^WB

Fig. 1. Experimental test bench.

69

G. Belforte et al /First MIT Conference on Computational Fluid and Solid Mechanics

(TP2) (Kistler 701a, FS 2.5 bar, linearity 0.5% FS, rise-time 6 |xs) for measuring pressure pulses are installed upstream and downstream of the line under test (TL) by means of appropriate adapters. The line includes a resistive load (ZL). In the present configuration, the hne is excited by switching valve (V) which connects it in alternation to two reservoirs which are maintained at different constant pressures by means of reducers (Rl) and (R2). Valve switchover is controlled by the signal generator (SG). Pressure signals from transducers (TPl) and (TP2) are acquired over time at a frequency of 10 kHz, after which the line's frequency response function (FRF) is calculated in terms of the modulus of the ratio of upstream to downstream pressure, expressed in dB.

3. Time domain analysis Line modehng in the time domain was carried out using the characteristics method [4] implemented in the MatlabSimulink environment. The equations of motion, assuming one-dimensional viscous motion with propagation of small isoentropic oscillations, have the following solutions for the internal points of the grid (Fig. 2): Pi(t + ^t) _ 1 Pi^iit) + P,+i(0 + Ze • (G/_i(0 - Qi+i(t)) ~ 2 2 AD

Zc • (Qi-i(t)\Qi-dt)\

Qiit + At) _ 1 ~ 2 e/-i(o + a+i(o _fAt_ IAD

+

-

Qi+i(t)\Qi+i(t)\)

time > <

Z - —

z. -

(2)

(3)

^

The investigation was carried out for a line with known upstream pressure. The downstream boundary conditions were calculated by combining the characteristics equations with the equation for flow through a nozzle, expressed by means of conductance C and critical ratio b as per ISO 6358 [5]. The characteristics method makes it possible to investigate line behavior in the time domain for different input pressure laws after identifying the following parameters: line geometry and characteristics (length, inside radius, friction coefficient), air characteristics (mean pressure, mean temperature, dynamic viscosity, specific heat ratio) and exhaust nozzle conductance and critical pressure ratio. The results of time simulation carried out on a line excited with a pressure step can then be post-processed to determine the FRF of the line under test. 4. Frequency domain analysis Line modeling in the frequency domain was carried out using the impedance method [6,7] implemented in the Matlab-Simulink environment. Upstream pressure and flow rate (Pi and Q\) and downstream pressure and flow rate (P2 and G2) are linked by a four pole equation:

(1)

p,_i(0-P.+i(0

(G/-i(Oia-i(OI + G/+i(OIG/+i(OI)

where Zc is the line's characteristic impedance:

Pi

cosh r

Zc sinh F

P2

Qi

l/Z^sinhr

coshr

Qi

(4)

where Zc is the characteristic impedance of the line, and F is the propagation operator of the line. On the basis of the assumptions made, the expressions for both Zc and F can be calculated in relation to frequency. Once load impedance Zi is known: ZL

=

(5)

02

the line's FRF can be calculated: t+2At t+At

yu

Pi

4

Pi-i(t)

t

P Z — = cosh r + — sinh F

Pi(t+At) Qi(t+At,

i-1

Pi^l(t) Qi.i(t)

i

i+l

> pipe axial position

Fig. 2. xjt grid of the characteristic method.

(6)

ZL

Simulating the line's dynamic behavior in the frequency domain calls for defining the following parameters: line geometry (length, inside radius, thickness), mechanical properties of the Hne (complex Young's modulus), air characteristics (mean pressure, dynamic viscosity, mean density, specific heat ratio) and load impedance.

G. Belforte et al. /First MIT Conference on Computational Fluid and Solid Mechanics

70

5. Results A number of graphs of line dynamic behavior are presented by way of example. The graphs refer to a poly amide line with length of 1 m, inside diameter of 4 mm and thickness of 1 mm connected downstream to a 0.5 mm diameter nozzle. Fig. 3 shows downstream pressure as measured experimentally and as calculated using the characteristics method with the line excited by an upstream pressure step. The values for the downstream nozzle's conductance C and critical pressure ratio h measured as per ISO 6358 are 3.5 x 10~'^ m^/(sPa) ANR and 0.4, respectively. The line's friction coefficient was considered to be independent of frequency and equal to 0.06. Despite the extensive simplifications introduced in modeling, there is a good degree of agreement between the experimental and calculated curves. The experimental step response curve shown in Fig. 3 was used to evaluate the line's FRF. In Fig. 4, this curve is compared with the curve calculated using the impedance method {ZL = 1/C = 2.8 x 10^ Pa-s/m^^) and with that cal-

3.4 experimental CM model

3.3

:\

h:

\l I \

/-V/V''^^

w

2.5 0.04

The theoretical and experimental method presented herein provides a simple means of analyzing line dynamic behavior in both the time domain and the frequency domain. Only the following parameters need be known in order to identify a line's dynamic behavior: the length, inside diameter, thickness, complex Young's modulus and friction coefficient of the line; the mean pressure, mean temperature, dynamic viscosity and specific heat ratio of the gas; and the conductance and critical pressure ratio or impedance of the exhaust nozzle. In particular, the load conductance and critical pressure ratio can be measured in accordance with ISO 6358, while load impedance as a function of excitation frequency can be measured on the same bench or estimated in subsequent simulations.

A b c C D

2.6

0.02

6. Conclusions

7. Notation

h--^-

0

culated by post-processing the time simulation performed with the characteristics method. As can be seen, the resonance peaks on both the experimental curve and that produced with the impedance method become smaller as frequency increases. This phenomenon was not modeled with the simplified characteristics method adopted for the investigation.

0.06

0.08

0.1 0.12 time [s]

0.14

0.16

0.18

0.2

Fig. 3. Time response of the line to a step pressure.

f

k P Q t Zc

f %.

ZL

:AT

At

r

\\. . . /

Po \ /:

300 Frequency [Hz]

m2

m/s

mV(s Pa) m

Pa m^/s s Pa-s/m^ Pa-s/m^ s kg/m^

References + —

200

line cross section area critical pressure ratio of the load propagation velocity conductance of the load line inside diameter friction coefficient specific heat ratio pressure volume flow rate time characteristic impedance load impedance time step propagation operator mean density

Experimental CM model IM model

400

Fig. 4. Comparison between experimental characteristics method (CM), and impedance method (IM) FRF of the line.

[1] Romifi A, Raparelli T. A simulation program for analysis of any type of fluid mechanical systems 'FLOWSIM'. Proceeding of 12th World Congress International Federation of Automatic Control, Sydney, 1993, pp. 523-530. [2] Edge K. Designing quieter hydraulic systems — some recent developments and contributions. Fluid Power, Forth JHPS International Symposium, Tokyo, 1999, pp. 3-27.

G. Belforte et al /First MIT Conference on Computational Fluid and Solid Mechanics [3] Sorli M, Franco W. Gas line pulse analysis. Flucome 2000, Sixth International Symposium on Fluid Control, Measurement and Visualization, Sherbrooke, 2000. [4] Streeter VL, Wylie EB. Fluid Transients. New York: McGraw-Hill, 1978. [5] Romiti A, Raparelli T. Rigorous analysis of transients in gas and liquid circuits and comparison with experimental data. J Fluid Control 1993;21(4):7-27.

71

[6] Stecki JS, Davis DC. Fluid transmission lines-distributed parameter models Part 1: a review of the state of the art. Proc Inst Mech Eng 1986;100:215-228. [7] Krus P, Weddfelt K, Palmberg JO. Fast pipehne models for simulation of hydraulic systems. Trans ASME J Dyn Syst Meas Cont 1994;116:132-136.

72

On some relevant technical aspects of tire modelling in general F. Bohm, A. Duda*, R. Wille Technical University of Berlin, Institute of Mechanics, Sekretariat MS 4, Einsteinufer 5/7, D-10587 Berlin, Germany

Abstract The study of pneumatic tire mechanics is divided into external tire mechanics that deals w^ith the effect of tires on the vehicle dynamics and internal tire mechanics that focus on the computation of stress-strain and heat states in tires. Internal tire mechanics employs models founded on physical understanding, but not on empirically obtained curves. The objective of this paper is to use the results of internal tire mechanics for improving the external tire models in vehicle model systems. These tire models are applied to rolling contact also on deformable ground. Keywords: Tire models; Rolling contact; Tire mechanics; Terramechanics

1. Comparison of different tire models

2. Stationary and transient rolling of tires

Deriving from the paper [1] presented on the 2nd International Colloquium Tyre Models for Vehicle Dynamic Analysis different tire models (Timoshenko type ring-beam, layered shell model, space continuum, multi-masspoint model) and their transitions one to another are investigated. The main focus was directed to composite shell models and to the application of the Bohm multi-masspoint approach on the rolling tire [2-4]. In order to treat the dynamic contact problems the pneumatic tire is described geometrically non-linear as a multi-layered anisotropic torus shell with low transfer shear stiffness. The membrane and bending deformations were assumed small and the cross-section will exhibit moderate rotation angles. The possibilities of describing the tire composite by different layer models are discussed. For practical tire calculations, which take into account the significant transfer shear deformation, the Timoshenko type shell model and 3- or 5-layer sandwich models with weak rubber layers are adequate. In order to reflect energy losses the visco-elastic behavior of rubber-cord-composite is taken into account. The investigations are based on results of Bohm [5], Duda and Wille [6], INTAS-RFBR [7], Kulikov et al. [8], and Belkin et al. [9].

The stationary rolling problem, quasi-static with friction, is investigated in a coordinate axis rotating simultaneously with the tire. The real dynamic behavior of the rolling tire is non-conservative and self-excited. Appropriate damping of cords and rubber is to be taken into account in order to stabilize the dynamic system. The static equilibrium and the equations of motion of a membrane/shell model are treated. This model is modified for the real structure of an agricultural tire with ribs. Data of 3D models are fitted from given design parameters. The parameters used for 2D masspoint models are extracted from measurements of tire section for variable inner pressure loads and from the eigenvalues of the tire. The non-linear and hysteretic system of Newton equations of this method is solved by explicit predictorcorrector integration with respect to time. The numerical integration procedure needs short time steps. The highest eigenvalue of masspoint model and the shortest relaxation time of the rheological models for tire material and for soil needs to be in correlation with Shannon criterion in order to achieve numerical stable solutions. New theoretical and numerical results and comparison with FEM-results, e.g. [10], will be discussed.

* Corresponding author. Tel.: +49 (30) 314-72411; Fax: +49 (30) 314-72433; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

F. Bohm et al /First MIT Conference on Computational Fluid and Solid Mechanics

73

3. Rolling contact on deformable ground

References

Tire models described above can be used for vehicle dynamics analysis. It can adequately be used for computing the tire-ground interaction in accordance to the technical demand for tire durability, road cover resistance and soil protection in agriculture and forestry. A new mechanical multi-point measurement technique for displacements in the inner of a rolling tire was developed [11]. For high frequency tire deformation and quick driving manoeuvres an acoustic measurement system is in development. The slip between tire ribs and ground cannot be assumed as constant because of the elasto-dynamical tire motions. At present only the 2D masspoint model has an acceptable amount of computational time on PC and is suitable for application in vehicle dynamics. The soil under the tire is described by different rheological laws of the ground surface behavior in the normal and tangential direction [12]. The soil models are tested for simple rigid and elastic tire models in rolling contact. Frohlich/Sohne approach is used for describing the soil compaction. The apphcability of this method is tested by a finite element computation on the basis of a critical state soil model.

[I] Belkin AE, Bukhin BL, Mukhin ON, Narskaya NL. Some models and methods of pneumatic tire mechanics. 2nd International Colloquium on Tyre Models for Vehicle Dynamic Analysis, 1997, pp. 250-271. [2] Bohm F. Dynamic rolling process of tires as layered structures. Mech Composite Mater 1996;32(6):824-834. [3] Tang T. Geometrisch nichtlineare Berechnung von rotationssymmetrischen faserverstarkten Strukturen. Dissertation, TU Berlin 1985. [4] Feng K. Statische Berechnung des Giirtelreifens unter besonderer Beriiksichtigung der kordverstarkten Lagen. Dissertation, TU Berlin 1995. [5] Bohm F Reifenmodelle und ihre experimentelle Uberpriifung. In: F. Bohm, K. Knothe (Eds.), Hochfrequenter Rollkontakt der Fahrzeugrader, Ergebnisse aus dem DFG Sonderforschungsbereich 181. Wiley-VCH 1998, pp. 80-115. [6] Duda A, Wille R. Mechanische Grundlagen des umweltvertraglichen Rad-Boden-Kontaktes. Zwischenbericht zum Projekt DFG - Bo 648/6-1, June 1999, 144 p. [7] INTAS Final Report: Mathematical models and solving methods of the static and dynamic stress-strain state in composite shell structures. INTAS-RFBR 95-0525, 18.04.2000. [8] Kulikov GM, Bohm F, Duda A, Wille R. Zur inneren Mechanik des Radialreifens. Teil 1 und Teil 2. Technische Mechanik 2000;20(1): 1-12,81-90. [9] Belkin AE, Narskaya NL, Bohm F, Duda A, Wille R. Dynamischer Kontakt des Radialreifens als viskoelastische Schale mit einer starren Stiitzflache bei stationarem Rollen. Technische Mechanik 2000;20(4):355-372. [10] Gleu U. Berechnung des nichtlinearen dynamischen Verhaltens des Luftreifens beim instationaren Rollkontakt mit einer Vielteilchenmethode und der Methode der Finiten Elemente. Dissertation, TU Berlin 2001. [II] Bohm F, Duda A, Wille R, Zachow D. Investigation of the non-stationary rolling contact of a tire on natural soils. Proc. 13th International Conference of the ISTVS, Munich, Sept. 14-17, 1999, pp. 353-360. [12] Wille R, Bohm F, Duda A. Rheologie und Hysterese beim dynamischen Reifen-Boden-Kontakt. Annual Scientific Conference GAMM 2-7 April 2000, Gottingen.

4. Conclusion Analytical and numerical analysis of different level tire models is an important pre-condition for suitable choosing of practical calculation schemes for tires and for better understanding of the rolling tire behavior. Investigations are aimed at applications in vehicle dynamics and in tire design. The Bohm multi-masspoint model was used for determining the rolling contact forces on a rigid and deformable ground. The later simulation is meant to avoid the negative effects of soil compaction in agriculture and road damage by truck tires.

74

An index reduction method in non-holonomic system dynamics Marco Borri *, Carlo L. Bottasso, Lorenzo Trainelli Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20158, Milan, Italy

Abstract We present a general methodology for non-holonomically constrained mechanical systems where the governing equations are reformulated employing differentiated multipliers and modified momenta. This procedure allows the algebraic and differential parts of the problem to be completely uncoupled, so that the two subproblems can be solved separately. Any suitable ordinary differential equation integration algorithm can be applied to solve the differential part, by-passing the need for a specialized differential-algebraic equation solver. The approach may be interpreted as a consistent index reduction from 2 to 1 that simplifies the numerical solution of the problem. Keywords: Differential-algebraic equations; Embedded projection; Index reduction; Constraint stabilization; Multibody dynamics; Non-holonomic systems; Constrained systems

1. Introduction A considerable effort within the scientific community has been devoted in the past years towards the development of efficient and reliable numerical methods for the simulation of constrained dynamical systems. These systems are usually cast in terms of sets of differential-algebraic equations (DAEs). Solving general DAE systems still represents an open field of research, since their intrinsic numerical difficulty has prevented to date from reaching the same degree of maturity achieved in the numerical treatment of ordinary differential equation (ODE) systems. This difficulty is usually measured by the differential index of the DAE problem, a concept discussed in [7,9,10]. While index 1 DAEs may be dealt with by using a variety of available numerical methods, for DAEs of index greater than 1 obtaining a good numerical solution may still prove to be a difficult task. In the present work, we are concerned with systems governed by index-2 DAEs, or systems subjected to nonholonomic constraints. It must be pointed out that these systems cannot, in general, be directly solved by applying a standard off-the-shelf ODE integrator, because of its inability to exactly solve algebraic equations. Here we seek a complete uncoupling of the DAE system into separate * Corresponding author. Tel: +39 (2) 2399-8399; Fax: +39 (2) 2399-8334; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

algebraic and differential parts. To this end, we introduce differentiated Lagrange multipliers and define a new variable, the 'modified momentum'. While the 'standard' momentum must obey the non-holonomic constraints imposed on the system, the modified momentum must not, and in this sense it represents a completely free (unconstrained) variable. This way, we obtain an ODE for this quantity that can be integrated using any suitable ODE solver. The original momentum is then recovered by means of an 'embedded projection' onto the constraint space. In general, this procedure allows the same order of accuracy to be attained for all the fields of a DAE problem (and, in particular, for the algebraic variables) that is provided by the chosen solver when applied to a purely ODE problem. This framework was presented originally for both holonomic and non-holonomic problems in [3,4] and its successful application to the parallel computation of the dynamics of general topology rigid multibody systems was reported in [11]. Apart from minor developments, the main novel contributions of this work are to be found in the interpretation of the procedure as a consistent index reduction and in the recovery of the reaction forces by a second 'embedded projection' onto the space defined by the constraint derivative. This process, which recovers even the multiplier derivatives with the same order of accuracy of the primary variables, indeed completes the whole picture, in close analogy to the methodology recently presented in [5,6]

M. Borri et al. /First MIT Conference on Computational Fluid and Solid Mechanics in the context of index-3, i.e. holonomically constrained, dynamical systems.

2. Lagrangian framework Let a generic dynamical system with n degrees of freedom, be characterized by a Lagrangian function £(q, q, 0 , where q G R" represents the vector of Lagrangian coordinates describing the system configuration, and let the system be subjected to m < n linearly independent nonholonomic constraints through a constraint function j/r, ^(q,q,O = 0^

(1)

We require that this function be linear in the Lagrangian velocities q, or f (q, q, t) = A(q, t)^q + a(q, t).

(2)

where the constraint matrix A := f^ e W''"' has full-row rank. In passing, we remark that, under suitable smoothness assumptions, f vanishes together with all its time derivatives. This obvious feature is not inherited by the numerical solution obtained via conventional methods, which, due to time discretization, satisfies only the velocity-level constraint (1). In the following, we show how a more consistent numerical solution can be obtained, exactly satisfying both Eq. (1) and its first time derivative, i.e. the acceleration-level constraint. It is well known that the governing equations for this system are given by the following augmented Lagrange equations - £ q - £ q = Q + A).,

(3)

together with Eq. (1). Clearly, X G E'" represent the vector of Lagrange multipliers, while Q G M" denotes the vector of Lagrangian external forces conjugated to q. The term AX accounts for the reaction forces associated to the constraints (1). It has been shown in [4] that equations equivalent to the set formed by Eqs. (1) and (3) may be derived from a variational statement by defining a modified Lagrangian function C*(q, q, fi,t) and a modified Lagrangian external force Q* as (4) (5) where fi e W^ is a. new multiplier vector. The EulerLagrange equations of the system are found as A/2* — /2* — O*

75

We note that the classical Lagrange multipliers k are related to our multipliers fi by /i = X. Furthermore, note that, with the substitution X = ft, the first equation of set (6) is exactly equivalent to Eq. (3), while the second simply expresses Eq. (1), being £* = 0^ and C^ = -f. Remarkably, in the case of integrable constraints f =^, i.e. those velocity constraints which correspond to the total time derivative of position constraints 0(q, 0» we get Q* = Q, since (d^q/d/ — ^q) vanishes identically. Therefore, the additional force Qnh := Q* - Q is peculiar to proper non-holonomic constraints. The gyroscopic nature of this quantity was analyzed in [4], where it was shown that it may be cast in the following form Qn/.(q, q, /^, 0 = B(q, iij)q

+ b(q, ti, t),

(7)

where B is a skew-symmetric matrix linearly depending on fi, while b := (aq — dA/dt) fi. From the preceding we infer that the power Wnh •= q • Qnh of this additional force on the Lagrangian velocities reads Wnh = q-h,

Vq.

(8)

This power clearly vanishes identically whenever b = 0^. In particular, when A is time-independent and a does not depend on the coordinates q.

3. Hamiltonian framework As shown, the introduction of new multipliers ft (the reaction impulses) instead of the classical X (the reaction forces) leads to an important theoretical result: the extension of Hamilton's variational principle to non-holonomic mechanical systems through the definition of a modified Lagrangian £* and a modified force Q*. In the following, we show that this procedure inspires analogous extensions in the Hamiltonian framework, where a modified Hamiltonian 1-L* can be defined accordingly. In this case, however, the interest of the proposed methodology is not limited to theoretical issues, but also possesses algorithmic implications on the numerical solution that shall become clear in the following. We switch to the Hamiltonian formulation by means of a standard Legendre transformation, defining the momentum p := £q, inverting this relation to find q as a function of p, or q = VH(P, q, 0 . and obtaining the Hamiltonian function H(p, q, t) as n = p-yH-jCH,

(9)

where >Cif(p, q, 0 •= >^(v//(p, q, 0 , q. 0- ^^^ following 'mixed form' canonical equations P + ' H q ^ Q + AX,

(10)

(6) govern the system together with Eq. (1). However, a critical

76

M. Borri et al. /First MIT Conference on Computational Fluid and Solid Mechanics

point in this process lies in the fact that the momentum p is intrinsically constrained by the algebraic equation ^//=0,,,

(11)

where f ^(p, q, t) := ^(v//(p, q, r), q, r), or ^ ^ ( p , q, 0 = A(q, 0 V ( P , q, 0 + a(q, 0-

(12)

At this point, we introduce the modified momentum p* := £*. Since £* = £q - fl ft, we get p* = p - A / i .

(13)

Now, coupling this equation with the algebraic constraint (11) we can solve for p and fi as functions of (p*, q, r), obtaining P =P//*(p*,q,0,

(14)

This enables us to get q = V//*(p*, q, t) and, performing a Legendre transformation on £*, to obtain a modified Hamiltonian 1-L*(p*, q, t) as ^ * = P* • v//* -

CH*

(15)

where £//*(p*,q, r) := £(v//*(p*, q, r), q, r). Now, the canonical equations governing the system can be found as P*+H; = Q*, q-n;.=On.

(16)

Note that, in contrast to the Lagrangian framework, in the Hamiltonian case, there is no appended constraint equation to the system (16), since the modified momentum p* adopted as the independent variable together with the vector of Lagrangian coordinates q, is an unconstrained quantity under all respects: it yields, by construction, a solution for the original momentum p which exactly satisfies the constraint equation (11). Therefore, the set of canonical equations (16) may be directly integrated in terms of (qp*) It is worth looking at an alternative form assumed by the governing ODEs (16), in view of its numerical implementation. In fact, the canonical equations are formally equivalent to the following set

q-n^ =o„,

(17)

provided that Eqs. (14) are understood in the dependencies of the terms (Tiq, Tip, Q, A, JJL) on (q, p*). However, these equations are much simpler than Eqs. (16) to implement and evaluate in the context of numerical integration since all the quantities involved are easily retrieved, the only additional burden being the knowledge of A when compared to a conventional integration method.

4. Consistent index reduction The differential system (17), explicitly cast in terms of (q, p*), may be directly integrated by means of any suitable ODE solver from consistent initial data q\tQ and p\f^ — p*|^Q. As an example, take a generic one-step integrator, such as a 5-stage Runge-Kutta method: the procedure calls for solving the problem composed of Eqs. (13) and (11) at each of the s internal stages. This, when a general quadratic form in q is assumed for the original Lagrangian £, turns out to be a linear problem for (p, fi). When (p//, fif^) are known, one solves the equations corresponding to the discretized ODEs (16) or (17) at that internal stage and moves on to the next. This shows the profound difference existing between this methodology and a conventional projection method, where the projection is performed only at the end of the time step. Such an approach, referred to as the (i-method or the modified phase space method, has been presented in [3,4]. Experience has shown that this formulation positively impacts the accuracy and stability of the numerical solution [11]. In fact, comparison with the widely adopted Baumgarte stabilization technique [2] has shown much lower constraint violations (for holonomic constraints imposed at velocity level) and a considerable robustness. However, we presently do not favor the treatment of holonomically constrained mechanical systems by imposing velocity-level constraints, since the 'drift' phenomenon cannot be completely eliminated. We presently recommend the approach presented in [5,6] for holonomic problems, and the present one for proper non-holonomic problems. It may be proved that the method oudined here is strictly equivalent to a process of reduction of the differential index of the problem. In fact, the original DAE problem corresponding to Eqs. (10) and (11) has index 2, while in the proposed framework the DAE problem given by Eqs. (16) or (17), (13), and (11) has index 1. It is worth noting that, in index 1 problems, the algebraic equation may always be interpreted as a definition of the algebraic variables rather than as a constraint acting on the state variables.

5. Preservation of accuracy In the approach followed in [3,4,11], recovering of the reaction forces (essentially, (i) was performed by numerical differentiation, thus loosing the chance of retaining the same order of accuracy for these quantities as that obtained for the primary variables (q, p*) and, consequently, for (P, l^) The following developments are carried out for the explicit purpose of overcoming such a limitation in accuracy and are closely related to the ideas presented in [5,6] in the context of holonomically constrained systems, with the

77

procedure termed the Embedded Projection Method. We consider the original equiUbrium equation (10a) and the time derivative of Eq. (11), both viewed as linear algebraic equations in the variables (p, //.): p + 'Hq^Q + A ^ , (18) By using eqs. (14), we can evaluate each term in the previous equations as a function of (p*, q, t) and solve for p and /t, giving P = ^i/*(p*,q,0,

(19)

In summary, these quantities are recovered by using the equilibrium equation and the acceleration-level constraint as an algebraic problem, just as (p, JLC) are obtained by using the modified momentum definition and the velocity-level constraint. This process has been termed the 'embedded projection'. It is clear that, within the context of exact mathematics, JT//* = Pif* and XH* = /i,^*. However, when dealing with time discretization processes, the present procedure allows to compute {KH*,XH*) independently from (Pi/*»/^H*)- This improves the consistency of the solution, and also allows the same accuracy for the algebraic variables (p, p, /t, (i) to be retained as for the independent variables (q, p*). In other words, the outcome of the methodology may be described as the retrieval of both the augmented state (p, q, /t) and its time derivative (p, q, /t) fully satisfying the constraints in the original and differentiated forms.

6. Concluding remarks In this work, we presented a general methodology for the consistent index reduction of the equations governing the dynamics of mechanical systems subjected to non-holonomic constraints. We showed how the governing equations may be split into uncoupled algebraic and differential parts. This process, which involves the definition of a modified, unconstrained momentum, leads to the formulation of an ODE which can be solved by any suitable standard numerical integrator, by-passing the need for specialized DAE solvers. The solution of a first algebraic subproblem allows to recover the original momentum, while a second one pro-

vides the reaction forces. The outcome of the method is a substantially enhanced accuracy, in particular with respect to reactions, plus an intrinsical gain in robustness due to the exact preservation of both the constraint and its time derivative. The methodology is closely related to the Embedded Projection Method recently presented in the context of holonomically constrained systems. Preliminary applications, not detailed in this work, have been implemented and tested, confirming the properties predicted in the analysis.

References [1] Ascher U, Chin H, Petzold LR, Reich S. Stabihsation of constrained mechanical systems with daes and invariant manifolds. J Mech Struct Mach 1995;23:135-158. [2] Baumgarte J. Stabilization of constraints and integrals of motion in dynamical systems. Comput Math Appl Mech Eng 1972;1:1-16. [3] Borri M, Mantegazza R Finite time element approximation of dynamics of nonholonomic systems. AMSE Congress, WiUiamsburg, VA, 1986. [4] Borri M, Bottasso CL, Mantegazza P. A modified phase space formulation for constrained mechanical systems differential approach. Eur J Mech, A/Solids 1992;11:701727. [5] Borri M, Bottasso CL, Trainelli L. An embedded projection method for constrained dynamics. NATO-ARW on Computational Aspects of Nonlinear Structural Systems with Large Rigid Body Motions, Pultusk, Poland, 2000. [6] Borri M, Trainelli L. A new formulation of constrained dynamical systems. 16th IMACS World Congress, Lausanne, Switzerland, 2000. [7] Brenan KE, Campbell SL, Petzold LR. Numerical solution of initial-value problems in differential-algebraic equations. New York: Elsevier Science, 1989. [8] Eich E. Convergence results for a coordinate projection method applied to constrained mechanical systems with algebraic constraints. SIAM J Numer Anal 1993;30:14671482. [9] Gear CW. Differential-algebraic equation index transformations. SIAM J Sci Stat Comput 1988;9(l):40-47. [10] Petzold LR. Order results for implicit Runge-Kutta methods applied to differential/algebraic systems. SIAM J Numer Anal 1986;23(4):837-852. [11] Sika Z, Valasek M. ParalleHzation of multibody formalism for rigid bodies using natural coordinates and modified state space. Eur J Mech, A/Solids 1997;16(2):325-339. [12] Yen J, Petzold LR. Convergence of the iterative methods for coordinate splitting formulation in multibody dynamics, TR 95-052, Tech Report, Dept of Comput Sci, University of Minnesota, July 1995.

78

Application of the LATIN method to the calculation of response surfaces p.A. Boucard * LMT Cachan, ENS Cachan, CNRS, University Paris 6, 94235 Cachan Cedex, France

Abstract The aim of the present work is to develop an apphcation of the LArge Time INcrement (LATIN) approach [6] to the calculation of response surfaces used for parametric analysis. The scheme followed was previously introduced to solve multiple-solution problems [2,3]. Here, applications concern elastic buckling and viscoelastic structures. Keywords: Non-incremental method; Multiple solutions; Response surface methodology; Parametric uncertainty

1. Introduction The solutions to deterministic problems are often calculated by finite element analysis (FEA). Incorporating system parametric uncertainties into the problem represents a challenge for structural engineers; yet, without this information, the structural response could not be analyzed accurately. These system parametric uncertainties include mechanical properties of the material (modulus and strength, etc.), geometric properties (cross-sectional properties and dimensions), boundary conditions, magnitude and distribution of loads, etc. Assessing the stability or the calculation of the limit states of structures taking these parametric uncertainties into consideration is much more difficult than the general parametric field problem because highly nonlinear structural behavior must be considered. To obtain such responses of structures, the perturbation method [1,7] is one of the important approaches. In recent years, many researchers have focused on the stochastic finite element method, in which the system parametric uncertainties mentioned above can be included. The response surface methodology (RSM) was developed initially by Veneziano et al. [9]. The RSM is already a widely accepted procedure in structural reliability analysis [5]. Schueller et al. [8] used the RSM to model the actual limit state function of large structures subject to static *Tel.: +33 (1) 4740-2186; Fax: +33 (1) 4740-2185; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

and dynamic loading. The calculation of response surfaces and, further, of the response of the structure along the whole loading path involves multiple solutions. Each set of data considered necessitates a separate, full-scale calculation. Consequently, a significant number of problems of the same type must be solved. The goal of the work presented here is to develop a strategy well-suited to multiple-solution problems. Thus, the choice of an appropriate and efficient computational method is of vital importance. The LATIN method [6] is non-incremental in nature and, consequently, would appear to be a promising approach, considering that its inherent principles tend to be more applicable than most conventional incremental algorithms. The strategy proposed is based on the LATIN method and, more specifically, on its capacity to reuse the solution to a given problem in order to solve similar problems [2,3]. It allows total computing costs to be minimized with respect to the determination of response surfaces.

2. Review of the LATIN method The principles of this method can be found in [6]. The method uses quantities (displacement, strain, stress and internal variables) defined over the space-time domain Q. X [0, r ] , where [0, T] is the time interval studied and Q is the domain occupied by the structure (assuming small displacements). It takes advantage of the remarkable properties of the equations. The procedure is iterative and

PA. Boucard /First MIT Conference on Computational Fluid and Solid Mechanics

79

FC/FCQ ratio

0.95

0.85

""'m,

10 rrX^i^o') 10 Htttba»°^ t^e^et^^

Cer/J^e

•'W'^'"' Fig. 1. Response surface.

Number of space runctioos

^-^*^^le^e^^^°^ Fig. 2. Number of space functions generated.

creates at each iteration an approximation of the displacement, strain, stress and internal variables over the spacetime domain Q x [0, T]. Each iteration consists of two stages. For simplicity's sake, one can say that in the first stage the constitutive relations are integrated; therefore, this is a local stage with respect to the space variable. In the second stage, a global, linear problem on ^ x [0, T] is solved. The direct solution of the global linear problems with time as a parameter required at the global linear stage can

lead to considerable computing times. Mechanics-based approximations of unknowns are introduced as a means of reducing these computing times. The separation of the functional dependencies both in time and in space yields satisfactory results for quasi-static loadings. Corrections are then sought by superimposing solutions of the radial loading type. Such solutions are recognized as good approximations of non-linear, quasi-static problem solutions.

80

PA. Boucard/First MIT Conference on Computational Fluid and Solid Mechanics Number of draws 400 300

200

-0.5

0

0.5

Amplitude of the perturbation Probability of collapse

Number of draws 700 600 500 400

123456789101112131415

max/Fco ratio

Position of the perturbation (element n°)

Fig. 3. Probability of collapse and distribution of perturbations. 3. Multiple-solution method The LATIN method leads to an approximation of the problem's solution in the form of a sum of products of both time and space functions. In this sense, the LATIN method builds an optimal basis for representing the solution. The idea is, therefore, to reuse this special basis in order to find the solution to a problem similar to the one for which it was built in the first place. The multiple-solution method uses the fact that the LATIN algorithm can be initialized with any solution which verifies the admissibility conditions (usually an elastic solution). Therefore, the idea here is to initialize the process associated with the similar problem (the 'perturbed' structure) using the results of the calculation carried out on the 'initial' structure. In this manner, a basis of space functions with a strong mechanical content is immediately available at the onset. In this case, the preliminary stage plays a vital role: it enables one both to verify that the basis of the space functions is well-suited to the target problem and to search for new time functions leading to the solution of the 'perturbed' problem. In the best-case scenario where the basis is sufficient, no new space function is generated and, thus, the solution to the problem is obtained at low cost.

Otherwise, new space functions are generated in order to enhance the initial basis. If the solutions to the 'initial' and 'perturbed' problems are close enough, the solution to the latter problem can still be derived at a significantly lower cost than using full-scale calculation.

4. Example The example presented here is the buckling of a cantilever beam. Additional details on the formulation used can be found in Boucard et al. [4]. The first example considers a straight beam built-in at one end and subject to a prescribed displacement at the other. The structural perturbation introduced consists of variations of the Young's modulus in different elements (15 in all) ranging from —50% to +50%. The influence of a particular perturbation on the value of the critical buckling load (Fc/Fco ratio) is examined. The results are presented on Fig. 1. Fig. 2 shows the number of space functions added at the initial basis level during the calculations (six groups of time-space functions). This number provides an indicator of the total computing cost, given that this phase is the most costly stage of the algorithm. It can be observed that no more than one space function is added in the majority of

PA. Boucard /First MIT Conference on Computational Fluid and Solid Mechanics the cases processed. Therefore, the basis of initial functions enables us to conduct many 'perturbed' calculations at a much lower cost than that of a full-scale calculation: in the cases presented here, the computing time necessary to obtain the solution on the 'perturbed' bar is between 10 and 20% of that of a full-scale calculation. This demonstrates the effectiveness of the method. Using these results, one can carry out a Monte-Carlo simulation using the response surface to determine the probability of collapse of the beam. In this case, we assume a normal distribution for the Young's modulus perturbation. The position of the perturbation is randomized on all 15 elements. Fig. 3 shows the results. Ten thousand draws were carried out to obtain the probability of collapse as a function of the ratio of the maximum loading force F^ax to the buckling force obtained on the initial beam Fco.

References [1] Benaroya H, Rehak M. Finite element methods in probabilistic structural analysis: a selective review. Appl Mech Rev 1998;41(5):201-213. [2] Boucard PA, Ladeveze R Une application de la methode LATIN au calcul multiresolution de structures non lineaires.

81

Rev Eur Elem Finis 1999;8(8):903-920. [3] Boucard R\, Ladeveze R A multiple solution method for non-linear structural mechanics. Mech Eng 1999;50(5):317328. [4] Boucard PA, Ladeveze P, Poss M, Rougee P. A non-incremental approach for large displacement problems. Comput Struct 1997;64(l-4):449-508. [5] Faravelh L. Response-surface approach for reliabiUty analysis. ASCE J Eng Mech 1989;115(12). [6] Ladeveze P. Nonlinear Computational Structural Mechanics — New Approaches and Non-Incremental Methods of Calculation. Springer, 1999. [7] Macias OF, Lemaire M. Elements Finis stochastiques et Fiabilite Application en mecanique de la rupture. Rev Fr Gen Civil 1997;1(2). [8] Schueller Gl, Bucher CG, Pradlwarter HJ. The response surface method, an efficient tool to determine the failure probability of large structural systems. Proceedings of the International Conference on Spacecraft Structures and Mechanical Testing, Noordwijk, The Netherlands, 24-26 April 1991. ESA SP-321, pp. 247-251. [9] Veneziano D, Casciati F, Faravelli L. Method of seismic fragility for complicated systems. Proceedings of the 2nd Committee on the Safety of Nuclear Installations (CSNI) Specialistic Meeting on Probabilistic Methods in Seismic Risk Assessment for NPP, Livermore, CA, 1983.

82

A unified failure approach for sheet-metals formability analysis M. Bmnet*, R Morestin, H. Walter Laboratoire de Mecanique des Solides LN.S.A, 20 Avenue A. Einstein, Villeurbanne, 69621, France

Abstract A macroscopic yield criterion for anisotropic porous sheet metals is first proposed to investigate failure of sheet metals under arbitrary strain paths. The hardening behavior of the matrix material combines isotropic and non-linear kinematic hardening. An inverse identification technique is proposed based on bending-unbending experiments on anisotropic sheet-metal strips. The void coalescence failure mechanism by internal necking is also considered by using a modified Thomason's plastic limit-load model. Finally, a plastic instability criterion coupled with damage is used here to predict failure in a sheet-metal forming analysis by finite element. Keywords: Damage; Plastic-hardening; Sheet-metal forming; Failure; Necking

1. Introduction Plastic deformation in sheet metal consists of four distinct phases, namely, uniform deformation, diffuse necking, localized necking and final failure. The last three phases are commonly known as non-uniform deformation. New sheet-metals such as aluminum alloys, titanium alloys and Ni-based superalloys, present from experimental evidence necking-failure behavior where the localized thinning is hardly visible. Plastic instability of these sheet-metals has been found to suffer material degradation which confirmed the need to properly characterize their forming limit using a theory of damage mechanics. Coupling the incremental theory of plasticity with damage and a plastic instability criterion, the new criterion can be used to predict not only the forming limit but also the fracture limit under proportional or non-proportional loading and then is suitable for sheet-metal forming simulation by finite-element analysis.

2. Yield criterion Most metallic materials contain different sizes and degrees of particles, including precipitates and inclusions, which may cause micro-defects including micro-voids and micro-cracks. As fracture in sheet-metals forming processes is mainly due to the development of ductile damage and to * Corresponding author. Tel.: +33 472 43 81 46; Fax: +33 472 43 85 28; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

represent the damage of anisotropic sheet-metals, an extension of the Gurson's model for anisotropic sheet-metals is used where an analytical formulation for plane-stress has been found by Liao et al. [1]. For all possible plane-stress conditions, the anisotropic yield function is approximate as: CD = ^ + 2 ^ 1 / * c o s h ( r*2.

l + 2 r 3p 6(1+F)^ (1)

-(l+^3/n=0

where 7 is the mean normal anisotropy parameter of the matrix material, and / * the effective void volume fraction. Consider x,y to be the 'rolling' and 'cross' directions in the plane of the sheet, z is the thickness direction. Based on Hill quadratic yield function, the macroscopic effective stress q in Eq. (1) is defined as [3]: q = {a -aj'^lMUa

M

(2a)

-a]

g + /i

-h

0

-h

f-\-h

0

0

0

2n

(2b)

where the relative macroscopic stress tensor with respect to the center of the current yield surface is defined as: f

Ox -Olx

\a -a]

= 1 ^.v -ay

1

[Oxy - a ^ j j

(3)

M. Brunei et al. /First MIT Conference on Computational Fluid and Solid Mechanics

83

In Eq. (1), /? represents the hydrostatic stress of the relative stress tensor of Eq. (3) and the size of the elastic range Oy is defined as a function of the equivalent plastic strain £^ : or, = ao + Goo(l-e-^^')

(4)

where (TQ is the yield surface size at zero plastic strain, and goo and b are material parameters that must be calibrated from cyclic test data. The evolution of the kinematic components of the model is defined as, [3]:

y{a}dF

[da] = C-^{G -a}-

(5)

where C and y are additional material parameters to be calibrated.

Fig. 2. Theoretical versus experimental stress-strain tensile-test curves.

4. Damage parameters identification 3. Constitutive parameters identification The initial anisotropy parameters (the r-values) are first determined independently with our Digital Image Correlation method (DIC) [2] by mean of uniaxial tests and to obtain the test data for the kinematic-hardening parameters identification, a bending-unbending apparatus has been built [3]. As an example. Fig. 1 depicts the moment versus curvature for one loading and reverse loading. The material is an aluminum alloy of strip thickness 0.8 mm, E = 69000 Mpa, Go = 137 Mpa, RQ = 0.71 and Rgo = 0.74. It can be seen, that very substantial agreement of experimental and simulated data is obtained with the converged values: C = 740.4, y = 4.167, G = 111.6 and ^ = 13.56 for the mixed hardening model. Fig. 2 compares the theoretical stress-strain curves to the experimental data for the case of the uniaxial monotonic tensile tests. Very good agreement for the stress-strain curves has been obtained due to the fact that the optimization is carried out both on the uniaxial monotonic curve and on the moment-curvature curve.

e -0^10

I

^.CB

^.06

-0.04

-OLOE

-0.20

Curvature (mm"^)

Fig. 1. Theoretical versus experimental moment-curvature curves.

The damage model can take into account the three main phases of damage evolution: nucleation, growth and coalescence. An optimization procedure could be also performed to match the experimental and numerical finite element results as regards the loads vs. displacement curve in a tensile test. However, the critical void volume fraction is not unique due to the fact that the void nucleation parameters are difficult to monitor in experiments and are usually arbitrarily chosen. To overcome this shortcoming, the void coalescence failure micro-mechanism by internal necking is considered by using a modified Thomason's plastic limit-load model, [4]:

Rz

X-Rx

+

t)l

f ^n

—

(6)

where F and G are constants, A^ and M are exponents, Rx, Rz are the radii of the ellipsoidal void and X denotes half the current length of the cell. What is interesting in the plastic limit-load criterion is that void coalescence is not only related to void volume fraction but also to void-matrix geometry, stress triaxiality and initial void spacing. By mean of a void spacing ratio parameter, the anisotropic nature of rolled sheet is better account for in the coalescence micro-mechanism, moreover this effect is more pronounced at low stress triaxiality [4]. The modified Gurson's model is used to characterize the macroscopic behavior assuming that the void grows spherically and to calculate the void and matrix geometry changes using the current strain and void volume fraction. Once the equality Eq. (6) is satisfied, the void coalescence starts to occur and the void volume fraction at this point is the critical value fc provided that the stress triaxiality is greater than 0.33 (1/3) which is always the case just after necking.

84

M. Brunet et al. /First MIT Conference on Computational Fluid and Solid Mechanics

5. Necking-failure criterion The strain ratio ^ = Aez/Asi has an evident influence on the internal damage of sheet metals. At the same level of deformation, it is generally noted that the damage increment is the greatest at plane strain such that Asji = 0 when the localized necking occurs, which requires a drift to the plane strain state and then an additional hardening. The formulation follows our previous work [2], the unified necking-failure criterion is formulated in terms of the principal stresses and their orientation with respect to the orthotropic axes leading to an intrinsic formulation including damage: q [dG\ dq dcFy ds Gy \_ dq day d'e ds\

dai dp' dp ds\

'. (y\

(7)

where an analytical form of the left-hand side has been formulated and implemented in our implicit and explicit FE codes suitable for sheet metal forming simulation.The deep-drawing of a square box has been conducted experimentally and numerically, the material is the previous analysed aluminum alloy. The failure of a critical point of the aluminum alloy in an FEM forming simulation (Fig. 3) is determined by using the failure prediction methodology describe above.

6. Conclusion In this work, a unified failure approach has been presented based on the theory of damage mechanics including the non-linear kinematic hardening of the matrix material and void coalescence by internal necking of the inter-void ligament. In sheet-metals, developing of damage makes the strain state gradually drift to plane strain, this fact leads to propose a unified instability criterion for localized necking and rupture.

References [1] Liao KL, Pan J, Tang SC. Approximate criteria for anisotropic porous ductile sheet metals. Mech Mater 1997; 26:213-226. [2] Brunet M, Mguil-Touchal S, Moresdn F. Analytical and experimental studies of necking in sheet metal forming processes. J Mater Proc Technol 1998;80/81:40-46. [31 Brunet M, Moresdn F, Godereaux S. Non-linear kinematic hardening identification for anisotropic sheet-metals with bending-unbending tests. In: ASME MED-12A Symp. on Advances in Metal Forming, IMECE 2000 Congress, Orlando, FL, USA, Nov 5-10, 2000. [4] Benzerga AA, Besson J, Pineau A. Coalescence-controlled anisotropic ductile fracture. J Eng Mater Technol 1999;121: 221-229.

85

Underground explosions: their effect on runway fatigue life and how to mitigate their effects John W. Bull* Department of Civil Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne NEl 7RU, UK

Abstract The detonation of an explosive device underneath a runway causes an underground void (a camouflet) to be formed. This paper describes how such a void can be detected, repaired and the fatigue life of the runway determined. Keywords: Underground explosion; Runway repair; Fatigue life; Finite element

1. Introduction This present paper assumes a detonation has formed an underground void as shown in the half section of Fig. 1. Around the void is a shell of highly compacted subgrade, with the disturbed subgrade above the void forming a cone, zones 2-5 of Fig. 1, that extended to the underside of the runway. The vertex of the cone is the detonation point, with the base of the cone being on the underside of the runway. The size of the void is linked to dimensional analysis, statistical reasoning and scahng laws [1-6]. Any linear dimension L, in metres, can be related to L/W^-^^ where W, in kg, is the mass of the explosive charge [7]. The factors determining crater size and shape are W, X^ and the subgrade. A-c is the detonation depth (in metres) divided by W^^^ The resulting crater being a camouflet if Ac < -1.388. When a camouflet is formed, in time, the walls of the void will collapse. Collapse is complete when one of three following conditions is satisfied: the height of the collapsed cone extends to the underside of the runway; the void is completely filled but the collapsed subgrade does not extend to the surface; and the material in the collapse path forms a stable dome. The first condition will cause immediate loss of runway support. Loss of subgrade support due to the second condition will take time to develop. For the third condition, once the void has been detected, it can be filled. *Tel.: +44 (191) 222-7924; Fax: +44 (191) 261-6059; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Fig. 1. Half section of the camouflet showing zones 1-8. 2. Camouflet location Experimental work shows that for no surface disturbance, the detonation depth requires a minimum of L39 ^0.333 j ^ ^Q 2.78 W^-^^^ m [7]. Detonations at these and greater depths present considerable repair difficulties. Laboratory tests have shown that a saturated clay subgrade subjected to superimposed cyclic loads has an initial set-

86

J.W. Bull/First MIT Conference on Computational Fluid and Solid Mechanics

ouflet was modeled by setting the cyUnder radius to 14.112 m. The axial length of the cylinder was 16.128 m. Four thousand and eighty three-dimensional isoparametric finite elements were used in the model of the camouflet with the polystyrene infill and 2940 for the camouflet without the infill. The effect of an aircraft was modelled by a single downward point load of 100 kN at a succession of 15 nodes, equally spaced between the boundary and the centre of the upper surface of the runway. Elastic analysis was used as it gives sufficiently accurate results [1-6].

tlement of between 60 and 80% of the total permanent settlement and is attained within the first 10 cycles of the loading [8]. This is followed by slower secondary settlement that continues for up to 20,000 load cycles, until equilibrium is reached. For runways, it is possible to obtain deflection and settlement measurements along the length of the undamaged runway to determine the runway's settlement stage. The introduction of a void changes the settlement conditions. Overrunning of the camouflet will show altered settlement readings enabling the repair team to identify the location and extent of the camouflet. Deflection data and cone penetration test results allow the repair team to determine the type of camouflet to be repaired [1-6].

5. The numerical model Following detonation, the Young's modulus of the runway; zones 1 and 8 did not change, but zones 2, 3, 4, 5 and 6 were introduced. The detonation depth was 8.354 m, with the void having a horizontal diameter of 6.246 m and a vertical diameter of 6.183 m. The outer radius of the compacted zone, the interface between zones 4 and 5 and 6 and 7 was 3.776 m. The radius of the interface between zones 3 and 4 was 5.149 m, with the radius of the interface between zones 2 and 3 being 6.601 m. The Young's modulus of subgrade zones were calculated using £ = 10 CBR(%) MPa [1-6].

3. Camouflet size and material requirements Bull and Woodford [1-6] describe the dimensions and material properties of the camouflet and the subgrade. That is a 213-kg explosive charge has created a camouflet in a previously homogenous, isotropic 9.5% California Bearing Ratio (CBR) subgrade. The loosened subgrade on the underside of the runway having a diameter of 16.128 m. A number of subgrade strengths between two extremes are considered. The first is when the detonation is contained within the outer diameter of the compacted shell; material set 1 of Table 1. The second is where significant changes have been made in the subgrade; material sets 2-9 of Table 1. Fig. 1 shows the eight zones. Table 1 gives the Young's modulus for the zones that are changed. In all cases, zones 1 and 8 had a Young's modulus of 36,000 MPa. The Young's modulus of the polystyrene void filler was 10 MPa, Zone 7 was 95 MPa and zone 6, 950 MPa with the exception of material set 10 where zone 6 was 95 MPa. The Poisson's ratio for zones 1 and 8 was 0.2, for the polystyrene, 0.1 and for zones 2-7, inclusive 0.3.

6. Filled and empty camouflets Material set 10, provided the benchmark displacements, stresses and fatigue life for the undisturbed subgrade and runway. The deflection results for the filled camouflet and for the unfilled camouflet, showed that for all material sets, the change in the corresponding displacements was no more than 0.01 mm. The fatigue life of the runway is found from A^c = 225,000[MR/ac]'^, where A^c is the aflowable number of overruns, MR the modulus of rupture of the concrete and Gc the principal tensile stress induced by the load [1-6]. The number of load repetitions A^s the subgrade can sustain is predicted using as the maximum downward vertical stress in the subgrade, the CBR and the equation, A^s = [[280 X CBR(%)]/crs]'^ [1-6]. Where a reduction in fatigue

4. The finite element model The finite element model was idealized within a circular cylinder with its axis lying vertically in the ground. The notional infinite nature of the ground surrounding the camTable 1 Young's modulus (MPa) for the 10 material sets Zone number

Material set 1

2

3

4

5

6

7

8

9

10

2 3 4 5

95 95 95 950

95 95 95 95

95 95 95 190

7 95 95 190

7 7 95 190

7 7 7 190

7 95 190 190

95 190 190 190

190 190 190 190

95 95 95 95

87

J.W. Bull/First MIT Conference on Computational Fluid and Solid Mechanics Table 2 Fatigue life of the filled and the unfilled camouflet Zone number

Material set 1

2

3

4

5

6

7

8

9

1 8 2 3 4 5 6 7

NC NC NC 87.2 55.5 I I NC

NC NC NC I 24.3 I I NC

NC NC NC 95.7 72.8 I I NC

38.4

26.3 I 1.0 0.1 I I I 8.6

31.5 I 1.0 0.2 0.04 I I 8.5

38.8

I NC 89.2 I 54.7 I I NC

I NC I I 79.9 I I I

1.0

8.9

life occurred, the difference between the corresponding filled and unfilled void was no more than 1.1%. Thus, both the filled and the unfilled camouflet are recorded as having the same fatigue life, as shown in Table 2. The remaining fatigue life is given as a percentage of the fatigue life of material set 10. Where there was no change or an increase in the fatigue life, this is indicated by NC or I, respectively. All nine material sets have a reduced fatigue life in the subgrade.

[2]

[3] 7. Conclusions The major cause of the large surface deflections is the weakening of zone 2. The extent of the surface deflection indicates inversely the fatigue life remaining in the pavement. The filling of the camouflet has little effect on reducing the runway deflections or on increasing fatigue life, although it does prevent the runway from collapsing completely. Once a camouflet has been identified, it should be excavated and refilled with the runway surface being cut back beyond the zone 1-8 interface.

[4]

[5] [6]

[7]

[8] References [1] Bull JW, Woodford CH. Computer simulation of explosion effects under concrete runways, B, Advances in Civil and

1.0

8.9

Structural Engineering Computing and Practice. In: Topping BHV (Ed), 4th International Conference on Computational Structures Technology. Edinburgh: Civil-Comp Press, 1998, pp. 369-376. Bull JW, Woodford CH. The effect on the fatigue life of an airfield runway when a large void beneath a runway is left unfilled or is filled. In: Seventh International Conference on Civil and Structural Engineering Computing, Oxford, UK, A, Computer Techniques for Civil and Structural Engineering, 1999, pp. 165-174. Bull JW, Woodford CH. The effect of camouflets on subgrade surface support, Comput Struct 1999;73:315-325. Bull JW, Woodford CH. The prevention of runway collapse following an underground explosion, Eng Failure Anal 1998;5(4):279-288. Bull JW, Woodford CH. Camouflets and their effect on runway support. Comput Struct 1998;69(6):695-706. Bull JW, Woodford CH. The effect of the tensile stress in the subgrades on the fatigue life of an airfield runway. In: Fifth International Conference on Computational Structures Technology, B, Computational Civil and Structural Engineering, Leuven, Belgium, 2000, pp, 265-274. Chadwick P, Cox AD, Hopkins HO. Mechanics of deep underground explosions, Phil Trans Roy Soc Lond Ser A Math Phys Sci, 1963-64:256;235-300. Das BM, Shin EC, Cyclic load-induced settlement of foundations on clay. In: Teeming MB, Topping BHV (Eds), Mouchel Centenary Conference on Innovation in Civil and Structural Engineering. Edinburgh: Civil-Comp Press, 1997, pp. 241-246.

Stochastic seismic analysis of R-FBI isolation system p. Cacciola, N. Impollonia, G. Muscolino * University of Messina, Dipartimento di Costruzioni e Tecnologie Avanzate, Salita Sperone 31, Vill. S. Agata, Messina 98166, Italy

Abstract The response of a structure isolated by a Resilient-Friction Base Isolator (R-FBI) subjected to a ground motion modeled as a stochastic process is studied. The moment equation approach is applied and the probability density function of the non-Gaussian response is evaluated adopting a C-type Gram-Charlier expansion. The results are compared with those obtained by means of Monte Carlo simulation. Keywords: R-FBI isolation system; Friction damping; Non-Gaussian response; Closure technique

1. Introduction In recent years considerable attention has been focused on the use of base isolation systems to protect structures against earthquake effects. The isolation system decouples the structure from the horizontal components of the ground motion by interposing a mechanism between the structure and the foundations. Several base isolation systems have been proposed and developed for various type of structures, and they are reviewed by Kelly [1]. The resilient-friction base isolator (R-FBI) system, proposed by Mostaghel and Kelly [2], is considered herein. The isolator combines rubber bearing and friction element in parallel and belongs to friction type systems. The simplest base isolators of this kind are pure friction base isolators. Generally it is assumed that the friction characteristics observe the Coulomb friction law. Consequently, the structure shding on a R-FBI system posses non linear behavior and equivalent linearization technique or stochastic averaging [3] can be resorted to determine the response with short computational time. In the present paper, an alternative method [4] evaluating the response by applying the moment differential equations approach is considered. A non-Gaussian closure technique is required due to non normality of the response process. Moreover, the use of the C-type Gram-Charlier expansion is proposed for the evaluation of the response probability

* Corresponding author. Tel.: -f-39 (90) 676-5618; Fax: +39 (90) 395022; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

density function which requires the knowledge of the statistical moments obtained solving a set of linear equations. The simple structural model considered refers to a rigid structure with a resilient-friction base isolator system under white noise excitation. As no closed form solution are available, in the numerical application the stationary response obtained with the proposed formulation will be compared with Monte Carlo simulation.

2. Mathematical formulation The rigid structure on friction devices is mathematically represented by a SDOF with viscous and Coulomb damping [2]. Under the assumption of high intensity base excitation, the stick phases do not occur so that the equation of motion is given as X -h l^cox -f o/x 4- Mg sgn(i) = —Xg(t)

(1)

where x is the displacement of the rigid structure relative to its foundation, oj is the natural frequency of the base isolator, ^ is the damping ratio, g is the gravity acceleration, sgn() is the signum function, /x is the friction coefficient and Xg(t) is the ground acceleration assumed to be a Gaussian, stationary white noise process, so that Xg{t) =

-W(t).

In order to evaluate the stochastic response, the statistical moments of the response have to be evaluated. To this aim Eq. (1) is converted into an equivalent first-order system and the Ito's differential rule [5] is utilized so that

R Cacciola et al. /First MIT Conference on Computational Fluid and Solid Mechanics

0.12
p(x)

0

\

O 0 O MCS CGC (N=4)

0.12 I

0.08 1

0.04

!
-20

i

\

^W^..,„..

-10

10

20

-20

10

0

-10

20

(b)

(a)

Fig. 1. Stationary marginal probability density function of the displacement (a) and the velocity (b) by C-type Gram-Charlier expansion (CGC) and Monte Carlo simulation (MCS).

the sought moment equations read

where the coefficients

EWi"^] :: lEix^-^x"^^^]

r]Q=

- l^comEWk"^] + \m{m -

+00

-

co^mEW+'x"^-^] sgn(i)]

- iigmEWi"^-^ l)E[x^x^-^]q

-00

(2)

where q = ITCSQ is the strength of the white noise. The latter equations are not closed, as the averages with the signum functions appear which have to be evaluated starting from the knowledge of all moments. In what follows an evaluation of these averages is performed in approximate form, by adopting a closure technique of the probability density function expressed by a A-type Gram-Charlier expansion and observing that all odd order moments vanish p(x,x)

=

po(x)po(x) &Po(x)

dJpo(x) dxJ

djc'

iJ-i—A 6 'J ' /+;=4,6

(3)

where r is the closure order and po(x), po(x) are the probability density functions of jc, x assumed as Gaussian ones PoM

=

1 V27ro-,

Poix) =

1

exp

(S?)' /-x^'

(4)

Then, in Eq. (2), the averages with signum functions become E[x'x'"sgn(i)]

= n',X'S+

E j,_L + ; i—A = 4 , 6A

rl-/i^'JCij[x,i] 'J '

x^po(x)dx,

(5)

+00

X'S = f x"^ sgn (x)po(x)dx, -00 +00

i&poix)

r^\ = {-iy I X

dx'

+00

xj = {-ly

sgn(x)

dx,

dJpoix) dxJ

dx

(6)

can be easily evaluated in closed form and the coefficients Cij[x,x] are related to the statistical moments of order equal or lower than (/ -h j) [4]. Substitution of Eq. (5) into Eq. (2) gives a set of non-linear differential equation where only statistical moments up to r-th order appear. However, if the variances a^ and a | are first obtained with enough accuracy, for example by means of Monte Carlo Simulation, than the system become a linear one and the evaluation of statistical moments up to order r is straightforward. The approximate response probability density function resulting from Eq. (3) posses some inconsistency, in particular the A-type Gram-Charlier expansion can lead to negative values around the tails. For these reason a C-type G r a m Charlier expansion is adopted p(x,x)

= A/'exp

tj"'i^A)

yj[x,x]

(7)

where A/" is a normalization constant, Hj l-^, f-j is the multi-dimensional Hermite polynomial vector and yj[x, x]

90

P. Cacciola et al /First MIT Conference on Computational Fluid and Solid Mechanics

is the y-th coefficient vector linear function of the statistical moments of the response, both of order 2^ [6]. The coefficient vector are linearly related to the coefficients appearing in the A-type expansion and can be evaluated by an efficient procedure [6]. Note that if Eq. (7) is utilized a closure of order r = 2A^ — 2 is needed.

tory accuracy for the displacement (Fig. la). On the other hand, a higher order closure is needed to approximate the stationary marginal probability density function of the velocity (Fig. lb) which is strongly non-Gaussian.

References 3. Numerical application An R-FBI isolator system with the following parameters has been considered: natural period 7 = 4 s, damping coefficient ^ = 0 . 1 , friction coefficient /x = 0.04. The ground acceleration is assumed to be a white noise with spectral density SQ = 55.44 cm^/s\ The stationary marginal probability density functions of the displacement and the velocity have been evaluated through Eq. (7) and reported in Fig. 1 along with those resulting from Monte Carlo simulation. Note that for the evaluation of the stationary characteristics the algebraic system arising from Eq. (2), where the left side is set equal to zero, has to be solved. The figure reveals that a low closure order (A^ = 4) produces satisfac-

[1] Kelly JM. Earthquake-Resistant Design with Rubber. London: Springer, 1996. [2] Mostaghel N, Kelly JM. Design procedure for R-FBI bearings. Report UCB/EERC-87/18, 1987. [3] Fan FG, Ahmadi G. Random response analysis of frictional base isolation system. J Eng Mech 1990;116:1881-1901. [4] Muscolino G, Pirrotta A, Ricciardi G. Non Gaussian closure techniques for the analysis of R-FBI isolation system. J Struct Control 1997;4(l):23-46. [5] Ito K. On a formula concerning stochastic differential. NagoyaMathJ 1951;3:55-65. [6] Muscolino G, Ricciardi G. Probability density function of MDOF structural systems under non-normal delta-correlated inputs. Comput Methods Appl Mech Eng 1999; 168:121133.

91

Geometric softening in geotechnical problems J.P.Carter*, e x . Wang University of Sydney, Department of Civil Engineering, Sydney, NSW 2006, Australia

Abstract An investigation is made of some of the circumstances under which softening of overall system response can occur in geotechnical boundary value problems, even when no material softening is permitted. It is demonstrated that a finite deformation formulation is required in order to capture this phenomenon in finite element computations. Comments are also made on the type of large deformation analysis likely to produce the most accurate results for footing penetration and plate uplift problems. Keywords: Large deformations; Finite strain; Footing penetration; Anchor uplift

1. Introduction Large deformation analyses of boundary value problems are not common in geotechnical engineering despite the fact that finite deformations may be important, particularly in problems involving penetration of relatively rigid bodies, such as footings, spud-can foundations, and in situ test probes, into much softer soil deposits. Although methods have been proposed for the numerical solution of this type of problem, detailed assessments of their capabilities and limitations are also rare in the literature. The purpose of this paper is to highlight a number of applications where a large deformation analysis is essential to capture some subtle but important aspect of soil behaviour. Boundary value problems involving footing penetration and anchor uplift are discussed, in order to demonstrate particular features that cannot be captured using conventional infinitesimal strain analysis. In particular, softening of the overall system response is identified as a possibility in some circumstances. A number of formulations for large deformation problems in geotechnical engineering have been published in the literature, e.g. [1-3]. Detailed discussion of similarities and key differences between these methods are given by Chen and Mizuno [2] and Wang [4]. Example problems solved using an updated Lagrangian approach published by

* Corresponding author. Tel.: +61 (2) 9351-22-99; Fax: +6\ (2) 9351-33-43; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Carter et al. [1] and the remeshing technique proposed by Hu and Randolph [3] are described in the following sections. Comments on the suitability of these published finite element formulations are also provided.

2. Penetration of a strip footing The problem of penetration of a smooth rigid strip footing of width B into purely cohesive soil has been analysed for both a homogeneous and a two-layered soil deposit. In all cases the material behaviour is characterised by an initial linear elastic response at small strains, followed by perfectly plastic behaviour. Yield is determined by the Tresca criterion and an associated flow rule, so that shearing occurs at constant volume. Undrained conditions have been simulated. 2.1. Homogeneous clay The normalised load-penetration curves for this case, obtained using the re-meshing technique of Hu and Randolph [3], are presented in Fig. 1. These curves indicate that at larger penetrations of the footing the mobilised penetration force is a function of the rigidity index of the soil {G/c). Generally, the stiffer the elastic response the greater the force required to cause a given penetration of the footing, even after the behaviour becomes dominated by plastic yielding. For these homogeneous soils, the curves continue to rise monotonically until an ultimate value is reached.

92

J.P. Carter, C.X. Wang/First MIT Conference on Computational Fluid and Solid Mechanics MESH

"(a)"

1950.0

-^-^-X^^x^>'^^~>^t^X^

\

"'*">< \

V'^-

:50.a i 100.0 n5o.D REMESH* Slnp tooting on the surlace of elastic-Mmple plastic soil

20

1200.0

(b)

30 Gs/Bc

Fig. 1. Normalised load-settlement curves for a strip footing on homogeneous clay {H/B = 1).

corresponding to the solution for a footing deeply buried in a half-space. Large deformation analyses were also conducted using an updated Lagrangian (UL) approach [1]. Deformed mesh plots from each analysis are shown in Fig. 2, for a footing displacement equal to 40% of the footing width. By comparing these plots, the relative advantage of the remeshing technique can be clearly seen. In the UL approach elements near the edge of the footing have become highly distorted at this footing displacement, and ultimately unfavourable element configurations will affect the accuracy of the numerical results. 2.2. Two-layered clay The bearing response of strip footings on a stronger clay layer of thickness H overlying a weaker clay deposit was also examined, and a comparison is made between the results given by the small and large deformation analyses. Various cases corresponding to H/B = 1, and C2/C1 = 0.1, 0.2, 1/3, 0.5, 2 / 3 and 1 (homogeneous soil) were investigated. For these particular analyses the effect of soil self-weight has been ignored, so that these results are strictly relevant in practice whenever yB <$C c i , where y is the unit weight of the soil, i.e., for relatively narrow footings or strong soils. Normalised load-displacement curves for a weightless soil are shown in Fig. 3, for cases where H/B = 1. Typically, the curve given by the small deformation analysis reaches an ultimate value after a relatively small footing penetration, and generally the load-displacement curve given by the large deformation analyses is quite different from that given by the small displacement analysis.

'

^C

AFENA * Strip Footing on Layered Soil

Fig. 2. Finite element meshes for penetration of homogeneous clay by a strip footing, (a) Mesh configuration using the remeshing method, (b) Mesh configuration using the updated Lagrangian method.

Large deformation Cg/Cisl

0.8

2+2n

0.8(2+2)t)|

2/3

2^3{2+2K)\

" ^

2/3 0.5(2+2rt)

Q2

Meyerhof(1951) anaiyticai solution lor deep footing 0.2(2+2K)

150 200 Gs/Bc, or Gs/Bcg

Fig. 3. Normalised load-settlement curves for a strip footing on layered clay (H/B = 1).

J.p. Carter, C.X. Wang/First MIT Conference on Computational Fluid and Solid Mechanics For cases where a stronger top layer overlies a much weaker bottom layer (e.g., C2/C1 = 0.1, 0.2, and 0.5), the overall response is characterised by some brittleness (softening), even though the behaviour of both component materials is perfectly plastic and thus characterised by an absence of softening. For these cases, the load-penetration curves given by the large deformation analysis rise to a peak, at which point the average bearing pressure is generally lower than the ultimate bearing capacity predicted by the small deformation analysis. With further penetration of the footing into the clay, it appears that the load-displacement curve approaches an asymptotic value. It is reasonable to expect that even footings exhibiting a brittle response should ultimately behave much like a deep strip footing buried in the lower clay layer, so that the ultimate value of the average bearing pressure should then be approximately (2 -h 2n)c2, where C2 is the strength of the lower layer. These theoretical limits for a deeply buried smooth footing are also indicated in Fig. 3. Curves obtained from the large deformation analysis appear to approach these limiting values at deep penetrations. It is also interesting to note that for this geometry, H/B = 1, and when ci/ci is greater than about 2/3, the large deformation curves appear to rise monotonically to their asymptotic ultimate values. For these cases the ultimate values are reached only when the footing has penetrated into the bottom layer and the top layer has separated into two distinct parts. Wang [4] has demonstrated that brittle behaviour of the footing tends to be suppressed as the self-weight effects become more significant, i.e., as yB increases relative to the strengths of the clay layers, ci and C2. Clearly this trend is to be expected, because with increasing penetration the surcharge effect of the soil to the sides of the footing becomes more significant. This aspect of penetration behaviour has also been demonstrated previously in the numerical solutions obtained by Hu and Randolph [5] for spud-can footing penetration into inhomogeneous soil. 3. Uplift of a rigid strip anchor The problem of a horizontal strip anchor embedded beneath the surface of a homogeneous, elastoplastic, purely cohesive half-space and pulled vertically upward has been investigated. Fig. 4 indicates predictions of the load-displacement behaviour of anchors at relatively shallow depths of embedment, i.e., at depths given by H/B = 0.5, 1, 2, where H is the depth of embedment and B is the width of the strip anchor. For all cases shown in this figure perfect bonding was assumed between the underside of the rigid anchor plate and the underlying soil. In addition, no limit has been placed on the tensile capacity of the soil. Solutions for both a weightless soil, which is a reasonable idealisation for

93

•Small deformation analysis " Large deformation analysis • Rowe & Booker (1979) elastic solution

__

z i 1 /

,

, ......

,

^

^

^

^

^

^

16

^

"

2 TH/C=O

sii^aji cjeformatlon analysis Large deformation analysis

(b) H/B=l

Rowe & Booker (1979) elastic solution

GSi^c

to

. , . . •r.JUi.^S^met-^'

' '' •'••^-*^*^^^^-^-^'

••• r,.v» - " •

16

^

7H/c=0

10 8 6 4

1

Small deformation analysis

1

Large deformation analysis

2 \

(c) H/B=2

Rowe & Boolter (1979) elastic solution

n 15

20 Gs/8c

Fig. 4. Load-deflection curves for fully bonded anchors.

cases of relatively shallow burial in relatively strong soils, and soils with significant self-weight are included in Fig. 4. It is clear from this figure that a softening response occurs for cases where the strength of the soil, c, is relatively large compared to the overburden pressure at the plate level, yH. Softening tends to become suppressed as the depth of burial and the self-weight effects increase. From a practical perspective it is also of interest to examine the case where separation of the rigid anchor from the soil immediately beneath it is allowed to occur. It was assumed that separation will occur and a gap under the plate will form once the initial total overburden pressure is offset by the uplift load applied to the plate. Small and large deformation solutions for this important case are presented in Fig. 5. Comparison of Fig. 5 with Fig. 4b reveals that bonding of the soil has a very significant influence on the mobilised uplift capacity. Indeed for the case where yHjc = 0, the difference between the ultimate capacities in these two cases is approximately the same as the reverse bearing capacity of a strip footing on a purely cohesive half space, i.e., (2 + 7t)c.

94

J.P. Carter, C.X. Wang/First MIT Conference on Computational Fluid and Solid Mechanics 12

the softening behaviour could only be predicted using an appropriate large deformation analysis; the small strain analysis could not capture this type of response. For the footing and anchor problems it was also found that selfweight of the soil medium tends to suppress the tendency for a brittle system response.

/

10

)——*—x^..,,^^

^-^•''>^-

f^r=m

' ^S^x""' ' •/H/c=6

1

YH/C=4

1

Acknowledgements

yH/c=2 Smalt deformation

The work described in this paper has been supported by grants from the Australian Research Council.

Large deformation 1/^

Rowe & Booker(1979) elastic solution

"~7H7C^O

10

O

Breakaway (large deformation)

X

Breakaway (smal! deformation)

15

20

25

30

35

References

Gs/Bc

Fig. 5. Load-deflection curves for strip anchors with separation (H/B = 1). As for the footing problem, ultimately unfavourable element configurations in the UL formulation will affect the accuracy of the numerical predictions. More reliable solutions at large displacement were obtained using the remeshing technique.

4. Conclusions Two boundary value problems have been examined using both small and large deformation analyses. In each case it was discovered that softening of the overall system response occurs under certain conditions. Furthermore,

[1] Carter JP, Booker JR, Davis EH. Finite deformation of an elastoplastic soil. Int J Num Anal Methods Geomech 1977;l(l):25-43. [2] Chen WF, Mizuno E. Nonlinear Analysis in Soil Mechanics, Theory and Implementation. Amsterdam: Elsevier, 1990. [3] Hu Y, Randolph MP. A practical numerical approach for large deformation problems in soil. Int J Num Anal Methods Geomech 1998;22:327-350. [4] Wang e x . Applications of Large Deformation Analysis in Soil Mechanics. PhD Thesis, University of Sydney, 2000. [5] Hu Y, Randolph MP. Deep penetration of shallow foundations on non-homogeneous soil. Soils Pound 1998;38(l):241-246. [61 Meyerhof GG. The ultimate bearing capacity of foundations. Geotechnique 1951;2(4):301-332. [7] Rowe RK, Booker JR. A method of analysis for horizontally embedded anchors in an elastic soil. Int J Num Anal Methods Geomech 1979;3:187-203.

95

A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates Song Cen ^, Yuqiu Long ^, Zhenhan Yao ^'* ^ Tsinghua University, Department of Engineering Mechanics, Beijing, 100084, China ^ Tsinghua University, Department of Civil Engineering, Beijing 100084, China

Abstract A simple displacement-based, quadrilateral 20 DOF (5 DOF per node) bending element based on the first-order shear deformation theory (FSDT) for analysis of arbitrary laminated composite plate is presented in this paper. This element is constructed by the following procedure: (i) the variation functions of the rotation and shear strain along each side of the element are determined using Timoshenko's beam theory; and (ii) the rotation, shear strain and in-plane displacement fields in the domain of the element are then determined using the technique of improved interpolation. The stress solutions are improved by a simple hybrid procedure. The proposed element, denoted as CTMQ, possesses advantages of both displacement element and hybrid element. Thus, very excellent solutions for both displacements and stresses, especially for the transverse shear stresses, can be obtained. Keywords: Finite element; Laminated composite plates; Timoshenko's beam theory; First-order shear deformation theory (FSDT); Hybrid-enhanced procedure

1. Introduction In the past 40 years, the formulation of robust plate bending elements based on FSDT (Reissner-Mindlin plate theory) has attracted the attention of many researchers. One of the best approaches is the mixed interpolation method, in which the displacement fields and the shear strain field are interpolated independently [1]. In this paper, a new similar method is proposed to construct bending element for analysis of laminated composite plates. Furthermore, a simple hybrid method is also presented to improve the stress solutions. Thus, good results can be obtained for both displacements and stresses, and no shear locking will happen even the thickness of plate approaches zero.

{UY = [Ui Vi Wi

\ll^i

fy2

The formulas of deflection ic, rotation f and shear strain y for the Timoshenko's laminated composite beam element, as shown in Fig. 2, are as follow: w = Wi(l - r) + Wjr + -{f,i ^

- xlf,j)F2 - - F d - 28)F3 (2a)

^Ir, = f,i{l - r) + ir.jr + 3(1 - 28)rF2

The nodal displacement vector for the quadrilateral 4-node element is (as shown in Fig. 1):

8= where

(2b) (2c)

Y =8T F = -l^-^i

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

: U2 V2 W2 fx2

2.1. Locking-free Timoshenko's beam element

2. Formulation of the new quadrilateral element

* Corresponding author. Tel.: -h86 (10) 6277-2913; Fax: +86 (10) 6278-1824; E-mail: [email protected]

ifyi

k = F2 =

+ ^ y ) ~ ^^i ~ ^^J

6X 1 + 12A

A

r(l-r)

F3 = r ( l - r ) ( l - 2 r )

(3)

S. Cen et al. /First MIT Conference on Computational Fluid and Solid Mechanics

96

Wx3 4< ^. H'

^x\ ¥x2 Fig. 1. Quadrilateral plate element.

r=— /

r={)

bi = yj-yk

i

11

/=1

0 0

V2 ^

til

- • ' • • • -

-

ITT

—

1

"^1

t

-^6

Ci=Xk-Xj

[r*] =

0

0 0 0

Di and C/ in Eq. (3) are the bending and shear stiffness of the beam, respectively. It can be proved that when the thickness r ^- 0, 5 ^- 0. No shear locking will happen. 2.2. Interpolation formulas for the shear strain fields

[A^?][Fj[r] [A^?] = [ <

A^o

^0

{uY =

(4)

[B,][uY

^0]

0

-ci(5i

bi8i

0

0

253

-C353

^3^3

0

0

0

0

0

0

0

-254

-C454

^4^4

0

0

254

-C454

^4-54

28i

-ciSi

bi8x

0 0

0

0

0

0

0

-2^2

-C252

^2^2

0

0

2^2

-C282

b2h

0

0

0

0

0

0

0

-253

-C3(53

b^h

0

0

0

0

0

0

0

(9)

where A^^ (/ = 1, 2, 3, 4) are the bilinear shape functions. 5/ are given by Eq. (3).

The rotation fields can be obtained using 8-node isoparametric nodal shape functions A^, (/ = 1, 2 , . . . , 7, 8): ^x = T^^ii^x,

bi b\C2 — bjci b3 — bi,C2

0

-b\C4

0

0

b2

0

b2CT, — byC2

bT,C4 — b^CT,

i=\

Since the rotations of the mid-side nodes can be expressed in terms of the nodal displacement vector by using Eq. (2), the rotation fields can be rewritten as follows:

bi

b^cx

bjcj,

0

0

0 0 0

(5)

b^ci — biC4 biC2 — bjci

0 0 -25i 0

(10)

0

[^.]-

0 0

2.3. Interpolation formulas for the rotation fields

From Eq. (Ic) and some simple geometrical relations, the shear strain fields can be obtained as follows:

[A^?][xj[r*]

0 0

0

0 0

Fig. 2. The Timoshenko laminated beam element.

(8)

(i,j,k=\,2,3,4)

(6) {V'} =

[N]{u}

(11)

2.4. Interpolation formulas for in-plane displacement fields of the mid-plane

b2,C4 — ^4C3

C4

[Ys} =

Z?4Ci

blC2

-biC4

— b2C\

b4Ci

b\C2

— b2C\ ^3

blC^

— b2,C2

-bxC4

0

^ 2 ^ 3 — Z73C2

0

(7)

The in-plane displacement fields are expressed by the bilinear shape functions. Finally, the stiffness matrix can be obtained by the standard procedure.

S. Cen et al. /First MIT Conference on Computational Fluid and Solid Mechanics 3. The hybrid-enhanced procedure for element stresses

ai - \(-xi

+ X2 + -^3 - -^4)

The bending moment field {M} and the shear force field {J} are only required to satisfy C~^-continuity between two elements based on FSDT [2]. Thus, [M] can be assumed as follows:

as = \(-xi

- X2 + ^3 + X4)

h = \(-yi

-yi

[M] = [M,

My

I ^ [PM]

{(XM)

=

M,yf

(12)

= [PM]{aM} 0

ri ^T] 0 0 0

0 0 0

0

I

0 0 0

0

0 0 0

^

0 0 0

0

Y] ^7] 0 0 0

— [Ofi 0^2 0^3 0^4 Qf5 a^

0

I

^

OCl Qfg Otg

0

(13)

T) ^T] Q^IQ

Otn

at (/ = 1, 2, 3 , . . . , 12) are 12 unknown parameters. From the equilibrium equation of a plate, the shear field {T] can be obtained: 9 M , ^ dM^y

r,r

dx

dy

dM^y

dMy

dx

dy (15)

= [PrUaM]

The membrane force field {A^} can be assumed as follows [3]: {A^} = [N,

[PN]

=

Ny

N^yf

1

0

0

0

1

0

0

0

1

=

air]

aibiY]

[PNMPN

(16)

a\^

t^

(18)

+ ys + yd

{M = [Pi ft ft ft ft]

(19)

ft (/ = 1, 2, . . . , 5) are 5 unknown parameters. Then the stresses of the element can be obtained by using Hellinger-Reissner variational principle and hybrid element method. Note this procedure doesn't influence the element stiffness matrix, it is only for improving stress solutions.

a^^

(14)

{T] = [r,

97

4. Numerical examples Several numerical examples are presented to evaluate the performance of the new element. One of them is showed in Fig. 3 and Table 1.

5. Conclusions The presented element, CTMQ, can pass all the patch tests, is free of shear locking and insensitive to mesh distortion. It possesses advantages of both displacement element and hybrid element: Relatively simple formulation, high accuracy for both displacements and stresses.

(17)

a^b^^

STl GKOMHrRY 1=1000.; ^=250,100,20,10,1,0.1 MATERIAL (orthotropic) Skins: Er^25.; Er'U Git^S; Gif^^S; Gw^l; pi2=^.25 STl: 0/90/0/90/0/90/0/90/0 symmeMc BOUNDARY CQNDmONS (simply-gupported: SS2) on AB: tt=w=\|r/=0; oa EC: 1^^^^ on CD: v=%=0 ; on DA: v=H?=^y^ LQADINe (doubly sinusoidal) . nx . nv Fig. 3. Square plates with 9 layers.

98

S. Cen et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Maximum deflection and stresses in 9-ply laminate L/h

Mesh and model

w

(iJQ) 4

10

50

100

100000

(L L ±lL) V2 ' 2 ' ^ 2 /

Oy (L L _|_2/i\ l 2 ' 2 ' ^ 5 >>

^x

"^xy

(Q^Q^I)

4x4 8x8 16 X 16 DST 10 X 10^ FSDT

4.283 4.252 4.244 4.242 4.242

±0.498 ±0.493 ±0.492 ±0.547 ±0.491

±0.494 ±0.489 ±0.487 ±0.419 ±0.487

4x4 8x8 16 x 16 DST 10 X 10^ 3D elasticity FSDT

1.529 1.524 1.523 1.526 1.512 1.522

±0.526 ±0.521 ±0.519 ±0.541 ±0.551 ±0.519

±0.461 ±0.456 ±0.455 ±0.425 ±0.477 ±0.454

4x4 8x8 16 X 16 DST 10 X 10^ 3D elasticity FSDT

1.021 1.021 1.021 1.020 1.021 1.021

±0.545 ±0.539 ±0.538 ±0.522 ±0.539 ±0.538

±0.438 ±0.434 ±0.433 ±0.447 ±0.433 ±0.432

TO.0209

=F0.0210

4x4 8x8 16 X 16 3D elasticity FSDT

1.005 1.005 1.005 1.005 1.005

±0.545 ±0.540 ±0.539 ±0.539 ±0.538

±0.437 ±0.433 ±0.432 ±0.431 ±0.431

4x4 8x8 16 X 16 FSDT CPT

1.000 1.000 1.000 1.000 1.000

±0.545 ±0.540 ±0.539 ±0.539 ±0.539

±0.436 ±0.432 ±0.431 ±0.431 ±0.431

References [1] Bathe KJ, Dvorkin EN. Short communication: A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int J Numer Methods Eng 1985;21:367-383. [2] Ayad R, Dhatt G, Batoz JL. A new hybrid-mixed variational approach for Reissner-Mindlin plates: The Misp Model. Int J Numer Methods Eng 1998;42:1149-1179.

=F0.0214 TO.0217 TO.0217

IFO.0217 =F0.0212 IFO.0214 :T0.0214

IFO.0233 TO.0215 =F0.0210 TO.0212 TO.0213

TO.0214 =F0.0213

TO.0212 TO.0213 =F0.0213 =F0.0213

TO.0212 =F0.0213 TO.0213 =F0.0213

ixz (Q^JQ)

'^yz

0.234 0.237 0.237 0.225 0.238

0.243 0.245 0.246 0.231 0.245

0.246 0.249 0.249 0.219 0.247 0.250

0.228 0.230 0.231 0.257 0.226 0.230

0.251 0.256 0.257 0.190 0.258 0.258

0.213 0.218 0.220 0.263 0.219 0.219

0.249 0.254 0.257 0.259 0.259

0.210 0.215 0.218 0.219 0.219

0.247 0.250 0.250 0.259 0.259

0.207 0.210 0.210 0.219 0.219

(f^Q^Q)

[3] Pian THH, Sumihara K, Rational approach for assumed stress finite element. Int J Numer Methods Eng 1984;20:1685-1695. [4] Lardeur P, Batoz JL. Composite plate analysis using a new discrete shear triangular finite element. Int J Numer Methods Eng 1989;27:343-359.

Simulating static and dynamic lateral load testing of bridge foundations using nonlinear finite element models S. Chakraborty ^'*, D.A. Brown^ ^ Wilbur Smith Associates, P.O. Box 92, Columbia, SC 29202-0092, USA ^ Auburn University, Civil Engineering Department, Auburn, AL 36849 USA

Abstract The response of bridge foundations to large amplitude lateral loads was the subject of a study conducted at Auburn University. As part of the study, static and dynamic load tests were carried out on two full-scale instrumented test foundations on the Pascagoula River at Pascagoula, Mississippi. The measured response was used to develop and calibrate nonlinear finite element models for a detailed analysis of the parameters that govern the lateral behavior of such systems. The results of this study have been summarized in this paper. Keywords: Bridge foundations; Lateral load testing; Finite element analysis; Soil-structure interaction; P-Y Nonlinear dynamic analysis

1. Introduction and background Bridge foundations are subject to dynamic lateral loads in the form of earthquakes and ship impact, which usually involve the transfer of large amounts of energy to the foundation in short periods of time. The response of the foundation-soil system under such conditions is usually highly nonlinear, and difficult to define mathematically. Since dynamic load testing on instrumented foundation groups can be extremely expensive, not much data are available to assess the reliability of existing modeUng techniques. This paper provides a brief description of a load test program carried out on the Pascagoula River in Mississippi [1]. The testing was conducted to provide guidelines for the design of a new bridge over the river at Pascagoula, which would replace an existing bridge. In addition to extensive in-situ and laboratory testing to determine soil properties, lateral static and dynamic load tests were carried out on two test foundations. The tests were designed to induce significant nonlinearities in the structural elements (piles and shafts) and the soil in which they were embedded. Displacements, strains and accelerations, along with appUed load, were monitored and recorded for further analysis. * Corresponding author. Tel +1 (803) 758-4643; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

curves;

The program SeaStar/CAP (PMB Engineering, [2]) was used to develop finite element models of both groups. The models were designed to account for the nonlinear behavior of the system under static and dynamic loads. The fundamental soil properties used in the model were established using the static load data, and the parameters governing dynamic response were studied using the measured dynamic response (displacements, and bending and axial forces).

2. Load test program The foundations along with the applied loads are shown in Fig. 1. The foundation on the east side consisted of six driven 0.76 m square prestressed concrete piles, four of which were driven on a 1:4 batter. The second group consisted of a pair of 2.13 m diameter reinforced concrete drilled shafts. Each shaft had a permanent steel casing extending from the bottom of the shaft for a length of 11.0 m. Both foundations were instrumented with displacement transducers, accelerometers, and strain gauge pairs placed at selected elevations down the length of the piles and shafts. A static lateral load test was conducted by pushing the foundations apart with a hydraulic jack (*S in Fig. 1). A series of five dynamic lateral load pulses at increasing amplitude were then applied to each foundation using a

100

S. Chakraborty, DA. Brown / First MIT Conference on Computational Fluid and Solid Mechanics *P - applied statnamic load *S - applied static load

N

PLAN

All dimensions in meters Not to scale

10.67 4.57

1.45

-^ I 1.52 Two 2.13m diameter drilled shafts

Six 0.76m square prestressed concrete piles

k 2.44

1:4 Batter

Fig. 1. Load test setup at Pascagoula, Mississippi. Statnamic device (*P). The Statnamic device is a loading mechanism that uses rapid bum fuel to generate gas pressure and accelerate a large mass away from the test foundation, thereby imparting an impulsive load to it. Most of the energy delivered to the foundation lies between 1 and 10 Hz, and the test simulates extreme event loads such as seismic, transient wind loading, and vessel impact. The peak dynamic loads recorded at Pascagoula ranged from 1.3 MN to 7.4 MN, with the ramp time (zero to peak load) decreasing from 0.25 to 0.13 s. The data were recorded using a Megadac data acquisition system at a sampling frequency of 2000 Hz. A detailed analysis of the measured data is available in Chakraborty [3].

3. Pile-soil interaction: the P-Y approach The lateral load displacement relationship between the foundation and soil is usually defined using the so called P-Y curves. P represents soil resistance per unit length of pile, expressed as a nonlinear function of its lateral deflection, F. The formulation of these curves are empirical, and based on the results of load test programs conducted in the 1970's and the 1980's. Details on the formulation of P-Y curves in different soil types may be found in Matlock [4], Matlock et al. [5], and Reese et al. [6,7], amongst others. The structural components (piles/shafts/cap) are usually modeled using linear or nonlinear beam elements, and the complex soil reactions are modeled using nonlinear lateral springs (P-Y curves), axial springs {T-Z curves), and tip springs (Q-Z curves), along with dashpots attached at the node points.

The P-Y model for the soil reactions combined with a finite element formulation of the foundation structure provides an effective solution to the problem of laterally loaded deep foundations. P-Y curves can be used to model short term static, cyclic (including degradation effects), and dynamic loading conditions. The effects of soil (internal) damping are accounted for during hysteretic cycling through the curve, while the effects of radiation damping are modeled through use of mechanical dashpots, which have been described in Dobry et al. [8].

4. Structural models The finite element model for each group was setup using a combination of linear and nonlinear beam elements, along with appropriate soil-springs and dashpots. The model for the pile group is shown in Fig. 2. The pile-cap was modeled using seven linear beam elements, with section and material properties simulating the rigidity and mass of the concrete block. Each pile was represented by twenty six beam elements, 1.219 m each in length. The topmost element (where significant cracking was observed at the larger loads) was modeled as nonlinear, and its flexural rigidity was computed as a function of the applied moment (M vs. EI) at the prestress load using the program STIFF 1 [9]. Lateral and axial soil-springs were attached at the pile nodes as shown in Fig. 2, along with axial springs at the pile toes to simulate resistance in end bearing. The loads were applied at the center of the pile-cap in the plane depicted by the front view, and out-of-plane displacements and rotations were eliminated in order to reduce problem

S. Chakraborty, DA. Brown/First MIT Conference on Computational Fluid and Solid Mechanics

Lateral and Axial Soil Springs

Front View

Side View

Total number of groups = 8 1 linear group representing the cap 1 nonlinear group - topmost element of each pile 6 linear groups - each pile from second to bottom element

Top View

Fig. 2. Finite element model of pile group. size. The forcing function used as input for each dynamic load case was derived by re-sampling the corresponding measured load vector at 400 Hz.

5. Verification and analysis Based on the soil-test data, the soil profile was described using four layers (top to bottom): - Sand: angle of friction, 0 = 30°, 2.44 m thick - Clay: undrained shear strength, c = 27.58 kPa, 4.57 m thick - Clay: c = 41.37 kPa (top) to 82.74 kPa (bottom), varying hnearly, 5.18 m thick - Sand: 0 = 38°, 28.0 m thick In addition to the fundamental properties (c, 0), the following parameters (amongst others) were used to control soil response (PMB Engineering, Inc. [2]): - Maximum Displacement Factor (A i) Ratio of the displacement at which the maximum spring resistance is mobilized, to the effective pile diameter (default/recommended = 0.1). - Rate Effect Parameter (P) A scaling factor for the P-ordinates (^ < I for creep effects, ^ > 1 for dynamic loading). - Shear Modulus of Soil (Gmod)

101

Used to define the radiation damping coefficient under dynamic loading. The models were verified and calibrated by comparing the computed response to the measured static and dynamic response (load test data). The following response parameters were used in the comparison: • lateral displacement/rotation of the cap • lateral/axial deformation profiles along selected piles/shafts • lateral/axial force profiles along selected piles/shafts The measured and computed static axial force profiles for the batter piles in tension and compression are shown in Fig. 3. The profiles have been plotted at two levels of lateral load. The measured axial forces were derived from the strain readings at each load level, and agree with the predicted axial load distribution for the piles in uplift and in compression. Fig. 4a plots the measured and computed static lateral load displacement response of the pile-cap. Fig. 4b shows the measured and computed lateral displacement time history of the pile-cap for the largest dynamic loading event (Statnamic load case 5). Figs. 3 and 4 are representative of the nature of the results obtained for the other load cases, as well as for the shaft group. A detailed discussion of the results has been presented in Chakraborty [3].

6. Summary and conclusions The program SeaStar incorporates a dynamic soilstructure interaction model, and was able to simulate the nonlinear response of the test foundations to lateral loading through the use of P - F , T-Z and Q-Z curves. It is believed however, that the performance of the program can be improved through the incorporation of a nonlinear material model for reinforced concrete. The results of the static load test were used quite effectively to establish the fundamental soil strength parameters and verify the model. The absence of damping and inertial forces reduced the complexity of the problem and the number of parameters that needed to be established initially. The shaft group exhibited a significantly higher degree of nonlinearity than the pile group. For the pile group, a time step ranging from 0.02 to 0.015 s appeared to produce acceptable results. For the shaft group, a much smaller time step was required to get the solver to converge, decreasing from 0.006 to 0.001 s from the first Statnamic load case to the last. The response of the pile in end bearing appeared to have a significant influence on the lateral stiffness of the foundations. For both groups, a large proportion of the compressive axial load in the piles and shafts was carried by the toe. The recommended value of the Maximum Displacement Factor, Ai appeared to be too conservative

102

5. Chakraborty, D.A. Brown/First MIT Conference on Computational Fluid and Solid Mechanics Batter Pile in Compression

Batter Pile in Tension

10

''

^ 15 E

20

20

E

25

30 h

o Measured ,

0

200

400

600

800

o Measured * Computed

* Computed

1000

35 -1000

-800

Compressive Load (KN)

-600

-400

-200

Tensile Load (KN)

Fig. 3. Axial force distribution in batter piles. 4000r 3000H S

2000

- I lOOOh

Measured Computedl 0.005

0.01

0.015

0.02 0.025 Displacement (m)

0.03

0.035

0.04

0.06r

Fig. 4. (a) Static lateral load displacement response — pile group, (b) Dynamic lateral displacement history — pile group.

and resulted in an under-prediction of the lateral stiffness. The inclusion of cyclic degradation and gap formation in the soil model caused the free vibration time period to elongate significantly, but did not affect the amplitude of the response.

References [1] Crapps DK, Brown DA. East Pascagoula river bridge test program. Project report prepared for Mississippi Dept. of Transportation, Vol. I and II, 1998. [2] PMB Engineering, Inc. SeaStar P3.20: Offshore analysis and

S. Chakraborty, DA. Brown/First MIT Conference on Computational Fluid and Solid Mechanics design software. User's manual, San Francisco, California, 1994. [3] Chakraborty S. Dynamic lateral load testing of deep foundation groups. Doctoral Dissertation, Auburn University, 2000. [4] Matlock H. Correlations for design of laterally loaded piles in soft clay. In: Offshore Technology Conference, Vol. 1, Paper No. 1204, Houston, Texas, 1970. [5] Matlock H, Ingram WB, Kelley AE, Bogard D. Field tests of the lateral load behavior of pile groups in soft clay. In: Offshore Technology Conference, Paper No. OTC 3871, Houston, Texas, May, 1980.

103

[6] Reese LC, Cox WR, Koop FD. Analysis of laterally loaded piles in sand. In: Offshore Technology Conference, Vol. II, Paper No. 2080, 1974. [7] Reese LC, Welch RC. Lateral loading of deep foundations in stiff clay. J Geotech Eng Div ASCE 1975;101(GT7). [8] Dobry RE, Vicente MJ, Roesset JM. Horizontal stiffness and damping of single piles. J Geotech Eng Div ASCE 1992;108(GT3). [9] Wang ST, Reese LC. STIFFl: Computation of nonhnear stiffnesses and ultimate bending moment of reinforced-concrete and pipe sections. For Ensoft, Inc. Austin, Texas, 1987.

104

Asymptotic analysis of the coupled model shells-3D solids D. Chapelle *, A. Ferent INRIA Rocquencourt, Projet MACS, BP 105, 78153 Le Chesnay Cedex, France

Abstract The purpose of this paper is to find a mathematical model for coupling a thin shell with a softer 3D elastic material. One of the main issues involved pertains to the treatment of interfaces. We have the choice of using or neglecting the rotations in the coupling conditions. We justify the use of one or the other strategy by an asymptotic analysis, the model with free rotations being the limit problem of the model with coupled rotations, when the thickness goes to zero. We also present some numerical results. Keywords: Shell; Linear elasticity; Asymptotic analysis; Penalized problem; Singularly perturbed problem

1. Variational formulation of the coupled problem We denote by Q^^ and Q^j the elastic body domains, by Q' the shell domain and by co the middle surface of the shell (Fig. 1). We also introduce ^5 and Qi as the domains occupied by one of the elastic bodies together with the superior and the inferior part of the shell, respectively. Hence: ^ ' - = (^s \ ^ s ) U ( Q / \ ^ )

and

co =

Qs(^^i-

and where z represents the distance to the midsurface o) counted positively in ^ 5 and negatively in Q/. The variational formulation of the coupled problem is given by: {Vt) : Find (w^, u\e\u\)

e V such that

tEsDiU^ e\ u^ ^0 + t^EsA{u\ 0\ v\ ^0 = F&s^ v\ ^^ v]),

W&s, v\ ^^ v'j) e V^

Let us introduce the following spaces: V, = {(u, ^) G H\OJ)^

^•a3=0,

X

V{Qi) = [v e H\Q,)\v^^i^ V(Qs) =

H\a))\

v\r^ = rj\r^ = 6 } , =0],

{veH\Qs)\v\ri,=0],

V = {(u^, v', ^\ v]) e V{Qs) xV,x

V(Q;)

which satisfies (C,) and rj' • ^3 = 0 ) , with

(C,)

on Q5 \ ^\

on Qf \ Q'j * Corresponding author. Tel: +33.1.39.63.57.56; Fax: +33.1.39. 63.58.82; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Fig. 1. A shell coupled with a soft elastic material.

(1)

105

D. Chapelle, A. Ferent/First MIT Conference on Computational Fluid and Solid Mechanics In Eq. (1), tEsD{u\ 0\ v\ ^ 0 + t^EsA{u\ 0\ v', ^ 0 represents the contribution of the shell to the internal virtual work, provided by the Naghdi linear shell model. We denote by t the shell thickness and by Es Young's modulus for the shell material. The contributions of the elastic bodies are given by the bilinear forms 5 ^ and B\ corresponding to a tridimensional linear elastic model defined on the volumes Q^^ and ^ ; , respectively. Finally, the linear form F represents the virtual work of external forces. We then have the following result [4]: Proposition 1.1 VF G L^(Qs)^ X L^(co)^ X L^(Qi)^ there exists a unique (u's, u',d\ u\) G V solution of Problem (Vt). Like in the asymptotic analysis of a shell alone, the space of pure bending displacements plays an important role. It is defined by: Vpb = {{V, ri)eVs\

D(v, ^, V, ^) = 0}.

The asymptotic behaviour of the shell is enced by whether or not this pure bending non-zero displacements. Of course, for the lem, this is also crucial. We then distinguish

strongly influspace contains coupled probthe two cases.

2. Non-inhibited pure bending A shell with non-inhibited pure bending has a bending-dominated behaviour, namely the membrane and shear energies tend to 0 with t [3]. In order to obtain a model with a real coupling, we need to assume that the dominant energies are of the same order of magnitude in the shell and in the elastic body. This leads to the following assumption: Assumption 1 EQ

= Est

where L^D represents a characteristic dimension of the 3D body. Under this additional assumption the solution of (Vt) remains uniformly bounded. Then (Vt) becomes a penalized problem and the constraint that we tend to impose is that membrane and shear energies vanish. Theorem 2.1 Under Assumption 1, the solution (u^^, u\0\ u\) of (Vt) converges strongly with respect to the norm of H^(Qs)^ x H\cof X H\Qj)\ as t goes to zero, to (u^^lP J^,iPj), the solution of the following problem: (V^^"")

: Find (ul, u\ e\ u^j) G V ^ such that

EQA(U\

e \ v\

= F(vl

7P) + Bs(u%

v\ ri\ 5^),

v\ ri\ S?) G V ^

yo = [(vl

v\ ri\ v1) G V(Qs)

X V. X

V(^i)

which satisfies (Co), ^^ -a3= 0, D(v^, ^^,v\

(2)

ri^) = 0},

with (Co)

3. Inhibited pure bending Unlike in the previous case, a shell with inhibited pure bending has a membrane-dominated behaviour and the bending energy can be neglected if the shell thickness is small. Then, the tridimensional elastic body must have an energy with the same order of magnitude as the shell membrane energy to obtain a coupled 3D-membrane problem, as t goes to 0, as in: Assumption 2 EQ = Est ^

E^oLsD-

Under this assumption, (Vt) becomes a singularly perturbed problem, where the perturbation corresponds to the shell bending energy. Since Vp, = {(0,0)}, D(', y^^ provides a norm on Vs and we can introduce aer ^

^ HH^S) ' + D ( - , - ) +

- HH^I) '

1/2

the corresponding norm on V^ As Vs is not a complete space for D(', ')^^^, we define V^ as its completion with respect to this norm. With this definition, V^ is less regular than Vs and the difficulty consists in establishing its exact nature. This nature depends on the boundary conditions and the shell geometry. As an example, we consider a situation where we can characterize the space V^, namely the case of an elUptic shell clamped along the whole boundary. In this case, we have V, = H^(o)) X H^(co) X LHCO) X 7^,, where IZs is the regularity space of the rotations, defined so that the shear strains are in L^(co). The global space V that takes into account the 3D parts is the following space: V = {(vl

v\ ri\ 5?) G V(Qs)

X V. X

V(Qi)

which satisfies (Co) and rp -a^, =0}.

C^) + fi/(w?, v^j)

^(vl

where

(3)

Note that, in the specific case of the clamped elliptic shell, the coupling conditions (Co) can be understood as holding in L^(a)). These conditions, however, can also be

106

D. Chapelle, A. Ferent/First MIT Conference on Computational Fluid and Solid Mechanics

20

40 60 number of elements

100

Fig. 2. The convergence of the transverse displacements as / ^^ 0.

used in the more general case where V, is a distribution space. In this general framework, we can show the following result [4]. Theorem 3.1 Under Assumption 2, if F e V, the solution (w^, u\6\ u)) of (Vt) converges strongly with respect to the norm • v, ^^-^ ^ goes to 0, to (M^, w^, 5^, i/^) the solution of the following problem: (p^-^D^

: Find (M^, U\ e\ w?) e V such that

EoD{u\

e\ v\ f)

inated asymptotic behaviour. In this respect, we point out that we used a locking-free finite element procedure for beam analysis [1]. We compared these solutions with the solution of the asymptotic problem (characterized in Theorem 2.1), as shown in Fig. 2. We thus observe how neglecting the thickness in the kinematical constraints on interfaces can introduce significant errors with respect to the limit model unless the thickness is very small.

References -h Bsiu'^s^ u?) + 5;(i?5, u«)

(4) 4. Numerical results Finite element simulations were performed in the case of a beam coupled with 2D plane stress linear elasticity. We obtained the solutions for several values of t, using the asymptotic assumption t^E,

L3DE3

Note, indeed, that a beam necessarily has a bending-dom-

[1] Bathe, KJ. Finite Element Procedures. Englewood Cliffs, NJ: Prentice Hall, 1996. [2] Bemadou, M. Finite Element Methods for Thin Shell Problems. John Wiley, New York, 1996. [3] Chapelle, D, Bathe, KJ. Fundamental considerations for the finite element analysis of shell structures. Comput Struct 1998;66(l):19-36. [4] Chapelle, D, Ferent, A., in preparation. [5] Ciarlet, PG. Introduction to Linear Shell Theory, Series in Applied Mathematics. Gauthier-Villars and North-Holland, 1998. [6] Lions, JL. Perturbations Singulieres dans les Problemes aux Limites et en Controle Optimal. Springer, 1973.

107

Some experiments with the MITC9 element for Naghdi's shell model D. Chapelle^ D.L. Oliveira*''*, M.L. Bucalem'' ^ INRIA-Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France ^ Laboratorio de Mecdnica Computacional, Departamento de Engennharia de Estruturas e Fundagdes, Escola Politecnica da Universidade de Sao Paulo, 05508-900 Sao Paulo, SP, Brazil

Abstract A nine-node mixed-interpolated shell element based on Naghdi's theory is presented and analyzed in the Hght of some fundamental considerations for the finite element analysis of shells. The element is based on the Mixed Interpolation of Tensorial Components (MITC) approach, but the assumed covariant strain fields are applied only for the membrane and shear components. The proposed element is used in the analysis of judiciously selected test problems to evaluate to what extent its behavior satisfies the ideal requirements for general shell analysis. Keywords: Locking; Shell element; Mixed interpolation; Mixed interpolation of tensorial components elements; Naghdi's model

1. Introduction A topic that continues to challenge researchers is the development of locking-free shell finite element. Much progress, however, has been made and there is a family of quadrilateral elements (MITC) [1-3] that has shown a good behavior both in membrane- and bending-dominated shell problems [5]. The MITC elements have been constructed from their displacement-based counterparts which are formulated using the degenerated solid approach. This approach is widely accepted as being the most attractive for engineering appUcations, as discussed in [2,3]. However, the way these elements are constructed — the shell behavior is introduced together with the degeneration process (see [4]) — does not provide the best setting for a mathematical analysis with respect to locking. One could mention that the interpolation of the geometry is inherent to the degenerated solid approach. Also, the bending energy is combined with the membrane energy. The objective of this paper is to summarize the formulation of a nine-node mixed element constructed for Naghdi's shell theory and report upon some numerical ex* Corresponding author. Tel. -f-55 (11) 3818-5246; Fax: +55 (11) 3818-5181; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

periments performed with this element, which is referred to as MITC9-N element, since it is formulated based on the same strain interpolation assumptions used for the MITC9 of the degenerated solid approach. We follow the numerical evaluation strategy suggested in [6].

2. The linear model of Naghdi The basic assumptions of the Naghdi model are that the material line normal to the midsurface in the original configuration remains straight and also unstretched during deformation and that the stresses in the direction normal to the midsurface are zero. We also assume that the material of the shell is elastic, homogeneous and isotropic. Using the Naghdi shell theory, the structural problem may be formulated in the form: Find Ut eU such that:

PA(Ut, V) + tD(Ut, V) = F(V)

yy

eU,

(1)

where } n BC (2) is the space of admissible displacements, v is the displacement vector of the shell midsurface, r) lists rota-

U=[V

= {v,r]), ve [H\Q.)f

, r]_ e [H\Q)f

108

D. Chapelle et al. /First MIT Conference on Computational Fluid and Solid Mechanics

tions of the sections (originally normal to the midsurface) and EC symbolically represents the essential boundary conditions imposed. The bilinear forms A(-, •) and D(-, •) = /)'"(•,) + /)'(•,•) are, respectively, the bending and membrane/shear strain energies. We refer to the work of Bemadou [7] for the detailed expressions of A(-, •), Z)(., •) and F(.).

• Full cylinder with free ends (a bending-dominated case); • Full cylinder with clamped ends (a membrane-dominated case). These problems were also analyzed in detail in [9]. We consider a cylindrical shell of uniform thickness t, length 2L and radius R loaded by an axially-constant pressure distribution p{(p) acting on the outer surface of the shell,

3. The finite element formulation

p((p) = Pocos(2(p)

A conforming displacement-based approximation is obtained if we consider a space of admissible discrete displacements U^' c U. Let us denote by f/J' the finite element solution, with h denoting a representative mesh size. The displacement-based finite element problem corresponding to Eq. (1) is: Find r/J' G U^' such that: t^AiU^l V^) + rD(^f, V'') = F{V^)

V y'' € UK

(3)

As proved in [6], the solution using the above approach is effective only when considering shell problems in which pure bending is inhibited. An effective approach to formulate reliable and quite efficient low-order shell elements is the use of mixed interpolation on strains and displacements. This mixed formulation can be written as the following discrete variational problem: Find U^l e W' such that:

where (p denotes the circumferential angle. We take L = R = I. Furthermore, the material is assumed to be homogeneous and isotropic with Young's modulus E = 200x 10^ and Poisson's ratio v = 1/3. At any point on the cylinder's midsurface, the formulation presented lead to the selection of the axial displacement Ml, circumferential displacement W2, radial displacement Ml and the rotations of the normal ^i about the tangent vector and ^2 about the longitudinal axis as the displacement variables. For the non-inhibited case, we also impose the essential boundary conditions MI = U2 = UT, = P\ = PI = 0 on the clamped ends. Finally, the scalings applied to the loading were chosen accordingly, i.e. (6) for the non-inhibited case and Po = t Po

(4)

where D* is obtained by considering mixed-interpolated membrane and shear strain fields. The actual form of the membrane and shear strain fields and the details of D* can be found in [8].

4. Numerical experiments Since a mathematical analysis is at present out of reach, we must resort to judiciously selected numerical tests in order to assess the convergence behavior of shell elements. The aim of such a selection of problems is to determine whether a finite element discretization is equally well applicable to both categories of shell behaviors (membrane- and bending-dominated) and whether its convergence properties are independent of the shell thickness. In other words, these requirements mean that locking must not occur (in bendingdominated cases) and consistency must not be lost in all terms (in particular, when a membrane-case is analyzed). With due regard to these considerations, a suitable numerical evaluation strategy is presented in [6], where some well-posed test problems are also given. We use here two of such problems for the convergence studies:

(5)

(7)

for the inhibited case, where po = 2 x 10^ is a constant independent of t. We consider here uniform NxN meshes, where A^^ is the number of subdivisions per side in the angular direction of the discretized domain, with element sides aligned with the principal directions of curvature. A mesh grading scheme must be considered to appropriately capture the effect of stress gradients in the boundary layer region. Based on the results presented in [9], we consider a boundary layer width of 2^/t in which A^ layers of elements are also placed. We use in ours tests A^ = 4, 8, 16 and 32. Since locking corresponds to a deterioration of convergence behavior as the shell thickness decreases, it is crucial to compare the results of the same discretization for different values of t. Hence, the sequence of meshes is repeated for each problem considered for values of dimensionless thickness parameter t/R ranging from 1/10 to 1/1000. We take, as the reference solution, a finite element solution obtained with a very refined mesh, since the solution of the mathematical model is not available. The sequence of proposed meshes is solved and we use E^ as the error measure, where r^ def

Er =

fl(t/Q-^f,t/°-t/f) a{Ul f/?)

(8)

109

D. Chapelle et al. /First MIT Conference on Computational Fluid and Solid Mechanics NON-INHIBITED CASE: FREE CYLINDRICAL SHELL - Q2 Element

INHIBITED CASE : CLAMPED CYLINDRICAL SHELL - t = 1/100

t / R = 1 / 1 0 -9— t / R = 1/100 -+--• t / R = 1/1000 -E3-ITHK)

Fig. 2. Convergence for the free cyhnder problem, Q2 element.

Fig. 1. Convergence for the clamped cylinder problem.

NON-INHIBITED CASE: FREE CYLINDRICAL SHELL - Ml TC9-N Element

and

t/R =1/10 -«— t/R =1/100 - + - - J t/R =1/1000 - Q - • h'Hk) •

" • • • • • • , .

\.

0.1

U^ is the reference solution and «(•, •) is a case-dependent symmetric bilinear form defined below. The error is then measured on an energy norm V \ = a(V,V). In the bending-dominated case, to render the error indicator Er independent of the thickness t, we define

\ 0.01

-

^^^ ^^:^

"^-f^^^©^

:

\

'V..^^^'"""---...

0.001

a(- •)'^d' A(-,.) + Z)(-,-)

(9)

where J is a characteristic geometric dimension of the problem other than the shell thickness t. For membrane-dominated problems, we recall that D(-, •) is a norm [6]. Therefore we use

^(•, •) = /)(•, ).

""•••.„

Fig. 3. Convergence for the free cyhnder problem, MITC9-N element.

(10)

The reference solutions U^ were calculated using the finest mesh (with A^ = 32), using the Q2 displacementbased element for membrane-dominated cases or using the MITC9-N element for bending-dominated cases. In Figs. 1-3 we plot the Er values vs. the number of elements per side N in the logarithmic scale. The aim is to estimate the magnitude of the constant c and the order of convergence k defined in \Er\=ch'

"••Q

(11)

and how these constants behave as we change the dimensionless thickness parameter t/R. Ideally, an element should have both constants c and k independent of the shell thickness t regardless of the nature of the problem (i.e. membrane- or bending-dominated). In addition, k should approach its optimal value 2, considering that the loading is sufficiently smooth and the meshes were designed to reflect the exact solution [9]. We translate these requirements to the convergence methodology assessment considered here, i.e.:

(1) Considering non-inhibited cases, we must have no shift of the error curves as the thickness t changes; (2) The curves obtained for the MITC9-N element must be close to those of the Q2 element when inhibited cases are analyzed; (3) All curves must approach the direction parallel to the dashed line, corresponding to ^ = 2. Hence, any deviations from what is prescribed above will imply: • locking, if (1) is violated; • a lack of consistency of the mixed element solution, if (2) is not observed or • that a reasonable convergence behavior is not attained, if (3) is not verified. Let us consider first the inhibited case. Fig. 1 shows the results obtained for both the Q2 and the MITC9-N elements when t/R = 1/100. As expected, the little shift of MITC9-N error curve (when compared to the corresponding Q2 curve) provides an evidence of the better performance of the displacement-based element in this

no

D. Chapelle et al. /First MIT Conference on Computational Fluid and Solid Mechanics

case, due to the consistency errors deriving from the use of the modified biUnear form /)*(-,). We recall that a better convergence behavior of the MITC9-N element than that for the Q2 element should not be expected in this case. We note, however, that both elements present an excellent convergence behavior: the convergence is very close to the asymptotic rate ~ /z^. A similar behavior is obtained for the other values of t/R and of course the superiority of the displacement-based element (i.e. the distance between the error curves) becomes more and more clear as the ratio t/R decreases. We chose t/R = 100, in particular, because this represents a rather realistic value in practical applications and neither the influence of other energy terms than membrane (as for r//? = 1/10) nor of round-off errors (as a result of a very fine mesh refinement for the boundary layer region when t/R = 1/1000) may significantly affect the results obtained. Let us consider now the non-inhibited case. Figs. 2 and 3 show the results obtained for the Q2 and the MITC9-N elements, respectively, for each one of the three values considered for the relation t/R. Again, there is no surprise in the displacement-based element behavior. In this case, as expected, its performance is strongly affected by locking effects, as it becomes clear from the shift of the curves as t decreases. We may also observe the deterioration of the order of convergence h accordingly. On the other hand, the MITC9-N element shows an excellent performance: the error indicator E^ is essentially of the same order regardless of the case being solved (although we may observe a slight deviation in the convergence behavior for t/R = 1/10). Even though these results are very encouraging, we must not expect the same performance for the MITC9-N element when either distorted or not graded meshes are used [8].

5. Concluding remarks A nine-node mixed-interpolated finite shell element based on Naghdi's theory was formulated using the MITC approach.

The evaluation of this element has shown that we may expect an efficient and sufficiendy reliable performance in shell analysis. The numerical tests suggest that a relatively locking-free behavior in the analysis of bending-dominated problems is achieved when the remarks concerning mesh properties are observed. In addition, consistency errors are kept in a reasonable magnitude for realistic small values of the ratio t/R. Finally, since the proposed element represents a connection between the easy-to-use general shell elements and a consistent 2-D shell theory, we consider that a valuable step was taken towards providing a mathematically oriented guidance to obtain reliable and improved finite shell elements for general use.

References [1] Dvorkin EN, Bathe KJ. A confinuum mechanics based four-node shell element for general nonlinear analysis. Eng Comput 1984;1:77-88. [2] Bucalem ML, Bathe KJ. Higher-order MITC general shell elements. Int J Numer Methods Eng 1993;36:3729-3754. [3] Bathe KJ. Finite Element Procedures, 2nd edn., Englewood Cliffs, NJ: Prenfice Hall, 1996. [4] Chapelle D, Bathe KJ. The mathemafical shell model underlying general shell elements. Int J Num Methods Eng 2000;48(2):289-313. [5] Bathe KJ, losilevich A, Chapelle D. An evaluation of the MITC shell elements. Comput Struct 2000;75:1-30. [6] Chapelle D, Bathe KJ. Fundamental considerations for the finite element analysis of shell structures. Comput Struct 1998;66:19-36. [7] Bemadou M. Finite Element Methods for Thin Shell Problems. New York: John Wiley and Sons, 1996. [8] Chapelle D, Oliveira DL, Bucalem ML. On the reliability of MITC elements based on Naghdi's model. To appear. [9] Malinen M, Pitkaranta J. A Benchmark Study of ReducedStrain Shell Finite Elements: Quadratic Schemes. To appear.

Ill

Frictional contact analysis of articular surfaces X. Chen*,T. Hisada The University of Tokyo, School of Frontier Sciences, Tokyo 113-8656, Japan

Abstract Finite sliding between articular surfaces occurs during the motion of loaded diarthrodial joint. In this work, an attempt is made for frictional contact analysis of articular surfaces by introducing convected coordinates and redefining the sliding term as a spatial vector in the reference configuration to deal with finite sliding. Keywords: Finite element method; Frictional contact problem; Finite sliding; Articular surface

1. Introduction Force transmission by contact between articular surfaces plays an important role in mechanically initiated osteoarthritis. Ateshian and Wang [1] indicated that the interstitial fluid pressurized articular cartilage supports most of the load and thus significantly reduces the friction coefficient of the articular surface. However, the protection due to interstitial fluid pressurization may become less effective in degenerative cartilage. Taking into account the friction effect is considered to be necessary for revealing the factors causing and advancing osteoarthritis. Diarthrodial joints generally undergo considerable motion and finite sliding between contact surfaces during cyclical loading. To preserve the objectivity of the friction law, Laursen and Simo [2] developed a finite element method based on the convected coordinate system for analyzing frictional contact problems. In this method, a difficulty arises in dealing with finite sliding that occurs over the element boundary where local coordinates are discontinuous. Chen et al. [3] proposed a procedure to overcome this difficulty by redefining the sliding term as a spatial vector in the reference configuration. In this work, we attempt to apply the procedure of Chen et al. [3] to the frictional contact in articular surfaces.

* Corresponding author. Tel/Fax: -F81 (3) 5841-6321; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

2. Formulation for frictional contact problems with finite sliding In the formulation of Laursen and Simo [2], based on the convected coordinates, the friction force is derived as Par = £

(ti

•''«'-^sn.

f

(1)

where ptt denotes the covariant component of the friction force vector p ^ and tt and t' indicate covariant and contravariant base vectors at the contact point with convected coordinate §', respectively, s is the penalty parameter for the stick state. The relative velocity of contact points is given by (tt • tj)^H\ It is noted that the left-hand side of Eq. (1) and the first term on the right-hand side are in the form of the Oldroyd rate of a vector. Additionally, the second term on the right-hand side includes a scalar and a product of the friction force vector with its norm. Thus, it is clear that the friction equation satisfies the objectivity requirement. Using the backward Euler integration for incremental analysis, the incremental form of Eq. (1) becomes Ptk{i+\)t (^i^i-^ =

Ptk(i)t(i+i)

+ s (tk -r^Oo'+DC^o'+i) - t ( o ) • AX

Pt_k{im+i)

t{i+\)\\

1 ^k

(2)

where subscripts (/) and (/ + 1) indicate the incremental steps. Considering Eq. (2), the increment of the convected coordinate appears as the result of incremental decomposition. In the finite element method, the contact surface is divided into elements and the local coordinate of every element is used practically as a convected coordinate. Thus, the increment of a convected coordinate cannot be com-

112

X. Chen, T. Hisada /First MIT Conference on Computational Fluid and Solid Mechanics

Current configuration

Reference configuration Fig. 1. Motion of contact surface.

puted if the contact point slides over the boundary of the element. To overcome this difficulty, the following approach is proposed. Defining incremental relative displacement as Ar^(^,.r,),+o(?^.^j^-?^^,)rf,^,^

(3)

and referring to the discretized contact surfaces shown in Fig. 1, Ar can be approximated using (4)

^^-
where x[^^^^ is the position vector of the contact point at increment / + 1, whereas x\-^ is the position vector, at increment / + 1 , of the material point where the contact point has been located at increment /. However, because of movement due to deformation, x\-^ is unknown at increment / + 1. Because this movement affects the tangent stiffness due to friction, it is not convenient to use Eq. (4) directly. Now a mapping of the contact point to the reference configuration, as shown in Fig. 1, is considered. Let ^y(/+i) be the covariant base vector at the contact point x\-^^^ in the current configuration (increment / + 1), and °f JC+D and °ff.^,j the covariant and contravariant base vectors, respectively, at the mapped point X[.^,j in the reference configuration. Use of the dyadic expression of the deformation gradient based on ^yo+i) and ^t\.^^^ gives Ar - (^,(,+1) . r.o-+i))(^4+i^ • ^'r)t{^^y,.

(5)

where

A°r^(t,^^,,-§^.,)%,,^„

(6)

The mapped contact point can be obtained from its element local coordinate in the current configuration and the node coordinate before deformation.

3. Numerical example Although the mechanical property of articular cartilage exhibits viscoelasticity and the friction coefficient between articular surfaces depends generally on load, time and relative velocity, a numerical example is carried out focusing on the treatment of frictional contact problems with finite sliding. The distribution of contact stress is computed for a two-dimensional simplified finite element model shown in Fig. 2. A hexahedral type of element with eight nodes is 66mm

E = 5GPa, v = 0.3

50mm

t.

is the relative displacement increment mapped to the reference configuration and can be calculated as AV

^ X'

- x\^

30mm

(7)

Finally, the use of Eqs. (3) and (5) with Eq. (2) results Pr/:(/+l)^(, + l) - /?rA:(/)^(/+i)

+ e (^^•^;)(/+i)('4+i)-A'r)-AX (8)

Fig. 2. Finite element model.

X. Chen, T. Hisada/First MIT Conference on Computational Fluid and Solid Mechanics

113

30

'^

OH

S

25 20

Vi

I 10 u

ju= 0.05

5

0

Coordinate X (mm) Fig. 4. Distributions of contact stress.

• U = 20 mm

\ F = 400N Fig. 3. Deformation of two contact bodies. used and the degree of freedom in the direction leading out of the plane is constrained to simulate plane strain condition. Three-dimensional analyses for real diarthrodial joints are currently under way. Fig. 3 shows the deformation of two bodies. By defining the relative movement of the contact points as a spatial vector in the reference configuration, the large amount of sliding of the contact node over the element boundary is successfully simulated. The distributions of contact stress with different values of friction coefficient fi are shown in Fig. 4. High contact stress occurs near the center of the contact area and is clearly affected by the friction coefficient.

4. Conclusions A finite element approach that enables the analysis of frictional contact problems with finite sliding was introduced in an attempt to perform frictional contact analysis

of articular surfaces. Finite shding of the contact node over the boundary of the element is treated by redefining the relative movement in the reference configuration. The applicability of the proposed procedure was investigated by using a simplified two-dimensional model. In further three-dimensional analysis, to reveal the relevance between the friction phenomenon and the cause of mechanically initiated osteoarthritis, it is necessary to consider viscoelastic behavior of the articular cartilage and the dependence of the friction coefficient on the load, time and relative velocity.

References [1] Ateshian GA, Wang H. A theoretical solution for the frictionless rolling contact of cylindrical biphasic articular cartilage layers. J Biomech 1995;28(11): 1341-1355. [2] Laursen TA, Simo JC. A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. Int J Numer Methods Eng 1993;36:3451-3485. [3] Chen X, Nakamura K, Mori M, Hisada T. Finite element analysis for large deformation frictional contact problems with finite shding. JSME Int J Ser. A 1999;42(2):201-208.

114

Finite element modeling of human head-neck complex for crashworthiness simulation H.Y. Choi^'*, S.H. Lee^ I.H. Lee^ E. Haug^ ^ Hong-Ik University, Mechanical Engineering Department, Seoul, South Korea ^ Hankook ESI, Seoul, South Korea ^ ESI Software, Rungis Cedex, France

Abstract A finite element human head-neck model is under development for the car occupant safety simulation. The model is constructed based on the precise anatomical geometry and currently under validation process. In this paper, structural and physiologic explanations of the human head-neck complex will be introduced as well as the modeling methodology. Some of the simulation results are also chosen to present major features of the model. Keywords: Human head-neck; Finite element model

1. Introduction A finite element model of the human head-neck complex has been developed in order to study the basic injury mechanisms due to the dynamics loading such as a car crash. The human head-neck complex is well exposed to the abrupt translational and rotational movement compared to the rest of the body parts during a crash accident. These kinds of movements often cause serious injuries of the head-neck complex even without direct contact with foreign objects. Relative movement of brain inside the skull, mainly caused by its inertia, could cause vascular injury on the connecting vessels and also may induce a negative pressure in subarachnoidal space, which result in axonal injuries. Soft tissues such as ligaments and muscles in the human cervical spine are easily injured when their connected bony parts undergo excessive motions in relation to each other. The causes of whiplash injury, for example, are known to be closely related to the damages of soft tissues located between the adjacent vertebrae that experience extraordinary movements in relation to each other. The understanding of the basic injury mechanisms of the human head-neck complex, however, is quite limited and many studies have been carried out, both experimen* Corresponding author. Tel.: +82 (2) 320-1699; Fax: +82 (2) 326-0368; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

tally [1-6] and analytically [7-12]. Recently, numerical simulations, especially using the finite element method, have been utilized to investigate the hypothetical theories based on experiments and clinical findings. Quite many numbers of finite element human head and neck models have been developed for this purpose and the evolution of computational models has been remarkable thanks to the advance of computational powers and the FE codes. The finite element model of the human head-neck complex to be introduced in this paper is one of the sub-modules for the H-Model family [12]. The H-Model consists of articulated rigid skins with flexible joints (HARB), and detailed local models for important internal body components (e.g. head, neck, thorax, pelvis, ankle, etc.). These local modules can be selectively added to the HARB model when needed. In practice, the H-Model is an assembly of the HARB model and the local modules that are selected depending upon the purpose of the investigation of the moment. The one with the distinguishing feature of the headneck model in this study would be the precise modeling of the fluid-solid interactions. The structural role of cerebrospinal fluid (CSF) occupying the subarachnoidal space in the brain is a cushioning and buffering role between the skull and the brain as well as for transmitting forces. The Mumaghan equation of state for a solid element has been employed to model the CSF layers in the head and dura sec of the cervical spinal foramen. The incompressible behavior of CSF in the head, which has a closed volume,

H.Y. Choi et al /First MIT Conference on Computational Fluid and Solid Mechanics also induces a 'cavitation' when the brain has sufficient relative motions inside the skull. The ideal gas equation is, thus, apphed in order to simulate this cavitational phenomenon. These new attempts in the head model produce more realistic results than the previous head models do. One-dimensional Hill type bar elements recently became available in Pam-Crash^^ [13] and were used to simulate neck muscle forces. Active muscle forces according to the various activation times and level, restrict neck motions and therefore have an important function in the injuries. Multiply segmented twenty-three neck muscles are included in the model.

2. Injury of the human head-neck complex due to dynamic loading Major head injuries are skull fractures and brain damages. Most of the skull fractures result from direct impact of a foreign object on the head; brain damage, on the other hand, is caused by secondary impact within the cranial space and/or relative motions between skull and brain. Brain injury is often classified into diffuse and focal injuries according to their causes and symptoms. Fig. 1 shows typical MR and CT images of brain injuries. Sublux-

115

ation with interconnecting ligamental rupture and vertebral burst fractures are typical injury patterns in the neck due to dynamic loading on the human neck (Fig. 2).

3. FE modeling of the human head-neck complex Data from Visible human projects and View point datalab^^ are used to construct the geometry of the finite element model (Fig. 3). Fig. 4 shows the FE model of the head-neck complex. Material properties assigned to bony components (Table 1), brain matter and soft tissues in the model are acquired from the Uterature [5-12].

4. Case studies: selected simulation results 4.1. Frontal pendulum impact on the head The experimental study using the cadavers performed by Nahum et al. [1] was used to vaUdate the head model for the case of Hnear acceleration loading. Since the neck was excluded in this simulation, a free-boundary condition was applied to the head-neck joint. This constraint is justified by the findings of Willinger et al. [9] and Ruan et al. [10]

Fig. 1. MR and CT images of (a) hypointense lesion inside corpus callosum (hemorrhagic type of DAI) and (b) hypointense left frontal lesion (non hemorrhagic type of DAI), (c) frontal extradural haematoma, (d) acute subdural haematoma shift, (e) parietal contusion with midline shift.

.^" !

1^

Fig. 2. Cervical vertebrae subluxation and burst fracture.

116

H. Y. Choi et al /First MIT Conference on Computational Fluid and Solid Mechanics

^

I \

--*

/t

^ni

Fig. 3. Process of building thefiniteelement model of the human head. about 45° to the Frankfort plane in the mid-sagittal plane (Fig. 5a). The mass and initial velocity of the pendulum were 6 kg and 5.9 m/s, respectively. Model responses were compared with the measured cadaver test data in terms of impact force, head acceleration (CG), and epidural pressures. We could observe the relative motion between the brain and the dura, posterior cavity, and flow of the CSF layer. High positive peak pressures appeared beneath the impact site in the frontal region and the greatest negative pressures were generated at the posterior fossa which, due to the inclination of the skull, was the area opposite the impact site. The coup-contrecoup pressures were considerably asymmetric. The head model predicted a maximum pressure of 250 kPa in the frontal region and a minimum pressure of —40 kPa in the contrecoup region. The overall trends of pressure histories from the calculation and test correlate quite well considering the possible geometric discrepancies between model and specific cadaver specimen. Fig. 5b shows the movement of head components and the pressure contour of the brain surface. The pressure gradients changed smoothly from the frontal to the posterior regions and a higher negative pressure, which representing

Fig. 4. Finite element head-neck model (left: skeleton with neck muscles, right: quarter sectional view of head). who showed that the neck does not influence the kinematic head response during the pulse duration. The impact was delivered by a pendulum along the axis inclined at Table 1 Material properties of the H-model Component Skull

Facial bone Mandible Dura mater Venous sinus CSF Falx Pia Tentorium Brain

E Outer table Inner table Diploe

B

G

7.3 X 10^ 7.3 X 10^ 2.02 X 10^

1.39 X 10^

7.3 X 10^ 7.3 X 10^ 3.15 X 10-*

V

P

0.22 0.22 0.22 0.22 0.22 0.45

3000 3000 1410 2700 2700 1133 1000 1000 1133 1133 1133 1040 1040 1000 1040

1 X 10^ 1 X 10^

7.96 X 10^ 1.27 X 10^

0.45 0.45 0.45 0.499 0.499

1.27 X 10^

0.499

3.15 X IC* 3.15 X 10* 3.15 X 10-* Gray matter White matter

Ventricle Cerebellum and brain stem

7.96 1.27 1X 1.27

X 10^ X 10^ 10^ X 10^

E = Young's modulus (kPa); B == Bulk modulus (kPa); v -- Poisson's ratio; p = mass density (kg/m^).

H.Y. Choi et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Impact pendulum

111

t = 0,0 ms

t = 2.5 ms

Frankfort plane (a)

t = 5.0 ms

Stress_press(«*e

B- -^^ • 1 _ _

-6.2fe-00S 3,75e-00S

« Z _ ^ ^ „ M

0:000125 0.000189 a.WfZi^ 0.000258

•IF ^'"^^

t = 7.5 ms

mn -0.34703 in SOLID ^201837 STftTE 3.50004 «asc 0.0203417 in SOLID 21200067 STftTE 3.50004

(b) Fig. 5. (a) Impact condition, and (b) coup and contrecoup pressure distribution on the brain due to the frontal pendulum impact. a probability of cavitation, occurred at the occipital and posterior areas. 4.2. Low-velocity rear impact on the head-neck complex Fig. 6 shows the extension of the head-neck model due to the rear impact. Horizontal linear acceleration with maximum 5 g for 100 ms was applied at the thoracic level, which is a typical condition for the real accident. Responses of head and each cervical vertebra were verified with cadaver and live human volunteer test results and showed a good correlation.

5. Conclusion The objective of the finite element human model including the head-neck model presented in this paper for car occupant safety simulation is to understand the basic injury mechanism and quantitatively assess the injury levels due to the dynamic loading. By applying the precise anatomic structures and material properties of each body component, constructed finite element model(s) could simulate the deformational behavior of the human body similar to the real event. In order to utilize these FE models and simulation results to predict

118

H. Y. Choi et al. /First MIT Conference on Computational Fluid and Solid Mechanics

.-^ . j

~^^^^H - VV

1 STATE1:G

:

STATE1:0

n«M!^:lMw»^ 11K«N$yi18|ll J

i^<M^ii^^.^MM«ii^ J ^ ^

--

i

STATEe:60i)(M)7

1

•'^ 1

STATE6:e0j0007

mmm^^^^^tmet wmmmm

wmm-^^e^^^t ^^^mumm ^ « I I M I N ; ^ l i f p ^

fW^%timmi^

dw^mm^m

IHJim^y^ll^ JIM)4<mC^|MRg

1

1

STATE 18:180.001

1

i

STATE 18:180.001

Fig. 6. Extension of head-neck due to the low-speed rear impact.

H.Y. Choi et al. /First MIT Conference on Computational Fluid and Solid Mechanics the detailed injury levels, a further validation procedure is needed.

References [1] Nahum M, Smith R, Ward CC. Intracranial pressure dynamics during head impact. In: Proceedings of the 21st Stapp Car Crash Conference, 1977, pp. 339-366. [2] Bandak FA. Biomechanics of impact traumatic brain injury. In: Proceedings of the NATO-ASI on Crashworthiness of Transportation Systems, 1996, pp. 213-253. [3] Donnelly R, Medige J. Shear properties of human brain tissue. J Biomech Eng 1997;119:423-432. [4] Koshiro Ono et al. Relationship between Localized Spine Deformation and Cervical Vertebral Motions for Low Speed Rear Impacts Using Human Volunteers. IRCOBI Conference, Spain, 1999, pp. 149-164. [5] Van der Horst MJ et al. The Influence of Muscle Activity on Head-Neck Response During Impact. SAE 973346. [6] Szabo TJ, Walcher JB. Human Subject Kinematics and Electromyographic Activity During Low Speed Rear Impacts. SAE 962432.

119

[7] Bandak FA, Eppinger RH. A three-dimensional finite element analysis of the human brain under combined rotational and translational accelerations. National Highway Traffic Safety Administration, 1994. [8] Claessens M, Sauren F, Wismans J. Modeling of the human head under impact conditions: a parametric study. In: Proceedings of the 41st Stapp Car Conference, 1997, pp. 315-328. [9] Willinger R, Kang HS, Diaw B. Three-dimensional human finite-element model validation against two experimental impacts. Ann Biomed Eng 1999;27:403-410. [10] Ruan JS, Khahl TB, King AL Finite element modeUng of direct head impact. In: Proceedings of the 37th Stapp Car Crash Conference, 1993, pp. 69-81. [11] Voo L, Kumaresan S, Pintar FA, Yoganandan N, Sances A Jr. Finite-element models of the human head. Med Biol Eng Comput 1996, pp. 375-381. [12] Hyung-Yun Choi, In-Hyeok Lee, Eberhard Haug, Advanced Finite Element Modeling of the Human Body for Occupant Safety; H-Model for the next Millennium. Proceedings of 5th HanPam, 1999. [13] Pam-Crash, Pam-Safe, Theory Notes Manuals, version 2000, ESI Software.

120

A constitutive model associated with permanent softening under multiple bend-unbending cycles in sheet metal forming and springback analysis B.K. Chun *, J.T. Jinn \ J.K. Lee Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210, USA

Abstract It is essential to model the Bauschinger effect correctly under cyclic bending for simulation of the sheet metal forming process and springback prediction. However, most cyclic plasticity models are designed so that the reverse flow stress always converges to the initial loading curve. These models cannot represent permanent softening which occurs in certain materials, such as high strength steel and aluminum alloys. Anisotropic nonlinear kinematic hardening rule is proposed to represent the Bauschinger effect including permanent softening under multiple bending-unbending cycles, which allows the bounding yield surface to grow at different rates for loading and reverse loading. Comparisons with the affordable tests, tension-compression test and drawbead test, show that this model can predict cyclic bending behavior of sheet metal more accurately. Keywords: Bauschinger effect; Permanent softening; Cyclic bending; Nonlinear kinematic hardening; Sheet metal forming; Springback

1. Introduction Cyclic behavior of metal sheet plays a very important role in the sheet metal forming processes. A material point in a blank may experience 1-3 cycles (tension-compression-tension) during the forming processes, which will influence springback. For example, bending-unbending on the die shoulder and rebending-unbending at the punch can be expected during a typical deep drawing process. Therefore the material model in the simulation of sheet metal forming should represent the proper behavior under multiple bending-unbending cycles. A reduction of yield stress due to reversal staining is known as the Bauschinger effect. It has been also observed that the Bauschinger curves (or reversal stress-strain curve) asymptotically approach or run parallel to the initial loading curve. For some high strength steels and aluminum alloys, offset of reversal flow * Corresponding author. Tel/Fax: +1 (614) 292-3566; E-mail: [email protected] ^ Present address: Scientific Forming Technology Company, 5038 Reed Rd., Columbus, OH 43220, USA. © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

curves have been reported in various references (see, for example, [1,2,17]). It is generally believed that three basic requirements for the material model to incorporate the Bauschinger effects in sheet metal forming are: • correct non-linearity of stress-strain loop; • reduced elastic limit at reversal staining; • permanent softening for some materials. Most material models for cyclic plasticity, including the Chaboche model [3-5,8], are designed so that the reverse flow stress always converges to the monotonic tensile stress curve. Therefore these models cannot represent permanent softening. The same issue has been investigated by Geng and Wagoner [12] and Wagoner et al. [17] with different approaches. An anisotropic nonlinear kinematic hardening model (ANK model) has been proposed to represent the Bauschinger effect including permanent softening [9,10]. By allowing the bounding surface evolve differently during the reversal straining in the nonlinear kinematic hardening rule, permanent softening can be expressed consistently over multiple bending cycles. In this way, the nonlinear evolution rule for the total back stress can be represented consistently during the whole deformation as in

B.K. Chun et al /First MIT Conference on Computational Fluid and Solid Mechanics the Chaboche model. This feature always produces correct non-linearity of the stress-strain loop at reversal straining.

2. Material models for sheet metal forming with multiple bending 2.1. Nonlinear kinematic hardening model A material model for cyclic plasticity in simulation of sheet metal forming has been developed by Chaboche and colleagues [3-8,15]. This model has been recently implemented into most commercial finite element packages, such as ABAQUS, Pam-Stamp and LS-DYNA 3D. The back stress vector is assumed to be a sum of Nk vector components,

a = J2^,

(1)

Each component of back stress is assumed to evolve independently as, di = —(a - a)s

- Yiats

(2)

where Ct and yt represent material parameters that can be obtained from a cyclic test. The y, term determines the rate at which the saturation value of kinematic hardening decreases with increasing plastic deformation. When Q and Yi are zero, the model reduces to an isotropic hardening model. The expansion of yield surface size is governed as below: ••Go +

(3a)

R

Nk

R=

KNc-'^''-s'-

ai = CiS

Ri = bi(Qi ~

Ri)^'

By superposing several backstress vectors which evolve individually. From the observation of experimental results [1,2,18], it is assumed that the amount of permanent softening can be expressed as the following; (6)

Aa = f{sPrs^ )

where s^ is a current equivalent plastic strain and s^* is a prestrain at initial loading. Eq. (6) can be extracted from the evolution equation by introducing the anisotropic kinematic hardening term; (7)

a = Oil + oi2

where oii and a 2 evolve differently during initial loading and subsequent reversal loading; •ays'

G^

yx{ax)s^

— (a - a)s Oil =

\

G""

3. Numerical examples

R=

(5a)

=Go-^R

K(l-Q-

^)-E^^

(9) for reversal loading

(3c)

If tension data is introduced, then Eq. (3c) can be modified, including the evolution of kinematic hardening as follows:

(5b)

(8)

for initial loading

G^

where Ci, y\ and C2 are material parameters.

(4)

(5c)

2.2. Anisotropic nonlinear kinematic hardening model

(3b)

where Go represents the initial yield stress; Qi, bt, k, n are material parameters. This evolution rule can be modified to utilize the uniaxial tensile data directly through a simple curve fit with the following equation: G=Go-^K{l-Q-'''')

• YiOiiS

where K and N represent material parameters obtained by curve fitting the tensile data and overhead bar is used to signify scalar quantities for the uniaxial tension data. Hence the parameters to be determined are Ct and yt after the tensile data fit. The combined case, Eqs. (1), (2) and (5), has different evolution of yield surface size compared with Chaboche model, Eqs. (1), (2), and (3). Thus this modification is referred to as Modified Chaboche model.

0

R=J:R^

121

3.1. Simulation of tension-compression test To evaluate the effects of material models on the stress calculation of a tension-compression test with AL6022-T4 and EGDQ, one element is tested. Strain history is input and the corresponding stress and backstress are output. The lack of compression information is a common problem with flat metal sheets because compressive loads are difficult to apply in the sheet plane [19]. Therefore, the identification of the proper material model is another issue on the application in FEA of sheet metal forming processes. Details can be seen in references [9,10,19-22]. The material parameters associated with the material models are shown in Table 1.

122

B.K. Chun et al /First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Associated material parameters Material

C (Mpa)

Material models

Ci

AL6022-T4

Modified Chaboche ANK Tensile data by OSU Modified Chaboche ANK Tensile data by OSU

EGDQ

R90

(MPa)

E (GPa)

R values C2

3500 3000

150 N/A 10600

8000

200 N/A

74.2 200 N/A 300 310 N/A

C^o

RO

R45

0.73

0.44

0.63

171

68

0.73

0.44

0.63

171

68

1.29

175

180

1.29

175

180

reduced yield stress at reversal loading, but the reversal flow curves always converged to initial loading. While ANK model showed the permanent softening as well as the reduced reversal yield stress. Both Modified Chaboche and ANK models always produce a smooth transient stressstrain relation at the beginning of reverse loading. 3.2. Drawbead test

0

0.04

0.08

0.12

0.16

Equivalent plastic strain

Fig. 1. Comparison of Bauschinger curves for AL6022-T4. 400

1 O

!

Measured compression data [1 ]

1

1

;

ANK i

tension curve ; ^

2 S

200

K

\ 1

^ H

(D

2

i

1

1 l

;

1

^ 0

0.04

—1

0.08

1

1

0.12

0.16

Equivalent plastic strain

Fig. 2. Comparison of Bauschinger curves for EGDQ. The generated stresses of Modified Chaboche and ANK models are obtained and compared with the measured tension-compress data by Balakrishnan [1] in Figs. 1 and 2. Considering the three features of Bauschinger effects as described before, Modified Chaboche model showed the

A drawbead simulator has been popularly used for evaluating the drawing forces during sheet metal forming process. Recently Jiang et al. [14] used this test for the predictability of springback. The same test is employed to observe the effects of the material models on springback prediction in this study. Two material models, Modified Chaboche and ANK models are implemented into ABAQUS/Standard through UMAT and compared with conventional isotropic hardening model. Normal anisotropic yield criterion is used for all materials models of EGDQ. The blank was drawn to a maximum of 165 mm in the rolling direction with 1.6 mm fixed gap condition. The specimen thickness is 0.8 mm. Total analysis is composed of four steps; move die for bead formation with displacement control, drawing the blank with fixed gap, pseudo step for remove dies, and springback. For more accurate calculation of internal stress, 25 integration points through thickness are used for a four-node shell element (S4R). Along the length direction, 100 elements are used for smooth change of contact history. One element is used for half of width with plane strain assumption. Friction coefficient between the specimen and dies are assigned as 0.138. Die shapes and dimension are shown in Fig. 3. The final deformed shapes after springback are highly dependent on the material models as shown in Fig. 4. Two reference lines separated by a distance of 102 mm are used for the calculation of radius as described in [14]. These two lines are marked as points in Fig. 4. The corresponding clamping force and drawing force are compared with measured ones by Jiang [14] in Table 2. Isotropic hardening model requires higher forces both clamping and drawing with fixed 1.6-mm gap, which makes a larger curvature. Even the Modified Chaboche model cannot predict

123

B.K. Chun et al /First MIT Conference on Computational Fluid and Solid Mechanics Table 2 Comparison of modified Chaboche model and ANK model for drawbead test: EGDQ

Radius ^ (mm) Clamping force (kN) Drawing force (kN)

Isotropic hardening

Modified Chaboche model

ANK model

Measured [14]

70 5.048 5.736

150 3.602 4.256

250 2.114 2.604

300 2.294 2.480

^ Approximate radius between two reference points in Fig. 4. C l a m p i n g Force

t

99 A

D r a w i n g Force

^^

1

Jf4 1

Die T Gap 1

.1

16-2

R=65

^R=6.5 Specimen

\

Unitimm

Fig. 3. Die shape and dimension for drawbead forming and springback.

models, Modified Chaboche model and ANK model, are proposed. Based on comparisons with tension-compression tests, the fundamental multiple bending behaviors are compared. It is shown that the ANK model can present permanent softening correctly, while the Modified Chaboche model does not. Therefore, the ANK model is very effective in calculating cyclic bending behavior. Finally, springback prediction of the proposed model is evaluated by using a drawbead test. From comparison, the accurate considering of the Bauschinger effects through the material model is essential for better springback prediction in the sheet metal forming processes.

Acknowledgements Isotropic hardening

d Chaboche

The authors would like to express sincere gratitude to the following: SPP committee for financial support; Ohio supercomputer Center for providing generous computing services; Professor R.H. Wagoner and Mr. Balakrishnan for providing the experimental data of the tension-compression test; Ms. Jiang for drawbead test results; Mr. Allen for providing tensile data and proofreading the manuscript.

References Fig. 4. Comparison of deformed shapes after springback: EGDQ. the measured curvature properly. The generated forces of these models can be overestimated due to no permanent softening during multiple bending actions. Only the ANK model can give us closer values compared to measured values. However, error between ANK model predictions and measured ones still exist. Assumptions for plane strain condition through width direction and normal anisotropic yield function may be possible sources of error.

4. Concluding remarks The effect of material models on springback prediction is discussed in this paper. To incorporate the Bauschinger effects of metal sheet under multiple bending, two material

[1] Balakrishnan V. Measurement of in-plane Bauschinger Effect in metal sheets, Master thesis, The Ohio State University, 1999. [2] Bate PS, Wilson DV. Analysis of the Bauschinger effect. ActaMetall 1986;34(6): 1097-1105. [3] Chaboche JL. Viscoplastic Constitutive Equations for the Description of Cyclic and Anisotropic Behavior of Metals. Bull Acad Polonaaise Sci Sevie Sc Techn 1977;25(1):33. [4] Chaboche JL, Dang-Van K, Cordier G. Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. SMIRT-5, Division L., Berlin, 1979. [5] Chaboche JL, RousseUer G. On the Plastic and Viscoplastic Constitutive Equations Based on the internal variables concept, SMIRT-6 Post Conf, Paris, TP ONERA no. 8-11, 1981. [6] Chaboche JL, RousseUer G. On the plastic and viscoplastic constitutive equations. J. Pressure Vessel Technol 1983;105:153-164. [7] Chaboche JL. Time independent constitutive theories for cycUc plasticity. Int J Plast 1986;2(2):149.

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B.K. Chun et al. /First MIT Conference on Computational Fluid and Solid Mechanics

[8] Chaboche JL. Constitutive equations for cyclic plasticty and cyclic viscoplasticity. Int J Plast 1989;5:247-302. [9] Chun BK, Jinn JT, Lee JK. Modeling the Bauschinger Effect for sheet metals, part I: Theory. Int J Plast, to appear. [10] Chun BK, Kim HY, Lee JK. Modeling the Bauschinger Effect for sheet metals, part H: Applications. Int J Plast, to appear. [11] Crisfield MA. Non-Linear Finite Element Analysis of Solids and Structures, Vol. 2. John Wiley and Sons, 1997. [12] Geng L, Wagoner RH. Springback analysis with a modified nonlinear hardening model, SAE2000-01-0410, 2000. [13] Geng L. Application of plastic anisotropy and non-isotropic hardening to springback prediction, Ph.D dissertation, The Ohio State University, 2000. [14] Jiang S, Garnett M, Liu S-D. Springback of sheet metal subjected to multiple bending-unbending cycles, SAE 2000-01-1112,2000. [15] Lemaitre J, Chaboche J-L. Mechanics of Solid Materials, Cambridge University Press, 1990, pp. 161-241.

[16] Takahashi H, Shiono I. Backlash model for large deformation behavior of aluminum under torsional cyclic loading. Int J Plastic 1991;7:199-217. [17] Wagoner RH, Geng L, Balakrishnan V. Role of hardening law in springback. Proceedings of Plasticity, 2000. [18] Wilson DV, Bate PS. Analysis of the Bauschinger effect, Acta Metall 1983;34(6): 1097-1105. [19] Zhao K. Cyclic stress-strain curve and springback simulation, Ph.D dissertation, The Ohio State University, 1999. [20] Zhao K, Lee JK. On simulation of bending/reverse bending of sheet metals. ASME, MED-Vol. 10, Manufacturing Science and Engineering, 1999, pp. 929-933. [21] Zhao K, Lee JK. Inverse estimation of material properties for sheet metals. Commun Num Methods Eng, in press. [22] Zhao K, Lee JK. Material properties for accurate simulation of springback, ASME Trans J Eng Mat Technol, submitted.

125

Non-linear wave propagation in softening media through use of the scaled boundary finite element method Roger S. Crouch*, Jens Fernandez-Vega Department of Civil and Structural Engineering, University of Sheffield, Sheffield SI 3JD, UK

Abstract This paper reports on the use of a novel Finite Element-based sub-structure method to model the dynamic far-field in Mode I localisation analyses. The requirement for accurate yet efficient representations of an elastic domain extending to infinity in wave propagation studies is discussed. In particular, the need to cope with arbitrarily oriented stress waves arriving at an interaction horizon is recognised. The work presented here forms part of a larger study into rate-dependent regularisation techniques which are designed to recover objectivity in fracture simulations using an equivalent continuum (smeared crack) approach. After briefly describing the Scaled Boundary Finite Element Method, the paper shows how this attractive scheme may be incorporated into a non-linear implicit dynamic FE code. The use of an element-by-element, non-symmetric, iterative solver is discussed and an example given of strain localisation using an advanced, generalised elasto-plasticity constitutive model for concrete. Keywords: Strain-softening; Wave propagation; Dynamic far-field; Element-by-element iterative solver; Generalised elasto-plasticity model for concrete

1. Introduction Considerable interest currently exists in identifying robust, efficient equivalent continuum methods of simulating fracture in concrete structures. To-date most localisation investigations (which are designed to explore the sensitivity of the solutions to the FE mesh density and mesh orientation) have been based on the use of simplified constitutive models. While findings from these preliminary studies are valuable, there is a strong need to undertake further analyses using more advanced constitutive models which are able to account for the brittle-ductile transition under increasing confinement. One difficulty in modelling dynamic fracture propagation (for example, when simulating the split-Hopkinson bar experiments) is that of extending the mesh sufficiently far away from the region of interest to prevent stress wave reflections corrupting the results. The additional degrees of freedom lead to high CPU-times. Use of local transmitting boundary methods can introduce errors when the wave strikes the boundary non-orthogonally. * Corresponding author. Tel.: -F-44 (114) 222-5716; Fax: -^44 (114) 222-5700; E-mail: [email protected] © 2001 PubHshed by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Recently, a highly innovative global (that is, spatially and temporally coupled) technique has emerged which mimics the response of unbounded domains in a rigorous manner. This paper briefly describes the method in Section 2, presenting the FE dynamic equilibrium equation. Section 3 identifies the form of the hardening-softening elasto-plasticity model adopted and subsequently reports on a Mode I localisation study. After presenting these FE results, some comments are given on the value of monitoring the evolution in the determinant of the acoustic tensor. Such a measure expresses the state of the material during softening and may be used to drive automatic re-meshing strategies.

2. The scaled boundary finite element method The Scaled Boundary Finite Element (SBFE) Method is constructed from an assumption of geometric similarity in the unbounded medium [2]. A scaling centre is identified and the unit response impulse matrices obtained by forming a relationship between two nested regions. The method converges to the solution in the Finite Element sense in the

R.S. Crouch, J. Fernandez-Vega/First MIT Conference on Computational Fluid and Solid Mechanics

126

tangential direction and is exact in the radial direction (satisfying the radiation condition at infinity). The technique (similar to the Boundary Integral Method) reduces the number of dimensions by one on the structure-unbounded domain interface but operates with fully-populated symmetric sub-structure matrices [ModThe dynamic forces at the structure-unbounded medium interface (representing the presence of the far-field) are discretised as

{V*} = -

1 i^(AO-

:[M]

PAt

+ ^[M]+^[>M.] + (1 - 2p) [M] 2p

'M^

'Ad'

I'd)

Af ( l - ^ ) [ ^ M . ] ^Ar(y-2^) 0 V'd] 2yS

'^11

(3)

convolution

-El ?+i,Moc]{!k,^}

f t-At

^11

(1)

where [M^] represents a piecewise constant acceleration unit impulse matrix with units the same as those of a damping matrix. Full details of how these are constructed is given by [2]. The velocity and acceleration approximations are taken as identical to those used in the Newmark algorithm (for example, see [6]). Incremental non-linear equilibrium is expressed as [K*]l'+'''8d''^'} = {8f*] ^K'

[K'] =

(2)

where the superscript k refers to the (Newton-Raphson) iteration number, [K] and [M] are the familiar system matrices [7] and {/im} and {/ext} are the internal and external forces, respectively. P and y represent the Newmark parameters and [d] identifies the nodal displacements (overdots refer to time derivatives thereof). The second equation in (2) and equation (3) may be assembled in an element-by-element approach allowing the first equation in (2) to be solved efficiently using an iterative scheme. In the work reported here, a GMRES stabilised, diagonally pre-conditioned, bi-conjugate gradient algorithm is used [8]. This solver routine can treat non-symmetric systems (which arise through a lack of normality in the plastic flow rule).

3. Dynamic localisation analysis

s^'

.c^^^

(a)

scaled boundary f i n i t e elements

Fig. 1. (a) 16-element mesh with 8 SBFE interface elements, (b) Equivalent extended mesh.

A prism comprising sixteen 20-noded isoparametric elements is used to represent a 0.05 x 0.05 x 0.005 m concrete specimen under dynamic tensile loading. Four SBFE elements are attached to the top face of the structure and four to the bottom. Two scaling centres are used in this novel analysis (each placed 10^ m away from the SBFE-FE interfaces; one below and one above). A ramped tensile load (in the form of a uniformly distributed pressure) is applied at the upper SBFE-FE interface (Fig. la). Fig. lb shows a portion of an equivalent extended mesh analysis. Note that the use of simple transmitting boundaries (local in space and time [1]) rather than the SBFE approach would have resulted in run-time savings, but errors would be introduced as the stress waves do not strike the interface normally, once localisation initiates. An advanced, generalised elasto-plasticity constitutive model is used to represent the concrete. This formulation includes non-linear, pressure-dependent hardening and fracture-energy-controlled softening [3]. Considerable care has been taken to provide a robust, accurate stress return algorithm in this model [4]. Extensions to include a form of Duvaut-Lions viscosity are reported elsewhere [5]. Here an inviscid simulation is given. The multiaxial hardening and softening surface is de-

R.S. Crouch, J. Fernandez-Vega /First MIT Conference on Computational Fluid and Solid Mechanics stagel

stagell

stagelll

H>j

4'-"

of the minimum determinant of the acoustic tensor versus the time-step number (for point A) shows a steady drop followed by a slight recovery in the final stages of the analysis despite continued softening. The acoustic tensor is calculated (using a hierarchical search algorithm) at each sampling point, at the end of each time-step, using

determinantof acoustictensor

Qjk=niD\.j^ini

0

50

100 150 200 2S0 300

axialstress(MPa)strain

Fig. 2. Acoustic tensor determinant maps (top), stress-strain response at point A (right, front) minimum determinant of acoustic tensor at point A (right, rear) cohesion contour (left, rear) strain profile (left, front). scribed by

i"-

+

mky -

+ ——^-cky

V3

111

-2

=0 (4)

where c represents a measure of cohesion (which degrades under increasing fracture strain) and k represents a measure of material hardening, p and ^ are the Haig-Westergaard deviatoric and hydrostatic stress invariants, respectively, and r provides a Lode angle dependent function. Fig. 2 shows two contour plots of the structure. The rear mesh illustrates the degree of softening achieved at the end of the analysis (the blue zone at the top indicates almost a complete loss of cohesion, whereas the red region at the base suggests almost no degradation). The mesh to the front of Fig. 2 gives the corresponding strain profile at the end of the run. Once localisation occurs (in a single row of Gauss points at the top of the structure) and axial stretching continues, the lower portion unloads elastically, as indicated by the blue zone. A representative axial stressstrain plot from the analysis (at the point A, identified by the white circle) is given in Fig. 2. A tensile strength of approximately 3.5 MPa is realised, whereafter softening occurs. The three circles on this stress-strain diagram refer to three stages in the analysis. Note that a softer element at the lower left-hand comer of the specimen was introduced to provide non-symmetry in the problem; thereby creating non-normal stress waves striking the SBFE interfaces. The contoured spheres at the top of Fig. 2 illustrate the directional variation of the determinant of the acoustic tensor at the three different stages in the analysis (I, pre-peak; II, just post-peak; and III, at the end of the run). The plot

(5)

where n is the search direction. The determinants resulting from the non-linear constitutive tangent tensors (Z)-y^/) have been normalised with respect to the linear elastic constitutive tensor (£>fy^/). Note that the spheres have been plotted in a skewed orientation. The black lines show the true axial directions. Dark red regions indicate that the non-linear acoustic tensor has changed little from the corresponding elastic tensor, whereas blue zones show where the determinant has become negative. The latter leads to a loss of well-possedness in the problem, creating inobjective results as mesh densities change (not shown here). The introduction of an effective material length into the constitutive formulation can recover objectivity. Note that the onset of localisation has been used by Pearce [9] as a monitoring device to trigger re-alignment of the element boundaries and refinement of the local mesh density. Preliminary use of the SBFE has illustrated its potential to produce useful dynamic strain softening simulations, avoiding spurious wave reflections.

References [1] Lysmer J, Kuhlemeyer RL. Finite dynamic model for infinite media. J Eng Mech ASCE 1969;95:859-877. [2] Wolf JP, Song C. Finite-Element Modelling of Unbounded Media. New York: John Wiley and Sons, 1996. [3] Tahar B. C2 Continuous Hardening/Softening Elasto-Plasticity Model for Concrete. PhD Thesis, Department of Civil and Structural Engineering, University of Sheffield, UK, 2000. [4] Tahar B, Crouch RS. Techniques to ensure convergence of the closest point projection method in pressure dependent elasto-plasticity models. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [5] Mesmar S. On the Use of Duvaut-Lions Viscosity as a Regularisation Technique in Softening Media. PhD Thesis, Department of Civil and Structural Engineering, University of Sheffield, UK, 2000. [6] Bathe K-J. Finite Element Procedures. Englewood Cliffs, NJ: Prentice Hall, 1996. [7] Zienkiewicz OC, Taylor RL. The Finite Element Method, 5th Edn. New York: McGraw-Hill, 2000. [8] Smith IM. General Purpose Parallel Finite Element Programming. 7th Annual Conference of the Association for Computational Mechanics in Engineering, Durham, UK, 1999, pp. 21-24. [9] Pearce CJ. Computational Plasticity in Concrete Failure Mechanics. PhD Thesis, Department of Civil Engineering, University of Wales, Swansea, UK, 1996.

128

Time integration for dynamic contact problems: generalized-of scheme A. Czekanski, S.A. Meguid* Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, ON, M5S 3G8, Canada

Abstract In this paper, we employ the generalized-a time integration scheme for treating elastodynamic contact problems. The criteria invoked for the selection of the time integration parameters are motivated by our desire to ensure that the solution is unconditionally stable, second order accurate, provides optimal high frequency dissipation and preserves energy and momentum transfer in dynamic rigid impact problems. The selected parameters help in avoiding the spurious high frequency modes, which are present in the traditional Newmark method. New closed-form expressions for the time integration parameters are determined in terms of a user-specified high frequency spectral radius. The dynamic contact problem is formulated in terms of the variational inequalities approach and solved using quadratic programming. In order to demonstrate the versatility and accuracy of the proposed time integration scheme, two numerical examples are examined. The results show a significant improvement compared to existing solution techniques. Keywords: Contact; Impact; Time integration; Generalized-a scheme; Modified Newmark

1. Introduction Dynamic finite element analyses usually employ time integration methods. The implicit schemes are unconditionally stable and involve larger time steps. Although the Newmark method is the most popular implicit scheme, its commonly used values {y = ip = 0.5) result in excessive numerical oscillations and is therefore unsuitable for contact problems. For such problems, the use of }/ = ^ = 0.5 is recommended [1]. These parameters result in second order accuracy and satisfy energy and momentum conservation during rigid impact. However, they also result in spurious high frequency modes for small time steps. Introduction of numerical dissipation in the Newmark scheme reduces these oscillations, but leads to a loss of second order accuracy. In this paper, the variational inequalities expressions representing the contact problems are solved through a sequence of mathematical programming problems. The generalized-Qf scheme is used for time marching. The four * Corresponding author. Tel.: -hi (416) 978-5741; Fax: -Hi (416) 978-7753; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

time integration parameters are selected to satisfy unconditional stability, second order accuracy, provide optimal high frequency dissipation and preserve energy and momentum transfer in dynamic rigid impact problems. The resulting values avoid numerical oscillations often present in impact and dynamic frictional problems.

2. Time integration scheme 2.7. Introduction The time integration scheme establishes a relationship between the acceleration, velocity and displacement fields at time t and r 4- Ar as follows: '+^'U = 'U + [(1 - yyt

-f y^+^^tJ]A^

'+^'U = ^U + ^UAr + [(0.5 -P)'t-^p

(la) ^+^^U]

(lb)

The use of the trapezoidal rule (y = ip = 0.25) with a fully implicit treatment of the contact constraints produces oscillations, which can be significant as the time steps and spatial discretizations are refined. Recently, the

129

A. Czekanski, SA. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics generalized-Qf method was developed for solving structural dynamics problems with second order accuracy even if numerical dissipation is presented [2]. In this method, the equation of motion is modified by introducing two new parameters ag and an'. jYI (f+Ao-as^ + C (^+^^)-«wu + K (^+^o-a//u _ a+Ao-an jr

(6)), and energy-momentum conservation criterion (Eq. (7)): otH = Poo/(poo - 1)

(8a)

an = (-2al + of^ - 1 + ^ 2 ^ 1 - 3 ^ 5 + 2^ / ( I - a^)

(2)

for - 0.5
where (r+AO-a//u = (1 _ an) '+^'U + au 'U a+AO-anu ={\-

an) ^+^'U + an 't

a+Ao-a^U = (1 - ae) '+^'U + aB 'U

(3)

(r+AO-«//p ^ (1 _ ^ ^ ) .+A.p ^ ^ ^ fp

2.2. Criteria for selecting time integration parameters

(4)

-as

In order to maintain unconditional stability, the following inequalities must be satisfied: - 1 < c^5 < «// < 0.5 and ^ > 0.25 + 0.5(0?^ - as) (5) It is also desirable to filter the high frequency components of the response. This condition is satisfied when [2]: yS = 0.25(1-^5+Of/,)'

(6)

For frictionless contact problems another criterion can be derived based on conservation of momentum and energy during rigid impact. This criterion is formulated based on the generic problem of two dissimilar stiffness-free masses in contact. The time integration scheme should ensure that the rebound velocities of the two point masses satisfy energy and momentum conservation. Furthermore, the contact should last for only one time step. The analytical solution is satisfied when ^ is given by [3]: P = 0.25 {-2al + a^O + 2^^) - 2)/{aB - 1)

(7)

subject to: P<0.5(aB-\)(l

+

(8c)

K = 0.5 + Of// - aB

(8d)

where poo is a user-specified high frequency spectral radius. Note that Eq. (5b) is not used.

3. Finite element implementation

A second order accurate solution is obtained for the generalized-a method when [2]: / = 0 . 5 +Of//

P = 0.25 {-2al -f 0^5(3 + 2aH) - 2) /{aB - 1)

2y)/{aB-2)

For elastic problems, a small amount of energy is lost during impact. This amount depends on the selected time integration parameters as well as the mesh size. 2.3. Optimal time integration parameters for contact problems The values of the time integration parameters can now be selected based on the following criteria: second order accuracy (Eq. (4)), optimal high frequency damping (Eq.

For each time increment, the solution algorithm can be summarized as follows: (i) evaluate the equivalent stiffness matrix and load vector using the generalized-of time integration scheme, and (ii) solve the current time-instant iteratively to obtain the displacement, velocity and acceleration fields as well as the current contact surface and contact forces. In conjunction with the generaUzed-of method, the reduced variational inequalities formulation is equivalent to solving the following minimization problem [4]: minfi^+^^AU^'+^'^K^+^^AU'+i - ^+A^AU''+IT^+A.^P+I ?+ A?ATTi + l + ^+^^AU''+^ TS'+^'Fy.} V'+^^AU

(9)

subject to: (10a) and §jr+Ar^U/+i < ST('+^'U^' - ^U)

(10b)

where (11a)

K = {\ ' aH)K + {\ - aB) —^M

^'Ar+^ = (1 - a„) '+""¥ +ttH' F - asM'U - auK'U + (l-Qf5)M *- i n t

;6Af2

fiM

\2fi

)i

(lib)

The first constraint (Eq. (10a)) represents the assembly of the kinematic contact conditions of the nodes on the candidate contact surface Fc- Eq. (10b) represents the assembly of the non-differentiable frictional constraint for the nodes on the candidate contact surface. In this expression, the matrix T extracts the discretized tangential displacement components from the global displacement vector. The

130

A. Czekanski, SA. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics 2.0-

sign matrix S, which allows to switch between two complementary frictional sub-problems, is unknown a priori, and is part of solution. 3.1. Spring-mass

o.i:

1^.

system

4.1. Impact of two identical bars

'rMiiiiiiiiiiiiinnii 10

1

1

[nn

10.02 4<

h

0.5

2.5x10'

Fig. 3. FE model of spring-mass system investigated.

this example, we consider the impact of two identical These bars were modeled using four-noded elements 1). Both bars were given initial opposing unit velocFig. 2 shows the time history of the contact force

I

t

Beam:E,= 10\ v, = 0, p, = 0.01 Rigid block: E,= 10', v,= 0, p, = 2.5

4. Numerical examples

In bars. (Fig. ities.

-H^O.4-*

' Y Y Y ^IB^IIIII 77TTyT?!T7T7TT77T!?

10

•

E=1000, v = 0 , p =0.001 Fig. 1. Impact of two identical bars.

for two time integration schemes and two time increments. The results show that the time increment strongly affects the contact force when using the classical Newmark approach. This scheme also fails to represent contact for the smaller time step. Superior results (displacement, velocity, acceleration and contact forces) are obtained using the generalized-a scheme with the newly selected parameters. In this example, we examine the spring-mass system. The spring and mass are modeled using 4-noded elements (Fig. 3). In order to model a nearly rigid mass and a flexible weightless spring, the material properties satisfied the 10

1 At=5xlO' At=8xlO'

—

c o o

o

E2

^

(b)

4 V 2 ..

A

^

^

0.01

0.04

0.04

0.03

0.02 Time

Fig. 2. Total contact force for colliding bars for two selected time increments using: (a) Newmark scheme {y = ^ = 0.5), and (b) generalized-a {y =2^ = an = OCB = 0.5). 0.5

0.5

(a)

imiiyii

llllli

E2

E2 -0.5

-0.5 1

2 Time,s

(b)

'J

^ 1

2 Time,s

Fig. 4. Total friction force for spring-mass system using: (a) Newmark scheme (y = ^ = 0.5), and (b) generalized-o? (y = 2^ = an = as = 0.5).

A. Czekanski, SA. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics following conditions: pi < p2 and Ei ^ E2. The mass was subjected to a constant prescribed vertical displacement 8. A horizontal step load F was then applied. The time histories of the total friction force shown in Fig. 4(a) and Fig. 4(b) were obtained using Newmark and generalized-Qf schemes. The newly proposed method experienced smaller numerical oscillations in contact forces compared to the classical Newmark scheme, when the system experiences transition from slip to stick.

5. Conclusions In this work, a generalized-^ scheme, with optimal contact parameters, was employed for the time integration of the dynamic frictional contact problem. The proposed technique leads to a significant reduction in numerical oscil-

131

lations in impact and dynamic frictional contact problems and is less sensitive to variations in the time increment.

References [1] Chaudhary AB, Bathe KJ. A solution method for static and dynamic analysis of three-dimensional contact problems with friction. Comput Struct 1986;24:855-873. [2] Chung J, Hulbert GM. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-a method. J Appl Mech 1993;60:371-375. [3] Czekanski A, El-Abbasi N, Meguid SA. Optimal time integration parameters for elastodynamic contact problems. Commun Numer Methods Eng 2000; submitted. [4] Czekanski A, Meguid SA. Solution of dynamic frictional contact problems using nondifferentiable optimization. Int J Mech Sci 2000; submitted.

132

Semi-analytical solution to a mechanical system with friction L. Dai * Industrial Systems Engineering, University of Regina, Regina, Sask. S4S 0A2, Canada

Abstract This investigation is devoted to the development of a novel semi-analytical solution for a nonlinear dynamical system involving frictional interaction. A piecewise-constant procedure is employed in developing the solutions which are continuous everywhere in terms of displacement and velocity. Keywords: Friction interaction; Piecewise-constant argument; Nonlinear dynamics; Oscillation; Approximate analysis; Brush-spring system

1. Introduction

expressible in the following form.

Friction is the primary source of oscillations in many mechanical systems. The friction and nonlinear damping forces produce nonlinearity in mechanical systems. Uncertainties are always presented in modeling the motions involving contacts and interfaces, and the interactions of friction and the overall system lead to the possibility of unstable and complex dynamic behavior. In this paper, the motion of a highly nonlinear dynamical system with frictional interactions is investigated. A semi-analytical solution of a nonlinear system is produced by a piecewiseconstant technique reported by Dai and Singh [1,2]. The solution developed is in a closed form and continuous everywhere. The numerical results based on the semi-analytical solution provide convergence with sufficient accuracy.

x//" -f IQ^xJ/' + ^ V = ^ [ ( K + M) sin lA - (1 - fjiy) cos ir] (1) where F = F-\-sin xj/ — y-\-y cos x// -\-2^xl/' cos x// — 2^ yxj/'sin xl;-\-xjf''cos xfr — xl^' sini//yxlf" sin xj; — yxjr' cos xjr

(2)

The corresponding motion is governed by the following equation if -^ < 0, xj; ^IQ^xj/'

+ Q^xl; = F[{y -\- /i) sin x// + (I -\- fiy) cos xj/] (3)

where F = F — sinxlf — y -\- y cos x// —l^x//' cos xj/

2. Governing equation and the corresponding semi-analytical solution Swayze and Akay recently investigated the behavior of a brush-spring system from a window lift electric motor [3]. A steady friction force excites the system. The oscillatory motion of the brush of the system is governed by the following equations of motion for the two conditions of positive and negative values of the angular displacement of the brush x//. For positive x//, the governing equation is *Tel.: +1 (306) 585-4498; Fax: +1 (306) 585-4855; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

— It; yxjf'sin xj/— xf;" cos-^-\-xjr' sinxjr — yxl/" sin xjr — yx//' sini/r

(4)

In the above equations, the prime ( )' represents the derivative with respect to a nondimensional time r; F, y, ^, IJL and ^ are parameters related to physical and geometrical measurements of the system. To approximately or numerically solve the governing equations (1) and (3), the piecewise-constant technique presented by Dai and Singh [1,2] can be employed. By the piecewise-constant technique, a nonlinear dynamical system is converted into a linear oscillatory system on an arbitrary time interval, [Nz]/N < r < ([A^r] ± 1)/A^, to obtain a continuous

L. Dai/First MIT Conference on Computational Fluid and Solid Mechanics solution. A^ is a parameter controlling accuracy of the solution. When N is large enough, the corresponding solution can be sufficiently accurate. As A^ approaches infinity, theoretically, the approximate solution produced by piecewise-constant technique becomes the accurate solution. For solving the nonlinear differential equations (1) and (3) by the piecewise-constant technique, the terms on the right-hand-side of the equal sign in equations (1) and (3) are considered as constants in a tiny time interval, [A^r]/A^ < T < ([A^r] ± \)/N, such that the nonlinear differential equations are converted to linear ordinary differential equations in the following form. f'; + 2Qi^lf[ + Q^^lft=Ai

(5)

For the case xjr > 0, At in Eq. (6) is considered as a constant Ai = Fi[(y + /x) sindi — (1 — /xy) cos J/]

(6)

and the iih interval is random, for r > 0, general solution of the problem can thus be obtained in the following form on the entire time range considered. NT]\

COS ( §r — ^-

N J . A JA^r]\ 1 . / [iVr]\ s m ( ^ ? r - ? — j J- ssin m(^^r? r - ? — j ^i

-Q^(r-[Nr]/N)

cos \^x — i

N

where and

^ < 1.

The matrix G in equation (7) is expressible as [Nr] Q ^

^-^^([Nr]/N)^Nr] 7=1

M/N

cos

sm — ^2

and the square matrix

R =

cos A^ •

+

^

sm —

Kf-)' 1

A^

1 . ? - sm — ? A^

'OK"

(9) cos — A^ ^ N Eq. (7) is an approximate solution to Eq. (1) in a closed form. The recurrence relations can be directly derived from Eq. (7) for numerically solving for governing equation (1). It can be seen from Eq. (7), the approximate solution is continuous everywhere in the time range r > 0. In numerically calculating for the motion governed by equations (1) and (3), the solutions developed through the recurrence relations provide results with sufficient accuracy in comparing with the fourth-order Range-Kutta method, except that the solution produced by Range-Kutta method is discrete. As can be seen from the discussion above, the motions of the nonlinear dynamical system involving frictional interaction are complex. With the help of the piecewiseconstant technique, continuous closed-form approximate solution for the nonlinear dynamical system is derived allowing further theoretical analysis, and a numerical simulation for the motion of the system can be conveniently carried out on the basis of the solutions. A^

References

(7)

^ = y/Q^ - ( ^ O ^

133

(8)

[1] Dai L, Singh MC. An analytical and numerical method for solving linear and nonlinear vibration problems. Int J Solids Struct 1997;34:2709-2713. [2] Dai L, Singh MC, On oscillatory motion of spring-mass systems subjected to piecewise constant forces. J Sound Vibrat 1994;173(2):217-232. [3] Swayze JL, Akay A. Effects of system dynamics on friction-induced oscillations. J Sound Vibrat 1994;173:599-609.

134

A novel displacement variational boundary formulation G. Davi*, A. Milazzo Department of Mechanics and Aeronautics, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy

Abstract This paper deals with a novel displacement variational formulation for elasticity. The mathematical model is obtained from the stationarity condition of a modified hybrid functional expressed in terms of displacements and tractions. The domain displacement field is approximated by suitable trial functions, whereas the boundary variables are expressed by using their nodal values. The final system is expressed in terms of nodal displacements only and it is symmetric and positive definite. Moreover, the domain integrals can be directly transformed into boundary ones to recover the boundary nature of the method. Keywords: Variational approaches; Quadrature methods; Numerical methods; Boundary methods

1. Introduction Fundamental properties of self-adjoint problems, as the symmetry and definiteness of discrete operators, play a crucial role from both theoretical and numerical point of view. The FEM possesses the above-mentioned requisites of the energy based domain discretization methods. On the other hand, the conventional BEM destroys the continuum properties, but leads to accurate results with some computational advantages compared to field methods. In this paper, a novel displacement variational formulation is derived basing on a hybrid variational formulation of BEM [1-4]. With such a formulation, the mathematical model involves nodal displacements only and its matrix operators preserve the symmetry and definiteness properties of the continuum. Additionally, these operators are computed running boundary integrations of regular kernels only with the consequent computational advantages.

ment and traction vectors. The functions u, u and t are assumed as independent variables. According to references [4-6], let us introduce the following modified functional d^

- f {u-uYidr- fu^lc r

(1)

r2

where e is the strain vector, p is the mass density, ii is the acceleration, f are the domain forces and t are the prescribed tractions on the free boundary r2. Assuming that the compatibility and constitutive equations and the kinematical boundary conditions are satisfied, the solution of the elasticity problem is given in terms of the functions u, u and t which make U stationary [3-5].

3. Discrete model 2. Modified variational principle The formulation proposed in this paper is based on a modified variational principle previously presented by the authors [2-5]. Let u be the vector of displacements in the domain Q and let again u and t be the boundary displace-

Let us consider the boundary of the body F discretized by boundary elements and some additional nodes within the domain Q, other than those introduced by the boundary discretization [5]. The domain displacement field is approximated by means of a superposition of trial functions u*

* Corresponding author. Tel.: +39 (91) 665-7110; Fax: +39 (91) 485439; E-mail: [email protected]

u = J2^*s

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

:U*S

(2)

G. Davi, A. Milazzo /First MIT Conference on Computational Fluid and Solid Mechanics The trial functions are elasticity solutions in an infinite domain, subjected to given body forces. The boundary displacement and traction variables are expressed as

where the stiffness matrix K and the mass matrix M are given by D„U*dr + // «U*^2^U*d^

K u = m = [Ni N2]

^2

(11)

(3) M = p^l f u*Tu^ d ^ ^ 2

t = ^p

(12)

(4)

where 5 and p are the nodal displacements and tractions, N and ^ are matrices of shape functions and the subscripts 1 and 2 refer to constrained and free nodal displacements, respectively. By substituting the expressions of u, u, and t in Eq. (1), the discretized form of the functional n is obtained. The stationarity conditions of n with regard to s, ^2 and p, after some manipulations, yield j j V^^VnWdT + j W^VV"" d^ j s + p /" U*TU*d ^ s

(5)

/ ^ N j ^ d r p - / N ^t d r = : 0

(6)

Notice that the model involves nodal displacements only and the matrices K and M are frequency independent, symmetric and positive definite [2-5]. Therefore, in the proposed approach, these two fundamental properties of the continuum, i.e. symmetry and definiteness of the structural operators, are preserved. Now, the idea is to associate the trial functions with the point load solutions, so that it results V (Pu*) = c*5 (P - Po)

(13)

where c*(5 (P — PQ) denote the Dirac function of amplitude c*, applied at PQ. By so doing, the trial functions are regular, as required by the formulation, and they enable us to transform the domain integrals that appear in the definition of the stiffness and mass matrix into boundary integrals. Indeed, let us consider a set of auxiliary functions W* (/ = 1,2...), which satisfy the following equation

r>w* = w* f ^ ^ u M r s - f ^^^Ndr^ = 0

(7)

where V is the static equilibrium differential operator and Vn is the boundary tractions operator. Eq. (7) is satisfied for every choice of ^ if it results U*s = N8

135

on r

(8)

The relations between the unknown parameters s and the nodal displacements 8 can be established according to [5]. Evaluating Eq. (8) at the boundary nodal points, by virtue of the properties of the shape functions, we directly obtain some relationships between s and the boundary nodal displacements. Further relationships between s and the domain nodal displacements can be established collocating Eq. (2) at the internal nodes. If the number of trial functions is equal to the number of nodal displacements and these functions are regular and linearly independent, one obtains s = U* ^8 = ^8 = [^1 ^2]

(9)

where ^ is the inverse of the collocation matrix U*. Premultiplying Eq. (5) by ^ 2 ' t>y using Eqs. (2) and (9), one obtains the dynamic model which can be written as M8' + K 5 2 - / N ^ t d r -

[N.l^dQ

(10)

(14)

where WQ — U*. The reciprocity theorem for the auxiliary functions and the trial functions provides CV*dQ

= f Upnyv*Y cv* - w f p„/:u*] dr

/

(©W*+i) P£U*dn

(15)

where C is any operator. Applying recursively Eq. (15), the domain integrals of the stiffness and mass matrices are transformed into boundary integrals, because at least we obtain a domain integral involving the Dirac function. In conclusion, the computation of the stiffness and mass matrix requires only boundary integrations of regular kernels and the pure boundary nature of the formulation is recovered. Additionally, the class of trial functions presented, which are associated with particular points, are well suited for computer implementation, since they can be generated using the same nodes as those defined for the model.

4. Numerical application To check the soundness of the proposed method the membrane vibration problem [7] has been solved. For this

G. Davi, A. Milazzo /First MIT Conference on Computational Fluid and Solid Mechanics

136

Table 1 Errors of the dimensionless frequency parameter for a simply supported square membrane Nodes

Error (%) Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

21 32 45 60 77 96

0.3398 0.0247 0.0112 0.0025 0.0023 0.0022

1.8661 0.2249 0.0284 0.0072 0.0056 0.0042

5.1729 0.7734 0.1260 0.0168 0.0083 0.0067

6.2993 3.1949 0.7489 0.1651 0.0443 0.0040

Exact A/ 4.4429 7.0248 8.8858 9.9346

8.6006 4.8248 1.1547 0.3160 0.0953 0.0335

13.0378 1.7752 4.0677 1.2684 0.4393 0.1691

11.3272 12.9531

problem, the static equilibrium operator T> coincides with the Laplacian operator and one has the following trial and auxiliary functions ' In r,

(16)

<-^i-A

Inn - -

(17)

wA'-D

(18)

cOiy/p/Ta^

(19)

where w^- is the kih column of W^ and r, = r, (P, PQ) is the distance between the generic point and the /th source point. By virtue of the operator properties, the resolving model becomes a linear algebraic eigenvalue problem, which can be solved by standard routines. Results are presented for a simply supported square membrane in terms of the dimensionless frequency parameter Xi =

where coi is the /th mode angular frequency, T is the surface tension and a is the membrane dimension. Table 1 lists the error of the dimensionless frequency parameter with respect to the exact value [7]. The results obtained

show the accuracy and the good convergence properties of the method.

5. Conclusion A novel variational formulation for elasticity problems has been presented. The model obtained involves nodal displacements only and it preserves the fundamental properties of symmetry and definiteness of the continuum. The model exhibits the same nature of the more popular finite element models and the standard numerical procedures available for FEM resolving systems can be used in the present approach. Moreover, the present method has significant computational advantages due to the reduction in dimensionality typical of boundary element formulation. The results obtained show that the method is efficient and accurate.

References [1] De Figueiredo TGB, Brebbia CA. A new hybrid displacement variational formulafion of BEM for elastostatics. In: Brebbia CA (Ed), Advances in Boundary Elements. Berlin: Springer, 1989, pp. 47-58. [2] Davi G. A hybrid displacement variational formulation of BEM for elastostatics. Eng Anal Bound Elem 1992; 10(3): 219-224. [3] Davi G, Milazzo A. A symmetric and positive definite variational BEM for 2-D free vibration analysis. Eng Anal Bound Elem 1994;14(4):343-348. [4] Davi G, Milazzo A. A symmetric and positive definite BEM for 2-D forced vibrations. J Sound Vibr 1997;206(4):611617. [5] Davi G, Milazzo A. A new symmetric and positive definite boundary element formulation for lateral vibration of plates. J Sound Vibr 1997;206(4):507-521. [6] Washizu K. Variational Methods in Elasticity and Plasticity. Oxford: Pergamon Press, 1968. [7] Rayleigh JSW. Theory of Sound. New York: Dover PubHcations, 1976.

137

Investigation about nonlinearities in a robot with elastic members Sergio A. David *, Joao M. Rosario

Faculty of Mechanical Engineering, State University of Campinas (UNICAMP), Cidade Universitdria Zeferino Vaz, Campinas, Brazil

Abstract The need for fast and precise robots in the industrial environment, capable of attending the productivity and quality demands and that allow a high volume of work, needs the usage of manipulators with flexible links. Besides this, aeronautic applications demand the usage of long and thin arms, which leads to remarkable structural changes. Therefore, the development of manipulators with structural flexibility and its automatic control has become an important research area [1,4,6,9]. The main goal of this work is to model the dynamic behavior of flexible manipulators. It is also presented a comparative study with rigid robots. It is possible to use the model for computer simulations to aid the development of efficient control. Keywords: Robotics; Dynamic modeUng; Nonlinear dynamics; Industrial robots

1. Introduction Most of the industrial applications that involve a manipulator robot use rigid links. The increase in the rigidity of the links has the main objective of avoiding structural vibration. For this reason the manipulators are designed with over-dimensioned cross-sections in order not to degrade the control accuracy. When flexible link manipulators are compared to rigid link ones, they need less material for their fabrication, are lighter, faster, can handle larger loads, show less power consumption, need smaller drivers and usually are easier to be transported. Because of those reasons, the usage of manipulators with flexible links is directly related to the optimization of the elements that comprise a robotics system.

of the concentration point and also for the orientation of the referential. The second part is the wrist. It is normally constituted of three rotational degrees of freedom and has the function of orienting the terminal referential. Not considering the deformation of the joints, the degree of freedom related to the movement of the base of the robot can be treated as rigid, as well as the three degrees of freedom related to the orientation. Thus, one notes that the flexibility of the system is related to the two degrees of freedom related to the movement of the two links of the manipulator, as shown in Fig. 1.

1.1. Problem description A robotics manipulator is a mechanical device that has the function of positioning and orienting its terminal element. This terminal element has the function of handling tools suitable to the work to be performed. Two main parts are to be considered in the design of a manipulator structure. The first part is the arm that comprises at least three degrees of freedom and is used for the positioning *Tel.: -M9 466-1172, E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Fig. 1. Links 2 and 3 (possible flexibility).

138

SA. David, J.M. Rosdrio /First MIT Conference on Computational Fluid and Solid Mechanics - Rigid S ^ t e m Re^ibie System (subtractedfleMbilltyIn t B term J^

-t6

^

47

s -13 -t9i

-%QO

02

04

06

OB

to

T!me(s)

- R i g d System ReMble System reduced lo a rigid one

-SIstemaR^pdo -SstemaRBduzidoaoR^do^F^rdoReBJvel

-1j6

^ ^1

-V

1

CM

06

I 04

-t8

h «^

-t9

-02

001 -041

-^,QO

•OBI

02

04

06

08

00

to

02

04

06

Tfme(s)

Fig. 2. Simulations.

For a flexible manipulator the structure presents a considerable flexibility and therefore an efficient control system must be developed. It is well known that in general a control problem consists of the manipulator dynamic model formulation [1]. This model is further used to establish the control laws that provide the desired performance. In this work the dynamic modeling is performed for a system that contains two flexible links and two rotational joints. For the sake of comparison, a rigid structure with two links and two rotational joints is also analyzed. In this case two degrees of freedom are defined. A convenient parameterization of the terms of the motion equations, which makes it easier to compare the simulation results for the rigid and for the flexible system is also developed. We outline the fact that the motion equations are treated with all nonlinearities taken into account, without the usage of any simplifying linearization procedure, as found in most of the works present in the literature. This linearization procedure may not to consider small contributions of

physical effects that are sunmied or superimposed and that may significandy influence the system behavior. For this reason, one of the tasks of this work is to treat the motion equations according to a general approach, without simplifying linearizations, and to assess the system behavior through controlled simulations.

2. Simulation and results This flexible system may be mathematically reduced to the rigid one by vanishing the terms related to system flexibility, which characterizes the possibility of finding a frontier between both systems. With this fact in mind, mathematical simulations are performed according to the following methodology: (1) initially the rigid system is simulated in a separate manner; (2) following, the flexible system is simulated with all its contributions taken into account;

S.A. David, J.M. Rosdrio /First MIT Conference on Computational Fluid and Solid Mechanics (3) after that, the effects are individually and cumulatively subtracted and the system behavior is analyzed; (4) the effects are subtracted until the limit condition in which the flexible system is reduced to a rigid one and the system response converges — as expected — for the case of the rigid system modeled separately. Some results are presented in this article correspond to simulations realized for the angular position Oi and 02 (Fig. 2). The other simulations may be found in [4]. 3. Conclusion The way in which the motion equations are treated in this paper may allow the monitoring of each contributing factor for the system flexibility. The flexible manipulator may be mathematically reduced to a rigid one by means of vanishing the flexibility related terms. The same procedure may be extended to the simulations, which makes it possible to find a frontier between both systems. It is also possible to consider the development of controllers that compensate the physical effects — which in accordance to dynamic simulations results is relevant for the system flexibility — in order to correct the response of the terminal element of the manipulator with respect to the signals from the control system.

139

References [1] Book WJ. Recursive Lagrangian dynamics of flexible manipulator arms. Int J Robot Res 1984;3(3):87-101. [2] Craig JJ. Introduction to Robotics: Mechanics and Control. Addison Wesley, 1986. [3] David SA, Rosario JM. Dynamic modeling and simulation of robot manipulator with twoflexiblelinks. Proceedings of Sixth Pan American Congress of Applied Mechanics, 1999. [4] David SA. Modelagem, Simula9ao e Controle de Robos Flexiveis. MSc Thesis, State University of Campinas (in Portuguese), 1996. [5] Farid M, Lukasiewicz SA. Dynamic modeling of spatial flexible manipulators. Comput Methods Exp Meas 1997;3:255-264. [6] Li CJ, Sankar TS. A systematic method of dynamics for flexible robot manipulators. J Robot Syst 1992;9(7):861891. [7] Nathan PJ, Singh SN. Nonlinear ultimate boundedness control and stabilization of aflexiblerobotic arm. J Robot Syst 1992;9(3):301-326. [8] Nayfeh AH, Mook DJ. Nonlinear Oscillations. New York: John Wiley, 1979. [9] Rosario JM. Modelisation Dynamique Dun Robot Industriel. Ministere de I'Education Nationale. Institut Superieur des Materiaux et de la Constmction Mecanique, France, 1987. [10] Schielen IW. Technische Dynamik. s.l.p.; s.c.p., s.d.p. 106 pp.

140

Virtual surgery simulation using a collocation-based method of finite spheres S. De*, J. Kim, M.A. Srinivasan Laboratory for Human and Machine Haptics, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract The method of finite spheres using moving least squares interpolants and point collocation as the weighted residual scheme is applied to the development of a virtual reality based training system for laparoscopic surgical procedures. The localization of approximation and the lack of numerical integration results in very high computational speeds required for real time simulation with graphical and haptic feedback. Keywords: Method of finite spheres; Meshless technique; Haptics; Medical simulation

1. Introduction The objective of this paper is to illustrate how the method of finite spheres [1] may be applied to develop a laparoscopic surgical simulator which will enable the user to interact with three-dimensional computer models of biological tissues and organs in real time, using both visual and haptic sensory modalities. As minimally invasive surgery is gaining popularity, the need to train medical students and also to provide surgeons with appropriate computer tools to experiment with new surgical techniques, without having to use cadavers or animals, is becoming increasingly important. The main challenge in real time virtual surgery is computational speed. For real time visual display an update rate of about 30 Hz is sufficient. To enable the user to interact with the computer models using the sense of touch we use a three degree-of-freedom haptic interface device called Phantom ^. For stable real time simulation, the haptic loop requires to be updated at a rate of about 1 kHz. A variety of simulation techniques, ranging from purely geometrical procedures without any physical basis to spring-mass-dashpot-based models, are found in the literature (see reference [2] for a summary of the existing techniques). Although the finite element technique [3] is a * Corresponding author. Tel.: +1 (617) 253-8503; E-mail: [email protected] ^ Developed by SensAble Technologies, Inc. © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

physically based procedure, it is computationally very slow since the entire domain needs to be meshed and numerical integration has to be performed. In this paper we develop a specialized version of the method of finite spheres based on moving least squares interpolants and point collocation for the purpose of real time surgical simulations.

2. The numerical scheme In our technique, A^ nodal points are sprinkled around the surgical tool tip (see Fig. 1). Moving least squares interpolants hj(x) = Wj{x)F(xfA-\x)F(xj)

J = l,...,N

(1)

are used to generate the local finite dimensional approximation spaces. In Eq. (1) A(x) = J]f=i W/(X)P(X/)P(X/)T. ^he vector P(x) contains polynomials ensuring consistency up to a desired order (in our implementation we have ensured consistency up to degree one). Wj is a compactly supported radial weighting function at node J (which we have chosen as a quartic spline function). We assume linear elastic tissue behavior. A point collocation technique is used to generate the discrete equations KU

(2)

where K is the stiffness matrix and f is the vector con-

S. De et al. /First MIT Conference on Computational Fluid and Solid Mechanics

141

rameters as U = [Utooitip U/,] where U^ is the vector of nodal unknowns which maybe obtained as Vt = —^hh K/,«U -i'a^toooltip The reaction force to be delivered to the haptic interface device is obtained as ftooitip = K^aUtooitip + ^ab^b-

3. Simulation demonstration

Fig. 1. A schematic showing the distribution of nodal points around a surgical tool tip. taining nodal loads. We note here the stiffness matrix K is nonsymmetric, but banded. For the purpose of surgical simulation, the tool tip may be modeled as having point interaction with the tissue (see Fig. 1). A node is placed at the tool tip and all other nodes are placed such that their spheres do not intersect the node at the tool tip (or do so only minimally to ensure the invertibility of A(jt:)). The node at the tool tip bears the applied displacement, Utooi tip at the tool tip. The stiffness matrix in Eq. (1) may be partitioned as K

(3)

corresponding to a partitioning of the vector of nodal pa-

Fig. 2 shows the deformation field computed using the technique described in the previous section when a tool interacts with a hemispherical object. Linear elastic tissue behavior was assumed. The undeformed surface as also the deformation obtained using ADINA with a finite element discretization of the object are presented for reference. The point collocation based method of finite spheres provides reasonable deformation fields near the tool tip but the errors are quite high further away. This technique is however very fast. Computational rate of about 100 Hz is obtainable for the example shown in Fig. 2 when 34 spheres are used for discretization. Real time rendering rates of about 1 kHz is obtained using a force extrapolation technique (refer to [2] for details).

References [1] De S, Bathe KJ. The method offinitespheres. Comput Mech 2000;25:329-345. [2] De S, Kim J, Srinivasan MA. The method of finite spheres in real time multimodal medical simulations. To appear. [3] Bathe KJ. Finite Element Procedures. Englewood Cfiffs, NJ: Prentice Hall, 1996. Undeformed Surface

MFS solution with 34 spheres Fig. 2. The deformation field obtained when MFS is used for the simulation of a surgical tool tip interacting with a hemispherical object is shown. The undeformed surface and the deformation field obtained using afiniteelement discretization are also shown.

142

Efficient analysis of stress singularities using the scaled boundary finite-element method Andrew J. Decks ^'*, John P. Wolf"' " Department of Civil Engineering, The University of Western Australia, Nedlands, WA 6907, Australia ^Department of Civil Engineering, Institute of Hydraulics and Energy, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne, Switzerland

Abstract The scaled boundary finite-element method is reviewed, and an adaptive implementation is applied to the classical elasto-static problem of an L-shaped domain. The method is shown to outperform a similar adaptive finite-element implementation, in terms of both computational time and memory requirements. Keywords: Scaled boundary finite-element method; Singularities; Adaptivity

1. Introduction The scaled boundary finite-element method is a semianalytical method that combines the advantages of the numerical and analytical approaches to solve linear partial differential equations. It also has appealing features of its own, such as the ability to model certain free and fixed boundaries without spatial discretisation. As an analytical solution is obtained in the 'radial' direction, the method is particularly useful in situations involving stress singularities. Stress recovery and error estimation techniques have recently been developed for the method, and these have allowed adaptive techniques to be implemented. This paper applies these techniques to a classical problem containing a stress singularity, and compares the efficiency of an adaptive scaled boundary finite-element procedure with the efficiency of a similar adaptive finite-element procedure. This is the first time a direct comparison of computational efficiency between the two methods is presented.

of two-dimensional plane stress elasto-statics. Omitting body loads, the governing differential equation can be represented as [L]V(x,^)}-{0}

(1)

where [L] is the linear operator, and the stresses {o{x, y)} are related to the strains {s{x, y)} by the elasticity matrix [/)], and in turn to the displacements {u{x, y)} {a(x, y)] = [D][e{x, y)] = [D][L]{u{x, y)}

The differential equation is subject to certain boundary conditions on displacements and surface tractions. The method defines a new coordinate system based on a scaling centre O within the domain, as illustrated in Fig. 1. The normalised radial coordinate ^ has zero value at the scaling centre, and unit value at the boundary. The circumferential coordinate s measures the distance around

2. The scaled boundary finite-element method Since the scaled boundary finite-element method [1-3] is not widely known, a brief summary of the method will be given here. The method will be discussed in the context * Corresponding author. Tel: +61 (8) 9380-3093; Fax: +61 (8) 9380-1018; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(2)

^^ = 0.5 Scaling centre (XQ, yo) Fig. 1. Scaled boundary domain with side-faces.

A.J. Deeks, J.P. Wolf/First MIT Conference on Computational Fluid and Solid Mechanics the boundary from some origin. The boundary may include two 'side-faces', 0 - P i and O-P2, as indicated in Fig. 1. The linear operator can be mapped to the scaled boundary coordinate system using standard methods. of

5

OS

(3)

where [b^{s)] and [/?^(5)] are dependent only on the boundary definition. The scaled boundary finite-element method seeks a solution to the differential equation in the form [Uh

(§,5)} = Y,Ni{s)uhi{^)

= [N{s)][uh{^)}

(4)

This represents a discretisation of the boundary with the n shape functions contained in [^(5)], where n is the number of nodal degrees of freedom on the boundary. The unknown vector {uh{^)} is a set of n functions of §. The method proceeds by first seeking the stiffness matrix of the scaled boundary domain with respect to the degrees of freedom on the boundary (without applying boundary conditions around the i'-boundary), and then solving for the nodal displacements on the boundary in the usual finite-element manner. The stiffness of the domain is obtained by applying the virtual work equation. Virtual displacement fields of the Galerkin form {8u{^,s)} = [N{s)]{8u{^)]

(5)

are used, where {5w(§)} contains n functions describing the variation of the virtual displacements in the radial direction. Substituting into the virtual work equation, and integrating terms in the internal virtual work integral by parts with respect to ^, the surface integrals cancel out and the work statement becomes 1

1 [uHm,^-[E\{uh(^)]]d^=Q where [E''] =

j[B\s)f[D][B\s)]\J\As

and | 7 | is the Jacobian at the boundary. The [E] matrices can be assembled element by element on the boundary. The governing equation will be satisfied exactly in the I direction when this equation is satisfied for any set of functions {8u(^)}, and so [E°]t2{„,(^)},jj + [[£0] + [£l]T - [£l]] X §{«.(?)},^ - [£']{«*(?)) = {0)

(7a)

(10)

This is the scaled boundary finite-element equation displacement. B y inspection, solutions are of the form {u{^,s)] = [N{s)]r^{cl>}

in (11)

Substituting this solution into Eq. (10) yields the quadratic eigenproblem [X\E'] - X[[E']^ - [E'i\ - [E^]] {0} = {0}

(12)

This equation can b e solved using standard techniques, yielding 2n modes. For a bounded domain only the modes with non-positive real components of X lead to finite displacements at the scaling centre. This subset of n modes will be designated by [i], where the vectors in the set form the columns of the matrix. Any particular solution of the differential equation will b e a weighted s u m of these modes. For each mode the approximate stresses on the boundary are determined. After transformation to surface tractions and integration with the shape functions along the boundary, the equivalent nodal forces for the modes follow as {q} = [[E']^-k[E'i\{(t>}

(13)

The subset of these modal force vectors corresponding to the n modes in [OJ is denoted as [Q\\. For any set of boundary node displacements {uh}, the modal participation factors required are {c} =

(6)

143

{^xT'iuh}

(14)

The equivalent nodal forces required to cause these displacements are [P] = [QiMc] = [QI][^I]-'{UH}

(15)

The stiffness matrix of the domain is therefore [£>

= j{B\s)V [D][B'(5)]|7|di

(7b)

[K] = [Qi]mr'

(16)

and the equilibrium requirement is reduced to

[^^] = / [[B\s)Y[D][B\sW\ds

(7c)

in which [B'(^)] = [Z)'W][iV(i)]

(8)

[B\s)] =

(9)

[b\s)][N{s)l,

[K]{uh} - {P} = {0}

(17)

Boundary conditions place constraints on subsets of {uh} and { P } , and the solution proceeds in the same manner as in standard finite-element analysis. However, unlike that method, only boundary degrees of freedom are present. The modal participation coefficients are then obtained using Eq. (14), and the displacement field is recovered as

A.J. Deeks, J.P. Wolf/First MIT Conference on Computational Fluid and Solid Mechanics

144 {Uh (§,j)}

w^^^^^

(18)

= [yv(5)]^c,r''{<^,}

A stress recovery technique has been developed by Deeks and Wolf [4]. A recovery-based estimator compatible with the widely used Zienkiewicz-Zhu [6] estimator has also been developed by the same authors [4], allowing implementation of an /z-hierarchical adaptive procedure [5]. This procedure can be compared directly with an /z-hierarchical adaptive finite-element implementation [7]. At the present time a general-purpose eigenvalue extraction procedure has been used in the scaled boundary finite-element implementation. Considerable improvement in efficiency may be expected when the solution routines are optimised. A fast active column solver with profile optimisation is used in the finite-element implementation.

i

^

\^

p-'M—' ^ 1^ L_!

^<

^

>

Fig. 2. Model of square plate with square hole under uniaxial tension. 10 7.5 h

;

^

5 ^

1

2.5 V

Scaled boundary finite-element —

Finite element ^ x x ^

3. Example

HZTTT"

0 1

The example represents a quarter of a square plate with a square hole under uniaxial tension, as illustrated in Fig. 2. Advantage is taken of the biaxial symmetry. The true stress field contains a singularity at the interior comer O. Poisson's ratio is taken as 0.3, and Young's modulus as 1000. This example has been used extensively in the adaptive finite-element literature (e.g. [5]). In the scaled boundary finite-element analysis the scaling centre is selected at O. No spatial discretisation is required on the side-faces 0-Pi and O-P2. The problem was analysed using both the adaptive scaled boundary finite-element procedure and the adaptive finite-element procedure with target error levels of 5%, 2% and 0.5%. The number of degrees of freedom, the solution time in milliseconds, and the displacement at point A in Fig. 2 were recorded for each target error. The results are presented in Table 1. The scaled boundary finite-element solutions and the finite-element solutions are in close agreement, as indicated by the displacements. The number of degrees of freedom (and hence the memory requirement) of the scaled boundary finite-element solution is significantly less than the equivalent finite-element solution at each error level. The time taken for the scaled boundary finite-element solution at the 5% error level is about 20% of the time taken for the

— — ^

v/

-2.5 p -5 ^ 1.0

1

,

0.8

0.6

^

0.4

0.2

0.0

Fig. 3. Stresses along the line BB at the 5% target error level. finite-element solution, representing a considerable saving. This advantage reduces as the target error is decreased, but the scaled boundary finite-element method still takes only about 50% of the time of the finite-element method to achieve a 0.5% error. The scaled boundary finite-element method yields a solution with a singular point at the interior comer, and the power of the singularity follows directly. In contrast, the finite-element method returns finite stresses at the interior comer. This is illustrated in Fig. 3, where the variation of all the stress components along the line designated BB in Fig. 2 calculated by the two methods for the 5% analysis is plotted. There is excellent agreement between the methods (which is expected since the error level is the same, and is only 5%), except in the vicinity of the singular point, where the scaled boundary finite-element method results are clearly superior.

Table 1 Superior performance of the scaled boundaryfinite-elementmethod Error target

5% 2% 0.5%

Scaled boundaryf finite-element

Finite element DOF

Time

Displacement

DOF

Time

Displacement

670 1774 4986

1805 6775 37136

-2.109 X 10-5 -2.113 X 10-5 -2.113 X 10-5

20 38 74

398 2565 18524

-2.114 X 10-5 -2.113 X 10-5 -2.113e X 10-5

A J. Deeks, J. P. Wolf/First MIT Conference on Computational Fluid and Solid Mechanics 4. Conclusions This paper shows that problems containing stress singularities can be solved accurately and efficiently using the scaled boundary finite-element method. The example shows that the cost in both computing time and memory usage is lower for the scaled boundary finite-element method than for the finite-element method at all target error levels. In addition, the stresses near the singularity are more accurately modelled. These results were achieved using general-purpose eigenvalue extraction routines, and considerable improvement in the scaled boundary finiteelement results can be expected when the solution routines are optimised.

References [1] Song Ch, Wolf JP. The scaled boundary finite-element method — alias consistent infinitesimal finite-element cell

[2]

[3] [4] [5] [6] [7]

145

method — for elastodynamics. Comp Meth Appl Mech Eng 1997;147:329-355. Wolf JP, Song Ch. The scaled boundary finite-element method — a semi-analytical fundamental-solution-less boundary-element method. Comp Meth Appl Mech Eng, in press. Wolf JP, Song Ch. Finite-Element Modelling of Unbounded Media. Chichester: Wiley, 1996. Deeks AJ, Wolf JP. Stress recovery and error estimation for the scaled boundary finite-element method. Submitted for publication. Deeks AJ, Wolf JP. An /z-hierarchical adaptive procedure for the scaled boundary finite-element method. Submitted for publication. Zienkiewicz OC, Zhu JZ. A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 1987;24:337-357. Deeks AJ. An adaptive /?-hierarchical finite element system. In: Advances in Finite Element Techniques and Procedures, 4th Int. Conf. Computational Structures Technology, Edinburgh 1998, pp. 115-124.

146

Relocation of natural frequencies using physical parameter modifications M.S. Djoudi*, H. Bahai Department of Systems Engineering, Brunei University, Uxbridge, Middlesex UBS 3PH, UK

Abstract An efficient relationship between physical properties of pin-jointed structures and their eigenvalues is established. The formulation allows the determination of the necessary modifications on the structural members to achieve the specified frequency. The calculations involved do not include any iteration or convergence and therefore it is computationally efficient. The modification can either be global or local. In addition to the modification of the existing structural elements the formulation can also be used to add new structural elements to achieve the desired natural frequencies. Although in the present paper only simple structures are considered the formulation can be applied to large and more complex structures. Keywords: Inverse problem; Structural modifications; Desired frequencies; Structural vibration; Eigenvalues; Pin-jointed structures; Cross-sectional area

1. Introduction Many engineering constructions such as highway bridges, aerospace structures and ship structures are frequently subjected to dynamic loads and thus, dynamic analysis is necessary to determine the vibration response of these structures. It is a common design requirement to ensure that all the natural frequencies are far away from the frequency caused by the exciting forces. The common industrial practise for optimising the design is to subject the proposed structure to a series of structural modifications based on the engineer's experience. Each series requires the analysis of modified structure, which is usually slightly different from a structure previously analysed. This complete reanalysis of the structure is often very expensive and a time consuming task. To eliminate the need to reanalyse the whole structure, more research effort was conducted towards developing new concept with sufficient information to find the exact modified parameters, which yield the required natural frequencies. Early work in this direction done by Wilkinson [1], Van Belle [2] and Vanhonacker [3] utilised the 1st order terms of Taylor's series expansion and

* Corresponding author. Tel.: +44 (1895) 274-000; Fax: +44 (1895) 812-556; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

is based on Rayleigh's work. Chen and Garba [4] used the iterative method to modify structural systems. Further research on structural modification was carried out by Tsuei and Yee [5-7] who presented a method of shifting the desired eigenfrequencies using the forced response of the system. More recently Kim [8] investigated the use of mass matrix modification to achieve desired natural frequencies. Sivan and Ram [9-11] extended further the research on structural modification by studying the construction of mass spring system with prescribed natural frequencies, they obtained stiffness and mass matrices using the orthogonality principles. However, the resulting stiffness or mass matrix may not be physically implemented. In reference [9] Sivan and Ram developed a new algorithm based on Joseph's work [12] which involves the solution of the inverse eigenvalue problem. In the last few years the work on the inverse problem done by Gladwell [13] started to be taken seriously by engineers and researchers interested in this field of engineering. The work is applied to both discrete and continuous systems. In this paper an efficient formulation between the geometric and material properties of structures and their eigenvalues is established. The formulation allows the shifting of the natural frequencies and solves for the required modification on chosen geometric and material properties.

141

M.S. Djoudi, H. Bahai/First MIT Conference on Computational Fluid and Solid Mechanics equation of motion becomes:

2, Theoretical consideration To construct a system with desired eigenfrequencies it is necessary to find a relationship between the structural parameters of the system and its eigenfrequencies. For a discrete system such as mass spring systems, and when only one or two degrees of freedom are involved. The formulation, which accounts for such relationship is easily obtained and hence the change of stiffness or mass required for shifting the eigenvalues can easily be evaluated. However, for systems with a large number of degrees of freedom and continuous systems special algorithms have to be developed. A contribution in this direction was made by Esat and Akbar [14]. They presented the stiffness of the system as a function of the desired eigenvalues and showed that the stiffness varies linearly with the eigenvalues. The formulation is very simple, however the resulting stiffness of the modified system cannot be physically implemented. For the new system to be constructed, the modification carried out on the structural properties of the system must have a physical meaning (realisable). For example in the case of truss structures both the elastic modulus and the cross-sectional area of the bars can be modified to shift the eigenfrequencies. Any modification on the elastic modulus would cause only stiffness modification of the structures. Whereas, a modification in the area parameter would result in both stiffness as well as mass modification. In the following section a formulation giving the crosssectional area modification as function of the required eigenfrequency is first developed. This formulation can then be used to obtain the elastic modulus variation as function of the desired eigenfrequency. For a pin-jointed truss structure both the stiffness and mass modifications can be given as functions of the area modification of any member in the structure. ^K = AA[K'] AM = AA[M']

(1)

where AK and AM are the variations or modifications on the system stiffness and mass matrices respectively, AA is the change in the area of the modified member and [K'] and [ M ' ] are the stiffness and mass matrices of the modified member where the area is taken as unity. The equation of motion for the free vibration of a dynamic system is given by: (K - XoM)8 = 0

(2)

where K is the stiffness matrix of the system, M is the mass matrix, 8 is the displacement vector and ko is the eigenvalue of the original system. If a modification A A is carried out on any member of the structure, this would result in modifications in both stiffness and mass matrices of the structure and hence the

(3)

XdM - XdAM)8 = 0

(K-\-AK-

where Xd is the new eigenvalue of the modified structure. Eq. (3) can be transformed to modal co-ordinates by putting 8 = ^u where 0 is the mass normalised modal matrix. Hence, (K-]-AK-

=0

XdM - Xd^M)^u

(/TO + AK^

- XdM^

(4)

- Arf A M O ) M = 0

(5)

If we pre-multiply the above equation by O^ and use the orthogonality characteristic of O with respect to K and M we obtain the following equation: iSl + ^^ AK^ - Xdl - Xd^'^AM^)u

(6)

=0

where Q is the diagonal eigenvalue matrix and / is the unity matrix. Eq. (6) can be written as: u = -(Q-

- Xd^^AM^)u

Xdir\^^AK^

(7)

By pre-multiplying both sides by O and rearranging the equation, we obtain ^u = - 0 ( ^ - Xdiy^<^^{AK

- XdAM)^u

(8)

By substituting for AK, AM and Ow by AA [K'], A A [M'] and 8 respectively we obtain: 8 = -AAcD(^ - Xdiy^^^(K'

- XdM')8

(9)

This can be written as {8} = -AA[F][G]8

(10)

= -AA[R]{8}

where [F] = <^(Q - Xdl)'^^^, [G] = [K' - XdM'] and [R] = [F][Gl Eq. (10) can be written in matrix form as: AA-^ + /?i,i /?2,1

Ri,2

Rl,n

A A - l + R2,2 •

Rn,2

R2,n

• • • AA-1 + Rn,n (11)

where the terms Rij are function of the eigenvalue Xd. The characteristic equation of the modified system for the eigenvalue Xd is given by: ^1,1

^1,2

AA-1 + R2

Rl,n Rl,n

• • . A A - 1 + Rn,n

= 0

(12)

148

M.S. Djoudi, H. Bahai/First MIT Conference on Computational Fluid and Solid Mechanics

Eqs. (12) are for global modification where all the bars are to be modified at the same time and in this case n is equal to the total number of unconstrained degrees of freedom. However, if this is not the case then only the terms corresponding to the nodes associated with the modified bars are retained. A solution for the above problem can be obtained by solving the characteristic equation (12) and obtaining A A. 3. Numerical examples:

-bar1 -bar9

g

3.2. Space truss structure The second example consists of the tower shown in Fig. 3. The dimensions and material properties are shown E=2xlO"N/m2 p=7860kg/m^ A=5xl0^m^for all bars

Fig. 1. Plane truss structure.

-

-bar3 •

-bar?

-bar 11 •

-bar 12

-

150 4

V, 100 o S

50

•5

0

^

The first example is a twelve bar truss cantilever as shown in Fig. 1. This example is used to illustrate the modification required on the cross sectional area of the bars to shift the lowest frequency. The addition of new bars is also considered in this example. The material properties and the cross sectional area of the bars are shown on Fig. 1. The lowest natural frequency of the structure has been increased by A / = 5% through steps of 0.5% and for each step the required change in the cross sectional area of each bar is obtained. These are shown in Fig. 2. It can be seen that while an increase in the cross sectional area of some bars, for example 1,2,3 and 7, is necessary to achieve the desired frequency, other bars require their areas to be decreased. This is due to the fact that the cross sectional area affects both the mass and stiffness matrices of the structure. It is also noticed that the fixed frequency may not be achieved by varying the area of some bars, for example in this case, a shift in the frequency by 2% cannot be obtained by modifying the cross sectional area of bars 1, 3 and 9 only. Therefore, if no restriction is made on which bar is to be modified to shift the frequency, the designer can compare the set of results and choose the structural member to be modified.

bar2

bar 10 -

200

>

3.1. Plane truss

•

-50

-100 % Variation of first frequency

Fig. 2. Variafion of first frequency with required modification on bars area.

E=2.1xlO"N/m2 p=7860kg/m3

Fig. 3. Space truss structure.

in the same figure. The cross sectional areas for each bar is given by: • A = 3 X 10""^ m^ for Ci and C2 bars (comer columns in bottom and top levels respectively) • A = 1.5 X 10""^ m^ for Bi bars (horizontal members in bottom level) • A = 0.8 X 10"'^ m^ for B2 bars (horizontal members in top level) • A = 0.8 X 10""^ m^ for Ti bars (diagonal members in bottom level) • A = 0.4 X 10"'* m^ for T2 bars (diagonal members in top level) The sensitivity of the lowest natural frequency to any modification on the cross-sectional area of the different bars is first investigated. Fig. 4 shows the percentage variation of the first natural frequency with the required percentage variation on the cross-sectional area of the bars. It is seen that the first frequency is most sensitive to bars Ci and C2.

M.S. Djoudi, H. Bahai/First MIT Conference on Computational Fluid and Solid Mechanics !-•—C1 bars - • — C 2 bars - A - B 1 bars - » ^ B 2 bars U-d—T1 bars —H—T2 bars

100 60 20

•I -20 r4^\ ^

;>

^

•

i

'—m

-60

-100 % Variation of first frequency

Fig. 4. Variation of first frequency with the required modification in the cross-sectional area of bars.

4. Conclusion In this paper a method for determining the required structural modification to achieve desired frequencies for pin-jointed structure is established. The formulation allows the determination of the necessary modifications on the material and geometric structural properties to shift any of the frequencies to desired positions. The approach can be used to increase as well as decrease the natural frequencies, and the structural modifications can also include the addition of new structural members. This approach provides the structural designers with efficient algorithm, which is formulated in such a way that no iterations or convergence are involved in the process and only few calculations are required to obtain the necessary modifications.

149

References [1] Wilkinson JH. The Algebraic Eigenvalue Problem. Oxford University Press, 1963, pp. 62-109. [2] Van Belle H. Higher order sensitivities in structural design. AIAA J 1982;20:286-288. [3] Vanhonacker P. Differential and difference sensitivities of natural frequencies and mode shapes of natural structures. AIAA J 1980;18:1511-1514. [4] Chen JA, Garba JA. Analytical model improvement using modal test results. AIAA J 1980;18:684-690. [5] Tsuei YG, Yee E. A method for modifying dynamic properties of undamped mechanical systems. J Dyn Syst Meas Control 1989;111:403-408. [6] Yee E, Tsuei YG. Modification of stiffness for shifting natural frequencies of damped mechanical systems. DE-Vol. 38, Modal Analysis, Modelling, Diagnostics and Control Analytical and Experimental ASME 1991, pp. 101-106. [7] Yee E, Tsuei YG. Method for shifting natural frequencies of damped mechanical systems. AIAA J 1991 ;29( 11): 19731977. [8] Kim Ki-ooK. A review of mass matrices for eigenproblems. J Comput Struct 1993;46:1041-1048. [9] Sivan D, Ram YM. Mass and stiffness modification to achieve desired natural frequencies. Commun Numer Methods Eng 1996;12:531-542. [10] Ram YM. Enlarging a spectral gap by structural modification. J Sound Vib 1994;176(2):225-234. [11] Sivan D, Ram YM. Optimal construction of mass-spring system with prescribed model and spectral data. J Sound Vib 1997;201(3):323-334. [12] Joseph KT. Inverse eigenvalue problem in structural design. AIAA J 1992;30(12):2890-2896. [13] Gladwell GML. Inverse vibration problems for finite element models. Inverse Probl 1997;29(4):421-434. [14] Esat II, Akbar S. Synthesis of multi-body systems for desired eigenfrequencies. ASME, ASIA976, Congress and exhibition, Singapore.

150

Fourier transformed boundary integral equations for transient problems of elasticity and thermo-elasticity F.M.E.Duddeck* Technical University of Munich, Lehrstuhl fiir Baumechanik,

Arcisstrasse

21, D-80333 Munich,

Germany

Abstract To overcome the restriction of actual boundary element methods (BEM) to cases where fundamental solutions are known, an alternative BEM-approach was presented in Duddeck and Pomp [6] and Duddeck and Geisenhofer [5]. This approach is based on new boundary integral equations (BIE) for the computation of the entries of the standard BEM matrices which are obtained by a spatial and temporal Fourier transform of the traditional BIE. In these equations, we only need the transform of the fundamental solution and not the fundamental solution itself. The former is always available as long as the underlying differential operator is linear and has constant coefficients. Here, this method is extended to dynamic problems. Transient problems can be tackled by a Galerkin time-step scheme. Keywords: Boundary element method; Galerkin-boundary integral equations; Fourier transform; Fundamental solution; Transient problem; Elasticity; Thermo-elasticity

1. Prototypic example: Fourier-BIE for the bar To illustrate the general principle of the new approach we start with the simple example of an elastic bar. The insights gained from this prototype are transferred later to isotropic and anisotropic elasticity and thermo-elasticity. The differential equation and its Fourier transform are {—EAd^.-\-pd^) u{x,t) = fix,t) -f> (EAP — pa)^)ii{x,co) = f{x,co) with EA,p as the stiffness and the mass density respectively, u is the displacement and / the volume force. (?) denotes a Fourier transformed quantity and 9^, dt are differentiations according space and time, a;, JC are the circular frequency and the wave number, respectively. The displacement boundary integral equations (BIE) are for vanishing initial conditions, e.g. Bonnet [2], K(x)u{x,t) = /

f(y,T)U(x

- y,t - T)d^vdr

+ 11 qiy,T)Uix

•y,t

u(x, t) ^ ^Uij(pi{x)(t)j{t),

'^J

(2)

Temporal and spatial weighting with 0^:, (pi leads to the Galerkin-BIE, cf. Barbier [1],

Jf-

(t)(piix)K(x)uix, t) dF^ dt =

R+Tv

= JlMt)(Piix)fjf(y,r) R+ ^

+ X ! / / ^k{t)(pi{x) I i qij(Pi{y)(pj(T) '•J R+ r.v

*Tel.: +49 (89) 2602-5472; Fax: +49 (89) 2602-5474; E-mail: [email protected]

R+ Tv

xU(x -y,t

(1)

R+Tv

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

^^qij(Pi{x)(l)j{t).

X U(x - y,t -r) dQy dr dF^ dt

-T)drvdT

u(y,T)Q(x-y,t-T)dr,dT.

q(x, t)

'•J

R+ r V

R+Tv

7/

with the traction ^ = £A9yM on the boundary F^ with outer normal y, the fundamental solution U, and the fundamental traction Q = EAd^U. K is the free term. The boundary quantities u,q are approximated by spatial (Pi{x) and temporal trial functions (j)j{t)

- r) dVy dr dF;, dt

~ X ! / / ^kiO(piM / / iiij(Pi(y)(pj(r) ^'j R+ r ,

R+ Fy

X Q(x -y,t

- r) dF^ dr dF;, dr.

(3)

FM.E. Duddeck/First MIT Conference on Computational Fluid and Solid Mechanics

L/4

All quantities are extended formally to M x M to apply spatial and temporal Fourier transform. We abbreviate by defining the convolution a ^ b = /RXE^(>'' '^)^(^ ~ y,t - T)d};dr and the scalar product {a, h) = f^^^ a(x, t)b(x, t) dx df. {(pk(pi, KU) = {(pkCPl, f^U)-\-

f(x,t) EA, p

Y^{(pk(Ph C\ij(Pi(l)j * U)

~ Yl^^^^^ ^ij^i(t>i * 2>-

151

(4)

•

^

^

L=2

Fig. 1. Geometry and loading of the bar. Due to the convolution theorem a * Z? ^ a^ and Parseval's theorem {a,h) = {2n)-^[a^,b) with a^ = a{-x, -0)) we get the equivalent Fourier-BIE, cf. Duddeck and Pomp [6],

test functions we have

{4>lvJ, Ku) = {4,li,J, fu) + jy>lvj, %UjU) ~T.{^I$!^^ij^i4>jQ)^

for w(L):

(pi{x) = 8(x — L)

forg(O):

(P2(x) = 8{x)

^f>

<^i(^) = e"'^^

^

(^2(^) = 1.

The linear temporal trial and test functions (pj for the yth time step were constructed by translations t ^ t — j dt in the original domain and by modulation with e''^^^"^ in the transformed domain of the reference element

(5)

where the factor {2ny^ was cancelled. The double integrations in the original domain are replaced by single intet -]-dt t -dt Po = —7—[Ha + dO H(0] + — grations in the Fourier domain. The transformed BIE lead d^ dt^ [ H ( 0 - H(/ - dO to the same matrices as the conventional BIE, for example 1 — COS (o; do {(l>k(ph qij(Pi(l)j^U) = -^[$T^T^ qij(pi$jU). Therefore, the 0 0 •• 2dtco^ ' processus after construction of the boundary element methWith homogeneous initial conditions and with boundary ods (BEM) matrices can directly be taken from standard 0,q(L,t) = 0 we get the displaceconditions u(0,t) = BEM algorithms. The Fourier transformed fundamental soments and tractions at w(L, 0 , ^(0, t), cf. Fig. 2. lution U as the response to / = 8(x)8(t) ^ / = 1 is obtained by simple inversion, i.e. U(x, co) = [EAp-pa)^]-K The traction BIE needed for symmetric BEM is es2. Extension to elasticity and thermo-elasticity tablished by derivation -dy of the BIE, we get with

S = dyQ^S

= ivxQ:

-{(l>k(pi, Kq) = -{(Pkcpi, f^Q) - Y^[(t)m, (\im(pj^Q) + Yl{^k(Ph yxij(pi(t)j^S).

(6)

In the full paper, we present the dynamic Fourier-BIE for arbitrary anisotropic elastic and thermo-elastic media. Due to the limited space here, we give only the crucial point of constructing the Fourier transformed fundamental solution

The Fourier transform of (6) leads to the Fourier traction BIE:

-(0[^f, Kq) = -(0j^f, fQ) - Y,[^lcpJ, q,j0,4>jQ)

u(L,t)

+ J2{^Iv!,^,jV'ihs)- (7) I

•

4

We regard as an example a transient volume force, cf. Fig. 1,

•

•

t-

- ^; I

• • ; - - •

'^ '

•

1

8

^0

fix, t) = 8(x - L/4)[H(0 - H(^ - 1/2)] _ f(x,co) = e

^-ixL/4_

-i(oL/A 100

(8)

The wave velocity is Cp = -s/EA/p = .JYfA [m/s], and the length of of the bar is L = 2 [m]. As spatial trial and

\\

q(0,t)

Fig. 2. Boundary displacement u(L,t) and boundary traction ^(0,0 for the bar.

152

for linearized thermo-elasticity. The four differential equations are (cf. Nowinski [7]): -fiUijj - (A -h /ji)ujji + yOj + pt^ijt = fi and —Ojj + ^^,r + Wjjt = PI^- 0, p sue the increment of the temperature and the heat sources, ^, y,r] are constants of heat conduction. The four-dimensional Fourier transform of these equations is P{x,co)u(x,a)) = —f(x, co) with the symbol —/xx"

-ex1^2

—

-iyx\

CX\XT,

+pa)^ -CX2X\

-/xx

— c.

-CX2X3,

-lyxi

+pap— CX2X\

-CX3X2

(9) -IJLX"-

-iyx3

-\-poJ'' r]X\co ,

,

r]X2C0 ^T

^

r]X3(o ^

^

-^

^

^

^

and the vectors u = [uuU2,u^,9],f = [/i,/2,/3, p/x]. The transformed fundamental solution is obtained by simple matrix inversion, i.e. U = [P]"'. This approach can be transferred to all linear and homogeneous differential operators. Anisotropic elasticity and thermo-elasticity problems can be treated in two or three dimensions.

References [1] Barbier D. Methode des potentiels retardes pour la simulation de la diffraction d'onde elastodynamique par une fissure tridimensionnelle. Ecole Polytechnique, PhD thesis, 1999. [2] Bonnet M. Boundary Integral Equation Methods for Solids and Fluids. New York: Wiley, 1999. [3] Duddeck F Funktional Analysis in Solid Mechanics — Spatial and Temporal Fourier Transform of Energy Methods (in German). PhD thesis, TU Munich, 1997. [4] Duddeck F. A general boundary element method for homogeneous differential operators — linear or non-linear. ECCOMAS 2000, Barcelona. [5] Duddeck F, Geisenhofer M. A general boundary element method for anisotropic plates. Comput Mech, submitted for publication. [6] Duddeck F, Pomp A. Calculation of BEM matrices by Fourier transform. Math Comput, submitted for publication. [7] Nowinski W. Dynamic Problems of Thermoelasticity. Leyden: Noordhoff, 1975.

153

A fluid-like formulation for viscoelastic geological modeling stabilized for the elastic limit Frederic Dufour*, Louis Moresi, Hans Miihlhaus CSIRO Exploration and Mining, Perth 6009, Australia

Abstract We present and discuss a new stabilization procedure for viscoelastic flow models of large deformation, such as geological folding. Viscoelastic equations are solved for an increment of observation time At^ different from the advection timestep A^ An averaging procedure for the stresses is needed over a number of advection timesteps. We study the relative values of the relaxation time a, the elastic timestep At^ and the advection timestep required to prevent any numerical instabilities and to obtain accurate results. Keywords: Viscoelasticity; Stabihzation; Folding; Large deformation; Deborah number

1. Introduction Within the geological record, there is evidence of numerous occasions where creeping flow of sohd crustal rocks dominates deformation. Strains are typically very high, strain-rates are low (10"^"* s~0, viscous, elastic and brittle effects influence the observed structures (for example, Fig. 1 (top) shows folds in Archean migmatitic gneiss). The particle-in-cell finite element scheme [1] was designed to deal with very large deformation geological problems including folding (Fig. 1, center). It uses an Eulerian mesh to solve modified viscoplastic equations of motion, and a Lagrangian set of particles which carry material history including stresses. As with other strain-rate based formulations for viscoelasticity, the method is optimized for the viscous rather than the elastic limit [2,3]. However, through a simple stabilization procedure, we are able to study problems where the relaxation time is longer than any characteristic timescale of the deformation.

2. Mathematical model We use a Maxwell viscoelastic constitutive relationship ao-\-a = 2r]D,

tr(D) -\- -p = 0 A,

(1)

* Corresponding author. Tel.: -f-61 (8) 9284-8463; Fax: +61 (8) 9389-1906; E-mail: [email protected]

where cr is the Cauchy stress tensor, a its Jaumann derivative, D is the stretching, p = - | t r ( a ) , X is a penalty parameter, /x and r] are the elastic shear modulus and the shear viscosity, respectively. Eq. (1) can be written in a finite difference form: ^t+At^

=_ '/eff 2Z)

/x \At'

orW

(2)

where At^ is the elastic timestep, W is the material spin tensor, a = r]//! is the relaxation time and y/eff is an effective viscosity defined by ^^£7^2.1. Stability in the elastic limit We need to choose a timestep which is both stable, and accurately represents the physics of the problem. The extent to which a Maxwell viscoelastic system behaves elastically depends greatly on the timescale of observation (see [2,3]), and so may depend subjectively upon what we consider worth resolving in the time evolution of a problem. By "elastic" problems, we refer to cases where the timescale of interest is small relative to the time over which stresses relax. As elasticity becomes more important, the representation of the material as a viscous fluid with additional stored stresses becomes less appropriate. Problems in the elastic range correspond to very soft effective viscosity, and a more explicit character to the solution strategy. Elastic displacements are calculated by integrating comparitively large velocities over short times. There are two related

Crown Copyright © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

154

F. Dufour et al. /First MIT Conference on Computational Fluid and Solid Mechanics times between the calculation of new velocity solutions, A^ In the update of internal stresses we now write:

•(--i(ip + Wa'

• 0»?eff I

- (T^W

-f(l-0V-^^

(3)

where A^ ... 0 = ^ (4) This amounts to a running average of the stress over a time Af. This procedure allows the choice of a physically relevant time to model elastic effects, independent of that required by other physical processes, mesh dependencies etc. Secondly, if we require that Ar is always less than Ar^ (in other words, requiring a maximum value of 0), the averaging is strongly stabilizing for elasticity dominated problems.

3. Application

0.05 Dimensionless time

The choice of At' and 0 is illustrated by the following example. We compress a viscoelastic compressible unit square block on one edge with a constant velocity (V = 0.1) to 90% shortening. The stresses are then relaxed without further shortening. The pressure is benchmarked (Fig. 2) against the analytical solution for a given material {a — 1.0), a given advection timestep (Ar = 0.0037) and different observation times {At'). As expected from Eq. (2), the smaller the elastic timestep the more accurate the result. An instablity occurs if the advection timestep is larger than a certain fraction of the observation time. In order to determine the value of this fraction, we repeated this study for a range of materials with different relaxation times. We plot (Fig. 3) T

Fig. 1. Quartzo-feldspathic layers (light colors) defining asymmetric folds in Archean migmatitic gneiss, Simo, northern Finland (top). A numerical simulation of a viscoelastic layer with a yield stress (centre). Stress measured at a point with the folding layer as a function of time (bottom). difficulties which may arise: (1) in the limit of elastic behaviour, this system is not well conditioned, and may be numerically unstable; and (2) if other physical processes, such as thermal diffusion, porous flow or chemical reaction, impose a very short timestep then we may be forced to consider an unstable, quasi-elastic solution when there is little physical reason to do so. We address both these problems by the following stabilization. First we consider that the timescale over which we differentiate the stress rate, At^, may be larger than the

o Theoretical solution - - Af^ = 10.0 --- M"" = 1.0 '•M" = 0.1 — At*^ = 0.0098

' 1 At*= = 0.0095

4 6 Dimensionless time Fig. 2. Stability and accuracy of the solution for different observation timesteps (Ar^) and for fixed relaxation time and advection timestep. 0

F. Dufour et al. /First MIT Conference on Computational Fluid and Solid Mechanics 0.55

20 10 15 Relaxation time Fig. 3. Linear regression on numerical values of the stability factor for different materials.

5

155

the less competent. Initially the competent layer is straight and axial stress increases with a constant shortening velocity, then the buckling occurs and leads to a drop in the stress (Fig. 1 (bottom)). The layer is broken by yielding concomitant with folding and the different parts of the beam straighten due to the elastic effect. The doublescale integration scheme presented solves accurately and effectively the model equations for Maxwell materials undergoing very large deformation. Although the code was initially designed for viscous fluids, this scheme is able to solve any problems even in the elastic limit for large a. Empirically established stability criteria for the two timesteps are t^f < a/100 and A^ < |Ar''.

References the stability factor 0 as a function of the relaxation time, for all computations we keep the ratio Ar^ = or/100 (constant Deborah number). In the limit of short observation times we find that the value of 0 required to stabilize the method is greater than 0.35. This result also holds for other values of the ratio between Ar^ and a. We apply the stabilization procedure with (p — 0.35 to a folding problem (Fig. 1) with two incompressible viscoelastic layers, the more competent layer embedded into

[1] Moresi L, Miihlhaus H-B, Dufour F. Particle-in-cell solution for creeping viscous flows with internal interfaces. Proceedings of the 5th International Workshop on Bifurcation and Localization, Perth, WA, Balkema, 2000. [2] Tanner RI, Jin H. A study of some numerical viscoelastic schemes. J Non-Newtonian Fluid Mech 1991;41:171-196. [3] Debbaut B, Marchal JM, Crochet MJ. Numerical simulation of highly viscoelastic flows through an abrupt contraction. J Non-Newtonian Fluid Mech 1988;29:119-146.

156

An Eulerian formulation for modeling stationary finite strain elasto-plastic metal forming processes Eduardo N. Dvorkin *, Dolores Demarco Center for Industrial Research, FUDETEC, Av. Cordoba 320, 1054, Buenos Aires, Argentina

Abstract Lagrangian formulations are suitable for modeling a material behavior that incorporates elasticity but are not specially appropriate for modeling stationary processes; on the other hand, the available Eulerian formulations are very appropriate for modeling stationary processes but fail to properly incorporate the elastic material behavior. In the present paper we outline a new solid mechanics Eulerian formulation that properly describes a finite strain elasto-plastic deformation process and therefore seems to be specially suited for modeling stationary elasto-plastic metal forming processes. Keywords: Metal forming; Finite elements; Stationary problems; Eulerian formulation; Finite strain; Elasto-plasticity

1. Introduction In previous publications [1-7] we presented the development of finite element models for simulating stationary metal forming processes under the assumption of rigid-viscoplastic material behavior. Those models were based on the flow formulation [8] and were implemented using an Eulerian description of motion via the pseudo-concentrations technique [9,10]. The resulting numerical model is equivalent to the one that describes the flow of an incompressible nonlinear fluid (at every point the viscosity is a function of the strain rate); in our formulation the free surfaces are described using the pseudo-concentrations procedure which does not incorporate the complications of the standard free surface algorithms that require shifting nodes and the use of remeshing procedures. When modeling certain metal forming processes it is not realistic to neglect the elastic deformations; for example, when modeling the cold rolling of thin steel plates; also, in some cases the model objective is to investigate phenomena that are governed by the elastic deformations such as spring back effects or the build up of residual stresses; in all of the above mentioned cases it is necessary to use an elastic-viscoplastic model rather than a rigid-viscoplastic one. * Corresponding author. Tel: -h54 3489-435302; Fax: -h54 3489435312; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Lagrangian formulations are suitable for modeling a material behavior that incorporates elasticity but are not specially appropriate for modeling stationary processes; on the other hand, the available Eulerian formulations are very appropriate for modeling stationary processes but fail to properly incorporate the elastic material behavior. In the present paper we outline a new solid mechanics Eulerian formulation that properly describes a finite strain elasto-plastic deformation process and therefore seems to be specially suited for modeling stationary elasto-plastic metal forming processes.

2. The Eulerian solid mechanics formulation Let us consider the solid in Fig. 1 that evolves from its reference configuration (/ = 0) to its spatial one {t). Using the standard multiplicative decomposition of the deformation gradient we can write [11-15], 'F-

(1)

For the strain rates we define in the spatial configuration the velocity gradient (^/) and the elastic velocity gradient (7 ); hence we can write the following material derivatives ^' F Dt

dt

vV'F='l-

(2a)

E.N. Dvorkin, D. Demarco /First MIT Conference on Computational Fluid and Solid Mechanics Reference Configuration

157

where ^a/y are the Cartesian components of the Cauchy stress tensor, ^dtj are the Cartesian components of the spatial strain rate tensor, ^Rt are the Cartesian components of the external loads and ^ V is the volume of the body in the spatial configuration.

Spatial Configuration

3. The pseudo-concentrations technique

Intermediate Configuration

Fig. 1. Multiplicative decomposition of the deformation gradient.

Dt

dt

+ 'v'YLL

='L 'iK

(2b)

where ^y_ is the material velocity field. In the intermediate configuration we define the viscoplastic velocity gradient CL ); its push-forward to the spatial configuration is [17],

-vp

(^-r.=t^L CM', t r ^

(3)

(4)

• d =vp

Using Perzyna's viscoplastic constitutive relation we can write in the spatial configuration [19], 'r,f = 2/x(?„,)(5„,),.

(5)

where ^T-^ are the Cartesian components of the deviatoric Kirchhoff stress tensor. Calling ^r_ the tensor we get by pulling-back the components Tij from the spatial configuration to the intermediate one we can write the following hyperelastic constitutive relation, using the elastic Hencky strain tensor ('^, = l n ( / l f ^ ) ) [ 1 8 ] , TAB = [!,^c*('^a(>)]^g T =a

: 'H

(6a) (6b)

At each point of the spatial configuration the stress tensor has to fulfil the relations (5), (6a) and (6b); also the velocity field has to fulfil the Principle of Virtual Work [20],

/

^aij 8dij ^ dv

'RiSvi

— hp'cdv DtJ "^

=0

(8)

and using Reynolds transport theorem [16] we get.

We can decompose lyp into a symmetric component (dyp) and an anti-symmetric one (coyp); for isotropic elasticity we assume cOyp = 0 [18]. Hence [18],

u =n

In 9^-^, at time t, we define a variable ^c such that the spatial configuration of the body is the locus of the set of points that have ^c > 0. If we assume a trial distribution of ^c we can use Eqs. (1-7) to determine the velocity field ^i; (for the points with ^c < 0 we consider "small" elasticity constants and a "small" viscosity, as compared with the points where actual material is present). Defining ^c as "pseudo-concentration per unit mass" we can postulate the conservation of ^c in a control volume V

(7)

d'p'c dt

+ v_-(;p'c'v) = o.

(9)

For a stationary process, and considering also mass conservation, we get. 'v'V'c

= 0.

(10)

Please notice that being the material elasto-plastic, the flow is not incompressible and therefore, incompressibility was not invoked for deriving the above equation. A new ^c-distribution is determined using Eq. (10) and afterwards the velocity field is updated. The iteration loop is followed until at two successive iterations the ^c- and ^i;-distributions are coincident, within prescribed tolerances.

4. Conclusions A new soUd mechanics formulation was developed for the modeling of stationary elasto-plastic metal forming processes. The new formulation is based on: • An Eulerian description of motion implemented via the pseudo-concentrations technique. • A sound description of finite strain elasto-plastic deformation processes, based on the multiplicative decomposition of the deformation gradient and on a hyperelastic constitutive equation for the elastic part. In a forthcoming paper we will discuss the finite element implementation of the proposed Eulerian formulation.

158

E.N. Dvorkin, D. Demarco /First MIT Conference on Computational Fluid and Solid Mechanics

References [1] Dvorkin EN, Petocz EG. An effective technique for modelling 2D metal forming processes using an Eulerian formulation. Eng Comput 1993;10:323-336. [2] Dvorkin EN, Cavaliere MA, Goldschmit MB. A three field element via augmented Lagrangian for modelling bulk metal forming processes. Comput Mech 1995;17;2-9, [3] Dvorkin EN, Goldschmit MB, Cavaliere MA, Amenta PM. On the modelling of bulk metal forming processes. In: Proc. Second ECCOMAS (European Community on Computational Methods in Applied Sciences) Conference on Numerical Methods in Engineering. Wiley 1996. [4] Dvorkin EN, Goldschmit MB, Cavaliere MA, Amenta MP, Marini O, Stroppiana W. 2D finite element parametric studies of the flat rolling process. J Mater Process Technol 1997;68:99-107. [5] Cavaliere MA, Goldschmit MB, Dvorkin EN. 3D modeling of bulk metal forming processes via the flow formulation and the pseudo-concentrations technique. In: Owen DRJ et al (Eds), Proceedings Fifth Int. Conf. on Computational Plasticity. CIMNE, 1997. [6] Dvorkin EN, Cavaliere MA, Goldschmit MB, Amenta PM. On the modeling of steel product rolling processes. Int J Forming Process (ESAFORM) 1998; 1:211-242. [7] Dvorkin EN, Cavaliere MA, Zielonka MG, Goldschmit MB. New developments for the modeling of metal rolling processes. In: Wunderlich W et al. (Eds), Proceedings European Conference on Computational Mechanics, Munich, 1999. [8] Zienkiewicz OC, Jain PC, Onate E. Flow of solids during forming and extrusion: some aspects of numerical solutions. Int J Solid Struct 1977;14:15-28. [9] Thompson E. Use of the pseudo-concentrations to follow

[10]

[11]

[12] [13]

[14]

[15]

[16] [17] [18]

[19] [20]

creeping viscous flows during transient analysis. Int J Numer Methods Fluids 1986;6:749-761. Thompson E, Smelser RE. Transient analysis of forging operations by the pseudo-concentrations method. Int J Numer Methods Eng 1988;25:177-189. Lee EH, Liu DT. Finite strain elastic-plastic theory with application to plane-wave analysis. J Appl Phys 1967;38:1727. Lee EH. Elastic plastic deformation at finite strain. J Appl Mech 1969;36:1-6. Simo JC, Ortiz M. A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput Methods Appl Mech Eng 1985;49:221-245. Simo JC. A framework for finite strain elasto plasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I: Continuum formulation. Comput Methods Appl Mech Eng 1988;66:199-219. Simo JC. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: Computational aspects. Comput Methods Appl Mech Eng 1988;68:1-31. Malvern LE. Introduction to the Mechanics of a Continuous Medium. Englewood Cliffs, NJ: Prentice-Hall, 1969. Marsden JE, Hughes JR. Mathematical Foundations of Elasticity. Englewood Cliffs, NJ: Prentice-Hall, 1983. Dvorkin EN, Pantuso D, Repetto EA. A Finite element formulation for finite strain elasto-plastic analysis based on mixed interpolation of tensorial components. Comput Methods Appl Mech Eng 1994;114:35-54. Perzyna P. Fundamental problems in viscoplasticity. Advances in Applied Mechanics, vol 9. Academic Press, 1966. Bathe K-J. Finite Element Procedures. Englewood Cliffs, NJ: Prentice Hall, 1996.

159

Effects of internal/external pressure on the global buckling of pipelines Eduardo N. Dvorkin, Rita G. Toscano * Center for Industrial Research, FUDETEC, Av. Cordoba 320, 1054, Buenos Aires, Argentina

Abstract The global buckling (Euler buckling) of slender cylindrical pipes under internal/external pressure and axial compression is analyzed. For perfectly straight elastic pipes an approximate analytical expression for the bifurcation load is developed. For constructing the nonlinear paths of imperfect (non straight) elasto-plastic pipes a finite element model is developed. It is demonstrated that the limit loads evaluated via the nonlinear paths tend to the approximate analytical bifurcation loads when these limit loads are inside the elastic range and the imperfections size tends to zero. Keywords: Internal pressure; External pressure; Axial compression; Euler buckling; Pipeline

1. Introduction

T = C + kpi

When a straight pipe under axial compression and internal (external) pressure is slightly perturbed from its straight configuration there is a resultant force, coming from the net internal (external) pressure, that tends to enlarge (diminish) the curvature of the pipe axis. Hence, for a straight pipe under axial compression, if the internal pressure is higher than the external one, there is a destabilizing effect due to the resultant pressure load and therefore, the pipe Euler buckling load is lower than the Euler buckling load for the same pipe but under equilibrated internal/external pressures; on the other hand when the external pressure is higher than the internal one the resultant pressure load has a stabilizing effect and therefore the pipe Euler buckling load is higher than the Euler buckling load for the same pipe but under equilibrated internal/external pressures. The analysis of the buckling load of slender cylindrical pipes under the above described loading is important in many technological applications; for example, the design of pipelines. In Fig. 1 we present a simple case, for which the axial compressive load (T) has a constant part (C) and a part proportional to the internal pressure (p/). That is to say.

where ^ is a constant depending on the particular application. In the second section of this paper we develop an approximate analytical expression for calculating the Euler buckhng load for elastic perfectly straight cylindrical pipes (bifurcation limit load) and in the third section we develop a finite element model to determine the equilibrium paths of imperfect (non straight) elasto-plastic cylindrical pipes. From the analysis of the nonlinear equilibrium paths it is possible to determine the limit loads of pipes under axial compression and internal/external pressure. Of course, this limit loads depend on the pipe imperfections; however, we show via numerical examples that, for the cases in which the bifurcation limit loads are inside the elastic range, the pipe limit loads tend to the bifurcation limit loads when the imperfections size tends to zero.

* Corresponding author. Tel.: +54 (3489) 435-302; Fax: -F54 (3489) 435-310; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(1)

2. Elastic buckling of perfect cylindrical pipes 2.1. Internal pressure In Fig. 1 we represent a perfectly straight slender cylindrical pipe, in equihbrium under an axial compressive load and internal pressure; let us assume that we perturb the straight equilibrium configuration getting an infinitely close

160

E.N. Dvorkin, R.G. Toscano /First MIT Conference on Computational Fluid and Solid Mechanics I T=c + k Pj

1^^^ p^^ ^j^-^ length has horizontal and vertical components that in our case {v'{x) <^\) are, qh{x) = q{x) cos [v'{x)],

qy{x) = q(x) sin [v\x)] . (5)

Using a series expansion of the trigonometric functions and neglecting higher order terms, we get:

qh(x) = -pi7Trfv\x),

qv(x)=0.

(6)

To analyze the equilibrium of the perturbed configuration, being this an elastic problem, we use the Principle of Minimum Potential Energy [1,2]. When only conservative loads are acting on the pipe, equilibrium is fulfilled if, in the perturbed configuration. (7)

sn = 0 where 77 is the potential energy,

n = u -V

(8)

U: elastic energy stored in the pipe material, V: potential of the external conservative loads. In our case we have to consider the displacement dependent loads (non-conservative) given by Eq. (6), therefore [3]: L

8{U Fig. 1. Cylindrical pipe under internal pressure and axial compression. configuration defined by the transversal displacement, v{x), of the points on the pipe axis. If for some loading level, defined by /?, and by Eq. (1), this perturbed configuration is in equilibrium we say that the load level is critical (buckling load) because a bifurcation of the equilibrium path, in the loads-displacements space, is possible. Due to the polar symmetry of the problem we consider that all the displacements v{x) are parallel to a plane. For a longitudinal fiber defined by the polar coordinates ix,r,0) (see Fig. 1) we have, for the case of small strains, (2)

£xx = -v'\x)rcosO

where e^^ is the axial strain and v"{x) = ^^^. On a differential pipe length, the resultant pressure force due to the pipe bending is normal to the bent axis direction (follower load) and its value is. q(x)dx = 2

COS 0(1 +£,.,)^/d0djc

(3)

where r, is the pipe inner radius.Using Eqs. (2) and (3) we get, q{x) = -piTtrfv'Xx)

(4)

which is the resulting force per unit length produced by the internal pressure acting on the deformed configuration. This

-V) - / qh^v (jc) djc

= 0

(9)

and [1], EI

f [v'\x)f dx,

(10a)

0 L

(10b) 0 L

I qh8v{x)dx=

j

0

0

—pi7Trfv"(x)8v{x)dx

(10c)

E: Young's modulus of the pipe material, /: inertia of the pipe section with respect to a diametral axis. Hence, introducing the above in Eq. (9) we get for the fulfillment of equilibrium, L

L

&]^^-j[v"(x)f<^x-'^j[v'(x)]'Ax

+ PiTzrf j v"(x)8v(x) djc = 0.

(11)

We search for an approximate solution of the above equation using the Ritz Method [1], therefore we try as an approximate solution, . nnx ^_^

E

n=l,2,...

a«sm-^.

(12)

161

E.N. Dvorkin, R.G. Toscano/First MIT Conference on Computational Fluid and Solid Mechanics

"P

An example of this case is the hydraulic testing of a pipe. In this case: C = 0, k = TT (r^ — rf). Hence, using Eq. (14b) we get,

rr

T"

_ Pier

F

Eljt

—

J- 2

2'

Obviously, if there are {n — I) intermediate supports we have,

cL"

n^EIn L^r]

H

tl

2.2. External

pressure

For the cases in which the pipe is submitted to external pressure we rewrite Eq. (6) as.

i

qhM

Hence, after some algebra we get for the equilibrium of the perturbed configuration,

Fig. 2. Simply supported pipe open on both ends under internal pressure.

L

dx 0

Introducing the proposed approximate solution in Eq. (11) and taking into account that the an are arbitrary constants we get for equilibrium, £ / n 4V^ 4

2^2 Tn^Tt

- Pin

an=0

n =

l,2,.. (13)

The above equations have two possible solution sets: • Un = 0 ; which corresponds to the unperturbed straight configuration. • [^T^ - ^ - Pi^f^] = 0; which corresponds to an equilibrium configuration different from the straight one. The second solution gives the location of the bifurcation point (critical loading), Tcr + PicrTCrf

Ccr -^kpicr

-\-

=

n^ElTT^ L2

PicrT^rf

(15)

qy(x) = 0.

= Pe7rrfv'\x),

(14a) (14b)

It is interesting to realize that the above equations predict that there is a critical (buckling) pressure also if there is no axial compression {T = 0) and even if there is axial tension on the pipe (T < 0). Let us consider the following cases: • Simply supported pipe, closed on both ends, under internal pressure. In this case, C = 0 and k = —Jtrf; hence, from Eq. (14b) it is obvious that the only possible solution is the straight configuration and no bifurcation is possible. • Simply supported pipe, open on both ends, under internal pressure (Fig. 2).

0 L

— PeTtrf j v'\x) 8v(x)dx

(16)

= 0

using as an approximation for the equilibrium configuration the one written in Eq. (12), we finally get, ^2^3-|

2^2 Tn^Tt

Eln'^Tt'^

+ Perf

0

n =

l,2,... (17)

therefore, for the nontrivial solution, PecrTtr;

o

=

n^EIic^

(18a)

L2

n^ElTt^ C c r H~ K^Pecr

Pecr^^i

—

(18b)

From the above equations it is obvious that the external pressure has a stabilizing effect on the pipe; that is to say, the axial compressive load that makes the pipe buckle is higher than the Euler load of the pipe under equilibrated internal/external pressures. Let us consider the following case: • Simply supported pipe, closed on both ends, under external pressure. For this case C = 0 and k = nr^ therefore from Eq. (18b) we get, Peer —

EiTt

LHrj - rf)

and if the pipe has (n ~ I) intermediate supports, n^EIn Peer — TT7~7

2\"*

162

E.N. Dvorkin, R.G. Toscano /First MIT Conference on Computational Fluid and Solid Mechanics

Comparing this result with the one corresponding to the pipe under internal pressure it is obvious that the pipe under external pressure can withstand a higher pressure without reaching the bifurcation load; hence, it is obvious the stabilizing effect of the external pressure.

3. Nonlinear equilibrium paths for non-straight elasto-plastic cylindrical pipes An actual pipe is not perfectly straight, and its random imperfections will have a projection on the buckling mode of the perfect pipe; hence, when analyzing the equilibrium path of a non-perfect pipe we shall encounter a limit point rather than a bifurcation point [4]. The load level of this limit point shall depend on the pipe imperfections, will be lower than the bifurcation load of the perfect pipe and will tend to this value when the imperfections size tends to zero. In order to analyze the nonlinear equilibrium paths of imperfect pipes we developed a finite element model using the general purpose finite element code ADINA [5]. Some basic features of the developed finite element model are: • The pipe behavior is modelled using Hermitian (Bernoulli) beam elements [6]. • The pipe model is developed using an Updated Lagrangian formulation with an elasto-plastic (associated von Mises) material model (finite displacements and rotations but infinitesimal strains) [6]. • Acting on the beam elements we consider a conservative load ( r ) and a deformation dependent load normal to

the pipe axis, that for the case of internal pressure is (see Eq. (6)), qh =

-piiTr^[v"{x)+^\x)]

where f (x) is the initial imperfection of the pipe axis. We simply calculate, in our finite element implementation,the second derivatives using a finite differences scheme. To provide a numerical example, we use the finite element model to analyze the following case: Pipe outside diameter Pipe wall thickness Pipe length Intermediate grips Pipe yield strength Hardening modulus

60.3 mm 3.9 mm 12,200 mm 4 38.70 kg/mm^ 0.0

under the loading defined by an internal pressure and,

C = 0,k =

n{rl-r}).

3.1. No clearance between the pipe and the grip We consider the following initial imperfection for the pipe axis, ^{x) = a 0 . 2

L

.

11

/57Tx\ I

looo^'U ;

(19)

which is obviously zero at the grips and is coincident with the first buckling mode predicted using the Ritz method (Eq. (12)). In Fig. 3 we plot the load-displacement equilibrium path for various values of a and in the same graph we plot the bifurcation limit load obtained using Eq. (14b).

Lateral displacement at the tube center [mm]

Fig. 3. Grips with no clearance. Load-displacement curves.

E.N. Dvorkin, R.G. Toscano/First MIT Conference on Computational Fluid and Solid Mechanics

]

163

Bifurcation limit load : 3.37 kg/mm2

f7 E

I

-•- Case with clearance at the grips - ^ Case with no clearance at the grips

V Lateral displacement at the tube center [mm]

Fig. 4. Clearance between grips and pipe body. Load-displacement curves.

We can verify from this figure that the limit load increases when the size of the imperfection (a) diminishes, and that it tends to the bifurcation limit load when a ^ 0. 3.2. Clearance between pipe and grips This is a more realistic case because, unless the grips are welded to the pipe body, there is usually some clearance between the pipe and the grips. We analyze the same case that was considered in the previous subsection but allowing for a clearance between the grip and the pipe body of 5 mm. We consider the following initial imperfection for the pipe axis, ^W = 0.2

( 5nx\ 1000

+ I 0.2-^ - 0 . 2 — 100 1000

)"(T)

(20)

and between the rigid grip and the pipe we introduce a contact condition. In Fig. 4 we plot the nonlinear equilibrium paths corresponding to the cases: • Clearance between grips and pipe body (initial imperfection as per Eq. (20)). • No clearance between grips and pipe body (initial imperfection as per Eq. (19) with a = 1.0). From the results plotted in Fig. 4 it is obvious that the only imperfection that has an influence on the pipe critical load is the imperfection that is coincident with the first pipe buckling mode.

4. Conclusions We derived an approximate analytical expression for calculating the Euler buckling load of a pipe under axial compression and internal/external pressure. This expression incorporates the destabilizing/stabilizing effect of the internal/external pressure. We constructed a finite element model to determine the nonlinear equilibrium paths, in the loads-displacements space, of imperfect (non-straight) elasto-plastic pipes. From the analysis of the nonlinear equilibrium paths it is possible to determine the limit loads of pipes under axial compression and internal/external pressure. Of course, these limit loads depend on the pipe imperfections; however, we showed via numerical examples that, for the cases in which the bifurcation limit loads are inside the elastic range, the pipe limit loads tend to the bifurcation limit loads when the imperfections size tends to zero.

Acknowledgements We gratefully acknowledge the financial support from SIDERCA (Campana, Argentina).

References [1] Hoff NJ. The Analysis of Structures. John Wiley and Sons, New York, NY: 1956. [2] Washizu K. Variational Methods in Elasticity and Plasticity. New York, NY: Pergamon Press, 1982.

164

E.N. Dvorkin, R.G. Toscano/First MIT Conference on Computational Fluid and Solid Mechanics

[3] Crandall SH, Kamopp DC, Kurtz EF, Pridmore-Brown DC. Dynamics of Mechanical and Electromechanical Systems. McGraw-Hill, New York, NY: 1968. [4] Brush DO, Almroth BO. Buckhng of Bars, Plates and Shells. McGraw-Hill, New York, NY. 1975.

[5] ADINA R&D. The ADINA System. Watertown, MA, USA. [6] Bathe KJ. Finite Element Procedures. Englewood CUffs, NJ: Prentice Hall, 1996.

165

On a new segment-to-segment contact algorithm Nagi El-Abbasi, Klaus-Jiirgen Bathe *

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Abstract A new contact algorithm is presented which satisfies both stability and the contact patch test. The segment-to-segment algorithm involves a contact pressure interpolation and an accurate integration of the contact constraints over the surfaces of the contacting bodies. Numerical integration is carried out over sub-segments based on the element topologies of both contacting surfaces. The algorithm is applicable to both linear and quadratic element surface interpolations. Keywords: Contact algorithm; Finite element solution; Stability; Patch test

1. Introduction

mization problem

To guarantee stability and optimal convergence, contact formulations, like other mixed formulations, should satisfy an ellipticity and an inf-sup condition [1,2]. Furthermore, the contact algorithm should satisfy a contact patch condition, which describes its ability to represent a state of constant normal traction between two flexible contacting bodies. However, a review of the literature indicates that current contact algorithms do not satisfy both, the stabiHty and contact patch conditions [3]. In this paper, we present a new contact algorithm, which satisfies both requirements. We classify the algorithm as a segment-to-segment procedure since it involves an accurate integration of the contact constraints over the surfaces of the contacting bodies, not just using values at the nodes. We describe the solution approach using 2D conditions but the theory is directly applicable to 3D conditions as well.

min[nA(v) + n5(v)]

(1)

where v represents any admissible displacement. Hi denotes the total potential of body I not accounting for contact effects, and K represents the set of functions satisfying the no-penetration contact constraint K = {\\\eV,

g(\) > 0 on F d

(2)

where g is the gap, V = {\\\eH\

v^OonFz)}

(3)

and H^ is the usual Sobolev space. Using a Lagrange multiplier to enforce the contact constraint, and assuming contact, the minimization problem is

Body A 2. Contact formulation Consider a system consisting of two bodies in contact (Fig. 1). Assuming infinitesimally small displacements, a linear elastic material and frictionless conditions, the contact problem can be expressed as a constrained miniBodyB * Corresponding author. Tel: +1 (617) 253-6645; Fax: +1 (617) 253-2275; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Fig. 1. Two bodies in contact.

166

N. El-Abbasi, K.J. Bathe/First MIT Conference on Computational Fluid and Solid Mechanics

r ('+!)*

sCV

Target / surface T T I

-•-

\ X> 1

• Target node o Contactornode

Contactor surface F^ f

X Integration point D Target point

Contact pressure distribution Fig. 2. Schematic of new contact algorithm.

converted to an unconstrained saddle point problem involving the following functional

We then assume that the discretized Lagrange multiplier space Q/j is

nz.(v, X) = n^Cv) + OfiCv) + nc(v, x)

Qh = [x, \ h e H-'^\

(4)

where nc(v,A) = y ^ g W d F c

(5)

and X is the contact pressure which can only be zero or positive. The variational form of the contact problem can be obtained by extremizing Eq. (4) with respect to the field variables v and X. Note that the constraint function method can be used to solve the contact problem without the need for distinguishing between active and inactive contact constraints [1].

3. New contact algorithm The algorithm involves a master-slave approach. One of the surfaces, Fc, is assumed to be the contactor, and the other, F j , is the target as shown in Fig. 2. The contact constraint is evaluated at the integration points (not necessarily the nodes) along Fc. Let the superscript / denote an integration point. For a point with coordinates x'^^^, the displacement v'^ can be interpolated from the nodal displacements on Fc as follows:

=j:h'^<

(6)

where h'^ is the interpolation function (evaluated at point /) relating the displacement of the contactor point to the displacements of the contactor nodes. For each integration point on the contactor surface Fc the displacement of the target point on Fr is interpolated as follows:

y^ = J2hH

(7)

X,\i^eP/{k)]

(8)

where P/ denotes a polynomial of degree j , with Ocontinuity between elements, and ^ is a reference contact segment. The polynomial degree j must be less than or equal to that of the element interpolation, and the segments K are defined on Fc . Thus, the Lagrange multiplier value at integration point / is obtained as follows: A^^ = ^ / f ^ X

(9)

where the A^ are the independent (usually nodal) multipliers on Fc and the interpolation function values /f^^ depend on the polynomial degree and inter-element continuity of the contact pressure field. The contact integral of Eq. (5) is then converted to a summation over the integration points (see Fig. 2) n c = J2^c^'^(^c

- V;) • N' + g'o']

(10)

where w' is the integration weight factor, N' is the unit normal vector to measure the gap, and g^^ is the initial gap width; all given at integration point /. It is important that we select a numerical quadrature rule that accurately evaluates the contact integral. This expression is piecewise continuous with possible discontinuities occurring at the nodes of either contact surfaces. Accordingly, any integration scheme involving integration points that are dictated by only one of the two surfaces cannot exactly evaluate Eq. (5) regardless of the number of integration points used. If, however, the integration intervals are based on 'sub-segments' corresponding to any two neighboring nodes regardless of their surface of origin, an exact evaluation is possible. This accurate integration feature enables the algorithm to pass the patch test for both linear and quadratic elements.

A^. El-Abhasi, KJ. Bathe/First MIT Conference on Computational Fluid and Solid Mechanics O Contactor node

4

4 ^eO-^<

•

Target node

4—^ i ><—r-^—K-O

x

4

167

Integration point

\.j<

®

(a)

4

A.'p

4

A.J'

4

^—^

(b)

Fig. 3. Location of integration points based on: (a) Gaussian quadrature, and (b) trapezoidal rule. Hence, the algorithm involves two main steps. In the first, the sub-segment boundaries are determined by projecting the nodes of the target surface onto the contactor surface (only the edge nodes need to be projected for quadratic and higher order elements). In the second step, the contact expression on each sub-segment is integrated using Gaussian or Newton-Cotes integration rules as shown in Fig. 3.

4. Stability and patch conditions for contact algorithms Contact algorithms should satisfy the stability and patch conditions. Stability is represented by an ellipticity and an inf-sup condition. Satisfying the ellipticity condition depends on the use of appropriate finite elements and boundary conditions, not on the contact formulation. The inf-sup condition for contact problems can be represented as follows [3] inf sup

frc^hg(yh)drc -i/2,r

V/j

>P>0

(11)

The inf-sup condition is satisfied if the constant P is independent of the element size. The stability of the new contact algorithm has been assessed numerically, and it was found that with linear elements it is best to use a Hnear continuous pressure interpolation, whereas with quadratic

elements the quadratic continuous pressure interpolation is optimal [3]. As mentioned above, the patch test is also passed by the algorithm [3].

5. Conclusions A new segment-to-segment contact algorithm was developed which accurately evaluates the contact constraints between the contacting bodies. The algorithm provides optimal performance by satisfying both the stability and the contact patch conditions, using linear or quadratic element displacement interpolations. While the theory given here is directly applicable to 3D contact problems, the actual detailed solution algorithm needs still to be developed.

References [1] Bathe KJ. Finite Element Procedures. Englewood Cliffs, NJ: Prentice Hall, 1996. [2] Brezzi F, Bathe KJ. A discourse on the stability conditions for mixed finite element formulations. Comput Methods Appl Mech Eng 1990;82:27-57. [3] El-Abbasi N, Bathe KJ. Stability and patch test performance of contact discretizations. Comput Struct, submitted.

168

Modeling 2D contact surfaces using cubic splines N.El-Abbasi,S.A.Meguid* Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, ON, M5S 3G8, Canada

Abstract A new algorithm for representing 2D contact surfaces is developed and implemented, based on C^-continous cubic splines. The new surface interpolation does not influence the element calculations, and possesses a local support characteristic, which simplifies the representation of the contact constraints. Consequently, it can be easily implemented in standard FE codes. A numerical example is used to illustrate the advantages of smooth representation of contact surfaces. The results show a significant improvement in accuracy compared to traditional piecewise element-based surface interpolation. The predicted contact stresses are also less sensitive to the mismatch in the meshes of the different contacting bodies. Keywords: Contact; Cubic splines; Surface approximation; Lagrange multipliers; Splines; Ring compression; Smooth surfaces

1. Introduction Most finite element based contact formulations rely on the element interpolation functions to describe the contact surface and to impose the kinematic contact conditions. Consequently, the contact surface is defined as a sequence of lines (or curves) connecting the FE nodes with only C°-continuity. In this case, the normal vector is not uniquely defined at the nodes. Even when higher order elements, such as the 8- and 9-noded elements, are used the contact surface is still non-smooth at the exterior nodes. In cases involving contact with a rigid target, analytical surface profiles and spline interpolation functions have been used to describe the rigid surface and its normal vector [1]. This approach has resulted in significant improvement in the solution accuracy, especially in metal forming applications [2]. However, it has not been used to describe the surfaces involved in contact between flexible bodies since the analytical surface profiles that describe the initial geometry cannot be used to describe the deformed one. In this paper, we develop an algorithm for smooth contact surface interpolation (Fig. 1). The contact surfaces * Corresponding author. Tel/Fax: +1 (416) 978-5741; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

are described using cubic splines passing through the FE nodes and possessing C^-continuity. The normal vector associated with the resulting surface profile is uniquely defined at all points. This interpolation is applicable to both rigid and flexible bodies and it can be easily implemented in finite element codes.

2. Spline interpolation Fig. 1 shows a parametric cubic spline segment connecting two FE nodes. The interpolation function passes through the end points po and p3, while the intermediate points pi and p2 dictate the shape of the curve. The end

Ns Normal to spline surface N- Normal to element a Spline surface Element surface

Fig. 1. Spline based surface interpolation and normal vectors

N. El-Abbasi, S.A. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics control points po and ps are located at the finite element nodes, while the intermediate points dictate the smoothness of the curve. Their location is selected based on the specific spline form adopted. C^ is the highest degree of continuity that is achievable using cubic splines. In this case, the location of the intermediate control points for all the spline segments are coupled. They can be obtained by solving a predominantly tri-diagonal matrix expressing the continuity equations [3]. When the location of the FE nodes changes, the matrix must be solved again for the new location of the intermediate points. The overhead associated with this process does not offset the advantages of second order continuity. By employing C^-continuity, however, simple and fast interpolation functions can be constructed where the intermediate control points can be obtained without resorting to matrix solution. In this case, the location of the intermediate control points is governed by a few nodes adjacent to the segment. This property is known as local support. Interpolation functions can be constructed to satisfy a prescribed tangential vector, a prescribed tangential direction or a prescribed normal direction [3]. However, these vectors are generally not available in standard FE meshes. Overhauser splines offer an alternative approach that ensures C^-continuity without requiring prescribed tangential or normal vectors [4]. Accordingly, they are the most suitable interpolation form for finite element contact problem involving flexible bodies. For each segment a, the spline curve can be considered as a linear blend of two parabolas q"~^ and q"', where each parabola passes through the two surface nodes pg and p" as well as a neighboring surface node (one from each side) x'^ (w) = {\-u)

0 < M < 1 (1)

q"-^ {u + \) + uct{u),

The spUne curve can be expressed directly in terms of the coordinates of the two nodal points defining the segment and their two adjacent surface nodes: — ^w' -\- u^ — \u

x«(„) = [p«-

pf

lu^ - fw^ + l \u'

0< w < 1

\u^ (2)

Two modified interpolation function are applied for spline segment at sharp comers and for those that intersect a lines of symmetry.

3. Contact search The use of high order polynomial functions to represent the contact surface can slow down the contact search pro-

169

cedure. To overcome this, the contact search is divided into two stages. The purpose of the first stage is to obtain a quick estimate of the proximity of a master node to a specific spline segment. In this stage, interference is checked between the master node and the control polygon of the spline segment. According to the convex hull property, the spline curve cannot exceed the geometric bounds of the control polygon [3]. If the master node is inside the search region, an accurate iterative contact check is performed in the second stage of the search. In this stage, the exact target point and gap/penetration are determined.

4. Solution strategy The contact can generally be expressed in the form of a variational inequaUty [5]. In this work, the active contact constraints are imposed using Lagrange multipliers. UnUke penalty-based methods, Lagrange multipliers satisfy the contact constraints exactly without any interpenetration between the contacting bodies. The solution to the saddle-point problem can be expressed in matrix form as: K

C

C^

0

(3)

where the C matrix is the assembled constraint matrix, and G is the gap vector. The active constraint set is modified after each iteration step and a full contact search is performed. More details on the solution algorithm are provided in Ref. [6].

5. Numerical example One numerical example was selected to assess the accuracy of the newly developed smooth surface interpolation technique. It involves a ring compressed between two beams. The following dimensions were selected (Fig. 2(a)): L = Vd, h = t — \ and /? = 8. In view of the symmetry condition, only one quarter of the model was discretized (Fig. 2(b)). An incremental vertical displacement da = 0 . 2 was applied to the symmetry surface of the ring. The beam was modeled using 40 x 5 four-noded elements as shown in Fig. 2(b), while a variable mesh of A/^ x 5 elements was used for the ring. Fig. 3 shows the contact stress distribution when the applied displacement is da = 2.8. The contact stresses were normahzed by the bending stiffness of the beam. The results reveal that using splines (A^ = 20 and 'N = 40) leads to a uniform contact stress distribution. The element interpolation results in unrealistic numerical stress oscillations. For A^ = 20, these oscillations lead to intermediate regions of non-contact between the beam and the ring. A higher number of elements results in more uniform contact stress profiles. However, even when N — 60,

170

N. El-Abbasi, 5.A. Meguid / First MIT Conference on Computational Fluid and Solid Mechanics

PI

B

PI

(a)

(b)

Fig. 2. Compression of a ring between two beams: (a) schematic, and (b) FE mesh

«

0.3

"S

0.2

Spline (N=20)

Spline (N=40)

Element (N=20)

Element (N=40)

Element (N=60)

o

0.0 0.05

0.1

0.15

0.2

through the finite element nodes to provide an accurate description of the contact surfaces. The selected splines were shown to possess a local support characteristic, which simplifies the representation of the contact constraints. The selected numerical example illustrates the advantages of the newly developed representation of contact surfaces. The results reveal a significant improvement in the prediction of contact stresses and contact area.

References

Contact length (x/L) Fig. 3. Contact stress distribution along ring for different ring mesh densities.

these stresses are still less accurate than those obtained using splines. Other numerical examples provided in Ref. [6] show that the predicted contact stresses are less sensitive to the mismatch in the meshes of the different contacting bodies.

6. Conclusions A new technique for interpolating the contact surface in 2D finite element problems was developed and implemented. Cubic splines with C^-continuity were interpolated

[1] Hansson E, Klarbring A. Rigid contact modelled by CAD surface. Eng Comput 1990;7:344-348. [2] Santos A, Makinouchi A. Contact strategies to deal with different tool descriptions in static explicit FEM for 3-D sheet-metal forming simulation. J Mater Proc Technol 1995;50:277-291. [3] Farin G. Curves and Surfaces for Computer-aided Geometric Design — A Practical Guide. Toronto: Academic Press, 1997. [4] Brewer JA, Anderson DC. Visual interaction with Overhauser curves and surfaces. Comput Graphics 1977;11:132137. [5] El-Abbasi N, Meguid SA. On the treatment of frictional contact in shell structures using variational inequalities. Int J Numer Methods Eng 1999;46:275-295. [6] El-Abbasi N, Meguid SA, Czekanski A. On the modelling of smooth contact surfaces using cubic splines. Int J Numer Methods Eng 2000, accepted.

171

Optimal triangular membrane elements with drilling freedoms C.A. Felippa * Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado, Boulder, CO 80309-0429, USA

Abstract The construction of optimal 3-node, 9-degrees of freedom triangular membrane elements with comer drilling freedoms is studied in some generahty. It is shown that all elements of this geometry and freedom configuration that pass the patch test can be generated through a template with six free parameters: one basic and five of higher order. The selection of optimal parameters that optimize in-plane bending behavior for arbitrary aspect ratios is shown to coincide with a triangle element published in 1991. A similar study isconducted for an optimal quadrilateral macroelement formed with four triangles. The macroelement assembly may possess internal degrees of freedom represented as the tangential displacement deviation at midpoints to further improve performance. Keywords: Finite element method; Membrane; Plane stress; Comer drilling degrees of freedom; Normal rotational freedom; Triangular element; Quadrilateral element; Shell element; Template; Free parameter; Macroelement; Optimal element

1. Summary The idea of including normal-rotation degrees of freedom at comer points of plane-stress finite elements — the so-called drilling freedoms — is an old one. The main motivations behind this idea are: (1) To improve the element performance while avoiding the use of midpoint degrees of freedom. Midpoint nodes have lower valency than corner nodes, demand extra effort in mesh definition and generation, do not fit the data stmctures of standard commercial FEM codes, and can cause modeling difficulties in nonlinear analysis and dynamics. (2) To solve the 'normal rotation problem' of smooth shells analyzed with finite element programs that carry six degrees of freedom per node. This is done by using the triangular element with drilling degrees of freedom as the membrane component of a facet triangular shell element with 18 degrees if freedom. (3) To simplify the modeling of connections between plates, shells and beams, as well as the treatment of junctures in shells and folded plates.

* Corresponding author. Tel: +1 (303) 492-6547; Fax: +1 (303) 492-4990; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Many efforts to develop membrane elements with drilling freedoms were made during the period 1964-1975 with inconclusive results. A summary of this early work is given in the Introduction of an article by Bergan and Felippa [1], where it is observed that Irons and Ahmadin in their 1980 book [2] had dismissed the task as hopeless. In fact, the subject laid largely dormant throughout the 1970s. It was revived in various publications [1,3-8] that appeared in the mid and late 1980s, and which present several solutions to this challenge. A three-part paper pubhshed in 1992 [9-11] presented a triangle that performs optimally as regards inplane bending for rectangular mesh units of arbitrary aspect ratio. In those papers, elements was derived with two different techniques: the Extended Free Formulation, and the Assumed Natural Deviatoric Strain formulation. Both formulations involved free parameters. The optimal elements provided by both formulations coalesced. The present paper studies the results from the point of view of finite element templates [12] and confirms that the 1992 optimal element is indeed unique for an individual triangle. The present study goes beyond that point in leaving tangential hierarchical midpoint freedoms in the triangle template. These freedoms are troublesome for individual triangles since they conflict with data structures of most general-purpose FEM codes. They are useful, however.

172

C.A. Felippa /First MIT Conference on Computational Fluid and Solid Mechanics

in the construction of quadrilateral macroelements, where tangential freedoms on internal edges can be eliminated by static condensation.

References [1] Bergan PG, Felippa CA. A triangular membrane element with rotational degrees of freedom. Comput Methods Appl Mech Eng 1985;50:25-69. [2] Irons BM, Ahmad S. Techniques of Finite Elements. Chichester: Ellis Horwood, 1980. [3] Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Comput Struct 1984;19:1-8. [4] Cook RD. On the Allman triangle and a related quadrilateral element. Comput Struct 1986;22:1065-1067. [5] Cook RD. A plane hybrid element with rotational D.O.F. and adjustable stiffness. Int J Numer Methods Eng 1987;24:1499-1508. [6] Bergan PG, Felippa CA. Efficient implementation of a

[7]

[8]

[9]

[10]

[11]

[12]

triangular membrane element with drilling freedoms. In: Hughes TJR, Hinton E (Eds), Finite Element Handbook series. Swansea: Pineridge Press, 1986, pp. 139-152. Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Int J Numer Methods Eng 1988;26:2645-2655. MacNeal RF, Harder RL. A refined four-noded membrane element with rotational degrees of freedom. Comput Struct 1988;28:75-88. Alvin K, de la Fuente HM, Haugen B, Felippa CA. Membrane triangles with comer drilling freedoms: I. The EFF element. Finite Elem Anal Des 1992;12:163-187. Felippa CA, Militello C. Membrane triangles with comer drilling freedoms: II. The ANDES element. Finite Elem Anal Des 1992;12:189-201. Felippa CA, Alexander S. Membrane triangles with corner drilling freedoms: III. Implementation and performance evaluation. Finite Elem Anal Des 1992;12:203-239. Felippa CA. Recent advances in finite element templates. In: Topping BHV (Ed), Computational Mechanics for the Twenty-First Century. Saxe Cobum Publications: 2000, pp. 71-98.

173

Numerical analysis of a sliding viscoelastic contact problem with wear J.R. Fernandez-Garcia^'*, M. Sofonea^, J.M. Viano^ ^ Departamento de Matemdtica Aplicada, Universidade de Santiago de Compostela, Facultade de Matemdticas, Campus Sur, 15706 Santiago de Compostela, Spain ^ Lahoratoire de Theorie des Systemes, Universite de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan, France

Abstract We consider a mathematical model which describes the sliding frictional contact with wear between a viscoelastic body and a rigid moving foundation. The process is quasistatic and the wear is modeled with a version of Archard's law. We present a summary of our recent results on the variational and numerical analysis of the model. Finally, we provide numerical results in the study of a one-dimensional test problem. Keywords: Viscoelasticity; SUding contact; Wear; Archard's law; Finite elements; Error estimates; Numerical simulations

1. Introduction Wear is one of the plagues which reduce the lifetime of modem machine elements. It represents the unwanted removal of materials from surfaces of contacting bodies occurring in relative motion. Wear arises when a hard rough surface slides against a softer surface, digs into it, and its asperities plough a series of grooves. Generally, a mathematical theory of friction and wear should be a generalization of experimental facts and it must be in agreement with the laws of thermodynamics of irreversible processes. A general model of quasistatic frictional contact with wear between deformable bodies was derived in Stromberg et al. [6] from thermodynamic considerations. This model was used in various papers (see, for example, Rochdi et al. [4,5]), where existence and uniqueness results of weak solutions have been proved. The present paper is devoted to the study of a quasistatic problem of sliding contact with wear. We model the process as in Stromberg et al. [6] by introducing the wear function which measures the wear of the contact surface and which satisfies Archard's law. The variational analysis of the model was provided in Ciulcu et al. [1], while the numerical analysis was performed in Fernandez-Garcia et al. [3]. Here, we summarize our main results and provide * Corresponding author. Tel.: -^34 (981) 563100; Fax: +34 (981) 597054; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

numerical simulation in the study of a one-dimensional test problem. The paper is organized as follows. In Section 2, we present the variational formulation of the mechanical problem and state an existence and uniqueness result, which shows that under a smallness assumption on the given data, the mechanical problem has a unique weak solution. In Section 3, we analyze a fully discrete scheme, using finite elements with implicit discretization in time. We also derive error estimates and, under appropriate regularity assumptions on the exact solution, we obtain optimal order error estimates. Finally, in Section 4 we present numerical results.

2. The problem of sliding frictional contact with wear The physical setting is as follows. A viscoelastic body occupies the domain ^ C M"^ (^ = 1, 2, 3) with outer Lipschitz boundary F, divided into three disjoint measurable parts Fi, F2 and F3, such that measTi > 0. Let [0, T] be the time interval of interest. We suppose that the body is clamped on Fi x (0, J ) , surface tractions act on r2 x (0, T), and a volume force acts in ^ x (0, T). On F3 x (0, T) the body is in contact with a moving rigid foundation, which results in the wear of the contacting surface. We assume that there is only sliding contact, which is always maintained. The wear is modeled with a simplified version of

J.R. Fernandez-Garcia et al. /First MIT Conference on Computational Fluid and Solid Mechanics

174

Archard's law. Moreover, we assume that the tangential displacements on the contact surface vanish, the process is quasistatic and we use a Kelvin-Voigt viscoelastic constitutive law. With these assumptions, the variational formulation of the mechanical problem of sliding frictional contact with wear is the following one (see Ciulcu et al. [1]).

the normal depth of the material that is lost. Since the body is in bilateral contact with the foundation it follows that

Problem P. Find a displacement field u : [0, T] -^ V, and a stress field a \[0,T]^ Q such that

3. Fully discrete approximation

cf{t) = ^e{u(t))

+ (5e{u(t))

{a(t),e(v))Q + j{u{t),v)

=

Wt e [0, T], {f(t).v)^ Vr € V, r e [0,7],

w(0) = MoHere V and Q denote the spaces V = {v e H\Q.Y \ v = 0 on Ti, r, == 0 on r3}, Q = [a = (Oij) \ aij = ajj e L^(Q) i, j = l,d], {•,-)Q represents the inner product on Q and (•, ')Y denotes the inner product on V given by (u,v)v = {e{u),e(v))Q where e : V ^ Q is the deformation operator. The operators 21 and 0 , related to the constitutive law, are defined on Q with the range in 2 , and the dot above represents the derivative with respect to the time variable. The element / : [0, 7] ^ V represents the body forces and tractions and UQ e V is the initial displacement. Finally, j denotes the functional j( u, v) =

P\Hv\Vv dfl

VM, V £ V,

where ^ is a given function related to the velocity of the foundation and u^, v^ denote the normal traces of the elements u and v, respectively. Under reasonable assumptions on the constitutive functions it follows that 21 is a Lipschitz continuous strongly monotone operator on Q and 0 is a Lipschitz continuous operator on Q. Moreover, under appropriate regularity assumptions on the body forces and tractions, it follows that / G C([0, 7], V). The well-posedness of this problem is given by the following result. Theorem 1. Assume that p e L^CFs) and there exists P^ such that P(x) > yS* > 0 a.e. x e Vj,. Then, there exists Po > 0 which depends only on Q, Fi, F3 and 21 such that problem P has a unique solution {«, a} if

-w -f wov

(2)

on F^ X (0, 7).

Eq. (2) allows us to obtain the wear of the contact surface, when the displacement field u is known.

Following Femandez-Garcia et al. [3], we now consider a fully discrete approximation of problem P. To this end, let V^ c V and Q^ C Q be finite element spaces to approximate the spaces V and Q. Here /? > 0 is a discretization parameter. Let ^QH : G ^ G^ be the orthogonal projection operator defined through the relation ("^Q^q, q')Q = (q, q')Q

V^ e Q, q'e

Q\

To discretize in time, we consider a partition of the time interval [0, 7]: 0 = ro < fi < • • • < r/v = 7, we denote the step size kn = tn - ?„-i for n = 1, 2 , . . . A^ and let k = max„ kn be the maximal step size. For a sequence {^n]n=o^ we denote 8wn = (Wn — Wn-i)/kn.ln this section no summation is considered over the repeated index n and, everywhere in the sequel, c will denote positive constants which are independent on the parameters of discretization h and k. The fully discrete approximation method is based on the backward Euler scheme. It has the following form: Problem P*^ Find u^'' = {wf }lo ^ ^^ and a^^ = {af }„% C Q' such that: a f = q3^.2le(5Mf) + ^ g / , 0 e ( M f ) (erf, eiw'))^

a.e. t e (0, 7),

+ j ( 5 " f , w') = ( / „ w'h

Vu;^ e V\

Here «Q G V^ is an appropiate approximation of MQWe have the following existence and uniqueness result. Theorem 2. Under the assumptions of Theorem 1, if (1) holds, then problem P^^ has a unique solution.

a e

In practice, the fixed point algorithm used in the proof of Theorem 2 is directly applied. To solve the semilinear equality obtained, a penalty-duality algorithm is suggested (see Femandez-Cara et al. [2]). In the study of the discrete problems, we have the following result.

Now, we recall that in our model, the wear function is identified as an increase in gap in the normal direction between the body and the foundation or, equivalently, as

Theorem 3. Let {u, a) e C^([0, 7], V) x C([0, 7 ] , Q) be the solution of problem P, and let {wf, (xf l^^Lo C V^ x g^ denote the solution of fully discrete problem P^^. Assume the conditions stated in Theorem 2 and ii e L~(0, 7, V).

I^IL^CFJ) < Po-

Moreover, the solution satisfies u e C\[0,T],V), C([0, 7], Q).

(1)

175

J.R. Femdndez-Garcia et al /First MIT Conference on Computational Fluid and Solid Mechanics Then we have the following error estimate: m^a^x^(|(r„ - a f Ig + \Un - uf\v)

4. Numerical results

< c( \uo - M^IV

+ ^l«lL~(o,r,v) + max ( inf \Un

-w^\v)].

l'eVh

y

From Theorem 3, we derive the convergence of the fully discrete method. Corollary 4. Assume the conditions stated in Theorem 3. Assume moreover that the initial value UQ is chosen in such a way that \UQ

- UQ\V -> 0 as /? - ^ 0, "iv e H^{QY

1(7 - ^eO(T)le -> 0 as /^ ^ 0,

Ti = {0},

r 2 = 0,

r\V,

UQ{X)

- w f |v) ^

0 as /z, A: ^

0.

The following error estimate is obtained as in Corollary

Corollary 5. Let the assumption in the above corollary hold. Assume, moreover, that the initial value UQ is chosen in such a way that

o{xj)

< ch,

and there exists c > 0 such that

|(/-^eO(r)lG
Wv eVnH^(QY,

]

VTGg.

J

Then the following error estimate is obtained: m^ax^(|a„ - a f Ig + |M„ - uf\y)

P = 10-4 N . s/m,

< c(h + k).

(3)

Wx e (0, 1),

0.01 - 9900

A

9900 - 0.005 The exact solution of the above problem, called problem n Z ) , is given by

u(x, t)

inf \v-w^\v
6 e = e,

= A — -\-x

4.

-UQ\V

r 3 = {i},

VT G e.

l
|wo

lOs,

99Ae-'vdx,

Then the fully discrete method converges, i.e. |M„

T =

1


max {\(jn -
Q = (0, 1),

^e = lOOe,

and there exist c,a > 0 such that inf \v-w^\v

In order to verify the accuracy of the numerical method described in the above section, some numerical experiments have been done in the study of one-dimensional test problems. In this section, we resume the numerical results obtained, which exhibit the performance of the algorithm. The test problem P has been considered for the following data:

(4+^)^"' 99 = - — {Ax +

l)e-\

By using the discrete method described in the above section, we have implemented the numerical method in a standard workstation. In Fig. 1, the displacement fields for several time values {t = 0.5, 1, 2, 4, 8 s), calculated with parameters h = 0.01 and k = 0.01, are drawn. Also, the difference with exact solution (4) at these time values is shown. Moreover, in Fig. 2, we show the evolution in time of the displacement of the points x = 0.25, 0.5, 1 provided by the algorithm and the corresponding error with the exact solution. Exact error

Displacement field

0.4

0.5

Fig. 1. Problem TID: displacement field and exact error for different time values.

0.6

0.7

0.8

0.9

176

J.R. Ferndndez-Garcia et al. /First MIT Conference on Computational Fluid and Solid Mechanics Displacement field

0,5 k

x 10*

Exact error 25 1 x-0 5 XKO

""

0.45

s

~^

0,4 ^

0.35-^

-,

— x=0.25 - ~ x=0.5 „ x=1

-.

^' ^ ~^

3

" "

-

^\.,^

_^

"•

" • ^ ^ - - ^

^

^~^—^_ 0.1

0.2

0.3

0.4

0.5

0.6

~

^

-

0.7

"

i

-

-^

0.1-

on?i-_—

^ ^

~ ^ -.. _^

" ..^ "~

^ ^ - ^ ^ ^

0,15-

x=1 1

n "-^

0.25-

0.2 -

i

^^

f 0,3.

L

^ -.._ ^

~^^^^---^..._ ~^---^-^,. 0.1

0.9

- .

^-^^^

-_ 0.8

• "

0.2

0.3

0,4

0.5

0.6

0,7

0.8

0,9

1

Fig. 2. Problem 7 I D : evolution of displacements of points x = 0.25, 0.5, 1 and corresponding scaled exact error. Evolution of the wear function through the time

In Table 1, the exact error values for several discretization parameters k and h are shown. From here, asymptotic behaviour (3) is obtained with c = 0.13874157, independent of h and k. Finally, from Eq. (2), the wear function can be obtained. Its evolution through the time is shown in Fig. 3.

References [1] Ciulcu C, Hoarau-Mantel TH, Sofonea M. Viscoelastic sliding frictional contact problems with wear. Submitted. [2] Femandez-Cara E, Moreno C. Critical Point Approximation through exact regularization. Math Comput 1988;50:139153. [3] Femandez-Garcia JR, Sofonea M, Viano JM. Numerical analysis of a quasistatic viscoelastic sliding frictional contact problem with wear. Submitted. [4] Rochdi M, Shillor M, Sofonea M. A quasistatic viscoelastic contact problem with normal compliance and friction. J Elast 1998;51:105-126. [5] Rochdi M, Shillor M, Sofonea M. A Quasistatic contact problem with directional friction and damped response. Appl Anal 1998;68:409-422.

Fig. 3. Evolution of the wear function through the time.

[6] Stromberg N, Johansson L, Klarbring A. Derivation and analysis of a generalized standard model for contact friction and wear. Int J Solid Struct 1996;33:1817-1836.

Table 1 Exact error values for several discretization parameters

ki 0.1 0.05 0.025 0.01 0.005 0.0025 0.001

h -^

0.1

0.05

0.025

0.01

0.005

0.0025

0.001

0.026276 0.027474 0.028167 0.028594 0.028738 0.028809 0.028853

0.013411 0.013749 0.014079 0.014291 0.014363 0.014399 0.014421

0.007236 0.006910 0.007041 0.007145 0.007181 0.007199 0.007209

0.004078 0.002862 0.002822 0.002858 0.002872 0.002879 0.002883

0.003392 0.001593 0.001422 0.01421 0.001436 0.001439 0.001442

0.003198 0.001061 0.000703 0.000715 0.000718 0.000719 0.000721

0.003142 0.000855 0.000349 0.000287 0.000287 0.000287 0.000288

177

Numerical fracture mechanics in 3D by the symmetric boundary element method A. Frangi'^'*, G. Novati'', R. Springhetti^, M. Rovizzi'' ^Department of Structural Engineering, Politecnico of Milan, Milan, Italy ^Department of Mechanical and Structural Engineering, University ofTrento, Trento, Italy

Abstract Three-dimensional linear elastic fracture mechanics problems are addressed by means of the symmetric Galerkin Boundary Element Method (SGBEM). The technique is first shown to be efficient and accurate with reference to the stress intensity factors evaluations for a non-planar crack; subsequently it is utiUzed to simulate a propagation process for an elliptical crack within a finite body. Keywords: 3D linear fracture mechanics; Fracture propagation; Boundary element method

1. Introduction In the numerical modelling of linear elastic fracture mechanics problems, boundary element methods have distinct advantages over domain approaches, especially when cracks are directly represented as displacement discontinuity loci and the traction integral equation is employed to enforce static conditions on the crack itself. The displacement discontinuity method, the dual BEM and the symmetric Galerkin BEM (SGBEM) share the above features and permit single domain formulations for problems with single or multiple cracks embedded in finite bodies or in the infinite medium. At difference from the other two techniques, the SGBEM (see the review paper by Bonnet et al. [1]) is based on a variational (weak) version of the integral equations, thus entaiUng double integrations, and, through the adoption of a Galerkin discretization scheme, leads to a symmetric linear equation system. The evaluation of the double surface integrals in the singular cases represents probably the main obstacle which has hampered the application of the method in the 3D context. However, recent results obtained by applied mathematicians have led to innovative algorithms which are now being adopted by the engineering BE community and have served as a basis for the fracture-oriented implementation of the SGBEM in 3D recently presented by Frangi et al.

* Corresponding author. E-mail: [email protected] © 2001 PubHshed by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

[2]. The approach is here extended to deal with a simple example of fracture propagation, in order to explore the potentialities of the SGBEM in this domain. Compared with the finite element method, the SGBEM appears to be a very attractive tool to carry out incremental crack extension analysis for two reasons: (1) the required re-meshing work is greatly reduced; and (2) SIFs can be accurately evaluated through extrapolation from the displacement discontinuity field even for rather coarse meshes.

2. Numerical examples 2.1. Spherical-cap crack Let us consider a spherical-cap crack bounded by a circular front and subjected to a remote stress 0-3^3 (see Fig. 1); a is the radius of the spherical surface and 2a is the subtended angle. For this problem, numerical results in terms of SIFs are given in [4] for a given range of a. The analysis has been carried out for three values of a (a = 15°, 30°, 45°) and v = 0.3, using three meshes with 40, 112 and 240 elements on the spherical surface. Fig. 1 gives a planar representation of the actual meshes adopted for the spherical-cap crack, obtained by prescribing that the polar coordinate p equals a(j). Results in terms of SIFs are presented in Table 1 (quarter-points elements are used along the crack front and the SIFs are evaluated through extrapolation from the displacement discontinuity field).

178

A. Frangi et al /First MIT Conference on Computational Fluid and Solid Mechanics mesh 2

mesh A A A A A A

meshZ Fig. 1. Spherical-cap crack: loading conditions and meshes adopted.

Table 1 Spherical-cap crack: computed SIFs {K ^ = {llTi)\l7ta sin a ) for different values of the subtended angle 20?} Mesh 1 2 3

Ki/K^ (15°)

(15°)

(30°)

(30°)

(45°)

(45°)

0.964 0.966 0.966

0.263 0.266 0.267

0.845 0.849 0.851

0.520 0.525 0.527

0.655 0.662 0.665

0.769 0.774 0.776

2.2. Fatigue-growth of an elliptical-shaped

crack

Let us now consider a cylinder of length h and radius R containing an elliptical shaped crack of major semi-axis a and minor semi-axis b {b/a = 0.5, R/a = 10, h/R = 6), positioned in the middle and inclined at an angle y = 45° with respect to the horizontal plane (Fig. 2). The fatigue crack growth of the crack is analyzed by adopting the same criteria for incremental propagation as in [3]. For each

point along the front, propagation occurs in the plane perpendicular to the crack front itself, along the polar angle ^o^ ^0 tan — = 2

-2Kij (1)

Kieff = Kj -{- B\Kiji\ The crack front extension a{l) (l being a curvilinear coordinate running along the front) is described by means of the generalized Paris law: Aa

= CK:,,,

(2)

AN and Aa{i) is scaled so that, at each step, the maximum value Aa^ax is equal to a prescribed value. Material parameters are chosen as follows: E = 100, 000 MPa, v = 0.3, C = 1.5463 X 10-^^ m = 3.88, B = 1; a cyclic loading (J^^i^) is applied to the cylinder bases (0-3^3 ^^ax = 100 MPa, ^33 min = 0 MPa). Fig. 2 illustrates the configuration of the crack after the first propagation steps which compares well with the results presented by Mi [3].

References

Fig. 2. Elliptical crack: initial geometry and crack front propagation.

[1] Bonnet M, Maier G, Polizzotto C. Symmetric Galerkin boundary element method. Appl Mech Rev 1998;51:669704.

A. Frangi et al. /First MIT Conference on Computational Fluid and Solid Mechanics [2] Frangi A, Novati G, Springhetti R, Cazzani A. On the numerical implementation of the symmetric Galerkin BEM in 3D fracture analysis. In: Atluri SN, Brust FW (Eds), Advances in Computational Engineering Sciences, Vol. 1. Tech Science Press, 2000, pp. 81-86. [3] Mi Y. Three-Dimensional Analysis of Crack Growth. Southampton 1996, Computational Mechanics Pubhcations.

179

[4] Xu G, Ortiz M. A variational boundary integral method for the analysis of 3-D cracks of arbitrary geometry modelled as continuous distributions of dislocation loops. Int J Numer Methods Eng 1993;36:3675-3701.

180

Spring back of automotive assemblies P.C. Galbraith^'*, D.N. Thomas ^ M.J. Finn^ ^ Metal Forming Analysis Corporation, 2579 Highway #2 E, Kingston, ON K7L 4V1, Canada ^ Centre for Automotive Materials and Manufacturing P.O. Box 8400, Kingston, ON K7L 5L9, Canada ^ Alcan International Ltd. P.O. Box 8400, Kingston, ON K7L 5L9, Canada

Abstract This paper presents results from a forming simulation of an experimental component that is representative of an automotive assembly. An inner panel (1.6 mm AA-5754 aluminum sheet) and an outer panel (0.93 mm AA-6111T4 aluminum sheet) are stamped from tooling to produce a square pan. After trimming and assembly by spotwelding, the component is allowed to spring back. The manufacturing process (including forming, trimming, assembly, and spring back) is simulated using LS-DYNA software. Results show that including contact between the inner and outer panel during spring back is important for obtaining realistic spring back predictions. Spring back of the assembly is shown to be a function of the spring back of its components and the method by which they are connected. Keywords: Spring back; Finite element analysis; Sheet metal forming; Assembly; Hemming

1. Introduction

2. Approach

Many authors have examined spring back of automotive panels with the finite element method [1-7]. Earlier papers focused on the spring back of the first draw panel [ 1 5]. In later years, as the technology for conducting finite element simulations increased, spring back analyses were conducted on panels that underwent subsequent forming operations such as re-striking, trimming, and flanging [6,7]. Spring back analysis is undertaken largely to determine the final shape of a component. For example, hood panels are often the subject of spring back analyses because hoods are highly visible products and prone to low spots. The analysis may reveal the tendency for low spots to appear, but these may be corrected after assembly with the inner panel. In accordance with the concept of the functional build [8], emphasis should be placed on tuning the shape of the assembly and not on the shapes of the unassembled components. The methods outlined in this paper allow computer simulations to assume a role in constructing a functional build.

2.1. Experimental

* Corresponding author. Tel: +1 (613) 547-5395; Fax: -Hi (613) 547-5397; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

An inner and outer panel were each stamped on an experimental press using tooling shown in Fig. la. For the inner panel (Fig. lb), a 470 mm square blank made of AA5754 aluminum sheet was first stamped into a square pan with a dome in the bottom. This panel was then trimmed to create a channel 200 mm wide with a dome in the bottom (Fig. Ic). The channel height and dome heights were each 38.1 mm. The outer panel (Fig. Id) was 19.05 mm deep and did not have a dome. The outer was formed from 0.93 mm thick AA6111-T4 aluminum sheet. After forming, the outer panel was trimmed to the same width, 200 mm, as the inner panel (Fig. le). When the panels are removed from the tooling, their shape will change due to spring back. The 'sprung' shape of the panels was determined after forming and after trimming. After trimming, the inner and outer panels were attached with spot welds. The shape of the new assembly was determined after it was removed from the spot welding fixture.

EC. Galbraith et al. /First MIT Conference on Computational Fluid and Solid Mechanics a)

181

b)

d)

Fig. 1. (a) Tooling for forming the inner panel. For forming the outer, the tools are inverted, and the backup is not used, (b) The inner panel, (c) The inner panel after trimming, (d) The outer panel, (e) The outer panel after trimming.

2.2. Modelling The experimental approach outlined above was simulated using LS-DYNA version 950d running on a COMPAQ XPIOOO workstation. A forming model was run for each of the inner and outer panels. The blank material was modelled as an isotropic material with a Von Mises yield criterion. Coulomb friction was implemented between the blank and the tooling. Only one quarter of the geometry was modelled due to symmetry conditions. Initially, the blank in each model contained only 400 'type 16' fully integrated shell elements, but due to adaptive meshing this increased to 11,401 elements for the outer panel and 13,333 for the inner panel (Fig. 2). The tools were constructed from 4,377 elements. Tool elements were considered to be rigid. After simulating the first draw process, LS-DYNA wrote out a file named 'dynain' that contained the final nodal

a)

locations, element connectivity, and adaptive constraints. As well, this file contained the effective plastic strain and the stress tensor for each integration point of each element. Seven integration points were selected for the blank elements in order to accurately map the through thickness stress distribution for the spring back calculation. The spring back analysis was conducted for each part individually as it came out of the forming tooling. In order to trim the excess material from the inner and outer panels, the dynain file was read into DYNAFORM [9]. Within DYNAFORM, the excess material was removed, and a new dynain file was created for each of the inner and outer panel, containing only those nodes and elements inside the trim line. Spring back predictions were obtained for the inner and outer panels after trimming. Also within DYNAFORM, the nodes in the region of the spot welds were identified. Nodal rigid bodies were created

b)

Fig. 2. The blank, (a) Initially the blank was made up of 400 elements, (b) After forming the inner, the blank had 13,333 elements, (c) the formed outer had 11,401 elements.

182

P.C. Galbraith et al. /First MIT Conference on Computational Fluid and Solid Mechanics

at 4 locations to simulate the effect of the spot welds, thus simulating the assembly process. The dynain file output by DYNAFORM was used for the spring back calculation of the assembly. The spring back predictions were obtained by using the implicit solver built into LS-DYNA. A BFGS solver, which is a modified Newton's method, was used with automatic time step control and artificial stabilization. Spring back predictions for the full assembly were obtained at either two or three intermediate time steps based on convergence rates and the automatic time step controls.

In Fig. 5, the assembled component is shown prior to spring back. After spring back (Fig. 5b) the flange of the outer panel has passed through the inner panel, indicating that contact between inner and outer panels should be enforced if a proper shape prediction is to be obtained. The addition of contact between inner and outer panels prevented the flanges from passing through each other. Obviously, the spring back prediction would differ for the two cases. In Fig. 6, the effect of the assembly operation on the outer panel is shown. The low spot at the centre of the outer panel is 0.25 mm higher after assembly.

3. Results 4. Discussion and conclusions The 'as formed' inner and outer panels are shown in Fig. 3 with contours of effective plastic strain calculated by LS-DYNA plotted on the deformed geometry. In Fig. 4, the panels are shown after trimming and spring back. The contours in Fig. 4 represent the amount of displacement in the z-direction that occurs during spring back. The z-direction corresponds to the direction of the normal to the blank prior to forming. In order to remove rigid body translations in the z-direction, the z-displacement at the location of one of the spot welds was set to zero, so all z-displacements shown are relative to this point.

a)

Currently, tool engineers attempt to make dimensionally accurate parts that can then be assembled with mating parts. Conversely, in a functional build, parts are evaluated by assembling them with their mating parts, and determining if the assembly meets its function. With a functional build, automakers can save time and cost on die tryout. The techniques outlined here allow the benefits of sheet forming simulations (particularly the evaluation of tooling designs prior to manufacturing a forming tool) to be applied to the functional build process. For example, if the analysis shows Effective Plastic Strain 0.35 T | 0.28 ^ 0.21 - a 0.14 ^"^ 0.07 ^

0.00 i Hi

Fig. 3. LS-DYNA model predictions of effective plastic strain after forming. The inner panel (a) has higher strains because it is drawn deeper than the outer panel (b) and has a dome stretched in the bottom. Z-disp acen (mm)

'

1.30 0.64 -0.02 -0.68 -1.34 -2.00

3 1

z-constraints Fig. 4. Spring back of the unassembled inner and outer panels. The outer panel (b) springs more than the inner panel (a) because it has higher yield strength, thinner gauge and lower strains. Z-displacements are relative to the displacement of the nodes pointed to by the arrows. These nodes were selected because they are the sites of the spot welds to be used during assembly. Note the low spot in the centre of the outer panel.

RC. Galbraith et al /First MIT Conference on Computational Fluid and Solid Mechanics

a)

183

b)

!!• Fig. 5. Assembly before and after spring back. After spring back (b), the flanges have passed through each other because contact between the inner and outer panel was not modelled. Z-displacement (mm)

Fig. 6. The effect of assembly on the outer, (a) The spring back shown in Fig. 4b is repeated here, showing the spring back of the outer prior to assembly, (b) The outer panel after assembly shows a reduced low spot in the centre of the panel, indicating that the spring back is less problematic in the assembly than in the outer panel alone for this example.

that the low spots in a hood outer are corrected by the assembly with the inner panel, further work need not be done to correct the shape of the outer. For these techniques to be widely applicable, it will be necessary to have a computer-based analogue to the inspection rooms commonly used for evaluating surface appearance of automotive assemblies. In these rooms, bright lights are used to search for any surface defects such as low spots or teddy bear ears. Presumably ray tracing techniques could be implemented in post-processors to achieve on the computer screen what is obtained from these inspection rooms.

References [1] Finn MJ, Galbraith PC, Wu L, Hallquist JO, Lum L, Lin T-L. Use of a coupled exphcit-implicit solver for calculating spring-back in automotive body panels. J Mater Pro Tech 1995;50:395-409. [2] Various Authors. Benchmark B3. 2-D draw bending. In: Makinouchi A, Nakamachi E, Onate E, Wagoner RH (Eds), Proceedings of Numisheet '93. 2nd International Conference: Numerical Simulation of 3-D Sheet Metal Forming Process, 1993. [3] Various Authors. Benchmark B2. S-rail benchmark problem. In: Lee JK, Kinzel L, Wagoner RH editors. Proceedings

of Numisheet '96. 3rd International Conference: Numerical Simulation of 3-D Sheet Metal Forming Process, 1996. [4] Suh YS. Virtual manufacturing applications to stamping and structural analyses. In: Sheh M (Ed), High Performance Computing in Automotive Design, Engineering, and Manufacturing. Proceedings of the 3rd International Conference on High Performance Computing in the Automotive Industry, 1996, pp. 499-522. [5] Wu L-W, Du C, Zhang, L. Iterative FEM die surface design to compensate for springback. In: Shen SF, Dawson PR (Eds), Simulation of Materials Processing: Theory, Methods and Applications. Rotterdam: Balkema, 1995, pp. 637-641. [6] Valente F, Traversa D. Springback calculation of sheet metal parts after trimming and flanging. In: Gelin JC, Picart P (Eds), Proceedings of Numisheet '99. The 4th International Conference and Workshop on Numerical Simulation of 3d Sheet Forming Processes, 1999, pp. 59-64. [7] Various Authors. Benchmark A. Forming of a front door panel. In: Gelin JC, Picart P (Eds), Proceedings of Numisheet '99. The 4th International Conference and Workshop on Numerical Simulation of 3d Sheet Forming Processes, 1999. [8] Hammett PC, Wahl SM, Baron JS. Using flexible criteria to improve manufacturing validation during product development. Concurr Eng Res Appl 1999;7(4):309-318. [9] Eta/DYNAFORM User's Manual, Version 3.1. Engineering Technology Associates, Inc. 2000.

184

Constitutive and finite element modeling of human scalp skin for the simulation of cutaneous surgical procedures L. Gambarotta*, R. Massabo, R. Morbiducci Department of Structural and Geotechnical Engineering, University ofGenova, Via Montallegro 1, 16145, Geneva, Italy

Abstract A constitutive and finite element model of human scalp skin is formulated for the simulation of reconstructive surgical procedures. The model is calibrated using experimental results of tests on in vivo scalp flaps. Keywords: Biomechanics; Constitutive modeling; Human skin; Large deformation analysis; Parameter estimation

1. Introduction The paper deals with the formulation of a numerical model for the simulation of the mechanical behavior of human skin. The model will be used inside a virtual reality environment for computer-assisted reconstructive and aesthetic surgery simulation. The software allows preoperative planning of surgery procedures concerning the reconstruction of skin defects resulting from trauma, bums or tumor resection. The theoretical work is part of a broad research program, which includes also a campaign of experimental tests on human skin. To formulate the theoretical model, the constitutive equations of the human skin are firstly defined. Different large deformation hyperelastic models are considered which are able to reproduce the stiffening phenomenon characterizing the behavior of the skin at large deformations. The constitutive model is then implemented into a finite element code. Finally, the parameters of the model are identified from indirect experimental measurements using an inverse procedure. The experimental measurements are load versus displacement curves on in vivo skin flaps obtained through the non-destructive technique designed by Raposio and Nordstrom [1,2].

* Corresponding author. Tel.: -h39 (010) 3532517; Fax: +39 (010) 3532534; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

2. Constitutive and finite element modeling of human skin It is generally accepted that the stress versus strain curve of the skin in uniaxial or biaxial tension is characterized by an initial low-stiffness region followed, on increasing the strain, by a dramatic increase in stiffness, 'locking', [3]. The typical stress state of the skin under normal working conditions falls into the first region. On the other hand, in reconstructive surgery procedures, where the requirement of maximum skin extension needs to be satisfied, the skin can approach stress-strain configurations which are close to the 'locking' condition. The model proposed here refers to both the low stiffness and the high stiffness ranges. Other important features of the mechanical behavior of the skin are the time dependency, a moderate hysteretic behavior and a more or less marked anisotropy [3]. The model proposed here is restricted to short-term and monotonic loading processes, such as those which are typically applied to the skin during reconstructive surgery. The present analysis is based on a phenomenological constitutive model. This choice requires only the identification of the empirical parameters of a response function chosen in order to satisfy the main features of the observed macromechanical behavior of the skin. The other possible choice, that of a mechanics-based model (e.g., [4]), would imply a micromechanical interpretation of the in vivo skin response based on assumptions that, in this phase of the work, would put undesired restrictions on the range of solutions.

L. Gambarotta et al /First MIT Conference on Computational Fluid and Solid Mechanics Following the above observations, a large deformation hyperelastic model is considered. Different hyperelastic models have been proposed in the past for materials exhibiting stiffening under large deformations, such as skin and rubber [3,5]. Several isotropic constitutive models have been formulated and calibrated for different soft biological tissues (e.g., lung tissue, cat skin, arterial walls, rabbit mesentery, . . . ) . On the other hand, only a few models have been formulated which account for the anisotropy of some soft tissues (e.g., [6]). The proposed numerical model focuses in this initial phase on the simulation of surgery procedures concerning the reconstruction of scalp skin defects. Experimental observations by Raposio and Nordstrom [1,2] seem to indicate that the scalp skin has no preferred material directions. Consequently the compressible forms of the isotropic model formulated by Ogden [7] and the isotropic version of the model formulated by Tong and Fung [6] are considered. The models, which assume strain energy functions having different mathematical expressions, are able to capture the different aspects of the stress versus strain response observed in the experiments. In the surgical procedures to be simulated the scalp skin is cut, undermined within a predefined region and loaded tangentially to the hull surface. Taking into account the weak curvature of the hull, which supports the skin during the test, the domain of the model is approximated as two-dimensional and discretized in finite elements. The finite element model is assumed to be totally constrained at the boundaries between the undermined region and the surrounding skin. The finite element procedure examines incremental loading processes in terms of prescribed displacements or applied forces. The solution is obtained at each loading step using an 'eulerian — updated lagrangian' formulation [8,9] coupled with the Newton-Raphson iterative technique. 3. Calibration of the model The numerical model is calibrated using the experimental results of a testing methodology designed by Raposio and Nordstrom [1,2] which can be applied, due to its simplicity, also during surgery. The testing methodology consists of: incision of the scalp skin; undermining of a predefined portion of the skin; measurements of the relaxed configuration of the undermined skin; application of two concentrated forces along the incision at a distance of a few centimeters by means of a suture fixed by a full thickness bite; measurements of the displacements at different points along the incision for different values of the applied loads; extension of the undermined region of the skin and repetition of the previous steps. In the calibration of the model the reference configuration is geometrically known and it corresponds to the

185

configuration of the undermined skin after the incision. This configuration is only partially relaxed and some of the stresses that are present in the skin in normal conditions, the in vivo initial stresses, are still active. This stress field must be evaluated together with the model parameters, on the basis of the experimental measurements previously described. A simplified approach is proposed here which requires the evaluation of the parameters of the model and the in vivo isotropic initial stress (before the incision). This can be done by simultaneously satisfying two conditions. The first is that the initial isotropic stress uniformly applied along the incision in the reference configuration must restore the virgin configuration. The second condition is that the numerical model must reproduce the experimental load versus displacement curves. The simplified approach assumes that the results of the process incision -j- undermining coincide with the results of the fictitious process undermining -\incision. The identification procedure is based on the minimization of the norm of the residuals between the experimental measurements and the theoretical predictions. The minimization problem is solved using classical algorithms (e.g., Levenberg-Marquadt method). The search of the unknown quantities will be facilitated by the utilization of more than one load versus displacement curve in the minimization problem. This can be done using load versus displacement curves corresponding to undermined regions of different sizes.

References [1] Raposio E, Nordstrom REA. Tension and flap advancement in the human scalp, 4. Ann Plast Surg 1997;39:20-23. [2] Raposio E, Nordstrom REA. Biomechanical properties of scalp flaps and their correlations to reconstructive and aesthetic surgery procedures. Skin Res Technol 1998;94-98. [3] Fung YC. Biomechanics. New York: Springer, 1984. [4] Bischoff JE, Armda EM, Grosh K. Finite element modeling of human skin using an isotropic, nonUnear elastic constitutive model. J Biomech 2000;33:645-465. [5] Ogden RW. Non-Linear Elastic Deformations. Mineola, NY: Dover Publications, 1984. [6] Tong P, Fung YC. The stress-strain relationships for the skin, J Biomech 1976;9:649-657. [7] Ogden RW. Elastic deformations of rubberlike solids. In: Hopkins, Sewell (Eds), Mechanics of Solids. London: Pergamon Press, 1982, pp. 499-537, . [8] McMeeking RM, Rice JR. Finite element formulations for problems of large elastic-plastic deformation. Int J Solids Struct 1975;11:611-616. [9] Crisfield MA. Nonlinear Finite Element Analysis of Sohds and Structures, 2. Chichester: Wiley, 1997.

186

Material modelling in the dynamic regime: a discussion N. Gebbeken^'*, S. Greulich% A. Pietzsch% F. Landmann'' ^ Institute of Engineering Mechanics and Structural Mechanics, University of the Federal Armed Forces, Munich, 85577 Neubiberg, Germany ^ Federal Testing Center {WrD52), Oberjettenberg, 83466 Schneizlreuth, Germany

Abstract Wherever the mechanical behavior of materials is of interest, mathematical material models are needed to describe the physical phenomena. The higher and/or shorter the dynamic loading, the less known and validated from experiments are the material properties. Therefore, the material behavior can only be postulated in the high dynamic regime. The physical properties, the mathematical description and the numerical application are discussed critically in the example of the inhomogeneous, compressible, brittle material concrete. Keywords: Macroscopic material modelling; High dynamic loading; Rate effect; Damage property; Experiment; Equation of state; Hydrocode

1. Introduction High dynamic loadings like explosions have the capability to release large amounts of energy within microseconds. This causes high pressures in the kilobar range (1 kbar = 100 MPa) and high strain rates up to 10^ s~' in the affected bodies. For the numerical simulation of high frequent stress wave propagations, hydrocodes have been successfully applied. They are based on Finite Difference Methods [1]. And, the conservation equations of mass, momentum and energy in addition to an equation of state (EoS) are solved simultaneously in time, whereas the EoS is a functional correlation of two unknown variables in the conservation equations. Its mathematical correlation gives an additional constitutive law and its data has to be determined from expensive experiments. Furthermore, material models are required characterizing the elastic, non-elastic and damage behavior. As a result of the high dynamic loadings, strain-rate effects have to be taken into account.

2. Physics, material modelling and numerical algorithms In order to model the microscopic mixture of materials like concrete, a homogenization hypothesis is adopted which enables the formulation of the constitutive equations on the level of macromechanics (see Fig. 1). Even though the numerical modeling of structures subjected to shock waves leads to element sizes in the millimeter regime, it is too complicated and computing time too consuming to approach on the micromechanical level. The material modeling of concrete is based on the macromechanical constitutive law of Ruppert and Gebbeken [2,3]. Characterizing the physical material behavior properly, the current stress state is depicted in the three-dimensional stress space, whereas the hydrostatic tensor can be separated from the stress tensor resulting in the deviatoric stress tensor. Illustrating the material behavior of concrete, the current material state will be explained by means of a loading, unloading and reloading path, see Figs. 2 and 3. 2.1. Elastic material behavior

* Corresponding author. Tel.: -H49 (89) 6004-3414; Fax: 4-49 (89) 6004-4549; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

In the elastic response (Figs. 2 and 3; loading, PO-Pl; unloading, P3-P4), the incremental form of Hooke's law has been adopted, whereas the stress state is divided into a hydrostatic and deviatoric part. The bulk modulus and the

A^. Gebbeken et ah /First MIT Conference on Computational Fluid and Solid Mechanics

real microscopic material

homogeneous microstructure

simplify the microstructure

classical continuum mechanics

^

^

187

^ ^

•i

material formulation on the level of micromechanics

microplane formulation

X

macroscopic models -theory of elasticity -theory of plasticity

-damage and fracture application to structural behaviour Fig. 1. Homogenization of a microscopic material to a macroscopic scale.

P =;^oct t

P

U

'S \ 1

P2

yC-Pl PO,P4

•eloadin2=unloading crush

"^oct

Fig. 2. Loading and unloading path in a 3D stress space and in a porous Hugoniot EoS. shear modulus are the only parameters. The nonlinearity occurring in compressible materials is taken into account with the bulk modulus, which is responsible for the volume change. 2.2. Equation of state The full EoS giving a complete material characterization is a three-dimensional function of a surface in space, called

P

7P

P

lock P4 ref Fig. 3. Loading and unloading path in a 3D stress space and in a porous Hugoniot EoS. the Mie-Grueneisen surface. The basis are any of the two unknown variables of the equation of state and the energy. Since the data are coming from large-scale experiments, the energy is considered implicitly. Considering the energy explicitly would be problematic because it is not possible to distinguish between different energy contributions in the tests. The three dimensional curve of the EoS, called the Hugoniot curve, is projected in the two dimensional plane of the remaining variables. For concrete, experimental results of different authors are depicted in Fig. 4 in a

N. Gebbeken et al /First MIT Conference on Computational Fluid and Solid Mechanics P.

w^

j^^W^

# Grady [1996]

,

• HJC [1993]

2.3. Strain-rate effects: enhancement of strength

A Eibl,Ockert[1997] • exp.data (Adiment) -H— - ^ R-average

1

#

1 [\ Yieldsuiface

ill

^

^

EoS . / •

• A ^

experimental data from explosive field tests, carried out at the WTD 52, provide assured data up to 40 kbar.

•

2,4 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 p[g/cm]3 Fig. 4. Experimental strain data to obtain an equation of state.

Experimental data for various materials have shown that their strength enhance by increasing strain-rates. For metals, it can be shown that there is a linear correlation between strength and strain-rate in a logarithmic scale. This is more complex for porous materials like concrete. Bischoff and Perry [4] have assembled experimental results from different authors using different concrete mixtures and no consistent testing devices (see Fig. 5). Furthermore, experimental data are only available up to e < 10^ s~^ High dynamic loadings lead to strain-rates over £ > 10^ s~^ Therefore, a reasonable enhancement function has to be postulated, which can be easily adapted to experimental results. 2.4. Invariant yield surface

pressure-density plot, which is an advantageous form for porous media. It is obvious that measurements in the high pressure regime are widespread. Here, a multi-linear approximation of the test results has been adopted. It contains an elastic path from tensile limit T to the Hugoniot elastic limit at Pcrush (Figs. 2 and 3; PO-Pl). Densities greater than Pcrush cause compaction and gradually, concrete converts into a granular kind of material (Figs. 2 and 3; P1-P3). Unloading and reloading are following the same path, which is interpolated between the slopes c^ and c]^^-^ (Figs. 2 and 3; P3-P4). But in order to develop a complete Hugoniot curve, a sufficient set of measurement data is needed for a wide range. It should be mentioned that own

Commonly, yielding of materials can be determined from experimental data, e.g. uniaxial tension tests for metals. The more complex the material the more difficult is the derivation of the yield surface in the three-dimensional stress state. Here, concrete is an anisotropic composite, it is brittle in tension as well as in shear, and ductile under high pressure. Fig. 6 shows the essential features of the yield surface. It was fitted to test results in the range of a^ < 10, where (To = (Toct/fc-'' and To = Tocr///^". Herein, aoct, roct are octahedral stresses and fc is the characteristic strength (see Fig. 7) [5]. Experimental as well as numerical simulations of high explosive loadings revealed that relative hydrostatic

- I (tanhlilogi* - 2) • 0.4])

Fig. 5. Enhancement for extreme strain-rates (based on Bischoff and Ferry [4]).

W^

+ iyw^

A^. Gebbeken et al. /First MIT Conference on Computational Fluid and Solid Mechanics

To = a

c-(Jo + Cc{sinl.he)\2.0

c = CticoslMy-^

deviatoric plane octahedral plane

a

9=60°CQinpressive meridian

189

^0

,

Fig. 6. Yield surface for concrete and its experimental results [5].

Versuche Mould & Levine-1987 • : q=0°,60°(fc:variiert) •*s^ : Ausgleichskurven Vers. Hanchak & Forrestal-1992 V : q=60Xfc=48MPa) *•••., : Ausgleichskurve

-7

-0

~B

-4

-a

-2

-1

O

1

Fig. 7. Yield surface for concrete and its experimental results [5]. pressures up to GO = 100 have to be expected. Consequently, this yield surface description has to be refined with data of further experiments. In high pressure region, the von Mises (J2) flow theory is adopted.

2.5. Yield curve shifting Strain-rate effects as well as damage cause an isotropic shifting of the strength and stiffness. The first part, ex-

190

N. Gebbeken et al /First MIT Conference on Computational Fluid and Solid Mechanics i^^

decrease in

| ^ stiffness and ^^strength

^^v.Mises ^!:^assoc

^aQ

^ ^ assoc.

Fig. 8. Loading, unloading and reloading in the uniaxial case (a-e-plane) and the yield surface in the triaxial case (cap, plastic flow and shift). plained in Section 2.3, is implemented in the normalization of Go and To, whereas the characteristic strength /,. increases with respect to the strain-rate (Formula in Fig. 5). The second part, depends on a damage parameter D [3]. The yield limit and static tensile limit parameters (see Fig. 8) starts at QQ, bo and ends at au b\. The subscripts 0 and 1 represent the initial (undamaged, D = 0) and the damaged (D = 1) parameters. Isotropic shifting is assumed, because cyclic loading is not taken into account in this case. For certain reasons, an additional yield surface part (called cap) can be used. This is of importance, especially if an EoS is not necessarily needed. Numerical problems as well as questions regarding the physical interpretation occur by using a cap. Essentially problems are: the transition between the standard yield function and the cap is not continuous, radial return vector is overestimated in accordance with a nonassociated flow rule, whereas a perpendicular (associated) return onto the cap surface would cause a negative dilatation which is nonphysical. Therefore, the yield surface is 'open' (Fig. 6), which is part of the classical theory of plasticity and physically correct. 2.6. Monotonia convergence in hydrocode simulations high dynamic loadings

It should be mentioned that other convergence studies have shown the same material independent problems in a critical range nearby an explosion.

3. Conclusion A physical problem can only be simulated satisfyingly within constitutive models if the appearing phenomena are known from experiments, and if they are adequately described by mathematical formulations. Up to now, the physical behavior of a large number of materials is well known from experiments concerning static or dynamic loading up to a strain rate s < 10^ s~K If strain rates exceed this value, gathering data is complicated and reliable results are rare. As it was shown, it is only possible to state

with

Finite methods are approximate methods. Hydrocode inherent problems of discretization in space (e.g. mesh size sensitivity) and in time (time-step) have to be detected in sensitivity investigations and their errors have to be reduced to an acceptable minimum. Within the scope of a convergence study [6], based on the convergence theorem [7], an explosive charge is initiated on a concrete structure (Fig. 9). The pressure is measured in different targets by scaling down the element size. The evaluation of targets #0 to #2 in Fig. 10 show, that in a limited region (target #0) convergence problems concerning pressure near the high pressure zone have occurred.

air

explosive charge > o o

1 ^Target#0 ^Target#l L^Tai^et#2

mm] _30

concrete Fig. 9. Physical problem and target points.

A^. Gebbeken et al /First MIT Conference on Computational Fluid and Solid Mechanics 6_Euler pressure 1 [mm]

[kbar]

0,1 0,5 1 1,5 2 4

132,0 90,0 79,2 88,0 51,0

43,0

|

[!H(30/5)

150 n

130110-

90 -

\

70-

1 50 -

0,5 1 2 3 4 5 6 7 8

1

2

3

4

5

[kbar] 80,8 76,5 74,0 57,6 68,8 44,3 46,3 40,0

49,3

80 70 60 50 40 30

^""^^v

X^<- ^ \\

References

11 \.r—V^,^t >

1

6_Euler pressure H 3 I (30/50) [mm] 0,5 1 2 3 4 5 6 7 8

[kbar] 7,5 6,9 6,9 6,6 6,1 5,3 5,7 7,5

A

a—,, *'^T-'*'

r r

decreasing element size, as is demanded for the numerical algorithms used in finite element formulations. This paper points out the enormous demand for research in the high dynamics field. Especially the measurement engineering is in charge to provide reliable data. Just then it will be possible to validate postulates and to extend the constitutive models.

^••(0/30)

6 Euler pressure 1 [mm]

11

\ .,._ ^ 1

30 •

()

1

V*"->rA, ^

191

J

Fig. 10. Pealc pressure vs. mesh fineness for different targets. postulates extrapolating experimental data. But, one should avoid the prediction of physical contradictions. Furthermore, the numerical tools, the hydrocodes, are not capable for certain regions of explosive loading. It was shown that for the region adjacent to the explosives, the Lagrangian formulation does not converge asymptotically for

[1] Benson DJ. Computational methods in Lagrangian and Eulerian hydrocodes. Comput IVlethods Appl Ivlech Eng 1992;99:235-394. [2] Ruppert M, Gebbelcen N. Material formulations for concrete, high strain-rates and high pressures, elasticity-plasticity-damage. 9. International Symposium on Interaction of the Effects of Munitions with Structures, Strausberg, 1999, pp. 397-405. [3] Gebbeken N, Ruppert M. A new concrete material model for high dynamic hydrocode simulations. Arch Appl Mech 2000;70:463-478. [4] Bischoff PH, Perry SH. Impact behavior of plane concrete loaded in uniaxial compression. J Eng Mech 1995;121(6):685-693. [5] Guo Z, Zhou Y, Nechvatal D. Evaluation of the multiaxial strength of concrete tested at Technische Universitat Munchen. DAfStb447. Berlin: Beuth, 1995, pp. 591-600. [6] Gebbeken N, Ruppert M. On the safety and reliability of high dynamic hydrocode simulations. Int J Numer Methods Eng 1999;46:839-851. [7] Bathe KJ. Finite Element Procedures. Englewood Cliffs, NJ: Prentice Hall, 1996.

192

Error estimation and edge-based mesh adaptation for solid mechanics problems G. Gendron*, M. Fortin, P. Goulet GIREF Research Center, Universite Laval, Quebec, Canada GIK 7P4

Abstract A simple error estimator based on a low-order finite element interpolation is described in details. The estimator is used to guide a mesh-adaptation procedure for solid mechanics problems. The overall procedure is illustrated and validated on a 2D elasticity and a plate-bending problem. It is shown that the procedure automatically generates well-adapted meshes for which the error is uniformly distributed and is thus very attractive in the context of complex structural analysis problems. Keywords: Error estimator; Mesh adaptation; Plate bending; Elasticity; Finite elements

1. Introduction It is well-established that the accuracy of finite element results strongly depends on the appropriateness of the mesh. Complex stress concentration areas that develop in real-life structural components can only be predicted accurately if appropriate mesh densities and element formulations are selected. To obtain an appropriate mesh, two main ingredients are required. These are an error estimator and a mesh adaptation strategy. In this study, the methodology proposed in [1] is reviewed in details, and applied to the design of finite element meshes for 2D elasticity and plate bending problems.

2. Error estimator By a now classical procedure [1], our error estimator is based on the use of a metric associated with the second derivatives of some scalar function g computed from the solution. The choice of g is delicate and problem-dependent. For the procedure to be successful, g must be sensitive to the features of the solution that must be predicted accurately. For CFD problems, the Mach number has been used [1]. For the structural mechanics problems presented in Section 5, one component of the nodal displacement is * Corresponding author. Tel.: +1 (418) 656-7892; Fax: +1 (418) 656-2928; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

chosen. The principles of the method are simple: (1) A local quadratic representation of the function can be built using, for example, a Taylor series expansion. (2) This quadratic function can be interpolated linearly on a triangular mesh using piecewise linear triangular elements. The interpolation error, E, then depends on the Hessian matrix. In fact, on an element edge, one has:

where h is the length of the edge, and d^g/d^^ is the second-order derivative of g along the edge. (3) Taking the absolute value of the Hessian (through its eigenvalue decomposition) the error on the edges can be seen as a length in a Riemannian metric. The problem with the evaluation of the estimate 1 is that the function g is not known and thus its Hessian H cannot be evaluated. This difficulty can be circumvented by replacing g by gh, its finite element approximation. However, to keep the procedure general and make the use of linear interpolation functions possible, a weak formulation is used to calculate H. The procedure is presented here for d^g/dxdy, it is identical for the other components of H. First, we write:

where Qi represents the domain formed by the elements connected to node /, and 0/ is any test function that is 0

G. Gendron et al. /First MIT Conference on Computational Fluid and Solid Mechanics

193

^

Material Properties E = 200000 z/ = 0.3

0.00613

0.00525

body force

100

XM

20

10

(a) Problem Definition

10

(b) Initial Grid - 100 elements

(c) Error

(d) Estimator

Fig. 1. Problem 1. Prismatic bar subjected to a uniform body force.

on 9Qj, the boundary of ^/. Eq. (2) can be applied to a piecewise linear approximation of g since the right-hand side involves only first-order derivatives. Also, we replace d'^g/dxdy by a constant on ^/, D^yt. With this assumption, the second-order derivative d^g/dxdy at node Xj, internal to ^/, is approximated by:

f ^xy,i

dg d(pi

dx dy

•dA

—

/

basis function (pi does not vanish on the boundary of the domain. Consequently, a line integral should be added to Eq. (3). Unfortunately, this integral is difficult to evaluate. Instead, for a boundary node, we choose to extrapolate the values of the second-order derivatives from neighboring internal nodes.

0/dA

(3)

Each node is successively processed to finally obtain a linear approximation of the second-order derivatives. An approximation to the Hessian matrix is thus defined at the nodes and it is easy to take its absolute value or interpolate it where needed. For a boundary node, the piecewise linear

3. Mesh adaptation strategies We define an optimal mesh as a mesh for which the error is approximately uniform on all edges. To obtain such a mesh, we start with an initial mesh and then tend to improve it by iteratively performing the following operations: OPl Refinement and coarsening (A-method). OP2 Reconnection. Based on the fact that an edge be-

194

G. Gendron et al. /First MIT Conference on Computational Fluid and Solid Mechanics )0i

^/^J %

0.007

0.007

0.00613

0.00613

0.00525

0.00525

0.00438

0.00438

10.0035

0.0035

^m

m 0.00263

0.00263

0.00175

10.00175

0.000875

0.000875

10

10

(a) Adapted Grid 19 elements

(b) Error

(c) Estimator

Fig. 2. Problem 1. Results.

tween two triangles is actually the diagonal of a quadrilateral, the orientation of the diagonal is such that the minimum internal angle of a triangle is maximized. OP3 Node relocation (r-method). Each edge is replaced by a spring, the stiffness of which is proportional to the value of the estimator on that edge. The complete algorithm is as follows: (1) Select a scalar function g and an optimal edge length {Lopt) in the space of the Riemannian metric. The value of Lopt allows the calculation of more or less refined meshes. Fixed values of L„pt will be used in Section 5. (2) Define an initial mesh and calculate a solution and the error estimator on this mesh. (3) Iteratively use OPl through OPS to define an adapted mesh. (4) Calculate a new solution on the adapted mesh.

(5) If necessary, calculate the error estimator based on the new solution and return to step 3.

4. Model problems Two model problems are considered. The first one corresponds to two-dimensional elasticity which is discretized using the well-known Constant Strain Triangular (CST) element. The second problem corresponds to a plate bending problem which is studied using the DST element [2]. This element is convenient for the modeling of moderately thick to thin plates. The element has nine degrees of freedom only: the displacements w and rotation of the normal in the (x, z) plane, )6^, and in the {y, z) plane, Py. Shear locking is avoided through the use of appropriate approximation fields.

G. Gendron et al. /First MIT Conference on Computational Fluid and Solid Mechanics

195

w = Px =0

w=

PY

=Q

\EX = EY = 13800 MPa GxY =Gxz — GYZ = 1870 MPa WxY = 0.12 ^ = 10 MPa \t— 12 mm

100 mm.

w=

/3Y

=0

X

100 TTim

w = /3x = 0 (a) Problem Definition

(b) Initial Grid - 200 elements

j%; J '''^^^^^^^^H ^m^'^i 1 ^' '^^^^H J ' ^^"^^^^Hl^^^^K.^ 1

^S^'^

(c) Error

wwx

0.09

W\\''-'J

W\

' \W \ kww KWW 1

-*v ,

' . C ^ ^

-.:

i

0.08 0.07 0.06 0.05 0.04 0.03 0.02

(d) Estimator

Fig. 3. Problem 2. Simply-supported square plate. 5. Numerical studies In this section, the results of two numerical tests are reported in order to validate the error estimator and demonstrate that it is suited to the design of meshes for which

the error is reduced and uniform over every element edge. These problems have been selected because closed-form solutions are known. This will allow the direct comparison of the exact error with the predicted estimator. In all cases, these quantities are calculated at the center of every edge.

196

G. Gendron et al. /First MIT Conference on Computational Fluid and Solid Mechanics )0-i

(a) Adapted Grid - 263 elements

(b) Error

(c) Estimator Fig. 4. Problem 2. Results.

5.7. Problem 1: prismatic bar subjected to a uniform body force Fig. la shows a prismatic bar made of isotropic material subjected to a downward constant body force. Symmetry boundary conditions are applied along the jc = 0 line, and consequently only half of the bar is modeled using 2D elasticity elements. The function gh used to estimate the error corresponds to the finite element approximation of the vertical displacement, Vh. The exact solution for the vertical displacement is a quadratic function of x and y [3]. The initial mesh is shown in Fig. lb. The exact error calculated on this mesh along with the error estimator are

compared in Fig. lc,d. It is seen that the estimator and the exact error both calculated at the center of each element edge present the same distribution. The average error is 0.0041 and its standard deviation is 0.0016. Fig. 2 shows the adapted mesh along with the distributions of the exact error and the estimator. Elements of identical size are obtained throughout the domain. This result could be expected since the Hessian of the exact solution is constant. For the adapted mesh, the average error is 0.0029 and its standard deviation is 0.0010. The procedure has thus allowed the design of a mesh with significantly less elements, for which both the average error and its standard deviation have been reduced.

G. Gendron et al. /First MIT Conference on Computational Fluid and Solid Mechanics

197

5.2. Problem 2: simply-supported square plate

6. Conclusion

A simply-supported square plate 100 mm x 100 mm under a uniform lateral pressure is shown in Fig. 3a. The plate is made of an orthotropic material with the property values indicated. The plate is simply supported (hard conditions) on all four sides. The plate thickness t is 12 mm which corresponds to a side-to-thickness ratio of 8. The pressure value q is 10.0 MPa. The function gh used to estimate the error corresponds to the finite element approximation of the transverse displacement Wh. For this problem, an infinite-series solution based on a first-order shear deformation theory has been derived by Reddy [4]. The initial mesh is shown in Fig. 3b. The exact error calculated on this mesh along with the error estimator are compared in Fig. 3c,d. The estimator and the exact error, both calculated at the center of each element edge, present the same distribution. The main differences are at the center of the domain where the estimator slightly underestimates the error. The average error is 0.040 and its standard deviation is 0.021. Fig. 4 gives the adapted grid along with the distributions of the exact error and the estimator. For this mesh, the average error is 0.018 and its standard deviation is 0.007. The procedure has thus allowed the design of a mesh for which the error is reduced and quite uniformly distributed over every edge. It could be reduced further by decreasing the value of Lopt. As it was the case for the initial mesh, the error and the estimator are in good agreement.

An error estimator based on a metric derived from the Hessian of a scalar function has been presented. Any scalar function that relates to the solution can be used. In this work, a nodal displacement component is proposed. The results presented herein confirm that the error estimator correctly predicts the value of the error. The estimator drives the adaptation process in such a way that the final adapted mesh presents a uniform distribution of the error. More work needs to be done to verify the applicability of the strategy to other structural problems. The choice of a displacement component to estimate the error also needs to be assessed.

References [1] Habashi WG, Fortin M, Ait-Ali-Yahia D, Boivin S, Bourgault Y, Dompierre J, Robichaud MP, Tarn A, Vallet MG. Anisotropic Mesh Optimization: Towards a SolverTndependent and Mesh-Independent CFD. VKI Lecture Series, 1996-06. [2] Batoz JL, Lardeur R A discrete shear triangular 9-dof element for the analysis of thick to very thin plates. Int J Numer Methods Eng 1989;28:533-560. [3] Timoshenko S, Goodier JN. Theory of Elasticity, 2nd edition. New York: McGraw-Hill, 1961. [4] Reddy JN. Mechanics of Laminated Composite Plates, Theory and Analysis. CRC Press, 1997.

198

Reliability-based importance assessment of structural members Emhaidy S. Gharaibeh^, John S. McCartney^, Dan M. Frangopol^* ^ University of Mutah, Department of Civil Engineering, P.O. Box 7, Mutah, Al-Karak, Jordan ^ University of Colorado, Department of Civil, Environmental, and Architectural Engineering, Boulder, CO 80309-0428, USA

Abstract When analyzing a structural system, it is often useful to identify critical members by quantifying the safety importance of individual members. In this process, several aspects have to be examined, including but not limited to the location of each member in the system, the safety level of each member, and the material behavior and stiffness sharing of each member. Two types of importance factors are formulated in this paper, the member reliability importance factor and the member post-failure importance factor. Each of the above factors has its area of application and may be of great significance in analysis, design and maintenance of structural systems. These factors measure the impact of each individual member on the performance of the overall system. Keywords: Critical members; Member importance; Member ranking; Reliability assessment; Sensitivity analysis; System performance; System reliability

1. Introduction In recent years, design codes have been continuously revised to include limit states based on probabilistic methods. In fact, the limit states design approach has been used in nearly all of the recent advances in codified design [1]. The use of structural reliability methods for design can lead to structures that have a more consistent level of risk [2]. However, most of the current assessment and design codes require safety checks at the member level only. This leads to either over-conservatism in the assessment of structural systems which are able to continue to carry loads after one member becomes damaged, or under-conservatism in the design of structural systems which are not able to redistribute loads [3]. To account for the system effect in structural assessment and design, safety importance of structural members must be quantified.

2. Model A simple idealized three-member series-parallel system model comprised of two subsystems in series (i.e. members * Corresponding author. Tel.: +1 (303) 492-7165; Fax: -\-\ (303) 492-7317; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

1 and 2 in parallel connected in series with member 3) is used to exemplify the proposed approach [3-5]. The system is subjected to a random load P with a mean of 0.5 and a coefficient of variation of 0.15. The data used to describe the series-parallel system can be found in Gharaibeh [4]. The failure path approach [6] is used to formulate the Umit state of the multi-member system, and the RELSYS software [7,8] is used to compute system failure probabilities for post-failure member behaviors ranging from perfectly britde (i.e. r]i = 0) to perfectly ductile (i.e. rit = 1).

3. System reliability analysis In reality, systems exist in any combinations of series and parallel subsystems. The process of finding the reliability of a complex structure made out of a combination of series and parallel subsystems can be simplified by introducing the concept of an equivalent system. The system can be represented by a series of equivalent subsystems, each of which represents a combination of either series or parallel components. These subsystems are broken down into simpler equivalent subsystems until the system is reduced to a single equivalent component [7,8]. The reliabiUty of the actual system can be assumed to be the same as the reliability of the single equivalent component.

Emhaidy S. Gharaibeh et al /First MIT Conference on Computational Fluid and Solid Mechanics 4. Reliability importance factor

4.25

Member reliability importance factors can be derived from the sensitivity of the system reliability to changes in the reliability levels of its members. Each member has its own impact on the system reliability level. This impact depends on many factors such as the correlation between resistances of individual members, the stiffness sharing factor of each member, the member reliability level, the member post-failure behavior, the system failure criterion model adopted, and the position of each member in the system (i.e. system topology). In general, system reliability is a function of its individual members, /^system = / ( P , il), where P and r] are vectors of member reliabilities and post-failure behavior factors, respectively. The reliability importance factor of member / is derived from the sensitivity of system rehability index, ySsystem, to changes in rehability of member /, p^nA [3-5]. This measure can be defined as the gradient of the system reliability, y^system, with respect to the member reliability, Prn,i^ as follows: 9^s, OPm,i

The associated normaUzed member importance factor is: /».. =

(2)

N

where A^ = number of members in the system. In calculating the importance factors a small change in member reliability level is imposed and the corresponding system reliability is evaluated.

5. Post-failure importance factor System reliabihty is usually very sensitive to the postfailure behavior factor 77, of its members. In order to quantify this sensitivity, another importance factor, called the post-failure importance factor, is defined as follows [4]:

I 4.00

Q

S

3.50

3

3.25

S

2.75

199

P,=3.0, P2= 3.0, p3=3.0 ; r| = 0.50 A : p,=3.0, P2= 3 0 ' P3=3.0 A - I I : p,=3.0, p2= 4.0, p3=3.0

A - I : Pi=4.0, p2= 3.0, P3=3.0 A - H I : p,=3.0, ^^= 3.0, p3=4.0

O.l 0.3 0.5 0.7 0.9 STIFFNESS SHARING FACTOR OF MEMBER 1, Cj

Fig. 1. Effect of member reliability level on system reliability. 6. Example Based on the three-member series-parallel model described in Section 2, Fig. 1 shows the reliability importance factor implementation for the default case of P]^ = ^2 = h = ^-0 and post-failure behavior factor r]i = r]2 = r]3 = 0.50. The reliability of one member was changed to 4.0 while the other members retained their default reliability indices. The system reliability index ^system associated with each case considered is plotted in Fig. 1 against the stiffness sharing factor of member 1. The differences in the associated system reliability of these cases compared to the default case show the impact of each member on the overall system performance. Fig. 2 shows the normalized reliability importance factors obtained for the default case according to Eq. (2). Fig. 3 shows the variation of the default reliability index of the series-parallel system described above with the stiffness sharing factor of member 1. Different combinations of extreme values of the post-failure behavior factors of members 1, 2, and 3 are assumed and the system reliabilities associated with each of these combinations are investigated. This figure shows that ySsystem is maximum and minimum for r]i = ri2 = r]3 = 1-00 and r]i = r]2 = r]3 = 0.0, respectively. It can be seen that y^system is not sensitive to rj^ due to its series system effect, while the post-failure behaviors of members 1 and 2 affect the system reliability. Finally, Fig. 4 shows the post-failure importance factors of the Ti = O.SO ; P, = 3.00 , p2 = 3-00 , P3 = 3.00

/..• = ,''system 5 ' 1

^system? ^0

(3)

where 7^,/ = importance factor with respect to the postfailure behavior of member /, ^system, ii = reliability index of the system given that member / has a perfectly-ductile post-failure behavior (i.e. rji = 1), and ^^system, io = rehability index of the system given that member / has a perfectly-brittle post-failure behavior (i.e. r]i = 0). The most important member with respect to its post-failure behavior is the member that has the maximum effect on the system reliability index.

Fig. 2. Normalized reliability importance factors for each member of a series-parallel system.

Emhaidy S. Gharaibeh et al /First MIT Conference on Computational Fluid and Solid Mechanics

200

O < T|, < 1

4.25

Acknowledgements

-

4.00 3.75

w

s

3.50 3.25 r|,= l , T i 2 = 0

3.00 2.75

n,=o, Ti^=i

^ ^ r|,= l . T , 3 = l

-

-

-

r | , = 0 , Ti2=0

This material is based upon work partially supported by the National Science Foundation under Grants CMS9506435, CMS-9522166, CMS-9912525 and the University of Mutah, Al-Karak, Jordan. This support is gratefully acknowledged. Opinions expressed in this paper are those of the writers and do not necessarily reflect those of the sponsoring organizations.

O.l 0.3 0.5 0.7 0.9 S T I F F N E S S S H A R I N G F A C T O R O F M E M B E R 1, C ,

Fig. 3. Effect of member post-failure behavior on system reliability. P, = 3.00 , P^ = 3 OO ' Ps = 3 OO

S ^

O.l 0.3 0.5 0.7 0.9 STIFFNESS S H A R I N G F A C T O R O F M E M B E R 1 , C,

Fig. 4. Post-failure importance factors for each member of a series-parallel system, r/ = 0 . 5 . series-parallel system for the default case. The post-failure importance factor is derived from the contribution of member post-failure factor to the overall system performance.

7. Conclusions This paper presents an approach to assess the reliability importance of members in any structural system modeled as a series-parallel combination of failure modes. The proposed approach takes into account the system reliability as a whole and identifies the contribution of individual members to the overall system performance. Along these lines, the importance of a member is defined as the impact of that member on the overall system reliability. The results are useful for assessment, design and maintenance of structures in an overall system reliability perspective.

References [11 Ellingwood BR. Reliability-based condition assessment and LRFD for exisfing structures. Struct Safety 1996; 18(23):67-80. [21 Zimmerman JJ, Corotis RB, Ellis JH. Structural system reliability considerations with frame instability. Eng Struct 1992;14(6):371-378. [3] Frangopol DM, Gharaibeh ES, Heam G, Shing PB. System reliability and redundancy in codified bridge evaluation and design. In: Srivastava NK (Ed), Structural Engineering World Wide 1998. Paper Reference T121-2, Elsevier: Amsterdam, 1998, 9 pp. on CD-ROM. [4] Gharaibeh ES. Reliability and Redundancy of Structural Systems with Application to Highway Bridges. Ph.D. Thesis, Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, CO, 1999. [5] Gharaibeh ES, Frangopol DM, Shing PB. Structural importance assessment of bridge members: A reliability-based approach. In: Dunaszegi L (Ed), Developments in Short and Medium Span Bridge Engineering'98. Canadian Society of Civil Engineering, Montreal, 2, 1998, pp. 1221-1233 (also on CD Rom). [6] Karamchandani A. Structural system reliability analysis methods. The John A. Blume Earthquake Engineering Center, Department of Civil Engineering, Stanford University, Stanford, CA, 1987, Rep. No. 83. [7] Estes AC, Frangopol DM. RELSYS: A computer program for structural system reliability analysis. Struct Eng Mech 1998;6(8):901-919. [8] Estes AC. A System Reliability Approach to the Lifetime Optimization of Inspection and Repair of Highway Bridges. Ph.D. Thesis, Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, CO, 1997.

201

Probabilistic life prediction for mechanical components including HCF/LCF/creep interactions Dan M. Ghiocel *, Hongyin Mao STI Technologies, Advanced Engineering Applications, 1800 Brighton-Henrietta, Rochester, NY 14623, USA

Abstract Stochastic life prediction of mechanical system components represents a difficult engineering problem involving modeling of multiple complex random phenomena. The paper presents a simulation-based stochastic approach for mechanical component life prediction under normal operating and accidental conditions. The paper addresses key aspects of stochastic modeling of component life prediction. Specifically, results computed for a generic aircraft jet engine blade are shown. The paper also discusses critical modeling issues that drastically impact on the component fife prediction. Keywords: Life prediction; Stochastic modehng; Crack initiation; Damage accumulation; Fatigue; Creep

1. Introduction A typical illustration of a jet engine life prediction problem is shown in Fig. 1. As shown in Fig. 1 for each critical location, the operational stress profiles and local damage accumulation are modeled as non-stationary stochastic processes [1,2]. Stochastic stress variation in a blade location is obtained by the superposition of a slow-varying

Loading History t Stress Amplitude

Operational (HCF, LCF, Creep, etc.) Extreme Events (Accidental impact, etc.)

0

5

10 15 20 25 30 35 40 45 50 55 60 65 70

Impact Occurence

TIME (Minutes)

Resonances . Blade Vibration Mode 2

Fig. 2. Flight stress profile.

Blade Vibration Mode 1

>• Time scale Damage Accumulation J)amage Larger Initial Defect

: Impact damage

-^^ Time scale

Fig. 1. Stochastic environment and damage. * Corresponding author. Tel.: +1 (716) 424-2010; Fax: +1 (716) 272-7201; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

stress component (pulse process with holding times) due to pilot's maneuvers with a fast-varying stress component (intermittent continuous narrow-band process) due to vibration under unsteady aero-forcing. The vibratory stresses occur when the excitation frequency is sufficiently close to blade natural frequencies a shown in Fig. 2. The random slow-varying stress cycles produce the low-cycle fatigue (LCF) damage and creep damage in the component, while the randomly occurring vibration stress cycle with lower amplitude produce the high-cycle fatigue (HCF) damage. Herein, the stochastic HCF/LCF and creep damaging interactive effects are studied using simulated stress histories in

D.M. Ghiocel, H. Mao/First MIT Conference on Computational Fluid and Solid Mechanics

202

engine blades. Both the crack initiation and crack propagation evolution stages are considered. However, the paper focuses on crack initiation modeling aspects.

The total damage being defined by the sum of damages of all closed hysteresis loops. 2.2. Strain-life approach

2. Stochastic life prediction models 2.1. Crack initiation models Generally, stochastic stress/strain tensor in a blade location can be expressed by the equation of motion: X(r,5)=:g(X(r,5),£(r,5),D(/,5))

Z)(r, s) = h{X{t, s), E(t, 5), D(r, s))

For evaluating the stochastic crack initiation life, a local strain-life approach with a randomized strain-life curve parameters is used. The local notch plasticity is introduced using Neuber's rule [3]. Stochastic strain-life curve Sa-Nf is described by s, = ^i2Nf)'

(1)

where t is time and s is space coordinate. The stochastic stress-strain vector process, X(t,s), the input environmental/material vector process, the E(t,s), and the scalar damage parameter, D(t,s), are fully coupled. Such an approach includes both changes in strength and constitutive model using damage parameter as an internal variable in the material constitutive model. The damage growth depends on stress amplitude and reciprocally the stress amplitude depends on damage level. However, currently in engineering practice the influence of damage on stresses and strains is not considered. The damage accumulation models describe the damage evolution as a function of loading stress-strain history, or more specifically as a function of stress-strain closed cycle sequence. A key modeling aspect is to reduce the spatial stress/strain state problem to a uniaxial tensile stress/strain state problem similar to the lab test conditions, with an alternating stress component and a mean stress component. Most often, the equivalent (Von Mises) stress is used to define the alternating stress component and the hydrostatic stress (in fact the first stress invariant) to define the mean stress component [3]. There is a significant modeling uncertainty associated with the idealization of the multiaxial stress/strain case by a simple uniaxial stress/strain case. This modeling uncertainty should be reduced in the future through the development of more accurate physics-based strain-damage models based on stochastic micro-mechanics. This issue is not further addressed in this paper. After stress/strain state reduction to a simple uniaxial lab test case, typically the rainflow counting procedure is used to determine the closed stress-strain cycle sequence. The total cumulated damage due to cyclic loading can be directly computed by the convolution of damage function, DiX^m, ^max) with cycle counting distribution

+ 8'fi2Nfy

(3)

where the quantities a^, b, e'^ and c are considered to be random material parameters. The mean stress effect (including temperature, static, residual stresses from previous damages or processing, etc.) is included using a randomized Morrow, modified Morrow and Smith-Watson-Topper (SWT) correction procedures. An important aspect of using the strain-life curve is that it is possible to handle the random effects coming from surface finish, fretting effects, temperature effects, creep, etc. The mean stress correction procedures adjust Eq. (3) as shown below: (a) Morrow correction include mean stress effects for both the elastic and plastic strain terms:

-l(-5)

c/b

{2Nff

+ E'^ 1 _

^

(2Nf)

(4)

(b) Modified Morrow correction removes the mean stress effect in plastic strain term: E

(c) Smiths-Watson-Topper approach strain-life curve expression as follows: \^a —

(5)

G r

{a'f?

{INff+a'e'AlNf)-

changes

the

(6)

An approximately inverse function of strain-life equations (3-6) can be used to get the cycle life for a given pair of alternating strain and mean stress. The modeling uncertainty associated with mean stress correction is extremely large even for simple uniaxial lab tests. There is a high need in industry to set these mean correction procedures on a more adequate physical basis including key stochastic micro-mechanics aspects. 2.3. Cumulative damage mechanics models

DT = I d(t)dt=

-If

J2

^i^i^^i) dv dw

NT {V, U)

dvdu

Theoretically, any cumulative damage process is defined by its first-order differential kinetic equation (2)

dD = dN

f{D,N,Nf{Sa,cr^),p)

(7)

203

D.M. Ghiocel, H. Mao/First MIT Conference on Computational Fluid and Solid Mechanics Nc,f

1.00 0.90 0.80

HCF CYCLES

//|

0.70 LOF CYCLES

0.60

y\

—

10**4

—

10**5

//J

10**6

0.50

10**7

0.40 —

0.30 0.20

10**8 10**9

0.10 0.20

0.40

0.60

0.80

1.00

3. Computed results 3.1. Stochastic HCF/LCF interaction Fig. 4 shows the HCF/LCF interactive damage for the simulated stress profile given in Fig. 2. It should be noted that for this severe flight profile the vibratory stresses are highly damaging. About 90% of the damage produced is due vibratory stresses. Large vibratory stresses occur randomly at the minutes 9, 31 and 38 of the flight, as illustrated by the big three steps in the damage evolution. 3.2. Stochastic LCF/creep interaction

LIFE CYCLES, Nf

Fig. 3. Damage curves for different life levels. where constant amplitude cycle life is a function are alternating strain and mean stress. The letter p denotes the parameters of damage model. Experimentally, it has been shown that a damage curve, Nf{Sa, Om), can be accurately constructed based only on two experiments for extreme amplitude levels, i.e. maximum and minimum life levels. The damage curve parameters are determined so that for any arbitrary life, Eq. (7) can be applied. The greater the ratio between the (two) extreme life levels is, the more severe damage interaction is and the more deviation from linear damage rule is noted [4]. Herein, stress/strain amplitude-dependent cumulative damage mechanics models, such as Damage Curve Approach, Double Damage Curve Approach [4] and Lemaitre-Chaboche and modified Rabotnov-Katchanov [5] models were comparatively used. These stress-dependent damage models or nonlinear damage rule-based models capture adequately the complex HCF/LCF/creep damage interactions. Stochastic damage models were obtained by randomizing the deterministic ones shown in Fig. 3. A key stochastic modeling aspect is that the damages produced by cumulative damage mechanisms of different nature such as LCF, HCF, creep, impact loading, etc. are not directly additive. Thus, appropriate stochastic adjustments of damage curves have to be used in addition to the randomization of the model parameters.

Fig. 5 indicates the effect of creep damage (at 700°C) on Ufe prediction. Results were computed for pure LCF damage and LCF/creep damage. Both crack initiation and propagation stages are included. As shown in Fig. 5 there is a significant Hfe reduction due to creep. It is interesting to note two modeling aspects: (i) the probability density of predicted life has a skewed shape for pure LCF damage and relatively symmetric shape for LCF/creep damage and (ii) the coefficient of variation is smaller for the LCF/creep damage. This last remark is due to the fact that the creep damage effects are drastic during the crack initiation stage reducing severely the statistical spread between the short and long LCF simulated lives. 3.3. Critical modeling issues Only two critical modeling issues are investigated in this paper: (i) the modeling uncertainty induced the selection of probability density function of cumulative damage and (ii) the modehng uncertainty introduced by mean stress correction procedures in crack initiation life prediction. Fig. 6 shows a simulated histogram of the fatigue damage cumulated after 1000 flights (crack initiation stage).

LOF+HCF

UJ

2.4. Crack propagation models For crack propagation, three stochastic fracture mechanics models derived using (i) Forman, (ii) SINH and (iii) MSE models. The random effects of cyclic loading frequency, stress ratio, holding time, and temperature are incorporated in these crack propagation models.

J 1 ; Sa"25ksi

9

(5 Ii

0.60

;

0

5

10

15

20

25

30

35

\

40

HCF

45

50

55

TIME (Minutes)

Fig. 4. Damage evolution per flight.

60

65

70

DM. Ghiocel, H. Mao/First MIT Conference on Computational Fluid and Solid Mechanics

204

1

(a)

1

'

III

0.1

i

0.08

sm 0.06

1

/ 1

f //

S o

/tu^fs^^^^^suuilft

^111

^J^^^^^m^y Jy

0,02

1

^

^

I

SQOO

Number of Flights

100QO

1SQ00

1

1

1'

1

1

1""

I

>

1

1^

1

1

1

i

aoel-

•

I

0.04

aQ2 1

n

''

(b)

an

1

o

0.04

0.12 r

1

^

1Q0O2QQD3QOO4OQOSQOOeQOO?O0OaQ0O9ODO Number of Rights

Fig. 5. Simulated stochastic life, (a) LCF damage; (b) LCF/creep damage. Mean Stress Effect

70 60 50

I 40

I 30

u.

20 10 0

o o o

I/) O

o

05 O

t

Morrow Modified Morrow

• • CO T-

en CM

r^

CN

en

d Damage Index

5000

10000

15000

20000

Number of FlighAs

Fig. 6. Simulated histogram of damage.

Fig. 7. Mean stress correction effects.

The probability density function appears to have a skewed shape with an extreme long tail. A typical analytical distribution type does not fit accurately the histogram on the entire damage value range. It can be observed that the largest damaging flight sequence is a clear outlier for lognormal and Weibull probability density function. This result is a consequence of the highly nonlinear relationship between alternating strain and the incremental and cumulated damage. An early damaging flight condition in the component life may have a great impact on the life length. This modeling aspect has a great influence on component failure risks. Thus, it needs further research attention. Fig. 7 indicates the effect of using different mean correction procedures. The results are computed for deterministic flight conditions and material. The predicted life is 5000 flights for Morrow and SWT and 15,000 flights for modified Morrow. This result indicates that there is more

uncertainty in the predicted life due to modeling assumptions than due to randomness in the loading and material behavior. The research need for going in depth in the micro-scale physics of fatigue phenomena is obvious.

4. Concluding remarks The paper presents a simulation-based stochastic approach for component life prediction. Specifically, the stochastic life prediction a typical aircraft jet engine blade is studied. Computed results show that the state-of-the-art of engineering tools for evaluating fatigue and creep effects did not reach yet the level of highly accurate fife estimates. Thus, it appears that the most rational approach to component life prediction is to compute stochastic lowerbounds and best-estimates of lives which include the mostconservative and the experimentally-best-fitted engineering

DM. Ghiocel, H. Mao/First MIT Conference on Computational Fluid and Solid Mechanics models, respectively. A key role of reducing the modeling uncertainty and increasing the accuracy of life estimates is played by the on-going micro-mechanics research developments.

References [1] Ghiocel DM. ProbabiHstic fatigue life prediction for jet engine components: stochastic modehng issues. ECOMASS 2000, Barcelona, September, 2000, pp. 11-14.

205

[2] Ghiocel DM. Factorable stochastic field models for jet engine vibration response. The 13th ASCE Speciahty Conference, Baltimore, June, 1999, pp. 13-16. [3] Dowhng NE. Mechanical Behavior of Materials — Engineering Methods for Deformation, Fracture, and Fatigue. Englewood Cliffs, NJ: Prentice-Hall, 1993. [4] Halford GA. Cumulative fatigue damage modeling — crack nucleation and early growth. The 1st International Conference on Fatigue Damage in Structural Materials, Hyannis, MA, September, 1996, pp. 22-27. [5] Lemaitre C, Caboche F. Mechanics of Sohds. Amsterdam: Elsevier, 1998.

206

A discretization error estimator associated with the energy domain integral method in Hnear elastic fracture mechanics E. Giner*, J. Fuenmayor, A. Besa, M. Tur Departamento de Ingenieria Mecdnica y de Materiales, Universidad Politecnica de Valencia, 46022-Valencia, Spain

Abstract The implementation of the EDI method through the FEM introduces a discretization error that is inherent in the mesh and type of element employed. In this work, an error estimator for the evaluation of G or / in linear elastic problems in fracture mechanics is proposed, which is based on shape design sensitivity analysis. The reliability of the estimator is then analyzed solving a numerical problem using an /z-adaptive process. Keywords: Finite element method; Fracture mechanics; Error estimation; EDI method; /-integral; Sensitivity analysis; Adaptive refinement

1. Introduction In the context of linear elastic fracture mechanics (LEFM) there are two distinct approaches to the analysis of crack problems: one is the local approach, which is based on the well-known concept of stress intensity factor K (SIF) as a single characterizing parameter of the state of stress in the vicinity of a crack tip. The other is the so-called global or energetic approach and takes the strain energy release rate G (SERR) as the characterizing parameter of the problem. Both are directly related and have been shown to be equivalent as can be found in any text on fracture mechanics (e.g. [1]). Closed-form solutions for K have been derived for a small number of simple geometries and load configurations. In those real cases where complex geometries are involved, numerical methods have become customary, specially the Finite Element Method (FEM). The application of the FEM in order to obtain fracture mechanics parameters, such as K, G or the more general 7-integral can be done through a great variety of post-processing techniques. Those techniques related to the global approach are called indirect methods and they yield a value for G or / (both refer to the same concept in LEFM) by means

* Corresponding author. Tel: -h34 (96) 387-7626; Fax: -H34 (96) 387-7629; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

of virtual crack extension and stiffness derivative methods [2,3], contour integrals like the /-integral [4], their equivalent domain integrals, e.g. EDI [5-8] or the modified crack closure integral [9,10]. One major consideration in applying the finite element method to fracture mechanics is the order of magnitude of the error made in the calculation, which is greatly due to the so-called discretization error. This kind of error is inherent in the nature of the FEM and basically depends on the mesh and type of element used. Obviously, when the FEM is applied to the calculation of K or G, the discretization error introduces an error in the results for K or G. Besides, the post-processing technique chosen may be another source of errors. Some computationally efficient estimators for the FE discretization error are currently available. In this work we will make use of an extension of the Zienkiewicz-Zhu discretization error estimator [11] in order to study the influence of the global discretization error on the calculation of G when the well-known Energy Domain Integral Method is employed. The indirect (or energetic) methods can be applied to the whole domain of the problem and therefore they lend themselves to an estimation of the global discretization error. To obtain an efficient error estimator for G, an alternative approach to the EDI method is needed. As explained below, this approach is given by the shape design sensitivity analysis as applied to a crack problem. The effectivity

E. Giner et al. /First MIT Conference on Computational Fluid and Solid Mechanics of the proposed error estimator is then checked by means of a numerical verification.

2. The EDI method as a shape design sensitivity analysis The Energy Domain Integral method is one of the most efficient methods for obtaining / in an elastic (not necessarily linear) problem. It is essentially a domain integral which results of applying the divergence theorem under certain assumptions to the J contour integral [5-8]. Thus, for a 2D elastic problem, in absence of body forces and tractions on crack faces and assuming that the crack propagates in a self-similar manner, / can be calculated as:

'^/(^^^•£-S£'^

(1)

with /, 7 = 1,2 and where ^* is a portion of the domain of the problem which completely surrounds the crack tip, Gij and Ui are the stress and displacement fields, V^ is the strain energy density, <5/y is the Kronecker's delta and qi is a sufficiently smooth function which must take values between 0 and 1, subject to the conditions (see Fig. 1) q\{xx,X2) =

0

if (jci,;c2) e Ti

1

if (xi, X2) e Fs

(2)

The outer contour Fi and inner contour F3 are arbitrary: Fi can be the external boundary of the body (excluding the crack faces) and F3 is often reduced to a point (the crack tip). Physically, the qi function may be interpreted as a weight function which scales the virtual extension 8x1 of any point in Q* between 0 and 8a through the expression 8x1 = qi8a. deLorenzi [5] established a relationship between this method and the concept of material derivative of Continuum Mechanics. Recently, it has been shown [12,13] that

Eq. (1) can be obtained under the more general approach of shape design sensitivity analysis as applied to LEFM. The key assumption in the appUcation of SDSA to a crack problem is to interpret the crack length a as a design variable, whose change of length impUes a modification of the boundaries (shape). Several procedures are available to carry out a SDSA [14]. In Saliba et al. [12] and Taroco [13] a continuum approach is used to show that the sensitivity of the total strain energy t/ of a cracked component in LEFM is given by U = — = fa:VudQo+ da J

/ [WI - ( V u ) V l : Vvd^o J •n\

where ^0 is the domain of the problem, a is the stress tensor, Vii is the gradient of the sensitivity of the displacement field, I is the identity matrix and v is the so-called velocity field, which exactly corresponds to the qi function described above. Assume that the prescribed tractions T on the problem are held constant and that crack faces are traction free. If V satisfies the above conditions, then the sensitivity of the external work done by T (denoted here by V) equals the first integral in Eq. (3) for any kinematically admissible field li. This permits to establish an equivalence between the second integral in Eq. (3) and the Eq. (1). To do so, the change in sign must be taken into account since -G = fl = tl - V in LEFM (where 77 denotes the sensitivity of the total potential energy). Moreover, the first integral in Eq. (3) equals exactly 2G and therefore Eq. (3) is also a way of calculating G and it will be employed in this work.

3. Error estimation The error estimator proposed here is based on the underlying principle behind the Zienkiewicz-Zhu estimator [11], i.e. the unknown exact fields for discontinuous magnitudes of the FE solution are replaced with improved fields, derived from the same FE solution. In this work, Eq. (3) forms the basis of the proposed estimator in G, which can be defined for a FE discretization with ne isoparametric elements as ne ^es(G)

p

{a, - fffe) : [(Vii), - (Vu)fe] UI dQ,

(ff.

- fffe) (e« -

Sfe)

[(Vu). - (Vu)fe]T (
207

I

: VvlJI dfi,

(4)

where Q^e is the local domain of the reference element, J is the Jacobian matrix and e is the infinitesimal strain tensor. The improved fields in Eq. (4) are denoted by ( )*

208

E. Giner et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Mesh 1

Mesh 2

Mesh 4

Mesh 6

Mesh 10

Fig. 2. Model for periodic array of collinear cracks (Mode I). Sequence of deformed meshes (quadratic elements). 1

F O

<

1

1

1

1 1 1

1

1

-

-

E

E ^ ^ V c v:

^^^^^^^"^"^"^•'•^^-^

^

LU 0.1

1

L^

^^^^\^

m

>

1

P

^^'^^^^^--...,^^^

" ^ '^r r|ex( G ) (linear elements)

p

" ^ ' %^es{G)

F

^^^^^^<;;-

^c T]^^^ ^ ^ (quadratic elements) (quadratic elements)

" ' 9c X] -

-

1

1

-

^^^'^<<..

(linear elements)

1

1

1

1

._ 1

1 1 1

E

= : J

100

3-10 D.O.F.

Fig. 3. Exact and estimated relative errors in G (%) for the sequences of /z-adapted meshes: linear and quadratic elements. in contrast to the FE solution ( )fe. For linear elements, the improved fields were obtained through nodal averaging whereas a SPR technique [15] was employed for quadratic elements. Note that a sensitivity problem must be solved to getu. Using this error estimator an improved solution for G is given by G^es =

G f e + ees(G)

(5)

where Gfe is obtained either through Eq. (1) or Eq. (3). The relative error can be estimated as ^es(G)

=

^es(G) Gfe + ^es(G)

(6)

An effectivity index to validate the error estimator when the exact solution is known is defined as follows ^(G)

'7es(G) ^ex(G)

(7)

which should be close to unity, being

'?ex(G) =

C/gx — G f e — <^ex

(8)

4. Numerical verification In order to check the validity of the error estimator and its convergence with refined meshes, an /i-adaptive procedure was used. The specific problem discussed here is an infinite array of collinear cracks of the same length 2a in Mode I, whose exact solution for plane strain is [1]

E

v \

Kiex = O^jTta

[

2b

/7za\

1/2

(9)

E. Giner et al /First MIT Conference on Computational Fluid and Solid Mechanics n—\—I

209

I I

- a - 9 c / - \ (linear elements) -G- Q(^. (quadratic elements) •^^9

J

\

(ref. [16,17], linear elem.) .

\ l_i-

D.O.F.

Fig. 4. Reliability of the error estimator: effectivity index for the sequences of /z-adapted meshes: linear and quadratic elements. Effectivity indexes calculated according to [16,17] are included for comparison. having used a = \, b = 1 (half distance between similar points of two consecutive cracks), E = 10'^, v = 0.333 and a = 100. The height of the FE model was taken large enough {h = 6) to assume this exact solution as valid for comparison purposes. Fig. 2 shows the discretized model after deformation and some of the adapted meshes. The results for the estimated relative error r/es(G) are given in Fig. 3 compared to the exact relative error r^ex(G), both for linear and quadratic triangular elements. Note that the number of dof. of the first mesh is larger than for the second mesh due to the adaptive procedure. The reliability of the proposed estimator seems to be fairly high, as it is emphasized by the Fig. 4, where the effectivity index given by another error estimator in G [16,17] is included for comparison. Further refinement would not be appropriate in this example, since actually the FE model converges to a slightly different exact solution (which is unknown) due to its finite height. Other numerical examples yielded similar results and therefore this estimator can be regarded as acceptable.

5. Conclusions In this work, an error estimator for G based on a SDSA for linear elastic crack problems when solved through a FE analysis has been proposed. This estimator implies solving a sensitivity problem as well as computing improved fields for the FE solution. Through a numerical example, its high reliability has been checked. It has also been shown that this error estimator improves notably other estimators available in the literature.

Acknowledgements This work was financially supported by CICyT in the framework of research project PB97-0696-C02-02.

References [1] Kanninen MF, Popelar CH. Advanced Fracture Mechanics. Oxford Engineering Science Series. New York: Oxford University Press, 1985. [2] Parks DM. A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Fracture 1974;10:487-502. [3] Hellen TK. On the method of virtual crack extensions. Int J Numer Methods Eng 1975;9:187-207. [4] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379-386. [5] deLorenzi HG. On the energy release rate and the /-integral for 3-D crack configurations. Int J Fracture 1982;19:183193. [6] deLorenzi HG. Energy release rate calculations by the finite element method. Eng Fract Mech 1985;21(1): 129-143. [7] Li FZ, Shih CF, Needleman A. A comparison of methods for calculating energy release rates. Eng Fract Mech 1985;21(2):405-421. [8] Shih CF, Moran B, Nakamura T. Energy release rate along a three-dimensional crack front in a thermally stressed body. Int J Fracture 1986;30:79-102. [9] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 1977;9:931-938. [10] Shivakumar KN, Tan PW, Newman JC, Jr. A virtual crack closure technique for calculating stress intensity fac-

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tors for cracked three dimensional bodies. Int J Fracture 1988;36:R43-R50. [11] Zienkiewicz OC, Zhu JZ. A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 1987;24:337-357. [12] Saliba R, Venere MJ, Padra C, Taroco E, Feijoo RA. Shape sensitivity analysis and energy release rate of planar cracks embedded in three-dimensional bodies. In: Idelsohn S, Ofiate E, Dvorkin E (Eds), Computational Mechanics: New Trends and Applications. Proceedings Congreso Buenos Aires. Barcelona: CIMNE, 1998. [13] Taroco E. First and second order shape sensitivity analysis in fracture mechanics. In: Idelsohn S, Ofiate E, Dvorkin E (Eds), Computational Mechanics: New Trends and Applications. Proceedings Congreso Buenos Aires. Barcelona: CIMNE, 1998.

[14] Haug EJ, Choi KK, Komkov V. Design Sensitivity Analysis of Structural Systems. Volume 177 of Mathematics in Science and Engineering. Orlando, Florida: Academic Press Inc., 1986. [15] Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part I: The recovery technique. Int J Numer Methods Eng 1992;33:1331-1364. [16] Fuenmayor FJ, Oliver JL, Rodenas JJ. Extension of the Zienkiewicz-Zhu error estimator to shape sensitivity analysis. Int J Numer Methods Eng 1997;40:1413-1433. [17] Fuenmayor FJ, Dominguez J, Giner E, Oliver JL. Calculation of the stress intensity factor and estimation of its error by a shape sensitivity analysis. Fatigue Fract Eng Mater Struct 1997;20(5):813-828.

211

Micromechanical analysis of two-phase materials including plasticity and damage C. Gonzalez *, J. Llorca Polytechnic University of Madrid, Department of Materials Science, E.T.S. de Ingenieros de Caminos, Madrid, 28040, Spain

Abstract A model is developed to compute the mechanical behaviour of two-phase materials including the effects of damage. The material is represented by an interpenetrating network of randomly distributed spheres, which are assumed to behave as isotropic elasto-plastic solids. The incremental self-consistent method is used to compute the effective response of the material as well as the elastic stress redistribution due to damage. As an example, the model predictions are compared with experimental results — previously reported — for a particle-reinforced metal-matrix composite, which presented damage by reinforcement fracture during deformation. Keywords: Self-consistent method; Plasticity; Damage; Effective property; Particle-reinforced composite

1. Introduction Structural materials are usually made up of two or more phases which exhibit a nonlinear mechanical behaviour. Classical models assume that the volume fraction of each phase is constant. While this is often true, there are materials which exhibit phase changes triggered by the inhomogeneous stress and strain fields generated during deformation. This is the case, for instance, of progressive damage in metal-matrix composites by either reinforcement fracture or interface decohesion, which is known to play a critical role in their ductihty and strength [1]. In all these situations, the evolution of volume fraction of each phase is one critical factor to simulate with accuracy the mechanical behaviour. This paper presents an extension of the classical self-consistent model to analyze the mechanical behaviour of a twophase material, where a phase change (due to damage, or any other physical process) may occur during deformation.

subindex u) stands for the behaviour of the undamaged phase, while the second phase (identified by the subindex d) represents the damaged phase. The model assumes that the behaviour of each phase in the material is adequately represented by an isotropic, elasto-plastic soHd following the incremental (J2) theory of plasticity. The volume fraction of the damaged phase is given by p. The effective response of the two-phase material can then be computed by integrating along the loading path the effective strain hardening rate, which is given by da

dF

dW a?

+

mi

(1)

where the first term in (1) stands for the hardening contribution without any phase transformation. The second term introduces the stress redistribution due to the damage of dp material when the prescribed boundary conditions are held constant. The following two sections are devoted to calculate these terms.

2. Model description and application

2.1. Deformation without damage

The material is made up of two spherical phases forming an interpenetrating network of randomly distributed spheres [2]. It is assumed that one of the phases (identified by the

The strain hardening rate for the two-phase material without damage is given by the effective tangent stiffness tensor, L, which can be computed as

* Corresponding author. Tel.: +34 (91) 336-6419; Fax: +34 (91) 336-6680; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

L = (l-p)L,Au+pLdAd

(2)

where Au, and Ad stand, respectively, for the fourth or-

212

C Gonzalez, J. Llorca/ First MIT Conference on Computational Fluid and Solid Mechanics

der strain rate concentration tensors corresponding to the untransformed and transformed phases. To compute L is just necessary to determine the strain rate concentration tensors, which depend on the material properties as well as on the volume fraction, shape and spatial distribution of each phase. This problem was solved by Hill [3] within the framework of elasto-plastic deformation for an isolated ellipsoidal inclusion of phase / [4].

600

a.

S L/i

The stresses and strains in the undamaged phase change after each strain increment and, as a result, a dp volume fraction of material is transformed. Two hypotheses are necessary to compute the associated stress redistribution. Firsdy, it is assumed that the damage occurs very rapidly (as compared to the strain rate) and thus that the prescribed boundary conditions remain constant. Secondly, the damage leads to an elastic stress relaxation in the effective material, J a , which can be calculated derivating its elastic constitutive equation given by a = L"€ei and thus — = U'—^-\€,, (3) dp dp dp where a and ?^/ stand for the effective stress and elastic strain prior to damage. The right expression in (3) is a set of equations in which the terms of da/dp and d?^//dp corresponding to prescribed boundary conditions are zero. The derivative of the overall elastic stiffness tensor is also computed using the same self consistent method.

Peak-aged condition

500

-

300

on r/i dJ UN

2.2. Stress redistribution due to damage

(a)

7,00

Self-consistent simulation

100

Experimental

f , , ,,

!

1

1

1

1

1

1

1

1

2

1

1

I

I

I

3

Strain (%) 600

Naturally aged condition

-(b)

500

-x

.400

300

Self-consistent simulation Experimental

2.3. Model application

1

0

The model developed in the previous section was used to compute the tensile stress-strain curve for a 2618 Al alloy

r

1

2

1

4

1

1

1

6

1

8

1

1

10

1

12

Strain (%) Fig. 2. (a) Model (dashed) and composite (solid) curves for the tensile stress-strain behaviour of the peak-aged metal-matrix composite, (b) Idem for the naturally aged composite.

COMPOSITE

INTACT

I

1

DAMAGED

Fig. 1. Geometric representation of a particle-reinforced metalmatrix composite as an interpenetrating network of intact and damaged regions.

reinforced with 15 vol.%. SiC particles [5]. It was found that the dominant damage mechanism during deformation was reinforcement fracture, the SiC particles being broken by cracks perpendicular to the loading axis. According to the model, the composite was represented as an interpenetrating network of two spherical phases, both formed by the metallic matrix surrounding either an intact or broken SiC particle (Fig. 1). The broken SiC particles contained a penny-shaped crack perpendicular to the loading axis. The constitutive equation for each region (intact or damaged) in the composite was determined through the finite element analysis. The fraction of broken particles was assumed to be governed by a Weibull statistic. The simulations of the tensile stress-strain curves were in reasonable agreement with the experimental results, as shown in Fig. 2(a) and (b).

C. Gonzalez, J. Llorca/First MIT Conference on Computational Fluid and Solid Mechanics

213

3. Conclusions

References

A model to compute the effective response of a twophase material including the effect of damage was developed. It was assumed that damage occurs instantaneously and leads to an elastic stress relaxation. The stress redistribution was computed by solving the nonlinear set of equations obtained by derivating the equations of the selfconsistent method, where the elastic values of the stiffness and strain concentration tensors are used.

[1] Llorca J, Gonzalez C. Microstructural factors controlling the strength and ductility of particle-reinforced metal-matrix composites. J Mech Phys Solids 1998;46:1-28. [2] Gonzalez C, Llorca J. A self-consistent approach to the elasto-plastic behaviour of two-phase materials including damage. J Mech Phys Solids 2000;48:675-692. [3] Hill R. Continuum micro-mechanics of elastoplastic polycrystals. J Mech Phys Solids 1965;13:89-101. [4] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc Roy Soc London 1957;A241:376-396. [5] LLorca J, Martin A, Ruiz J, Elices A. Particulate fracture during deformation of a spray formed metal matrix composite. Metall Trans 1993;A24:1575-1588.

214

Accurate analysis of shell structures by a virtually meshless method K. Goto^'*, G. Yagawa% T. Miyamura''

" University of Tokyo, Department of Quantum Engineering and Systems Science, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ^ The Institute of Physical and Chemical Research, Materials Fabrication Laboratory, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

Abstract A three-node triangular shell element is developed for the Free Mesh Method (FMM), which is a virtually meshless method modified from the node-by-node finite element method. To apply the FMM to the analysis of general shell structures, it is important to employ an accurate three-node triangular shell element. For this purpose, the discrete Kirchhoff triangular element is improved by introducing the mixed method to the membrane stiffness. Finally, an illustrative example is presented. Keywords: Finite element method; Free mesh method; Meshless method; Shell structures; Three-node triangular element; Mixed method

1. Introduction

2. Free mesh method

The Free Mesh Method (FMM, [1]) is a virtually meshless method, which is a kind of the node-by-node finite element method. In the FMM, elements are automatically created around each node in a local manner, and then a conventional node-by-node finite element analysis is conducted with those elements. Because the processes from the local mesh generation to the construction of equations are seamless and independent in every node, the FMM can be easily implemented on parallel environments [2]. A three-node triangular element is, however, desirable as a local element used in the FMM. It is known that the accuracy of membrane behavior of three-node triangular elements is poor in comparison with that of four-node quadrilateral elements. Hence it is important to develop the three-node triangular element that is accurate enough in membrane deformations to apply the FMM to the analyses of general shell structures. In this research, the three-node discrete Kirchhoff triangular element [3] is used with the FMM, and its membrane behavior is improved by using mixed method [4].

In the FMM, global mesh is not necessary as input data, but only the nodes distributed in the analysis domain and the boundary conditions are used as input data. Fig. 1 shows the conceptual figure of the FMM. First, a node is selected as a central node and nodes within a certain distance from the central node are selected as candidate nodes. This distance is usually decided from the prescribed density of the distribution of nodes. Then satellite nodes are selected from the candidate nodes, which form the local elements around the central node. For each local element,

* Corresponding author. Tel.: -h81 (3) 5841-7005; Fax: -hSl (3) 5841-6994; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

• Central node ® Satellite nodes © Candidate nodes o Othernodes

Lx)cal area

Temporary local elements

Fig. 1. Concept of FMM.

K. Goto et al. /First MIT Conference on Computational Fluid and Solid Mechanics the element stiffness matrix is constructed in the same way as the FEM. Then, the rows of the matrix concerned with the central node are stored in memory. Thus, the stiffness concerned with the central node is evaluated. The above procedures are carried out for all nodes.

3. Three-node triangular shell element To make the procedures in FMM independent, it is preferable that the element is triangular without mid-nodes. In this point of view, the three-node discrete Kirchhoff triangular (DKT) element is adopted in this study. The DKT element is suitable for thin shell analysis and has six degrees of freedom per each node. The total element stiffness matrix is formulated by superimposing a plane stress membrane stiffness, a bending stiffness and an in-plane rotational stiffness. The membrane stiffness is the constant strain plane stress stiffness of a three-node element. In this study, the membrane behavior of the three-node DKT element is improved by using the mixed formulation. In the conventional displacement formulation, only the displacement is treated as the problem variable that will be independently approximated. When the displacement formulation is used for three-node element, the stress and the strain that are the function of the differential of the displacement are constant within an element, and are approximated discontinuously in the analysis domain. When the mixed formulation is used, it is possible to approximate independently not only the displacement but also the stress and the strain. In this case, the stress and the strain are approximated continuously in the analysis domain.

4. Illustrative example The Scordelis-Lo roof shown in Fig. 2 is analyzed to demonstrate the present element. It is loaded vertically by its uniformly distributed dead weight of intensity of

215

Table 1 Normalized displacement of Scordelis-Lo roof Model

3-Node DKT with mixed formulation (FMM)

4-Node quadrilateral (MARC)

2x2 4x4 6x6 8x8 16 X 16

1.231 1.030 0.993 0.983 0.985

1.186 0.925 0.941 0.953 0.971

6.2055 X lO^Pa. In this problem most part of the strain energy is due to membrane deformation. The geometrical and material data of the problem are: radius R = 0.635 m, length L = 1.27 m, thickness h = 6.35 mm, arc AB = 40°, Young's modulus E = 2.979 x lO^^Pa and Poisson's ratio V = 0. The exact value of vertical displacement at point C in a steady state is -7.838 mm [5]. Because of the symmetry of the geometry and the load, only one quarter of the roof is analyzed. Table 1 shows the vertical displacement at point C normalized by the exact value. The models are labeled as / X j where the integers / and j indicate numbers of nodal spacing along arcs AB and DC, and sides BC and AD, respectively. It is compared with the result obtained with a four-node quadrilateral element using MARC, which is a commercial FEA code. It is observed that almost the same accuracy is achieved by introducing the mixed method to the membrane stiffness with the same node distribution.

5. Concluding remarks To apply the FMM to the analysis of shell structures, it is important to develop an accurate three-node triangular shell element. For this purpose, the DKT element was improved by introducing the mixed method to the membrane stiffness. As a result, almost the same accuracy as the four-node quadrilateral element was attained with the present 'meshless' scheme.

References

rigid diaphragm Fig. 2. Scordelis-Lo roof

[1] Yagawa G, Yamada T. Free mesh method: a new Meshless finite element method. Comput Mech 1996;18:383-386. [2] Yagawa G, Fumkawa T. Recent developments of free mesh method. Int J Numer Methods Eng 2000;47:1419-1443. [3] Bathe KJ, Ho LW. A simple and effective element for analysis of general shell structures. Comput Struct 1980;13:673-682. [4] Zienkiewicz OC, Kui LX, Nakazawa S. Dynamic transient analysis by a mixed, iterative method. Int J Numer Methods Eng 1986;23:1343-1353. [5] MacNeal RH, Harder RL. A proposed standard set of problems to test finite element accuracy. Finite Elements Anal Des 1985;1:3-20.

216

An implicit time integration strategy for use with the material point method J.E. Guilkey^'M.A. Weiss'' ^ Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA ^ Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA

Abstract An implicit integration strategy for use with the Material Point Method (MPM) is described. This strategy uses an incremental-iterative solution strategy based on a Newton method to solve the equations of motion and Newmark integration to update the kinematic variables. An example problem was used to compare the implicit integration scheme to the traditional explicit integration scheme used with MPM, as well as with integration methods used with the Finite Element Method. Keywords: Implicit integration; Material point method

1. Introduction The Material Point Method (MPM) as described by Sulsky et al. [1,2] is a particle method for structural mechanics simulations. The method uses a regular structured grid as a scratchpad for computing spatial gradients. The grid also functions as an updated Lagrangian reference frame, moving with the particles during advection, then being reset to its original position at the end of a timestep. In addition to avoiding Eulerian diffusion, the method avoids the mesh entanglement problems frequently encountered with large deformation finite element calculations. Additionally, contact algorithms do not require searches for contact surfaces [3]. The use of the regular grid has also been exploited for doing fluid-structure interaction problems [4]. By sharing the grid with a multimaterial CFD code, tight coupling between the two phases can be achieved, while each phase still enjoys the benefits of its traditionally preferred frame of reference. One limitation of this approach has been that the stable timestep sizes for explicit time integration for the solid and fluid are often disparate by several orders of magnitude, with the solid phase requiring the smaller

* Corresponding author. Tel.: +1 (801) 585-5145; Fax: +1 (801) 585-9826; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

timestep to maintain the conditional stabiUty of the explicit integration scheme. An implicit integration strategy was implemented to alleviate the small timestep required by the explicit integration strategy for the solid phase. The approach borrowed heavily from the strategies traditionally used in implicit Finite Element Method (FEM) calculations. The many similarities between these two methods will allow improvements to the algorithm described herein based on the large amount of work which has been done with implicit FEM codes.

2. Implicit time integration algorithm 2.1. Incremental-iterative solution of the linearized equations of motion A derivation of the linearized equations of motion in matrix form can be found in any standard finite element textbook (e.g., [5]). Linearization of the matrix form of the equations of motion about the current time t yields KK^-^ (r + dr) • du^ = Fext^(t -h dO - Fint^"^ (t + dr)

- M , - a ^ - H r + dr).

(1)

Here KK„ is the stiffness matrix, du„ is new estimate of the

J.E. Guilkey, J. A. Weiss / First MIT Conference on Computational Fluid and Solid Mechanics incremental displacements, Fextg is the vector of external forces at the new time t -\-dt, Fint^ is the vector of internal forces to to the stress divergence, Mg is the mass matrix and 2Lg is the acceleration vector, k is the current iteration number, t the current time, and dt the increment in time. By solving Eq. (1) for the current estimate of incremental displacements du^, a new estimate for the displacement was obtained via

n\{t + dt) = n\-\t + dt) + du\

(2)

With the new total displacements, the other kinematic variables, stiffness matrix, and internal forces were updated. On the background grid, note that u^ {t + dO is not the total displacement, but is the displacement from t to r + dr. The effects of the total displacement of the material were contained in the positions and total deformation gradient of the particles. Iteration continued until convergence is achieved, as determined the following criteria: du^„

du^

< €d

and

(3)

duOQ^

where Q^ is the right hand side of Eq. (1). 2.2. Kinematic update via trapezoidal

rule

Once the nodal displacements u^(f + dt) were determined, the trapezoidal rule was used to find the nodal velocities, \g{t + dt), and accelerations, a„(r + dt): dt u,(t + dt) = - ( v , ( 0 + VgCr -h d o ) .

(4)

217

Interpolate to the grid: Particle data is interpolated to the grid to obtain M^, \g{t) and Fextg(r + dr): Mi = ^Sipmp,

\i =

Mt

(7)

¥exti=j:^SipFexip, where / refers to individual nodes of the grid. Sip is the trilinear shape function of the ith node evaluated at x^. Iterative solution of equations of motion on grid: The linearized equations of motion on the grid are solved iteratively using Newton's method. (1) Initialization: For the first iteration ( k = l ) , assume: u ^ - i a + dO = 0,

¥intl-\t-]-dt)=Finig(t), KK'-\t^dt)

= KKg(t).

Note that since Ug refers to the displacement of the grid between t and t-^dt, the first of these corresponds to resetting the grid back to its original undeformed configuration. The material points remain in their deformed locations. (2) Solve for du^: For iteration k, invert Eq. (1) to get the current estimate for the displacement increment. (3) Update kinematics on the grid: Using Eqs. (2) and (4), solve for u^(f + dt) and v^(r -h dt). (4) Update stress divergence and tangent stiffness on the grid: The total deformation gradient PQ^^^ is computed via a recursion relation, P^^'^' = F;+^'F[). Vvi'p(t + dt) = Gpul{t + dt) F'pit + dt) = (Vu^(r + dt) + I) F ^ ( 0

dt y^it + dO = v g ( 0 + - (ag(0 + ag(r + dO)

(5)

( / ( r + dO is determined from F^ and any relevant history variablesD^(r -h dt) follows from o^

Eq. (4) can be solved for \g{t + dt), and, when (4) and (5) are combined with Eq. (2), the acceleration for the current iteration k at time t + dt can be approximated in terms of known quantities at time t and estimates at time t -hdt from the previous iteration k — 1:

Here, G^ is the gradient of the interpolation functions evaluated at x^, and D^(r -\-dt) is the spatial elasticity tensor. Integrate to get the internal force vector and the material and geometric stiffness on the grid.

<^' + ^^) =dt^;^ K~'(^ + ^^) + K) - xv.(0 - ag(o-

Fint^(r + dO =J2f

^^p^^ + ^^^ "^^

Kmsitl(t-{-dt) =

^l(t + dt)Bldv J2j^lK

Kgeo*(/ + dt) =J2f

B L < ( ^ + dOBj;^ dt;

(6)

This value for a^C? -h dO is used in Eq. (1). 2.3. Computational

algorithm

Known quantities: At the beginning of each implicit timestep, including the initial one, the following particle quantities are known at time t\ mass m^, volume Vp, position Xp(0, velocity yp{t), deformation gradient ¥p{t) and Cauchy stress cfpit). The known quantities on the grid are: Fintg(r),KKg(Oandag(0.

KK^ (t + dO = Kmat^ (t + dO + KgeoJ (t + dt) 4 d^2

^

Here, B [ is the linear strain displacement matrix at x^, and B ^ ^ is the non-linear strain displacement matrix at

J.E. Guilkey, J.A. Weiss/First MIT Conference on Computational Fluid and Solid Mechanics

218

3. Numerical example: pressurization of a cylinder

Xp. J2^ represents the standard finite element assembly operation, performed in this case on the regular grid mesh. (5) Convergence criteria: Convergence is checked using Eq. (3). Once a converged solution has been reached:

One scenario of interest to our research group is the response of a steel container filled with an energetic material (explosive) to a pool fire. Phase change of the contents results in pressurization of the container. A simplified problem is used here for demonstration. A one-quarter symmetry, plane strain model of a long cylindrical container with properties p = 7.86 x 10"^ kg/cm^ ^ = 1.66 x 10^ N/cm^ and G = 7.70 x 10^ N/cm^, subject to pressurization via the load curve F = 71.1 x 10^ N/cm^ x time, was modeled with implicit and explicit MPM, and FEM (using NIKE3D and DYNA3D). For the explicit analyses, timestep size was 1 X 10"^ s, while for the implicit analyses a timestep size of 4 X 10""^ s was used. 2720 particles were used for the MPM calculation (Fig. 1, top left), while 340 trilinear finite elements were used for the FEM calculations (Fig. 1, bottom left). This provided approximately equivalent resolution since the material points function as integration points and the finite elements had eight integration points per element. Contours of von Mises stress at time 0.006 s demonstrated that differences between the explicit and implicit time integration schemes were small within a com-

Save ¥p{t + dr), Fint^Cr + dr), KK^(r + dt). Compute Sig{t + dt) using Eq. (6). Interpolate Ug{t -\- dt) and ag(r + dr) to the particles: Up(t + dt) = J2^ipUi{t^dt),

(8)

ap(t -^dt) = J2 S'P^ii^ + dr).

(9)

Update the particle position and velocity: Xp(t + dr) = Xp(t) + Up{t + dr),

(10)

\p{t + dr) = \p{t) + 5 (a^(r) + a^(r + dr)) dr.

(11)

Continue to the next timestep. Otherwise, return to step 2 and continue the iterations. j 1 i i i i-jr: i i i i i MM! j I I M I [1 |!|j:W4UT^ j j M M M h j i i 1 M M

'•^fj|||i/|7%^

/jfj^^y^jj 1 1 j h j j i 1 I i MfIJlffffi^ i^M^^fflfe.! i 11! h i h

WMii T T' ^ ^ 555*W^

J M 1 1 1 i i

'] 1 j

M

MM M i •''(^A^}t^'/Z'(^S4''^U4i! MM i

:: Mn m I w ^SPm i j i ^i -

MM MT^^^fgi^m j 'h MlwM^i^ i Ml ^^$S$%F

1 11 lill ll^g^P 1 ' i j l j M ' - J - - J4iU4444>^ 20

25

30

35

0

5

10

15

20

25

20

30

25

30

35

6.0 X 105 N/cni2

Fig. 1. Cylindrical container subject to pressurization at fime r = 0.006 s. Soludons via Material Point Method (top) and Finite Element Method (bottom). Contours indicate von Mises stress distribution. Left column - computational grids. Center column - results for implicit time integration.

J.E. Guilkey, JA. Weiss /First MIT Conference on Computational Fluid and Solid Mechanics

219

Table 1 Quantitative comparisons of displacement and stress at inner, middle and outer radial locations on the cylinder

Inner Middle Outer

MPM Disp. (cm)

FEM Disp. (cm)

MPM von Mises Stress (N/cm^)

FEM von Mises Stress (N/cm^)

2.30 2.17 2.03

2.13 1.96 1.85

1.61 X 10^ 1.40 X 10^ 8.08 X 10^

1.72 X 10^ 1.34 X 10^ 1.08 X 10^

putational technique (Fig. 1). However, the results for the MPM analyses had larger circumferential and radial variations in von Mises stress than the FEM analyses. This can be attributed to the use of a rectilinear computational grid for the calculations. Although the algorithm can readily handle non-rectilinear grids, our current implementation requires a rectilinear grid. The variations decreased with increasing grid resolution. The rightmost frames show results from the explicit codes. The asymmetry of the stress distribution is more pronounced for the explicit MPM results. Explicit MPM is known to have difficulty in situations involving quasistatic loading, being better suited to highly dynamic problems. The implicit version clearly performed better for this particular situation. Quantitative comparisons of von Mises stress and radial displacement between the two implicit methods demonstrated generally good agreement (Table 1).

entanglement and the ability of the method to be coupled with CFD calculations. Because of the similarities between MPM and FEM, the implicit solution strategy approach can be easily modified to accommodate quasi-Newtonian solution methods. The BEGS method introduced by Matthies and Strang [6] is an obvious choice as it has proven to be robust for a wide range of nonlinear problems in solid mechanics.

Acknowledgements This work was supported by the U.S. Department of Energy through the Center for the Simulation of Accidental Fires and Explosions, under grant W-7405-ENG-48.

References 4. Conclusions An implicit integration strategy was developed and implemented for MPM. The algorithm accommodates much larger timesteps than the explicit version of MPM without any apparent loss in accuracy for the problem presented here as well as other test problems. Timesteps several thousand times larger than the CFL condition have been used successfully. Additionally, the implicit method performs far better for quasistatic loading scenarios. Solution differences between the MPM and FEM can be attributed to the use of a nonconforming computational grid for the MPM calculations. Although this may appear to be a disadvantage of the method, the limitation is counteracted by the ability to treat extremely large deformations without mesh

[1] Sulsky D, Chen Z, Schreyer HL. A particle method for history dependent materials. Comput Methods Appl Mech Engrg 1994;118:179-196. [2] Sulsky D, Zhou S, Schreyer HL. Application of a particle-in-cell method to solid mechanics. Comp Phys Commun 1995;87:236-252. [3] Bardenhagen SG, Brackbill JU, Sulsky D. The material-point method for granular materials. Comput Methods Appl Mech Engrg 2000;187:529-541. [4] Kashiwa BA, Lewis MW. Fluid-structure interaction modeling. LA-13255-PR 1997;1:283-295. [5] Bathe K-J. Finite Element Procedures. New Jersey: PrenticeHall, 1996. [6] Matthies H, Strang G. The solution of nonlinear finite element equations, Int J Numer Methods Eng 1979; 14:16131626.

220

Computation of reliability of stochastic structural dynamic systems using stochastic FEM and adaptive importance sampling with non-Gaussian sampling functions Sayan Gupta, C.S. Manohar* Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India

Abstract The problem of computation of reliability of randomly excited linear structural dynamical systems with stochastic parameter uncertainties is considered. The statistical fluctuations in the system properties are modeled as non-Gaussian random fields with bounded ranges. The displacement fields are discretized using frequency dependent shape functions and the random fields using covariance dependent shape functions. An adaptive importance sampling scheme that uses non-Gaussian sampling functions is developed to evaluate failure probabilities. Specific non-Gaussian sampling distribution functions, that account for the bounded range of system property random fields, are constructed by invoking principle of maximum entropy. Numerical results illustrative of successful application of methods developed are presented. Keywords: Stochastic finite element; Maximum entropy method; Reliability; Failure probability; Adaptive importance sampling

1. Introduction

2. Dynamic stiffness of stochastic curved beams

A simulation based method for the computation of reliability of stochastically parametered curved Timoshenko beams under random loadings is developed. This study is in keeping with the current research interest in the vibration analysis of structures with parameter uncertainties [1,2]. The proposed method is based on evaluation of stochastic dynamic stiffness of the beam elements. Subsequently, Monte Carlo simulations are performed for computing the failure probabilities. The following are the salient features of this study: (a) discretization of the displacement fields using frequency and damping dependent shape functions [3], (b) modeling the system properties as non-Gaussian random fields with bounded ranges thereby allowing for strict positivity of the physical parameters, (c) use of random field discretization scheme that retains the non-Gaussian nature of the random fields [4] and (d) estimation of failure probabilities using a newly developed adaptive importance sampling scheme which employs non-Gaussian sampling functions.

The problem of evaluation of the dynamic stiffness matrix of curved Timoshenko beams with randomly varying Young's and shear moduli, mass density, damping coefficients and cross-sectional dimensions has been studied recently by the present authors [5]. In this study, the system properties have been modeled as jointly homogeneous random fields. The information available on these random fields is taken to be limited to their range, mean and covariance functions. The range of these random fields are constrained to ensure the strict positivity of the physical parameters. This automatically implies that these fields are non-Gaussian. The partial information available on these random fields has been complemented, by first invoking the principle of maximum entropy to construct the first order probability density functions (pdf), which are then combined with the information on the covariance functions to arrive at Nataf's models. This leads to marginal density functions of the form

* Corresponding author. Tel. +91 (80) 309 2667; Fax: +91 (80) 3600 404; E-mail: [email protected]

where, the unknowns Ai, A2 and A.3 are determined by solving the following set of equations

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

f{x) = Aiexp[-A.2X - X^ix -

fif]

(1)

S. Gupta, C.S. Manohar / First MIT Conference on Computational Fluid and Solid Mechanics D

f

fix) Ax = 1.0

(2)

xfix) dx = fi

(3)

a b

/

scheme depends on the choice of the sampling function ^z(l)- The procedure for selecting the sampling density function, as proposed by Bucher [6], involves the generation of samples according to the original density function /x(f). which are employed to evaluate the conditional moments (Z),

a b (x

I

- ii)^f{x)dx

= a^

(4)

Here, a, b denote the range and /x and a^ are, respectively, the mean and variance. The study further employs frequency and damping dependent shape functions to discretize the displacement fields. The system property random fields have been discretized using covariance dependent shape functions. The system equilibrium equation in frequency domain has been shown to be of the form \y{(D, Xo)YM = F.

(5)

Here, D((X>, Xo) is the stochastic dynamic stiffness matrix with Xo being the A/^-dimensional vector of non-Gaussian random variables resulting from discretizing the random fields and F is the vector of amplitudes of harmonic excitations, which could be random. The focus of this paper is on evaluating probability of failure with the performance function given by g(X):

a-

max

(|D(a;, Xo)~^/C|)

(6)

where X = {Xo, F} is the extended vector oi N -\-\ random variables with joint pdf /x(f). The probability of failure Pf can be computed by evaluating the A^ + 1 dimensional integral

^/

S '^ (l)df.

(7)

hzih

•

(^ 11 G g(M) < 0) /x(^)

In this study, we propose to evaluate this probability of failure by using adaptive importance sampling simulation procedures.

3. Adaptive importance sampling using non-Gaussian sampling function

(9)

(zz')/.z(i) = (^^' I ^ ^ sib
(10)

Here, (•) denotes the mathematical expectation. This is followed by the formulation of N -\- I dimensional normal density with mean and covariances computed from Eqs. (9) and (10). This normal pdf is chosen to be the importance sampling density function. In our studies, we encountered difficulties in evaluating failure probabilities below a certain level when this sampling density function was used. This difficulty has been attributed to the small variance associated with the sampling density function. To circumvent this difficulty, we propose to use Nataf's model for the sampling density function. To realize this, we first estimate the first order pdfs of samples in failure region by invoking maximum entropy principle. This leads to first order pdfs of the form as given in Eq. (1). The parameters of this pdf are now estimated by using the conditional mean and variances as given in Eqs. (9) and (10). Subsequently, the sampling density function according to Nataf's model is obtained as hz,...zAl)=(t>V,...Vn(^U---,^nAp])

(l>Vi(^l)--'(t>Vn(in)

(11)

where, Vi,... , K are standard normal variates obtained by the transformations on Zi, . . . , Z„ given by ^i =
g(^)
111

i= h

, n.

(12)

Here, hz^,...,Zn = d/d§/{Hz^.(fj)} is the marginal probability distribution function and ^y^ .y^(^i,... , fn, [/)]) is the multivariate normal probability distribution function with zero mean, unit standard deviation and unknown correlation coefficient matrix [p] [7]. These correlation coefficients pij are expressed in terms of the correlation coefficients ^tj of Z through the integral relation

For a importance sampling function /iz(f), the probability of failure is well known to be given by

= y"%(f)
(8)

where, I is an indicator function taking values of unity if ^(1) S 0 and 0 otherwise and IZ spans the range of the random variables. The efficiency of the importance samphng

a a

i,j = l,...,n. These equations are solved iteratively to obtain py.

(13)

S. Gupta, C.S. Manohar / First MIT Conference on Computational Fluid and Solid Mechanics

222

/ K =100 N / m / /\ A A f(t) = F„e' / _y V M 0 = 2 0 k g / / / / / / / / / / /^/ / / /

_u

o o

Fig. 1. Example 1. Single degree of freedom system with random mass, stiffness, damping and excitation.

,-^ ; ^^

'

1 .O

1.4

i-; I..

1" 2^0.4 0.2

3

; 2 ^^r'

1

,-''

n

^-"'

--^V. \

Fig. 2. Example 1. Marginal probability density function of the random perturbation on mass: (1) parent density, (2) a = 0.06 m (3) a = 0.07 m. -•- Importance Sampling i - e - Direct Simulation |

^.

and 0.07 respectively. The fact that this distribution is bounded between dzVs x0.05 must be noted. The estimates of probability of failure are shown in Fig. 3 as a function of threshold values a. This figure also shows results from extensive Monte Carlo simulations (with sample size of 10"^). The mutual agreement between the two results is found to be good. 42. Example 2 A harmonically driven curved Timoshenko beam with randomly inhomogeneous mass density is considered next (Fig. 4). The mass field is modeled as m(0) = m^Ll + 6^/(0)] where t/(0) is a zero mean Nataf random field with samples bounded in the region ± ^ 3 and covariance function of the form R{T) = exp[—yr^] with y = 13. This random field is discretized using optimal linear expansion that leads to six random variables. The moment F is modeled as a Gaussian random variable with mean 10 kNm and standard deviation 10 x 0.05 kNm respectively. The performance function is as per Eq. (6) with a taken to range from 0.0017-0.012 rad. The initial Monte Carlo simulation run was done for threshold value a = 0.0017 rad with 1000 samples. The estimation of probability of failure subsequently employed 500 samples as per density given in Eq. (11). Fig. 5 shows the resulting estimates of probability of failure.

[1.H0"

m = 2850 kg/m^ 0

Q-10"

11

Eo=2.1 X 10 N / m 160 Ns/m

0 10-^

•

— 0.35 m * ^0.05

0.06

0.07 0.08 0.09 Threshold a, m

L = 100 m

0.11

Fig. 3. Example 1. Estimates of probability of failure using importance sampling and Monte Carlo simulations.

Fig. 4. Example 2. Curved Timoshenko beam with random mass variafion; radius of the beam = 82.03 m.

4. Numerical examples and discussion 4.1. Example 1 Fig. 1 shows a harmonically driven single degree of freedom system. Here, the nominal values of stiffness and damping are perturbed by random variables which have a range in ±y/3 x 0.05. The excitation amplitude is assumed to be Gaussian with unit mean and standard deviation of 0.05. The procedure described in the previous section is employed to compute probability of failure as a function of the threshold value a. Fig. 2 shows the marginal pdf of the perturbation on the mass variable associated with the importance sampling density function for a = 0.05, 0.06

0

0.002

0.004 0.006 0.008 Threshold a, rad

0.01

0.012

Fig. 5. Example 2. Failure probability using adaptive importance sampling.

S. Gupta, C.S. Manohar /First MIT Conference on Computational Fluid and Solid Mechanics 5. Conclusions A frequency domain stochastic finite element analysis is combined with an adaptive importance sampling simulation procedure to compute the probability of failure of randomly parametered curved beam structures that are excited by harmonic loads with random amplitudes. The procedure outlined handles successfully the non-Gaussian nature of beam property random fields both in stochastic finite element analysis as well as in importance sampling computations. Limited numerical results that are presented show successful application of the proposed method.

References [1] Manohar CS, Ibrahim RA. Progress in structural dynamics with stochastic parameter variations: 1987-1998. Appl Mech Rev ASME 1999;52(5): 177-197.

223

[2] Schueller GI (Guest Editor). A state-of-art report on computational stochastic mechanics. Probab Eng Mech 1997;12(4):198-321. [3] Adhikari S, Manohar CS. Dynamical analysis of framed structures with statistical uncertainties. Int J Numer Methods Eng 1999;44:1157-1178. [4] Li C, Der Kiureghian A. Optimal discretization of random fields. ASCE J Eng Mech 1993;119(6):1136-1153. [5] Sayan Gupta, Manohar CS. Dynamic stiffness method for circular stochastic Timoshenko beams: Response variability and reUability analysis. J Sound Vib, submitted. [6] Bucher CG. Adaptive sampling — an iterative fast Monte Carlo procedure. Struct Safety 1988;5:119-126. [7] Der Kiureghian A, Liu PL. Structural reliability under incomplete probability information. J Eng Mech ASCE 1986;112(1):85-104.

224

Accuracy of analytical approaches to compressive fracture of layered solids under large deformations Igor A. Guz *, Costas Soutis Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK

Abstract Based on the results obtained within the scope of the model of piecewise-homogeneous medium and 3-D stability theory (i.e. the most accurate approach), the accuracy of the continuum theory of compressive fracture is examined for layered solids undergoing large deformations. The investigation is carried out for the cases of uniaxial and biaxial compression as applied to compressible and incompressible, elastic and elastoplastic, isotropic and orthotropic layers. For all these cases, the asymptotic accuracy of the continuum theory is rigorously proved. The influence of the type of loading, layer thickness and their stiffness on the continuum theory accuracy is illustrated by several numerical examples for the particular linear and non-linear models of materials. Keywords: Composite; Instability; Compression; Fracture; Non-linear; Large deformation; Homogenization

1. Introduction The wide usage of the continuum theory in solid mechanics, due to its simplicity, puts into consideration the question of its accuracy and of its domain of applicability. The answer may be given only by comparison of the results delivered by both the continuum theory and the most accurate approach (i.e. the piecewise-homogeneous medium model). Indeed, the approach based on the model of piecewise-homogeneous medium (Fig. la), enables the investigation of the mechanical response in the most rigorous way at the microstructural level (exact solution). However, due to its complexity, this method is restricted

to a very small group of problems. This makes the continuum theory more attractive since it involves significant simplifications (Fig. lb). The continuum theory may be applied when the scale of the investigated phenomenon (for example, the wavelength of the mode of stability loss /) is considerably larger than that of a material structure (say, the layer thickness h), i.e. I ^ h. The results obtained by the continuum theory must follow from those derived using the model of piecewise-homogeneous medium when hl~^ -> 0. If this is the case, the continuum theory can be considered as an asymptotically accurate one.

2. Investigation of accuracy of the continuum theory matrix

fibre (layer)

Fig _-^. 1. (a) Model of piecewise-homogeneous medium; (b) Continuum approximation. * Corresponding author. Tel.: +44 (20) 7594-5117; Fax: +44 (20) 7584-8120; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

This paper is devoted to substantiation of the continuum theory applied to predict compressive fracture of layered solids (composites or rocks) with periodical structure. Within the scope of this theory, the moment of stability loss in the structure of material (internal instability according to Biot [1]) is treated as the beginning of the fracture process [2]. In the past, investigations of the continuum theory accuracy in relation to the model of piecewise homogeneous medium were performed only for other physical phenomena (for example, for the problems of wave propagation) or for other layer models [2-5]. But, there are not yet

225

LA. Guz, C. Soutis /First MIT Conference on Computational Fluid and Solid Mechanics investigations of the influence of biaxiality of loading for problems of stability loss in solids under finite (large) deformations. This paper attempts to fill the gap. Along with the exact approach (i.e. the continuum theory or the model of piecewise-homogeneous medium which are based on the 3-D stability theory [6]), there are also approximate approaches to the considered problems proposed by Rosen [7] and by many other authors. A detailed review of the approximate models was given, for example, by Soutis [8]. However, the approximate approaches do not describe the phenomenon under consideration even on the qualitative level. It is proved [2-5], that they give a significant discrepancy in comparison with the exact approach and with experimental data.

1st (shear) mode, cannot be described by the continuum theory. Estimation of accuracy of the continuum theory can be obtained by comparison with the critical values, calculated using the model of the piecewise-homogeneous medium [4,5,10]. The influence of the biaxiality of loading, layer thickness and their stiffness on the continuum theory accuracy was studied for several particular linear and non-linear (including elastoplastic) models of materials. Special attention was given to calculation of the continuum theory accuracy for composites when the layers were assumed to be hyperelastic and the simplified version of Mooney's potential, namely neo-Hookean potential, may be chosen for their description in the following form (2)

c|>-2Cio/i(4) 3. Asymptotic analysis and numerical results The investigation was carried out for the cases of uniaxial and biaxial compression as applied to compressible and incompressible, elastic and elastoplastic, isotropic and orthotropic, linear and non-linear models of layers under finite (large) deformations (Fig. 2). For all these cases, characteristic determinants were derived for the plane and for non-axisymmetrical 3-D problems [2,9,10] using the model of piecewise-homogeneous medium and 3-D stability theory (i.e. the most accurate approach) for four modes of stability loss. To perform the asymptotic analysis, the condition of applicability of the continuum theory hl-

0

where Cio is a material constant, and l\{s) is the first algebraic invariant of Cauchy-Green strain tensor. This potential is also called Treloar's potential, after the author who obtained it from an analysis of model of rubber regarded as a system of long molecular interlinking chains [11]. The accuracy of the continuum theory A (i.e. the ratio of the results obtained in the context of the most accurate approach and continuum theory expressed in percentage) is given in Figs. 3-5 for different models of layers (including the above-mentioned hyperelastic) and different values of layer thickness ratio, hrjhyn. These dependencies have a strongly non-linear character proving the importance of

(1)

was applied to all formulae and the limits are calculated analytically under this condition. As a result of such manipulation, the long-wave approximation was obtained and the characteristic equations were reduced to a much simpler form. It was rigorously proved that the results of the continuum theory follow as a long-wave approximation from those for the 1st mode of stability loss obtained using the model of piecewise-homogeneous medium. It was also shown that modes of stability loss, other than the

00-1

[

99-

/

98 -

97-

—^96-

h Am = 0.2\

hAm = o.n hAm = 0A2

95 -

94-

93 -

/>

^

;^

Fig. 2. The co-ordinate system and applied loads for the cases of biaxial compression.

1

1

1

1

1—— 1

1

1

20

1

1

1

1—

25

Fig. 3. Values of parameter A plotted against the ratio of the material constants of layers CIQ/C'I'Q for the case of Treloar's potential (uniaxial compression).

226

LA. Guz, C. Soutis /First MIT Conference on Computational Fluid and Solid Mechanics

100

4. Conclusions The asymptotic accuracy of the continuum theory of compressive fracture is established for composites consisting of compressible and incompressible, elastic and elastoplastic, isotropic and orthotropic layers. Following the general 3-D approach developed in this paper, the accuracy of the continuum theory as applied to laminated solids with other properties of layers or other kinds of loads can also be investigated. 0

20

40

60

Fig. 4. Values of parameter A plotted against the ratio of the material constants of layers C\Q/C^Q for the case of Treloar's potential (biaxial compression). 100

biaxial compression uniaxial compression 84 20 40 60 80 100 120 Fig. 5. Values of parameter A plotted against the ratio of Young's moduli of layers Er/E,n for the case of linear elastic layers. vspacel.5pt

taking into account the materials' non-linearity. One can also see that the larger the ratio hr/h„j, the higher is the accuracy of the continuum theory. It means that the increasing volume fraction of the stiffer layers has a strong impact on the application of the continuum theory making it more accurate.

References [1] Biot MA. Mechanics of Incremental Deformations. New York: Wiley, 1965. [2] Guz AN. Mechanics of fracture of composite materials in compression (in Russian). Kiev: Naukova Dumka, 1990. [3] Guz lA, Soutis C. Continuum fracture theory for layered materials: investigation of accuracy. Z Angew Math Mech 1999;79(S2):S503-S504. [4] Guz lA, Soutis C. A 3-D stability theory applied to layered rocks undergoing finite deformations in biaxial compression. Eur J Mech A/Solids, to appear. [51 Soutis C, Guz lA. On analytical approaches to fracture of composites caused by internal instability under finite deformations. In: Soutis C, Guz lA (Eds), Impact and damage tolerance modelling of composite materials and structures. Proc. of Euromech Colloquium 400, London: Imperial College, 1999, pp. 51-58. [6] Guz AN. Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies. Berlin: Springer, 1999. [7] Rosen BW. Mechanics of composite strengthening. In: Fiber Composite Materials. Metals Park: American Society of Metals, 1965, pp. 37-75. [8] Soutis C. Failure of notched CFRP laminates due to fibre microbuckling: a topical review. J Mech Behav Mat 1996;6(4):309-330. [9] Guz lA. Spatial nonaxisymmetric problems of the theory of stability of laminar highly elastic composite materials. Sov ApplMech 1989;25(12):1080-1085. [10] Guz I A. Internal instability of laminated composites with a metal matrix. Mech Comp Mater 1990;26(6):762-767. [11] Treloar LRG. Large elastic deformations in rubber-like materials. In: Proceedings of lUTAM Colloquium, Madrid, 1955, pp. 208-217.

227

Computational elasticity based on boundary eigensolutions A.R. Hadjesfandiari, G.R Dargush * Department of Civil Engineering, State University of New York at Buffalo, Amherst, NY 14260, USA

Abstract The theory of fundamental boundary eigensolutions for elastostatic problems is applied to formulate methods for computational mechanics. This theory shows that every elastic solution can be written as a linear combination of some boundary orthogonal deformations. One finds that the traditional boundary element method and finite element methods are largely consistent with this theory, but do not harness its power. Use of the new theory permits, for example, the systematic solution of non-smooth problems. Keywords: Finite element method; Boundary element method; Non-smooth problem; Eigenvalue problem

1. Introduction The general theory of fundamental boundary eigensolutions is presented in Hadjesfandiari [1]; Hadjesfandiari and Dargush [2-4]. Here we present application of this theory to computational mechanics, and more specifically to the development of boundary element and finite element methods for elastic bodies. The major traditional methods of computational mechanics do not have a common means to enforce boundary conditions. For an elastic boundary value problem, the traditional finite element method uses lumped nodal forces to model the tractions in a very approximated manner, but as a result generates a symmetric stiffness matrix. On the other hand, the standard boundary element method uses tractions as primary variables, but generates non-symmetric matrices. The theory of fundamental boundary eigensolutions not only gives a new common view to both methods, but also directs us in modifying these methods and in understanding the source of some ill behavior. The computational methods based on this theory are completely consistent with the theory of elastostatic boundary value problems, including all of those problems that are classified as non-smooth. This theory shows that the resulting computational methods are indirectly a general discrete Fourier analysis. The introduction of a weight function

* Corresponding author. Tel: +1 (716) 645-2114/2405; Fax: +1 (716) 645-3733; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

alters the underlying orthogonal basis functions, thus enabling us to solve non-smooth problems systematically.

2. Theory of fundamental boundary eigensolutions The fundamental boundary eigenproblem for elastostatic problem can be defined as follows: Find the non-trivial displacement u such that in the domain V ^ijJ = CijkiUk,ij — 0

(la)

and on the boundary S (lb)

ti = X(pij Uj

In Eqs. (1), a, t and C represent the stress tensor, traction vector and elastic constitutive tensor, respectively, while X is the eigenparameter. Furthermore ^ is a positive definite, integrable tensorial weight function defined on the boundary S. Notice that this definition permits cpij to be discontinuous and even singular at some points. The eigensolutions of Eqs. (1) have a number of interesting and useful properties. The most important properties include the following: (1) All of the eigenvalues are real. (2) All non-zero eigenvalues are positive. (3) The sequence of eigenmodes are orthogonal on the boundary with respect to 0 . Thus, (m)

/

(n) 1 c

(fijU- Uj ^ dS ••

(2)

228

A.R. Hadjesfandiari, G.F. Dargush /First MIT Conference on Computational Fluid and Solid Mechanics

by assuming normalized eigenmodes. (4) The system of eigenfunctions is complete. As a result, these fundamental eigensolutions provide a basis for solutions to elastostatic boundary value problems in the form of generalized Fourier series or fundamental eigen-expansion

This problem has an infinite number of eigensolutions (A„,M^"^) which are boundary orthogonal with respect to In terms of u and ^^, the boundary integral representation Eq. (7) reduces to Cij(^)uj(^) + j Fij(^, x)uj(x) dS(x)

(3)

f

and on the boundary

= /

OQ

t = ^-J2^nKu^"^

on 5

(4)

with A„= f u 0

i/<"^ dS=

f cpijUiU^"^ dS

(5)

We assume that in physical problems u is continuous everywhere, but that t can be piecewise continuous. This allows t to exhibit discontinuities, and even singularities. With the present approach, we attempt to choose 0 such that the weighted traction ^^ is piecewise regular. Thus f^, defined by the relation

still may have discontinuities, but it now remains bounded everywhere on S. Then, the expansion for f^ is DC

/^ = ^A„A,M^">

on 5

(6)

3. Boundary element methods The boundary integral representation for the elastostatic problem without body force can be written Cij(^)uj(^) + j Fiji^, x)uj{x) dS(x)

I

= /

Gij{^,x)cpjk(x)t;^{x)dS(x)

Following [5], by discretizing the boundary into a finite number NE of elements, utilizing low-order polynomial shape functions within the elements and collocating at the nodes, we obtain a system of algebraic equations that can be written (10)

FU = G^r^

where U and T"^ represent nodal values of displacement and weighted traction, respectively, while F and G^ are system matrices formed through an assembly process. By using the fundamental boundary conditions, the boundary element version of the fundamental eigenproblem is FU

XG U

(11)

While G^ in Eq. (10) is in general a rectangular matrix to allow for discontinuity in weighted traction T^, the matrix G for the eigenproblem Eq. (11) is a square version of G^ due to the continuity requirement inherent in the fundamental boundary condition T"^ = XU. We expect boundary orthogonality of the eigenmodes with respect to 0 in closed-form from Eq. (2). In discretized form this becomes U(m)^S'^UM : 0

m^ n

(12)

where 'NdS

Giji^,x)tj(x)dS(x)

(9)

(13)

(7) with shape function matrix N(x). Since S depends on the boundary discretization and weight function 0 we call it the weighted boundary matrix.

where G(^,x) and F{^,x) are the elasticity kernels and c(^) is a tensor that characterizes the local geometry at 5 [5]. By substituting the fundamental boundary condition tj(x) — X(pjk{x)uk{x) into Eq. (7), we obtain the fundamental eigenproblem in integral form as

4. Finite element methods

Cij{^)uj{^) + j Fij{^, x)Uj(x) dS(x) s

The formulation can be derived from the principle of virtual work or weak formulation in the form

= xj

Giji^, x)(pjk{x)u,(x) dS(x)

(8)

/ ajj hSij dV = j (pijt'J huj dS

(14)

A.R. Hadjesfandiari, G.F. Dargush /First MIT Conference on Computational Fluid and Solid Mechanics Discretizing the domain and boundary, and interpolating weighted traction on the boundary, we obtain

/

W^B^CBUdV

(15)

s where C represents the elastic constitutive tensor in matrix form and B is the usual matrix of shape function derivatives [6]. Introducing the usual stiffness matrix K and the new matrix 5*^ from Eq. (13), this can be written (16) Finally, since W"^ is arbitrary, we establish KU =

Partitioning the left-hand side of Eq. (17) to correspond with the right-hand side, we obtain KBI

Kl

Ku

(18)

where V B and U i are the vectors of nodal displacement for boundary and interior nodes, respectively. In terms of boundary nodes, we can write 'KBBUB

(19)

= S^T^

where KBB is the boundary stiffness matrix defined by KBB

= Kt

KBIKJJK^J

(20)

The corresponding generalized fundamental eigenproblem can also be formulated strictly in terms of boundary nodes and written as KBBUB

— ^S

Mode

4 8 15 23 25 40 60 80 100 150 190

Exact

BE

FE

0.76923

0.76923 1.2821 2.1376 3.0769 3.4263 5.3853 8.4469 11.532 16.637 27.914 34.581

0.76933 1.2821 2.1370 3.0770 3.4237 5.3867 8.4859 12.084 19.845 47.074 126.47

3.0769

5.3846 8.4615

-

= 0

f/(m)T5V^"^ = 0

on the boundary. Both traction-oriented finite element and boundary element methods are investigated. A FEM mesh with 1345 nodes and 432 quadrilateral elements has been used. The number of nodes on the boundary is 96, thus forming 48 quadratic boundary elements. The eigenvalues for some eigenmodes are listed in Table 1. The modes with exact eigenvalues are completely shear deformations. Closed-form expressions were obtained in [4]. It is seen that for lower modes, FEM has reasonably good eigenvalues similar to those of BEM. For higher modes, the eigenvalues in FEM become less accurate. However, increasing the number of internal nodes in FEM improves the eigenvalues and eigenmodes toward those obtained via BEM. This clearly shows why BEM can often solve problems more accurately for a given boundary discretization. In practice for FEM we usually increase internal and external nodes together. In this way with an

(21)

UB

Because KBB and S are symmetric, the eigenproblem associated with this traction-oriented finite element method has real eigenvalues and eigenvectors, which are orthogonal with respect to KBB and S U^'^^^KBBU^''^

Table 1 Boundary eigenvalues for unit disc

(17)

0

KB

229

for m y^ n formT^n

undeformed • deformed-60

(22) (23)

Solutions U of Eq. (19) implicitly utilize the eigenvectors of Eq. (21) as a basis.

5. Numerical examples 5.1. Eigenmodes of unit circular disc Consider an elastic circular disc with radius a = I. Here we generate the fundamental eigenmodes for the plane strain case with E = 1 and v = 0.3, assuming cp = I

Fig. 1. Generalized Eigenproblem for FE.

230

A.R. Hadjesfandiari, G.F. Dargush /First MIT Conference on Computational Fluid and Solid Mechanics

I

I

to

I

5.2. Plate with edge notch

f

E=1, v=0.3

W

^

^

^

^

t t

Fig. 2. Notched plate.

FEM approach we increase the number of eigenmodes and improve the lowest ones. The FEM eigenmode 60 is shown in Fig. 1. This deformation is in good agreement with the closed form solution.

We now apply the new boundary element and finite element methods for plane strain loading of a plate with an edge V-notch. Here we consider the geometry and boundary conditions shown in Fig. 2. Let h = 5, w = 5, a = 1 and to = I, while 2a = 270° where a is the included half-angle at the notch. Material properties are E = 1 and v = 0.3. For stress analysis at the notch tip we can use a multi-region method, but here we use half-symmetry and model only the upper portion of the plate. From the asymptotic expansion of Williams [7] we know the singularity of stresses for free-free edges is r^~^ where y = 0.544484. Then the weight function cp = r^~^ is used on the cut line. On the rest of the boundary, we take (p = 1. In all cases, (pij = (p8ij. In the numerical analysis, a mesh with 200 boundary nodes and 100 quadratic boundary elements is used. Meanwhile, the finite element domain model consists of 600 eight-noded quadratic elements. Fig. 3 provides the numerical solutions for the weighted traction t'^ versus horizontal distance from the tip of the notch. Solutions away from the tip are converged. However, Gibbs' phenomenon is clearly visible in the vicinity of the notch. The boundary element solutions show much lower amplitude oscillation. This can be attributed to the improved resolution of the higher fundamental eigenmodes obtained with the BE formulation. Discontinuity induces

2.00

1.50

^J^

1.00

0.50

0.00 0.00

0.20

0.40

0.60

r Fig. 3. Notched plate. Weighted normal traction.

0.80

A.R. Hadjesfandiari, G.K Dargush /First MIT Conference on Computational Fluid and Solid Mechanics participation from higher modes, and thus requires better accuracy of those modes to resolve the boundary variable. We should emphasize that in the FE formulation utilized here, the traction, or in this case weighted traction ff, is a primary variable that is interpolated to the same level as the displacement u. The traction component ^J is related to the general stress intensity factor Ki defined for the notch. Recent research has shown that the value of Ki is a controlling parameter for failure analysis of some materials [8,9].

6. Conclusion The theory of fundamental eigensolutions gives a new view to the theory of elastostatic boundary value problems and their numerical solution. The numerical formulations based upon boundary element and finite element methodologies that have been developed here remain valid even for non-smooth problems associated with notches, cracks and mixed boundary conditions. Most mathematical models of practical engineering problems are non-smooth. For example, mixed boundary conditions may be specified, reentrant comers may be present or bi-material interfaces may exist. In non-smooth problems, using the proper weight function (p to make ^*^ piecewise regular has several advantages. Most importantly, calculations are then based on bounded functions. However, t"^ may be discontinuous. This results in oscillations associated with Gibbs' phenomenon. Additionally, the Fourier coefficients decrease faster for higher modes. This means that the participation of higher modes are less important than for the case with cp = I. Consequently we may expect higher quality solutions for a given mesh when cp is chosen properly.

231

Acknowledgements Support for the work described in this paper was provided in part by the Multidisciplinary Center for Earthquake Engineering Research under a cooperative agreement from the National Science Foundation (Grant EEC-970147 1). The authors gratefully acknowledge this support.

References [1] Hadjesfandiari AR. Theoretical and computational concepts in engineering mechanics. Ph.D. dissertation, State University of New York at Buffalo, 1998. [2] Hadjesfandiari AR, Dargush OF. Theory of boundary eigensolutions in engineering mechanics. J Appl Mech ASME, in press. [3] Hadjesfandiari AR, Dargush OF. Computational mechanics based on the theory of boundary eigensolutions. Int J Numer Meth Eng 2001;50:325-346. [4] Hadjesfandiari AR, Dargush OF. Boundary eigensolutions in elasticity. I. Theoretical development. Int J Solids Struct, in press. [5] Banerjee PK. The Boundary Element Methods in Engineering. London: McGraw-Hill, 1994. [6] Bathe KJ. Finite Element Procedures. Englewood Cliffs, NJ: Prentice Hall, 1996. [7] WilHams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech ASME 1952;19:526-528. [8] Carpinteri A. Stress singularity and generalized fracture toughness at the vertex of re-entrant corners. Eng Fract Mech 1987;26:143-155. [9] Dunn ML, Suwito W, Cunningham S. Stress intensities at notch singularities. Eng Fract Mech 1997;57:417-430.

232

Stochastic response of nonlinear structures A. Haldar'''*, S.Y. Lee^ J. Huh'' " University of Arizona, Department of Civil Engineering and Engineering Mechanics, Tucson, AZ 85721, USA ^ Yonsei University, Department of Civil Engineering, Seoul 120-749, Korea

Abstract A finite element-based reliability evaluation procedure is proposed to evaluate the risk of linear and nonUnear structures subjected to static and short-duration time-varying loading including seismic loading. It is parallel to the deterministic finite element method, except that it can incorporate information on the uncertainty in the variables present in the problem. It is capable of capturing any special features that can be handled by the finite element method, making it a robust reliability evaluation technique. Keywords: Reliability analysis; Finite element analysis; Nonlinear analysis; Stochastic finite element analysis; Seismic loading; Response surface method

1. Introduction The analytical procedures to calculate the nonlinear deterministic response of structures to both static and dynamic loading have matured significantly in recent years. It is not difficult now to track the load path to failure considering complicated geometric arrangements, realistic connection and support conditions, and various sources of nonlinearity. Since it is not possible to avoid the uncertainty in the load and resistance related variables, the focus has shifted to incorporating uncertainty into deterministic computational algorithms. Finite element analysis is a very powerful tool commonly used by many engineering disciplines to analyze simple or complicated structures. The word 'structure' is used in a broad sense to include all systems that can be discretized using finite elements. With this approach, it is easy and straightforward to consider complicated geometric arrangements and constitutive relationships of the material, realistic connection or support conditions, various sources of nonlinearity, and the load path to failure. It gives good results for a set of assumed values of the variables while ignoring the uncertainty in them. On the other hand, many of the available reliability methods are able to account for

* Corresponding author. Tel.: +1 (520) 621-2192; Fax: +\ (520) 621-2550; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

the uncertainties, but fail to represent the structural behavior as realistically as possible, and can be computationally challenging when the performance function is not available in an explicit form [1]. If the basic variables are uncertain, every quantity computed during the deterministic analysis is also uncertain, being a function of the basic variables. The currently available reliability methods can still be used if the uncertainty in the response can be tracked in terms of the variation of the basic variables at every step of the deterministic analysis. The finite element method (FEM) provides such an opportunity, and this concept forms the basis of the stochastic finite element method [2]. With the advances in computer technology, it is quite appropriate to develop a finite element-based reliability analysis technique, parallel to the deterministic analysis procedure. Most engineers will have a tool to estimate the risk or reliability of simple or complicated systems considering all major sources of uncertainty and nonlinearity as realistically as possible. The authors have developed a finite element-based algorithm to estimate the reliability or probability of failure of structures, capturing the nonlinear behavior just before failure. The authors call it the stochastic finite element method (SFEM) or probabilistic finite element method (PFEM) [2]. It will be of interest to researchers working to advance the deterministic finite element concept. It will also be of interest to the general risk and reliability research community, since it is a powerful and robust reliability method

A. Haldar et al. /First MIT Conference on Computational Fluid and Solid Mechanics that can be used for both imphcit and explicit performance functions.

2. Concept In general, nonlinear complicated structural systems are expected to have implicit performance functions when subjected to static and dynamic loadings. Several computational approaches could be pursued for the reliability analysis of structures with implicit performance functions. They can be broadly divided into three categories, based on their essential philosophy [2], as: (1) Monte Carlo Simulation; (2) response surface approach; and (3) sensitivity-based approach. The sensitivity-based approach can be implemented in the context of the first- or second-order reliability method (FORM or SORM) and the finite element method. In the application of FORM or SORM, only the value and gradient of the performance function at each iteration are required in the search for the design or checking point. The value of the performance function can be estimated from deterministic structural analysis. The gradient can be calculated using sensitivity analysis. In the case of explicit performance function, the gradient is calculated simply by analytical or numerical differentiation. For the implicit performance function, several approximate methods can be used to compute the gradient of the performance function, e.g. finite difference, classical perturbation, and iterative perturbation methods. Combining the iterative perturbation and the finite element approaches, an SFEM-based reliability evaluation procedure is discussed next. The concept is applicable to both linear and nonlinear problems.

3. Methodology SFEM-based reliability evaluation procedures for static and dynamic loadings are discussed briefly and separately. 3.1. Static loading The reliability analysis procedure for static loading is based on FORM. The formulation requires an expression for a limit state function G(x, u, s), where vector x denotes the set of basic random variables pertaining to a structure (e.g. loads, material properties and structural geometry), vector u denotes the set of displacements involved in the limit state function, and vector s denotes the set of load effects (except the displacement) involved in the limit state function (e.g. stresses, internal forces). The displacement u can be expressed as u = QD, where D is the global displacement vector and Q is a transformation matrix. In general, x, u and s are related in an algorithmic sense, for example, a finite element code. The algorithm evaluates the performance function deterministically, with the corre-

233

sponding gradients at each iteration point. It converges to the most probable failure point (or checking point or design point) and calculates the corresponding reliabihty index p. The following iteration scheme is used to find the checking point: G(y,) y;+i = y;«. + |VG(y;)|

(1)

where AG(y) =

Oli =

-

9G(y) dyx

9G(y)

AG(y,) |AG(y,)|

(2)

(3)

and \ 9G,D I JD,X + -r~ j;.i (4) ) ^^. In Eq. (4), J^y are the Jacobians of transformation and j / ' s are statistically independent random variables in the standard normal space. The evaluation of the quantities in Eq. (4) will depend on the problem under consideration (linear or nonlinear, two- or three-dimensional, etc.) and the performance functions used. The essential numerical aspect of SEEM is the evaluation of three partial derivatives, namely, 9G/9s, aG/9u and dG/dx, and four Jacobians, namely, J,,;^, J^,^, JD,X, and J3;,;,. These are briefly discussed next. AG =

9G, {^dG •hx + 1 Q

3.1.1. Performance functions and partial differentials The safety of a structure needs to be evaluated with respect to predetermined performance criteria. The performance criteria are usually expressed in the form of limit state functions, which are functional relationships among all the load effects and resistance-related parameters. Two types of limit state functions are commonly used in the engineering profession: the Hmit state function of strength (axial load, bending moment, combined axial and bending moment, etc.), which defines safety against extreme loads during the intended life of the structure, and the limit state function of serviceabiHty (lateral deflection, interstory drift, etc.), which defines the functional requirements [1]. 3.1.2. Evaluation of Jacobians and the adjoint variable method To evaluate the gradient VG, the evaluation of the three partial derivatives on the right-hand side of Eq. (4) is necessary. They are easy to compute since G(x, u, s) is an exphcit function of x, u and s, as discussed in the previous section. The next task is to evaluate the four Jacobians in Eq. (4). Because of the triangular nature of the transformation, J^^ and its inverse are easy to compute. Since s is not an explicit function of the basic random variables x, J^,;^ = 0. The Jacobians of the transformation Js,D and JD,X, however, are not easy to compute since s.

234

A. Haldar et al. /First MIT Conference on Computational Fluid and Solid Mechanics

D and x are implicit functions of each other. The adjoint variable method [3] is used here to compute the product of the second term in Eq. (4) directly, instead of evaluating its constituent parts. It is accurate and computationally efficient when a large number of basic random variables are involved in a problem. An adjoint vector X can be introduced such that (5)

du ds The adjoint vector X depends on the limit state function being considered. It is not possible to derive all these equations due to lack of space; however, it will be discussed in detail during the presentation. The reliability of linear and nonlinear two- and three-dimensional structures can be evaluated using the concept. Special features like partially restrained connections or support conditions are incorporated in the algorithm in addition to geometric and material nonlinearities. It is expected that any features that can be modeled by the finite element algorithm can also be incorporated in the algorithm. The accuracy of the algorithm is established by comparing the information on risk estimated by the algorithm with the Monte Carlo simulation technique. Several examples on trusses, frames, frames with infilled shear walls, etc., will be given during the presentation to show the application potential of the concept to various types of structures. 3.2. Dynamic loading Section 3.1 discusses the SFEM-based reliability analysis procedure for static, time-invariant loads. Many engineering systems are subjected to both short and long duration time-variant loadings. Short duration loading, particularly seismic loading, is of considerable interest to engineers since it has enormous damage potential. Thus, the SFEM-based algorithm needs to be developed for short duration time-variant loadings. In general, the reliability analysis of nonlinear structures in the time domain is very difficult. Recently, Huh [4] suggested a method. The algorithm intelligently integrates the concept of the response surface method, the finite element method, and FORM. Since the performance function of a nonlinear dynamic structural system is implicit, the response surface method is used to approximately generate

the performance function and FORM is used to calculate the corresponding reliability index, the coordinates of the most probable failure point, and the sensitivity indexes for the random variables involved in the problem. It cannot be presented here due to lack of space, but will be discussed in detail during the presentation with the help of examples.

4. Conclusions A finite element-based reliability evaluation procedure is proposed to evaluate the risk of linear and nonlinear structures subjected to static and short duration time-varying loads. It is parallel to the deterministic finite element method except that it can incorporate information on the uncertainty in the variables present in the problem. It is capable of capturing any special features that can be handled by the finite element method. The concept appears to be robust and accurate.

Acknowledgements This paper is based on work partly supported by the National Science Foundation under Grant CMS-9526809. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsor. References [1] Haldar A, Mahadevan, S. Probability, Reliability and Statistical Methods in Engineering Design. New York: John Wiley and Sons, 2000. [2] Haldar A, Mahadevan, S. Reliability Assessment Using Stochastic Finite Element Analysis. New York: John Wiley and Sons, 2000. [3] Ryu YS, Haririan M, Wu CC, Arora JS. Structural design sensitivity analysis of nonlinear response. Comput Struct 1985;21(l/2):245-255. [4] Huh J. Dynamic reliability analysis for nonlinear structures using stochastic finite element method, Thesis, Department of Civil Engineering and Engineering Mechanics, University of Arizona, 1999.

235

A continuum mechanics based model for simulation of radiation wave from a crack Sixiong Han^'*, Mingkui Xiao^ ^Research Laboratory of Geomechanics, Etowa-ru Tokorozawa 301, Kitaakitsu 885-3, Tokorozawa 359-0038, Japan ^ Department of Civil Engineering, Chongqing Jianzhu University, Chongqing, China

Abstract This paper proposes a numerical model for the description of the mechanical phenomenon of radiation wave field due to dynamic crack-propagation. It is shown that the mechanical effect of crack-propagation can be reduced to a set of equilibrating forces acting at the position of cracking if we use the finite element method. In the paper, the formulations for this approximation are derived in displacement-controlled wave field. Both Mode-I and Mode-II crackings are considered in this study. Keywords: Dynamic cracking; Equivalent nodal force; Cracking mode; Wave propagation

1. Introduction The properties of waves due to dynamic crackings have been widely applied with success to a variety of engineering problems. However, the fundamental mechanisms responsible for the radiating wave phenomena caused by the dynamic cracking are not yet thoroughly understood. In order to make wave information quantitative and to utilize such waves in material research, basic studies to clarify generation mechanisms of radiation waves due to dynamic crack-propagation are required and some more sophisticated analytical methods are expected to be developed to describe the dynamic crack-propagation problem. The aim of this paper is to establish a mathematical model providing a numerical approximation to describe the mechanical phenomenon of dynamic crack-propagation. This model is established based on Betti's reciprocal principle and the discretization technique of the standard finite element method. Instead of modeling the crack directly, the mechanical effects of the radiation waves due to the crack-propagation are formulated by a set of equilibrating nodal forces acting at the positions of cracking based on the rigorous mechanics theory. The methodology of this procedure is midway between the conventional theoretical analyses and numerical models. There are two significant * Corresponding author. Tel/Fax: +81 (42) 996-5338; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

characteristics of the proposed model that are different from the conventional theoretical and numerical analyses on this type of problem. The first one is that, until now, the dynamic crack-propagation is usually modified as the traction releasing process ahead of the crack-tip, and treated in a stress-controlled wave field. In this study, we consider the cracking as a displacement loading process and treat the crack-propagation in a displacement-controlled wave field. The second one is that the cracking domain is formulated through a singularity function and this operation could avoid the treatment of the mathematical discontinuity in the Euclidean space. This procedure leads to the advantage of the independence between the mesh division and crack configuration. It is shown that the cracking problem can be treated in the framework of continuum mechanics and the radiation waves due to crack-propagation may be obtained easily by the proposed method. By carrying out a numerical simulation of a dynamic cracking, the radiation waves by Mode-I (opening) and Mode-II (sliding) crackings are obtained and studied.

2. Modeling for dynamic crack-propagation 2.1. Numerical formulation In a homogeneous linear elastic body D with the domains U^(x, t) which is respected with the crack domain.

236

S. Han, M. Xiao/First MIT Conference on Computational Fluid and Solid Mechanics X2 investigation point

'• nC

\

04

(a)

o

Fig. 1. Mechanical effect of cracking in element, (a) Cracking state in element, (b) Equivalent nodal forces for Mode-II.

we consider the crack as a displacement gap in a continuous medium and describe it through a singular function [2]. By some mathematical operations, and the techniques of the finite element method, it shows that the mechanical effect of the cracking in the material can be evaluated by a set of equivalent nodal forces acting at the position with respect to the cracking domain. One can obtain the wave equation as: MU + KU = P*, in which, U is the nodal displacement vector, M is the mass matrix and K is the stiffness matrix. The vector P* represents the equivalent nodal force vector induced by the crack-propagation, and Vj, dS ^

Te

AZe

<3=>

(1)

^=^1^

where, p is the mass density, N is the shape function and ^1 is a unit vector lying on the crack in the ^-th element. Ze and AE^ are the regions with respect to the initial and propagating cracks in the ^-th element, respectively. J2e(^e) = Z, E . ( ^ ^ ^ ) = ^ ^ ' and Z U AZ = E\ V in the above equation is a known parameter contains the information of the material properties and crack configuration [2]. To demonstrate the performance of the proposed model, let us consider a simple case of a 4-node square isoparametric element with the side length h in a. linearly cracking state as shown in Fig. la, in which the shadow area represents the magnitude of the cracking displacement along ^i-axis for both cracking modes. Without loss of the generality, we only consider the case when cracking crossed the

Xi - cracking domain original crack

Fig. 2. Mesh for numerical calculation.

element with a unit magnitude of the maximal displacement gap. The components of the equivalent nodal forces are calculated as shown in Table 1 in which the parameters A = Cs/Cp, Cp and C^ represent the velocities of longitudinal and transverse waves, respectively. One can understand from the results that in Mode-I, the mechanical effect of cracking is equivalent to four couples of tensile force acting at the nodes of the element. In Mode-II, the mechanical effect of cracking is equivalent to four couples of shear force. The result for Mode-II is illustrated in Fig. lb. The properties of those results can be proved to have a generality. 2.2. Numerical

example

We simply consider the case that crack propagates along the jci-axis with the velocity of the value of half of the transverse waves. The numerical calculation model for the problem shown in Fig. 2 is a rectangular body with the size 65.0 x 65.0 cm. The origin of the coordinate system is on its gravity center. The elements discretized for calculation are all square with a size of 1.0 x 1.0 cm, and the crack lying on xi-axis is centered at the origin of coordinate system. The material constants are fellows: Young's modulus E = 5.67 x 10^ MPa, mass density p = 2.1 t/m^ and Poisson's ratio v =

Table 1 Equivalent nodal forces

^h^C^ Mode-1 I ^

Mode-II

h^Cl

Node 1

Node 2

Node 3

Node 4

(^1,^2)

(^1,^2)

(^1,^2)

(^1,^2)

(1 - 2 7 1 ^ 2 / 3 )

{-\+2A\4/3)

(-1+2^12,-4/3)

(1 - 2A^ -2/3)

(2/3, 1)

(4/3,-1)

(-4/3,-1)

(-2/3, 1)

S. Han, M. Xiao/First MIT Conference on Computational Fluid and Solid Mechanics

direction. The radial displacement changes abruptly at the arrival of transverse wave. Furthermore, the response in the circumferential direction changes its phase at ^ = 45°.

0.000B0.0006 —

E Z3

"Q. C TO 0)

E o i5

k

j

fin 1/1

O.OOCQ

jl

5 ,•'

«;:

W-f JV

MVJ pi^

0

«

ft A in

AAn

I

M

V

I V'

i :

g--0.0006-

K

1 ' ''' ''' '

0

1

1

I

1

237

1 '''

' ' '' 1

0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035

time (s)

3. Conclusions A mathematical model to describe the mechanical phenomenon of dynamic crack-propagation is proposed. The conclusions are as follows: the effect of dynamic cracking in material can be evaluated as equivalent nodal forces in a numerical procedure; the formulation to evaluate the equivalent nodal force is presented, and it is shown that the mesh divisions are independent of crack and cracking configurations if the finite element method is used; the mechanical effects of Mode-I and Mode-II crackings are equivalent to several couples of tensile (or compressive) and shear forces acting on the elements, respectively.

Fig. 3, Displacement responses for Mode-II. References 0.25. The initial crack length RQ = 7.0 cm, and the final accumulative length of the crack-propagation ARQ = 2.0 cm. The calculated results are plotted in Fig. 3, in which the black line represents the responses in the radial direction and the broken line represents the responses in the circumferential direction. Due to the limited space of the paper, we only give the responses measured at the point (r = 27.5 cm, 0 = 45°), in which r is the distance and 0 is the angle as shown in Fig. 2. Fig. 3 shows the displacement responses measured for Mode-II cracking. It can be seen that the displacement response rapidly increases just after the arrival of the longitudinal waves. The same phenomenon is also observed in the displacement in the circumferential

[1] Freund LB. Crack propagation in an elastic solid subjected to general loading-I. Constant rate of extension. J Mech Phys Solids 1972;20:129-140. [2] Han S. Evaluation of reservoir crack based on equivalent effect of scattering waves due to crack-propagation. Int J Rock Mech Min Sci 1997;34(3/4):Paper No. 118. [3] Lo CY, Nakamura T, Kushner A. Computational Analysis of Dynamic Crack Propagation along a Bimaterial Interface. Int J Solids Struct 1994;31(2): 145-168. [4] Nishioka T, Atluri SN. Numerical analysis of dynamic crack propagation: generation and prediction studies. Eng Fract Mech 1982;16:303-332. [5] Rose LRF. Recent theoretical and experimental results on fast brittle fracture. Int J Fract 1976;12(6):799-813.

238

Large strain, large rotation boundary integral multi-domain formulation using the Trefftz polynomial functions M. Handrik*, V. Kompis, P. Novak Faculty of Mechanical Engineering, University ofZilina, Velky diel, 010 26 Zilina, Slovakia

Abstract In this paper, a multi-domain formulation based on reciprocity relations, which is a combination of the finite element method and the boundary element method is presented [2-4]. The total Lagrangian formulation for large displacement and large rotation and Hook material law is used. The formulation is the weighting residual form, which leads to a non-linear equation system. The nonlinear equations system is solved by incremental Newton-Raphson procedure. Keywords: Total Lagrangian formulation; Trefftz function; Large displacements and rotations; Boundary integral multidomain method

1. Introduction In this paper, Trefftz polynomials (T-polynomials) [1] are used for the development of multi-domain (MD) based on the reciprocity relations. Such reciprocity principles are known from the boundary element formulations, however, using the Trefftz polynomials in the reciprocity relations instead of the fundamental solutions yields the non-singular integral equations for the evaluation of corresponding sub-domain (element) relations. A weak form satisfaction of the equilibrium is used for the inter-domain connectivity relations. For linear problems, the element stiffness matrices are defined in the boundary integral equation form. In non-linear problems, the total Lagrangian formulation leads to the evaluation of the boundary integrals over the original (related) domain evaluated only once during the solution and to the volume integrals containing the non-linear terms. Also, Trefftz polynomials can be used in the post-processing phase of the MD computations for small strain problems. By using the Trefftz polynomials as local interpolators, smooth yields of the secondary variables (strains, stresses, etc.) can be found in the whole domain (if it is homogeneous). This approach considerably increases the accuracy of the evaluated yields while maintaining the same rate of convergence as that of the primary yields. * Corresponding author. Tel.: +421 (89) 5132974; Fax: +421 (89) 5652940; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Now a stress smoothing procedure for large displacements will be presented as well. Considering the examples of simple tension, pure bending and tension of fully clamped rectangular plate (2D stress/strain problems) for large strain-large rotation problems, the use of the initial stiffness, the Newton-Raphson procedure, and the incremental Newton- Raphson procedure will be discussed.

2. The total Lagrangian formulation for finite deformation problems Equilibrium equation for this problem in undeformed (initial) configuration in the integral week form

fl{Sufu),k-^b'l\UidQ

=0

(1)

Applying integration by parts, the Gauss' theorem, substituting displacement gradient for deformation gradient to Eq. (1) we obtain f tfUi df + /" b^Ui dQ-

- f{SijUu)UidQ

=0

f Sij Uij dQ (2)

The strain tensor can be split into the elastic and plastic

M. Handrik et al. /First MIT Conference on Computational Fluid and Solid Mechanics parts and because of the linear dependence between the elastic part of Green strain tensor and the 2nd PiolaKirchhoff stress tensor, the reciprocity relation can be found in the form. [ t^Ui d r + / b^^Ui dQr

^ -

/ \uk,iUk,jT^ij

^(N-l)

,(N)

(3)

Eq. (3) is applied for the computation of the relation between the boundary displacements u and the tractions t^ for each sub-domain (element). The inter-domain tractions continuity j dui {ti - u) dr, + / dui (tf" - t^) dVi

= / dui ti dVe - / dui Ti dre=0

(4)

Fe

is used to the weak satisfaction of inter-domain equilibrium.

3. Linearization of resulting equations for large strain problems The resulting discretized and linearized equations are in the form (K + K^L)u(N)^p(N-i)

(6)

and the displacements in the N-th iteration steps are

dQ - f^ SijUi^kUi dQ

;.E,,d^ = 0

Te

are evaluated only once (in the zeroth equilibrium iteration step). On the other side, the nonlinear volume integrals are evaluated in the first and further iteration steps only. In the Newton-Raphson procedures, the increments are computed

( uj Ti d r r

239

^3^

where K corresponds to the linear part of Eq. (3) and K^^ to its non-linear part, which is linearized for each iteration step and p^^~^^ denotes the configuration dependent load corresponding to the configuration of the previous iteration step. The linear matrix K and thus, the boundary integrals

,i(N-l)

(N)

+ Au'

(7)

The iteration is stopped if the quadratic norm of the last displacement increment related to the quadratic norm of the displacements is less than the specified value e > ||Au (N)|

i(N)|

(8)

4. Examples The examples of simple tension, pure bending and tension of fully clamped rectangular plate (2D stress/strain problems) for large strain-large rotation problems, the use of the initial stiffness, the Newton-Raphson procedure, and the incremental Newton-Raphson procedure and the accuracy will be discussed.

References [1] Trefftz E. Ein Gegenstuck zum Ritzschen Verfahren. Proceedings of the 2nd International Congress of Applied Mechanics, Zurich, 1926. [2] Zienkiewicz OC, Taylor RL. The Finite Element Method, vols. I-II, 4th ed. New York: Wiley, 1989/1991. [3] Bathe K-J. The Finite Element Procedures, Englewood CHffs, NJ: Prentice Hall, 1996. [4] Balas J, Sladek J, Sladek V. Stress Analysis by Boundary Element Method. Amsterdam: Elsevier, 1989. [5] Kompis V, Jakubovieova L. Errors in modelling high order gradient fields using isoparametric and reciprocity based FEM, submitted for publication.

240

About linear and quadratic 'Solid-Sheir elements at large deformations M. Hamau, K. Schweizerhof * University of Karlsruhe, Institute for Mechanics, 76128 Karlsruhe, Germany

Abstract Efficient computation in sheet metal forming or car crash analysis is obtained only by using shell elements instead of fully three-dimensional solid elements. However, many requirements in the investigations, in particular when looking at edges and special situations like large stretching and bending with small radii as strains and stresses in thickness direction and general three-dimensional material laws, cannot be provided by shell elements even if they are based on the well-known degeneration concept. Therefore, in [10] a so-called 'Solid-Shell' formulation, following similar suggestions in [4,12,14], was proposed. For the biquadratic-linear as well as for the trilinear elements different locking effects appear, see also [3]. Different schemes to overcome the locking problems are used and an almost locking-free element formulation can finally be presented. However, as a consequence problems occur in the large deformation regime, such that under some types of loading the trilinear elements [7,17] as well as the biquadratic-linear elements show artificial instabilities, indicated by negative eigenvalues of the tangential stiffness matrix. This topic is discussed in detail. Keywords: Solid-Shell elements; Large deformations; Volumetric locking; Mixed interpolations; Trapezoidal locking; Numerical instabilities

1. Introduction With the 'Solid-Sheir concept [4,10,12,14] a shell element formulation was proposed, to overcome some limits of the well-known degeneration concept. Using nodes at upper and lower surface and using only displacement degrees of freedom allows general three-dimensional material laws to be implemented, thus strains and stresses in thickness direction can be properly computed. As a consequence also applications for large deformation problems become possible without artificial restrictions, see also [3,11]. In addition, the treatment of rotations can be avoided completely and the transition to full 3D-continuum parts is directly possible. The originally developed 'four-node type' elements with bilinear inplane shape functions have been extended to 'nine-node type' elements with biquadratic in-plane shape functions [9] expecting a geometrically better approximation for curved and heavily deformed structures. An as* Corresponding author. Tel: +49 (721) 608-2070; Fax: +49 (721) 608-7990; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

sumed natural strain (ANS) method as proposed in Refs. [5,6] is used for the 'four-node type' (8 node) elements to avoid transverse shear locking, and it is also used for the 'nine-node type' (18 node) elements to avoid, firstly, transverse shear locking and, secondly, the additionally appearing membrane locking for elements with higherorder shape functions. The problem of thickness locking is resolved by enhancing the normal strain in thickness direction with a linear extension using the EAS-method [4,14], or alternatively by increasing the order of interpolation for the displacements in thickness direction over the thickness using an additional degree of freedom [8]. Considering nearly incompressible material behavior, like rubber elasticity or metal plasticity, the problem of volumetric locking appears. A rather efficient possibility to overcome this problem is to use a lower order of integration for the volumetric parts of the stress tensor and the tangent moduli tensor, the so-called selective reduced integration (SRI) [11]. The selective reduced integration of volumetric parts indeed presumes that an isochoric-volumetric material behavior is considered. Another locking effect known for elements with linear and quadratic shape functions is the problem of so-called

M. Harnau, K. Schweizerhof / First MIT Conference on Computational Fluid and Solid Mechanics trapezoidal [15] or curvature thickness [3] locking. This effect is only found in structures where the vectors from the lower to the upper nodes at the element edges are not vertical to the mid-layer. A method to resolve this problem is using an assumed strain in-plane interpolation of the normal strain in thickness direction as proposed in [2].

2. Numerical instabilities To investigate the effects of numerical instabilities under certain loading conditions a study with a single 'four-node type' element under a homogeneous compressions/tension state is performed in analogy to [1]. Because large deformations are treated in this example, a material of the Neo Hookean type is used. The geometrical and material data

Fig. 1. Geometry, material data and loading of the investigated element. Geometry: I = 2, t = 2; Neo Hooke material: K = 1.0 • 10^, yit = 20. Uniform displacement v in y-direction.

241

are shown in Fig. 1. All nodes are fixed in the z-direction, thus a plane strain case is generated. The upper four nodes are linked together in the j-direction and as loading a uniform displacement v is prescribed for these nodes. As a consequence of the loading and the boundary conditions shown in Fig. 1 the number of degrees of freedom for the whole system is reduced to four. Therefore, only four eigenmodes (Fig. 2) are possible for the system, with the fourth eigenmode being the volumetric deformation mode. In this simple example the eigenvalues belonging to the eigenmodes shown in Fig. 2 can be derived analytically as a function depending on the displacement u. The results for these investigations are shown in some diagrams in Fig. 3. There it can be seen that the pure displacement formulation DISP3D remains always stable. The ANS3DL element is the displacement formulation combined with the ANS-method. It is clearly visible that for this element formulation the eigenmodes 1 and 2 become unstable in the case of very large compressive strains. But it must also be noted that this state of about 90% compression is hardly found in a realistic problem. If the inplane strains are enhanced using the E AS-method, as it is done for the EAS3DEAS element, the well known hourglass mode [17] appears at a compression of about 45%. Similar observations have been made for the biquadratic elements. As a conclusion it must be noted that all mixedtype enhancements of the low-order interpolated solid-shell elements lead to artificial element kinematics under homogeneous loading in the large deformation regime. For plane elements proposals to improve the element behavior are given by Wall et al. [16] for rectangular elements, by Reese [13] and by Armero [1] in a very detailed

Fig. 2. Eigenmodes of 8-node-element, e.g. mode 1, mode 2, mode 3.

1000 8001-

I 600

11 4 0 0

1000 mode 1 • mode 2 mode 3 •

mode 1 mode 2 mode 3

200

I 600

11400

200

0

-2-1.5-1-0.5 0 0.5 1 1.5 2 Displacement U a)

mode 1 mode 2 mode 3 -

800

-2-1.5-1-0.5 0 0.5 1 1.5 2 Displacement u b)

IZ^

-2-1.5-1-0.5 0 0.5 1 1.5 2 Displeicement u c)

Fig. 3. Eigenvalues of eigenmodes 1, 2 and 3 as a function of the deformation v in };-direction; (a) D1SP3D element, (b) ANS3DL element, and (c) EAS3DEAS element; -\- = tension; - = compression.

242

M. Harnau, K. Schweizerhof / First MIT Conference on Computational Fluid and Solid Mechanics

Study for arbitrarily shaped elements. A further, rather simple possibility to achieve a stable element formulation is to regain the stiffness matrix of the displacement formulation A^^-^p multiplied with a factor (p on the given element stiffness matrix Ke'. ke = (l~
+ cpKl^^^.

(1)

The factor (p must be adjusted to a value between one and zero depending on the type and the grade of deformation. The value of (p can even be equal to one for structures under a pure homogeneous stress state, where the displacement formulation delivers proper results without any locking effects. The current investigations are directed towards the proper automatic adjustment for non-rectangular element shapes and not fully homogeneous loading avoiding any overstiff behavior.

References [1] Armero F. On the locking and stability of finite elements in finite deformation plane strain problems. Comput Struct 2000;75. [2] Betsch P, Stein E. An assumed strain approach avoiding artificial thickness straining for a non-linear 4-node shell element. Common Numer Methods Eng 1995; 11:899-909. [3] Bischoff M, Ramm E. Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 1997;40:4427-4449. [4] Braun M. Nichtlineare Analysen von geschichteten elastischen Flachentragwerken. Bericht Nr. 19, Institut fur Baustatik, Universitat Stuttgart, 1995. [5] Bucalem EN, Bathe KJ. Higher-order MITC general shell elements. Int J Numer Methods Eng 1993;36:3729-3754. [6] Dvorkin EN, Bathe KJ. A continuum mechanics based four-node shell element for general nonHnear analysis. Eng Comput 1989;1:77-78.

[7] Glaser S, Armero F. On the formulation of enhanced strain finite elements in finite deformations. Eng Comput 1997;14(7):759-791. [8] Gruttmann F. Theorie und Numerik diinnwandiger Faserverbundstrukturen. Bericht Nr. F96/1, Institut fiir Baumechanik und Numerische Mechanik, Universitat Hannover, 1996. [9] Hauptmann R, Doll S, Harnau M, Schweizerhof K. 'SolidShell' elements with linear and quadratic shape functions at large deformations with nearly incompressible materials. Submitted for publication, 2000. [10] Hauptmann R, Schweizerhof K. A systematic development of solid-shell element formulations for linear and nonlinear analyses employing only displacement degrees of freedom. Int J Numer Methods Eng 1998;42:49-70. [11] Hauptmann R, Schweizerhof K, Doll S. Extension of the solid-shell concept for large elastic and large elastoplastic deformations. Accepted by Int J Numer Methods Eng 2000;49:1121-1141. [12] Parisch H. A continuum-based shell theory for non-linear applications. Int J Numer Methods Eng 1995;38:18551883. [13] Reese S, Wriggers P. A stabilization technique to avoid hourglassing in finite elasticity. Report No. 4/98, Institute of Mechanics, TU Darmstadt, 1998. [14] Seifert B. Zur Theorie und Numerik finiter elastoplastischer Deformationen von Schalenstrukturen. Bericht Nr. F96/2, Institut fiir Baumechanik und Numerische Mechanik, Universitat Hannover, 1996. [15] Sze KY, Yao LQ. A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part I. Solid-shell element formulation. Int J Numer Methods Eng 2000;48(4):545-564. [16] Wall WA, Bischoff M, Ramm E. A deformation dependent stabilization technique, exemplified by EAS elements at large strains. Comput Methods Appl Mech Eng 1998;188:859-871. [17] Wriggers P, Reese S. A note on enhanced strain methods for large deformations. Comput Methods Appl Mech Eng 1996;135:201-209.

243

Skull mechanic simulations with the prototype SimBio environment U. Hartmann^'*, F. Kniggel^, T. Hierl^ G. Lonsdale % R. Kloppel'* ^ C&C Research Laboratories, NEC Europe Ltd., Rathausallee 10, 53757 St. Augustin, Germany ^ Max-Planck-Institute of Cognitive Neuroscience, Stephanstrafie 1, 04103 Leipzig, Germany Department of Oral and Maxillofacial Plastic Surgery, University of Leipzig, NUmberger Strafie 57, 04103 Leipzig, Germany '^Department of Diagnostic Radiology, University of Leipzig, Liebigstr 22, 04103 Leipzig, Germany

Abstract The SimBio project will produce a generic simulation environment for advanced clinical practice designed for execution on parallel and distributed computing systems. This paper deals with the specific appHcation of current SimBio software components for the study of a skull mechanics problem relating to maxillo-facial surgery. In addition to a demonstration of physical results, performance characteristics of the bio-mechanical finite element code on parallel platforms is given. Keywords: Finite element model; Computer tomograph; SimBio; Computational biomechanics; Maxillofacial surgery; Head model

1. Introduction The objective of the SimBio project [1,2] financed by the European Commission's Information Societies Technology (1ST) programme is the improvement of clinical and medical practices by the use of numerical simulation. This goal is achieved by developing a generic simulation environment that enables users to develop application specific tools for many medical areas. The potential impact is demonstrated for specific areas through the SimBio evaluation and validation applications. A key feature in the SimBio project is the possibility to use individual patient data as input to the modelling and simulation process — in contrast to simulation based on 'generic' computational models. In order to meet the computational demands of the SimBio applications, the compute-intensive environment components are implemented on high performance computing (HPC) platforms. This paper presents an initial study for bio-numerical support of maxillo-facial surgery planning. The medical background to this study is discussed in Section 2. Selected software components under development within the SimBio

* Corresponding author. Tel.: +49 (2241) 925242; Fax: +49 (2241) 925299; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

project are discussed in Section 3. Section 4 of the paper illustrates preliminary results of numerical simulations and covers performance issues. Finally, steps towards a more accurate modelling are discussed.

2. Bio-mechanical simulation supporting facial-surgery planning One of the target applications of the SimBio framework deals with pre-surgical studies in the field of head biomechanics. In particular, this refers to the modelling of the deformations emerging during and/or induced by surgical interventions. Thus, simulation supports the optimisation of operation procedures and the planning of therapeutical strategies. Currently, a study is underway to investigate the mechanical consequences of the forces that occur during the sequence of interventions to remedy inborn deformations of the human face (mainly cleft lip and palate). In order to adjust deformed parts of the midface a metal frame (a so-called halo, see Fig. 1) is tightly fixed to the head using screws. After cutting the midfacial bone along exactly defined lines, this device exerts forces on the bone structure to be relocated. The distraction path length governed by the externally applied forces amounts to a length of 10-30

244

U. Hartmann et al. /First MIT Conference on Computational Fluid and Solid Mechanics In this paper, we first present the results of phase 1. Software tools used to model the skull response are described in the next section.

3. Overview of the software solutions 3.1. Pre-processing: segmentation and meshing

Fig. 1. Halo frame for maxillo-facial surgery mounted to a skull model. mm (1 mm/day), depending on the application site and duration, which is typically in the order of a few weeks. We divided the finite element (FE) modelling of this surgical intervention into two phases: (1) In a first step, skull deformations induced by the halo screws (see Fig. 2) are calculated. Exact knowledge about the mechanical consequences of the surgical device is important for the surgeon mounting the halo. (2) The goal of the second phase of the modelling process is to gain pre-surgical knowledge about the relation between the magnitude and the direction of the applied distraction forces and the resulting rearrangement of the bone structures and the surrounding soft tissues.

Fig. 2. A CT slice of the human head showing the halo fixed with screws.

The geometric description of our model is based on 3D medical images of individual patients acquired with a computer tomograph (CT). Spiral CT scans achieve a spatial resolution of 0.5 mm. Raw data are pre-processed by registering time-series scans to the first time point and are segmented into background, soft tissue, bone and halo. This segmentation forms the basis for mesh generation. A fast and high quality mesh generator creates hexahedral or tetrahedral meshes of user-defined spatial resolution [3] (see Fig. 3). 3.2. HEAD-FEM The finite element (FE) code for biomechanical problems (called HEAD-FEM) is based on linear solvers provided in the AZTEC library [4] and is parallelised using the Message Passing Interface (MPI) library. HEAD-FEM enables linear static and dynamic FE analyses [5]. Simulations presented here were carried out using the static version of HEAD-FEM. Input to the FE module is a distributed mesh partitioned by a modified recursive co-

Fig. 3. A hexahedral FE mesh of the human head divided into 16 partitions.

U. Hartmann et al /First MIT Conference on Computational Fluid and Solid Mechanics

245

Table 1 HEAD-FEM execution times and speed-up factors on the NEC Cenju-4 for different numbers of processors Processor no.

Time (s) Speed-up

Fig. 4. Skull deformation as predicted by the simulation. Inward deformations correspond to yellow-red colours, outward deformations to green-blue colours.

ordinate bisection (RGB) algorithm implemented in the DRAMA library [6] (see Fig. 3). To overcome some of the restrictions imposed by sequential FE codes, this FE tool enables simulations based on meshes with a spatial resolution about five times higher than that of previous models. The high spatial resolution guarantees: • a precise FE representation of head anatomy; and • a high numerical accuracy of the results obtained in reasonable calculation time. 33.

Postprocessing

The nodal displacements for the whole head are calculated and mapped onto a triangular surface mesh of the skull and visualised using the BRIAN software package [7] (see Fig. 4). A specific version of BRIAN will become the visualisation module of the final SimBio environment.

4. Results HEAD-FEM has been installed on the 64 processor NEC Cenju-4 supercomputer (MIPS RIOOOO in a multistage inter-connection network). An example input is a distributed hexahedral head mesh whose elements have an edge length of 3 mm (see Fig. 3). The equation system based on this mesh has about half a million unknowns and is solved by a preconditioned conjugate gradient solver provided by the AZTEC library. Table 1 lists execution times for a full HEAD-FEM analysis (data input, matrix

8

16

32

64

291 1.00

165 1.76

84 3.46

44 6.61

assembly, equation solving). These figures demonstrate that the code scales well and that a full FE problem is solved in less than a minute. Fig. 4 depicts the skull deformation produced by the screws of the surgical frame. Besides the expected focal inward deformation at screw positions, an outward protrusion of the skull at peripheral concentric areas is observed (see arrows). This result is in full agreement with clinical findings.

5. Concluding remarks We presented a surgical application of the FE method using initial components of the generic SimBio environment. Results obtained in phase 1 of our modelling process (see Section 2) are already considered to be clinically relevant. HEAD-FEM needs to be extended for phase 2 — surgical planning. That requires the implementation of: • geometrically nonlinear FE techniques, such as the Newton-Raphson method; • additional material models (e.g. visco-elastic material behaviour); and • a contact algorithm. Another important aspect of the SimBio project, inevitable for performing clinically valid simulations, addresses the measurement of realistic material parameters. Combining highly resolved FE models based on individual scan data, efficient HPC-based solver technology, simulations using reliable material parameters, the SimBio project is expected to deliver a software environment that offers the chance to provide safe predictions in clinical routine.

Acknowledgements The support of the European Commission (Project 1ST V-10378) is gratefully acknowledged.

References [1] Lonsdale G, Grebe R, Hartmann U, Hose DR, Kruggel F, Penrose JMT, Wolters C. Bio-numerical simulations with SimBio: project aims and objectives. Proceedings of

246

U. Hartmaim et al. /First MIT Conference on Computational Fluid and Solid Mechanics

the Symposium on Computational Biomechanics 2000 at RIKEN, Saitama, Japan, pp. 187-196. [2] SimBio Project Web-site, http://www.simbio.de [3] Hartmann U, Kruggel F. A fast algorithm for generating large tetrahedral 3D finite element meshes from magnetic resonance tomograms. Proceedings of the IEEE Workshop on Biomedical Image Analysis 1998. ISBN 0-8186-8460-7, pp. 184-192. [4] Hutchinson SA, Shadid JN, Tuminaro, RS. Aztec User's Guide: Version 1.1 (1995). Sandia National Laboratories Technical Report SAND95-1559.

[5] Hartmann U, Kruggel F. Transient analysis of the biomechanics of the human head with a high resolution 3D finite element model. Comput Methods Biomech Biomed Eng 1998;2(l):49-64. [6] DRAMA Project Web-site, http://www.ccrl-nece.techno park.gmd.de/~drama/drama.html [7] Kruggel F, Lohmann G. BRIAN (Brain Image Analysis) — a Toolkit for the multimodal analysis of brain datasets. Proceedings of the International Symposium on Computer and Communication Systems for Image Guided Diagnosis and Therapy, 1996. Amsterdam: Elsevier, pp. 323-328.

247

An analysis of a bilinear reduced strain element in the case of an elliptic shell in a membrane dominated state of deformation V. Havu*, H. Hakula Helsinki University of Technology, Institute of Mathematics, 02015 Hut, Finland

Abstract We consider a bilinear reduced-strain element formulation for a shallow shell of Reissner-Naghdi type. We show that under favorable circumstances the reduced formulation produces convergent solution also in the membrane dominated states of deformation. Keywords: Finite element; Reduced-strain; Shallow shell

1. The shell problem Our study is concentrated on the Reissner-Naghdi shell model where the (scaled) variational formulation of the problem is given by: Find ueUM such that v) + Am(u, v) = Q(v)

AMin, H) = t^Abiu,

yveUM.

(1) where u — (u, v, w, 0, V^) is the vector of three translations and two rotations and UM is the energy space which we take to be [H\Q)]^ with periodic boundary conditions ai y = 0, H and with the constraints u = v = w = 0 = \l/ =0 2iix = 0, L.WQ assume that Q(u) defines a bounded linear functional on [H^{^)f. The bilinear forms Ab(u, v) and Am(u, v) arising from the bending and membrane energies are given by Ab(u,

V)=

V(KU-\-

K22)(U)(KU

+ f<22)(v)

2 + (1 - V) ^ K i j (u)Kij (v) dx dy and Am(u^V)

(2)

''^-' = 6}/(l

v) / {Pi(M)Pi(lL) +

P2(u)P2(v)}dxdy

+ 12 J{v(fti + ft2)fe)(Al + k2Xv) + (1 - v)J2Pij(!i)^ij(y)}^^y

dO

du %i = -— dx

-^aw

ATii

=

=

h bw

/<:22

=

9y

du

dy

\

dv\ \ + cw

dx)

dx dy

1 /96> K\2

df\

(4)

••

and P\

dw

dw

dx

dy

(5)

The integration is taken over the midsurface ^ of the shell which we assume to occupy the rectangular region (0, L) X (0, H) in the xy coordinate space satisfying d~^ < L/H < d for some constant d > 0. We are considering the shell to be shallow so that the parameters a, b and c defining the geometry can be taken constants. We further note that the condition ab - c^ > 0 makes the shell elliptic.

2. The reduced-strain FE scheme

(3)

* Corresponding author. Tel: +358 (9) 451-3018; Fax: +358 (9) 451-3016; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

where v is the Poisson ratio of the material and y is a shear correction factor. Here, Kij, Pij and pi represent the bending, membrane and transverse shear stresses, respectively, depending on w_ as follows

We consider the following numerical approximation to the variational problem (1). Assume that Q is divided into rectangular elements with node points (x^,y"), k = 0, . . . , Nx, n = 0, . . . , Ny and a constant mesh spacing

248

V. Havu, H. Hakula / First MIT Conference on Computational Fluid and Solid Mechanics

hy in the _y-direction and that the aspect ratios of the elements satisfy d~^ < h'^Jhy < d for some d > 0 where /zj = x^+^ — x^. On this mesh each field is represented by a piecewise continuous bilinear approximation. Then the FE space is Uh = [Vh]^ where Vh is the standard biUnear space with appropriate constraints. This space will be denoted by UM.h' We consider the case where the membrane and transverse shear stresses are given by reduced expressions leading to the bilinear form ^milL^y)

= 6 / ( 1 -V)

IP\(U)P\{V) -\-

P2iu)P2iv)W^y

Q

where Ayv = [X e A\ — Tt < Xhy < 71 when A^^ is odd, or — TT < Xhy < 71 when A^^ is even}. Here (px{y) is the interpolant of (px{y), so that we are in fact considering a discrete Fourier transform of i^ e UM,h •

3. Consistency error in the membrane dominated case We start by giving a stability result for UMM • Lemma 3.1. Let ]i e hlM,h- Then

Q

y 1 < ct~^ \v \M,h

2

+ (1 - y) ^

A; (K)A7(v)](^dy

(6)

where ^jj = R'-' Ptj, Pi = R^ pi with suitable reduction operators R'^ and R\ We choose these operators for ^ij and pi to be 3ii=n^)Sn,

^22 — ^hP22

P\ = nipi

P2 = nlp2

Pn = n^^i2,

UM.H

(7)

Our main concern is the consistency error component given by {AM

-A%){U,V)

sup -"'' veUM,h

^ ' ^ — 1^ \M,h

(9)

where | • |M./J = ^ / ^ ^ C - , •) is the modified energy norm. The main tool of our analysis will be the Fourier transform where we write XeA

,-iv

A=\x=—,ve

n(y)±^M

= ^ XeA]\/

ay

+I

IL2

—+ —

dx 11^2

dy

hi^^y)


and thus by the Kom inequality (10)

lA 1 < Ct-^ \V \M,h-

Also, the definitions of the membrane strains fitj (4) imply du II

I ;,,. f du

II dv II

19^11^2

dv\\

||aylL2

< C( \V \M,h +

L2

W il)

and by [4] we have

I du

dv I

II dy

dx 1^2

(

II .V / du

dv

l|9«i

+

dx

II du II

+ b" + L2

resulting in

\dv — ay

L2

du

+

—

dy

-j-

dv \ —

dx 1L2

\dv\\

\

l^yWiJ

||9.^|IL2

< C( \V \M,h +

W il)

where from again by the Kom inequality W1 +

V X
W il).

(11)

The transverse shear strains pt (5) together with (10) imply in turn that dw

a7

L2

+

dw II T
27rv

making use of the periodic boundary conditions at y = 0, H. For functions in the FE space we write analogously AeA^r

df

+

XeA

f

(p,{y) = e''^\

^(x, y) = ^

L2

(8)

Remark 1. The modification introduced in [3] differs slightly from our choice, but similar or even little better results can be proved using forthcoming techniques.

ec,M(^u)=

do — dx

^ 1 +

where Ul and Ul are orthogonal L^-projections onto spaces VV^; and W^' consisting of functions that are constant in X an piecewise linear in y or constant in y and piecewise linear in x, respectively, and n,,^ is the orthogonal L^-projection onto elementwise constant functions. Thus, our finite element scheme solves the problem: Find Uf^ e UMJX such that

Vi; €

Proof. By (8) we have that for y_ = (u, v, w, 0, ^) e UM,h

\V \M,h

L2-\-

x/r Li) (12)

so that (10)-(12) together with the Poincare's inequality imply V I
^ \V \M,h

(13)

Next it is necessary to consider the lower modes separately.

V Havu, H. Hakula/First

MIT Conference

Lemma 3.2. Let d_^ = ^x^^ = (px(ux,yx,Wx,Ox, ifx) € UM,h- Then, ifb^Owe have for X such that \X\h < c < TT that 1+

^xUx

(pxVx

1+

(fxWx

L^
\^-A \M,h-

Vx

249

Mechanics

2

^k+l

i^-mx\. < C\X\-'e

(x'+')

-"I

^^ i2(o,L)+ ^A i2(o,L) < ^ i ^ r ' ^>^

(x')

Fxity'dt

and consequently

+ hlF^

^TkM

(15)

where r^ = 2 ^ tan(^A/z^)

+ Ff

M =i

(16)

from where it follows that ^X L2(0,L) +

"1

L2(0,L) -

and ^x^ =

(19)

L2(0,L)-

Also, (15) gives the relation

«A>

=

Fluid and Solid

Here a > fi > 0 and since iix (0) = vx (0) = 0 we obtain when X ^ 0

(14)

Proof. The translation components iix and Vx of ^^ satisfy the difference equation (cf. [1]) (x^+i) -

on Computational

-

cosi^khy)

^ l ^ l ^ ( ^ ^ L2(0,L) +

^ ^

L2(0,L)

^

^^

L2(O,L)-

+ • u^

L2(0,L)

(20)

Combining (19) and (20) gives (21)

^xux 1 + (pxvx 1 < C \^^ \M,h (17) Here ^^^(^^+1/2)

^

^-'•"^^^(y^n(i,))|(,.+i/2,,«)

f^^ix')

=

^-'•^"+^/^^''^(fe(i,))l(.^,«+l/2)

/l^2(^'+^/')

=

^-''^"+^/'^''^(A2(i,))l(.^+l/2,,«.l/2).

since Fx L^ < C \d_^ |M,/..

To consider Wx we note that (cf. [1]) -2/ wx(x'') = -j£- tan {\Xhy)dx(x^) + f22(^') bcos {^Xhy) bh and thus ,~

Due to the constraints at ;^f = 0, L we may without loss of generality consider only the exponentially decreasing solution of (15) starting from ;c = 0. Then the standard theory for A-stable difference schemes gives us the bound when \X\hy < c < TT

ix'^') < e -a.\X\x'^'

(0)

I

<^A^A

where F.=

is the Euclidean norm of vectors in K^ and

L2(0,L) < C

\d_^

(22)

\M,h-

The claim for A 7^ 0 follows from (21) together with (22). When A = 0 we have from (18) and from wo(x^) = If22(x')ih2it I +

(PoVo

1+

(poWo

L^ < C

\±Q\M,h'

With the help of the stability estimates given in Lemmas 3.1 and 3.2, we can now bound the consistency error.

Fx(t) e -a\X\t dt

i

/22 L2(O,L))

leading to

(Polio

-PlXlix'^+^-t)

2

^ ^ L(0,L) - ^(I'^I^ ^^ i2(o,L) +

(18)

Theorem 3.3. Assume that b ^ 0. The consistency error ec,M defined in (9) satisfies

/ ?^fn-'i-f, f\ 2c ,,22

ec,M
V /n

provided that u_ e [H^{Q>)]^ and Ci(t, s, u) < Ct~^ u_ 2+5 and C2(t,u) < Ct~^(^Yli IP/(i£)li) are finite.

.f27

Ci(t, s, u)h^^' + C2(t, u)h^,

s >0

250

V Havu, H. Hakula/First MIT Conference on Computational Fluid and Solid Mechanics

Proof. Write u = E . ^ A i ? , e WM and ^ = Y.x.!., h e UM.I, • Then by the orthogonahty of the discrete and continuous modes (cf. [1]) (AM

- Al,){u,

V)

= {A^ - AiKu,

V)

=

iA^-At)(Tl,T.^>)

=

(A„,-A';„)(Y^&,,J2^)

Table 1 Numerical results for the Morley shell with the reduced-strain formulation showing the square of the total deformation energy as a function of the degrees of freedom Degrees of freedom

Deformation energy squared

80 225 1425

2.8956 2.9800 3.0062 3.0107 3.0122

3625

6825 ^|Al>Ao

ij

M>Xo

'

ij

WA

1+

w^A 1 2 )

|A|>AO

+ cEE(^'(^;^^ - '"'(^A). P/(i.) - A(i;.)) /•

AeA

IA7(i?A)li IIA IM./,

-'^^Y.Y.

+ CIX0I-E E l^l'(A7(i?.)-4(i?x).A7(i.))

/•

AO

by Lemma 3.1 so that summing up (AM

- A\j){u,

v)
+ Ch'-^'t-'

u 2\v U,/,

U 2+s \V\MM

+ Ch't-'(y2\pi(ii)u] for any s

As a numerical example on the performance of our reduced-strain formulation (8) we take the Morley hemispherical shell as in [2] with clamped boundaries and uniformly distributed pressure load. We parameterize the problem by the angles i> and 0 and use a uniform rectangular mesh with respect to these parameters and lei R = \0, t = 0.04, y = 1/3 to define the geometry and material. The problem is essentially one dimensional, but since we are looking for two-dimensional effects, the computations were done exploiting less symmetry using one eighth of the shell. The results shown in Table 1 indicate rapid convergence of total deformation energy confirming our theoretical predictions.

A G A

by Lemmas 3.1 and 3.2. Since X^h < c < TT v/c have that

'7

10-2 10-2 10-2 10-2 10-2

4. Numerical example

l>-|
X X X X X

>0,veUh.

\y\M.H

(23)

References [1] Havu V, Pitkaranta J. Analysis of a bilinear finite element for shallow shells I: ApproximaUon of inextensional deformations, Helsinki University of Technology Institute of Mathematics Research Reports A430, 2000. [2] MacNeal RH. Finite Elements: Their Design and Performance. New York: Marcel Dekker, 1994. [3] Malinen M. On geometrically incompatible bilinear shell elements and classical shell models, to appear. [4] Pitkaranta J, The first locking-free plane-elastic finite element: historia mathemafica. Helsinki University of Technology Institute of Mathematics Research Reports A411, 1999.

251

Recent developments in nonlinear analysis of shell problem and its finite element solution Adnan Ibrahimbegovic * Ecole Normale Superieure de Cachan, Laboratoire de Mecanique et Technologie 61, avenue du president Wilson, 94235 Cachan, France

Abstract In this article we review some recent and current research works attributing to a very significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. Keywords: Shell problem; Nonlinear analysis; Finite elements

1. Overview of recent advancements In this review we have chosen to focus on only the very recent achievements in the formulation and numerical implementation of shell theories capable of handling finite rotations. Several points which, we believe, merit especially to be re-emphasized are: (i) Classical shell theory is reformulated [1] so that it becomes capable of handling finite (unrestricted-in-size) three-dimensional rotations. This feature is in sharp contrast with the classical developments on the subject (e.g., see [2-4]), where rotations are always of restricted size (linear, second order, etc.). (ii) Optimal parameterization of finite rotations is addressed in detail, with several competing possibilities being examined [5,6]. One possibility, which corresponds to the extension of the classical shell theory, leads to two-parameter representation constructed by exploiting equivalence between the unit sphere and a constrained group of proper orthogonal tensors [42,35]. Another possibility to parameterize finite rotations, which is used to construct a nonclassical shell theory with so-called drilling rotations, leads to the intrinsic rotation parameterization in terms of the proper orthogonal tensor. The orthogonal tensor parameterization of finite rotations can in some cases be replaced by so-called rotation vector parameterization. (iii) In recent works several enhanced finite element interpolations for shell elements have been proposed. Al* Tel.: +33 (0) 147 40 22 34; Fax: +33 (0) 147 40 22 40; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

though some of them, such as hybrid and mixed interpolations, assumed strain method, under-integration with stabilization, have met with considerable success, the particular issue of the optimal interpolation scheme for shells has not been definitely setded yet. What has been shown, however, is that a well-performing finite element interpolation [7]) can be rendered even more powerful if placed in a proper theoretical framework and when care is taken to preserve the salient features of the theoretical formulation [8]. In passing we note that certain aspects of the subsequent numerical approximation can be introduced up front in order to simplify the shell theoretical formulation; the case in point is the use of local Cartesian frames. Thus, there is a two-way relationship between the shell theoretical formulation and its numerical implementation, which should be exploited to obtain an optimal result [9,10]. (iv) The consistent linearization procedure in the geometrically exact shell theory is intimately related to the choice of parameters adopted for three-dimensional finite rotations. In the case of intrinsic parameterization with orthogonal tensor the issues in the consistent linearization become rather subtle for we have to deal with the differential manifold in the shell configuration space [11]. In the opposite case for the rotation parameterization based on the rotation vector, the consistent linearization simplifies with respect to the former case, for it can be performed by the directional derivative [5]. (v) The geometrically exact shell theory provides the enhanced performance in the buckling and post-buckling analysis of shells, and improved result accuracy.

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(vi) A very useful by-product is obtained in terms of a consistently derived linear shell theory, which is obtained by the consistent linearization of the governing nonlinear theory at the reference configuration. When combined with the proper choice of finite element interpolations such a linear shell theory leads to the excellent results in all demanding benchmark problems [12].

2. Summary of current research What has been extensively researched over last several years and presented in this review is, in essence, the finite rotation version of the classical shell theory (or one-director Cosserat surface) and its modification which can account for the third rotation component [38]. Only the simplest linear elastic constitutive model for stress resultants was considered. Hence, in current effort of trying to take these considerations a step further, several directions appeared worthy of further explorations. (i) Generalizing the set of stress resultant constitutive equations to other than linear elastic case, within the framework of nonlinear (geometrically exact) kinematics. Some work in that direction is already initiated by Simo and Kenedy [13] and Crisfield and Peng [14] on elasto-plastic stress resultant shell model, but with crucial limitation being that of small elastic strains. This limitation is removed in a recent work of Ibrahimbegovic [9] which considers the stress resultant constitutive model for finite deformation elasto-plasticity based on the multiplicative decomposition of the deformation gradient, where both elastic and plastic deformations can be finite. This, however, has been accomplished only for membrane shell theory, and it is by no means trivial to furnish an extension that accounts for the effects of bending. Other recent approaches consider the possibility to include three-dimensional constitutive equations and perform numerical integration through the shell thickness to provide the corresponding replacement for the stress resultants (e.g., see [15-17]). (ii) Other stress resultant model which assumes the large elastic strains has been provided for rubber-like shells (e.g., see [18,19]). In this case one can no longer justify the assumption of director inextensibility, and should rather take into account the change in the shell thickness. One immediate consequence of introducing the through-the-thickness stretch is the occurrence of numerical sensitivity in the limit case of thin shells, which manifests itself as an additional locking phenomenon. Several works have already dealt with this problem. One strategy, advocated by Hughes and Camoy [18], is to postpone the thickness update to the subsequent iteration, which simplifies the implementation but increases the number of iterations. Simo et al. [43] propose multiplicative decomposition of the director field combined with the exponential update for the through-thethickness stretch, the strategy which is well suited for the

limit case of thin shells although it increases the computational effort with respect to the standard update procedure. On the other hand, Buechter et al. [20] simply add the enhanced strains in through-the-thickness direction, which appears to be sufficient to alleviate the pertinent locking phenomena in the standard update procedure. We note in passing that the enhanced shell kinematics which accounts for the through-the-thickness stretching is especially well suited for analysis of shells made of composite materials (e.g., [21-23]). (iii) More work is needed on providing robust finite element interpolations. One area which is certainly yet unsettled is the research into high performance three-node shell element with finite rotations. Some attempts in that directions are the works of Bergan and Nygard [24] which relies on the co-rotational formulation, the works of Felippa and co-workers (e.g., see [25], and references therein) on providing enhanced finite element interpolations for a triangle and recent work of Carrive-Bedouaniet al. [26]. Even for a four-node shell element, which is already rather fine-tuned and performs quite well as shown in this review, there are still some weak points. Case in point is the oscillation of the computed shear force values clearly identified for 4-node assumed shear strain interpolations in somewhat more simplified setting of plates [27]. The higher-order finite element interpolations for finite rotation shell elements have not been much researched, although it appears that one should be able to benefit from the successful developments on the pertinent subjects such as in Park and Stanley [28], Belytschko et al. [29] and Bucalem and Bathe [30]. (iv) The complete mathematical analysis of convergence for different finite element spaces for nonlinear shell problem is not provided yet. Partial results which are very useful in treating the special cases are given in Brezzi et al. [31] and Stenberg [32] for plates and Leino and Pitkaranta [33] for membrane locking of shells. Another important goal of the mathematical analysis is to provide the error estimates for the nonlinear shell problem, so that the adaptive mesh refinement can be used in a more meaningful manner. (The benefit of the latter is briefly illustrated in this review for the linear shell problem.) This area of research appears to be strongly related and could certainly benefit from the search for a proper definition of the nonlinear shell problem by means of the asymptotic analysis of three-dimensional continuum (e.g., see [34], and references therein). (v) Shell dynamic analysis is a natural setting for many nonlinear problems, most notably, multi-body dynamics and snap-through of shells. The major obstacle to tackling that problem, the dynamics of finite rotation group, has already been addressed (e.g., see [19]). Some follow-up works treating the dynamics for shell theories with finite rotations are given in Simo and Tanrow [35], Brank et al. [36], orBranketal. [37].

A. Ibrahimbegovic /First MIT Conference on Computational Fluid and Solid Mechanics References [1] Simo JC, Fox DD. On a stress resultants geometrically exact shell model. Part I: Formulation and optimal parameterization. Comput Methods Appl Mech Eng 1989;72:267304. [2] Budiansky B. Notes on nonhnear shell theory. J Appl Mech 1968;35:393-401. [3] Naghdi PM. The theory oh shells and plates. In: Flugge S (Ed), Encyclopedia of Physics. Berlin: Springer, 1972. [4] Reissner E. Linear and nonlinear theory of shells. In: Fung YC, Sechler EE (Eds), Thin Shell Structures: Theory, Experiment and Design. Englewood Cliffs, NJ: Prentice-Hall, pp. 29-44, 1974. [5] Ibrahimbegovic A, Frey F, Kozar I. Computational aspects of vector-like parameterization of three-dimensional finite rotations. Int J Numer Methods Eng 1995;38:3653-3673. [6] Ibrahimbegovic A. On the choice of finite rotation parameters. Comput Methods Appl Mech Eng 1997;149:49-71. [7] Bathe KJ, Dvorkin EN. A formulation of general shell element — The use of mixed interpolation of tensorial components. Int J Numer Methods Eng 1986;22:697-722. [8] Ibrahimbegovic A. On assumed shear strain in finite rotation shell analysis. Eng Comput 1995;12:425-438. [9] Ibrahimbegovic A. Stress resultant geometrically nonlinear shell theory with drilling rotations — Part I: A consistent formulation. Part IT. Computational aspects. Comput Methods Appl Mech Eng 1994;118:265-308. [10] Ibrahimbegovic A. Finite elastoplastic deformations of space-curved membranes. Comp Methods Appl Mech Eng 1994;119:371-394. [11] Simo JC. The (symmetric) Hessian for Geometrically Nonlinear Models in Solid Mechanics: Intrinsic Definition and Geometric Interpretation. Comp Methods Appl Mech Eng 1992;96:189-200. [12] Ibrahimbegovic A, Frey F. Stress resultant geometrically nonlinear shell theory with drilling rotations — Part III: Linearized kinematics. Int J Numer Methods Eng 1994;37:3659-3683. [13] Simo JC, Kenedy JG. On a stress resultants geometrically exact shell model. Part V: Nonlinear plasticity, formulation and integration algorithms. Comput Methods Appl Mech Eng 1992;96:133-171. [14] Crisfield MA, Peng X. Stress resultant plasticity criterion. In: Owen DRJ et al. (Eds), Proceedings COMPLAS III. Pineridge Press, 1992, pp. 2035-2046. [15] Brank B, Peric D, Damjanic FB. On large deformation of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element. Int J Numer Methods Eng 1997;40:689-726. [16] Miehe C. A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains. Comput Methods Appl Mech Eng 1999;213:12331267. [17] Eberlein R, Wriggers P. Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: Theoretical and computational aspects. Comput Methods Appl Mech Eng 1998;199:340-377. [18] Hughes TJR, Carnoy E. Nonlinear finite element shell formulation accounting for large membrane strains. Comput Methods Appl Mech Eng 1983;39:69-82.

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[19] Simo JC, Rifai MS, Fox DD. On a stress resultants geometrically exact shell model. Part VI: Conserving algorithms for nonlinear dynamics. Int J Numer Methods Eng 1992;34:117-164. [20] Buechter N, Ramm E, Roehl D. Three-dimensional extension of nonlinear shell formulation based on the enhanced assumed strain concept. Int J Numer Methods Eng 1994;37:2551-2568. [21] Ba§ar Y, Ding Y, Schultz R. Refined shear deformation models for composite laminates with finite rotations. Int J Solids Struct 1993;30:2611-2638. [22] Braun M, Bischoff M, Ramm E. Nonhnear shell formulations for complete three-dimensional constitutive laws including composites and laminates. Comp Mech 1994; 15:118. [23] Gruttmann F, Khnkel S, Wagner W. A finite rotation shell theory with application to composite structures. Eur J Finite Elem 1995;4:597-632. [24] Bergan PG, Nygard MK. Nonhnear shell analysis using free formulation finite elements. In: Finite element method for nonhnear problems, (Eds PG Bergan et al.). SpringerVerlag, Berlin, 1985, pp. 317-338. [25] Felippa CA, Militello C. Developments in variational methods for high performance plate and shell elements. In: Analytical and computational models of shells, (Eds AK Noor et al). ASME Publ., CED-vol 3, 1989, pp. 191-215. [26] Carrive-Bedouani M, Le Tallec P, Monro J. Finite element approximation of a geometrically exact shell model. Eur J Finite Elem 1995;4:633-662. [27] Lyly M, Stenberg R, Vihinen T. A stable bilinear element for the Reissner-Mindhn plate model. Comput Methods Appl Mech Eng 1993;110:343-357. [28] Park KC, Stanley G. A curved C° shell element based on assumed natural-coordinate strain. J Appl Mech 1988;108:278-290. [29] Belytschko T, Wong BL, Stolarski H. Assumed strain stabilization procedure for the 9-node Lagrangian shell element. Int J Numer Methods Eng 1989;28:385-414. [30] Bucalem ML, Bathe KJ. Higher order MITC general shell elements. Int J Numer Methods Eng 1993;36:3729-3754. [31] Brezzi F, Bathe KJ, Fortin M. Mixed-interpolated elements for Reissner-Mindhn plates. Int J Numer Methods Eng 1989;28:1787-1801. [32] Stenberg R. A new finite element formulation for the plate bending problem. 1993, preprint. [33] Leino Y, Pitkaranta J. On the membrane locking of h - p finite elements in a cylindrical shell problem. Int J Numer Methods Eng 1994;37:1053-1070. [34] Ciarlet PhG. Plates and junctions in elastic multi-structures: An asymptotic analysis. Mason, Paris, 1991. [35] Simo JC, Tanrow N. A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int J Numer Methods Eng 1994;37:2527-2549. [36] Brank B, Briseghella L, Tonello M, Damjanic FB. On non-linear implementation of energy-momentum conserving algorithm for a finite rotation shell model. Int J Numer Eng 1998;42:409-442. [37] Brank B, Mamouri S, Ibrahimbegovic A. Finite rotations in dynamics of shells and Newmark implicit time-stepping schemes. 2000, submitted. [38] Ibrahimbegovic A. Geometrically exact shell theory for

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finite rotations and its finite element implementation. Eur J Finite Elem 1997;6:263-335. [39] Ibrahimbegovic A, Brank B, Courtois P. Stress Resultant Geometrically Exact Form of Classical Shell Model and Vector-Like Parameterization of Constrained Finite Rotations. Int J Numer Methods Eng, 2001, in press. [40] Peng X, Crisfield MA. A consistent co-rotational formulation for shells: Using the constant stress/constant moment triangle. Int J Numer Methods Eng 1992;35:1829-1847. [41] Sansour C. Large strain deformations of elastic shells: Con

stitutive modelling and finite element analysis. Comput Methods Appl Mech Eng 1998;161:1-18. [42] Simo JC, Fox DD, Rifai MS. On a stress resultants geometrically exact shell model. Part III: The computational aspects of the nonlinear theory. Comput Methods Appl Mech Eng 1990;79:21-70. [43] Simo JC, Rifai MS, Fox DD. On a stress resultants geometrically exact shell model. Part IV: Variable thickness shells with through-the-thickness stretching. Comput Methods Appl Mech Eng 1990;81:91-126.

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Modeling of friction pendulum bearings for the seismic analysis of bridges Tim J. Ingham * Associate T.Y. Lin International, 825 Battery Street, San Francisco, CA 94111, USA

Abstract The modeling of friction pendulum bearings using contact surf'aces is compared with the modeling recommended by codes and design guidelines, using bilinear hysteresis loops. The advantages of the contact surface model for the seismic analysis of bridges are discussed. The model is illustrated by the analysis of the Aurora Avenue Bridge in Seattle, Washington. Keywords: Friction pendulum bearings; Seismic analysis; Bridges

1. Introduction Friction pendulums bearings [5] are intended for the seismic isolation of structures. They have been installed for this purpose in several buildings and they have recently been installed in two bridges [3,8]. They are particularly well suited to bridge applications because they are insensitive to temperature over the range -40°F to 120°F [2]. Fig. 1 is a schematic drawing of a friction pendulum bearing. The bearing consists of a stainless-steel concave dish and a stainless-steel articulated slider surfaced with a composite liner. During an earthquake the slider moves back and forth on the concave dish; the spherical surfaces of the slider and the dish define a motion similar to that of a pendulum. The composite liner produces a frictional

force that is 5-7% of the vertical force acting on the bearing. A friction pendulum bearing isolates a structure from an earthquake through pendulum motion and absorbs earthquake energy through friction.

2. Code modeling The lateral response of a friction pendulum bearing can be described by the force-deformation relationship F = —D + R

fiNisgnD)

(1)

where F is the lateral force, N is the vertical force acting on the bearing, R is the radius of curvature of the bearing

A r t i c u l a t e d Slider

Deformation Concave Dish

^Conposite Liner

Fig. 1. Friction pendulum bearing and idealized bilinear hysteresis loop. *Tel.: +1 (415) 291-3781; Fax: +1 (415) 433-0807; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

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T.J. Ingham/First MIT Conference on Computational Fluid and Solid Mechanics

surfaces, D is the lateral deformation, /x is the coefficient of friction, and D is the velocity of the top half of the bearing relative to the bottom half. The first term in Eq. (1) represents the restoring force due to the curvature of the bearing; the second term represents the frictional force opposing the relative motion of the bearing. Eq. (1) is for motion in a single direction. For bi-directional motion, the restoring force acts towards the center of the bearing and the frictional force acts according to Coulomb's law of friction. Most codes and design guidelines [5,6] recommend that the vertical force acting on the bearing be taken as the structural dead load supported by the bearing, W. The lateral force is then W F = — D + M^(sgnD), (2) R the sum of a term proportional to the displacement — a stiffness term — and a force constant in magnitude but dependent on the direction of motion. This relationship is equivalent to the bilinear hysteresis loop shown in Fig. 1. The simplification of Eq. (1) to Eq. (2) ignores the variation with time of the vertical force acting on the bearing. This variation arises from overturning of the structure and from response of the structure to vertical ground motion. In buildings supported on many bearings, and interconnected by a horizontal diaphragm, the effects of overturning tend to cancel since the lateral force induced in each bearing is proportional to the vertical force acting on it. The total response of the building can then be adequately predicted by 'summing over' Eq. (2). Assuming that the vertical motions are uncorrelated with the horizontal motions, and that the building is vertically rigid, the analysis may be performed with upper and lower bounds [6] N = W(1±0.30C)

(3)

where C is a seismic coefficient (sometimes, vertical motions are just ignored). These simplifications may not be justified for large bridge structures, however, because the bearings in a large bridge act independently — it is often necessary to compute the forces in the critical connections of each bearing — and because bridges respond dynamically to vertical motions. Also, near active faults, the assumption that the vertical and horizontal motions are uncorrelated may be incorrect.

3. Finite element modeling Fig. 2 shows a model of a friction pendulum bearing based on contact surfaces with friction. The modeling was implemented using the ADINA [1] general-purpose finite element program. The dish is modeled with a spherical mesh of contact segments that together constitute a contact surface. The contact segments may be formed on the surface of shell elements, or in ADINA, they may be

aider- Contact Surface aider-Contact Fbint

Cfeh - Contact ajrface Fig. 2. Contact surface model of a friction pendulum bearing. defined as a rigid surface without any underlying finite element mesh. The slider is effectively modeled with a single contact point. For practical reasons, this point exists on a contact segment (surface) that lies on one face of a solid finite element. The opposing contact surfaces are defined as a contact pair with a coefficient of friction equal to that specified for the bearing. This modeling faithfully reproduces the force-deformation relationship given in Eq. (1). Both the restoring force and the frictional force are proportional to the instantaneous vertical force acting on the model. Furthermore the modeling properly reflects the two-dimensional behavior of the bearing. The model builds upon the work of Mutobe and Cooper [4], who developed a model with a flat contact surface and restoring springs. The correctness of the modeling was verified by analyzing some special cases. For example, the period of vibration of a frictionless slider was found to depend on the radius of curvature of the bearing in the same way that the period of vibration of a pendulum depends on its length. Fig. 3 shows the computed response of a slider on a flat surface with 5% friction subjected to horizontal and vertical earthquake motions representative of a stiff soil site. This case, of a rigid body on a flat surface, can also be analyzed using the sliding block method of Newmark [7]. The response computed by the Newmark method (using Mathcad) is also shown in Fig. 3. The two solutions agree reasonably well.

4. Application to bridge analysis Fig. 4 shows an ADINA model of the Aurora Avenue Bridge across Lake Union in Seattle, Washington. Exclusive of its approaches, this cantilever steel truss bridge is 1875 feet long and has a main span of 800 feet. It was designed and built between 1929 and 1931. The concrete substructure of the bridge is very lightly reinforced and vulnerable to large earthquakes. For the Washington State Department of Transportation a study was made of the effectiveness of retrofitting the bridge with friction pendulum bearings. The study assumed replacing each of the twelve pin bearings supporting the bridge with a friction pendulum bearing. Each of these was modeled using the contact surface model described in this paper; a typical bearing is shown in Fig. 4. The bearing shown has a radius of

T.J. Ingham /First MIT Conference on Computational Fluid and Solid Mechanics

257

1.5 n 1.0 H ADINA

Time, s Fig. 3. Analysis of a rigid body sliding on a flat surface.

Fig. 4. ADINA model of the Aurora Avenue Bridge. 400

•Bilinear Model -600 Deformation, ft

Contact Surface Model

Fig. 5. Transverse direction bearing response for the Aurora Avenue Bridge. curvature of 20 feet and a coefficient of friction of 5%. The model has 5 contact segments in the radial direction and 36 segments around its circumference. For comparison each bearing was also modeled using the bilinear hysteresis loop recommended by codes. The transverse direction force-deformation hysteresis loops for one of the main span bearings are shown in Fig. 5

for both the contact surface model and the bilinear model. The results for the contact surface model deviate significantly from the idealized hysteresis loops produced by the bilinear model. The contact surface model predicts a peak force of 592 kips whereas the bilinear model predicts only 424 kips. The peak radial displacement predicted by the contact surface model is 0.97 feet versus 0.80 feet predicted

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T.J. Ingham /First MIT Conference on Computational Fluid and Solid Mechanics

by the bilinear model. In part, these differences reflect the large variation in axial force acting on the bearing throughout the earthquake. This varies between 3580 and 8620 kips from an initial dead load of 5700 kips. This variation is fully accounted for by the contact surface model, but ignored by the bilinear model. Considering all of the bearings in the structure, the bilinear model usually predicted both smaller forces and smaller displacements than did the contact surface model. The improved performance of the contact surface model is at some cost, however. For 2000 time steps, the analysis time increased from 10.4 h for the bilinear model to 19.5 h for the contact surface model.

5. Conclusions A contact surface model of friction pendulum bearings has been developed for the seismic analysis of bridges. For a reasonable increase in computational effort, this model is significantly more accurate than the modeling recommended by codes and design guidelines.

References [1] ADINA Theory and Modeling Guide. ADINA R&D Inc., 1999. [2] Evaluation Findings for Earthquake Protection Systems, Inc. Friction Pendulum Bearings. Highway Innovative Technology Evaluation Center, 1998. [3] Imbsen RA. Seismic modeUng and analysis of the Benicia-Martinez Bridge. In: Structural Engineering World Wide. Amsterdam: Elsevier, 1998. [4] Mutobe RM, Cooper TR. Nonlinear analysis of a large bridge with isolation bearings. Comput Struct 1999;72:279292. [5] Naeim F, Kelly JM. Design of Seismic Isolated Structures. New York: John Wiley and Sons, 1999. [6] NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of Buildings. Building Seismic Safety Council, 1997. [7] Newmark NM. Effects of earthquakes on dams and embankments. Geotechnique 1965;14(2):139-160. [8] Zayas VA, Low SS. Seismic isolation for extreme low temperatures. 8th Canadian Conference on Earthquake Engineering, Vancouver, 1999.

259

MITC finite elements for adaptive laminated composite shells Riccardo lozzi, Paolo Gaudenzi * Universita degli studi di Roma La Sapienza, Dipartimento Aerospaziale, Via Eudossiana 16, 00184 Rome, Italy

Abstract The formulation of the MITC shell element is extended to active laminated shells. An active layer — made by a piezoelectric material or a similar active medium — is assumed to be included in the stacking sequence of a laminated shell. The actuation capability of the layer is represented by a given inplane strain field that can be thought of as being produced by the converse piezoelectric effect or other induced strain actuation mechanism. In this way, the actuation mechanism is included in the formulation of shear deformable shell element that has been demonstrated not to suffer of shear locking effects. The MITC four-node element has been selected for the preliminary investigation. Several comparisons have been performed to verify the accuracy of the formulation and to check the predicting capability of the element in comparison with both numerical and experimental results of the recent available literature. Keywords: Composite shell; Piezoelectric material; Finite element method

1. Introduction The use of active materials, like piezoceramics or shape memory alloys, has been recently proposed for developing actuation and sensing capability of structural systems. In this framework, laminated shells have been selected as a possible candidate typology of structural systems for including such materials at the level of one or more layers of their stacking sequence. Several models have been proposed in the recent literature for the analysis of active laminated plates and shells, since the studies by Crawley and Lazarus [1], in which the classical laminated plate model is extended to include the actuation mechanism produced by active piezoelectric layers. An analysis based on a CLT theory that included not only the piezoelectric, but also the thermoelastic effect was proposed by Tauchert [2]. First-order shear deformable active plate theories were also proposed and implemented in a finite element model by Han and Lee [3], Saravanos [4], Chandrshastra and Agarwal [5], and by Suleman and Venkayya [6]. In those cases, displacement based approaches were used by the different authors, but only the last one explicitly mentions the need for a proper integration of the stiffness matrix. In fact, it is well known that first-order shear deformable shell theories * Corresponding author. Tel.: +39 (6) 4458-5304; Fax: +39 (6) 4458-5670; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

suffer from the so-called shear-locking problem that should be overcome either by means of a reduced integration or by a mixed interpolation approach as illustrated by Bathe in his textbook [7]. The paper aims at generalizing the finite element modeling produced by Bathe and Dvorkin [8] in their MITC plate and shell models, to include the presence of active layers. In this way, a sound theoretical and numerical basis, capable of modeling the transverse shear deformation without the occurrence of the shear locking problem, will be made available for a class of advanced structural elements. In fact, to the knowledge of the authors, only Kirchhoff plate models or displacement-based Mindhn plate models (with shear locking problems) were proposed for active shells in the recent literature, as previously cited.

2. Finite element formulation The formulation of the four-node active shell element presented here (Fig. 1), represents an extension of the MITC-4 shell element proposed by Bathe and Dvorkin [8], and, therefore, the same notation as in those references will be used. The procedure is based on a different interpolation of the transverse shear strains with respect to the one used for inplane components. The finite element equilibrium equations are derived by first considering the expression of

260

R. lozzi, P. Gaudenzi /First MIT Conference on Computational Fluid and Solid Mechanics

C^^^.^r "^^

TT , • 1 T • X 1 cu 11 through-the-thickness Hybnd Laminated Shell Applied Electric Field Displacements

Fig. 1. Definition of the four-node shell element and of the active laminated shell, where some piezoelectric layers have been included to obtain actuation capability. containing, respectively, the piezoelectric and the thermoelastic constants of layer n:

the total potential energy: (1) where m is an index spanning the total number of elements in which the structure has been subdivided, and n is the index spanning the number of layers A^^, composing the laminated shell at element m. The appropriate constitutive law must then be used: ,^ij(m)

n P^ijkl^'"" ~(m)

(2)

where, to achieve the expression of the fourth-order contravariant constitutive tensor "C'^^^""* in convected coordinates Vi starting from the known constitutive law in the local Cartesian system of orthonormal base vector e,, / = 1, 2, 3, the following transformation is used:

(3) The induced strains have been represented introducing into Eq. (1) the 'piezoelectric equivalent stress', accounting for the presence of some active piezoelectric layers in the laminated shell, and the 'thermoelastic equivalent stress', accounting for thermoelastic effects:

-5E/(

- / / - ) ^ -/;•/'<-'

,n..^^

+r

n^^-^_n^^-.^(.)^y.

0

E],

^r = 0

0

0

0

0

"t

0

0

0

0

-d^f

0

0

0

0

"32

_ "31 .^(m)T

^

Wr

"2

"33

"af^

0

0

o|

(7)

Both the piezoelectric and thermoelastic equivalent stresses are obtained using the following transformation: (8)

^^/'^^«f/-(Og^-.e")(V-e:)

Invoking the stationarity of the total potential energy U, we finally obtain the finite element equilibrium equation, in matrix form: KU -RT-RV

(9)

= R

where K is the stiffness matrix, R is the mechanical force vector (due to applied forces) Rj is the 'thermoelastic equivalent force vector' (due to thermal actions) and Rp is the 'piezoelectric equivalent force vector' (due to applied voltages), of the entire system. The elemental expressions for those quantities are: nc'^^gim)

(4)

(5) (6)

where E is the electric field applied in the thickness direcand "a^"*^ are the matrix

dV,

Rm ^ I fj(n.>^fB"-> dv + /• H'''"'''fT AS, y{m)

tion, A T = r — To is the temperature variation from the reference temperature TQ, "d

={0

dV-W

The linear electromechanic coupling law, as well as the thermoelastic one, are known in the local Cartesian system of orthonormal base vector C/, / = 1, 2, 3:

^_„^(™),^(.)^

E

Am) Nrr,

RT--T.J

.

I<m)'H=<"

fiW'-fP'-'dV,

(10)

R. lozzi, P. Gaudenzi /First MIT Conference on Computational Fluid and Solid Mechanics

^1

1

i4

261

0.05 0.04S

Piezoceramics

A

oII

X

^

X

0.04

D

Present FEM o Experiment [Crawley, 1991] FEM[HaetaI,1992]

T

A—

r.=0,292

fit

0,0083m0,0025m

X10-*

5

a D

,/

/ -

4

/'

^"

a

D , - ' ' •

2

"

, , ' - '

. ^

1

^_ p , - ' • '

^^:tr"^.a-' ---Q"^ 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 2. Comparison of the present analysis to the experimental measurements and to previous FEM. „x10'*

[0/90/90/0/p] Present FEM [0/90/p/90/0] Present FEM [p/0/90/90/0] Present FEM [0/90/90/0/p] Sarav. FEM

Case 1: [0/90m/0/p]

Fig. 3. Comparison of the present FE analysis to the Saravanos' solution and deformed configuration for the first case analyzed [0/90/90/0/p].

262

R. lozzi, P. Gaudenzi / First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Results for laminated, simply supported Tauchert's square plate Configuration

Electric field (V/m)

[GE:079070790°]s [GE:079070790°]s/[PVDF:0°] [GE:079070790°]s/[PVDF:0°]

0 3 X 10^

where B^^^ is the strain-displacement matrix obtained from the MITC formulation, H^""^ is the displacement interpolation matrix, "f^*"" and "f^""' are the arrays containing the components of the thermoelastic and piezoelectric equivaB(m) lent stresses f'^ for the layer n, f and / / are the volume and surface force vectors externally applied.

3. Numerical results 3.1. Tauchert piezothermoelastic composite plate Tauchert [2] investigated the piezothermoelastic response of a laminated, simply supported, rectangular plate, subjected to a linear temperature variation with 80 and 0°C temperature increases on the upper and lower surfaces respectively, trying to eliminate thermally induced deformation through the addition of a PVDF layer to the original laminate. Attention is given to an eightlayer, graphite-epoxy, symmetric, cross-ply square panel ([079070790°]s) and to a nine-layer hybrid laminate with an additional PVDF layer located at the bottom surface ([(079070790°)s//7]). Both the geometrical and the mechanical properties of each layer are supplied in [2]. The transverse displacement at the center point of the plate is reported in Table 1 where the extended CLT solution by Tauchert is compared to the results provided by the current FE analysis. 3.2. Crawley and Lazarus cantilever plate The specimen used by Crawley and Lazarus [1] in their experiments, consists of a graphite/epoxy (ASA/3501) cantilevered plate on whose surfaces thirty G1195N piezoceramics are symmetrically bonded, as shown in Fig. 2. The mechanical properties used for the materials involved are those reported in the work mentioned. Mx, M2 and M3 are the out-of-plane displacements measured at y = C/2, y = 0 and y = —C/2, respectively. Wi, W2 and WT, are nondimensional quantities representing, respectively, longitudinal bending, lateral twisting and transverse bending: Wi = M2/C,

W2 = (M3 - Mi)/C,

Center deflection Extended CLT (Tauchert)

Present FEM (mesh: 32 x 32)

0.00257 0.00100 0.00008

0.00258 0.00101 0.0000860

H^3 = - [ M 2 - ( M 3 - h M i ) / 2 ] ,

(11)

Fig. 2 shows a fairly good agreement between the present FEM results and those presented in previous works. 3.3. Dimitris A. Saravanos [0/90]^ cylindrical panel Saravanos studied the response of a hybrid graphite/ epoxy simply supported 90° cylindrical panel with a continuous piezoelectric layer (PZT-4) subjected to a uniform electric field, E3 = —400 kV/m, applied in the thickness direction. Three stacking sequences have been considered corresponding to a different thickness location of the piezoelectric actuator: (1) [0/90/90/0//?]; (2) [0/90//?/90/0]; (3) [/7/O/9O/9O/O], where p indicates the piezoelectric layer. The geometry and the mechanical properties of the materials used are the same mentioned by Saravanos [4]. The comparison between the current FE analysis and those by Saravanos, reported in Fig. 3, shows the good prediction capability of the FEM presented here, in the analysis of adaptive laminated composite shells.

Acknowledgements The financial support of the CNR PFMSTA-II Project 99.01797.PF34 and of the MURST cofin.99 cap.7109 are gratefully acknowledged.

References [1] Crawley EF, Lazarus KB. Induced strain actuation of isotropic and anisotropic plates. AIAA J 1991;29(6):944951. [2] Tauchert TR. Piezothermoelastic behaviour of a laminated plate. J Therm Stresses 1992;15:25-37. [3] Han JH, Lee I. Active damping enhancement of composite plates with electrode designed piezoelectric materials. J Intell Mater Syst Struct 1997;8:249-259. [4] Saravanos DA. Mixed laminate theory and finite element for smart piezoelectric composite shell structures. AIAA J 1997;35(8):1327-1333. [5] Chandrashekhara K, Agarwal AN. Active vibration control of laminated composite plates using piezoelectric devices:

R. lozzi, P. Gaudenzi /First MIT Conference on Computational Fluid and Solid Mechanics a finite element approach. J Intell Mater System Struct 1993;4:496-507. [6] Suleman A, Venkayya VB. A simple finite element formulation for a laminated composite plate with piezoelectric layers. J Intell Mater Syst Struct 1995;6:776-782.

263

[7] Bathe KJ. Finite Element Procedure. Englewood Cliffs, NJ: Prentice-Hall, 1996. [8] Bathe KJ, Dvorkin EN. A formulation of general shell dements — the use of mixed interpolation of tensorial components. Int J Numer Methods Eng 1986;22:697-722.

264

Tire tread pattern analysis for ultimate performance of hydroplaning Isam Janajreh ^'*, Ali Rezgui ^, Vincent Estenne ^ " Michelin North America R&D Corp., Greenville SC 29602, USA ^ Michelin France, Center de Technologies, Ladoux 63040 Clermont-Ferrand Cedex 9, France

Abstract A brief introduction on the literature in hydroplaning modeling is presented. We have conducted an external flow study over two sets of tread patterns, elementary and analytical, and we have observed the magnitude of the drag force. Then, we have compared the ranking based on the computed indicator, namely the drag force and the measurements of the loss of tire contact area obtained at Michelin glass pit. The correlation was found to be in good agreement. Keywords: Hydroplaning; Local sculptural analysis; Glass pit; Drag force

1. Introduction At certain wet driving conditions over a road with a given surface texture and with a particular tire tread pattern made of a specific rubber compound, the available horizontal traction force is dramatically reduced. These conditions hinder the steering and braking capabilities of the driver. Under these conditions the vehicle is said to be experiencing hydroplaning. The loss of traction is due to an intervening fluid film characterized by high hydrostatic pressure, which separates part of the tire contact patch from the road surface asperities. Tire designers seek a tread pattern that allows maximum drainage capabilities and deep tread for efficient fluid expulsion to decrease the potential of a progressive hydrostatic pressure build up. In this paper, we present local sculptural analysis of the tire footprint by computing the tread pattern drag force and comparing the ranking with the glass pit results.

2. Review of tlie state of the art in hydroplaning Due to the lack of the essential computational resources, earlier hydroplaning simulation attempts utilized a simpli* Corresponding author. Tel.: +1 (864) 422-4336; Fax: +1 (864) 422-3508; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

fled scheme of the governing physical equations. Daughaday and Tung [1] were amongst the first to conduct analytical treatment of tire hydroplaning. They have used the concept of the boundary layer solution in matching the flow in two regions, namely the inviscid and the viscous regions. They started from the Navier-Stokes equations and used a perturbation technique that reduced the governing equations to the Euler equation and the Reynolds equation. The former governs the thick inviscid region denoted as hydroplaning region and the latter governs the thin viscous region denoted as viscoplaning. The solutions of the two equations are then matched at the interface by satisfying the same pressure and velocity components. The complexity of tire shape precluded an analytical solution and merged hydroplaning studies into the computation fluid dynamic field as presented in the work by Brown and Whicker [2], Aksenov [3], Grogger and Weiss [4,5], Sata et al. [6], and Okano and Koishi [7]. Nowadays, ranking tires can be targeted with minimum tread design and architectural modifications. This assurance stems from the available tools such as computational fluid dynamics (CFD) and finite-element modeUng (FEM). Thus, in this paper we utilize the above two simulation tools. First we perform FEM analysis to obtain a realistic tire contact patch, and second we perform CFD on the contact patch and tread patterns. The second aspect is the focus of this article by computing the fluid drag force and comparing it to glass pit experimental measurements.

/. Janajreh et al. /First MIT Conference on Computational Fluid and Solid Mechanics 3. The utilized CFD code The code used for our simulation is Euranus of Numeca Inc. It is a structured finite volume code based on the Jameson scheme. This scheme was initially constructed to handle compressible aerodynamics problems and has been later adapted using preconditioning to handle slightly compressible flows with a very small Mach number. The Jameson scheme is explicit, but it became implicit if we add the residual smoothing of Lerat. This allows the use of a high CFL number essential for steady or slowly unsteady flows. The code uses structured mutiblocks with a body-fitted mesh (to handle the meshing of complex geometry) and uses a multigrid to speed up the conversions.

4. Flow and experimental setup The computation domain consists of a parallelepiped, {x in the cross-stream, y along the stream and z perpendicular to the flow stream) built around the tire contact patch that is slightly non-symmetrical and is calculated by in-house FEM code. The domain is set up as follow: two mirror faces (one is the ground and the second is the base of tire treads), one upstream inflow; and three outflows (two at the sides and one at the downstream). The upstream flow velocity is 10, 15 and 25 m/s. The Reynolds number is about 10^ based on the groove depth (10 mm) which indicates a turbulent flow regime. A Baldwin-Lomax turbulence model was utilized since it produces a similar result to the classical KE within the grooves during initial testing and because it provides a shorter computation time. The constructed tread pattern meshes have 50,000 to 68,000 cells and the time step is controlled by a Courant number of 1.5.

265

The experimental measurement is conducted at the MicheHn glass pit. The glass pit is a hydroplaning performance measuring tool that evaluates the reduction of tire contact patch with respect to vehicle speed. This is due to the intrusion of the water film underneath the tire contact that separates part of the initial contact from the ground. There has been a good ranking correlation between the glass pit tire hydroplaning and wet skid tire testing that have been verified by tire industry [8]. This loss of contact is attributed to a poor fluid expulsion. This consequently results in a higher tire drag force. During the experiments a free rolling vehicle tire passes over the glass pit prism that is flooded with a fixed water film. The prism is equipped with a high-speed shutter camera underneath that snaps the image of the passing tire contact patch. The image is post-processed and the remaining tire contact area is computed. These areas are normalized and ranked accordingly amongst each other or against a targeted reference tire.

5. Results and discussions 5.7. Elementary tread patterns Our hypothesis is classical where we utilize the drag force as the indicator of the tread pattern quality in evacuating the encountered fluid. Therefore a higher drag level leads to an earlier hydroplaning situation. The addition of grooves should delay the onset of hydroplaning and thus should reduce the drag force. We compute the velocity field and the drag forces of three elementary tread patterns including a slick tire, and tires with 3 and 5 longitudinal grooves. The results agree well with our intuition as shown in Figs. 1 and 2 where the drag force is inversely proportional to the number of grooves and it follows the expected

magnitude v 40 f

magnitude V

30:

n

25: 20: 15:

y

T

W

io:

Fig. 1. Smooth contours of the velocity magnitude at 25 m/s, for the elementary slick (left) and 5-grooved (right) tire.

266

/. Janajreh et al. /First MIT Conference on Computational Fluid and Solid Mechanics 600 500 5. 400

1 300 a»

2 200 o

100 0 18 Velocity (mfe)

Fig. 2. Drag force evolution with the additions of tire grooves at 10, 15 and 25 m/s with their parabolic fit. parabolic trend versus the velocity. We observe the presence of two pronounced vortices in the front of the slick tread pattern that collapse with the addition of grooves. Due to the symmetry between the ground and the tread, we have observed insignificant pressure variation in the z direction that suggests a 2D computation can be sufficient. 5.2. Analytical tread patterns (longitudinal and lateral grooves) Two set of examples are presented where the flow configurations are similar to the elementary tread patterns discussed above, except that the analysis are conducted at one speed of 25 m/s. In the first example, Fig. 3 depicts the flow field on two analytical solutions denoted as sol. 1 and sol. 2. The objective is to determine whether solution 1 or 2 will perform better in hydroplaning. The computed drag force over sol. 1 (659 N) is higher than the computed drag force of sol. 2 (553 N) which suggests that sol. 2 is a better candidate than sol. 1. The glass pit measurements confirm the drag force ranking since it produces 36% improvement magnitude V

of sol. 2 over sol. 1. It is worth mentioning that the sol. 1 sculpture has a void ratio of 36% while the sol. 2 sculpture has a void ratio of 39%, and the increase of the void ratio is a classical trend of tire designers in attempting to improve the hydroplaning tire performance. In the second example, Fig. 4 depicts the computation of the flow field and gives the drag force of try. 1 and try. 2 sculptures where both have the same void ration of 39%. The computations of the drag force suggest that try. 2 will exhibit a better hydroplaning performance than try. 1 which are confirmed by the glass pit. A summary of the magnitude of the drag force and glass pit ranking are given in Table 1.

6. Conclusion The emergence of CFD in analysis of tire hydroplaning has become more evident. In this work we have conducted drag force sensitivity analysis over elementary sculptures and have observed that the addition of the grooves results magnitude V

Fig. 3. Smooth contours of the velocity magnitude at 25 m/s, for sol. 1 (left) and sol. 2 (right) of analytical tires.

/. Janajreh et al /First MIT Conference on Computational Fluid and Solid Mechanics

267 magnitude V 30:

Fig. 4. Smooth contours of the velocity magnitude at 25 m/s, for try. 1 (left) and try. 2 (right) of analytical tires. Table 1 Analytical sculptures drag force and glass pit comparison summary, sol. 1 versus sol. 2 and try. 1 versus try. 2 Tire 195/65/R15 Void ratio

(%) Sol.l Sol.2 Try.l Try.2

(36) (39) (39) (39)

Computed drag force (N)

Computed drag force

(%)

Measured glass pit area loss index (%)

659 553 496 464

100 116 100 106

100 136 100 104

in the reduction of the drag force and consequently an improvement in the hydroplaning performance. Two examples w^ere conducted over four analytical sculptures that suggest implementation of the drag force as a criteria to rank a set of sculptures for their hydroplaning performance. In this work we have shown that the ranking of the sculptures based on the computed drag force and the experimental measured contact area are in agreement for sculptures with a set of sculptures having different void ratio and another set with that have the same void ratio.

References [1] Daughaday H, Tung, C. A mathematical analysis of hydroplaning phenomena. Technical Report, Cal. No. AG2495-S-l, Cornell Aeronautical Laboratory, Jan. 1969. [2] Brown, Whicker D. An interactive tire-fluid model for dynamic hydroplaning, friction interaction of tire and pavement. Meyer/Walter, ASTM Special Technical PubHcation 793, pp. 130-150. [3] Aksenov A. Numerical Simulation of Car Tire Aquaplaning, CFD 96. John Wiley and Sons Ltd, 1996 [4] Grogger, Weiss. Calculation of the 3D free surface flow around automobile tire. Tire Sci Technol 1996;24(Jan-Mar). [5] Grogger, Weiss. Hydroplaning of automobile tire. Tire Sci Technol 1997; 27 (Jan-Mar). [6] Sata et al. Hydroplaning analysis by FEM and FVM: effect of tire rolling and tire pattern on hydroplaning. Tire Sci Technol, in press. [7] Okano T, Koishi M. A new computation procedures to predict transient hydroplaning performance of a tire, FEM/FVM. Tire Sci Technol, in press. [8] Yeager RW. Tire hydroplaning: testing, analysis, and design. In: Heys Browne (Ed), The Physics of Tire Traction, Theory and Experiment. New York: Plenum Press, 1974, pp. 25-57.

268

Design and framework of reduced instruction set codes for scalable computations for nonlinear structural dynamics R. Kanapady, K.K. Tamma * Department of Mechanical Engineering, 111 Church Street S.E. 125, University of Minnesota, Minneapolis, MN 55455, USA

Abstract vspace-3ptA general framework and avenues towards the design of a unified integrated computational technology for nonlinear structural dynamics encompassing a wide variety of new and unexplored, and existing time integration operators is now possible employing the so-called Reduced Instruction Set Codes (RISC) via a unified family of generahzed integration operators [GInO] towards scalable computations on massively parallel computing platforms. Whilst the RISC paradigm has a critical impact on the scientific code design and development time and efforts, it simultaneously increases the functionality of the scientific codes by many folds by providing a variety of choices to the analyst. A unified scalable computational approach towards such a computational technology is desirable for large-scale structures and large processor counts employing a message-passing paradigm (using MPI), graph partitioning techniques, and Lagrange multiplier based domain decomposition methods. Here, the focus is on the scalability analysis conducted via an integrated unified technology for [GInO] with emphasis on the family of optimal non-dissipative and dissipative algorithms for structural dynamics in conjunction with large deformation, elastic, elastic-plastic dynamic response. For geometric nonlinearity a total Lagrangian formulation, and for material nonlinearity elasto-plastic formulations are employed. This is the first time that such a general framework and capability is plausible via a unified technology and the developments further enhance computational structural dynamics areas. Keywords: Nonhnear structural dynamics; Time integration; Parallel computing; Lagrange multiplier based domain decomposition; Numerical scalability; RISC; MPI

1. Introduction It is being recognized that the pressing need for improved solution times and feasibility to conduct large-scale practical analysis accurately for nonlinear structural dynamics on modem computing platforms as the general goal. Hence, many of today's attempts to speed up solution and computational procedures center on optimization of codes for specific computing platforms. Computing platforms could be a single processor, high-performance computers or parallel computers. In the single-processor situation, optimization is performed by restructuring of the code to take advantage of the memory hierarchy and compiler technology and the like. In parallel computing realm, restructuring the code is done to take advantage of inherent parallelism in the formulation and the parallel architecture under consid* Corresponding author. Tel.: -Hi (612) 626-8102; Fax: +1 (612) 624-1398; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

eration. While these methods do indeed produce significant results in reducing the solution time, the efforts which are both time and development intensive, will constantly follow the development of new computer hardware having extremely short life cycles both in the serial and parallel computing realms. To handle such a wide variety of situations, a general framework and design encompassing the Reduced Instruction Set Code (RISC) based paradigm is described both for serial and parallel computations. Here, reference to the research efforts are not made to the differences between programming languages, nor to differences between the multitudes of parallel extensions to specific programming languages. The concern is more about the impact of a given parallel hardware architecture on the software design, and sometimes, on the solution algorithm itself. For scientific computations encompassing transient/dynamic analysis encountered in engineering, mathematical and physical sciences, the design of computational algorithms accounting for time-dependent phenomena plays a

269

R. Kanapady, K.K. Tamma/First MIT Conference on Computational Fluid and Solid Mechanics

critical role in a variety of applications. Scientific largescale simulations accounting for time-dependent phenomena of many industry and DoD relevant applications are encountered in process modeling and manufacturing studies, computational fluid dynamics, computational heat transfer applications, computational structural dynamics, multibody dynamics and the like. In this research, attention is focused towards scalable computations for nonlinear structural dynamics applications. Attention is restricted here to computational structural dynamics. Computational algorithms and solution techniques for structural dynamics systems of equations have indeed matured over the years. Of the various transient algorithms available in the literature for structural dynamics computations, the so-called direct time integration techniques continue to be popular in commercial codes. Recently emanating under the umbrella and explained via a generalized time-weighted philosophy, a formal theory of development/ evolution, characterization, design and implementation of a wide variety of computational structural dynamics algorithms is described in [1] for linear and in [2] for nonlinear situations, respectively, via a unified methodology. For the first time, such features now permit Reduced Instruction Set Codes to incorporate a unified computational technology with a wide variety of choices of new and existing algorithms to the analysts in conjunction with graph partitioning techniques and domain decomposition methods.

2. Computational algorithms Most of the traditional approaches we are familiar with, including new computational algorithms which inherit excellent algorithmic attributes in contrast to all existing approaches and which have not been explored and/or exploited to-date, are indeed an integral part of the present framework. Summarizing, for simplicity the so-called generalized integration operator [GInO] for nonlinear dynamic situations can be stated as follows [1,2]. Let W{T) =Z WQ-\- wiT + W2T^ + wsz^; r e [0, At] be the weighted time field approximation employed for enacting the time discretization process of the semi-discretized equations of motion. Then, the resulting family of generalized integration operators [GInO] for nonlinear situations are given by Mu„+i +p(u„+i,u„+i) = F

(1)

where

1

Wi =

r

At'

^ 3 .

,

.

. . .

I =

1,2,3

(3)

W3,o

(4)

Wr

with the following design updates Un+l =Vin+

>.lUn A^ + ^2Un Af^ + X^At'^ (Un+i - U„)

u„+i =\Xn+ X^yinAt + XsAt (ii„+i - ii„)

(5)

As such, the associated Discrete Numerically Assigned [DNA] algorithmic markers which comprise of both the weighted time fields w{x) and the imposed conditions on the dependent field variable approximations, uniquely lead to the design and characterization of various time discretized operators via: (i) specially assigned marker coefficients for the weighted time fields; and (ii) the corresponding imposed conditions upon the dependent field variable approximations in the semi-discretized system. The specific DNA markers (if/, A,, Xt) for the [GInO] optimal energy preserving and the family of optimal dissipative algorithms [3] for structural dynamics which are second-order accurate and unconditionally stable, and possess only zero-order displacement and velocity overshooting behavior [UO, VO] (in contrast to all other existing dissipative schemes which are at a minimum [UO, VI] and only restricted to first-order accuracy of load) and which also possess minimal dissipation and dispersion for any given Poo value, where Poo is the spectral radius of the time integration method described as CO At -^ OQ, are given as Weighted time field: Wo = \,w\

— — 5 , W2 = 5,W3

=0

GInO Optimal energy preserving: Ai = 1, A2 = 1/2, A3 = 1/4, A4 = 1, A5 = 1/2, A6 = 1, Ai = \,X2 = 1/2, A3 = 1/4, A4 = l,Xs = 1/2 GInO Optimal dissipative methods: Ai = 2/(1 + Poo), A2 = 2/(1 + Poo)', A3 = 2/(1 + Poo)', A4 = (3 - P o o ) / ( l + Poo), A5 = (3 - P o o ) / ( l +

Poof,

Ae = 2(2 - poo)/(l + Poo), Ai = 1, X2 = 1/(1 + Poo), A3 = 1/(1 + P o o ) ' , A4 = 1, A5 = 1/(1 + Poo),

The remainder of the [DNA] markers contained in [GInO] for most of the practical and so-called time integration methods are described in [1,2].

u„+i = u„ + AeWi (Un+i - iin) + A4WiUnAt

=Un

Un

=Un-\-AiWiUnAt-hA2W2UnAt^ -{-A3W3 (Un+l - iin)

F

3. Scalable computations

+ A5W2 (U„+i - U„) At

Un

=(l-Wi)f„-hWiUi

At^

(2)

The next generation parallel computers will consist of thousands (computers having processor counts greater than 10,000) of high-performance processors connected via a

270

R. Kanapady, K.K. Tamma/First MIT Conference on Computational Fluid and Solid Mechanics

high bandwidth interconnection network. The critical component of effective utihzation of such systems for computational structural dynamics involves design and development of efficient and scalable parallel formulations and computational models on a seamless programming environment, which is independent of program size, problem size, number of processors and HPC platforms. The scalability of the formulation can be characterized by three distinguishing properties: (i) numerical scalability; with modest to no convergence degradation of numerical algorithms for arbitrary large problem size and processor counts; (ii) parallel scalability; ability of the parallel algorithm to deliver larger speedups for arbitrary large number of processors; and (iii) scalability of computer memory utilization with increase in the problem size and the number of processors. With the parallel computer architectures evolving continuously and the availability of various HPC platforms, the biggest challenges lie in the substitution of the key selected algorithms in an application program with redesigned algorithms, which exploit the new parallel computer architecture. The unique features of the overall framework includes: a unified family of generalized time integration operators [GInO] described previously which encompass both the traditionally advocated explicit and implicit time integration (dissipative and non-dissipative) schemes, and new computational algorithms which provide optimal algorithmic properties (dissipative and non-dissipative) that have not been explored and/or exploited to-date in conjunction with Reduced Instruction Set Code enabled coarse-grained parallel computational models which employ the messagepassing paradigm (using MPI), graph partitioning and Lagrange multiplier based domain decomposition techniques. And, it now permits for the first time the general nonlinear and linear structural dynamics analysis for large-scale realistic engineering analysis in a single analysis code via an integrated computational technology.

4. Initial results Initial results and the unique features of the present integrated computational technology employing generalized integration operators, [GInO] in a RISC enabled single analysis code for serial and scalable parallel computations are presented here. To handle the complex finite element meshes on the HPC platforms, domain decomposition is employed using MPI-based ParMetis [4]. To achieve both numerical and parallel scalability, sub-domain interfacing via Lagrange multiplier based domain decomposition techniques are employed. One such robust unified framework for the predictor multi-corrector incremental [GInO] representations for nonlinear dynamics has been developed and its corresponding sub-domain interface Lagrange multiplier solutions for second- and fourth-order elasto dynamics via preconditioned conjugate gradient algorithm (PCG) is

solved in conjunction with the present unified formulations which provide a wide variety of choices to the analyst. First, the results pertaining to serial computations, the calculated predictions of the iso-parametric degenerated shell element formulation are compared with experimental results obtained from a cylindrical panel subjected blast load. Fig. la shows the layout and details of the geometry of the shell. Fig. lb and c show the simulated results and experimental results are in excellent agreement. Next, numerical scalability performance results for nonlinear elastic/elasto-plastic implicit computations are presented in Tables 1 and 2 for a second-order elasticity cantilever beam problem discretized using 8-node brick elements and in Table 3 for a fourth-order elasticity cylindrical panel subjected blast problem discretized using 4-node shell elements. Note that Tables 1-3 show the total number of iterations of the PCG algorithm pertaining to the sub-domain 'interface' problem to converge for the 'fixed-work-perprocessor', the 'fixed-problem-size' and 'fixed-storage-perprocessor' scaling problems, respectively. The results show that the RISC technology is indeed numerically scalable

Table 1 Numerical scalability results 'fixed-work-per-processor' scaling employing dual domain decomposition method for typical implicit [GInO] methods Mesh size h (eqns)

Total iterations/time step

1/2(540) 1/4 (3,000) 1/6 (8,820) 1/8 (19,440)

Table 2 Numerical scalability results of 'fixed-problem-size' scaling employing dual domain decomposition method for typical implicit [GInO] methods No. of subdomains

Total iterations/time step

16 32

Table 3 Numerical scalability results 'fixed-storage-per-processor' scaling employing primal-dual domain decomposifion method for typical implicit [GInO] methods No. of subdomains

Mesh size h (eqns)

4 16 64

1/50(12,001) 1/100(50,001) 1/200(200,001)

Total iterations/time step

R. Kanapady, K.K. Tamma /First MIT Conference on Computational Fluid and Solid Mechanics

271

5. Conclusions Blcivf Inpflecf

re^ir,

The design and analysis of a general framework towards RISC for unified scalable parallel computations for nonlinear structural dynamics was presented. Parallel performance was illustrated on: (i) numerical scalability, (ii) linear speedups, and (iii) parallel scalability.

\ \

Acknowledgements

O -^7 ...A----

Experimental Newmark (y= 1/2, p-1/4) Newmark (7=1/2,|3 = 0) Undeformed shape

0.5

(b)

The authors are very pleased to acknowledge in part by Battle/U.S. Army Research Office (ARO) Research Triangle Park, North Carolina, under grant number DAAH0496-C-0086. The support of the A R L / M S R C and the IMT activities and additional support in the form of computer grants from Minnesota Supercomputer Institute (MSI) are gratefully acknowledged. The support in part, by the Army High Performance Computing Research Center (AHPCRC) under the auspices of the Department of the Army, Army Research Laboratory (ARL) cooperative agreement number DAAH04-95-2-0003/contract number DAAH04-95-C0008 is also acknowledged. The content does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. Additional thanks are also due to X. Zhou, D. Sha, Dr. A. Mark and Dr. R. Namburu, Prof. G. Karypis and Prof. V. Kumar for relevant technical discussions.

1 1.5 X direction (in.)

References

0.0004

Time (sec)

0.0006

0.0008

Fig. 1. Cylindrical panel subjected to blast loading; geometry, displacement of cross-section of panel, deformation history for various algorithms via a code employing RISC paradigm, (a) Geometry; (b) cross-section displacement; (c) displacement history. for computational structural dynamics via [GInO]. Thus, demonstrating the potential to provide highly scalable parallel computations via an integrated computational technology for both explicit and implicit structural dynamics.

[1] Tamma KK, Zhou X, Sha D. A theory of development and design of generalized integration operators for computational structural dynamics. Int J Numer Methods Eng 2001 ;50: 1619-1664. [2] Kanapady R, Tamma KK. A unified family of generalized integration operators [GInO] for non-linear structural dynamics: implementation aspects. Adv Eng Software 2000; 31(89): 639-647. [3] Zhou X, Tamma KK, Sha, D. Linear multi-step and optimal dissipative single-step algorithms for structural dynamics. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [4] Karypis G, Kumar V. ParMETIS: parallel graph partitioning and sparse matrix ordering library. University of Minnesota, Department of Computer Science, Version 2.0, 1998. [5] Kanapady R, Tamma KK. Parallel computations via a single analysis code of a unified family of generalized integration operators [GInO] fornon-Unear structural dynamics. In: 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Atianta, GA, April 3-6, 2000. [6] Kanapady R, Tamma KK. A general framework and integrated methodology towards scalable heterogeneous computations for structural dynamics on massively parallel platforms. In: 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Seattie, WA, April 16-19, 2000.

272

Dof splitting p-adaptive meshless method M.-S. Kang,S.-K.Youn* KAIST, Mechanical Engineering Department, 373-1, Gusung, Yusung, Taejon, 305-701, Korea

Abstract In this paper, error estimator and p-adaptive refinement scheme for hp-clouds method called dof splitting meshless method (DSPMM) are proposed. The error estimator detects the difference of computed stress and projected stress. The essence of the DSPMM is to construct the p-refined equations with newly appended higher order nodal dof s. The p-refined equations are solved to minimize the residual of the unrefined solution. In refinement procedure, higher order dof's are appended only on the selected nodes determined in the error estimation procedure. Therefore the size of the p-refined equations is considerably smaller than that of unrefined equations. The DSPMM improves the solution with minor additional computational costs especially for large problems. Keywords: hp-Clouds method; Error estimator; p-Adaptive refinement; Dof splitting meshless method; DSPMM; Meshless method

1. Introduction Meshless methods such as element-free Galerkin (EFG) method [1] and hp-clouds method [2] are attractive for h-adaptive analysis because they do not need mesh structure that restricts the position of nodes in the formulation. Thus nodes can be easily added and deleted without consideration of the mesh structure. Hp-clouds method also has good features to adopt p-adaptive scheme since the method allows the addition of nodal dof's. Several error estimators for meshless method have been developed. Liu et al. [3] developed adaptive scheme using edge detection technique for reproducing kernel particle method (RKPM). Duarte and Oden [4] presented the error estimator derived in terms of residuals. Chung and Belytschko [5] estimated the error denoted by the difference of computed stress and projected stress. This paper presents an error estimator and dof splitting meshless method (DSPMM). The error estimator is based on the work of Chung and Belytschko [5]. In hp-clouds method, the projected stress cannot be directly obtained as in the Chung's method for EFG method because the number of shape functions is generally greater than that of nodes. Thus the projected stress in hp-clouds method * Corresponding author. Tel.: +82 (42) 869-3034; Fax: +82 (42) 869-3201; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

is obtained by additional formulation and matrix solution procedure. DSPMM uses the shape functions of hp-clouds method however the formulation is modified. In DSPMM, the adaptive procedure is formulated only with additional nodal dof's. The solution of DSPMM is used to update the solution of unrefined problem. This procedure allows the solution to be enhanced with minor additional computational costs. The error estimator and DSPMM are verified through numerical examples.

2. Error estimation Chung et al.'s error criterion is defined by the difference of projected stress and approximated stress. The projected stress is obtained by the linear combination of nodal stress and shape functions. The radius of influence used for the projection of stresses is smaller than that of shape functions in approximation. In hp-clouds method, the number of shape functions is greater than that of nodes. Thus the projection of nodal stress cannot be directly achieved. The stress error e^(x) in hp-clouds method can be defined as follows:

e^(x) = a'(x) - aP(x) = cj'{x) - J ^ E ^ / W o ^ ;

(1)

where a'^ix) the computed stress, a^(x) the projected

M.S. Rang, S.-K. Youn/First MIT Conference on Computational Fluid and Solid Mechanics stress, (j)\ /th shape function at node /, and a\{x) /th the expanded stress for node /. To obtain the expanded stress cf\{x) we introduce weak formulation and the resulting equations are as follow:

C I ] / ^/(•^)^/(^) d^^i = I
KYJ=

f

BfDB'jdQ

273

(10)

(11)

(2)

(12)

The pointwise estimated error is calculated by substituting the obtained expanded stress aj(x) from Eq. (2) into the Eq. (1). F ; = f (P'jtdr-^ f (p'jbdQ

(13)

Consider the trial function space U and the test function space V.

F ; = j (j)]~tdT+ f (p'jbdQ

(14)

U = {uG

3. Dof splitting p-adaptive hp-clouds method

H\Q),U

V = {veH^(Q),v

= U on Fu}

(3)

Now, we can write the equation Eq. (8) as the two coupled equations.

= 0 onT,}

(4)

^oo^o

A p-refinement procedure in hp-clouds method adds higher order shape functions to the selected nodes determined by error estimator. We can write the p-refined solution uP{x) as the sum of the solution of unrefined problem u'^ix) and the correction of the solution u''(x) obtained by the adaptive analysis. uP(x) = 8vP(x) =

u%x)-\-u'(x)

(5)

8v'(x)-\-8v'(x)

(6)

In Eq. (5), essential boundary conditions are already satisfied by the unrefined solution u'^ix). Thus the unrefined solution u'^ix) is in the space U, but the correction of the solution u''(x) is in the space V. The function space V is easily obtained by adopting the kinematically admissible meshless shape functions [6]. The variational formulation for the p-refined problem using the expressions of Eqs. (5) and (6) is 8n=

fv,(v''

+

- f 8{v' -f v'f

v'f:((j'-\-a')dQ :tdr-

f 8(v' + v'f

K''u'

^fO_

F' -

f.or^r

(15)

K''u'

(16)

Higher order shape functions in hp-clouds method are generated by the multiplication of partition of unity functions and proper basis functions. That is, the shape functions used in unrefined analysis are not affected by the generation of higher order shape functions in p-refinement procedure. Thus, the matrix K'''' and the vector F"" in Eq. (15) is the same as those in unrefined analysis. Furthermore, the errors in analysis are localized in some critical region. In refinement process, error estimator detects the critical regions, and the solution on the critical regions is mainly improved. The changes of solution on the critical regions have an important role for improving the solution. DSPMM improves the solution only on the critical regions. The unrefined solution u^ is fixed in the refinement process. Therefore Eq. (15) is not needed because the solution u^ is already determined, only Eq. (16) is solved with the unrefined solution u"". In most adaptive analysis added dof's are not many, thus we can improve the solution with minor computational cost increments.

: b dQ 4. Numerical examples

Vw^ G U, Vu' e V, Vu^ e V, Wv' G V.

(7)

The matrix form of the discretized p-refined equations is (8)

DB'^jdQ

(9)

In the numerical examples, error estimator and DSPMM have been verified for the problem of infinite plate with a hole under uniform lateral tension. The infinite plate is modeled as a finite quarter plate and analytical stress values are imposed as boundary conditions as denoted in Fig. 1. The dimensions of the plate are the length of square quarter plate L = 5, the radius of a hole d = \, uniform lateral tension G = 10, Young's modulus E = 1000, and Poisson's ratio V = 0.3.

274

M.S. Kang, S.-K. Youn/First MIT Conference on Computational Fluid and Solid Mechanics Fig. 2 compares the exact error and the estimated error for the stress component a^^ • Fig. 3 shows the computational costs and p-convergence of general hp-clouds method and DSPMM. The both p-adaptive analysis incorporate the presented error estimator to detect the analysis error, and modified weight functions [6] are used to impose essential boundary conditions. The first points that have the same number of dofs in Fig. 3(a) denote the same result since two methods are the same since that is the results of unrefined problem. The equations of DSPMM for the second, third and fourth points are solved only for the newly appended dofs and the unrefined solution is modified with these solutions. As shown in the p-refined results, the solution time is minor but the accuracy is comparable with hp-clouds method. In general, the accuracy of usual hp-clouds method is slightly better than that of DSPMM since DSPMM restrict the unrefined solution to be fixed. However, DSPMM requires lesser increase of computational cost.

14- 4^ 4* 4* 4* 4^fT 14-

Fig. 1. Modeling of infinite plate with a hole

li^^B \

(b)

(a)

Fig. 2. Contour plot of error in Oxx stress, (a) Exact error, (b) estimated error.

1000-

f [=

--n-- hp-Clouds

».,

3.5x10^

-DSPMM

'""-q

3x10^100

o *— o •—

Solving ByisihpOouds Solving Bcins:DSPMM Fomning Eqns:hp-Clouds Fomning Ecjns:DSPMM

25x10*

\ •—— (b)

1000

1500

1

2000 No. Of dofs

2500

3000

3600

1000

^

1

1500

1

1

2000

'

T

2500

"~~—'• '

—1

3000

'

1

3500

No. Of dofs

Fig. 3. Numerical results of general hp-clouds method and dof splitting p-adapdve meshless method, (a) Computation time, (b) p-convergence.

M.S. Rang, S.-K. Youn/First MIT Conference on Computational Fluid and Solid Mechanics

275

5. Conclusions

References

The error estimator for hp-clouds method and dof splitting meshless method (DSPMM) are presented. The numerical example shows that the error estimator is a good measure for the approximation of errors in hp-clouds method. This error criterion is used for p-adaptive hp-clouds method and DSPMM. DSPMM incorporates the results of unrefined solution in the formulation of p-refined analysis. The resulting equations are solved only for newly added nodal dof's. Thus, the solution can be improved with minor increase of computational costs.

[1] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. Int J Numer Methods Eng 1994;37:229-256. [2] Duarte CA, Oden JT. Hp clouds — a meshless method to solve boundary-value problems. Technical Report 95-05, TICAM. University of Texas at Austin, 1995. [3] Liu WK, Hao W, Chen Y, Jun S, Gosz J. Multiresolution reproducing kernel particle methods. Comput Mech 1997;20:295-309. [4] Duarte CA, Oden JT. An hp adaptive method using clouds. Comp Meth Appl Mech Eng 1996;139:237-262. [5] Chung HJ, Belytschko T. An error estimate in the EFG method. Comput Mech 1988;21:91-100. [6] Kang MS, Youn SK. Kinematically admissible meshless approximation using modified weight function. Int J Numer Methods Eng, in press.

276

Modelling of friction in metal-forming processes Stefan Kapinski * Institute of Machine Design Fundamentals, Warsaw Technical University, Narbutta 84, 02524 Warsaw, Poland

Abstract The predetermination of friction forces in metal-forming processes is essential. Investigations have been carried with this aspect in mind. This has resulted in the elaboration of a new graphics model of friction and modernization of instruments for the forming of materials. The friction model estimates the physical phenomena for contact conditions, such as: variable states of contact and friction; the difference in the quality of friction surfaces; and the velocity of friction. The results will help optimize such metal-forming processes as: the deep drawing process and the extrusion process [ 1 - 6 ] . Keywords: Friction; Graphic model; Metal forming; Instrument; Surface; Velocity

1. Introduction The predetermination of the rising frictional forces as well as of the forces required for metal forming is of utmost importance for the pre-judgement of failures. Only then is a specific optimisation of the metal-forming process possible, for instance by tests of the frictional forces in the region of contact of the forming material with the surface of the tool. In this way, the final frictional forces, which are of very great significance for tool abrasion, can be determined, whereby the influence of different lubricants, forming material and tool metals on frictional behaviour can be taken into consideration. The predetermination of the frictional forces by a mathematical and graphical model is very useful for metal-forming processes.

2. Investigation of friction in metal-forming processes Friction depends on the quality of the surfaces, the types of materials, the direction of movement, and the velocity and pressure of materials. Fig. 1 is an exemplary scheme of real contact materials and mechanics of friction. A variety of macroscopic phenomena are associated with frictional contacts between microscopically rough surfaces. Properties such as the apparent dependence of the frictional forces on relative velocities, quality of surfaces, adhesive *Tel.: +48 (22) 660-8682; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

forces and stick-slip oscillations are consequences of the combined influence of the large-scale mechanical makeup of the system and the microscopic interaction at the surface interface. The surface structure (Fig. 1) is created by technological processes and, for this reason, friction depends on direction of movement. Of fundamental importance to the properties of frictionally interacting surfaces is the tendency of microscopic localized slip to occur on macroscopic global relative sliding between the surfaces. More specifically, if a tangential shear force is applied to two bodies compressed against each other, there will be small relative displacements of elastic character, i.e. such that no residual displacement remains after unloading. However, experimental observations indicate that some plastic deformation occurs with a resulting residual displacement and consequent energy dissipation. If balance is not attained between the vertical forces, the surface separation will evolve with time. In particular, for sufficiently large separations and zero initial velocity, the pressure force will be outweighed by the remaining forces and thus the separation will tend to decrease. Similarly, when the separation is initially small, the normal force will act to separate the surfaces leading to an increase in friction. The geometrical interpretation of friction stresses is presented in Fig. 2. The parameters c, A. are connected with geometrical parameters of contact surface and value k is the material parameter.

S. Kapinski /First MIT Conference on Computational Fluid and Solid Mechanics

111

b)

a) Pressure

yy^-<>v Fig. 1. Exemplary surface structure (a) and dependence of friction forces on the direction of movement (b).

"5

T

°

Fig. 2. Graphical interpretation of the friction stresses. The tangential stress may be obtained from the formula (Fig. 2) (1) The value r^ may be expressed as r^ = tanAicr^

(2)

and O^ — 0rC0sA2

(3)

Substituting in Eq. (1) the values which are expressed by formulas (2) and (3), the tangential stress may be written as follows t = c + a sin A-2

(4)

The parameters c, Xi are connected with geometrical parameters of contact surface and value k is material parameter. The values of friction forces T for different situation of movement direction (p, quality of surfaces (parameters of surface and materials m) and pressures N may be described as functions (Fig. 3): • for a situation when movement of friction surfaces v does not exist (surface A): T^ = F[Ai(m, (^), A/']; • for a situation when displacement of friction surfaces is really (surface B): % = F[X2(m, cp, v), N]\

Fig. 3. Graphical interpretation of the surface friction. • for a situation when microstructures of local friction surfaces are cutting (surface C): Tc = FlX^im, v), N]. For a situation when the movement of friction surfaces V does not exist, the tangential stress is equal to the force of reaction with reference to the contact area. For any part of the geometrical model, the shape of the model surface of friction is dependent on the quality of the material, structure of the surface friction, direction of friction (Fig. 4) and the velocity of friction. Friction creates temperature and this fact must be taken into consideration when modifying the model. The influence of temperature is presented in Fig. 5. From this model, it can be seen that temperature changes the force of friction. A high temperature decreases the force of friction and friction depends on velocity (vi < V2 < vs). This situation has been experimentally demonstrated.

278

S. Kapinski / First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 4. Dependence of the force friction reference on the direction of friction {(p) (cross-section of surface B in Fig. 3).

for predicting frictional behaviour in the deep drawing process. In this way, the frictional forces at the die radius and in the flange alongside the forces for bending and loss-free forming can be determined. Even at the construction stage of a deep drawing part, assessments can be made about the behaviour of different lubricants, sheet materials and tool metals. The deep-drawing process of cylindrical elements is an optimal model for the prediction of the distribution of stress and strain or frictional behaviour and frictional forces for stamping drawpieces. The frictional forces at the die radius, type of lubricate used and the forces for bending should also be exactly determined. Therefore, a new concept model of friction is presented. The model of friction may also be useful for the predetermination of friction forces for other metal-forming processes.

References

Fig. 5. The space model of friction demonstrating influence of temperature.

3. Conclusions The predetermination of parameters and realisation of sheet forming is an essential aspect for the optimisation of the deep drawing process for the shaping of automobile chassis. Investigations have been carried out to demonstrate this. This has resulted in the proposition of a friction model

[1] Doege E, Schulte S. Design of deep drawn components with elementary calculation methods. J Mater Process Technol 1992;34:439-447. [2] Kapinski S. Influence of the punch velocity on deformation of the material in deep-drawn flange. J Mater Process Technol 1992;34:419-424. [3] Kapinski S. The forming of autobody panels. WKi£, Warszawa 1996 (in Polish). [4] Kapinski S. The analysis of forming process for bimetal materials. Third International Conference on Contact Mechanics — Contact Mechanics III. Madrid 30 June-3 July, 1997, pp. 217-226. [5] Kapinski S. Analysis and modelling of friction in deep drawing process. 4th International Conference on Advances in Materials processing and Technologies. Kuala Lumpur, Malaysia, 24-28 August, 1998, pp. 569-576. [6] Kapinski, S. Model of friction for sheet metal forming processes. 4th International Conference on Contact Mechanics — Contact Mechanics IV. Stuttgart, Germany, 3-5 August, 1999, 15-24.

279

Modelling of intra- and interlaminar fracture in composite laminates loaded in tension Maria Kashtalyan*, Costas Soutis Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK

Abstract Fracture process of multidirectional fibre-reinforced composite laminates under tensile loading involves sequential accumulation of intra- and interlaminar damage in the form of matrix cracks and delaminations. In this paper, local delaminations growing from matrix crack tips in angle-ply laminates are analysed using fracture mechanics concepts. Closed form expression representing strain energy release rate associated with crack tip delaminations as a linear function of the first partial derivatives of the effective elastic properties of the damaged layer with respect to the delamination area is derived. Parameters controlling the value of the strain energy release rate are established. Keywords: Crack tip delamination; Local delamination; Matrix crack; Strain energy release rate; Angle-ply laminate

1. Introduction Fracture process in multidirectional fibre-reinforced composite laminates subjected to in-plane static or fatigue tensile loading involves sequential accumulation of intra- and interlaminar damage in the form of matrix cracks that appear parallel to the fibres in the off-axis plies and matrix cracking induced edge and/or local delaminations. Formation and growth of crack induced local delaminations in angle-ply [O2/O2/ — ^2]^ carbon/epoxy laminates under quasi-static and fatigue tensile loading has been reported by O'Brien and Hooper [1] and O'Brien [2]. The present paper is concerned with analysis of local delaminations in angle-ply symmetric [0i/02]^ laminates using the approach suggested by Zhang et al. [3], earlier applied to modelling transverse crack tip delaminations in [±^^/90„]^ laminates.

2. Fracture analysis Fig. 1 shows a schematic of an angle-ply symmetric [0i/02L laminate subjected to in-plane tensile loading and damaged by matrix cracking in the (02) layer and delaminations growing from the tips of matrix cracks at the (01/02) interface. The laminate is referred to the global * Corresponding author. Tel.: +44 (20) 7594-5117; Fax: +44 (20) 7584-8120; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

xyz and local x[^^X2^-'x^^^ co-ordinate systems, with the jc{^^ axis directed along the fibres in the (^2) layer. Matrix cracks are assumed to span the whole width of the laminate and be spaced uniformly at a distance 2s^. Local delaminations are assumed to be strip-shaped, with a strip width 21^. Since delamination growth occurs in a self-similar manner, it can be analysed using fracture mechanics concepts. The total strain energy release rate G^"^ associated with local delaminations is equal to the first partial derivative of the total strain energy U stored in the damaged laminate with respect to the total delamination area A^"^ provided the applied strains {s} are fixed and the matrix crack density C = (25^)~^ remains unchanged G'^ = -

dU

aA^^

(1) [e},C

The strain energy release rate can be effectively calculated if instead of the damaged laminate one considers the 'equivalent' laminate, in which the damaged layer is replaced with an 'equivalent' homogeneous one with degraded stiffness properties, Zhang et al. [3]. The residual stiffness matrix [Q] of the 'equivalent' layer is a function of the relative delamination area Z)^ = ^^l/Sf^ and the relative crack density Dll"^ = h2/s^. The total strain energy stored in the laminate element with a finite gauge length L and width w is U = ^wL{s}^[A]{-s}

(2)

280

M. Kashtalyan, C. Soutis/First MIT Conference on Computational Fluid and Solid Mechanics

(p delamir^ations

matrix cracks

(^,)

(p

I

- i2)

2/i,

-

^

Fig. 1. Front and edge views of a [(t)\/(t>2]s laminate subjected to in-plane tensile loading and damaged by matrix cracks in the inner (02) ply and local delaminations growing from crack tips at the (0i/(/>2) interface. where [A] = X^,[2]/^/ is the residual extension stiffness matrix of the 'equivalent' laminate. Noting that the area of a single crack tip delamination is a^^ = 2i^w/\ sin021, Fig. 1, the total delamination area is equal to A ' ^ = la'^CL = 2LWD'^/\ sin021. Then the strain energy release rate, calculated from Eqs. (1) and (2), is

Calculation of the residual in-plane axial stiffness Q^x using Eq. (5) and transformation formulae [4] yields the strain energy release rate associated with local delamination in terms of IDEFs and Qjf ^ as l(M)2

G"(£.

2

(3)

+ 2e;^^ sin' 02 cos' 02 + Gl^^ sin^ 02

(4)

I sin 021

The residual in-plane stiffness matrix [Q] of the 'equivalent' layer in the global co-ordinates is related to the residual in-plane stiffness matrix [Q^^^] in the local co-ordinates by the well-known transformation formulae, Jones [4]. The residual in-plane stiffness matrix [Q^^^] of the 'equivalent' layer in the local co-ordinates is related to the in-plane stiffness matrix [Q^^^] of the undamaged material via the introduced in Zhang et al. [5] In-situ Damage Effective Functions (IDEFs) Af/ = A^J'/iD';;', Djf), J = 2, 6 as ^Qi,)] = [g^^)]

\ 3 ^

7 9^;f

Under uniaxial strain, Eq. (3) simplifies to G^'is.

5^22

Qtl^'^

Q'^2^'^2

0

0

0

Q^JA

(5)

+ 4066'sin'02 cos'02

dA in) dD\^

I sin 021

(6)

3. Stiffness analysis Substituting the residual stiffness matrix [2^^^], Eq. (5), into the constitutive equations for the 'equivalent' layer in) A'^^' /i(/^) :in terms of {a^'^} = [Q^^^W^} gives the IDEFs A'^^, the lamina macrostresses {a^^^} and macrostrains {£^^^} as vl^^^ -— 11 yi22

At' = 1 -

VM)

^ + a22 ^22

-r(2)

G^^^n?

(7)

M. Kashtalyan, C. Soutis /First MIT Conference on Computational Fluid and Solid Mechanics

281

are (2)

deiamination

M)

^ii

^M

Fig. 2. A quarter of the representative segment of the damaged laminate. To determine the IDEFs as functions of the damage parameters Z)^^ Z)Jf, micromechanical analysis of the damaged laminate has to be performed. Since cracks and delaminations are spaced uniformly, a representative segment of the laminate, containing one matrix crack and two crack tip delaminations, may be considered. The representative segment can be segregated into the laminated and delaminated portions. Due to the symmetry, the analysis can be confined to its quarter. Fig. 2. Let alj denote the in-plane microstresses in the damaged layer (i.e. stresses averaged across the layer thickness). In the delaminated portion, we have 0-22^ = G[^ = 0. In the laminated portion, the in-plane microstresses may be determined by means of a 2-D shear lag analysis. The equilibrium equations in terms of microstresses take the form dx (n)

(10)

-

E ,,£r,^hhlzlSl^c,>-^^>,,. J^m Xkh2 Uj

(11)

The macrostrains in the 'equivalent' layer [s^^^} are calculated from the constitutive equations for both layers and equations of the global equilibrium of the laminate, assuming {^(^>} = {8^'^}

(12)

Thus, the lamina macrostresses, Eq. (11), and macrostrains, Eq. (12), are determined as explicit functions of the damage parameters D^^, Djf. Consequently, first partial derivatives of IDEFs, Eq. (7), involved into the expressions for the strain energy release rate, Eq. (6), can be calculated analytically.

4. Results and discussion

-1^=0,

J = h2

(8)

By averaging the out-of-plane constitutive equations, the interface shear stresses Xj in Eqs. (8) are expressed in terms of the in-plane displacements u\j^ and u\f, averaged across the thickness of, respectively, (0i) and (^2) layers, so that Tj =

10

Here dx is the applied stress, X^ are the roots of the characteristic equation, and Akj and Cj are constants depending on the in-plane stiffness properties of the intact material [Q^^^], shear lag parameters Ku, K22 and K12 and angles 01 and 02The lamina macrostresses {a^^^}, involved in Eq. (7), are obtained by averaging the microstresses, Eqs. (10), across the length of the representative segment as explicit functions of the relative crack density D^^ and relative deiamination area D^f aj2

^^

•

k=l

(01)

matrix crack

/ v ^ A cosh A^ (4^^ -'^z^) , r^ I coshXkiSf, -if,)

Kjiiu (2)

• u^^^) + Kj2(uf^

..f)

(9)

The shear lag parameters Ku, K22 and Ki2(= K21) are determined on the assumption that the out-of-plane shear stresses in the damaged layer and outer sublaminate vary linearly with x^^^. Substitution of Eqs. (9) into Eqs. (8) and subsequent differentiation with respect to ^2^^ lead to the equilibrium equations in terms of microstresses and microstrains (i.e. strains averaged across the layer thickness). To exclude the latter, constitutive equations for both layers, equations of the global equilibrium of the laminate as well as generalised plane strain conditions are employed. Finally, a system of coupled second-order non-homogeneous ordinary differential equations is obtained, solutions to which

As an example, predictions of strain energy release rate G^^ associated with matrix crack induced local delaminations in angle-ply [02/^21^ laminates are presented. Properties of the AS4/3506-1 graphite/epoxy material system used in calculations are as follows [1]: £"11 = 135 GPa, £22 = 11 GPa, Gn = 5.8 GPa, vn = 0.301, single ply thickness t = 0.124 mm. Fig. 3 shows the normalised strain energy release rate G^'^/sl^ for [O2/252], laminate, calculated from Eq. (7) and plotted as a function of relative deiamination area D^"^ for two matrix crack densities C = I crack/cm and C = 2 cracks/cm. These crack densities are equivalent to crack spacing of approximately 80 and 40 ply thicknesses (s = 40r, s = 20t). It may be seen that the present approach gives the strain energy release rate for local deiamination that depends both on the crack density and deiamination length. The value of G'^^/e^^ decreases with increasing deiamination length and increasing crack density. It is worth noting that closed-form expression, suggested by O'Brien [2], gives strain energy release rate for

282

M. Kashtalyan, C. Soutis/First MIT Conference on Computational Fluid and Solid Mechanics

1 crack/cm 2 cracks/cm

0.32 2

4

6

8

10

Relative delamination area, %

Fig. 3. Normalised strain energy release rate G'"^ /s^^ associated with local delamination in a cracked [02/252]5 AS4/3506-1 laminate as a function of relative delamination area D'^ . local delaminations in [O2/O2/ - ^2]. laminates that does not account for matrix cracking and is independent from delamination length. Dependence of the normalised strain energy release rate G^"^ l^lx ^^ the ply orientation angle 0 is shown in Fig. 4 for two relative delamination areas: D^"^ = 0 (delamination onset) and D'"^ == 0.1. Matrix crack density in both cases is equal to 1 crack/cm. For the given lay-up [02/^2]^, normalised strain energy release rate reaches the maximum value at approximately 0 = 68°.

Acknowledgements Financial support of this work by the Engineering and Physical Sciences Research Council (EPSRC/GR/L51348) and Ministry of Defence, UK, is gratefully acknowledged.

15

30

45

60

75

Ply orientation angle, degrees Id I Fig. 4. Normalised strain energy release rate G'^/eJ^ associated with local delamination in a cracked [02/^2]5 AS4/3506-1 laminate as a function of ply orientation angle 0\ matrix crack density C = 1 crack/cm.

References [1] O'Brien TK, Hooper, SJ. Local delamination in laminates with angle ply matrix cracks: Part I Tension tests and stress analysis. NASA Technical Memorandum 104055, 1991. [2] O'Brien TK. Local delamination in laminates with angle ply matrix cracks: Part II Delamination fracture analysis and fatigue characterisation. NASA Technical Memorandum 104076, 1991. [3] Zhang J, Soutis C, Fan J. Strain energy release rate associated with local delamination in cracked composite laminates. Composites 1994;25(9):851-862. [4] Jones RM. Mechanics of Composite Materials: 2nd ed. Philadelphia, PA: Taylor and Francis, 1999. [5] Zhang J, Fan J, Soutis C. Analysis of multiple matrix cracking in [ib^;;,/90„]5 composite laminates Part 1: In-plane stiffness properties. Composites 1992;23(5):291-298.

283

Implicit integration for the solution of metal forming processes Marek Kawka^'*, Klaus-Jiirgen Bathe ^ ""ADINA R&D, Inc., 71 Elton Avenue, Watertown, MA 02472, USA ^ Massachusetts Institute of Technology, Mechanical Engineering Department, Cambridge, MA 02139, USA

Abstract The simulation of metal forming processes is performed using implicit integration analysis procedures. The approach is based on reliable and efficient solution procedures, uses the actual physical simulation parameters (that is, no adjustment of the tool velocity or work piece density is employed) and enables to achieve accurate results of the loading and spring-back processes in a single solution run. In the analyses performed, the solution times were not far from (and frequently less than) those required in explicit time integration analyses. Keywords: Metal forming; Implicit integration; Static and dynamic analysis; Spring-back

1. Introduction The finite element analysis of forming processes continues to represent significant challenges [1]. The problems are highly nonlinear, because, in general, large strains, contact and highly nonlinear material conditions are encountered. To simulate sheet metal forming processes, in addition, the metal piece to be formed is thin, which introduces also the difficulties encountered in the analysis of shells [2,3]. For the analysis of metal forming processes, effective finite element procedures are needed, and as more efficient procedures become available, increasingly more complex problems can be realistically simulated. At present, metal forming analyses are usually conducted using explicit analysis procedures. With an expHcit code, the solution is performed using an incremental dynamic analysis approach without forming a stiffness matrix and without iterating for equilibrium at the time step solutions. Hence, the solution effort per time step is relatively small. However, for the solution to be stable, the time step size has to be smaller than a critical time step, which requires many solution steps for the complete simulation. To obtain efficiency, usually finite elements are used that in a 'fast' dynamic analysis (such as a crash simulation) are tuned to obtain a good response prediction, but these ele* Corresponding author. Tel: -\-l (617) 926-5199; Fax: +1 (617) 0238; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ments are unstable in a 'slow' dynamic or static analysis. Hence, for overall stability of the solution, the time step has to be sufficiently small and the inertia forces need to be sufficiently large. For an analysis demonstrating these requirements (see [4]). It has long been recognized that an implicit dynamic solution based on equilibrium iterations in each solution step and reliable 'non-tuned' solution procedures would be preferable for many forming analyses provided the solution is computationally effective. The physical process is then more accurately modeled, in particular also the spring-back process in sheet metal forming problems. The objective of this paper is to present the effective implicit solution procedures available in ADINA to solve metal forming processes. We briefly summarize the solution approach and procedures used, and present some solution results.

2. Implicit integration solution The basic equations solved in an implicit integration are well-known, see for example [2],

and t+At^(i) ^ t+At^{i-l) _^ ^ u ( 0

(2)

284

M. Kawka, K.J. Bathe/First MIT Conference on Computational Fluid and Solid Mechanics • Effective finite elements; we use the u/p elements for fully 2D and 3D solid element models and the MITC4 shell element for shell models [2,5]. These elements have a strong physical and mathematical basis. • An efficient large strain inelastic analysis algorithm; we use the effective-stress function procedure [2,5]. • A robust and efficient contact solution technique; we use the constraint-function method [2,5] • An efficient equation solver; we use a sparse solver developed specifically for the program ADINA; the solver includes parallel-processing capabilities [5].

where M is the mass matrix, C is the damping matrix, K is the tangent stiffness matrix, R is the load vector, F is the nodal force vector corresponding to the internal element stresses, U is the displacement vector, the superscript r +Ar denotes the time at which the equations are formulated, and the superscripts (/) and (/ — 1) denote the current and previous iterations. An unconditionally stable implicit time integration scheme, for example, the trapezoidal rule, is used to discretize Eq. (1) in time. The equations given above do not explicitly show the contact conditions, but these can be imposed as described in [2]. We note that with Eqs. (1) and (2) iterations are performed until the equilibrium is satisfied at each time step (to a reasonable convergence tolerance). Of course, if a static analysis is pursued, simply the inertia and damping effects are not included in the solution. An effective implicit integration solution provides several advantages over explicit integration. Most importantly, there is no need to manipulate the metal forming technological parameters (such as the tool velocity or material density) in order to achieve the solution. Therefore, the calculated results are much more reliable than obtained in explicit integration. This situation is easily observed in the analyses of processes in which the spring-back must also be simulated: the implicit integration solution provides good results in a single run simulating the loading and spring-back conditions. An effective solution of Eq. (1), including contact conditions, must be based on reliable and efficient solution procedures. We list here briefly the techniques used in ADINA.

The individual advantages of the above-mentioned procedures were discussed in earlier publications, see references, but of course, for an overall effective analysis, these procedures need to work efficiently together and this has been achieved in the ADINA program. 3. Sample solutions The objective in this section is to present the results of some sample analyses. We consider cases that indicate some important features of the analysis capabilities available. All results were obtained using the implicit solution approach described above. 3.1. 2-D draw bending problem A very simple 2-D draw bending benchmark problem from the Numisheet '93 Conference (see Fig. 1) tests the initial blank

after stamping

after spring-back

110 105 ^

100

®

95

®

90

^

85

02

80

Fig. 1. Numisheet '93 draw bending test for high tensile steel and high blank holding force [6]. (a) Shape of metal sheet at subsequent stages of deformations, (b) Measurement of spring back angles 0i and 02. (c) Comparison of experimental data (circles) and simulation results (dashed lines).

M. Kawka, K.J. Bathe/First MIT Conference on Computational Fluid and Solid Mechanics ability of the software to predict accurately the springback after stamping [6]. Surprisingly, at the time of the Numisheet '93 Conference (of course, about eight years ago) most of the commercial software could not be used to produce reliable results. In our simulation, 105 nine-noded u/p elements were used (with only one layer through the thickness), and results very close to the experimental data were obtained (see Fig. 1; for brevity, the results for the high tensile steel are presented only).

Panel

Material

Elements

outer

mild steel, 0.8 mm

90 u/p

inner

high strength steel, 0.8 mm

54 u/p

285

i j . I j,j,„i:i "ij„ I, i'''i:jj...iJi..jjj..:i'''i:..i:..iJ.:i:''r:i:::i:ri::n'"i::i'''iTTT'i,i 11 ri'i 11 \i

5.2. Hemming problem The solution of this problem tests the stability and efficiency of contact algorithms. The simulation software has to deal with two types of contact conditions: the 'deformable body to deformable body' condition and the 'deformable body to rigid surface' condition. In the hemming problem (see Fig. 2), large strain conditions need also be modeled, and therefore the problem is an excellent test for finite element software. In our simulation, 90 nine-noded u/p elements were used for the outer panel and 54 elements were employed for the inner panel. A total of 1800 incremental solution steps were used in the simulation. Despite the large deformations in the bent section of the outer panel (up to 100% strains were measured) and the continuously changing contact conditions between the inner and outer panels, excellent convergence with an average of only four iterations per step in the incremental solution was observed.

I I I I I I I I I I .1

3.3. Deep drawing of an oil pan This industrial problem of a deep drawing of an oil pan [7] requires a powerful simulation code and versatile shell elements able to deal with the complex deformation path. In our simulation 16,922 MITC4 shell elements were used to represent the metal sheet and 16,500 rigid elements were employed to define the tool surfaces. The simulation was performed on a UNIX workstation using parallel-processing, a HP-J5000 workstation was employed. The results of the simulation compare very well with experimental measurements (see Fig. 3).

4. Conclusions The objective of this paper was to briefly present some solution capabilities for the simulation of metal forming processes. The implicit dynamic (including static) analysis capabilities developed in ADINA for metal forming processes and specifically sheet metal forming processes were summarized and some solution results given. The procedures are computationally effective when compared to explicit techniques now in wide use and allow the more realistic modeling of many metal forming processes.

Fig. 2. Plane strain deformation of outer and inner panels during successive stages of hemming process, (a) Pre-hemming, outer panel is bent 90°. (b-e) Hemming, outer and inner panels are attached.

References [1] Numisheet '99. Proceedings of the 4th International Conference and Workshop, Besan9on, France, September 13-17, 1999. [2] Bathe KJ. Finite Element Procedures. Prentice Hall, Englewood Cliff, NJ, 1996. [3] Chapelle D, Bathe KJ. Fundamental considerations for the finite element analysis of shell structures. Comput Struct 1998;66(l):19-36. [4] Bathe KJ, Guillermin O, Walczak J, Chen HY. Advances in nonlinear finite element analysis of automobiles. Comput Struct 1997;64(5/6):881-891. [5] ADINA R&D. Theory and Modeling Guide, Report No. ARD-00-07, Watertown, MA, 2000.

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M. Kawka, K.J. Bathe/First MIT Conference on Computational Fluid and Solid Mechanics

Assessment line A-B Fig. 3. Deep drawing of an oil pan [7]. (a) Tool geometry: punch, blank holder and die. (b) Shape after deformation (simulation results) and initial flat blank, (c) Definition of assessment line A-B, and comparison of experimental data and simulation results for various commercial finite element codes. [6] Numisheet '93. Proceedings of the 2nd International Conference and Workshop, Isehara, Japan, August 31-September 2, 1993.

[7] Metal Forming Process Simulation in Industry. Proceedings of the International Conference and Workshop, BadenBaden, Germany, September 28-30, 1994.

287

Computation of stress time history using FEM and flexible multibody dynamics H.S. Kim^ HJ. Yim'''*, C.B. Kim'^ ^ Commercial Vehicle Test and Research Team, Hyundai Motor Company, Kyoung-gi, Korea ^ Graduate School of Automotive Engineering, Kookmin University, Seoul, Korea ^Division of Mechanical, Aerospace and Automotive Engineering, Inha University, Incheon, Korea

Abstract Dynamic stress time history calculation deals with spatial, constrained mechanical systems that undergo nonsteady gross motion and small elastic deformation. A hybrid method that employs stress superposition as a function of constraint loads and component accelerations that are predicted by flexible body dynamic simulations is presented. A numerical example is given for stress time history evaluation of the vehicle structural component. Keywords: Dynamic stress; Flexible multibody dynamics; Component mode; Mode superposition; Fatigue life prediction

1. Introduction Recently, computational methods for dynamic stress time history have been developed to speed design cycle [1,2]. The two conventional methods — 'Modal Stress Superposition Method'; and the 'Flexible Multibody Dynamic Simulation and Quasi-static Method' are combined to form a hybrid method that improves the accuracy of dynamic stress prediction. The hybrid method is defined as a computational dynamic stress analysis method that employs stress superposition as a function of constraint forces and component accelerations that are predicted in terms of the assumed deformation modes from flexible multi-body dynamic analysis. Deformation modes such as Ritz modes, Craig-Chang modes, or Craig-Bampton modes may be used. In the case of using the finite element model of a large-scaled structural component, it is apparent that the large model is often inefficient for flexible body dynamic analysis and stress time history calculation. This motivates an efficient method, utilizing flexible dynamic analysis connected with the super-element method. The reduced model may have modal and static characteristics correlated with that of the original model. By use of the super-element method connected with the component mode synthesis

technique, the deformation modes for the reduced model of the structural component are used to implement the hybrid method.

2. Dynamic stress analysis In order to improve the efficiency and accuracy of conventional methods for stress recovery, the hybrid superposition method [2] is developed with the use of the super-element method. The method efficiently recovers the dynamic stress time histories by applying the principle of linear superposition of the mode acceleration method or static correction method. In this paper, the hybrid superposition method obtained from the mode acceleration method is briefly described. Dynamic stress (7(0 with the assumption of infinitesimal elastic deformation can be written

y ^ (sikPikiO + ^ckPckiO) - Y] ( — ^ I

^nqnit)

k=l

-E4 Nk

* Corresponding author. Tel.: -h82 (2) 910-4688; Fax: -^82 (2) 910-4839; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

SKmqKmit)

(1)

288

H.S. Kim et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Finite Element Model of si flexible body Reduced Model

Detail Model

Finite Element Eigenvalue Analysis

Finite Element Static Analys is

Vibration normal modes

Static correction modes

Modal stress coefficients

Static stress coefficients

Component Mode Synthesis Flexible Multibody Dynamic Analysis Time histories of Modal coordinates. Dynamic loads and Gross body motion

Superposition

I

Dynamic Stress Time History

Faligue Life Fig. 1. Conceptual procedure and data flow. where Ns and Nk are the number of components of surface loads and vibration modes, respectively, psi(t), PGjiO, PikiO and pckiO are /th component of surface force, 7 th component of gravity force, kih component of D'Alembert inertia and Coriolis force, respectively. qKiiO and qKmit) are the velocity of /th modal coordinates and acceleration of mth modal coordinates, respectively. S5, (/ = 1 , . . . ,Nh), SGJ U = 1 , . . . , 3 ) , Su- and Sck (k = 1 , . . . , 6) are static stress coefficients that are the contribution to the stress vector cr due to a unity of psi{t). PGjiO, PI kit) and pckiO, respectively. On the other hand. SKI and SKm Q and m = 1 , . . . , Nk) are the modal stress coefficients due to a unit displacement of modal coordinates qKi(t) or qKmit), and ^KI and COKI (/ or m = l , . . . , Nk) are the modal damping factor and the natural frequency, respectively. Dynamic loads and modal coordinates are time-dependent terms, which are obtained from flexible multi-body dynamic simulation. Static stress coefficients and modal stress coefficients are time-independent terms, which are obtained from FE static stress analysis. Fig. 1 shows the conceptual data flow for the proposed method.

3. Numerical example and conclusions The durability of a prototype vehicle has been traditionally estimated in accelerated test environments, such as the Belgian mode. In this paper, stress time history is obtained for the prototype vehicle, which is shown in Fig. 2. Fig. 3 shows the displacement time histories that were measured from the durability test of the Belgian road. Using the displacement information, flexible multi-body dynamic

Fig. 2. FE model for the example.

H.S. Kim et al. /First MIT Conference on Computational Fluid and Solid Mechanics O.Q320Z

Magnitude(m)

CH1_RS_DIS_FIL DAC

Magnltude(m)

CH2_RS_DIS_FIL.DAC

Magnitude(rn)

CH3 RS DIS FIL DAC

Magnitude(m)

CH4

289

-0.03S00

0.03©3-«-

-0.03©32

0.03«»©

0 . 0 3 0 0 0

-0.0321

RS_DIS_FIL.DAe

e

Fig. 3. Displacement time histories. References [1] Ryu JH, Kim HS, Yim HJ. An efficient and accurate dynamic stress computation by flexible multibody dynamic system simulation and reanalysis. KSME Int J 1997;ll(4):386-396. [2] Kim HS. Dynamic Stress Analysis of a Flexible Body in Multibody System for Fatigue Life Prediction, Ph.D. Thesis, The University of Inha, Korea, 1999.

40.0 60.0 Time (sec) Fig. 4. Dynamic stress time history at the roof measured at the axle spindle for the Belgian mode. simulation for the prototype vehicle is implemented and stress time history is obtained as in Fig. 4. Accuracy of the stress time history has been validated in the durability evaluation for the vehicle with the test result.

290

Probabilistic models for predicting the failure time of deteriorating structural systems Jung S. Kong, Ferhat Akgul, Dan M. Frangopol*, Yunping Xi Department of Civil, Environmental, and Architectural Engineering University of Colorado, Boulder, CO 80309-0428, USA

Abstract This paper presents some of the work that has been performed on the probabiUstic models used to predict the time to failure of deteriorating structural systems considering both no maintenance and maintenance options. Probabilistic models are proposed to take into account the uncertainties involved in this prediction. The time to failure is defined as the time at which the reliability of the system down-crosses a prescribed target reliability level. Implementation of the proposed models in computer programs is discussed. A computer program being developed for probabilistic modeling of structural systems with pre- and post-processing capabilities is introduced. Keywords: Probability; Structures; Reliability; Maintenance; Deterioration

1. Introduction In modem management of deteriorating structural systems, lifetime system performance has to be considered [1-3]. In order to consider the lifetime performance of an individual structure or a group of similar structures, different methods may be used in analysis, design, and maintenance [3-6]. The use of reliability based methods in predicting the lifetime performance of deteriorating structures is generally recognized [7]. The type of maintenance interventions can have a significant effect on the lifetime performance of deteriorating structural systems. These interventions can be classified into several types, however, essential and preventive maintenances are the most fundamental ones [8,9]. This paper presents some of the work that has been performed on the probabilistic models used to predict the time to failure of deteriorating structural systems considering both no maintenance and maintenance options. Probabilistic models are proposed to take into account the uncertainties involved in this prediction. The time to failure is defined as the time at which the reliability of the system down-crosses a prescribed target reliability level. Implementation of the proposed models in computer programs is discussed. A computer program being devel* Corresponding author. Tel.: +1 (303) 492-7165; Fax: -Hi (303) 492-7317; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

oped for probabilistic modeling of structural systems with pre- and post-processing capabilities is introduced.

2. Probabilistic models The probabilistic models developed originated as a result of research related to bridge reliability-based maintenance analysis. However, these models may be appHed to failure time prediction of any deteriorating structural system or group of similar structures. The model for no maintenance option uses a bilinear function for the time-variant reliability index profile of the deteriorating system. The time-variant reliability index profile function P(t) of a system is modeled as follows:

{

Po

for 0 < r < ti

(1) Po-itti)a for t > ti where ^o, «' and ti are random variables representing the reliability index of the structural system or group of similar structural systems as constructed (i.e., initial rehability index), the annual rate of reliability loss, (A)S(Oioss/year), and the time [i.e., age (years)] at which deterioration is expected to start, respectively. Fig. 1 shows the program implementation and pre- and post-processing phases of the probabilistic model for the simulation of failure time of deteriorating systems without

J.S. Kong et al. /First MIT Conference

on Computational

Fluid and Solid

291

Mechanics

U.UCJU LU

MEAN = 51.5 YRS

hh-

A

QC U.

u.

•^ I M LI

I U-

».| First T i m e to Failure) Program Program

I

O

LU

0.020

K,. = 4.6

t^ CO = !

z LL < UJ

Q o >- i h-^ 0.010

Monte Carlo Simulation Routine

STD. DEV = 26.9 YRS

/ \

\

CQ

< o CD

cc

CL

"" '"^^---.

/'

50

Function Selection Routine

100

ISO

TIME (YRS)

Fig. 1. Computer implementation of the probabilistic model for the simulation of failure time of deteriorating systems without maintenance.

maintenance. The probability density functions (PDFs) of the three basic input variables are shown. Also shown is the first time to failure (also referred as the rehabilitation time (rate)), ^R, based on a given target value Aarget- Time to failure, ^R, is the time at which the reliability index starts falling below the target reliability index, yStargetThe second option includes preventive maintenances performed during the lifetime of the deteriorating structural system or group of similar structures. This option includes five additional basic random variables in addition to the three random variables described earlier, as follows: the time of first application of preventive maintenance fpi, time of reapplication of preventive maintenance fp, duration of preventive maintenance effect on reliability /pD, annual reliability loss during preventive maintenance effect 0, and improvement in reliability index (if any) immediately after the application of preventive maintenance y. The assumed

1^1^

PDFs of y^o, h. oi, fpi, fp, ^PD, 0, and y are shown in Fig. 2. This figure also shows the program implementation and pre- and post-processing phases of the probabilistic model for the simulation of failure time of deteriorating systems with maintenance. The maintenance action is referred as preventive maintenance due to the fact that it is applied during the period at which ^{t) > target- Similar to the earlier case, the PDF of the failure time, fpR, based on a given target value target, is displayed. This is the time at which reliability index starts falling below the target reliability index, y^target, after preventive maintenance is applied. Consider the event that, at a specified time t, the reliability index of the system ^{t) is less than the target reliability index y3target, i-e., P{t) < target- The probability of this event P{P(t) < Aarget) cau bc cvaluatcd using Monte Carlo simulation or it can be approximated using the assumption that the distribution of ^(t) is normal. If the normal distribution

MEAN = 82.4 YRS W

STD.DEV = 43.4 YRS

0.015

u_ O m

First Time to Failure Program

• W ^

J ^\^

0.010

Q O

J 9

Monte Carlo Simulation Routine

f

iia

i

.

P...rge, = 4.6

\

0.005

/ J , 0 20 40 60 80 100 120 140 160 180 200 TIME (YRS)

Function Selection Routine

Fig. 2. Computer implementation of the probabilistic model for the simulation of failure time of deteriorating systems with maintenance.

292

J.S. Kong et al. /First MIT Conference on Computational Fluid and Solid Mechanics

approximation is used, this probability is:

,(„„<^.,)=*(^=^;^)

where E[p{t)], a[y6(0], and 0 ( ) are the mean value of ^(0, standard deviation of jS(r), and the standard normal probability, respectively. Both, simulation and normal approximation are used in the computation of the first time to failure of deteriorating structural systems with or without maintenance.

|«it«&4Lognormal3 \

7.50 |

120|

1

1

I J JLognormagj |

15.0 |

5.00 1

j

, - : l i ^ (uniform ' ^ j 0 - 0 0 2 0 |

0.10 1

1

1 1 "M5ao I I ""I 0.10

, ,

• ^:.','•/]-

-;'.,uiu,ifSI;J'

1o"l

40.0 1

UJ|-i|;^?,

Fig. 3. A sample pre-processing input screen. 3. Computer implementation As shown in Fig. 1, the necessary information related to the basic random variables such as probabilistic distributions, main descriptors, etc., are entered into the program at the initial stage. The program loops over time at specified intervals and computes the probability distribution of ^(r) at each time interval. For no maintenance case, p{t) is equated to the function shown in Eq. (1). The probability distribution of ^(r) is simulated using the Monte Carlo simulation method. The main program interacts with a separate routine that performs the Monte Carlo simulation for

Pit).

The algorithm used for the preventive maintenance case has the same structure as that used for the no maintenance case. The program follows the same algorithm except the

function calls are made to five additional time-variant functions of random variables at different points in time within the lifetime of the deteriorating structural system. p{t) is equated to a multi-linear equation. To accelerate the research in reliability based maintenance studies, to establish integration and linkage between separate reliability software applications, and last but not least, to obtain the capability for performing quick and reliable computations, a program has been developed to perform the above mentioned tasks in a graphical user interface platform. The program guides the user through successive input screens to ease the pre- and post-processing phases. Fig. 3 shows a sample pre-processing input screen and Fig. 4 displays the output through the post-processor. •^jgixi

Efe fyp^fm jjgwiKw- ^ifift

Fig. 4. Post-processing output screen.

J.S. Kong et al. /First MIT Conference on Computational Fluid and Solid Mechanics 4. Conclusions This paper proposes a reliability-oriented simulationbased modeling for failure time prediction of deteriorating structural systems or groups of similar structures with or without maintenance. The probabilistic modeling described can be adopted for reliability-based management of structural systems. Much work still remains to be done in this area but the procedure provides a first step toward developing such management systems.

Acknowledgements The partial financial support of the U.S. National Science Foundation through grants CMS-9506435, CMS9522166, CMS-9872379, and CMS-9912525, and of the U.K. Highways Agency is gratefully acknowledged. The opinions and conclusions presented in this paper are those of the writers and do not necessarily reflect the views of the sponsoring organizations.

References [1] Frangopol DM. Life-cycle cost analysis for bridges. In: Frangopol DM (Ed), Bridge Safety and Reliability. Reston, VA: ASCE, 1999, pp. 210-236.

293

[2] Ang AH-S, De Leon D. Target reliability for structural design based on minimum expected life-cycle cost. In: Frangopol DM, Corotis RB, Rackwitz R (Eds), Reliability and Optimization of Structural Systems. New York: Pergamon, 1997, pp. 71-83. [3] Wen YK, Kang YJ. Design based on minimum expected lifecycle cost. In: Frangopol DM, Cheng FY (Eds), Advances in Structural Optimization. New York: ASCE, 1997, pp. 192203. [4] Estes AC, Frangopol DM. Repair optimization of highway bridges using a system reliability approach. J Struct Eng 2000;125(7):766-775. [5] Enright MP, Frangopol DM. Reliability-based condition assessment of deteriorating reinforced concrete bridges considering load redistribution. Struct Safety 1999;21(2):159195. [6] Kong JS. Optimum Planning for Maintaining Reliability of Deteriorating Structures. Ph.D. Thesis, Department of Civil Engineering, University of Colorado, Boulder, CO, 2000, in progress. [7] Das PC. New developments in bridge management methodology. Struct Eng Int 1998;8(4):299-302. [8] Wallbank EJ, Tailor P, Vassie P. Strategic planning of future maintenance needs. In: Das PC (Ed), Management of Highway Structures. London: Thomas Telford, 1999, pp. 163172. [9] Frangopol DM, Das P C Management of bridge stocks based on future reUability and maintenance costs. In: Das PC, Frangopol DM, Nowak AS (Eds), Current and Future Trends in Bridge Design, Construction, and Maintenance. London: Thomas Telford, 1999, pp. 45-58.

294

PRESTO: impact dynamics with scalable contact using the SIERRA framework J.R. Koteras^'*, A.S. Gullerud% V.L. Porter % W.M. Scherzinger\ K.H. Brown' " Sandia National Laboratories ^ Computational Solid Mechanics and Structural Dynamics, P.O. Box 5800, Albuquerque, NM 87185-0847, USA ^ Sandia National Laboratories, Material Mechanics, P.O. Box 5800, Albuquerque, NM 87185-0847, USA ^ Sandia National Laboratories, Computational Physics Research and Development, P.O. Box 5800, Albuquerque, NM 87185-0819, USA

Abstract PRESTO is a three-dimensional transient dynamics code with a versatile element library, nonlinear material models, large deformation capabiUties, and scalable contact. It is built upon the SIERRA framework, which provides a data management framework in a parallel computing environment that allows addition of capabilities in a modular fashion. Keywords: Transient dynamics; Parallel computing; Scalable contact

1. Introduction Resolving the effect of dynamic loading events on engineering components represents a vital part of modem design. Problems of interest often include significant nonlinear behavior such as complicated material response, large deformation, and complex interaction of components in contact. PRESTO, a three-dimensional transient dynamics code, has been designed to provide a computational tool to solve such problems. The code is implemented within the SIERRA framework [1], which provides support for massively parallel computation and a modular approach to adding new capabilities. Extensive capabilities have already been added to PRESTO and more are forthcoming.

2. Computational procedures PRESTO discretizes the equations of motion for a body and solves the resulting system of equations using a central difference time integrator [2]. The equations of motion for * Corresponding author. Tel.: -\-l (505) 844-8624; Fax: -\-\ (505) 844-9297; E-mail: [email protected] ^ Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

the body are

S/T-pu

+

pf^=0

(1)

where T is the 'true' stress in the deformed configuration, p is the mass density per unit volume, u is the acceleration of the material point, and / g is a specific (force per unit mass) body force vector. The solution to Eq. (1) must satisfy u = d(t) on the boundary Su where kinematic boundary conditions are prescribed and T • n = s(t) onthe boundary Sj where the traction boundary conditions are applied. For surfaces 5c in contact, the jump conditions at the contact discontinuities must approximately satisfy

(2)

(T+-{-T-)-n=0

where the superscripts -h and — denote different sides of the contact surface. For the discretized set of equations, the quantity ii at any time t is computed with

ii, = (/f^^ - /r^)/M,

(3)

where /f^^ and f^^ are the external and internal nodal forces, respectively, and M is the nodal point lumped mass. With the central difference method as implemented in PRESTO, the displacements u are related to the accelerations by {Ut-

Arpid

• M,)/(Afold) - (Wr -

Wr+Arnew)/(^^new)

(A^old + A^new)/2

(4)

J.R. Koteras et al. /First MIT Conference on Computational Fluid and Solid Mechanics

295

and the velocities u are related to the displacements u by {Ut - W,_A,^jJ/(A^old) + iUt-

•

Ut)/(Atnc^)

(5)

where At in general denotes a time step increment. To provide objective stresses/strains under large deformation, the element formulations utilize polar decompositions of the current deformation gradient to evaluate the material models in an unrotated configuration, which is equivalent to the Green-Nagdhi stress rate [3]. Small-strain formulations for the material models can then be used within a large deformation environment.

^2500

c I 2000 w U

§ 1500

H

o '% 1000

500 •

0

3. Code structure Codes which use the SIERRA framework are organized into a series of modules called 'mechanics'. Mechanics can be nested inside each other to create a rational structure for computation. The isolation of code pieces into mechanics permits dynamic addition or removal of features as needed in an analysis. The core level of PRESTO is the region mechanics. This module is responsible for the solution of the discretized equations of motion for a single time increment. The region contains a number of nested mechanics which conduct computations for elements, boundary conditions, etc. The PRESTO region incorporates a two state architecture, in which known quantities at time t are stored as the 'old' state and quantities obtained by projecting ahead A^new to time t H- A^new are stored as the 'new' state. Velocities are computed at times t - At^x^/l and t + Arnew/2, and are stored in states old and new, respectively. This convention makes it easy to update velocities and displacements, as well as other values, to the new state. The processing of elements occurs within element mechanics, which are nested inside the region as needed. Element mechanics are responsible for computing the response of a set of elements by taking the corresponding nodal displacements and returning nodal forces. The current element library includes an under-integrated eight-node hexahedral element, four- and eight-node tetrahedral elements, and under-integrated four-node Key-Hoff shell and membrane elements [4]. All of the current elements have both a midpoint-incremental and strongly objective formulation to compute strain rates. The SIERRA framework allows for the straightforward implementation of most elements, and current plans for PRESTO include wedge, beam and spring elements, as well as fully integrated formulations. Boundary conditions also exist as mechanics nested within the region. A large number of kinematic and traction boundary conditions have already been implemented in PRESTO. Included in the boundary condition set are some specialized boundary conditions such as silent boundary, cavity expansion, and periodic boundary conditions. Within

2000 elements per processor

w •

100 1000 Number of Processors

Fig, 1. Scaling for increasing mesh size.

the SIERRA framework, implementation of the boundary conditions has been an easy process even for the specialized conditions. Material models are mechanics which nest inside the elements. Currently, six material models have been implemented. This material library will be expanded to meet demands from various analyses. All of the material models return a sound speed for each element, which provides a uniform approach to handling both equation of state models and some of the more typical engineering models (e.g., elastic-plastic model with hardening). Contact detection and enforcement are also part of the region mechanics. PRESTO uses the ACME (Algorithms for Contact in a Multi-physics Environment) package [5], for both contact detection and enforcement. PRESTO accesses ACME through a separate SIERRA interface which maps data structures in the SIERRA framework to the ACME library and back again. Much of the work used to develop scalable, parallel contact in the explicit dynamics code PR0NT03D [6], has been used as a basis for ACME. To drive the solution of a problem over time, the PRESTO region is nested within a procedure mechanics. A procedure mechanics, which may contain multiple regions for a multi-physics coupling, is responsible for advancing time, executing the region(s), and transferring data between regions as needed. The procedure is also responsible for updating state variables after executing the region(s), by copying all data of state 'new' to state 'old'. The current implementation contains a PRESTO procedure which only holds one PRESTO region. The PRESTO procedure provides analyses which only include transient dynamic response. However, a TEMPO procedure has also been created which couples PRESTO with the quasi-static structural code ADAGIO [7]. An analysis using a TEMPO procedure can use ADAGIO to compute pre-stress conditions for a part, and then pass the pre-stress data as initial conditions into PRESTO. The SIERRA framework provides extensive capabilities for transferring data between regions.

J.R. Koteras et al /First MIT Conference on Computational Fluid and Solid Mechanics

296 100000

ized functions — can be added to PRESTO in a modular manner. This should make code maintenance easier, and it should be able to incorporate new capabilities in a timely manner when the need arises. Future developments include the addition of crack growth capabilities, h-adaptivity, and a number of new elements and material models.

• - • ASCI Red Times - - Perfect Speed-Up

10000

1.024 million elements

^ '%

100

References

1000

10000

Number of Processors Fig. 2. Scaling for fixed mesh size. 4. Example problem The scalability of PRESTO on a massively parallel machine has been studied by using a simple impact problem. An impact load (a time varying pressure load) is applied to one end of a finite length, thin walled tube. Analytic results for this problem can be obtained from simple one-dimensional wave propagation problems. By modeling sectors of various sizes, models with different number of elements are created with the same element size, and, hence, the same time step. The largest model in the studies had four million elements; 2048 processors were used in this analysis. Results for increasing mesh size and fixed mesh size scalability are shown in Figs. 1 and 2, respectively. The results of this study show excellent scalability in PRESTO.

5. Future development Experience to date indicates that new capabilities — elements, boundary conditions, material models, special-

[1] Edwards HC, Stewart JR. SIERRA: a software environment for developing complex multi-physics applications. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [2] Bathe KJ. Numerical Methods in Finite Element Analysis. New Jersey: Prentice-Hall, 1976. [3] Johnson GC, Bammann DJ. On the analysis of rotation and stress rate in deforming bodies. Int J Solids Struct 1984;20(8):725-737. [4] Key SW, Hoff CC. An improved constant membrane and bending stress shell element for explicit transient dynamics. Comput Methods Appl Mech Eng 1995;124(l-2):33-47. [5] Brown KH, Glass MW, Gullerud AS, Heinstein MW, Jones RE, Summers RM. ACME: a parallel library of algorithms for contact in a multi-physics environment. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [6] Taylor LM, Flanagan DP PR0NT03D: A Three-Dimensional Transient Dynamics Program. Albuquerque, NM: Sandia National Laboratories, 1989. [7] Mitchell JA, Gullerud AS, Scherzinger WM, Koteras JR, Porter VL. Adagio: non-linear quasi-static structural response using the SIERRA framework. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001.

297

Layered higher order concepts for D-adaptivity in shell theory Wilfried B. Kratzig *, Daniel Jun Institute for Statics and Dynamics, Ruhr University Bochum, Universitdtsstr 150, 44780 Bochum, Germany

Abstract Problems of solid mechanics are basically formulated in tensor notation in the 3-dimensional Euclidean space. But engineering praxis favors — as far as possible — reduced dimensional representations, mainly in order to describe deformation processes in its most natural way by surface- and line-like geometries, and for the sake of easier error control. The present paper will systematically transform the set of basic mechanical conditions of a 3-dimensional soHd of arbitrary material into corresponding 2-dimensional sets of so-called higher order shell equations. Since modem surface-like structures often have a layered structure or are computed — in case of inelastic materials — by use of such an idealization, this transformation will be combined with a layered representation. Such models admit the simulation of rather arbitrary shell responses including all kinds of perturbations like thickness jumps, material cracking and crushing as well as internal damage phenomena. Keywords: Higher-order shell theory; Laminated shells; 3D-adaptivity

1. Basic transformations The derivations thus will start from an arbitrary 3dimensional body in the E3, described in 3-dimensional representation by a set of convected curvilinear co-ordinates 0 ' , / = 1,2, 3; ^ = 0. Herein 0", a = 1,2, describe parameter lines of the later reference surface of the shell, and 0^ denotes the transverse co-ordinate. We start with a global statement of energy conservation, the so-called rate-of-energy equation

V

(1) valid for the complete shell continuum at time r. In (1), the following abbreviations are used: dV represents a material volume element of the continuum considered, dS is its free surface element, and U* the internal energy density per unit mass Q*. f* abbreviates the vector field of body forces per unit mass in dV, and t* the vector field of tractions on the * Corresponding author. Tel.: 4-49 (234) 32-29064; Fax: +49 (234) 32-14149; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Du* denotes the velocity field of the shell surfaces, v* shell continuum. In order to transform the basic statement (1) into that one of a multi-director continuum (d„, 0 < w < oo), based on the reference surface 0^ = 0, it is localized by assumption of integrand continuity, multiplied by an arbitrary but complete set of scalar functions O, and then transformed into the weak integral form again. Next, the special choice of

0 = 0^,

/ = 0 , 1 , . . . (50

(2)

is selected. By such Fourier-integral-transformation of the original conservation of energy statement (1), sets of central moments (0 < n < oo) of all force variables are formed, always related to the reference surface [1,2]. Spatial invariance requirements in connection again with a localization concept of the transformed weak statements finally lead to infinite sets of interior dynamic equations, still representing as a whole the original continuum in its multi-director representation. In detail, these infinite sets of equations are the mass conservation (3), the set of equations of motions (4), some set of symmetry conditions (5) and finally the transformed residual energy statements (6), all valid for / = 0, 1,2, ...00 [3]: Q-{-Q(p'^^=0 for / = 0,

)(:^ = 0 for / = 1,2, ...oo, (3)

298

W.B. Krdtzig, D. Jun/First MIT Conference on Computational Fluid and Solid Mechanics

m^'^U-/m'+p^=0,

(4)

Y^ {n m'+" X d, + m^+"'' x d„,„) = 0,

(5)

As remarked earlier, all vector and tensor fields are decomposed with respect to the basis of the actual reference surface at time f, in detail the directors and their derivatives: d « — "Ai/3 5

QU^ -Y^(n

m'+" . v„ + m'+"'^ • v,,,) - 0.

(6)

In these equations, k^ are moments of the mass distribution in thickness direction, m^", m^ and p^ are moments of the in-plane stresses, the transverse stresses and the loading variables, where the order of the moments is given by the letter /. (7' represents moments of the internal energy, and the v„ abbreviate terms in the Fourier series of the velocity field. In correspondence to the dynamic conditions (4, 5), also the kinematic equations of the original 3d continuum have to be transformed likewise into infinite 2-dimensional sets of equations, a step which not will be repeated here.

2. Component decomposition with respect to the actual reference surface Up to this point the question treated is of purely physical nature. To de-couple physics from all kinematic considerations, the energy conservation has been used in the form of the rate-of-work statement. This starting point delivers an intrinsic rate-formulation of the shell equations, in which all tensor components of an arbitrarily deformed state are — as a next step — decomposed with respect to the base vectors of this actual state. Thus the derived equations are completely free of all kinematic transformations; they form a comprehensive linear set, valid for arbitrary, even large deformation processes. In detail we find the following vector-decomposed sets of equations: The equations of motion (7) and (8) in the directions of the base vectors a^, as of the actual reference surface at time t, the corresponding component forms (9) and (10) of the symmetry conditions and again the set of energy expressions (11). 3^^_/^/^+^^^0,

.^«^i

m'"3 L + m'^^Kp -Im''

+p^ = 0,

(7) (8)

n=l,2 CX)

(1=0,1

= 0,

(10) oo

^n,a

(12)

— ^nia^ ->

as well as all moments of the sets of internal as well as external force variables: P^ = / a , .

(13)

In its present form, the derived equations are applicable for the analysis of isotropic, an-isotropic and layered shells of elastic or inelastic material behavior. For the latter purpose, the derived equations can be reduced to low order classical shell theory and then applied to a single layer, or remained as higher order theories to approximate a package of layers. We again remind the reader that also the kinematic relations have to be decomposed in the same manner, a series of transformations skipped here for reasons of shortness.

3. Classical shell theories and corresponding sandwich concepts The 0th and 1st order variables and equations Q,n = 1,2) of the above given infinite sets of variables and equations, which can be gained by mapping of these complete sets on a Cosserat surface, will describe the classical shell theory. In this context, the displacement field u* generally will be represented by 2 vector fields on the reference surface, Uo and Ui respectively: U* = Uo + 0Ui

(14)

Uo always represents a 3-dimensional vector field (UQI , / = 1, 2, 3). Ui generally is assumed as a field of surface vectors tangential to the reference surface. If by application of the Kirchhoff-Love hypothesis Ui depends on UQ, we end up with 3-parameter shell theories (MQI, J^oi, ^03). in the case of independence of Ui with 5-parameter theories (woi, W02, W03, wii, W12). Exceptionally both vector fields Uo, Ui possess 3 components each, then 6-parameter theories appear (MOI » "02, "03, Wll, Uu, W13). We do not intend to elaborate in detail on this classical context, and rather point the interested reader to a variety of adequate literature [4,5]. Such classical shell theories are widely used in order to derive models for laminated shell structures in layered formulation [6,7]. However, such models are principally unable to correctly map arbitrary 3-dimensional responses even in a very dense package, since the transverse strains and the transverse shear strains are approximated only by constant terms. This consequently leads to discontinuities of the stress fields a"^, a^^ on all layer boundaries, since such classical theories are at best capable to achieve

W.B. Krdtzig, D. Jun/First MIT Conference on Computational Fluid and Solid Mechanics Co-continuity for their stress fields a^^. In order to cure these deficiencies, at least quadratic terms have to be considered in the displacement field representation for u*.

Finally, the rate of energy expression (11) reads as follows: •Vo .

QU'

;:,0(a;8)

oiiaf}) + m^^'^dia + m^^dis + m^''^ki^ai '^

+ m'''\2d2a + i i s J + 2m^^ 4 + m'^^^^X2^,. 4. Higher order shell theories and corresponding sandwich theories For this purpose, we now use a quadratic displacement field approximation as follows: : Uo + 0 U i + 0 ^ U 2 .

(15)

In order to derive the corresponding higher order shell theory from the basic general sets (7) to (11) of the multi-director continuum, we now evaluate from equations (7) and (8) the pertinent set of equations of motion with 0th, 1st and 2nd moments: (16) (17) (18) (19) ^2(a^)

l^-2m'^'

+ p^^=0,

n'^"^^b,^-2m''^p''=0,

^0«;S

5. Numerical example

(21)

5.1. Tensioned cylindrical shell with discontinuity

the membrane stress resultant tensor, the transverse shear stress resultant vector, the moment tensor, the transverse moment vector, the transverse normal force, the bi-moment tensor, the transverse normal moment.

AaP

^2aP

m^^

(27)

All further equations and functionals of this (higher order) quadratic shell theory will be presented in the conference lecture. In this context we are able to draw an interesting comparison to classical shell theories {l,n = 0, 1). From the point of view of Cosserat surface mechanics, that means classical shell mechanics, all their classical variables and equations describe load-induced states of stresses and deformations, as far as loads on and deformations of a surface are concerned. On the other hand and beyond classical shell theory, all sets of higher order variables and equations (l,n = 2) represent self-equilibrating states of stresses and deformations as constraints to classical shell theory, in order to overcome its deficiencies and to match 3dimensional mechanical processes with Q-continuity also in 3-dimensional direction. Obviously and as will be shown in detail in the conference lecture, for layered shell theories these self-equilibrating states play an important role in a more correct modeling of response properties, compared to packages of classical shell models.

(20)

In these 9 conditions (a = 1, 2) we find the following force variables of this theory:

299

This example deals with a thin tensioned cylindrical shell with a geometrical thickness discontinuity. Fig. 1 shows the dimensions and loading as well as the material parameters. The cylindrical shell has been analysed with an automatic adaptive computation procedure using an error

The symmetry conditions (9) and (10) deliver the following results:

-n^(-^)=m'"^-^m'^'bl

(22)

R^ = R2 = 5,0 m hi = h2 = 10,0 m

•A(a^) _

^1«/

= m'"^ + m

2(aA)7^

(23)

^2(a^)^^2a^^^2(a^)^

(24)

^Uc.3^^1a_^^U3^a

(25)

ti = 0,1 m t2 = 0,2 m q = 1,0 kN/m2 E = lO^kN/m^ V = 0,0

(26) which have been considered already in the above given equations of motion.

Fig. 1. Tensioned cylindrical shell with geometrical discontinuity.

300

W.B. Krdtzig, D. Jun/First MIT Conference on Computational Fluid and Solid Mechanics 0S4O9

2.9139

4.7547

18.8314

16.4963 2C.4101 243240 28.2378

14.2169

32.1516 30.1144 36900

35596

485144

18.2825 2330054

•577.1742 1691.8972 1806.6201

1317.4603

Steps

362.2847

Step 4

§921.3430

Fig. 2. h-Refined meshes with error distribution in L2-norm in circumferential direction.

y

3.6900

35596

4S.5144

U l 182825

93.3387

1233.0054

138.1630

347.7284

182.9873

•462.4513

[227.8117

1577.1742

272.6360

•691.8972

1317.4603 362.2847

1806.6201

Step

I92I.343O

Step 4

Fig. 3. h-Refined meshes with error distribution in L2-norm in thickness direction. estimator according to Zienkiewicz/Zhu in 0 " as well as in 0^-direction [8,9]. At the geometrical discontinuity the pure shell theory is no longer valid because of a dominating 3-dimensional state, and thus one can expect a required refinement also in thickness direction at the discontinuity. Fig. 2 shows the first four refined Finite Element meshes with the error distributions in the Li-norm. Due to sym-

metry of the shell only a quarter of the structure is shown. The h-refinement procedure obviously took place towards the geometrical discontinuity. Fig. 3 shows the corresponding error distributions of the 3-dimensional adaptive refinement steps — zoomed at the discontinuity — with the expected refinement steps in thickness direction.

W.B. Krdtzig, D. Jun/First MIT Conference on Computational Fluid and Solid Mechanics 6. Outlook Higher order shell theories have been derived by Fourier transformation of a 3-dimensional solid, from which a quadratic approximation has been truncated. The gained equations have been discretized and used for a sandwich concept in the finite element software FEMAS. In the lecture, several examples will demonstrate the general concept of applying such truncated sets of equations for the analysis of 3-dimensional problems. From these examples, simplifications of the derived sets of variables and equations can be filtered out by tensor norm estimates. References [1] Green AE, Laws N, Naghdi PM. Rods plates and shells. Proc Camb Philos Soc 1968;64:895-913. [2] Naghdi PM. The theory of plates and shells. In: Fliigge S (Ed), Handbuch der Physik, volume VI, A2. BerUn: Springer Verlag, 1972, pp. 425-640.

301

[3] Kratzig WB. "Best" transverse shearing and stretching shell theory for nonlinear finite element simulations. Comput Methods Appl Mech Eng 1993;103:135-160. [4] Ba§ar Y, Kratzig WB. Theory of Shell Structures. Number 258 in 18. Dusseldorf: VDI Verlag, 2000. [5] Ba§ar Y, WB Kratzig. Mechanik der Flachentragwerke. Friedr. Braunschweig/Wiesbaden: Vieweg and Sohn, 1985. [6] Ba§ar Y, Ding Y, Schultz R. Refined shear deformation models for composite laminates with finite rotations. Int J SoHds Struct 1993;30:2611-2638. [7] Reddy JN. A simple high-order theory for laminated composite plates. J Appl Mech 1984;51:745-752. [8] Ba§ar Y, Hanskotter U, Jun D. Error-controlled nonUnear simulation of shell structures. In: Proceedings of Fourth International Colloquium on Computation of Shell and Spatial Structures (lASS-IACM 2000), Crete, Greece, 2000. [9] Jun D, Hanskotter U, WB Kratzig. Adaptive strategies for the nonfinear simulation of shell structures. In: Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering (Eccomas 2000), Barcelona, Spain, 2000.

302

Superelement based adaptive finite element analysis for linear and nonlinear continua under distributed computing environment C.S. Krishnamoorthy *, Vr. Annamalai, U. Vinu Unnithan Department of Civil Engineering, Indian Institute of Technology Madras, Chennai, 600 036, India

Abstract With the availabiUty of cost effective high performance computing on Network of Workstations (NoWs), it is possible to provide accurate and rehable solutions to complex problems of engineering industry through Adaptive FEA (AFEA). The paper presents superelement-based domain decomposition suited for parallel implementation. A computational framework is presented for nonlinear stress analysis. The paper emphasises the need for further developmental work for industrial application of AFEA on NoWs. Keywords: Adaptive Finite Element Analysis; a posteriori error estimation; Automated meshing; Nonlinear analysis; Distributed and high performance computing; Domain decomposition

1. Introduction The developments in the last 10 years in a posteriori error estimation techniques in FEA provide valuable tools for quality assurance and quality control in engineering analysis and design. The error estimators coupled with adaptive mesh refinement strategies serve to control the quality of the finite element solution and provides the engineer, solutions within a prescribed tolerance. The advances in parallel computing technology offer the opportunity to provide accurate solutions in a cost-effective computing environment. It is in this context that adaptive FEA has gained importance. The essential ingredients of an adaptive scheme are: • Error estimator. • Refinement strategy. • Mesh generator. The publication of paper by Zienkiewicz and Zhu [1] gave the much needed practical and computer implementable approach to error estimates and refinement strategies. Since then, a very large number of papers have been published and research is in progress on many areas of application. In an adaptive FEA environment based on error estimation and refinement strategy, there is a need for complete automation of the mesh generation process for * Corresponding author. Tel.: +91 (44) 445-8286; Fax: +91 (44) 2545/445 8281; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

reanalysis. In a parallel-computing environment to meet the computational demands of AFEA of real world problems, the most important issue to be addressed is the efficient partitioning of a given unstructured mesh into A^^ subdomains. The paper presents superelement based domain partitioning and AFEA in distributed computing environment. In addition, the issues and needs of AFEA for industrial application are examined.

2. Superelement based adaptive meshing Parallel mesh generation can be divided into the following stages: • Creation of Np number of subdomains. • Subdomain mesh generation. • Refinement or repartitioning of subdomains. The initial decomposition is done by various techniques [2]. In the present work, the domain is divided using a medial axis decomposition and it is called Meshing by Successive Decomposition (MSD). In MSD, the superelements are generated using Approximate Skeletal Method (ASM) and meshes are generated by transfinite interpolation [3]. The MSD is highly suited for parallel implementation as the decomposition is done only once for the whole adaptive analysis. The process can be divided into: • Domain decomposition (superelement generation) and partition to form subdomains.

C.S. Krishnamoorthy et al. /First MIT Conference on Computational Fluid and Solid Mechanics • Parallel smoothing for well formed meshes. • Parallel attribute generation and repartition. In parallel implementation, the problem of evenly distributing the load to all the processors is complex since the number of elements in a 'superelement' is not known a priori. A detailed investigation has been carried out for load balancing using six algorithms and spectral bisection with Kemighan-Lin Heuristic, which has been found to be efficient in distributing the superelements to form as subdomains in different processors and also repartition them after mesh generation to form substructures for AFEA [4].

3. Distributed high performance computing (HFC) environment Almost all organisations have high-speed local area networks, connecting a cluster of workstations and PCs and it has been shown that this NoWs form a cost-effective highperformance computing environment needed for AFEA. In order to help the developer a library PAVE (Parallel Virtual Environment) has been developed which can be used as a layer over PVM and MPI to take care of all message passing and communication tasks [4].

303

Convergent Error Estimators [1] have been used. As a typical example, the Butt Strap problem Fig. 1 reported in [5], was analyzed. The domain partitioning and FE mesh are shown in Fig. 2. As the process of 'Mesh generation' and FEA are two separate tasks with different computational complexities, two plots are presented in Fig. 3 for performance evaluation.

5. Nonlinear finite element analysis Nonlinear analysis becomes important in the FE simulation of manufacturing processes like forging, metal casting and in the safety analysis such as crash worthiness of vehicles. These types of analyses require consideration of both kinematic and material nonlinearities. The most computationally intensive process in a geometricallly nonlinear process is the computation of the tangent stiffness matrix KT as K T = K L + KLD + K ,

(1)

KL is the linear stiffness matrix, K,

= j BlCB, dV

(2)

KLD is the large displacement matrix. 4. Linear finite element analysis

KL

Using the explicit decomposition based on MSD, AFEA program has been developed under distributed computing environment. The Zienkiewicz and Zhu and other Super

Q

/

BlCB^^dV +

f BLCB.dV

j Bl,CB^,dV

o (a)

(b)

Fig. 1. (a) Butt strap: problem definition, (b) Butt strap: superelements.

Fig. 2. (a) Domain partitioning, (b) FE mesh.

(3)

304

C.S. Krishnamoorthy et al /First MIT Conference on Computational Fluid and Solid Mechanics

^\

70 ^

s

60

^^-^ ,'-""""

Iso n ^ |40

/ -X'^.---•' p - -•;--.

^

'

^

• •

/ *

.,.-•"

'

• . .

^4

/

• " • ^ - #

if2 30

r

20

/ / - • • '

/

10,

BUTT: Speedup-FEA Linear Speedup Prob Size: 8004 Nodes Prob Size: 4735 Nodes

BUTT

80

/

Cn

/

'

- • - • * '

1 // f

^

Greedy-KL Graph-KL Spectral-KL

^ -

•

—"— -

•

•

•

4 5 6 No of processors

2

3 4 No of processors

Fig. 3. Performance evaluation. K(j is the geometric stiffness matrix. llZ.2

K„ = / G ^SGdV

= (^f{p*-p^y6vY

(7)

(4)

where S is the Second Piola-Kirchhoff stress matrix. In the case of material nonlinearity, the constitutive matrix C is not constant. In addition, for large deformation problems, the material may yield and elasto-plastic analysis has to be performed. The yield stress analysis and stress update are carried out by iterative procedures like Backward Euler Integration schemes. For nonlinear analysis a number of solution techniques are available for tracing full load deflection path. 5.7. Error estimation Once convergence has been reached in a load step, the structure is checked for global error. Though considerable amount of research has been carried out in the linear analysis, error estimation for nonlinear problems is still an active area of research. However, many investigators have used ZZ type estimators in nonlinear analysis [6]. More recently, a Super-Convergent Path Recovery (SPR) is found to be more efficient and gives more accurate results [6]. To highlight the computational intensive process, a typical step for error in elasto-plastic analysis is described here briefly. Adopting the ZZ-approach, the pointwise error in accumulated plastic strain '/?' may be approximated by [6]. (5)

However if the error tolerance is violated at a certain load step, the FE mesh is refined based on the computed elemental error indicators. 5.2. Computational framework The assemblage of the stiffness matrix utilizes over 70% [7] of the computational time and since stiffness components have to be updated for every iteration and increment, points to the severity of the computation involved. With the inclusion of adaptive techniques in nonlinear FEA, the additional task of checking errors locally and globally and remeshing for every load step, phenomenally increases the computational process. Thus the need for its implementation in a parallel environment becomes essential. Superelement based domain decomposition described in the earlier section is very well suited and the frame work being developed for the Nonlinear AFEA under the distributed computing environment is shown in Fig. 4. The subdomain distribution to various processors is carried out using spectral bisection algorithm. GNL-MNL Kernel, is executed by the master and the error is indicated to the different processors for every load step. Mesh refinement is done on the superelement by the corresponding processor. The whole process is repeated until a globally converged solution has been obtained satisfying the specified error percentage, at every load step.

1. The L2-norm error in stresses can be written as (6) 2. The L2 norm in accumulated plastic strain:

6. Conclusion For highly numerical techniques, like FEM coupled with adaptivity, distributed computing on an existing Local Area

305

C.S. Krishnamoorthy et al /First MIT Conference on Computational Fluid and Solid Mechanics

^

Ev^yationof stiffness components like Ki + KLO -•• K0 and Residual Force Vectcsr

f Traisf^fjg D ^ f mm the Slava^ Viaribies from

*_^

^ .^^

^ Evaluation c^stfUhess ^ componerts like Kl + KLD-f Key and Reskfeial Force I Vector /

Next Lpaei Step

to tie mas^r Ibf /s^sem^

mesh

Msemi^c^the $iffr>ass mafeix and kimr^ Force Sector.

Not WitNn limits.

Mesh Re^nemantj b^ed on MSD cr olh@f algorithms.

WitNn

Fig. 4. Framework of nonlinear AFEA in distributed computing environment. of Workstations and PCs is the most economically viable solution. Meshing of the problem domain by successive Superelement decomposition techniques has an inherent parallelism and is ideally suited for parallel implementation of AFEA. To solve real life practical problems, Nonhnear AFEA provides the ultimate key. A framework is also proposed in distributed computing environment for its efficient implementation. However, areas like Superelement based mesh generation for surfaces and 3D solid elements need further development. To make all these development processes serve the industrial needs, the adaptive processes need to be integrated with the Finite Element packages on HPC platform for providing the much needed reliability to analysis and design in engineering industry.

[3]

[4]

[5]

[6]

References

[7]

[1] Zienkiewicz OC, Zhu JZ, A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 1987;24:337-357. [2] Owens S, A survey of unstructured mesh generation tech-

[8]

nology, available on the Internet at World Wide Web \JB1. http://www.andrew.cmu.edu/user/sowen/survey/softref.html Krishnamoorthy CS, Raphael B, Mukherjee S, Meshing by successive superelement decomposition (MSD) - a new approach to quadrilateral mesh generation. Finite Elem Anal Des 1995;20:1-37. Annamalai Vr. Parallel mesh generation and adaptive twodimensional finite element analysis on distributed computing environment. MS thesis submitted in the Department of Civil Engineering, IIT Madras, India, 1999. Klaas C, Niekamp R, Stein E, Parallel adaptive finite element computations with hierarchical preconditioning. ComputMech 1995;16:45-52. Mathisen KM, Hopperstad OS, Okstad KM, Berstad T. Error estimation and adaptivity in explicit nonlinear finite element simulation of quasi-static problems. Comput Struct 1999;72:627-644. Fahmt MW, Hamini AH, A survey of parallel nonlinear dynamic analysis methodologies. C&S 1994;53:1033-1043. Gangaraj SK, A posteriori estimation of the error in the finite element solution by computation of the guaranteed upper and lower bounds. Ph.D. Dissertation submitted to the Office of Graduate Studies of Texas A&M University, 1999.

306

Multibody system/finite element contact simulation with an energy-based switching criterion Lars Kubler *, Peter Eberhard Institute of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstr. 5, 91058 Erlangen, Germany

Abstract The analysis of contact problems using hybrid multibody system (MBS)/finite element (FE) simulation is presented. An important problem within the hybrid simulation approach is the computation of the required mechanical information for the transitions between the two approaches. Kinematical relations and balance laws for momentum and angular momentum are utilized for this purpose. Especially the transition from FE to MBS modeling requires great care, as here the intrinsic information of the discretized body has to be reduced to the smaller amount of rigid body information. Furthermore, the problem arises, when to switch back to the multibody system method after contact separation. An energy-based criterion to automatically propose the appropriate switching time is presented. Keywords: Contact; Energy-based switching criterion; Hybrid simulation; Multibody system; Non-linear finite element method; Transition

1. Introduction The analysis of contact problems is an important technical problem which unfortunately always involves great computational effort. If several potentially colliding bodies are under consideration, it has been shown to be efficient to compute the motion of non-colliding bodies by the multibody system (MBS) method and the motion and deformation of colliding bodies with the nonlinear finite element (FE) method. This makes it possible to combine the advantages of both methods, i.e. the efficiency of the multibody system approach and the possibility to describe and compute deformations correctly with the nonlinear finite element method. For the example of three moving bodies which are colliding successively and the corresponding MBS/FEM transitions see Fig. 1. Several aspects have to be considered to enable a reliable hybrid contact simulation. In this paper, the nontrivial problem when and how to switch back and forth between the different modeling approaches is investigated. It will be shown how the required quantities are computed based on kinematical relations and balance laws for momentum and angular momentum. Finally, a criterion is discussed * Corresponding author. E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to propose an appropriate mesh deletion time after contact separation. While for the contact of rigid bodies a lot of work has been done during the last decade, see e.g. the text book by Pfeiffer and Glocker [1], and also finite element contact has gained great maturity, see e.g. the review paper by Wriggers [2], there seems to be few material in literature concerning mixed or hybrid multibody system/finite element contacts. The relevant algorithms which are required as well as a detailed description of the mechanical transitions are included in Eberhard [3]. This paper focuses additionally on the determination of a sound switching time.

2. Transition from rigid bodies to discretized bodies After contacts between bodies are detected they are discretized for the finite element computations using onthe-fly created high-quality meshes with quads or triangles [3]. Since it is not known in advance which bodies collide and how long they remain in contact, sophisticated administration schemes have been developed. The transition from rigid to discretized deformable bodies is illustrated in Fig. 2. The motion of rigid bodies during planar multibody system simulation is defined by

L. Kiibler, P. Eberhard / First MIT Conference on Computational Fluid and Solid Mechanics

307

(d)

Fig. 1. Example for a hybrid MBS/FEM simulation.

t

7

T

T

T

T

T

T

T

T

T

T

1 4

1

•

i^j I

rJk-^

4

Fig. 2. Transition from rigid to discretized bodies. the position rocit) and angular rotation a{t) of a coordinate system Kc in an arbitrary body fixed point C, as well as the respective velocity Vc and angular velocity co. In order to create the finite element mesh, the positions roPi =roc -\- re Pi of the boundary polygon points Pi with respect to the inertial coordinate system Kj are given to the meshing algorithm, that computes the mesh [4]. With the position roKj of the nodes Kj of the finite element mesh the node velocities VK can be calculated 0 VKJ = Vc + {roKj

-roc)^

0 CO

0 = Vc-\- rcKj X

0 (0

(1)

These nodal velocities complete the determination of kine-

matic quantities of the discretized body, used then as initial conditions for the further dynamic simulation.

3. Transition from discretized bodies to rigid bodies A more complicated topic within the hybrid MBS/FEM simulation is the transition from the finite element description to the multibody system, where some approximations are necessary. Discretized bodies possess much more degrees of freedom than rigid bodies. Therefore, a reduction of the available information is necessary for the transition FEM to MBS. The position roc and the velocity re of the rigid body, see Fig. 1, and its angular rotation and velocity a

308

L. Kiibler, P. Eberhard / First MIT Conference on Computational Fluid and Solid Mechanics

and CO must be determined appropriately from the positions ToKj and velocities VKJ of the n nodes of the finite element mesh of this body. After separation, a body moves freely with its center of mass being on a parabola. The center of mass of the body should remain at the same position before and after the deletion of the mesh and the transition to the rigid body. The position ros of the center of mass for a homogeneous planar body with finite element displacement shape functions A^, mass matrix M, density p, mass m, node reference positions X and nodal displacements U is ros = - f pNdQ'iX m J

E,, M,

+ U),

(2)

with R := X -\-U - Xs md Xs := [xs ys Xs ys ...] e 1^2/1 xi j ^ follows after some transformations [5], for the moment of momentum of the discretized body Dp

= R M

V,

(7)

\ Eqs. (6) with V := [v\y Vlx V2y - V2x ••] and (7) allow the determination of the angular velocity _

DFE,S _

R

M

V

(8) JfE.s ~ R M R Finally ros and Vs are converted to the body fixed reference point C [5]. This also requires the determination of rc5, which is based on the 'Ear-Cutting' algorithm by O'Rourke [6].

or with the auxiliary matrix A 4. Determination of the switching time m 1

0

1 0

0

1 0

1

...

1 0

...

0

1

(3)

The angular rotation of the rigid body is determined approximately by averaging of appropriate edge rotations or by an suitably formulated Least Squares problem, see Eberhard [3] or Kubler [5]. The basic idea for determining the velocity Vs is the conservation of linear momentum before and after mesh deletion. The momentum of a rigid body can be calculated by 75 = mvs, whereas for a discretized body the momentum follows as

= / pNdQ-V

= A'M

'V.

(4)

With Js = JFE the velocity follows 1 Vs = —A- M m

V.

(5)

The angular velocity co of the rigid body is determined similarly by conservation of angular momentum. Whereas CO is the same for each point of a rigid body, the calculation of the angular momentum requires a reference point. A favorable reference point is given by the center of mass 5. The moment of inertia of a discretized body with reference to its center of mass can be determined as follows JpE,s = jNRNRpdn

= R MR,

(6)

Another problem that arises for the hybrid simulation is how long bodies should be computed by the finite element method after separation. It is desirable to delete the mesh as soon as possible after the separation in order to switch back to the more efficient multibody system simulation. However, if the mesh is deleted too early, frequent (nonphysical) contact/separation transitions occur and the total simulation efficiency decreases. The conservative approach to overcome this problem is, to keep the mesh for many time steps after the separation, even if the efficiency suffers. A more advanced and efficient approach to deal with this aspect uses estimates for the ratio of the internal elastic energy before mesh deletion and the kinetic energy of the rigid body after mesh deletion. As an example for this approach, a simple system of two elastic bodies, as shown in Fig. 3, is investigated. Both bodies approach with same velocity. Fig. 4 shows computed curves for the kinetic energy of the bodies for different material damping coefficients at. After the collision the discretized bodies decelerate almost to rest. The kinetic energy decreases to a minimum when the maximum deformation is reached, where it is almost fully transformed into potential or strain energy. The kinetic energy does not reach exactly zero as some of the nodes are still in motion because of wave effects within the elastic bodies. Then the bodies expand again and their velocity increases. After their separation the bodies remain meshed, e.g. for 100 time steps. The kinetic energy of the discretized bodies after the separation slightly varies because of their eigenvibrations.

Fig. 3. Mesh creation and deletion.

L. Kubler, P. Eberhard/First MIT Conference on Computational Fluid and Solid Mechanics

309

14 12 10 c

I

0.1

0.15

0.25

time Fig. 4. Kinetic energy for different damping coefficients. A certain amount, which is quite low in this example, of the total energy of the bodies oscillates between kinetic and strain energy. The remaining amount of the total energy is the kinetic energy resulting from the free motion of the body. This part corresponds to the kinetic energy of the rigid body after mesh deletion. The strain energy decreases because of material damping in a non-conservative system. For the automatic determination of the mesh deletion time a certain ratio p of the strain energy to the kinetic energy ^ E^^ ^ UKU ^ Ekin V M V ^^ can be used as a threshold criterion. Special care is required if the center of gravity of a body remains at rest after separation.

5. Conclusions An important topic within the mixed MBS/FEM contact simulation is given by the transitions between the two modeling approaches. While the transition from multibody systems to finite elements is usually not problematic and also during the reverse transition the computation of the velocities and angular velocities from balance of momentum and angular momentum requires no approximations, the computation of the position and orientation of the rigid bodies from the node positions requires a lot of care and approximations cannot be avoided. Special attention is further required after the contact

separation to decide when to switch from the discretized bodies in the FE model back to the rigid bodies of the multibody system. One idea described in this paper is to use a threshold of the ratio of the internal elastic energy before mesh deletion and the kinetic energy of the rigid body after mesh deletion. The proposed procedure allows efficient simulations of contact problems without neglecting mechanical soundness during contacts. Hopefully, it may contribute to the simulation of large-scale systems with many moving bodies and multiple simultaneous contacts.

References [1] Pfeiffer F, docker C. Multibody Dynamics with Unilateral Contacts. New York: Wiley, 1996. [2] Wriggers P. Finite element algorithms for contact problems. Arch Comput Methods Eng 1995 ;2(4): 1-49. [3] Eberhard P. Kontaktuntersuchungen durch hybride Mehrkorpersystem/Finite Elemente Simulationen (in German). Habilitation. Aachen: Shaker, 2000. [4] Nowottny D. Quadriliteral mesh generation via geometrically optimized domain decomposition. Proceedings of the 6th International Meshing Roundtable, 1997, pp. 309-320. [5] Kubler L. Zur hybriden Simulation von Kontaktvorgangen mit Mehrkorpersystemen undfinitenElementen (in German). Stud-173. University of Stuttgart, Institute B of Mechanics, 1999. [6] O'Rourke J. Computational Geometry in C. Cambridge: Cambridge University Press, 1998.

310

Consistency of damage mechanics modeling of ductile material failure in reference to attribute transferability A. Laukkanen * VTT Manufacturing Technology, Technical Research Centre of Finland, 02044-VTT, Espoo, Finland

Abstract Damage mechanics formulations of fracture phenomena are qualified on the basis of transferability, i.e. how ample is the margin of applicability and the level of precision they can produce with minimal investment to the various intrinsic material parameters. Current work addresses the behavior of the Gurson-Tvergaard-Needleman (GTN) model when subjected to assessment of damage formation in different dimension fracture mechanics test specimens. The model predictions and the overall response are inferred by a comparison with experimental trends and fracture mechanics scaling estimates. The applicability and limitations of the model are considered by assessing the constraint description of the GTN model in the investigated cases, and the scaling of the damage evolution description is found to be principally different to those commonly applied in fracture mechanics pre-eminently in reference to experimental results. Keywords: Damage mechanics; Ductile failure; Constraint; Transferability; Damage evolution

1. Introduction

2. Methods and theory

Ductile fracture and crack propagation in local approach are modeled by void growth models, which are in some cases implemented with void nucleation and coalescence properties. Particularly since the most common solution method of the governing field equations is the finite element method (FEM), the numerical works in relation to ductile crack propagation have adapted the computational cell approach. The issues that have arisen concerning the modeling have been especially the parameters and the generality of the GTN approach. The micromechanical interpretations and the quantitative predictive properties of the GTN model are still under debate. Current study focuses on the quality and nature of the computational estimates for ductile crack propagation, when the investigation is carried out over a range of specimen sizes of single edge notched bend type and the results are interpreted in coincidence to experimental and fracture mechanical assessment procedures of the related micromechanical fracture phenomena.

Numerical simulations were performed incorporating the computational cell approach for ductile crack propagation with finite strains and the GTN model utilizing the WARP3D research code. The associative flow potential and the damage evolution equation of the used implementation of the GTN model were [1-3]:

*TeL +358 (9) 456-5538; Fax: +358 (9) 456-7002; E-mail: anssi.laukkanen @ vtt.fi © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

0 f =

-~ -\-2qif cosh (\-f)s,jGij

'q2crkk\ K 2oro /

(1+^3/'),

(la) (lb)

In Eqs. (la,b) a^ = ^/3siJsi~/2, Stj = dtj - akkGij/3, Gtj are the components of the metric tensor in a Cartesian frame, / is the scalar valued damage variable, qt are the constants introduced in [1], sf' = Jlkfjefj /2i is the effective plastic strain rate, Sij the logarithmic strain tensor, a/y the Cauchy stress tensor, CTQ the flow stress and A the normal distribution parameter for void nucleation as given in [2]. The used numerical formulation was Lagrangian with isotropic incremental plasticity description (A533B steel: yield strength = 400 MPa, strain hardening exponent = 1 0 , computational cell size = 100 iim). Different calibration procedures for the constants of the GTN model were ap-

A. Laukkanen /First MIT Conference on Computational Fluid and Solid Mechanics

140012001000r^ Jii^

T-

A^

/

J"

1J

.•*^

.•^ IF* 25-50 mm

400 200 H

fg = initial void volume fraction f^ = fraction of nucleating voids e^ = average void nucleation strain

-H—-« A T«r-q,=1.5, q^=1, q3=2.25 fQ=0.0015, f^=0.02,£^=0.3 2D Plane Strain

800 H 600

T"

311

-3-4-27 mm -10-10-55 mm^ -10-20-1 GO mm^

110-20 mm^ 10-10 mm^

-25-50-225 mm^

Aa

2

3

4

5

Aa [mm] Fig. 1. Simulated fracture resistance curves for bend specimens of different size. Included are standard validity bounds following ASTM E1737. plied and evaluated and some of these observations are given in the course of current work. Values of the damage variable / were transformed to crack propagation by defining the crack tip to be located at a point within the cell elements where the damage was 3/4 of that required for element extinction (which was taken as / ^ = 0.2). In addition to the GTN model, scaling predictions for overall toughness effects were performed following a small-scale yielding type of a correction for fracture toughness given in form (for similar references, see [4]):

/'

(P\^

0r

(2) d(Afl + ^,)

NCi >>^cro

where the integration domain carries over the near crack tip region satisfying the specified condition for the first principal stress with the value of A, = 1, the criterion overall following a principle of unified stressed volume. In Eq. (2) / is the J-integral, D the size of the computational cell, Afl the crack growth and (0, (p) the complete and specific mappings of oi over the criterion specified volume.

3. Results and discussion The computational fracture resistance curves for bend specimens of different size are presented in Fig. 1. The

GTN model was calibrated for 10-10-55 mm^ 10-20100 mm^ and 25-50-225 mm^ specimens, of which the specimen dependent results are given for the 10-10-55 mm^ calibration. The results of Fig. 1 demonstrate the overly conservative nature of standard set validity bounds in relation to the predictions of the GTN model. To illustrate specimen size related effects to crack tip constraint, the local hoop stress field ahead of a propagating crack is presented in Fig. 2. The results of Fig. 2 present clear specimen dependent differences as a function of applied loading, specimen size and crack propagation. The fracture toughness prediction according to the scaUng criterion of Eq. (2) is given in Fig. 3. The results of Fig. 3 rely on the properties of the near crack-tip conditions as presented in Fig. 2. This fracture mechanics prediction is in very limited agreement with the outcome of the GTN model, as can be seen by comparing the results to those of Fig. 1. As such, the relationships in the scaling treatment of field variables and the response provided by the constitutive equations of the GTN model differ. To illustrate this effect, results pertaining the sensitivity of damage rate on geometric size are presented in Fig. 4. The parameter introduced as abscissa in Fig. 4 can be in approximate sense understood as the equivalent deformation rate of different specimens. It is seen that the differences between specimen sizes start to become significant quite early on when compared to the fracture resistance curves. There is also a moderate scale effect, i.e. the predicted process zone of fracture is larger the larger the specimen. To describe the local response within the process zone, the GTN model predictions can be presented as dependent on the levels attained by the field variables. This is given in Fig. 5.

312

A. Laukkanen /First MIT Conference on Computational Fluid and Solid Mechanics J = constant

q^=1.5.q^=1.q3=2.25 f =0.0015, f =0.02, £^=0.3

lAa approx 0.6 mm

r = distance from crack tip b = initial specimen ligament

Fig. 2. Dependency of near crack tip hoop stressfieldon loading level, crack propagation and specimen size. -H e A A-q,=1.5, q^=1. q3=2.25 f =0.0015,1 =0.02,6=0.3 -3-4-27 mm -10-10-55 mm' -10-20-100 mm' -25-50-225 mm'

^ ^ ^ ^ T l r * * * *

0.025

0.050

0.075

0.100

0.125

0.150

Fig. 3. Scaling fracture resistance criterion for toughness differences between specimens. Fig. 5 illustrates that the near crack tip cells exhibiting the highest states of dilatational deformation do not contribute to the overall damage formation rate with such a difference that the results would pertain size dependencies particularly when the scale effects are noted. Since the damage evolution equation is basically strain-controlled, and even though the material outside the immediate damage zone experiences higher loading conditions in larger specimens, this does not affect the fracture toughness development due to the range of void growth and since the voids grow several orders of magnitude before final failure. As such, the predictions between damage and fracture mechanics have a scale difference due to the differences in parameters and the locality of the applied formulations.

4. Conclusions Numerical and analytical studies were carried out to evaluate the characteristics and properties of the GTN model in relation to consistency and behavior when subjected to different constraint conditions. The results of the study can be concluded as follows. (1) According to damage mechanics based analyses, the standard specified validity restrictions to fracture resistance determination appear overly conservative. (2) Overall, the GTN model provides a characteristic description to experimental results, but the generality of the material model parameters poses restrictions. (3) The predictions of fracture mechanical analysis methods for domain-related effects can be greatly different to those of local approach particularly for ductile rupture, leading to overshoot in constraint corrections.

A. Laukkanen /First MIT Conference on Computational Fluid and Solid Mechanics over domain

313

over cells

1.6x10"

-~«—3-4-27 mm , (Aa+3D) —©—10-10-55 mm^ (Aa+3D) - A - 1 0 - 2 0 - 1 0 0 mm^ (Aa+3D) 25-50-225 mm', (Aa+3D) 3-4-27 mm', x a - all 10-10-55mm', I D . - a l l 10-20-100 mm', ZDj-all 25-50-225 mm', I D - all

10

15

20

J/(CT„D)

Fig. 4. Scale-dependent damage rate of the GTN model results as dependent on crack propagation and level of crack driving force.

Fig. 5. Dependency of damage rate of the GTN model on state of stress-triaxiality and damage. (4) The scale of the interpretation in toughness transferability is greatly different, the GTN model focusing on the immediate near tip damage conditions leading to relative insensitivity to boundary conditions. (5) The strain-controlled formulation of the damage evolution equation in the GTN model makes it more independent of near crack tip region stress fields and also connects to the range of damage formation all the way to element extinction. (6) Considering the fracture resistance predictions, the GTN model predicts a mild ligament-controlled effect for resistance and a greater applicability of miniature specimen testing techniques in harmony with experimental results.

References [1] Tvergaard V. Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 1981;17:389-407. [2] Chu CC, Needleman A. Void nucleation effects in biaxially stretched sheets. J Eng Mater Technol 1980;102:249-256. [3] Tvergaard V. Material failure by void growth to coalescence. Adv Appl Mech 1990;27:83-151. [4] Dodds RH, Tang M, Anderson TL. Numerical modeling of ductile tearing effects on cleavage fracture toughness. In: Kirk Mark, Bakker Ad (Eds), Constraint Effects in Fracture Theory and Applications: Second Volume. ASTM STP 1244. Philadelphia: American Society for Testing and Materials, 1995.

314

A model of deteriorating bridge structures K.H. LeBeau*, S.J. Wadia-Fascetti Northeastern University, Civil and Environmental Engineering Department, Boston, MA 02115, USA

Abstract A structure is a system comprised of components and elements, each having a unique deterioration pattern. The interaction of the degenerating elements influences the system performance. A fault tree model of a structure appropriately represents the element and component interrelationships. This modeling approach offers a qualitative disassembling of the deterioration of the structure revealing the critical failure paths and significant elements. This paper presents a fault tree model of a bridge structure that is useful in the area of bridge management. Keywords: Fault tree; Element interaction; Bridge deterioration; Bridge management; System performance

1. Introduction

2. Current bridge management options

A structure is an assemblage of load-bearing and connective components and elements. For example, a bridge has three main components as shown in Fig. 1: (1) deck, which carries traffic, (2) superstructure, which supports the deck, and (3) substructure, which upholds the superstructure. Each component is comprised of a number of elements. A superstructure has girders, the main structural members, and bearings, which transmit loads from the superstructure to the substructure. Each element deteriorates in a unique manner. For example, a steel girder is susceptible to corrosion causing section loss and compromising its strength. A steel girder is also vulnerable to fatigue which may lead to sudden brittle fracture. On the other hand, a reinforced concrete deck deteriorates through cracking which allows chloride contamination that leads to delamination of the reinforcement and spalling of the concrete. The inter-connectedness of the elements into a system relates the deterioration of one element to the deterioration of another. The leaky joints of a bridge introduce corrosion to the girders and bearings. The malfunctioning of bearings induces stress on the beams and deck. This phenomenon of element interaction accelerates the deterioration of the component, which in turn increases the overall deterioration of the bridge.

The inevitable deterioration of a structure can be controlled through the monitoring of its system performance. In the case of bridges owned by states and municipalities, biannual inspections are conducted that are primarily visual. The numerical inspection data along with expert elicitation serve as input data for bridge management software packages that act as decision-making tools in the prioritizing of maintenance, repair and rehabilitative projects [5,6]. Prediction models that are Markovian in nature and implemented by these tools, are applied on an element-byelement basis at the population level (see Fig. 2). Although attempts are made to incorporate the interrelationships of

* Corresponding author. Tel.: +1 (617) 373-3987; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

f

s«»ERsmucrryi?E *Gk4mik • Braringd

DECK " D«ck MatttHil " JoJntB

SS^STHUCTtmE *MmmmrM • P'mr

V-. D«Ndk

\

IV. ^aiarins

Gk49t

AlHtlnMM«t

C

1

Fig. 1. Components and elements of a bridge system.

^ «-/

315

K.H. LeBeau, SJ. Wadia-Fascetti/First MIT Conference on Computational Fluid and Solid Mechanics

Asstssmtnt of Bridgts at Population Ltv^l PopirtaM^n 09 hM^mi

c, Sutrlnfisl

8^ B^anclBj 84anaB, B^m4B^ 84 e^aridBs 8^ e^andS^ e^amlB^ B^m^B^ B^andB, 8| 81

B2

Bj

B^ B4

8,

B4

Bg 84

Bj

Bj B3

B|

B§

tefend:

Fig. 2. Current bridge management systems. elements through the declaration of environmental effects categories or the linking of element deterioration models with the respective protection systems, this methodology fails to realistically represent element interaction.

3. Fault tree model of element interaction A fault tree is a viable approach to modeling a structure and provides a mechanism that evaluates the failure probability of the system [8,7]. It identifies failure paths and critical elements with the advantage of unveiling logical interrelationships of a system through graphical depiction and Boolean algebra. See Table 1 for explanation of the symbols used in a fault tree. The fault tree presented in this paper takes the failure event. Deterioration of Bridge Performance (F), and qualitatively explains the different failure paths, which are combinations of condition states of different elements (See Fig. 3). The top event is the consequence of deterioration of the three components: deck (Ci), superstructure (C2) or substructure (C3) as described in Eq. (1). F = Ci U C2 U C3

(1)

Further disassembly relates the deterioration of each Table 1 Symbolic notation used in fault trees (Ang et al. [1]) Symbol

Name

Usage

1

Event

Top and intermediate positions

Basic Event

of tree Bottom positions of the tree

0

n

0

OR Gate

Representing the union of two or more events

AND Gate

Representing the intersection of two or more events

Oetefioratioii of bridge peffwmance Deterioration of deck componwt Deterioration of syp^strudure a^mpcment Det^oratiort of sybstmdure component Deterioration of deck material Oet^oratiofl of Joints Deterioration of tjearings Oet^loratkjn of girders Oet^oratloii of abutments/f^ers

Fig. 3. A fault tree model of bridge deterioration. element to each of the three components described in Eq. (1). Poor deck material condition (Bi) directly contributes to the deterioration of a deck. However, the joints (^2), bearings (^3), and girders (^4), also have an influence over the condition of the deck. A leaky joint allows intrusion of water and deicing salts into the concrete promoting decay. Malfunctioning bearings that are 'frozen' induce stress on the deck. Girders with extensive corrosion and section loss are unable to support the loads of the deck and accelerate its deterioration. These interrelationships are reflected in Eq. (2): Ci = BiU (Bi n B2) U (Bi n B3) U (Bi n B4) = Bi

(2)

The laws of Boolean algebra reduce the probability of the deterioration of the deck to the probability of the condition of the deck material itself, rendering the interactions of the bearings, joints and girders with the deck inconsequential [4,2]. Superstructure deterioration can be directly attributed to the condition of the girders (primary structural members). Also, poor deck material (Bi) that is cracked and spalled introduces debris, water and other agents that induce corrosion and fatigue. Leaky joints (B2) are culprits of rust and section loss of girders and corrosion of bearings (B3). Bearings that are badly decayed no longer function properly placing undue stress on the girders. An abutment or pier that is in poor condition (B5), for example a pier cap that is cracked and spalled to the degree that the bearing is undermined, compromises the structural integrity of the superstructure. The element interactions pertaining to the superstructure are shown in Eq. (3):

C2 = ^4 u (B4 n Bi) u (B4 n B2) u (^3 n B4) u (B2 n 53) u (^3 n B5)

(3)

316

K.H. LeBeau, S.J. Wadia-Fascetti/First MIT Conference on Computational Fluid and Solid Mechanics quantitative fault tree provides an objective tool to compare maintenance alternatives based on the probabilities of the basic events reflecting the condition of the elements. A numerical fault tree also suggests element weight factors that a bridge inspector could utilize in determining a component or overall bridge rating.

The laws of Boolean algebra reduce Eq. (3) to the following:

C2 = ^4 u {B2 n 53) u (^3 n Bs)

(4)

The deterioration of the substructure is determined by the condition of the abutments and piers (B5). Also contributing are bearings that have allowed excessive movement of the superstructure resulting in stress on the substructure from the unbalanced load. In addition, water from leaky joints initiates decay of the abutments and piers. These interactions are represented by Eq. (5):

C3 = ^5 u (^3 n Bs) u {B2 n B5) = Bs

4. Conclusions

(5)

Combining the expressions for the three components, the failure event in terms of basic events takes the form:

r = 5i u ^4 u ^5 u (^2 n 53) u (^3 n

B^)

(6)

Eq. (6) can be simplified to Eq. (7). While the interaction between the joints and the substructure is a significant contributor when considering the deterioration of the superstructure, it is redundant information when the structure is considered as a whole.

r = fi, u ^4 u ^5 u (^2 n ^3)

(7)

Therefore, the deterioration of bridge performance is directly attributed to the condition of the deck material, girders and abutment/piers. However, the joints and the bearings together are also significant contributors. This acknowledgement of critical elements is beneficial in the arena of bridge management. Information on the importance of elements is helpful to field inspectors when evaluating the condition of bridges. This demonstration that the deterioration of bridge performance is the union of the conditions of the deck material, girders, abutments/piers and the interaction between the joints and bearings enhances the existing software through the suggestion of links between the element deterioration models to evaluate an overall assessment of the structure. A fault tree also has the advantage of being used in a quantitative aspect to obtain the probabilities of the failure events. Once the probabilities of the basic events are acquired, the Boolean algebra of the tree can be executed resulting in the probabilities of the intermediate events and ultimately, the failure event. The probabilities of the basic events of the fault tree demonstrated in this paper could be elicited from experts, calculated from existing inspection data, or obtained from analytical reliability models. A

Structures are systems made up of components and elements. The deterioration of one element affects other elements. A fault tree can properly model the structure as a system including the various element interactions. The logic of the fault tree following the laws of Boolean algebra reveals the critical failure paths and significant elements. Fault trees applied to bridge structures enhance current techniques in bridge management.

Acknowledgements Support from NSF Award No. CMS-9702656 is appreciated.

References [1] Ang AH-S, Tang WH. Probability Concepts in Engineering Planning and Design: Vol II. New York: Wiley, 1984. [2] Aven T Reliability and Risk Analysis. London: Elsevier, 1992. [3] Bridge inspector's training manual/90. Rep. No. FHWA-PD91-015. Washington, D.C.: Federal Highway Administration, 1991. [4] Dai S-H, Wang M-0. Reliability analysis in engineering applicafions. New York: Van Nostrand Reinhold, 1992. [5] Golabi K, Thompson PD, Hyman WA. Pontis technical manual. Tech. Rep. No. FHWA-SA-94-031. Cambridge, MA: Optima Inc. and Cambridge Systematics, Inc., 1993. [6] Hawk H, Small E. The BRIDGIT bridge management system. Struct Eng Int 1998;8:309-314. [7] Johnson P. Fault tree analysis of bridge failure due to scour and channel instability. ASCE J Infrastruct Syst 1999;5(1):35-41. [8] Sianipar P, Adams T. Fault-tree model of bridge element deterioration due to interaction. ASCE J Infrastruct Syst 1997;3(3): 103-110.

317

Analysis of 2-D elastostatic problems using radial basis functions Vitor M.A. Leitao * Departamento de Engenharia Civil, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

Abstract The work presented here concerns the use of radial basis functions for the analysis of stretching and bending plates. The basic characteristic of the formulation is the definition of a global approximation for the variables of interest in each problem (the deflection for the plate bending problem and the stress function for the stretching plates) from a set of radial basis functions conveniently placed (but not necessarily in a regular manner) at the boundary and in the domain. Depending on the type of collocation chosen, non-symmetric or symmetric systems of linear equations are obtained. Keywords: Collocation technique; Hermite collocation; Meshless; Radial basis functions

1. Interpolation using RBFs Radial basis functions have initially been used by mathematicians working on scattered data fitting and general multi-dimensional data interpolation problems. The basic idea of scattered data interpolation is described in detail in the works of Kansa [1] and Fasshauer [2], for example. An RBF interpolant is assumed in the form of: s{x) = 22^J^^

^ ~ ^j )

(1)

7=1

where 0( x - Xj ) = J(x — Xj)^ -^cj (the multiquadric RBF for example) and Cj 7^ 0 is an adjustable parameter. This equation is solved for the aj unknowns from the system of A^ linear equations of the type: N

s(Xi) = f{Xi) = ^ a ; 0 ( Xi - Xj )

(2)

;=i

where the field to approximate is known at A^ points. 2. PDEs solution using RBFs The application of the interpolation technique described above to the analysis of PDEs arising in computational mechanics was first presented by Kansa [3]. *Tel.: +351 (21) 841-8234; Fax: +351 (21) 849-7650; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Consider an elliptic PDF (hyperbolic and parabolic PDEs are formulated similarly, see Kansa [1]) with interior LI and boundary LB operators domain: (3)

Cu = T

where C^ = [LI LB] and T^ = [FI FB] is the righthand side vector. Assume an approximation Uh(X) to the PDF in the form, that is, by using radial basis functions: N

i^hM = y^aj(p(

X•

)

(4)

7=1

where Xj, fj, j = 1 . . . N, define a data set. The unknown coefficients aj are determined by solving the system of N linear equations formed by applying (that is, by collocating) the operators LI and LB to the approximation defined in Eq. (4) at N selected points. This form of collocation gives rise to an asymmetric system of equations and is therefore known as the asymmetric collocation method or Kansa's approach. Fasshauer [4], motivated by previous works on scattered Hermite interpolation, presented a method to obtain an approximate PDE solution which leads to inherently symmetric and non-singular systems of linear equations. The basic characteristic of this method is that the operators are applied twice for each pair of collocation point-RBF center that is being evaluated.

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V.M.A. Leitdo/First MIT Conference on Computational Fluid and Solid Mechanics

3. Analysis of elastostatic problems In this work, two types of problems are analyzed, namely, plane states and plate bending. For each problem, a global approximation for the appropriate variable (that is the deflection for the plate bending problems and the stress function for the plane states) must be obtained based on the approximate satisfaction of the boundary conditions and of the governing equations. This global approximation is constructed with radial basis functions of the type shown earlier, the multiquadrics, but many more families of RBFs are available. It is also possible, although not considered in this work, to use compactly supported radial basis functions, see Wendland [5]. Tests are carried out on stretching or bending plates subjected to different loading and boundary conditions. Comparisons are made with other results available in the literature.

4. Conclusions The results obtained so far show good agreement with reference solutions. The implementation is very straight-

forward and easy to apply to new problems. For typical problems, the resulting systems of equations are of reduced dimension and, thus, of quite fast solution. This formulation seems very attractive for several other types of problems. Further research on this subject must be pursued to extend its capabilities.

References [1] Kansa EJ. Motivation for using radial basis functions to solve PDEs. http://rbf-pde.uah.edu/kansaweb.pdf, 1999. [2] Fasshauer GE. Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv Comput Math 1999;11(2-3):139-159. [3] Kansa J. Multiquadrics — a scattered data approximation scheme with applications to computational fluid-dynamics — II: Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 1990;19:149-161. [4] Fasshauer GE. Solving partial differential equations by collocation with radial basis functions. In: LeMehaute A, Rabut C, Shumaker L (Eds), Surface Fitting and Multiresolution Methods. Nashville, TN: Vanderbilt University Press, 1997. [5] Wendland H. Piecewise polynomial, positive definite and compacfly supported radial basis functions of minimal degree. Adv Comput Math 1995;4:389-396.

319

An explicit three-dimensional finite element model of an incompressible transversely isotropic hyperelastic material: application to the study of the human anterior cruciate ligament G.Limbert*, M.Taylor Bioengineering Sciences Research Group, School of Engineering Sciences, University of Southampton, Highfield, Southampton, SO 17 IBJ, UK

Abstract A fully three-dimensional (3D) incompressible transversely isotropic hyperelastic material was implemented into a commercial explicit finite element (FE) code in order to achieve realistic numerical simulations of the mechanical behaviour of human Hgaments. As an appHcation, the present study focused on studying the mechanical behaviour of the anterior cruciate ligament (ACL) when the knee is submitted to a passive flexion. The natural pre-stressed state of the ligament was integrated into the FE formulation and its relevance was demonstrated. New insights into the stress and strain distributions within the ACL confirmed experimental observations. Keywords: Finite element; Explicit; Hyperelasticity; Anisotropic; Incompressible; Soft tissue; Ligament; Anterior cruciate ligament; Biomechanics

1. Introduction

2. Materials and methods

The ACL is essential for the stability of the knee by preventing anterior displacement of the tibia relative to the femur and hyperextension of the joint, and is the most commonly injured ligament of the body. In order to gain a better understanding of the mechanisms of injury within the ACL it is necessary to assess the magnitude and the distribution of stress within this ligament. Ligaments are dense connective tissues consisting of parallel-fibred collagenous tissues embedded in a solid matrix of proteoglycans. The preferred orientation of the collagen fibres induces the transversely isotropic symmetry of the ligament. Very few 3D FE continuum models of human knee ligaments have been developed [2,5,7,8,12]. Isotropic models fail to capture the essential anisotropy characteristics and lead to unrealistic results [6]. To overcome these shortcomings, an incompressible transversely isotropic hyperelastic material model was implemented into an explicit FE code. A 3D FE analysis of the ACL was performed in order to simulate its behaviour during a passive knee flexion.

2.1. Constitutive modelling and finite element implementation

* Corresponding author. Tel.: +44 (2380) 597665; Fax: +44 (2380) 593230; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Due to their natural composite structure, ligaments can be accurately described by the Theory of Fibre Reinforced Composites at finite strains, Spencer [10]. This postulates the existence of a strain energy function ^ which depends on /i, h and /s, the first three principal invariants of the right Cauchy-Green deformation tensor C and a fourth invariant, AHQ, corresponding to the stretch in the fibre direction characterized by the unit vector no. This unit vector can be defined globally or pointwise and X^^ is defined by Eq. (1): (X„j2 = no • (Cno) = h

(1)

where U is an invariant firstly introduced by Ericksen and Rivlin [4]. The mechanical behaviour of ligaments is assumed to be governed by a function ^m, representing the mechanical contribution of the matrix (Mooney-Rivlin material) and by a function f{X) representing the contribution of the

320

G. Limbert, M. Taylor/First MIT Conference on Computational Fluid and Solid Mechanics

Resultant force within the ACL

200

No initial stretch at full extension - Initial stretch of 1.09 at full extension •Wascheretal. (1993)MIN - Wascher et al. (1993) MAX

175 150 q^

u

125 -

U<

O ^^-H

c

a

100-

31/3

75-

pi:

50-

OJ

• ^

2 5 - \r.J 0-

j

cij^^^S

0

10

30

45

60

90

110

120

Angle of flexion (degrees) Fig. 1. Resultant force within the ACL during a passive knee flexion. Numerical results are presented along the experimental measurements made by Wascher et al. [11] on a sample of 18 cadaveric knees. fibres to the strain energy function: ^m(/l, /2, /3) = Cdh - 3) + C2(/2 - 3) + gih)

which, after development, leads to: (2)

S = 2 ( C , + / i C 2 ) l - C 2 C - h ^ 4 T ^ n o < ) no + pC dX

(3)

where 1 represents the second-order unit tensor, C~^ the inverse of C, p is the hydrostatic pressure appearing as a kinematic reaction to the incompressibility constraint and '(g)' denotes the tensor product. The constitutive model was implemented into the commercial explicit FE code PAM-CRASH™ (PAM Systems International, Rungis, France). To prevent the appearance of the 'locking phenomenon', well known in nearly incompressible FE analysis, it was decided to use uniform reduced integration 8-node brick elements with hourglass control that offer a good compromise between computational cost and accuracy.

^

V l / ( / , , / 2 , / 3 , A ) = Vl/^ + / ( X )

Ci and C2 are material parameters and gil^) is a simple penalty function used to enforce the kinematic condition /3 = 1, corresponding to total incompressibility:

8(h) ^

"'(i-')

+

a2(h-lf

(4)

where ofi and 0^2 are determined from material parameters. Collagen fibres do not support a significant compressive load along their longitudinal direction and structures that are composed of mostly collagen are prone to buckle under very small compressive forces (Eq. (5a)). The tensile stress-stretch relation for collagenous tissues such as ligaments and tendons can be well approximated by an exponential function (Eq. (5b)). These observations guide the choice of the following function /(A.) [12], such that: dX

= 0,

A < 1

(5a)

C3 (g^4(X-l) _ 1)

dx

X

X> 1

(5b)

Ci, C2, C3 and C4 are material parameters. The second Piola-Kirchhoff stress tensor S is defined as:

ac

(6)

(7)

2.2. Application to the modelling of the human ACL 2.2.1. Geometrical model of the ACL The 3D geometry of the insertion sites of the ACL were obtained from an experiment performed on a cadaveric knee specimen. Several markers were placed along the contours of the ACL at the tibial and femoral insertion sites in order to track their 3D location during the passive knee flexion tests. The geometrical model includes the non-planar insertion areas and respect the natural orientation of the fibres. 2.2.2. Finite element model of the ACL The solid volume the ACL was reconstructed with the knee at full extension and was meshed with 8-noded hexahedron elements using Patran® v9.0 (MSC, Palo Alto, CA,

G. Limbert, M. Taylor/First MIT Conference on Computational Fluid and Solid Mechanics

Full extension

10 degrees of flexion

60 degrees of flexion

90 degrees of flexion

30 degrees of flexion

110 degrees of flexion

321

45 degrees of flexion

Full flexion

Fig. 2. Deformations of the ACL (medial view) along a passive knee flexion.

USA). The mesh was constituted of 3297 elements and 3784 nodes. 2.2.3. Mechanical properties Material data of the ACL were extracted from literature, Pioletti [7], and adjusted to fit the anisotropic hyperelastic model. 2.2.4. Initial stress-initial stretch The ACL has no stress free state at any of the knee flexion angles [3], and the resultant force within the ACL at full extension is highly variable according to the experimental studies performed by Wascher et al. [11] (2130 N (18 ACLs)) or Roberts et al. [9] (100 N ( i 14)).

The capacity to apply initial stretch was implemented into PAM-CRASH™ by performing a special treatment of the deformation gradient [13]. 2.2.5. Boundary conditions As described in Section 2.2.1, the experimental kinematics tests were input into the FE model and used as boundary conditions. These passive flexion-extension tests were done with the knee in the neutral position (no internal or external rotation) for a flexion angle ranging from 0 to 120°. The tibia was fixed and the femur was free to move in the flexion plane. The nodes of the femoral insertion area were displaced and those of the tibial insertion area were fixed.

322

G. Limbert, M. Taylor/First MIT Conference on Computational Fluid and Solid Mechanics

3. Results

References

After prestressing the ligament by applying a initial stretch of 1.09, corresponding to a resultant force of 130 N, the maximal von Mises stresses (4.15 MPa) are localized in the anterolateral part of the ACL. Between 10 and 90° of flexion the maximum stresses do not vary significantly (4.15-5.45 MPa). At 30° of flexion, the stresses are maximal in the midsubstance of the anteromedial band of the ACL. This was observed experimentally by Yamamoto et al. [14]. At all the flexion angles, the maximal stresses were never located at the tibial insertion site on the lateralposterior part of the ACL as previously reported in numerical studies of isotropic models [6,7]. Applying an initial stretch affects significantly the resultant force within the ACL (Fig. 1) and it was shown, for the first time, to the best of our knowledge, that a FE model of the ACL can predict qualitatively the experimental force measurements. High stress values were found at full flexion, essentially due to the large sagittal plane rotation of the femoral insertion area (Fig. 2). The present ACL model also shows that the anterior-medial band of the ACL carries the maximum load during the flexion cycle, as observed experimentally by Butler et al. [1].

[1] Butler DL, Guan Y, Kay MD, Cummings J, Feder S, Levy M. Location-dependent variations in the material properties of the anterior cruciate ligament. J Biomech 1992;25:511518. [2] Daniel W. Three dimensional orthotropic viscoelastic finite element model of human ligament. Presented at Fifth US National Congress on Computational Mechanics (USNCCM99), Boulder, Colorado, USA, 1999. [3] Diirselen L, Claes L, Kiefer H. The influence of muscle forces and external loads on cruciate ligament strain. Am J Sports Med 1996;23:129-136. [4] Ericksen J, Rivlin R. Large elastic deformations of homogeneous anisotropic materials. J Radon Mech Anal 1954;3:281-301. [5] Hirokawa S, Tsuruno R. Three-dimensional deformation and stress distribution in an analytical/computational model of the anterior cruciate ligament. J Biomech 2000;33:10691077. [6] Limbert G, Taylor M. Three-dimensional finite element modelling of the human anterior cruciate ligament. Influence of the initial stress field. In: Middleton J, Jones ML, Pande GN (Eds), Computer Methods in Biomechanics and Biomedical Engineering, Vol 3, Gordon and Breach Science Publishers, in press. [7] Pioletti D. Viscoelastic properties of soft tissues: application to knee ligaments and tendons, Departement de Physique, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 1997. [8] Puso M, Weiss J. Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation. J Biomech Eng 1998;120:62-70. [9] Roberts CS, Gumming JF, Grood ES, Noyes FR. In-vivo measurement of human anterior cruciate ligament forces during knee extension exercises, presented at 40th Orthopaedic Research Society, February 21-24, New Orleans, USA, 1994:84-115. [10] Spencer A. Continuum Theory of the Mechanics of FibreReinforced Composites. New York: Springer, 1992. [11] Wascher DC, Markolf KL, Shapiro MS, Finerman GA. Direct in vitro measurement of forces in the cruciate ligaments. Part I: The effect of multiplane loading in the intact knee. Am J Bone Joint Surg 1993;75:377-386. [12] Weiss J, Maker B, Govindjee S. Finite element implementation of incompressible transversely isotropic hyperelasticity, Comput Methods Appl Mech Eng 1996;135:107-128. [13] Weiss JA, Maker BN, Schauer DA. Treatment of initial stress in hyperelastic finite element models of soft tissues. Presented at ASME Bioengineering Conference, 1995, BED-29:105-106. [14] Yamamoto K, Hirokawa S, Kawada T. Strain distribution in the ligament using photoelasticity. A direct application to the human ACL. Med Eng Phys 1998;20:161-168.

4. Discussion It is now widely accepted that the ACL is composed of two main fibre bundles. These bands have different lengths and mechanical properties. This issue has not been addressed in the present model, but may have a significant influence on the pattern of deformation, and thus, the stress distribution within the ACL. Due to the reduction in length of the posterior side of the ACL from 0 to 60° of flexion, the isotropic models of the ACL encountered in the literature, generate high compressive stresses at the tibial insertion site in the posterior part of the ACL, instead of producing buckling of the ligament. The model proposed here was able to address this issue. Prestressing of the ligament model was demonstrated to be essential for realistic FE simulations as shown by replicating the resultant force curve as a function of flexion angle.

Acknowledgements The authors would like to thank Dr. S. Martelli, Dr. V. Pinskerova, and Mr. M.A.R. Freeman for providing the experimental data. The company ESI Group is acknowledged for its technical support.

323

Simulation of the explosive detonation process using SPH methodology G.R. Liu'''*, M.B. Uu\ K.Y. Lam^ Z. Zong^ ^Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore ^ Institute of High Performance Computing, 89C Science Park Drive, 02-11/12, Singapore Science Park 1, Singapore 118261, Singapore

Abstract This paper presents the successful application of smoothed particle hydrodynamics (SPH) methodology to simulate the complicated explosive detonation process. The combination of meshless and Lagrangian nature inherent in the SPH methodology makes it very attractive in treating large deformations and large inhomogeneities in the detonation process. The detonation process of a one-dimensional TNT slab is simulated and presented in details as a numerical example. Numerical results show the SPH method can give good predictions for both magnitude and form of the detonation wave. Keywords: Smoothed particle hydrodynamics; Detonation; Detonation wave; Meshless method; Numerical method; Computational fluid dynamics

1. Introduction The detonation process involves a violent chemical reaction which converts the original high energy explosive into gas at very high temperature and pressure, occurring with extreme rapidity and releasing a great deal of heat. During the detonation process, a very thin reaction zone divides the domain into two inhomogeneous parts and produces large deformations in the detonation gas. Though many attempts [1,2] have been made in modeling the detonation process, simulation of such complicated progress is generally still difficult for traditional numerical methods. Traditional Lagrangian techniques such as finite element methods are capable of capturing the history of the detonation events associated with each material particle. It is, however, practically difficult to use since the severely distorted mesh may result in very inefficient small time step, and may even lead to the breakdown of the computation. Traditional Eulerian techniques, such as finite difference or finite volume method, can well resolve the problem due to the large deformations in the gross motions, but it is very difficult * Corresponding author. Tel.: +65 874-6481; Fax: +65 7791459; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to analyze the flow details because of the lack of history and smearing of information as the mass moves through the fixed-in-space Eulerian mesh. Recently growing interests have been focused on meshless methods as alternatives for traditional numerical methods to simulate solid or fluid flow problems [3-5]. Among the meshless methods, smoothed particle hydrodynamics (SPH) methodology [6,7] is unique in the computational fluid dynamics. SPH method is a meshless, particle-oriented method of pure Lagrangian nature. Since its invention to solve astrophysical problems in three-dimensional open space, SPH has been extensively studied and extended to dynamic response with material strength as well as dynamic fluid flows with large deformations. In SPH methodology, smoothed particles are used as interpolation points to represent materials at discrete locations, so it can naturally obtain the history of the fluid particles, and thus easily trace material interfaces, free surfaces and moving boundaries. The meshless nature of SPH methodology overcomes the difficulties due to large deformations since SPH uses particles or points rather than mesh as computational frame to interpolate. These nice features of SPH make it fairly attractive in simulating the detonation process.

G.R. Liu et ai /First MIT Conference on Computational Fluid and Solid Mechanics

324

This paper presents the successful appUcation of SPH method to simulate the explosive detonation process. A one-dimensional TNT slab detonation process is presented in details with consideration of the effects of different particle resolutions. The one-dimensional test does not lose generality, since numerical analyses of detonation process are generally based on the assumption that explosive charge is in the spherical shape and detonates from the charge center. The simulation results of the numerical example show that the present SPH methodology can successfully simulate the detonation process at fairly good accuracy with comparatively less computational effort.

2. Numerical method-SPH basics In SPH methodology, the state of a system can be represented by a collection of arbitrarily distributed particles while forces are calculated through interparticle interactions in a smoothed fashion. For a function / , multiplying / with the smoothing kernel function, and then integrating over the computational domain can approximate its function value at a certain point. This is the concept of kernel approximation in the SPH methodology. Using {/> to denote the kernel approximation of / yields {/U)>

-I

f{x')W(x

-x\h)dx'

(1)

where W(x — x\h) is the smoothing kernel function that must satisfy some requirements. In this paper, the cubic spline function [8] is used, jc and x' are the position vectors at different points, h is the smoothing length representing the effective width of the smoothing kernel function. According to the concept of particle approximation in SPH, Eq. (1) is estimated by summing over all the nearest neighboring particles that are within the region of the influence for a given particle / at a certain instant. Using fiXi) = fi gives ifi)

s(^)

X fj

X

Wij

(2)

where, rrij and pj are the mass and density of particle j ; N is the total number of particles that have effect on particle /. In SPH methodology, the approximation of spatial derivatives can also be obtained in the same way as the function / itself, and is usually devised through integration by parts to transform the operation on function / into operation on the smoothing kernel as follows (3)

The detonation process can be simulated by Eulerian equation coupled with the suitable equation of state. The concepts of kernel approximation and particle estimation

lead to the following generally used SPH equations to model the detonation process.

^

=

i2'^j(v.-Vj)-^.W, ;=i

-E"4^J + | + n„)v,

Wii

7=1

(4)

??=^E-45+a+"'^i("'-''^)-^'^'^ Dt PI PJ ;=i

DXi

_

^ "DT ~ ""' where, p, v, u and p are density, velocity vector, internal energy and pressure of the particle; 11/; is the standard artificial viscosity in SPH which can be used to stabihze the numerical scheme, prevent particle penetration and capture shock waves [8], -acijfiij + Pill

n, =

Vij • Xij

<

0

Vij • Xij

>

0

pij

0 hjjVij

'Xij

IJ^ij

Cij =

-^{Ci + C y ) ,

(5)

(6)

\{p. + Pj) Vij = Vi - Vj,

hij = \ {hi + hj)

(7)

where a, p, rj are constants that are set 1, 10 and 0.1/i respectively; c, and Cj represent the speed of sound for particle / and j . 3. Numerical simulation of one-dimensional TNT slab detonation The simulation of one-dimensional TNT slab detonation does not lose generality since early analyses based on the assumption of spherical charge detonating from the charge center can be also simplified into one dimension. Due to its particle nature, SPH methodology can be easily extended to three dimensions and can simulate various detonation scenarios, e.g. arbitrary charge shape, different detonation orientation, multiple charges and so on. In the numerical test, a 0.1 m long TNT slab is detonated along one end. Shin and Chisum [2] ever simulated the same case by using coupled Lagrangian-Eulerian analysis. The same assumptions and parameters as [2] are used in this simulation for the sake of comparison. The TNT is assumed to behave as a Jones-Wilkins-Lee (JWL) high energy explosive with the equation of state.

'-('-i?)'-'—(-f)

(8)

where the parameters are A = 3.712 x 10^^ Pa, B =

G.R. Liu et al. /First MIT Conference on Computational Fluid and Solid Mechanics 0.0321 X 10^^ Pa, Ri = 4.15, R2 = 0.95, co = 0.30, ri = p/po, po is the reference density of 1630 kg/m^, E is the initial specific internal energy of 4.29 x 10^ J/kg. The detonation velocity of 6930 m/s is used. In Shin's simulation, the wall boundary conditions were used to forbid material transport from everywhere. While in this simulation, the symmetric condition is used. This makes the detonation of the 0.1 m long slab from one end to the other end equivalent to the detonation of a 0.2 m long slab from the middle point to one end. Before detonation, particles are evenly distributed along the slab. The initial smoothing length is one and a half times the particle separation. After detonation, a plane detonation wave is produced. According to the detonation velocity, it takes around 14.4 |xs to complete the detonation to the end of the slab. In order to investigate the effects of different particle resolutions, analyses are carried out using 250, 500, 1000, 2000 and 4000 particles along the slab. Figs. 1-3 show the

1 1

1 i 1tI 1i1

xlO

2.5

0.02

0.04 0.06 0.08 Distance along the TNT slab (m)

325

0.1

Fig. 3. Velocity profiles along the TNT slab during the detonation process.

C-J pressure

2

^1.5 3 U)

,^

CO

a> "^ 1

0.5

n

0

0.02

/

'>

0.04 0.06 0.08 Distance along the TNT slab (m)

-

0.02

0.1

Fig. 1. Pressure profiles along the TNT slab during the detonation process. 2400 2300 2200

1i1Im,ii1i

"^2100 [

^'

^1900 [ 1800 1700 1600

(

1500 I 1400 I 0

/

/ / /

^2000 I

0.02

(V

/

y/

0.04 0.06 0.08 Distance along the TNT slab (m)

0.1

Fig. 2. Density profiles along the TNT slab during the detonation process.

0.04 0.06 0.08 Distance along the TNT slab (m)

0.1

Fig. 4. Peak pressures at 1 |xs intervals with the complete pressure profiles at 7 and 14 |xs for different particle resolutions. pressure, density and velocity along the slab at 1 |JLS interval from 1 to 14 |xs by using 4000 particles. Fig. 4 shows the peak pressures at 1 |xs with the complete pressure profiles at 7 and 14 |xs for different particle resolutions. The dashed lines in Figs. 1 and 4 represent the experimentally determined C-J detonation pressure, which is, according to the Chapman and Jouguet's hypothesis, the pressure at the tangential point of the Hugoniot curve and the Rayleigh line, and represents the pressure at the equilibrium plane at the trailing edge of the very thin chemical reaction zone [1]. For this one-dimensional TNT slab detonation problem, the C-J pressure is 2.1 x 10^^ N/m^. It can be seen from Figs. 1 and 4, with the process of the detonation, the detonation pressure converges to the C-J pressure. The detonation shock is resolved within several smoothing lengths. More particles along the slab result in sharper pressure profiles with bigger peak pressures. Figs. 1 and 4 are quite accurate and comparable to the results obtained by Shin. Though the number of particles is more than the number of elements

326

G.R. Liu et al /First MIT Conference on Computational Fluid and Solid Mechanics

that Shin used, the resulted detonation shock fronts are much sharper.

4. Conclusions This paper presents the application of a pure Lagrangian meshless method to the simulation of explosive detonation process with a numerical example of one-dimensional TNT slab detonation. The method is based on the smoothed particle hydrodynamics methodology, which is robust, easy to apply, and computationally efficient. The Numerical results show the presented method can give good predictions for both magnitude and form of the detonation wave.

References [1] Mader CL. Numerical Modeling of Detonations, University of California Press, 1979.

[2] Shin YS, Chisum JE. Modeling and Simulation of Underwater Shock Problems Using a Coupled Lagrangian-Eulerian Analysis Approach. Shock Vib 1997;4:1-10. [3] Atluri SN, Zhu T. A New Meshless Local Petrov-Galerkin (MPLG) Approach in Computational Mechanics. Comput Mech 1998;22:117-127. [4] Belytschko T, Lu YY, Gu L. Element-Free Galerkin methods. Int J Num Methods Eng 1994;37:229-256. [5] Liu GR, Gu YT. A point interpolation method for twodimensional solids. Int J Numer Methods Eng [6] Lucy L. A numerical approach to testing thefissionhypothesis. Astron J 1977;82:1013-1024. [7] Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Not R Astron Soc 1977;181:375-389. [8] Monaghan JJ. Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 1992;30:543-574.

327

MFree2D®: an adaptive stress analysis package based on mesh-free technology G.R.Liu*,Z.H. Tu Center for Advanced Computations in Engineering Science, c/o Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Abstract MFreelD is a numerical package based on mesh-free technology for stress and strain analysis in solids and structures. It consists of three processors: MFreePre, MFreeApp and MFreePost. MFreePre is a preprocessor to define and model a problem; MFreeApp performs computations and gives the numerical solutions which are then fed to MFreePost for visualization. These three processors are integrated using a graphical platform, enabling an analysis to be completed at one run; they can also be separated to work independently when necessary. The main features of this package are being automatic, adaptive, accurate and easy to use. It has remarkable value in speeding up design process, simplifying computational modefing and simulation, and reducing manpower cost for mesh creation. Keywords: MFree2D; Element-free method; Meshless method; Adaptivity; Stress analysis; Numerical package

1. Introduction Conventional mesh-based numerical methods have been well developed and seen great success in engineering applications. They are well commercialized and dominate the market of numerical analysis. However, these methods have inherent limitations: they are bothered by mesh-related difficulties when dealing with problems of extremely large deformation and crack propagation. To overcome this, various meshless methods have been developed, e.g. the element-free Galerkin (EFG) method, meshless local PetrovGalerkin method (MLPG) and point interpolation method (PIM) [1]. They are formulated entirely based on a set of scattered nodes and hence eliminate the mesh-related problems. As the nodes are not constrained using grid, these methods are also very appealing for adaptive analysis. Moreover, there is remarkable ease and flexibility in the pre- and post-processing with meshless methods. As a result, the mesh-free technology has tremendous potential applications in industry and engineering. However,

* Corresponding author. Tel.: +65 874-6481; Fax: +65 7791459; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

presently there is no commercial numerical package based on mesh-free technology in the market. The Center for Advanced Computations in Engineering Science (ACES) is spearheading the endeavors in developing and commercializing the mesh-free technology. One product of the effort is MFree2D, a commercial numerical package based on the mesh-free technology, for stress and strain analysis in solids and structures. This paper describes briefly the processors of MFree2D and the crucial aspects in implementation of mesh-free technology. Numerical testing is also carried out to verify the validity of the package.

2. Processors of MFree2D MFree2D is programmed based on an object-orientated approach for software reusability, extendibility and maintainability. There are three processors: MFreePre, MfreeApp and MFreePost. MFreePre is a preprocessor to formulate the input required by MFreeApp; the latter performs computations and yields the output results which are then fed to MFreePost for post processing. The processors are integrated in a graphical platform equipped with detailed instructions and help information; they can also be separated to undertake their work independently.

328

G.R. Liu, Z.H. Tu/First MIT Conference on Computational Fluid and Solid Mechanics

2.1. MFreePre and MFreePost MFreePre is used to define and model a numerical problem for meshless analysis, from creation of geometrical model and meshless model to definition of material model, initial and boundary conditions, and solution control. One salient feature is that troublesome and time-consuming mesh generations are no longer necessary. In discretization of a problem domain, users do not need to work on the geometrical model part by part, but simply set a desired average nodal density for the entire domain. MFreePre automatically identifies the geometry and discretizes the domain using scattered nodes. This saves significantly manpower cost from mesh creation as engineers usually spend much more time on mesh generations than other things in a numerical simulation. MFreePre allows an analysis to be customized with its open environmental setting system while providing default settings for new users. It is safe to say that MFree2D has fewer requirements for users than many of the existing mesh-based packages. The ultimate version of MFreePre will not be limited to being a preprocessor for MFreelD, but also be a convenient and powerful tool for computer-aided design. MFreePost provides a convenient graphical user interface for visualization of numerical solutions, e.g. initial and deformed domain displaying, field contouring, vector viewing, section projecting and surface and curve plotting. In addition, it allows animation of dynamic process or refinement process. 2.2. MFreeApp MFreeApp is the meshless code for numerical analysis. It reads and checks the input data from MFreePre, conducts analysis and generates result files for MFreePost. The main considerations in the design of MFreeApp are: accuracy of solution, effectiveness, robustness and efficiency of numerical procedures, and availability of computing resources. The major aspects in MFreeApp consist of: interpolation of field variables, integration of governing equations, enforcement of boundary conditions, solution of nonlinear equations and implementation of adaptivity. For interpolation of field variables, several meshless schemes are provided for selection, e.g. the moving least square method (MLS), point interpolation method (PIM) and partition of unity method, with MLS being the default. The built-in interpolation basis comprises monomial functions and radial functions, choice of which depending on the problem type and user's requirements. Users can also code their own basis functions via a user subroutine interface. In terms of weight function, the exponential, cubic spline and quadratic spline weight functions are provided. There is also a subroutine interface for this. In construction of shape functions, there are occasions that inverse of moment matrix does not exist or is ill-valued; MFreeApp cures

this problem by redistributing nodes locally. A relay model is developed for construction of shape function in highly irregular domains [2]. For integration of the variational form of governing equations, there are basically two approaches: Gauss integration and nodal integration. The former requires a background mesh and generates much better results than the latter. Therefore, the present implementation uses the Gauss integration approach based on triangular background meshes by default. Enforcement of essential boundaries is a crucial topic in meshless methods. As shape functions constructed by meshless schemes usually do not possess the Kronecker delta function properties, special techniques, e.g. collocation, Lagrange multiplier, penalty [3] and constrained moving least square methods [4] have been developed to solve this problem. Selection of these techniques depends on the requirement of accuracy, efficiency and effectiveness. By default, the penalty method is used. For nonlinear problems, the system equations are discretized in an incremental form and the load increments are determined automatically. A modified Newton-Raphson iteration solver is used to solve the nonlinear equations. Several nonlinear material models are incorporated, e.g. the Von-Mises elastoplasticity and the Duncan-Chang EB model for soil materials. Users can also define their own material model. For problems with singularity and stress concentration, adaptive analysis is usually required to capture these characteristics. The adaptive procedure incorporated into MFree2D uses three types of error estimates: stress projection error estimate, strain gradient error estimate and cell energy error estimate [5]. The first is constructed based on the difference between the projected stress and raw stress. The projected value is calculated in a way similar to that in FEM. The second utilizes the fact that gradients of stresses and strains may be calculated throughout the problem domain with a high accuracy. Its drawback is that it necessitates computations of the second derivatives which are quite expensive. The third examines error in each background cell and uses cell energy error as the basic measure. To achieve high efficiency in domain refinement, local domain refinement techniques [5] are developed to obviate refinement of global domain. The iterative solvers incorporated into MFree2D comprise the Gauss-Siedel method, Gauss-Jacobi method and conjugate gradient method. The refinement process is terminated when the desired accuracy is achieved or the solution is convergent.

3. Sample computations MFree2D are tested with numerous sample computations, among which three examples are presented here. The first is a square plate with a hole at the center subjected to a

329

G.R. Liu, Z.H. Tu/First MIT Conference on Computational Fluid and Solid Mechanics

^ ™ 2 000a00e+001

I rc4?8i?+ooi

__=;>_ 64 nodes 3<

" O - 145 nodes

I

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_

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^

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-

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Fig. 3. Distribution of stress Oxx in a rectangular plate with two close cracks.

^^^4.,^

1 .

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. 4 3;ll700e~001

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1

15

20

25

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1

30

35

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45

19806-001

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Fig. 1. Distribution of stress Oxx at x = 0 for the square plate with a hole problem.

^732e^000 10496-^000 ^366e+000

|

1010200e*000

i683e*000

.!.694986e^000

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Fig. 4. Distribution of principal stress 02 in a working gear.

!.064558e*000 • 7493448+000

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.434130e-^000 ;118916e+000 l,03T020e-001

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i

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Fig. 2, Distribution of stress o^x at the third refinement step (204 nodes) for the square plate with a hole problem.

unit horizontal tensile traction. The second is about the interference between two close cracks in a rectangular plate. The third is a gear loaded by distributed forces along the teeth. The materials properties for all problems are Young's modulus £^ = 3.0x 10^ and Poisson's ratio v = 0.3. For the first problem, there is stress concentration around the hole and the adaptive refinement procedure is invoked to capture the stress field. The numerical solution is very accurate at the third refinement step (Figs. 1 and 2). In the second problem, the two crack-tips are most refined as they are singular points. The stress field (Fig. 3) is depicted with a sharp resolution, clearly showing the interference between the two cracks. For the gear problem, the stress patterns (Fig. 4) reveal the most vulnerable locations under working conditions. These sample computations have shown that MFreelD is reliable, effective and efficient.

MFreelD is a numerical package based on mesh-free technology with its own graphical pre- and post-processors. With MFree2D, there is no mesh generation and no meshrelated difficulty. As a result, the package has remarkable value in speeding up design process, simplifying computational modeling and simulation, and saving manpower cost from mesh creation. Moreover, the desired accuracy of a solution can be achieved through automatic adaptive analysis. Rigorous numerical testing has shown its reliability, effectiveness and flexibility. It is believed that this package will evolve to be a numerical tool alternative to the existing FEM packages.

References [1] Liu GR, Gu YT. A point interpolation method. Int J Numer Methods Eng, accepted for publication. [2] Tu ZH, Liu GR. A relay model for meshless approximations in domains with irregular boundaries. Int J Numer Methods Eng, submitted for publication. [3] Liu GR, Yang KY A penalty method for enforcing essential boundary conditions in element free Galerkin method. Proceedings of the 3rd HPC Asia, Singapore, 1998, pp. 715-721. [4] Liu GR, Yang KY. A constrained moving least square method in meshless methods, submitted for publication. [5] Tu ZH, Liu GR. An adaptive procedure based on background cells for meshless methods. Comput Methods Appl Mech Eng, submitted for publication.

330

Energy estimates for linear elastic shells C. Lovadina* Dip. di Ingegneria Meccanica e Strutturale, Universita di Trento, Via Mesiano 77, 1-38050 Trento, Italy

Abstract The Koiter shell problem is considered. The asymptotic behavior of the shell energy (as the thickness tends to zero) is investigated by means of the Real Interpolation Theory. A result concerning the percentage of the total elastic energy that is stored in the bending part is also provided. Keywords: Shell; Elastic energy; Interpolation theory; Inhibited shell; Problem order; Intermediate state

1. The shell problem

^ e (0, 1) and p e[l, +oo] (cf. Lions et al. [3] and Bergh et al. [4], for instance).

When the Koiter shell problem with thickness s (cf. [1]) is considered, one is led to solve the variational problem 2. Main results

Find Ue eV such that (1) Above a"'{', •) is the membrane bilinear form, a^(-, •) the bending bilinear form and V is the admissible displacement space, which also takes into account the kinematical boundary conditions imposed to the structure. Moreover, / represents the loads applied to the structure, and we will suppose that / e V\ V being the topological dual space of V. We will not detail the precise form of the bilinear forms involved in Eq. (1), for which we refer to Ciarlet [1], for instance. We only recall that «'"(•, •) and a^(-, •) are both y-continuous and positive semidefinite. Furthermore, the sum <2'"(-, •) -\- a^(-, •) is V-coercive. In this note, we will suppose that a^'iv^v) =0 <=^ v = 0, i.e. we will consider the so-called inhibited shells (cf. Sanchez-Hubert et al. [2]). It follows that the membrane bilinear form defines a norm over V. We thus set W as the completion of V with the norm a'^iv, v)"^ := u vv- We notice that, by construction, V C W, with continuous and dense inclusion. As a consequence, for the dual spaces the continuous and dense inclusion W C V holds true, so that it makes sense to consider the real interpolation spaces {W, V')e,p for each

We are interested in studying the asymptotic behavior (as s -^ 0) of the elastic energy, defined by Eie) := sa^'iue, Ue) + s a (w^, Us).

(2)

To begin, we consider the energy functional of order p defined by E{£, p) := s^Eis) = s^+^a^'ius, M.) + s^+^a^Us, u,), (3) and we will say that the Koiter Problem (1) is of order a if a = 'm{{p I E{s, P) € L^(0, 1)}.

(4)

It is not hard to show that if a Koiter problem is of order a, then 1 < a < 3 (cf. Blouza et al. [5]). The following Theorem, proved in Baiocchi et al. [6], establishes a strict connection between the problem order (and therefore the behavior of E(6)) and the regularity of the datum / (regularity which is measured by means of the spaces {W\ V%.oc)Theorem 1. Fix / € V and consider the problem Find Ue e V, such that

*Tel.: 4-39 (461) 882524; Fax: +39 (461) 882599; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

sa^'iu,, v) 4- s^a\u,,

v) = (/, v)

Vi; G V.

(5)

C. Lovadina / First MIT Conference on Computational Fluid and Solid Mechanics 1. If f € {W, yOe,oo for some 0 < 6 < 1, then problem (5) is of order a given by (y = inf{2^ + l :

fe(W\V%,,o,

Moreover, for the function R(s) defined by s^a^ius, Us) ^7T^' E(£) we have the following

(7)

Theorem 2. Consider problem (5) and suppose that there is an a such that there exist -\-0Q > \im.e-^oE{s, a) > 0, and linie-^o s"'^^a^{Ue, Ue) > 0. Then it holds lim/?(£) = ^ ^ - .

intermediate

0 < ^ < l } . (6)

2. / / / ^ (W, V')e,oo for any 0 < 0 < I, then problem (5) is of order a = 3.

^^ ^ R(s)'-=

to perform a detailed analysis of the so-called states (cf. Piila et al. [7]).

331

(8)

The above result partly answer in a positive way a question raised by Sanchez-Palencia. Our Theory can be used

References [1] Ciarlet PG. Introduction to Linear Shell Theory. Paris: Series in Applied Mathematics. Gauthier-Villars, 1998. [2] Sanchez-Hubert J, Sanchez-Palencia E. Coques Elastiques Minces. Proprietes Asymptotiques. Paris: Masson, 1997. [3] Lions JL, Peetre J. Sur une classe d'espaces d'interpolation. Pubbl IHES 1964;19:5-68. [4] Bergh J, Lofstrom J. Interpolation Spaces: An Introduction. Berlin: Springer, 1976. [5] Blouza A, Brezzi F, Lovadina C. A New Classification for Shell Problems. Pubblicazioni lAN-CNR 1999; no. 1128. [6] Baiocchi C, Lovadina C. A shell classification by interpolation, submitted for publication. [7] Piila J, Leino Y, Ovaskainen O, Pitkaranta J. Shell deformation states and the finite element method: a benchmark study of cylindrical shells. Comput Methods Appl Mechan Eng 1995;128:81-121.

332

On the finite element analysis of flexible shell structures undergoing large overall motion I. Lubowiecka^'*, J. Chroscielewski ^ ^Department of Structural Mechanics, Technical University of Gdansk, Faculty of Civil Engineering, ul Narutowicza 11/12, 80-952 Gdansk, Poland ^Department of Bridges, Technical University of Gdansk, Faculty of Civil Engineering,, ul. Narutowicza 11/12, 80-952 Gdansk, Poland

Abstract The general, dynamically and kinematically exact, six-field theory of branched shell structures, extended to nonlinear problems of shell dynamics also involving the large overall motion is discussed. The generalized Newmark algorithm on the proper orthogonal group SO(3) with Newton's iterations is proposed. The numerical simulations of the behavior of the elastic T-shaped shell structure in forced and free large overall motion are presented. Keywords: Nonlinear dynamics; Shell structure; Large rotation

1. Introduction Various formulations of nonlinear dynamics of flexible shell structures undergoing finite deformations have been discussed in [1-6]. The aim of this report is to develop a time-stepping algorithm for transient dynamic analysis of branched shells using the six-field shell model, and to perform numerical simulations of the behavior of a branched T-type elastic shell in forced and free large overall motion. The complete set of equations describing an arbitrary motion of the branched shell structures was derived in [6,8,9,11]. There are many time-stepping schemes proposed in the literature, where stability and accuracy are most discussed properties of the algorithms (see e.g. [10,12]) for structural dynamics and [3] for shell dynamics). In our shell model, containing g G S0(3) as an independent field variable, standard time-stepping schemes cannot be directly applied. The algorithm used here is based on ideas suggested in [13-15] and developed in [6]. We propose in the iterative process, an exact calculation scheme of the incremental, relative rotation vector, whose material representation plays a crucial role. * Corresponding author. Tel.: -h48 (58) 347-2238; Fax: +48 (58) 347-1670; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The theoretical considerations are limited here to the weak formulation of the shell problem (Section 2) and the modified Newmark algorithm (Section 3).

2. Weak formulation of the six-field nonlinear theory of shells The general motion of the irregular shell structure in time t can be described by two fields ([6-9,11]): the displacement vector field u{x,t) = y{x,t) — x, where y(x,t) is the position vector of the deformed reference surface X e M, and the proper orthogonal tensor field Q(x,t) representing the mean rotary motion of the shell cross sections. In this report, we consider shell structures with geometric irregularities. We assume that the kinematic fields y(x,t) and Q{x,t) are continuous during the motion, and yr(xr, t) = y{x, t)\r, Qri^r, 0 = Q(x, Olr, where the curve JCr € r C M represents common parts of boundaries of any regular element of the reference surface. We assume simple constitutive kinetic relations for the translational p(x,t) = mov = PQHQV and rotational m(x,t) = IQ(O = {pohl/l2)(o momentum vectors suggested in [10], where po(x) is the initial mass density, ho{x) is the initial shell thickness, v{x,t) = yix,t) = u(x,t) is the velocity vector, and o){x,t) is the angular velocity vector

/. Lubowiecka, J. Chroscielewski / First MIT Conference on Computational Fluid and Solid Mechanics (ada> = QQ^, ad : JE"^ -> so(i)) in the spatial representation. For hyper-elastic shells, there exists a 2D strain energy function W{ep,Kp,x) of the shell strain vectors Kp = ad~H6,^ 6^)- Then defined by Sp = y,p-Qx,p, the constitutive relations of the shell material are given by n^ = dW/dep, m^ = dW/dK^, where n^(x,t)md m^(x,t) are the internal stress and couple resultant vectors, respectively. When expressed in the weak form, the initial-boundary value problem for the branched shell-like structure can be formulated as follows [6]. Given the external resultant force and couple vector fields f(x,t) and c{x,t) on X e M \ F, n*(x,t) and m*(x,t) along dMf, firixj) and mr(x,t) along the curve F c M, and the initial values Uo(x), Qo(x), Uo(x), QQ(X) at r = 0 find a curve u(x,t) = (u(x,t), Q(x,t)) e VA on the configuration space C(M, E^ x 50(3)), VA C C, such that for any continuous, kinematically admissible virtual vector field w{x) = (v(x), w(x)) e VA(M, E^ X E^) we have G[u

= 11 [fnoi) • V + IQCO ' w] da M\r

(f-v-\-c-w)da

-

(n* V ^m* w)ds dMf

- \ {Pr •'^r +rnr • Wp) ds = 0, r

Since external forces have physical sense only in spatial representation, a weak form of the momentum balance equations is formulated in this representation. Furthermore, the linearized dynamic equations (like in statics [7-9]) are written in an instant iterative configuration y^'li G U. It eliminates the relation Y„ gi+i • typical for e.g. [13-15], beT^ii) SO(3) -^ TQJO(3) cause here Y^'|j gj^'l^ = 1. Velocities and accelerations at different time steps can be directly added only in the material representation. Therefore, temporal approximation of them is done in material representation. Let [tnJn+i] C I = [0,T] C M+ be a typical time interval, with At = tn+\ —tn, and let the data available from converged solutions at the previous time step tn e I he: y„ = (Un, Qn) - generaHzed displacement, ^„ = (M„, (o„) - generaHzed velocity (material representation). (2) Bn = (Un,Sin) — generalized acceleration (material representation). The basic problem concerning the discrete time-stepping update may be formulated as follows. Having the data Un, w„ and a^ from the previous step at time r„ e / C M+, we search for y^+i, ^n+i and a„+i in the next moment tn+i = r„ + Ar e / C M+, so that they should be consistent with the problem equations and numerically stable. With known Aw^^'l^ = (Aw^7/\ Aw^^/^) we update the rotation tensor in the spatial representation as follows.

M\r

-II

333

(1)

where Vp = v\r,Wr = w | r - I n ( l ) i t i s implicitly assumed that the kinematic boundary conditions u(x,t) = u*(x,t) and Q(x, t) = Q\x, t) are satisfied on dMd = dM\dMf. The solution of (1) is achieved by an iterative procedure which reduces the problem to a sequence of solutions of linearized problems. Each linearized problem is formulated at discrete values of both temporal and spatial variables. The main difficulty of the solution procedure is associated with the structure of the configuration space C{M, E^ X 50(3)) involving 50(3). As a result, the solution procedure requires special techniques for temporal and spatial approximation, parameterization, and accumulation of the 50(3)-valued fields.

3. Modified Newmark algorithm The procedure applied for integration on 50(3) uses a single time-stepping implicit scheme through generalization of the classical Newmark method with Newton's iterative strategy.

fi«,=exp(AWl7")elT^ AWiV-;>=ad(A«;iV-/'),

0)

QZ - Qn We calculate then in the material representation the total incremental rotation vector Aw^i> = ad-i(AW^^^),

exp(AW^i^) =

filfil^i(4)

corresponding to the transition from y„ to y^'j^ configuration. According to the Newmark approximation, the angular velocity and acceleration in the material representation are: ^

P(Atr

[Awi'-^^ - Af(0, - (Atf

(i - fi) a , ] , (5)

,0-1) .

After transformation of (5) into the spatial representation .0-1)

n+l

^n +1 '

,0-1)

^O-DjjO-l)

(6)

it is possible to write the problem equations in the spatial representation as well from which a new correction Sw^^l^ = (^M^'IP 5w;^'|i) of the generalized displacements is calculated.

334

/. Lubowiecka, J. Chroscielewski / First MIT Conference on Computational Fluid and Solid Mechanics

4. Numerical simulation Following the twisted ribbon problem discussed in [16], a similar example for the branched T-shape structure is considered (Fig. 1) with the following data: L = 50, H = \0, B = 14.0112, ho = 0.25, a = 90°, E = 2x 10^, y = 0.3, poho = 1. The cross-section parameters B/H = 1.40111855316 have been matched so that J,, = JXX Uzx = 0). Two concentrated forces are applied at the points a and b, (Fig. 2) by the ramp time function from 0 to lOOCX) in 1 s and back down to 0 in another 1 s. After the 2 s, the shell is free from external loading and moves freely

Fig. 3. Deformed configurations in time. in the space. The shell is discretized by (B -\- H) x L ^ (4-h2) X10 = 60 16-node Lagrange (displacement/rotation) shell elements containing six degrees of freedom in each node (totally 3534 degrees of freedom). The full integration of the element matrices is used and the time integration step is Ar = 0 . 1 s for the modified Newmark scheme with p = 0.25, y = 0.5. Fig. 2 indicates the motion trajectories of points a and b. Several stages of the motion at specified time instants depicted in Fig. 3 show very large deformations of the

Fig. 1. Geometry of twisted T-section panel.

240^ / = 26y».

220200-

(a)

/=

(b)

180-

23.

\ ^^^V^^^-'

t^20rA

160-

V^^^'^^'^-AP''^ t =

140120-

t ==

60t

\

p

/ \

8

()

\ \ \

= 5^

A

10000-

^

1 2 i

IP

gb)

z,w

/ = 2_ '^\ r-0 (a)

0-20-

^

t

80-

20-

'/v"^^^

/=14\

100-

40-

17

\x,u r == 0 ^

-20

' r 0

1

20

^~T

—T

40

[

60

I

1

80

1

r

n—'—r"^—r- T

1

1

1

.

^

y

1

100 120 140 160 180 200 220^240

Fig. 2. Motion trajectories of points a and b.

/. Lubowiecka, J. Chroscielewski / First MIT Conference on Computational Fluid and Solid Mechanics

335

vjuvjmju

/ 250000 ~

pot kin

200000 -

+E,

E

k m

pot

|l50000 ^

J

100000 50000 -

n

L

0.0

1

2.5

'

1

5.0

'

1

'

7.5

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'

1

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12.5

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15.0

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1

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Fig. 4. Energy histories of twisted T-section panel.

flexible shell structure. The energy plots given in Fig. 4 contain time history curves of the potential, kinetic and total energies, the latter being conserved throughout the simulation time. This example indicates the complex 3D motion of the shell involving large elastic deflections and multiple turns of the shell. In this problem, the central mass point should always remain at the same straight line through the simulation time, which we have obtained.

Acknowledgements This work was supported by the Polish State Committee for Scientific Research under Grants 7 T07A 041 18.

References [1] Simo JC, Rifai MS, Fox DD. On a stress resultant geometrically exact shell model. Part VI: Conserving algorithms for non-linear dynamics. Int J Numer Methods Eng 1 1992;34:117-164. [2] Simo JC, Tamow N. A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int J for Numer Methods Eng 1994;37:2527-2549. [3] Kuhl D, Ramm E. Constraint energy momentum algorithm and its application to non-linear dynamics of shells. Comput Methods Appl Mechan Eng 1996;136:293-315. [4] Madenci E, Barut A. Dynamic response of thin composite shells experiencing non-linear elastic deformations coupled with large and rapid overall motions. Int J Numer Methods Eng 1996;39:2695-2723. [5] Brank B, Briseghella L, Tonello N, Damjanic FB. On non-linear dynamics of shells: implementation of energymomentum conserving algorithm for a finite rotation shell model. Int J Numer Methods Eng 1998;42:409-442.

[6] Chroscielewski J, Makowski J, Pietraszkiewicz W. Large overall motion of flexible branched shell structures. In: Ambrosio JAC, Kleiber M (Eds), Computational Aspects of Nonlinear Structural Systems with Large Rigid Body Motion. NATO ARW, Pultusk (Poland), July 2-7, IDMEC Lisboa 2000:201-218. [7] Chroscielewski J, Makowski J, Stumpf H. Genuinely resultant shell finite elements accounting for geometric and material non-linearity. Int J Numer Methods Eng 1992;35:6394. [8] Chroscielewski J. Family of C^ finite elements in six parameter nonlinear theory of shells (in Pofish). ZN PG BL 1996;540(LIII):1-291. [9] Chroscielewski J, Makowski J, Stumpf H. Finite element analysis of smooth, folded and multi-shell structures. Computer Methods in Applied Mechanics and Engineering 1997;141:1-46. [10] Bathe KJ. Finite Element Procedures in Engineering Analysis. Englewood CUffs: Prentice-Hall, 1982. [11] Libai A, Simmonds JG. The Nonlinear Theory of Elastic Shells, 2nd ed. Cambridge: Cambridge University Press, 1998. [12] Kuhl D, Crisfield MA. Energy-conserving algorithms in non-linear structural dynamics. Int J Numer Methods Eng 1999;45:569-599. [13] Simo JC, VuQuoc L. On the dynamics in space of rods undergoing large motions a geometrically exact approach. Comput Methods Appl Mechan Eng 1988;66:125-161. [14] Cardona C, Geradin C. A beam finite element non-finear theory with finite rotations. Int J Numer Methods Eng 1988;26:2403-2438. [15] Simo JC, Wong KK. Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum. Int J Numer Methods Eng 1991;31:19-52. [16] MacNeal RH, Harder RL. A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Design 1985;1:3-20.

336

A numerical investigation of chaotic motions in the stochastic layer of a parametrically excited, buckled beam Albert C.J. Luo* Department of Mechanical Engineering and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA

Abstract In the paper, the chaotic motions in the stochastic layer of a simply supported, geometrically nonlinear, planar buckled rod under a parametric excitation is investigated through the energy spectrum method. The resonance-characterized chaotic motions for the parametrically excited buckled beam are simulated through the symplectic scheme. Keywords: Buckled beam; Stochastic layer; Parametric vibration; Energy spectrum

1. Introduction The chaotic motion in a harmonically excited elastic beam was investigated [1,2] through the Melnikov method, the perturbation approach and Lyapunov exponent method. In 1995, the dynamical potential for the nonlinear vibration of cantilevered beams was discussed in [3]. In 1999, Luo and Han [4] presented the analytical conditions for chaotic motion of a periodically driven rod through the modified Chirikov overlap method. Luo et al. [5] developed an energy spectrum method for numerical predictions of resonance in the stochastic layer. The chaotic motion in the stochastic layer in parametric systems is still unsolved to date. In 2000, Luo [6] investigated the chaotic motions in the stochastic layer of the parametrically excited pendulum. The resonant characteristic between the parametrically and periodically driven systems is completely different. Therefore, in this paper, the chaotic motion in a parametrically excited beam will be investigated.

As in [4], the governing equation of motion for the buckled beam is derived: pAw -\-^w-^{P

-]- PQCOSQt)w^^^(^

~ 9/ / ^5c ^ ) 0

/

/

- EAw,,, — / u;^^ dx -f £•/ w;,;cxxx l l - — 0

h^l^) 0

-^2WrxxW^j,jcW,

The boundary conditions for a simply supported rod in Fig. 1 is u = w = u .y = w .r = 0,

aix = 0,1.

(2)

Based on the foregoing equation, we assume w(x,t) = Emit) sin(m7tx/l) for a specified m, and application of the Galerkin method to Eq. (1) yields / + [Oil + Qo cos Qt]f -h a2f

^ 0,

(3)

2. Equations of motion Consider the planar, nonlinear vibration of a simply supported, initially straight, slender rod experiencing large deformation, and this rod is subjected to an axially compressive force P -\- PoCosQt at one end, shown in Fig. 1. *Tel.: -hi (618) 650-5389; Fax: -hi (618) 650-2555; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

P-tPQCOsQt

Fig. 1. A nonlinear, planar rod subjected to the compressive force P + Pocos^t.

A.C.J. Luo/First MIT Conference on Computational Fluid and Solid Mechanics

I

337

o

> c W

Excitation Frequency Q

Displacement

Fig. 2. The energy spectrum for the parametrically driven beam (go = 0.2 and ai = -1.0, ^2 = 1.0).

where fit)

= mnFmiO/l,

ai = {Pmcr -

a^ = (EA ~ 4P^,r + a = pAl/m7T,

P)/a, (4)

P)/4a,

Pmcr = {mTtfEI/f,

2o = P^/a

For the buckled beam with specific m, the buckled case Qfi < 0 and ^2 > 0 is investigated.

3. Energy spectrum In 1999, Luo et al. [5] developed the energy spectrum method to predict the resonance in the stochastic layer. To achieve the energy spectrum, the Poincare mapping section for the parametrically driven beam is defined as ^ = { ( / f e ) , / ( % ) ) I satisfying Eq. (1), 2Nn and t^ = —— -\-to, A^ = 0, 1 , . . . ,

(5)

where fit^) = fn, f{tN) = /N and f(to) = /o, f(to) = fo dtt = to are the initial conditions. The Poincare map is: IJ -^ S. The energy for each Poincare mapping point of the parametric beam, the minimum and maximum energies are: 1 1 7(A^) ^o^^^^ = ; ^ / ^ - ; 7 « i / ^ + > 2 / ; , ^max — niax [//({''>) and £ „ , i n = m i n j / / o ^ ^ ' j

(6)

Fig. 3. The stochastic layer of the parametrically driven beam {QQ = 0.2, ^ = 3.68 and ai = -1.0, ^2 = 1-0). Using the above definition, the maximum and minimum energy spectra are shown in Fig. 2 for QQ = 0.2 and (ofi = —1.0, Qf2 = 1.0) in Eq. (3). The critical values of excitation frequency are Qcr approximately 2.76, 3.72 and 4.44 for the ( 2 : 1 ) , ( 3 : 1 ) and ( 4 : 1 ) inner resonance and ^cr approximately 2.60, 3.84 and 4.40 for the ( 2 : 1 ) , ( 4 : 1 ) and ( 6 : 1 ) outer resonance. If ^ is greater than such critical values, the corresponding resonance will not involved in the resonant layer. The chaotic motion in the separatrix is simulated with Q approximately 2.68 very close and less than ^^f-^^ and ^^^"^-^K as illustrated in Fig. 3. The ( 3 : l)-inner and ( 4 : l)-outer resonances are embedded in the separatrix band.

References [1] Holmes PJ, Marsden J. A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch Ration Mech Anal 1981;76:135-166. [2] Maewal A. Chaos in a harmonically excited elastic beam. ASME J Appl Mech 1986;53:625-631. [3] Berdichevsky VL, Kim WW, Ozbek A. Dynamics potential for nonlinear vibrations of cantilevered beams. J Sound Vib 1995;179(1):151-164. [4] Luo ACJ, Han RPS. Analytical predictions of chaos in a nonlinear rod. J Sound Vib 1999;227(3):523-544. [5] Luo ACJ, Gu K, Han RPS. Resonant-separatrix webs in the stochastic layers of the Duffing oscillator. Nonlin Dyn 1999;19:37-48. [6] Luo ACJ. Resonance and stochastic layers in a parametrically excited pendulum. Nonlin Dyn 2000.

338

Limit analysis usingfiniteelements and nonlinear programming A.V.Lyamin, S.W.Sloan* Geotechnical Research Group, Department of Civil, Surveying and Environmental Engineering, University of Newcastle, NSW 2308, Callaghan , Australia

Abstract This paper describes finite element formulations of the plastic limit theorems using nonlinear programming. The methods are based on simplex finite elements, provide rigorous upper and lower bounds on the collapse load, and are applicable to problems in one, two and three dimensions. Keywords: Limit analysis; Finite elements; Nonlinear programming

1. Introduction The upper and lower bound theorems of Drucker et al. [1] are powerful tools for predicting the limit load of many problems in mechanics. They are based, respectively, on the notions of a kinematically admissible velocity field and a statically admissible stress field. A kinematically admissible velocity field is simply a failure mechanism in which the velocities satisfy both the velocity boundary conditions and the flow rule, whilst a statically admissible stress field is one which satisfies equilibrium, the stress boundary conditions, and the yield criterion. The applicability of the bound theorems can be greatly enhanced by combining them with finite elements. The resulting formulations, which are called finite element bound techniques or rigid plastic finite element methods, inherit all the benefits of the finite element approach and are thus very general.

2. Lower bound formulation Examples of 2D lower bound finite element formulations include those of Lysmer [2] and Sloan [3]. These procedures both use linear triangular elements with a linearised yield surface to give a linear programming problem. The nodal unknowns are the stresses and the objective function of the optimisation problem corresponds to the * Corresponding author. Tel.: +61 (2) 4921-6059; Fax: +61 (2) 4921-6991; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

collapse load (which is maximised). To avoid nonlinear constraints on the former, the yield surface is linearised using an internal polygonal approximation and each nonlinear yield inequality is replaced by a series of linear yield inequalities. In addition to these inequalities, linear equality constraints on the nodal unknowns must also be enforced to satisfy equilibrium over each triangle, equilibrium along the stress discontinuities, and the stress boundary conditions. A key feature of lower bound formulations of this type is the incorporation of statically admissible stress discontinuities between the sides of adjacent triangles, which greatly improves the accuracy of the lower bound solution. Lower bound finite element formulations based on linear programming have proved successful for a wide range of 2D stability problems (see, for example, Sloan and Assadi [4] and Ukrichton et al. [5]), but are unsuitable for 3D geometries because the linearisation of a 3D yield surface inevitably generates huge number of inequalities. A superior alternative for formulating a lower bound scheme is to combine linear finite elements with a nonlinear programming solution procedure. This approach uses the yield criterion in its native nonlinear form and, apart from their additional geometric complexity, 3D stress fields present no special difficulty. One effective nonlinear programming formulation, which uses linear stress triangles in 2D and linear stress tetrahedra in 3D, has been developed recently by Lyamin [6] and Lyamin and Sloan [7]. This method has the same features as the linear programming schemes described above but is much faster and more general. After assembling the various objective function coefficients and equality constraints for the mesh, and imposing

A.V Lyamin, S.W. Sloan/First MIT Conference on Computational Fluid and Solid Mechanics the nonlinear yield inequalities at each node, the lower bound formulation of Lyamin [6] and Lyamin and Sloan [7] leads to a nonlinear programming problem of the form Maximise

c^a

Subject to

Aa = b fi(a)<0

(1) i=

{l,...,N}

where c is a vector of objective function coefficients, cr is a vector of unknowns (nodal stresses and possibly element unit weights), c^a is the collapse load, A is a matrix of equality constraint coefficients, b is a vector of coefficients, fi is the yield function for node /, and A^ is the number of nodes. For large scale finite element grids this optimisation problem may be very large, with hundreds of thousands of constraints, and its inherent sparsity must be fully exploited. One very effective solution strategy, described in Lyamin [6], uses a two-stage quasi-Newton algorithm to solve the Kuhn-Tucker optimality conditions directly. This approach is most attractive since it rarely requires more than about 30 iterations, regardless of the problem size.

3. Upper bound formulation In addition to the lower bound methods described above, a variety of upper bound finite element procedures have been proposed. The bulk of these assume a linearised yield surface and find a kinematically admissible velocity field by solving a linear programming problem (see, for example, Sloan and Kleeman [8]). Although effective for 2D problems of moderate size, this type of approach is unsuitable for large-scale 3D analysis. Recently, Lyamin and Sloan [9] have proposed an upper bound finite element method based on nonlinear programming which is fast, robust and general. In their formulation plastic flow is permitted to occur in the continuum as well as along specified velocity discontinuities. Linear elements are used to model the velocity field and each element is also associated with a constant stress field and a single plastic multiplier rate. Fortunately, it is unnecessary to include the plastic multipliers as variables, even though they are used in the derivation of the formulation. This is because the final optimisation problem can be cast in terms of the nodal velocities and element stresses alone. To ensure kinematic admissibility, flow rule constraints are imposed on the nodal velocities, element plastic multipliers, and element stresses. In addition, the plastic multiphers are constrained to be non-negative, the velocities must match the specified boundary conditions, and the element stresses must satisfy the yield criterion. To facilitate plastic deformation in the discontinuities, additional discontinuity variables are required to describe their velocity jumps. These variables are related to the nodal velocities via a set of equality constraints and are used to

339

compute the dissipated power. For a prescribed set of velocities, the finite element formulation works by choosing the set of velocities, element stresses and discontinuity variables which minimises the dissipated power. This power is then equated to the power dissipated by the external loads to yield a strict upper bound on the true limit load. After assembling the various objective function coefficients and constraints for the mesh, the upper bound formulation of Lyamin and Sloan [9] leads to a nonlinear programming problem of the form Maximise

a^Bu -h c^u + cjd

on a

Minimise

a^Bu -h c^u + cjd

on (u, d)

Subject to

AuU -f Add = b fi(
(2) i=

{l,...,E}

d>0 where u is a global vector of unknown velocities, d is a global vector of unknown discontinuity variables, a is a global vector of unknown element stresses, Cu and Cd are vectors of objective function coefficients for the nodal velocities and discontinuity variables, Au and Ad are matrices of equality constraint coefficients for the nodal velocities and discontinuity variables, B is a global matrix of compatibility coefficients that operate on the nodal velocities, b is a vector of coefficients, ft is the yield function for element /, and E is the number of finite elements. In Eq. (2), the objective function a^Bu + c^u + cjd corresponds to the total dissipated power, with the first term giving the dissipation in the continuum, the second term giving the dissipation due to fixed boundary tractions or body forces, and the third term giving the dissipation in the discontinuities. As discussed in Lyamin and Sloan [9], this optimisation problem can be solved using a two-stage quasi-Newton scheme which is very similar to that used for the lower bound case. Even for large scale problems, it rarely requires more than 50 iterations.

4. Application To illustrate an application of the upper and lower bound methods, we consider the collapse of a long square tunnel as shown in Fig. 1. In this example, failure is caused by

plane strain L_ (J -J

cohesion = C friction angle = 0' unit weight = y

Fig. 1. Plane strain square tunnel in cohesive frictional soil.

340

A.V. Lyamin, S.W. Sloan/First MIT Conference on Computational Fluid and Solid Mechanics )h

a„ = r = 0

^^^^

>-

^

C-^B/2

'

11

c1

/^ 1 o|

II 1

T = n ^WfflfffW B

H bM)^'I)^^(M<M)a)^^^^)^

B

1=0 ^ l l l l l l l l l l l l ^ ^

^1

^1 ^i1

II 1

s1

Ix

1U

u = V = 0 Quantity nodes triangular elements extension elements discontinuities

Quantity nodes triangular elements discontinuities

Lower bound 13692 4452 88 6763

Upper bound 13248 4416 6568

(ii)

(i)

Fig. 2. FE meshes for (i) lower bound and (ii) upper bound limit analysis of square tunnel with C/B = 1. the action of gravity and is resisted by the internal tunnel pressure o^. If downward velocities, downward body forces, and compressive tunnel stresses are defined as positive, the power dissipated by the external loads may be written as = -o"? / Vn^A + y / V dV

(3)

where i;„ is the outward normal velocity on the tunnel face. At is the area of the tunnel face, and V is the volume of material undergoing plastic deformation. Eq. (3) can also be expressed in terms of the dimensionless load parameters —Ot/c' and yB/c' according to

These two parameters provide a concise means of summarising the stability of the tunnel. For a given soil profile and tunnel geometry, the limiting value of -Ot/c' may be expressed conveniently in terms of the quantities yB/c', (j)' and C/B.

Typical lower and upper bound meshes for the case of C/B = \ are shown in Fig. 2i and ii, respectively. The lower bound grid has stress discontinuities at all shared element edges and generates 13, 692 x 3 = 41, 076 unknown nodal stresses. The extension elements around the mesh perimeter ensure the stress field can be extended throughout the semi-infinite layer without violating equilibrium, the stress boundary conditions or the yield conditions, and thus provide a rigorous lower bound on the true collapse load. The lower bound analysis is performed by fixing values of yB/c\ (j)' and C/B and then solving Eq. (1) to find a statically admissible stress field which maximises a uniform tensile stress over the tunnel face. The upper bound grid contains velocity discontinuities at all shared element edges and gives rise to 13, 248 x 2 = 26, 496 unknown nodal velocities, 4, 416 x 3 = 13, 248 unknown element stresses, and 6, 568 x 4 = 26, 272 unknown discontinuity variables (66,016 unknowns altogether). To perform an upper bound analysis, values of yB/c', (p' and C/B are fixed, the tunnel face is subject to the loading condition f^ VndA = 1, and Eq. (2) is solved to find a

A.V. Lyamin, S.W. Sloan/First MIT Conference on Computational Fluid and Solid Mechanics Table 1 Stability bounds on -Gt/c' for square tunnel with C/B = 1

cp' 10° 2(f 30° 40° 45°

yB/c' = 0

yB/c' = 1

yB/c'

LB

UB

LB

UB

LB

UB

1.87 1.70 1.44 1.13 0.98

1.93 1.76 1.48 1.15 0.99

0.99 1.07 1.03 0.86 0.74

1.07 1.15 1.08 0.87 0.75

0.08 0.44 0.60 0.55 0.48

0.19 0.53 0.66 0.57 0.49

=2

yB/c' = 3 LB -0.84 -0.21 0.16 0.25 0.22

UB -0.71 -0.10 0.22 0.27 0.24

341

yB/c' = 0,(1)' = 10° and a coarse mesh, is shown in Fig. 3. This plot indicates a 'roof and sides' mode of failure. For deep tunnels, collapse is typically governed by a simple roof mechanism that does not propagate through to the ground surface.

5. Conclusions Two powerful new procedures have been described for computing lower and upper bound limit loads using finite elements and nonlinear programming. The methods furnish the limit load directly, have an inbuilt error indicator, and can be used for both 2D and 3D geometries.

References

Fig. 3. Deformation pattern for tunnel with C/B = \, yB/c' = 0 and (f)' = 10°.

kinematically admissible velocity field which minimises the dissipated power (minus the power dissipated by gravity). This quantity is then equated to -atf^VndA = -Gt to find the limiting tunnel pressure. Lower and upper bounds on —Ot/d for the case of C/B = 1 are shown in Table 1. In general, the finite element bounds bracket the exact collapse pressure to within a few percent, with the best solutions being obtained for cases where yB/c' < 1. On SL Pentium III 800 MHz processor, the lower bound method used an average of 90 s of CPU time, while the upper bound analyses used an average of 150 s. The collapse deformation pattern obtained from an upper bound calculation, for the case with C/B = 1,

[1] Drucker DC, Greenberg W, Prager W. Extended limit design theorems for continuous media. Q J Appl Math 1952;9:381389. [2] Lysmer J. Limit analysis of plane problems in soUd mechanics. J Soil Mech Found Div ASCE 1970;96:1311-1334. [3] Sloan SW. Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Methods Geomech 1988;12:61-67. [4] Sloan SW, Assadi A. The stabihty of tunnels in soft ground. In: GT Houlsby, AN Schofield (Eds), Predictive Soil Mechanics: Proceedings of Peter Wroth Memorial Symposium, Oxford: Thomas Telford, London, 1992, pp. 644-663. [5] Ukrichton B, Whittle AJ, Sloan SW. Undrained limit analyses for combined loading of strip footings on clay. J Geotech Geoenviron Eng ASCE 1998;124(3):265-276. [6] Lyamin AV. Three-Dimensional Lower Bound Limit Analysis Using Nonlinear Programming, PhD Thesis, University of Newcastle: Australia, 1999. [7] Lyamin AV, Sloan SW. Lower bound limit analysis using nonlinear programming. Int J Numer Methods Eng, submitted for publication. [8] Sloan SW, Kleeman PW. Upper bound limit analysis with discontinuous velocity fields. Comput Methods Appl Mech Eng 1995;127:293-314. [9] Lyamin AV, Sloan SW. Upper bound limit analysis using linear finite elements and nonlinear programming. Proceedings 5th International Conference on Computational Structures Technology (CST2000), Leuven: Belgium, 2000, pp. 131145.

342

On degenerated shell finite elements and classical shell models M. Malinen^'*, J. Pitkaranta^ ^^ Laboratory for Mechanics of Materials, Helsinki University of Technology, P.O. Box 4100, FIN-02015 Hut, Finland ^Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 Hut, Finland

Abstract We report on recent advances in the direction of mathematical error analysis of the shell finite elements based on the so-called 'degenerated 3D approach'. The key idea of the analysis is to build a connection from the degenerated 3D FEM to the dimensionally reduced shell model of Reissner-Naghdi type, so that the error analysis can be done within the classical 2D shell theory. In particular, we consider here the bilinear degenerated 3D FEM showing a nearly equivalent formulation within the classical 2D framework. Having found the connection between the models, we examine how parametric error amplification, or locking, effects are treated within the lowest-order formulation. The results of preliminary error analysis of the MITC4 element are reported. Keywords: Finite element; Shell; Shell model; Locking; Degenerated shell finite element

1. Introduction In the current engineering practice, the computational modelling of shells is largely dominated by the so-called degenerated 3D approach. However, there are few studies where the degenerated shell finite elements are analysed mathematically. In this short paper, we report on recent results [7-9] in this direction. The classical 2D shell theories, which were developed mostly before finite element methods, provide a natural framework for finite element formulations for shells. However, this 'classical' approach is not taken so often in engineering computations, but the finite element methods are typically based on geometrically incompatible approaches where a rather crude geometry approximation is involved. The very first finite element models of this type were based on the use of flat plate elements, but later on, this technique was largely superseded by the degenerated 3D approach [1] where the usual dimension reduction hypotheses and the effects arising from isoparametric geometry approximation are coupled in a rather unclear manner. As the resulting numerical shell model has not been well understood, to tackle the error analysis has not been a straightforward task. Chapelle and Bathe [5] have recendy studied the shell model underlying the degenerated 3D formulation. Yet, the full explana* Corresponding author. E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

tion of the approach appears to still be missing, especially in case of the lowest-order elements, some of which can show relatively robust and locking-free behaviour in well-known benchmark problems, cf. e.g. [3]. Somewhat mysteriously, the straightforward higher-order generalizations of such elements have not proven to perform as well. The key idea of our study is to reformulate the degenerated 3D FEM nearly equivalently within the classical 2D shell theory in such a way that the resulting finite element formulation does not involve geometry approximation. It appears that this implicit relation in general exists. An obvious benefit of the geometrically compatible interpretation is that the finite element error analysis with well-established principles is made possible. In particular, we study here the bilinear degenerated 3D FEM showing the connection to the classical 2D shell model of Reissner-Naghdi type (cf. [10]). We assume the framework of the classical linearized elasticity theory, while we do not presume any specific shell geometry. A more detailed discussion than what is possible here can be found in the full-length papers [7-9].

2. The two-dimensional shell model To begin with, we consider dimensionally reduced shell models as formulated in the classical shell theory (cf. [10]).

M. Malinen, J. Pitkdranta / First MIT Conference on Computational Fluid and Solid Mechanics Our primary aim is to introduce a simplified 2D shell model where shell geometry is contained as localized geometric parameters. We assume that the shell is of constant thickness d and that material is homogeneous and isotropic with the Young modulus E and the Poisson ratio v. In order to derive an expression for the deformation energy of the shell, we assume that the middle surface of the shell is represented by charts of type s = xe^ + yey + z(x, y)ez.

x=ai,

y = ot2, z = z{oLi,a2).

(2.2)

Further, let the xj-plane be tangent to the middle surface at the origin, so that the quadratic Taylor approximation of z may be written as 1 -Aa\ + Caxa2 + -Bal +

0{hl),

(2.3)

where ha, < diam(<w) and A, B,C are coefficients related to the curvatures of the surface at the origin. Starting from the assumed representation of the exact middle surface, the deformation energy of the ReissnerNaghdi shell model (cf. [10]) over co can be simpUfied as (see [9]) W,,,

Ed ^d 2(1

f

,

+ 4(1

f

+

,

+ y2^3]daida2 -\-2vKuK22 -\-K' + 2(l-v)/Ci^2]daida2,

(2.4)

where ^^^ and /c„^ are the (simplified) 2D membrane strain and bending strain tensors and y„3 are to include the effect of transverse shear deformation. These depend on the displacements of the middle surface u,v,w and on the so-called rotations 6,(1) SLS du Ai = ~ Aw;, aa\ dv P22= BW, _ oa2 1 / du dv \ 2 \9Qf2

9^1 /

dO dai

/ dv \dai

K22=^+C ^12 = r

(^

2 \da2 B

Cw - APn - 2CPr. - Cw] - Bh2 - 2CPu,

da\ J

(£-)

2 \9a2

/

(2.6)

and Ki3 = — - + Aw + Ci;-6>, da\ y23 = ^ + Cu + Bv-(j). (2.7) oa2 The relative error of the above approximation is formally of 0(ha)). For later use, we express the deformation energy as W(^ = ^Aaj(u, u) where u = (u, v, w, 0, 0) and Aaj(-, •) is the bilinear form defined appropriately. The simplified 2D model is found to be useful in connection with low-order FEM where each element serves as a restricted part of the shell over which the energy may be computed according to Eq. (2.4), with the strains expanded as in Eqs. (2.5)-(2.7). Obviously, a more accurate approximation of the usual Reissner-Naghdi model would result when a higher order polynomial approximation of z were used. Anyhow, the quadratic approximation is sufficient so as to obtain the leading terms of the deformation energy. 3. The bilinear degenerated 3D FEM

+ 2(l-v)ySf2]daida2 :d

and

(2.1)

where e^,, Cy and e^ denote the unit vectors along rectangular Cartesian axes x, y and z, (x, >;) G co C M^ and z{x,y) is a smooth function. In connection with representation (2.1), we define curvilinear coordinates on the surface by the parameterization

343

In order to compare the bilinear degenerated 3D FEM to the schemes obtained by discretizating the classical dimensionally reduced formulation, we first write the deformation energy used in the degenerated 3D FEM in the form that resembles the usual 2D energy. In the degenerated 3D formulation we basically follow the procedure proposed in [6], i.e. the geometry approximation is based on the nodal interpolation of both the middle surface and the normal vector. In addition, it is assumed that the transverse normal stress in the direction of the interpolated normal vector vanishes. The key idea in the derivation of the underlying 2D model is to write the degenerated 3D formulation by using the elementwise-defined Cartesian coordinate system, the origin of which lies on the exact middle surface. Assuming the tangent plane parameterization of Section 2, we write the interpolated middle surface over the element as s = aiejc + oi2ey + z(ai,a2)ez,

(2.5)

(«i, 0^2) ^ K,

(3.1)

where K is the projection of the geometrically incompatible element on the tangent plane, with nodes naturally associated to it by the projection, and where z e Qi(K)

344

M. Malinen, J. Pitkdranta / First MIT Conference on Computational Fluid and Solid Mechanics

interpolates z. Here Q\{K) is constructed in the usual way as the space of isoparametric bilinear functions on K. Starting from representation (3.1) and making use of expansion (2.3), we conclude that within relative errors of 0(/i/j:), the deformation energy may be put in the 2D form (see [9]) Ed

w\

f

2

2

2(1

+ 2(l-y)^f2.Jda,dQf2 4(1

+

—[ K

E(P

f

, -\-2vKu.hK22,h

ing error analysis [7,11] is limited, being based on strong assumptions on the problem setting and on the mesh. Moreover, only the approximation of inextensional deformations has been studied so far. Anyhow, the extension of the theory in different directions should be possible. Finally, we point out that the lowest-order case appears unique among the degenerated formulations as higher-order generalizations do not seem to hide numerical modifications of the leading terms of the usual 2D strains [9]. Thus, in case of higher-order generalizations the interpretation of the 'tricks' proposed in the literature appears more straightforward than in the lowest-order formulation where the nature of the underlying modifications is not so visible.

2 -\-K-22./?

+ 2(1 - v)/c-,2/JdQfidQf2,

(3.2)

where ^u.h approximates Pu{Uh), etc, with i//, G [Q\{K)]^ denoting the approximation of M. A S above, we rewrite the energy as W*}. = ^Ah,K(Uh,Uh). The bilinear form Ah^Ki',-) may naturally be considered to approximate AK(- ,') as defined in Section 2, so that further analysis of bilinear degenerated shell elements may be done within the classical 2D framework.

4. On the analysis of the MITC4 element The main difficulty in the finite element modelling of shells is to avoid the various parametric error amplification, or locking, effects [4,11-13]. In the literature on degenerated 3D formulations, various remedies have been proposed to avoid the problem. Still, the question of how locking effects are actually treated in the resulting formulations has been somewhat an open problem due to the poorly understood shell model. Assuming that in Eq. (3.1) K is of nearly rectangular shape, we have analysed in detail how the usual 2D strains are approximated in the bilinear degenerated 3D FEM [9,8]. We have also considered the MITC4 formulation [2] where the additional modifications of the transverse shear strains are used. We are able to show that the basic bilinear degenerated 3D FEM indeed results in the proper approximation of the 2D Reissner-Naghdi model. However, within the classical framework, the resulting numerical scheme is far from the standard one. In particular, when smooth inextensional deformations are approximated, the membrane strains of the bilinear degenerated shell element become close to certain reduced expressions, i.e. the membrane strains are obtained from the usual two-dimensional ones by introducing elementwise averaging operators. It appears that these hidden reductions, together with the MITC4 shear reductions, may resolve the shear-membrane locking problem in the case of smooth inextensional deformations, at least under favourable conditions. So far, the exist-

5. Conclusions We claim that the classical 2D shell theories serve as a natural framework for the analysis of degenerated shell finite elements. The first step in the analysis is to construct the connection from the degenerated 3D formulations to the dimensionally reduced shell models. Then the error analysis can be based on the standard principles of finite element error analysis. We have applied this technique to the bilinear degenerated shell elements. It appears that our approach reveals the essential 'secrets' that relate to the treatment of locking effects within this formulation.

References [1] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finite elements. Int J Numer Methods Eng 1970;2:419-451. [2] Bathe KJ, Dvorkin EN. A formulation of general shell elements — the use of mixed interpolation of tensorial components. Int J Numer Methods Eng 1986;22:697-722. [3] Bathe KJ, losilevich A, Chapelle D. An evaluation of the MITC shell elements. Comput Struct 2000;75:1-30. [4] Chapelle D, Bathe KJ. Fundamental considerations for the finite element analysis of shell structures. Comput Struct 1998;66:19-36. [5] Chapelle D, Bathe KJ. The mathematical shell model underlying general shell elements. Int J Numer Methods Eng 2000;48:289-313. [6] Cook RD, Malkus DS, Plesha ME. Concepts and Applications of Finite Element Analysis. New York: Wiley, 1989. [7] Havu V, Pitkaranta J. Analysis of a bilinear finite element for shallow shells I: approximation of inextensional deformations. Helsinki University of Technology, Institute of Mathematics Research Reports, 2000, A430. [8] Malinen M. On geometrically incompatible bilinear shell elements and classical shell models. Helsinki University of Technology, Laboratory for Mechanics of Materials Research Reports, 2000, TKK-Lo-30. [9] Malinen M. The classical shell model underlying the bilinear degenerated 3D FEM. Helsinki University of Tech-

M. Malinen, J. Pitkdranta /First MIT Conference on Computational Fluid and Solid Mechanics nology, Laboratory for Mechanics of Materials Research Reports, 2000, TKK-Lo-31. [10] Naghdi PM. Foundations of elastic shell theory. In: Sneddon IN, Hill R (Eds), Progress in Solid Mechanics, Vol. 4. Amsterdam: North-Holland, 1963. [11] Pitkaranta J. The problem of membrane locking in finite element analysis of cylindrical shells. Numer Math 1992;61:523-542.

345

[12] Pitkaranta J, Leino Y, Ovaskainen O, Piila J. Shell deformation states and the finite element method: a benchmark study of cylindrical shells. Comput Methods Appl Mech Eng 1995;128:81-121. [13] Pitkaranta J, Matache AM, Schwab C. Fourier mode analysis of layers in shallow shell deformations. Comput Methods Appl Mech Eng, submitted for publication.

346

A fictitious domain method for linear elasticity problems Janne Martikainen, Raino A.E. Makinen, Tuomo Rossi, Jari Toivanen * Department of Mathematical Information Technology, University of Jyvaskyld, P.O. Box 35 (Agora), FIN-40351 Jyvaskyld, Finland

Abstract A fictitious domain method based on boundary Lagrange multipliers is proposed for linear elasticity problems in two-dimensional domains. The solution of arising saddle-point problem is obtained iteratively using the MINRES method with a positive definite block diagonal preconditioner which is based on a fast direct solver for diffusion problems. Numerical experiments demonstrate the behavior of the considered method. Keywords: Linear elasticity; Fictitious domain method; Lagrange multipliers; Iterative methods; Preconditioning; Fast direct solvers; Saddle-point problems

1. Introduction Fictitious domain methods have been primarily used for scalar partial differential equations. The purpose of this paper is to demonstrate the potential of these methods for vector valued linear elasticity problems. More extensively fictitious domain methods for vector valued problems have been considered in [12]. The presented theory and methods are based on the observation that the linear elasticity operator and the vector diffusion operator — A + / are spectrally equivalent. The most important result in order to accomplish this is the Kom inequality [13]. The arising linear problems are solved using iterative methods with preconditioners based on fast direct solvers [16,18]. The basic idea of fictitious domain methods is to extend the operator and the domain into a larger, simple-shaped domain in which it is more easy to construct efficient preconditioners. The two most important ways to do this are algebraic and functional analytic approaches. In the algebraic fictitious domain methods, the problem is extended, typically in algebraic level, in such a way that the solution of the original problem is obtained directly as a restriction of the solution of the extended problem without any additional constraints (see for example [11]). In this paper, we study the functional analytic approach in which constraints are used to ensure that the solution of the extended problem * Corresponding author. Tel: +358 (14) 260-2761; Fax: -1-358 (14) 260-2771; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

coincides with the solution of the original problem. Here, we consider the use of boundary Lagrange multipliers (see for example [2,10,12,15]). Another possibility would be to use distributed Lagrange multipliers (see [8] and references therein).

2. Setting of the problem We consider linear elasticity problems in a two-dimensional domain Q for homogeneous and isotropic materials (see for example [13]). The boundary is decomposed as follows: dQ = Fo U Fi U r 2 . On FQ and Fi Dirichlet boundary conditions are imposed and on F2 the tractions are prescribed. The Dirichlet boundary conditions on Fi are enforced using boundary Lagrange multipliers [3,5]. We define the function spaces Vu = [ne

[H\n)f

I u\r, = 0} and V, = [//"^/^(FOf. (1)

Now, the weak formulation of the linear elasticity problem reads: Find u e Vu and ^ e V^ such that a(u, v) + {ri, v)r, = F(v)

{lu)r,={lg)r,

Vi; G K

(2)

VIGV,,

where {rj, v)r^ is the dual pair between [H ^ / ^ ( F I ) ] and [//^/^(Fi)] . The bilinear form a(u,v) and the linear form

/. Martikainen et al /First MIT Conference on Computational Fluid and Solid Mechanics AAA/\AAAAy\Y/lA^AAA/V\A/ AAAAAAAAA AAMAAYA\A/\A^ AAAAAAAAA A^\M.AyVVVVWV\A\AAA''VVV AAAAM AAAAAAAAA

suitable iterative method for solving the problem. In order to improve the conditioning of the problem, we propose a positive definite block diagonal fictitious domain preconditioned It is based on three key observations. The first one is the well-known result by Astrakhantsev [1] for fictitious domain preconditioners for problems with Neumann boundary conditions. When this result is combined with a general finite element extension theorem [19] the following result is obtained.

A/yW\AA/\A AAAAAAAAA ^/fff/fi^wppyPi/M/i/bvr/pyM/i/i/i/i/i/i/i/i

4M4MM4M4W^Svnl'^ AAAAAAAAAAAAJ>^ ^-^NAAAAAAAAAAA AAAAAMAAA^ AAAAAAAA/N/ AWWVYVV 4/W\WA/ AAAWvVy AAAAAAA\ AMAMAH AAAAAAAA AVVVYYYX

^w /yyyyyyyy] ^ AAAAAAAAA \\JAAAAAAAA AAAAAAAAA f^AiAAMAA \AAAAAAAA \AAAAAAAA

4Wi4yhh AMAWrh MMMA/fK

AAAAAAAAA NA/\A]AAAA / AAAAAAAAA AAAAAAAAA \Y\AY\\A\\^ \ /y\/\/\/\/\A/\/\ AAAAAAYA AY\A^(\^A\A4^>p=r~. .--r^fyyy ^VVWVWVVVWWVTi^WVV AAAAAAAAA AMAAAMAAAA/AAAAM AAAAAAVVV] /yyyyyyylA/lAAMM^ yyOm^^ Ayyyyy\A/A AAAAAAAAA Ay[\yCyiy AAAAyyy\A^ MAAA/AAWWiAMAMAM/

Fig. 1. A triangulation for a rectangular domain with a circular hole.

Theorem 1. Let U be a domain such that ^ C 11 and let Tn be a regular triangulation based on Lagrangian finite elements such that a triangulation T^ for the domain ^ is attained as a subset ofTw Let the matrices ^11

F{v) are given by

I A21

a{u, v) = I T(U) : €(v)dx A- cr I u • vds F(v) = I f ' vdx A- I p -vds A-cr I g • vds, v

347

(3)

where r and e are the linear stress and strain tensors, respectively. The terms with a positive coefficient a are added into the weak form (2) in order to have a K-elliptic bilinear form a(- , •) when TQ = 0 and Fi ^ 0. For the finite element discretization, a triangulation is constructed from a rectangular mesh by locally moving nodes near the boundary in order to obtain a good approximation for the boundary. This kind of approach for constructing triangulations is described in [4]. An example of mesh is show in Fig. 1. The finite element spaces yield naturally the discrete counterpart for the space of displacements K • The discrete spaces for the Lagrange multipliers corresponding to V^ are obtained as the traces of the space of displacements on Fi. Thus, this space consists of piecewise linear functions on Fi and we may replace the duality pairings in (2) by L2-inner products. The finite element discretization leads to the saddle-point problem (4)

The matrix A in (4) is symmetric and positive definite. Furthermore, when the unknowns on Fi are numbered first, B has the block structure B = {M O), where M is the lumped boundary mass matrix on Fi.

^1:

and A,

A22J

be obtained by discretizing the diffusion operator — A + / with Neumann boundary conditions in FI and Q, respectively, using the triangulations Tn, T^ and the corresponding finite element spaces. Then there exist positive constants c\ and C2 such that Cl

u" (All

^12^22^21)"
<M^Aiii/^

• ^n^ii

^21 j ^

Vw.

The second key observation by Yuri A. Kuznetsov, 1990, is that the block diagonal matrix with the diagonal blocks A and 5A~^i5 is a spectrally equivalent preconditioner for the saddle-point problem (4). A generalization of this is given by the following result (see [9,17] for details). Theorem 2. Let the matrices A, BA~^B^, D, and S be symmetric and positive definite. Furthermore, let there be positive constants C3, C4, Cs and c^ such that C3 u^Du < u^ Au < C4 u^Du

Wu

cs r]^Sr} < r]'^BA-^B^r] < ce r)^Sr) Then there exist positive constants cy and c^ such that Cl < \M < cg holds for the eigenvalues X of the generalized eigenvalue problem

(:

: > - ( : : ) -

3. Preconditioning

The third key observation is the spectral equivalence of the linear elasticity operator and the vector diffusion operator - A + /. The following result is based on Korn inequality, Friedrich's inequality and trace theorem (details are given in [12]).

The matrix in the saddle-point problem (4) is symmetric and indefinite. Therefore, the MINRES method [14] is a

Theorem 3. There exist positive constants cg and cio depending only on the material parameters, the geometry

348 ofQ,

J. Martikainen et al /First MIT Conference on Computational Fluid and Solid Mechanics where M is the diagonal lumped boundary mass matrix in B and

To, Fi and Fi such that

cg j (Vw :Wu -\-u ' u)dx

/2

< a(u, u)

-1

Cm / (Vw: Vw -\- u • u)dx

-1

-A

2

(6) -1

for all u e Vu, where the space V„ and the bilinear a(- , •) are defined by (1) and (3), respectively. 3.1. Preconditioner for

form

displacements

For the construction of preconditioner Z), the domain and the vector diffusion operator —a A + a / are extend to a rectangle n . The discretization of the extended operator yields a matrix having the same block structure as the one in Theorem 1. The preconditioner is then the Schur complement ^11 — A,2A^2 ^21- Although this preconditioner is spectrally equivalent to the discretized elasticity operator even without the scaling by a , the scaling improves the conditioning of the preconditioned saddle-point problem. This preconditioner can be efficiently implemented using any of fast direct Poisson solvers [18], We employ the partial solution variant of cyclic reduction [16]. 3.2. Preconditioner for Lagrange

multipliers

The preconditioner S is based on the properties of the Laplace operator and it belongs to the so-called A"'/^ family of preconditioners. Due to Theorem 3 and results in [6,7,10], S can be shown to be spectrally equivalent with BA'^B^. By grouping the displacement components into two groups and by numbering them within each group in counterclockwise order, the preconditioner has the form

M

^aA2 -\-ahI [

0

0 aA^ -\-ahI i

M,

(5)

y-1

-1

2 /

is the operator —d/ds^ with periodic boundary conditions discretized using an uniform mesh with the mesh step size h and, then, scaled by h. The solution with S can be performed efficiently using FFT, since the matrix A is circulant. For details see [6,7,15].

4. Numerical results The domain Q for both test problems is the square [ - 1 , 1] X [ - 1 , 1] with a circular hole with the diameter equal to one. A 33 x 33 mesh for this domain is shown in Fig. 1. In all experiments, the Young modulus E is 100 and the coefficient a in the additional boundary terms in (2) and in the preconditioners is chosen to be A -|- 2/x, where X and fi are the Lame coefficients. The stopping criterion for the iterative methods is that the norm of residual vector is reduced by the factor of one million. The computations are performed on a HP9000/J280 workstation. In the first test problem, homogeneous Dirichlet and natural boundary conditions are posed o n F o = {—l}x[—1,1] and F2 = 9Q \ Fo, respectively. There is no Dirichlet boundary conditions which would have been enforced using Lagrange multipliers. The arising system of linear equations is solved using the preconditioned conjugate gradient method. The structure corresponding to Q is exposed to a gravity-like force. The scaled displacements are shown in the left-hand side of Fig. 2. Also in the preconditioner based on the vector diffusion operator, the

Fig. 2. The solutions using a 33 x 33 mesh for the test problems.

349

/. Martikainen et al. /First MIT Conference on Computational Fluid and Solid Mechanics Table 1 The results for the test problems with varying mesh Mesh

33 X 33

Problem

it

CPU

it

CPU

it

CPU

it

CPU

it

CPU

1 2

36 53

0.33 0.41

38 60

1.39 2.10

34 61

5.73 9.19

35 64

23.92 41.50

35 66

109.21 191.47

6 5 ;x 6 5

513 X 513

257 X 257

129 X 129

Table 2 The results for the test problems with varying Poisson's ratio V

0.2

Problem

it

CPU

it

1 2

30 55

5.30 8.38

34 61

0.49

0.4

0.45

CPU

it

CPU

it

CPU

it

CPU

5.73 9.19

49 79

7.39 11.36

66 107

9.28 14.82

140 225

17.47 29.45

0.3

Dirichlet boundary condition is posed on the left-hand size of n = [ - l , l ] X [ - 1 , 1 ] . In the second test problem, homogeneous Dirichlet and natural boundary conditions are posed on Fi = {jc e MM -^ = 1/2} and T2 = dQ\Tu respectively. Thus, the Dirichlet boundary conditions are now enforced using Lagrange multipliers. The arising saddle-point problem is solved using the preconditioned MINRES method. There are pressure boundary conditions on [ - 1 , - 0 . 5 ] x {-1} and [0.5, 1] X {1}. The scaled displacements are shown in the right-hand side of Fig. 2. For the displacements, we use the same preconditioner as in the first test problem except, now, the Neumann boundary condition is posed on the left-hand size of n . The preconditioner for the Lagrange multipliers is given by (5). In Table 1, we report for both test problems the number of iterations and the CPU-time in seconds when the mesh is varied and the Poisson ratio is 0.3. In this table, the first line gives the size of the rectangular mesh on which the locally adapted mesh in Q is based. In Table 2, we study how the Poisson ratio effects the iterative solution. These results are computed using a 129 x 129 mesh in Q. The first line of the table gives Poisson's ratio v.

[3] [4]

[5] [6]

[7] [8]

[9]

[10]

Acknowledgements

[11]

The authors are grateful to Professor Y.A. Kuznetsov for fruitful discussions. This work was supported by the Academy of Finland, grants #43066 and #66407.

[12]

References [1] Astrakhantsev GP. Method of fictitious domains for a second-order elliptic equation with natural boundary conditions. USSR Comput Math Math Phys 1978;18:114-121. [2] Atamian C, Dinh GV, Glowinski R, He J, Periaux J.

[13]

[14]

[15]

On some imbedding methods applied to fluid dynamics and electro-magnetics. Comput Methods Appl Mech Eng 1991;91:1271-1299. Babuska I. The finite element method with Lagrangian multipliers. Num Math 1973;20:179-192. Borgers C. A triangulation algorithm for fast elliptic solvers based on domain imbedding. SIAM J Num Anal 1990;27:1187-1196. Bramble JH. The Lagrangian multipher method for Dirichlet's problem. Math Comp 1981;37:1-11. Bramble JH, Pasciak JE, Schatz AH. The construction of preconditioners for elliptic problems by substructuring, I. Math Comp 1986;47:103-134. Chan TF. Analysis of preconditioners for domain decomposition. SIAM J Num Anal 1987;24:382-390. Glowinski R, Kuznetsov Y. On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrande multiplier method. CR Acad Sci Paris Ser. I Math 1998;327:693-698. Iliash Y, Rossi T, Toivanen J. Two Iterative Methods for Solving the Stokes Problem. Tech. Rep. 2, Laboratory of Scientific Computing, Department of Mathematics, University of Jyvaskyla, 1993. Kuznetsov YA. Iterative analysis of finite element problems with Lagrange multipHers. In: Bristeau M-O, Etgen G, Fitzgibbon W, Lions JL, Periaux J, Wheeler MF (Eds), Computational Science for the 21st Century. New York: Wiley, 1997, pp. 170-178. Marchuk GI, Kuznetsov YA, Matsokin AM. Fictitious domain and domain decomposition methods. Sov J Num Anal Math Modelling 1986;1:3-35. Martikainen J. Fictitious Domain Methods with Separable Preconditioners for Vector Valued Partial Differential Equations. Licentiate thesis. Department of Mathematical Information Technology, University of Jyvaskyla, 2000. Necas J, Hlavacek I. Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. Amsterdam: Elsevier, 1981. Paige CC, Saunders MA. Solution of sparse indefinite systems of linear equations. SIAM J Num Anal 1975; 12:617629. Rossi T. Fictitious Domain Methods with Separable Pre-

350

J. Martikainen et al /First MIT Conference on Computational Fluid and Solid Mechanics

conditioners. PhD thesis, Department of Mathematics, University of Jyvaskyla, 1995. [16] Rossi T, Toivanen J. A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension. SIAM J Sci Comput 1999;20:1778-1796. [17] Silvester D, Wathen A. Fast iterative solution of stabilised Stokes systems, II. Using general block preconditioners. SIAM J Num Anal 1994;31:1352-1367. [18] Swarztrauber PN. The methods of cychc reduction and

Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Rev 1977;19:490-501. [19] Widlund OB, An extension theorem for finite element spaces with three applications. In: Hackbusch W, Witsch K (Eds), Numerical Techniques in Continuum Mechanics. Proceedings of the 2nd GAMM-Seminar, Kiel, January 1719, 1986. Braunschweig: Vieweg, 1987, pp. 110-122.

351

Thick shell elements with large displacements and rotations p. Massin ^'*, M. Al Mikdad ^ ^ Departement de Mecanique et Modeles Numeriques, Electricite de France/Division Recherche et Developpement, 1 Avenue du General de Gaulle, 92141 Clamart, France ^ Samtech France s.a. 14 Avenue du Quebec, SILIC 618, 91945 Courtaboeuf, France

Abstract In the present paper we discuss a total lagrangian formulation for shell elements under large displacements and rotations to perform nonlinear geometrical analyses. The formulation we use is based on a 3D continuum approach in which we introduce the kinematics description of a plane stress hypothesis. The measure of deformation we take is the one of Green-Lagrange related to the second Piola-Kirchhoff tensor for the stresses by a linear material law. Linear buckling is treated as a limit case of the nonlinear geometrical analysis. Keywords: Shell; Large rotation; Lagrangian formulation

1. Introduction A method based on matrix combination of rotations of SO(3) was proposed a few years ago by M. Al Mikdad and A. Ibrahimbegovic [1] to deal with large rotations of beams. We suggest an extension of that method to the case of shell elements. Numerical developments are realized within the computation Code-Aster. Large transformations for shells are characterized with large displacements and rotations of fibers initially normal to the middle surface. This can be taken into account exactly in the kinematics of shell elements. The rotational degrees of freedom we have chosen are the components of the iterative spatial rotation vector. Between two iterates it identifies with the infinitesimal rotation vector acting on the deformed configuration. Due to this choice we obtain a non-symmetric tangent stiffness matrix. A new selective integration scheme is proposed for this tangent stiffness matrix in order to avoid membrane and transverse shear locking. The expression for the kinematics is first given and introduced in the weak formulation of the equilibrium to get the tangent stiffness matrix. We also show how the linear buckling can be treated as a limit case of the nonlinear geometrical analysis presented so far. A few * Corresponding author. Tel.: +33-1-47-65-45-08; Fax: +33-1-47-65-41-18; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

classical examples from the Hterature are then treated to validate our method.

2. Formulation 2.1. Geometric description The three-dimensional shell is represented by the volume ^ of Fig. 1 occupied by the points Q (§3 ^ 0). These points are located around the middle surface co formed by the points P (^3 = 0). At each Q point a local orthonormal frame [^1(^1,^2, §3) : ^2(^1, §2, ^3) : w(?i, ^2)] can be built. The vector w(|"i, §2) is normal to the middle surface co. §3 is the coordinate in the normal direction to the middle surface and §1, §2 two curvilinear coordinates which describe the middle surface. On the initial configuration, the position of a 2 point located on a normal to the middle surface is given with the help of the position of its projection P on the middle surface as follows: ^2(^1, ?2, ?3) = Xp{^u^2) + §3-w(§l,

fe)

(1)

2.2. Kinematics equations The position XQ and displacement UQ (we denote w, v and w the components of the displacement) of point Q on

352

P. Massin, M. Al Mikdad/ First MIT Conference on Computational Fluid and Solid Mechanics where [ 0 x ] is the anti-symmetric operator corresponding to the rotation vector 0 : 0

-0,

0,

©z

0

-0

-0,

©.

0

[0x]

(6)

and [ 0 (g) 0 ] the symmetric operator given by [ 0 0 0 ] = 00'^. Details concerning the treatment of large rotations and its numerical implementation can be found in Al IVlikdad [2]. 2.3. Constitutive law We consider an hyper elastic behavior. The constitutive law relates the second Piola-Kirchhoff local stresses to the local Green-Lagrange strains: {5} = [D]{E]

(7)

where the symbol indicates that quantities are expressed in the local orthonormal frame [^i(^i, §2, ?3) * ^2(?i,?2,?3) • w(^i,^2)]- We use a vectorial expression of the Green-Lagrange strain for which a transformation from the local frame to the global one is proposed by [3]: 6x^1

6><66xl

E = H E

(8)

The expression for E is: l(ul + vl+wl) Fig. 1. Geometry of 3D shells. Local frames on the reference configuration. the deformed configuration of Fig. 2 can also be given as a function of the position of its projection P on the initial middle surface: 4(?l'?2,?3)=JC:^(?l,?2)+?3^/in?,,?2)

(3)

(4)

A is the orthogonal operator of the large rotation of angle 0 about the vector 0 . Its expression is given by: A = exp[0x] = cos6'[/] + ^ ^ [ 0 x ] 0 1 — cos 6 -[0(8)0] 6*2

I

^(ul + v^ + wl)

V,

E = W,y +

Yx

v.

+

5 («'z+ « ' + " ' ' ) UjcU^y + V^x'^,y +

^,x'^,y

Yx yUyU^z

(2)

where n*^ is the unit vector resulting from the large rotation of the normal vector n. n"^ is usually not normal to the deformed middle surface due to the transverse shear. It is related to the initial normal vector by the relation: «^ = A(?i,fc)n

Eyy I

\

+ ^,v^,z + ^,y'^,z

J

(9)

m.

which can be reduced to [4]: du E = Q+JA 3x

(10)

3. Virtual work The virtual work of internal forces can be written on the reference geometry as: STTU

=

j{{8E]-{S}) dQ

(11)

Its iterative variation is such that: (5)

A57r.nt = f{{8E] • {AS} + {A8E} • (S)) d^

(12)

p. Massin, M. Al Mikdad/First MIT Conference on Computational Fluid and Solid Mechanics

^3

'

353

^

Fig. 2. Geometry of 3D shells. Large transformation of afiberinitially normal to the middle surface. where we need the virtual variation of the global GreenLagrange strain tensor: (5E:

Q + A

dx)

8UQ{^U

d8u ~dx

=Ar^^^+ \ dx J dx

dA8u dx

A8E = A

(14)

While the first term of the above expression is classical in 3D continuum, the second one, which usually doesn't exist, takes into account the influence of large rotations. To calculate the virtual variations of the displacement gradient vector and its iterative we need the virtual variation of the large rotation matrix A:

8n = 0

(17)

AUQ(^U §2, ?3) = Aupi^u h) + ^3~Aw(^u ?2) x n ^ ^ i , ^2), An = 0

(18)

The variation for that iterative displacement is also: A8UQ(^U ^2, h) = h:^8w(^u Hi) X (Au;(^i, ^2) x«^(^i, ^2)) (19) which disappears for 3D continuum.

(15)

where [8wx] is the anti-symmetric operator related to the virtual spatial rotation 8w such that: [5wx]b=:(5wAb

^2) XW^^i, §2),

which expression is very similar to the one obtained for the iterative displacement:

Non-classical term.

8A = [8wx]A

§2, ?3) = SUpi^u §2) + ^3-8w(^u

(13)

and its virtual iterative variation:

Classical term

placement is expressed as:

VbG/?^

(16)

Then we get the several differential variations of the displacement which are needed in (12). The virtual dis-

4. Implementation The kinematics described so far is implemented on seven nodes and nine nodes heterosis shell elements. In order to avoid an explicit calculus of curvatures, we choose to interpolate the normal to the initial middle surface, rather

354

P. Massin, M. Al Mikdad / First MIT Conference on Computational Fluid and Solid Mechanics

than interpolating rotations. Displacements or positions are interpolated on six and eight nodes {NBl), while the normal is interpolated on seven and nine nodes (NB2). Hence the interpolation of any point Q on the reference or current configuration can be written as:

(A X^^,,?2,?3)=

J^N\'\^u^2)

5.3. Buckling of a beam under axial compressive load [6] and large displacements of a simply supported cylindrical panel under concentrated load [7] In both cases, good results are obtained up to the first limit load point. Arc-length methods are currently being developed to go beyond these points.

y 6. Conclusion

/nA + ?35E<^(?-^^)

vv

(20)

The interpolation for the positions, being alike on the reference or current configurations, remains valid for the displacement. A selective integration scheme is chosen to avoid membrane and transverse shear locking. Since the tangent stiffness matrix is singular with respect to the rotational component around the transformed normal «*^ we define a potential energy associated to this rotation. A constant torsion stiffness is related to the rotation. Internal forces due to that potential energy are taken into account. Since no deformation can be associated to that energy, it must remain negligible; this can be controlled by the user who can set the value of the stiffness from 10"^ to 10~^ times the stiffness associated to the other rotational components. The linear buckling is treated as a limit case of the nonlinear geometric analysis. It is based on the hypothesis of a linear variation of the displacement and stress fields with respect to the load level. Its consequence is the linear development of the tangent stiffness matrix with respect to the load level. The stiffness matrix of the linear buckling analysis is then obtained as limit of the geometric part of the tangent stiffness matrix — second part of (12) — when the deformed normal n^ is identified with n.

5. Validation 5.1. Beam clamped at one end under linear moment With this classical test of the literature [5] we reach a rotation of n for the free end of the beam. Though we should be able to go as far as In we did not obtain that result. 5.2. Linear buckling of a T-shaped beam under transverse loading and of a cylinder under external pressure Critical load and pressure are recovered within 3% of relative error with respect to the analytical solutions [6].

The seven (triangle) and nine (quadrangle) node large transformations shell elements we have described so far are based on a 3D continuum approach in the weak formulation of the equilibrium of which we have introduced HenckyMindlin-Naghdi plane stress kinematics hypotheses. This weak formulation is purely lagrangian. The strain tensor of Green-Lagrange is linearly associated to the second PiolaKirchhoff tensor for the stresses. Goods results are obtained for linear buckling analyses and analyses with large rotations. However, beyond limit loads, analyses are still not possible due to the lack of arclength method control which is currently being developed.

References [1] Al Mikdad M, Ibrahimbegovic A. Dynamique et schemas d'integrafion pour modeles de poutres geometriquement exacts. Rev Eur elements finis 1997;6(4):471-502. [2] Al Mikdad M. Stafique et dynamique des poutres en grandes rotations et resolufion des problemes d'instabilite non lineaire. PhD thesis, Universite de Technologie de Compiegne, 1998. [3] Bathe KJ. Finite Element Proceedings in Engineering Analysis. Englewood Cliffs, NJ: Prentice Hall, 1982, p. 258. [4] Crisfield MA, Non-Linear Finite Element Analysis of Solids and Structures. Vol. 1: Essentials. Chichester: Wiley, 1994, p. 117. [5] Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part III: Computational aspects of the non linear theory. Comput Meth Appl Mech Eng 1990;79:21-70. [6] Timoshencko SP, Gere JM. Theorie de la stabilite elastique. Paris: Dunod, 1966, deuxieme edition. [7] Jaamei S. Etude de differentes formulations lagrangiennes pour r analyse non lineaire de plaques et coques minces elasto-plasfiques en grands deplacements et grandes rotations. These de Doctorat, Universite de Technologie de Compiegne, 1986.

355

Adaptive ultimate load analysis of shell structures K.M. Mathisen^'*, I. Tiller ^ K.M. Okstad'' ^Norwegian University of Science and Technology, Department of Structural Engineering, N-7491 Trondheim, Norway ^ FEDEM Technology AS, Prof Brochs gt. 6, N-7030 Trondheim, Norway

Abstract This paper presents an investigation on automatic adaptive ultimate load analysis to geometrically non-linear shell-type problems involving elastic as well as elastoplastic history-dependent materials. The paper deals with different aspects of the transfer of solution variables between the successively refined meshes. The paper also briefly reviews an adaptive non-linear solution procedure, error estimators for purely elastic and history-dependent materials and an /z-adaptive mesh refinement strategy. Two numerical examples are presented to illustrate some practical features of the adaptive solution procedure and the efficiency of the error estimators adopted herein. Keywords: Adaptive methods; Error estimation; Rezoning; Ultimate load analysis; Shell analysis

1. Introduction

2. Adaptive non-linear solution procedure

An /z-adaptive mesh refinement procedure applied to geometrically non-Hnear shell analysis has been presented in [1]. The primary goal of this work has been to develop effective methods to assess ultimate loads by automated adaptive numerical solutions to geometrically non-linear shell-type problems involving elastic as well as elastoplastic history-dependent materials. In this paper, we will focus on: (1) error estimation, i.e. development and study of reliable error estimates for shell-type problems involving linear and non-linear material models; (2) mapping of solution variables, i.e. transfer of solution variables between successively refined meshes; and (3) computational aspects, i.e. development and implementation of efficient computational procedures for traversing critical points, involving limit, turning and bifurcation points, safely. Several examples that illustrate the merit and potential of the approach will be presented. In particular, the geometrically non-linear pear-shaped cylinder problem presented in [2], involving instability and bifurcation, will be studied in detail.

In non-linear finite element (FE) computations, the principal mechanisms for controlling an adaptive process are path controls (selection of step length and local parameters for corrector adjustment) and discretization controls (modification of the FE mesh and element type). The present study focuses on integration of these two mechanisms. The key ingredients of an adaptive meshing procedure controlling the discretization error inherent in the solution, are the following: (1) error estimation; (2) mesh refinement; and (3) mapping of state parameters. The spatial error in the FE solution at a certain load increment is based on comparison of the FE solution with an improved, so-called smoothed, solution field. In this work the smoothed solution field is obtained through local smoothing, applying the wellknown superconvergent patch recovery procedure by Zienkiewicz and Zhu [3]. An unstructured meshing approach based on use of the finite octree mesh generator [4] is adopted for mesh re-generation. The procedure adopted for transfer of state parameters between successive FE meshes is described in detail in [1] and [5]. This paper will focus on some of the fine points and implementation aspects of the integrated adaptive procedure for path control and discretization control. In particular, an adaptive procedure for traversing critical points will be described in detail. This paper will also examine implementation aspects of the enhanced superconvergent patch

* Corresponding author. Tel: +41 73 59 46 74; Fax: -^47 73 59 47 01; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

KM. Mathisen et al /First MIT Conference on Computational Fluid and Solid Mechanics

356

recovery procedure presented in [3], when applied to shelltype problems. The quadrilateral Mindlin/Reissner shell element based on the geometrically exact stress resultant model with mixed finite element interpolation of the membrane, bending and transverse shear fields, as proposed by Simo et al. [6,7], has been adopted herein.

3. Numerical examples 3.1. Compressed pear-shaped cylinder The simply supported pear-shaped cylinder shown in Fig. la [2], subjected to uniform end shortening, is considered to demonstrate the applicability of the proposed procedure to traverse critical points. One quarter of the cylinder is analyzed using symmetry boundary conditions. Prescribed displacement control in 45 equal steps is used to increase the end shortening up to 1.8 times the wall thickness of the cylinder. Fig. lb shows a plot of the total axial load versus the outward normal deflection w\%^. Fig. Ic shows the mesh sequence obtained in order to satisfy a prescribed limit on the global discretization error of 5% throughout the entire analysis.

Raxlius Length Thickness Elasticity

R L t E V

= = = = =

3.2. Closed pinched hemisphere The second example considered is a closed hemisphere bounded by a free edge, under the action of two inward and two outward pinching forces 90° apart, and characterized by an isotropic hardening material law. This example problem was also analyzed by Simo and Kennedy [7] and their results are used as a reference solution herein. The geometry and material properties are given in Fig. 2a. As indicated, one quadrant of the hemisphere is modeled using symmetry boundary conditions. The problem is analyzed using the proposed adaptive regime where the global error limit is set to 10% in the energy rate energy norm error estimator (see e.g. [5]). As described in [8] a tighter tolerance is used when computing the refinement indicators in order to avoid too frequent mesh refinements. In the two adaptive analyses carried out here we have applied local tolerances corresponding to respectively 70.0% and 80.0% of the prescribed global tolerance when computing the refinement indicators. Fig. 2b shows the applied load versus the radial displacement under each respective load. In Fig. 2c the accumulated CPU-time is plotted as function of the load level, while the sequence of adaptively refined meshes obtained in order to satisfy a prescribed limit on

1.0 0.8 0.01 10^ 0.3

W\^QO Mesh 1: 56S elonoits Step : 21 - 38 z/uu

1

2250 •

1—

Adaptive 5%

.gl800

^

ll350 .

1 ^

J

900

450

b)

/

A> 1

r"^

—

0.01

1

0.02

0.03

Nornud outward displacement

0.04

c)

Fig. 1. The pear-shaped cylinder problem: (a) geometry and material properties; (b) load-deflection plots for the different mesh configurations; and (c) sequence of meshes used in the adaptive analysis.

KM. Mathisen et al /First MIT Conference on Computational Fluid and Solid Mechanics

Radius : R Thickness : i Elasticity : E V

Plasticity :

(TY

K'

MeshO: 141 elonoits Stq> : 1

= = = = = =

357

10.0 0.5 10.0 0.3 2.0-10 9.0

Mesh 1: 303 elements S t ^ : 1-10

Single mesh • A31-10/7% • A31"1()/8% -

b) AccumulatBd CPU-time

cooo Single mesh - » A31-ia7% - 1 - -

A

y^

5000

4000

y^

•

Mesh 3 : 688 elements Steps : 23-30

y^

3000

2000

Mesh 2 : 468 elemrats Stqw : 10-23

-

y^

1000

0

—#-=

c)

d)

Mesh 3 : 688 elements Final configuration

Fig. 2. The closed pinched hemisphere problem: (a) geometry and material properties, (b) load-deflection plots, (c) accumulated CPU-time consumption; and (d) sequence of meshes used in the adaptive analysis. the global discretization error of 10% throughout the entire analysis is shown in Fig. 2d. Compared to the results obtained by Simo and Kennedy in [7] we see from Fig. 2b that the adaptive analysis follows the reference solution almost exactly.

References [1] Okstad KM, Mathisen KM. Towards automatic adaptive geometrically nonlinear shell analysis. Part I: Implementation of an /z-adaptive mesh refinement procedure. Int J Num Methods Eng 1994;37:2657-2678.

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[2] Hartung RF, Ball RE. A Comparison of Several Computer Solutions to Three Structural Shell Analysis Problems. Technical Report AFFDL-TR-73-15, U.S. Air Force, 1973. [3] Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int J Num Methods Eng 1992;33:1331-1364. [4] Shephard MS, Georges MK. Automatic three-dimensional mesh generation by the finite octree technique. Int J Num Methods Eng 1991;32:709-749. [5] Tiller I, Mathisen KM, Okstad KM. On the mapping of solution variables within adaptive analysis of nonlinear shell-type problems. Proc 3rd ECCOMAS Conf Numerical Methods in Engineering. Barcelona, September 2000, volume on CDROM.

[6] Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory. Comput Methods Appl Mech Eng 1990;79:21-70. [7] Simo JC, Kennedy JG. On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity: formulation and integration algorithms. Comput Methods Appl Mech Eng 1992;96:133-171. [8] Mathisen KM, Tiller I, Okstad KM, Hopperstad OS. On adaptive non-linear shell analysis. Proc 4th World Congr Computational Mechanics, volume on CD-ROM. Buenos Aires, June/July 1998.

359

Boundary stress calculation for two-dimensional thermoelastic problems using displacement gradient boundary integral identity T. Matsumoto *, M. Tanaka, S. Okayama Faculty of Engineering, Shinshu University, 4-17-1, Wakasato, Nagano City, 380-8553, Japan

Abstract In this paper, boundary stress components for two-dimensional thermoelastic problems are calculated by using the direct boundary integral representation for the displacement gradients. The effectiveness of the present formulation is demonstrated through some numerical examples by comparing the results by the conventional method. Keywords: Thermoelasticity; Boundary stress; Displacement gradients; Hypersingularity

1. Introduction One of the advantages of the boundary element method is that the gradient components such as stresses can be calculated directly by using boundary integral identities. Although such identities are hypersingular, much efforts have been devoted so far to obtain the formulations which are numerically tractable [1-5].

1 Clklmn = 7r~hmhn{^ ZTT

2. Boundary integral representation for displacement gradients

ClklmnUk,l{y) + hmnUkiy)

where Q is the domain, T is the boundary of the body; Ui, ti, 0, and q are the displacement, the traction, the temperature and the heat flux, respectively. f/*^„, TJ^^, ®mn' Qlin ^^^ related to Kelvin's fundamental solution and their orders of singularities are 0 ( l / r ) , 0{\/r^), O ( l n r ) , and 0 ( l / r ) , respectively, aumn, hmn and Cmn are coefficients related to the quantities defined in the neighborhood of the collocation point, and aumn can be written as follows:

8(1 —E

+ C^n^(j)

- CijkiUk,i(y)ni + 2G ^ _

2 — (P )

1 + 2v - 4v2

• 2(1 + v ) 5 ^ , r > ; •8v

+ / ^jmnyj

1

^^aO{y)njj

dr

-8kmry,

•4v

+ hir:n<

_

+ hmr:X

J

3-

+

•4v

^imr'kK hnr%^l

+ 4rV;,r;r;

r^ur^ -

1

I r

Tjmn (";• - ^jiy)

fklmnir)

- rkUj,k{y)) d r

/ (1)

r^ur^ * Corresponding author. Tel.: +81 (26) 269-5122; Fax: +81 (26) 269-5124; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

-

fklmn(^)

dr +

•AL(0)lnr^ + /Ln(0)lnr-l

fklmn (r) - fklmn (0)

dr (2)

In discretization, we use two neighboring elements adjacent to the collocation point j for f^ and f ^. After we obtain the boundary displacements and tractions over the boundary by the standard boundary element

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T. Matsumoto et al /First MIT Conference on Computational Fluid and Solid Mechanics

D Conventional interpolation

.1

^

4

© Present interpolation

b I

O-ll^fl^teftroW^^ p, w^rescribed -2

A 1

2

3

4

5

6

^1

B

C

Node

DE F

G

Fig. 2. Results for the example problem.

Fig. 1. Example problem. References analysis, we can calculate the boundary displacement gradients by using (2).

3. Numerical example A boundary shown if Fig. 1 is discretized with quadratic conforming elements. For the elements f and f , we employed the interpolation function which satisfies C^ continuity at the collocation point. We show the relative errors of the results for a component of the stress over the boundary in Fig. 2. The results are compared with those obtained by the method which uses the constitutive equation and the relationship between the tangential derivative and the displacement gradients.

[11 Krishnasamy G, Schmerr LW, Rudolphi TJ, Rizzo FJ. Hypersingular boundary integral equations: some applications in acoustics and elastic wave scattering. ASME J Appl Mech 1990;57:404-414. [2] Sladek V, Sladek J. Regularization of hypersingular integrals in BEM formulations using various kinds of continuous elements. Eng Anal Boundary Elem 1996;17:5-8. [3] Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. ASME J Appl Mech 1992;59:604-614. [4] Young A. A single-domain boundary element method for 3-D elastostatic crack analysis using continuous elements. Int J Numer Methods Eng 1996;39:1265-1293. [5] Mantic V. Existence and evaluation of the two free terms in the hypersingular boundary integral equation of potential theory. Eng Anal Boundary Elem 1995;16:253-260.

361

Adagio: non-linear quasi-static structural response using the SIERRA framework John A. Mitchell *, Ame S. Gullemd, William M. Scherzinger, Richard Koteras, Vicki L. Porter Sandia National Laboratory, P.O. Box 5800, MS 0847, Albuquerque, NM 87185, USA

Abstract Adagio is a quasistatic nonlinear finite element program for use in analyzing the deformation of solids. It is massively parallel, built upon the SIERRA finite element framework [1], and employs the ACME library [2] for contact search algorithms. The mechanics and algorithms in Adagio closely follow those previously developed in JAC2D by Biffle and Blanford [3] as well as JAS3D by Blanford et al. [4]. Adagio assumes a quasistatic theory in which material point velocities are retained but time rates of velocities are neglected. Sources of nonlinearities include nonlinear stress-strain relations, large displacements, large rotations, large strains, and frictional/frictionless contact mechanics. Quasistatic equilibrium is found using a nonlinear solution strategy which includes nonlinear conjugate gradients. This paper briefly describes quasistatic equilibrium, kinematics of deformation, stress updates and the nonlinear solution strategy used in Adagio. In addition, we briefly describe how Adagio is implemented within the SIERRA architecture. Finally, we demonstrate Adagio's massively parallel capabilities on an example problem. Keywords: Nonlinear solid mechanics; Quasistatics; Conjugate gradients; Object oriented

1. Quasistatic equilibrium Quasistatic equilibrium in Adagio is based upon the principle of virtual work in a rate form. We start by writing down a nonlinear functional representing the power input to the body in the current configuration. By taking the first variation of the power input, and integrating by parts, we arrive at the weak form: j T :8idV - f pb-8vdV

-

f t -SvdA = 0

(1)

where Q corresponds to the volume of the body in the current configuration, 9 ^ is the boundary of the body in the current configuration, T is the Cauchy stress tensor, v is the material point velocity, 8s is the symmetric part of the virtual velocity gradient, t is an applied surface traction, p is mass density, and b is a body force vector. The extent to which Eq. (1) is not satisfied is a measure of the force imbalance and lack of quasistatic equihbrium. This force * Corresponding author. Tel.: +1 (505) 844-3435; Fax: -f-1 (505) 844-9297; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

imbalance is called the residual and quasistatic equilibrium is defined according to how close the residual is to zero. Adagio solves for quasistatic equilibrium over a set of time increments A^ = t^+i - tn defined by a sequence of times tn n = 0,1,2, A force imbalance occurs at tn+i due to loads, temperatures, or kinematic boundary conditions that are parameterized by time. Quasistatic equilibrium is assumed to exist at f„. The solver searches for a suitable equilibrium configuration at tn+i through a sequence of trial velocities that give rise to ever decreasing residuals (force imbalance). Equilibrium is satisfied when the force imbalance reaches a user specified tolerance for convergence.

2. Updated Lagrangian The solver finds velocity vectors Vn^\ for a load step at discrete times r„+i by solving the nonUnear problem

362

J.A. Mitchell et al. /First MIT Conference on Computational Fluid and Solid Mechanics 6 = Arv. n+ 1

Fig. 2. Kinemafics of the deformafion. 4. Stress rate and hypoelastic stress updates

Fig. 1, Updated Lagrangian schematic. implied by the weak form (1). The current position y„+i of material points is updated via the formula: yn+\

X + i/„ + Arii;„+i

(2)

where X are the material coordinates at r = 0, w„ is the total displacement at the last converged step r„, and Ar = r„+i — r„ is the time step size taken for the load step. This updated Lagrangian approach is depicted in Fig. 1.

3. Kinematics of deformation In order to manage a variety of constitutive models as well as large rotations in conjunction with objective stress rates, we calculate a total deformation gradient F = dyn+]/dX and an incremental deformation gradient F = dyn+]/dyn. However, we usually work with the inverse deformation gradients F~^ = I — (9M„+I/9J„+I) for purposes of computational efficiency since we need to evaluate the internal force vector that requires gradient and divergence operations in the current configuration (see Eq. (1)). Using the polar decomposition theorem on the deformation gradients, we calculate rates of strain, total stretches, and rotation operators. The polar decomposition on the deformation gradients is defined as F = RU and for the inverse deformation gradients it is defined as F~' = R^V~\ where R is an orthogonal rotation operator, and U and V are the corresponding stretch tensors. The incremental deformation gradients are similarly decomposed and are used for purposes of calculating rates of strain at material points for hypoelastic material models. Total deformation gradients are used for managing tensors in unrotated and rotated configurations as shown in Fig. 2 as well as in hyperelastic constitutive models which require a measure of the total strain.

Most material models in Adagio are hypoelastic so that stress rates are integrated forward in time over the time step Ar = t„+\ - tn to find the stress at tn+\. In order to develop this methodology, we first define an unrotated cauchy stress (configuration BU)'G = WTR. The unrotated cauchy stress rate which is analogous to the Green-Naghdi stress rate is objective and defined abstractly by a = /(Ar, J, a ) , where f{d,'a) represents the incremental form of the constitutive model specifics. The Green-Naghdi stress rate is defined as:

o = RaR' = T -QT

-h^T

(3)

where Q = RR^, and T is the cauchy stress tensor. Our algorithm for updating stresses is given as: (1) compute strain rate D = - ^ In V~^ (2) de-rotate strain rate using d = R^ DR\ (3) integrate constitutive model W = f(At,d,'a) to find W; (4) rotate a to current configuration a = R^TR. Note that V~Ms the incremental left stress tensor.

5. Solution strategy: nonlinear PCG The primary method for finding quasistatic equilibrium in Adagio is the preconditioned conjugate gradient method (PCG) [5]. The solver is configured in an object oriented way and consists of the following abstract plugins: preconditioner, line search, and residual operator. Just prior to running the solver for each loadstep, a loadstep predictor is invoked. The predictor runs a line search with the velocity vector from the last converged loadstep as the search direction. 5.7. Preconditioning The process of solving a loadstep with PCG is an incremental solution strategy and is conceptually very similar to

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J.A. Mitchell et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Newton-Raphson. We construct a preconditioner B that is an approximation to Kt~^ (inverse of tangent stiffness). In addition, the solver requires that the output of the preconditioner satisfy the homogeneous form of kinematic boundary conditions as well as contact constraints. Currently, Adagio has a nodal preconditioner which consists of a three-bythree block diagonal stiffness for each node in the mesh. These stiffnesses may be computed by probe or through an analytical formula. 5.2. Line search Both the PCG solver and the predictor in Adagio use a line search object as a plugin. The PCG algorithm assumes that the line search produces a new velocity such that the resulting residual will be orthogonal to the current search direction. This corresponds to an exact line search. However, this is usually a multi-step process and can be expensive. Adagio currently performs an inexact line search by using one step of a secant line search algorithm. This line search is used in both the predictor and solver. 5.3. Residual operator Quasistatic equilibrium is fundamentally based upon the residual/force imbalance. The PCG solver, predictor and line search objects in Adagio all use residual operators. In Adagio, the residual operator is responsible for managing geometry, external forces, internal forces and reaction forces on surfaces where kinematic boundary conditions are applied.

1 \

6. Code architecture The algorithms in Adagio described above are implemented within the SIERRA framework. SIERRA-based codes consist of mechanics modules which can be nested inside each other to provide a rational code hierarchy. Fig. 3 depicts the overall architecture of Adagio. The highest level of control in Adagio is Agio_Procedure, which manages time stepping. Nested inside of Agio_Procedure is Agio_ Region, which is responsible for orchestrating all calculations required for a particular time step. Agio_Region contains a set of mechanics modules that perform individual algorithms and are dynamically loaded at run time based upon user input. Fig. 3 shows several examples of these mechanics. They include: Agio_KinBC, which computes the effects of boundary conditions; Elements, which conducts element computations; Agio_NonlinearPCG, which drives the solver; and Agio_Predictor, which provides a predicted first guess for the solver. The nesting continues within the solver and the predictor. Any of the mechanics shown in Fig. 3 can be replaced by a different mechanics module as long as it conforms to the minimal interface as shown. The runtime behavior of Adagio is closely tied to the construction, scoping, and registration of algorithms on mechanics modules. The concept of scope in SIERRA is somewhat analogous to that of C-h+. For example, in Fig. 3 Agio_Fe_Operator exists in two locations: Agio_ LineSearch and Agio_NonlinearPCG. These two operators are at different scope and may have totally different implementations. Mechanics algorithms in SIERRA are invoked via a constant interface and algorithms run according to whether they are in the current scope. For example.

Agio_Pro :edure initialize(); execute 0;

N

J

Agio_Region ^ initializeO; compute_timestep_size(); executeO;

4gio_KinBC ^ apply_kinematics(); adjust_residual(); adj ust_gradient_direction(); V extract_reactions();

1 (^El sments j

f'Mpo_SecantLineSearch Agio Fe Operat or ^ 'v^ compute_resi( ualO;^

1

1

(^ gio_NonlinearPCG ''\ solveO;

Agio_Predict

,

y Agio LineSearch compute_alpha();

,/Agio_LineSearch ^ V compute_alpha(); '

/ Agio_Fe_Operator compute_residual();

Agio_Fe_Operator , compute_residual(); J /Agio_Pc N ' \ compute_pc_stittness(); \ action();

^

_y

Fig. 3. Schematic of Adagio mechanics algorithms.

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J.A. Mitchell et al. /First MIT Conference on Computational Fluid and Solid Mechanics

during the solution process, Agio_NonlinearPCG invokes the algorithm "compute_residual" once per iteration. Any mechanics nested within Agio_NonlinearPCG which has the algorithm "compute_residual" will have its algorithm executed — in this case, only Agio_Fe_Operator has an algorithm that will be executed. The Agio_Fe_Operator within Agio_LineSearch will not get executed. Furthermore, if Agio_Fe_Operator is not registered within Agio_ NonlinearPCG, then nothing will happen when "compute_ residual" is called. These mechanisms provide significant power for Adagio to connect modules with similar but not identical behavior, and to create a logical code structure where functionality can be selected easily.

[2]

[3]

[4]

[5] References [1] Edwards C, Stewart JR. SIERRA: a software environment for developing complex multi-physics applications. In: First

MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. Brown KH, Glass MW, GuUerud AS, Heinstein MW, Jones RE, Summers RM. ACME: a parallel library of algorithms for contact in a multi-physics environment. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. Biffle JH, Blanford ML. JAC2D: A two-dimensional finite element computer program for the nonlinear quasi-static response of solids with the conjugate gradient method. Sandia National Laboratories, Albuquerque, NM. SAND93-1891, 1994. Blanford ML, Heinstein M, Key SW. JAS3D: A multistrategy iterative code for solid mechanics analysis. Users' Instructions, Release 1.6. Sandia National Laboratories, Albuquerque, NM. To be published as a SAND report, 2000. Shewchuk JR. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 1994. Edition 1.25.

365

An object-oriented finite element implementation of large deformation frictional contact problems and applications M. Moubarack Toukourou, A. Gakwaya*, Amir Yazdani Department of Mechanical Engineering, Laval University, Sainte-Foy, Que. Canada

Abstract An object-oriented finite element implementation of large deformation multi-body contact problems is presented. The simulator can address arbitrary shaped models. Contact search algorithms and generic frictional contact element usable as a surface element for any finite element model have been implemented as C-I-+ classes in Diffpack. The interfacial constitutive law is of Coulomb's friction type that handles sticking and sliding contact. The non-linear solution process is based on variational penalty formulation with consistent tangent matrices. An example demonstrates the efficiency of the methodology. Keywords: Object-oriented programming; Contact finite element; Hyper-elasticity; Search algorithm; Friction; Penalty formulation

1. Introduction Although a large body of literature and computer codes have been developed to handle contact problems, experienced users still complain about longer computer time or unsatisfactory results when contact problems are considered in analysis, thus demonstrating the need for better and more efficient solution processes [1]. This class of kinematically constrained boundary value problems still remains one of the most challenging in computational mechanics, mainly because the contact region is not known a priori and its shape, size and strain distribution may vary considerably with load. Also, since contact constraints are not permanently active, classical solution techniques may appear not as good as expected. In practical situations, the problem becomes even more complicated when interfacial friction is considered. Formulations and solutions of such problems involve a variety of mathematical, physical and programming issues. Using recent computational algorithms, discussed, e.g. in reference [1], this paper presents an implementation of a large deformation frictional contact simulator within an object-oriented finite element environment that allows the user * Corresponding author. Tel.: -hi (418) 656-5548; Fax: +\ (418) 656-7415; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to concentrate on developing application classes and functions rather than the usual 'first master the code' philosophy. The widely used Diffpack library [6] of C + + classes has been selected for the adaptation of our algorithm. Although familiarity with Diffpack libraries structure may be useful, it is not, however, essential for understanding the present developments. Section 2 reviews the Diffpack environment used as a development tool and presents the UML structure for the contact simulator. Section 3 presents the classes developed and implemented and finally an example validating the approach is presented.

2. Non-linear finite element programming in Diffpack environment The development of classes, appropriate for large deformation contact problems, must consider four main items: (1) geometric and kinematics description of large deformation; (2) contact search algorithms; (3) a variational formulation and associated algorithmic treatments for deriving consistent contact stiffness and residual arrays; and (4) interfacial constitutive law. Using class hierarchies, virtual functions and smart pointers, coupling of hyper-elastic and contact simulators can be done easily within framework provided by the Dijfpack package that includes classes for

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M.M. Toukourou et al. /First MIT Conference on Computational Fluid and Solid Mechanics

linear algebra operations, finite element based classes and a secure way for pointer declaration, etc. For instance, the framework for defining scalar and vector fields is provided, respectively, by classes 'FieldFE' and 'FieldsFE' that allow for the calculation of field values at any point in the domain, etc. Objects of these classes are binded with a particular object of class 'GridFE' that contains geometric information of the FE mesh as well as information specific to a given boundary value problem (boundary indicators). All basic finite element calculations at Gauss points are handled by class 'Finite element'. The class 'FEM' provides the algorithmic framework for performing element stiffness matrices and load vectors calculations and for assembly of the linear system of equations. Particular problem dependent calculations are defined in a simulator class, derived from 'FEM' where, for example, pure virtual functions 'FEM:integrands()' can be used to perform computations of element matrices and vectors. Variable integration order is possible through the class 'ElmltRules' and the virtual function 'integrand4sides' can be used to handle configuration dependent surface load. The structure for selecting various linear or non-linear algebraic solvers is provided by classes 'LinEqAdm' and 'NonLinEqSolver'. To select from a list of available methods or to define user's own solver, a menu is provided via class 'MenuUDC. Finally, class 'TimePrm' provides for basic time management required by time-integration algorithm. The UML structure of developed 'CONTACT' simulator is shown in Fig. 1. The existence of such a pre-defined library is useful in writing shorter and better-structured programs.

3. Classes for handling frictional contact problems A hyper-elastic simulator was first developed in order to introduce classes and functions related with large deformation geometry and kinematics as well as with nonlinear material behavior. Two and three dimensional 'integrands' based on consistent linearized tangent matrices were used. Contact search algorithms were based on works by Zhong et al. [3] and Oldenburg et al. [4] combined with the closest point projection algorithm. Complete details can be found in Moubarack Toukourou [2]. The master class is class 'CONTACT' and represents the simulator class. This class inherits publicly from class FEM (finite element programming algorithms) and class 'NonLinEqSolverUDC (userdependent code for nonlinear systems). For multi-body contact problems, indexed objects, such as domains, contact surfaces, contact pairs and contact elements, were created using class 'VecSimplest' of Diffpack. Class 'Domain' allows to designate each body in the system; class 'Frontier' uses 'Domain' and allows for definition of candidate contact surfaces. Class 'Contact_Pair' uses 'Frontier' and allows for definition of each pair of contact with its associated contactor and target surfaces. Class 'Contact_Element'

Table 1 Objects indexed to help "solving" mulfi-body contact problems Members variables Class contact Field variables GridFE Coupled arrays Hyper-elasdcity Inifialization Integrands Linear/nonlinear solver State variables Contact IniUalizafion Friction, penalty Search flags Contact_Domain, Contact_Frontier, Contact_Pair Contact_element Class domain Grid information Contact interaction Linear system Finite element fields configuration ...

Member functions Menu initialization TimeLoop Hyper-elasticity Non-linear kinematics Compute stiffness and residual Material model History variables Update contact element Prepare report

Update progress control

Class frontier Domain Surface elements list Surface node list Position code number list Box numbers array Surface element node array

Set domain Set element Set node Progress control

Class Contact_Pair Contactor, target surface Interaction type, ... Contact territory data Geometric data Global search parameters Local search parameters

Global contact search. Position code, contact-territory Local contact search Closest point projection

Class Contact_Element Contact element Target element Contactor node Local to global transformation Geometric characteristics Contact pair, attached FE Displacements, forces Friction state, contact status Numerical integration database Residual, tangent stiffness arrays Configuration

Contact pair Contact database Global/local numbering Set shape functions ... Compute local frame and geometric properties Compute constitutive parameters Compute tangent stiffness and residual

367

MM. Toukourou et al /First MIT Conference on Computational Fluid and Solid Mechanics Simcase FieldFormat NonLinEqSolverUDC

SaveSimRes

DegFreeFE

FieldsFEatltgPt

X

VecSimplest(Domain)

GridFE

VecSimplest( Frontier)

LinEqAdmFE

VecSimplest(Contact^air)

FieldFE FieldsFE

VecSimplest(Contact_element) NonLinEqSolver_prm

NonLinEqSolver

Field

Fields

TimePrm

Fig. 1. UML structure of class Contact. Bold, classes related to abstractions already existing in Diffpack's library; italic, classes related to the resolution of contact interaction; simple line, composition ('has a') relation; arrow, inheritance relation.

Von Mises stress /, \

,-%;;

™^v .

• 1.46e4 HSOe 1.25e4

605

(c-1) jJ,:IJjl:,LLl

LiilHEHi

1.04e4 1403

i,J^^»^^i^-'^?K^^5^*^^55^"?~?™^

(a)

8.31e3

i

202

I

J6.20e3 |0.209

(C-1) (c-2) (b)

(c-2)

Fig. 2. Rigid punch case, (a) Finite element mesh (x-z plane), (b) Deformed mesh, (c) von Mises stress distribution. Table 2 Numerical performance of the Rigid punch case Iteration

1

Error

le + 020

0.383

1.25e-3

uses 'Contact_Pair' and allows for the definition of contact element formed by a contactor node and associated target element. Their main attributes and methods are given in Table 1.

3.42e - 5

1.60e-5

1.29e-5

1.06e-5

4. Numerical experiments We consider the contact of a rigid punch with an hyperelastic foundation under the condition of frictional stick

368

M.M. Toukourou et al./ First MIT Conference on Computational Fluid and Solid Mechanics

or slip (Fig. 2a for FE grid). The punch of dimensions 30 X 30 X 5, is modeled with 18 tri-linear elements and the foundation of dimensions 200 x 200 x 20, is modeled with 200 elements. A hyper-elastic Neo-Hookean material model was used with parameters: Epunch = 10^ (assumed rigid), Vpunch = 0; Efound = 10^ Vfound = 0.3; penalty coefficients: 6N = 2.10^, €T = 2.10"^; friction coefficient: jj. = 0.6. A constant pressure of p = 1000 is applied on the top of the punch with fixed foundation bottom. The results of Table 2 indicate quadratic convergence and agree with Ju and Taylor [5]. Fig. 2b shows the deformed mesh with a magnification factor of 260 and Fig. 2c shows the stress distribution.

5. Conclusions An object oriented finite element implementation of frictional contact problems was presented. In addition to Diffpack based finite element classes, hyper-elasticity and contact simulators were developed in the form of classes. Use of smart pointers, virtual functions and field objects allowed an easy and reliable coupling and testing of various parts of the code. The qualities of a robust code allowing

for generality, expandability, maintainability and FEM code reusability were greatly appreciated. References [1] Parisch H, Lubbing C. A formulation of arbitrarily shaped surface elements for three-dimensional large deformation contact with fricuon. Int J Numer Methods Eng 1997;40:3359-3383. [2] Moubarack Toukourou M. Modelisafion et simulation par la MEF du contact avec frottement dans les precedes de mise en forme des metaux. Master's thesis, Laval University, Quebec, December 2000. [31 Zhong ZH, Nilsson L. A contact searching algorithm for general contact problems. Comput Struct 1989;33(1):197209. [4] Oldenburg M, Nilsson L, The posifion code algorithm for contact searching. Int J Numer Methods Eng 1994;37:359386. [51 Ju J-W, Taylor RL. A perturbed Lagrangian formulation for thefiniteelement solution of nonlinear frictional problems. J Theor Appl Mech 1988;75;7240. [61 Langtangen HP. Computational Partial Differential Equations: Numerical Methods and Diffpack Programming. Springer, 1999.

369

Parallel simulation of reinforced concrete column on a PC cluster J. Nemecek^'*, B. Patzak^ Z. Bittnar^ " Czech Technical University in Prague, Department of Structural Mechanics, Faculty of Civil Engineering, Thdkurova 7, 166 29 Prague, Czech Republic ^ Swiss Federal Institute of Technology at Lausanne, Department of Civil Engineering, Laboratory of Structural and Continuum Mechanics, 1015 Lausanne, VD, Switzerland

Abstract This paper discusses the applicability and the efficiency of the parallel computing apphed to the analysis of reinforced concrete structures. In particular, the analysis of reinforced concrete column, adopting the microplane model as a constitutive relation for concrete, will be presented. The problem is solved using exphcit time integration. An efficient parallel algorithm is proposed. Keywords: Parallel computation; Explicit dynamics; Microplane model; PC cluster

1. Introduction Detailed analysis of engineering structures is usually based on a three dimensional finite element models involving advanced nonlinear material models and fine computational grids. These models can be computationally very demanding and can yield to a very large nonlinear problems, which are very time consuming when to be solved on a single processor workstation. Parallehzation of the problem can reduce the computational time and, in some cases, it can be the only way of how to solve the problem at all. Several types of computer hardware architectures are available. They can be based on shared-memory or massively parallel (multiple instruction and data) concepts. Another way is to connect office workstations (PCs) together into a computer cluster. This workstation network can provide a powerful parallel machine. Despite of variety of platforms available, the message passing is a common communication tool, available on all platforms.

2. Modeling of reinforced concrete structures Modeling of reinforced concrete structures typically leads to a very complex set of problems. These problems are solved using complicated nonlinear constitutive * Corresponding author. Tel.: +420 (2) 2435-5417, Fax: +420 (2) 2431-0775. E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

laws and nonlinear analysis. Microplane model for concrete [1,2] belongs between models describing the material on the high level of accuracy. However, this accuracy is counterbalanced by an increase of computational effort. The concept of the microplane model is relatively simple. But from numerical point of view, it is computationally extremely demanding. Computation of the stress tensor in a single integration point involves the strain projection to microplanes, evaluation of local microplane constitutive laws on each microplane and homogenization procedure for computing the overall stress tensor. Moreover, the tangent stiffness matrix cannot be directly obtained for the latest model formulation. Therefore, the use of implicit methods, which require the stiffness matrix, is cumbersome, due to extremely slow iteration process. An efficient computational scheme is based on an exphcit algorithm. If damping is expressed in a special form (for example one may use Rayleigh damping), one does not need stiffness matrix. Typically, the non-equilibrated forces are applied as loading in the next time step. The use of diagonal mass matrix leads to a very efficient computational scheme, which can be parallelized in a straightforward way.

3. Parallelization strategy The adopted parallelization strategy is based on a mesh partitioning technique. The two dual partitioning concepts exist. With respect to the character of the cut, the node-cut

370

/ Nemecek et al /First MIT Conference on Computational Fluid and Solid Mechanics

and element-cut techniques can be distinguished (see [3,4]). The node-cut concept leads to duplication of finite element nodes on neighboring partitions. On the other hand, the element-cut approach leads to duplication of elements divided by the cut. Since the computational demands associated with elements computations are typically superior to that for nodes, the node-cut approach is computationally more efficient. The ratio of element versus nodal computational demand is even higher for microplane model. Due to these facts, the node-cut approach has been used in this study. All necessary communication between partitions is buffered by the FE code. Explicit buffering leads to an effective memory usage and data exchange. A nonblocking point to point communication has been used, allowing to potentially use the specialized hardware and leading to more scalable code. The proposed approach was successfully implemented into an existing object oriented finite element environment. The MPI message passing library mpi was used. 3.1. Node cut concept In this approach, the cut runs through element sides and corresponding nodes. The nodes lying on partition boundaries are marked as shared nodes. These nodes are shared by all adjacent partitions. On each partition, the shared nodes have assigned unique local code numbers. The elements are uniquely assigned to particular partitions. In order to guarantee the correctness of the solution of the partitioned problem, a modification of the single processor algorithm is necessary. Central difference node-cut algorithm can be formulated in the following steps: Mass contribution exchange for shared nodes: while not finished loop (i) assemble load vector Rj; (ii) compute local real nodal forces Ft; (iii) exchange real nodal force contributions for shared nodes; (iv) solve displacement increment 8rt\ (v) compute acceleration and velocity fields; (vi) update displacement vector r,+A/ = ^r + 5r^; (vii) increment time r = r -h Ar; endloop; The equilibrium equations at local partition nodes are solved without any change. However, at shared nodes, one is confronted with the necessity to assemble contributions from two or more adjacent partitions. The correctness has been enforced by exchange of contributions of shared node internal forces between partitions. Each partition has to add the contributions received from neighboring partitions to the locally assembled shared node internal force and to send its shared node contributions to neighboring partitions. Since the partitions contain only the local elements, the

Fig. 1. FE model of reinforced concrete column. correct mass matrix has to be established by an analogous data exchange operation before the time-stepping algorithm starts. The process of mutual exchange of internal nodal force contributions must be repeated for each time step to guarantee the correctness of the solution. In order to Table 1 Details of FE model of reinforced concrete column Material region of the column

Type of elements

No. of elements

Linear elastic concrete (end parts) Microplane model M4 (middle part) Linear elastic steel (end plates) Elastoplastic steel (reinforcement)

Linear space Linear space Linear space 3D beams

4032 3453 864 744

Total number of nodes, 9971; total number of elements, 9096; total number of DOFs, 30525.

/. Nemecek et al /First MIT Conference on Computational Fluid and Solid Mechanics

371

80 r

2 4 6 8 Number of processors

0

2 4 6 Number of processors

Fig. 2. Speedups and real time consumed for 1-8 processors. efficiently handle this exchange, each partition assembles its send and receive communication maps for all partitions. While the send map contains the shared node numbers, for which the exchange, in terms of sending the local contributions to a particular remote partition, is required, the receive map contains the shared node numbers, for which the exchange, in terms of receiving the contributions from a particular remote partition, is required.

4. Example Reinforced concrete column (see Fig. 1) was modeled in order to simulate experimental results. The finite element mesh included several material models. Concrete was modeled by microplane model [1]. Elastoplastic behavior with isotropic hardening was used for steel reinforcement. The structure was analyzed by an explicit time integration. Details of FE model are summarized in Table 1. Good agreement with experimental results was obtained. However, the analysis took a very long time (70.13 h) on a single processor workstation. The proposed parallel algorithm was applied in order to reduce the computational time. The time was significantly reduced and an excellent parallel efficiency was achieved (see Fig. 2).

5. Conclusions The microplane model is a constitutive model that can be successfully used for 3D modeling of concrete structures. On the other hand, it is computationally very de-

manding. Therefore, parallelization of the analysis is very useful in this case. An efficient parallel algorithm was proposed and implemented into an existing object oriented finite element environment. It was applied on the analysis of reinforced concrete column. A significant reduction of the computational time and an excellent parallel efficiency was achieved.

Acknowledgements This work has been supported by the Ministry of Education of Czech RepubHc under Contract 104/98:210000003.

References [1] Bazant ZP et al. Microplane model for concrete. I: stress-strain boundaries and finite strain. J Eng Mech 1996;122(3):245-254. [2] Bazant ZP et al. Microplane model for concrete. II: Data delocalization and verification. J Eng Mech 1996;122(3):255262. [3] Krysl P, Bittnar Z. Parallelization of finite explicit dynamics with domain decomposition and message passing. Int J Numer Methods Eng, submitted for publication. [4] Patzak B, Rypl D, Bittnar Z. Explicit parallel dynamics with nonlocal constitutive models. In: Topping BHV, Kumar B (Eds), Developments in Analysis and Design using Finite Elements Methods. Civil-Comp Press, 1999. [5] Message Passing Interface Forum, MPI. A Message-Passing Interface Standard, University of Tennessee, 1995.

372

Application of ALE-EFGM to analysis of membrane with sliding cable Hirohisa Noguchi^*, Tetsuya Kawashima^ " Keio University, Department of System Design Engineering, Yokohama 223-8522, Japan ^ Keio University, Graduate School of Mechanical Engineering, Yokohama 223-8522, Japan

Abstract In the analysis of a cable-reinforced membrane structure, there are several complicated problems, such as fold of the membrane by cable, the sliding of the cable on the membrane surface and so on. As FEM can hardly be used to analyze these problems, the authors have applied a meshless method based on EFGM for the analysis of membrane structures with cable reinforcement. ALE formulation and the patch technique are adopted to enhance EFGM and to overcome these problems. A numerical example is demonstrated to show the validity of the method. Keywords: Element free Galerkin; Arbitrary Lagrangian-Eulerian; Moving discontinuity

1. Introduction For the design of membrane structures, several kinds of analyses must be carried out, such as form finding analysis, stress analysis and cutting analysis. In the finite element method (FEM), a different model with appropriate mesh in each analysis is required. While in the meshless method, a set of analyses can be carried out by using only one model, because it has no elements. Cables to reinforce strength often tense large membrane structures. In order to analyze cable-reinforced membrane structures, the folded membrane by cable, which yields discontinuity of slope, has to be taken into account. Furthermore, these cables are often attached in a way that permits them to slide over the surface of the membrane so that the cables can find equilibrium under the applied gravity load, wind load and so on. In the conventional FEM, discontinuity of slope can only be treated at the boundary of elements, so re-meshing or special development of the element which allows folding is necessary to model this moving discontinuity. Bearing this in mind, the authors have developed a meshless system for the analyses of membrane structures [1]. In the meshless method, it is not necessary to subdivide * Corresponding author. Tel.: +81 45-566-1737; Fax: +81 45566-1720; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

the model to be analyzed into elements and, therefore, the fold can be modeled at arbitrary points on the membrane surface and it can move freely on the surface by redefining the nodal arrangement. In the proposed model, a patch is defined to model the part surrounded by cables and the moving least squares interpolation is only defined in each patch and CO continuity condition is imposed at patch boundary by the penalty method. In order to analyze the sliding cable on the membrane, the patch interface must be moved with the cable. Therefore, the arbitrary LagrangianEulerian (ALE) method [2] is combined with the element free Galerkin method [3] (EFGM) to model the sliding cable. ALE formulation is used for stationary of potential energy to consider the effect of friction and the deformation of membrane and sliding between cable and membrane can be represented as a different displacement component.

2. Numerical example Fig. 1 shows an analysis model of a membrane structure represented by bi-quadratic function. The arrangement of nodes is shown in Fig. 2. This model is separated by two patches at JC = 0, and the number of nodes at one patch is 6 X 11. The radius of the domain of influence is 1.1c. The section stiffness Eh is equal to 6.0 x 10^ N/m, where h denotes the thickness, and Poisson's ratio is 0.267. The

H. Noguchi, T. Kawashima /First MIT Conference on Computational Fluid and Solid Mechanics

tensed cable (u is fixed)

373

0.5r^ 0.5r'

o.i{(rO-i}{(rf-i} -l
-0.25

0.00

0.25

-l
0.50

Fig. 1. Analysis model of membrane with sliding cable.

Ay /Eulerian nodes 0.5L,

-0.50

-0.25

0.00 X

0.25

0.50

Fig. 3. Configuration after deformation along line j = 0.

patch 1

patch 2

Fig. 2. Nodal arrangement. Lagrangian displacement is fixed along line jc = 0, because of the tensed cable for reinforcement. 'Eulerian nodes' in Fig. 2 show the location of fixed cable, so they only have the Eulerian displacement. The Eulerian displacement shows the slide between the cable and the membrane surface. In this analysis, friction is not taken into account for brevity. The pressure load, 5.0 x 10^ Pa is applied to the z-direction only for patch 2. Then, the analysis result under the condition that the cable is fixed on the membrane (u = 0) is compared with the case that the cable is free from the membrane (u: free). The configuration along line >' = 0 after deformation on each condition is shown in Fig. 3. In the case in which the cable is fixed on the membrane, the right side of membrane is swollen and the left side is not deformed. On the other hand, in the case in which the cable slides, the right side of membrane is more swollen and the left side becomes straight. The locations of nodes on the cable are coincided in each condition, so movement of only material points can be represented.

3. Conclusion The proposed ALE-EFGM is applied to the analysis of a membrane structure with a sliding cable. Additionally, by using the patch technique, the discontinuous slope of the membrane surface can be represented. In future research, analysis taking into account the friction and stiffness of cable for reinforcement should be carried out.

References [1] Kawashima T, Noguchi H. The analyses of membrane structures with cable reinforcement by element free method. Proceedings of the Fourth Asia-Pacific Conference on Computational Mechanics 1999;2:1003-1008. [2] Haber RB. A mixed Eulerian-Lagrangian displacement model for large-deformation analysis in solid mechanics. Comput Methods Appl Mech Eng 1984;43:277-292. [3] Belytschko T, Lu YY, Gu L. Element free Galerkin methods. Int J Numer Methods Eng 1994;37:229-256.

374

Modeling residual stresses at the stem-cement interface of an idealized cemented hip stem Natalia Nuiio *, Guido Avanzolini DEIS, Facolta di Ingegneria, Universita di Bologna, 40136 Bologna, Italy

Abstract The magnitude of the residual stresses caused by the cement curing at the stem-cement interface was determined experimentally. Using a finite element model, the effect of the magnitude of the residual stresses combined with the coefficient of friction at the interface, on the load transfer of an idealized stem-cement interface was investigated. The results show that the stress distributions are affected by the magnitude of the residual stresses; at least for an early post-operative situation, the residual stresses should be included in the analyses. Keywords: Cemented hip implant; Residual stresses; Coefficient of friction

1. Introduction

6 imposed

Loosening of cemented hip implants is one of the major causes of late failure of the arthroplasty. Debonding at the stem-cement interface and cracks within the cement are the primary cause of initial loss of fixation of the implant (e.g., [1,2]). In the present study, the load transfer of a cemented hip implant was numerically investigated to understand the mechanical behavior of the stem-cement interface. At first, an experiment was conducted to determine the residual stresses caused by the cement curing.

stram gauges 140

2. Experiments 2.7. Materials 65

An experiment was conducted to determine the residual stresses due to cement curing of an idealized stem surrounded by a PMMA cement mantle (Fig. 1). The diameter of the stem was slightly larger than the inside diameter of the cement mantle resulting in a press-fit problem. The Ti-6A1-4V alloy stem was 70 mm long with a 20 mm diameter. One end of the stem was fixed to the material testing machine (Instron 8502, Instron Ltd., USA). The * Corresponding author. Tel.: +39 0521 905 896; Fax: +39 0521 905 705; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

t T

i

46

Fig. 1. Schematic representation of the experiment conducted to measure the hoop strains on the outer surface of the cement mantle press-fitted over a stem. All dimensions are in mm.

N. Nuho, G. Avanzolini /First MIT Conference on Computational Fluid and Solid Mechanics PMMA mantle was 140 mm long, and had 20 mm and 30 mm for inside and outside diameters, respectively. Four superimposed triaxial rosettes with a gauge length of 1 mm were placed on the outer surface of the cement mantle; two on the medial and two on the lateral sides (Fig. 1). These strain gauges were connected to a multi-channel acquisition system (SAl, ADA S.R.L., Bologna, Italy) and the hoop strains were recorded. The strains did not directly measure the residual stresses due to curing since the gauges were placed on the cement after polymerization; they measured the deformation of the mantle in the tangential direction, indirectly measuring the residual stresses. Four stem-cement systems were tested. 2.2. Finite element model The geometry of the stem-cement system used in the experiments was based on [3]. The material properties were assumed linearly isotropic and homogeneous. The Ti6A1-4V stem and PMMA mantle were assigned Young's modulus of 110000 MPa and 2700 MPa, respectively, and Poisson's ratio of 0.3 and 0.35, respectively. The distal end of the stem was completely fixed. The three-dimensional finite element model was solved using Ansys 5.4. By symmetry, half of the system consisting of 6663 elements was modeled. The PMMA mantle and Ti-6A1-4V stem were modeled with solid elements, while the debonded interface consisted of 462 nonUnear contact elements, 3-D node-to-node elements, using elastic Coulomb friction behavior. The press-fit effect of the stem surrounded by a cement mantle was simulated by interferences varying between 0.01 and 0.04 mm assigned to the contact elements generating stresses at the stem-cement interface. 2.3. Determination of the residual stresses In the experiments, the cement mantle was compressed when press-fitted over the stem. Thus in the FE model, after modeling the press-fit effect, an axial displacement was applied to the mantle's proximal end to simulate relative sliding between the surfaces. Then, the axial displacement was removed, and it was verified from the FE analysis that slipping between the stem and mantle occurred. For each stem-cement system tested experimentally, an average value of the four hoop strains measured was computed. The numerical hoop strains, obtained at the corresponding locations of the experimental strain gauges, were recorded for the different interferences and plotted as a function of the interference. By comparing the experimental and numerical results, interferences varying between 0.021 and 0.037 mm simulated the hoop strains measured experimentally, thus indirectly determining the residual stresses exerted at the stem-cement interface. The hoop strains being related to the radial strains, the ra-

375

dial compressive stresses could be obtained from the FE analysis and ranged between 1.8 and 3.5 MPa.

3. Finite element analysis The interface characteristics on the load transfer of a cemented hip implant subjected to bending was numerically investigated for an early post-operative situation. Mann et al. [4] concluded that the behavior of the pushthrough-stem tests and the simulated FE model gave the best agreement when friction as well as residual stresses were included in the analysis; however, recent investigations do not include the residual stresses (e.g., [1,5]). The question was how are the results affected by not including the residual stresses. In addition, the effect of different surface conditions, simulated by coefficients of friction, was investigated. The new geometry of an idealized cemented hip stem surrounded by bone, shown in Fig. 2, was similar to [5]. In the FE model, the stem-cement interface was debonded, while the cement-bone interface was bonded since retrieved femoral components have shown that this interface is well fixed [2]. All materials were assumed linearly isotropic and homogeneous. The material properties of the stem and mantle were described earlier. The cortical bone mantle was 105 mm long and 7 mm thick, had Young's modulus of 15 500 MPa and Poisson's ratio of 0.28. The distal ends of the cement and the bone mantles were completely fixed. 3.1. FE analyses The press-fit effect was also simulated by assigning an interference to the contact elements from zero to 7.5 \im to generate compressive radial residual stresses at the interface varying between 0 and 3.6 MPa. The coefficient of friction varied between 0, 0.2 and 0.4. The half FE model was subjected to a transverse load of 300 N (Fig. 2). The stresses were determined in the cement at the stem-cement interface on the medial and lateral sides. The location of gap at the stem-cement interface was also investigated. 3.2. FE results On the medial side, for no residual stresses (0 |xm interference) away from the peak radial compressive stresses Gr that occurred at the proximal end because of the abrupt change in geometry, the stresses were zero (or even tensile) in the middle section of the interface. An increase in the interference from zero to 7.5 |xm had up to a 4-fold increase from 0.9 MPa to 4.4 MPa in the distal section. Friction had little effect (<6% increase). On the lateral side (Fig. 3), an increase in the interference had up to 50% increase in the peak stresses in the middle section and a 3-fold increase

N. Nuho, G. Avanzolini / First MIT Conference on Computational Fluid and Solid Mechanics

376

lateral side

proximal end

transverse load=300 N

distal end

36

medial side

Fig. 2. Three-dimensional FE mesh of the cemented hip implant analyzed. All dimensions are in mm. t

1 . -2

t

V

E

\

[

\

y

\ 1

t

0

//

A1 \

1

V

//

^^" y

^ .Xy^ 0^111, ;y-0.2 0 |im,/i=0.4 — X — 7.5)im,/v=0 —^— 7.5^m,/i=0.2 — ^ — 7.5 urn, /i=0.4

10

20

30

40

50

60

70

0^m,//=0.2

vf

0 i^m,//=0.4 7.5 ^m,//=0 7.5 ^m,//=0.2 7.5^m,//=0.4

1 80

Axial position, z (mm)

Fig. 3. Radial stresses Or in the cement at the interface on the lateral side for 0 |im and 7.5 |im interferences and coefficients of friction of /x = 0, 0.2, 0.4 versus the axial position z. Transverse load of 600 N. in the distal section, while an increase in the coefficient of friction decreased the stresses up to 20%. Zero or tensile stresses indicated a possible gap at the interface, i.e., separation of the stem-cement interface. An increase in the interference significantly reduced the gap extents, from zero to 7.5 iim interference (0 to 3.6 MPa compressive radial residual stresses) decreased from 53% to 5% the open status, even closing some gaps (Fig. 3). Friction had little effect; the gap extent decreased up to 4%.

30

40

50

Axial position, z (mm)

Fig. 4. Hoop stresses OQ in the cement at the interface on the lateral side for 0 |xm and 7.5 |xm interferences and coefficients of friction of /x = 0, 0.2, 0.4 versus the axial position z. Transverse load of 600 N. Peak hoop tensile stresses OQ (^16 MPa) occurred at the proximal end on the medial side caused by the abrupt change in geometry. The stresses were a little smaller on the medial side, and a little larger on the lateral side (Fig. 4) than the magnitude due to the press-fit effect related to the interference {OQ = 1.7 MPa): increasing from zero to 7.5 |xm caused a 10-fold increase from 0.2 MPa to 2 MPa on the lateral side. An increase in the coefficient of friction increased the stresses up to 20%.

A^. Nuno, G. Avanzolini /First MIT Conference on Computational Fluid and Solid Mechanics

111

4. Conclusions

References

This study accounted for the residual stresses and friction for an idealized cemented hip surrounded by bone and subjected to bending. Failing to include residual stresses underestimated the radial and hoop cement stresses at the interface. Since there is no chemical bond at the interface between the stem and the cement, the interface resistance depends on friction thus radial compressive stresses at the interface developed by the cement curing play a direct role. The residual stress magnitudes may vary with different implant designs, and although creep will decrease these stresses they will not disappear completely. Residual stresses reduced the gaps at the stem-cement interface as the load was distributed more uniformly. This study showed that the stress distributions are affected by the magnitude of the residual stresses; at least for an early post-operative situation, the residual stresses should be included in the analyses.

[1] Harrigan TP, Harris WH. A three-dimensional non-linear finite element study of the effect of cement-prosthesis debonding in cemented femoral total hip components. J Biomech 1991;24:1047-1058. [2] Jasty M, Maloney WJ, Bragdon CR, O'Connor D, Haire T, Harris WH. The initiation of failure in cemented femoral components of hip arthroplasties. J Bone Joint Surg 1991;73B:551-558. [3] Rohlmann A, Cheal EJ, Hayes WC, Bergmann G. A non-linear finite element analysis of interface conditions in porous coated hip endoprostheses. J Biomech 1988;21:605-611. [4] Mann KA, Bartel DL, Wright TM, Ingraffea AR. Mechanical characteristics of the stem-cement interface. J Orthop Res 1991;9:798-808. [5] Lu Z, McKellop H. Effects of cement creep on stem subsidence and stresses in the cement mantle of a total hip replacement. J Biomed Mater Res 1997;34:221-226.

378

Nonlocal numerical modelling of the deformation and failure behavior of hydrostatic-stress-dependent ductile metals H. Obrecht*, M. Brunig, S. Berger, S. Ricci Lehrstuhl fur Baumechanik-Statik, Universitdt Dortmund, August-Schmidt-Str. 8, D-44221 Dortmund, Germany

Abstract The paper deals with the numerical modelling of the large strain elastic-plastic deformation and failure behavior of metals which are sensitive to hydrostatic stresses. The model is based on an efficient logarithmic strain theory taking into account a generalized macroscopic yield criterion as well as a non-associated flow rule. In addition, the influence of nonlocal plastic variables is discussed. Numerical analyses show the influence of the geometry and boundary conditions of metal specimens as well as of various model parameters on the deformation behavior and failure modes of hydrostatic-stress-dependent metals. Keywords: Elastic-plastic metal; Hydrostatic stress dependence; Logarithmic strain; Nonlocal plasticity; LocaUzation; Finite element analysis

1. Introduction

2. Elastic-plastic continuum model

An important problem in computational plasticity is the modelling and prediction of the macroscopic finite elasticplastic deformation and localization behavior of metals subjected to complex loading conditions. Most numerical simulations in metal plasticity are based on isochoric theories using J2 yield criteria associated with non-dilatant normality flow rules. Experimental studies on the effect of superimposed hydrostatic pressure on the deformation behavior of iron based materials [1] have shown, however, that the flow stress depends approximately linearly on hydrostatic pressure. These effects are generally ignored in computational metal plasticity, but have been included in an extended theory and corresponding finite element analyses presented by Brunig [2]. Furthermore, nonlocal effects are important when deformation mechanisms caused by microscopic phenomena as well as scale effects have to be considered in order to explain and predict a number of experimentally observed critical phenomena [3-5].

A consistent large strain elastic-plastic theory is presented which relies on the introduction and the multiplicative decomposition of the mixed-variant metric transformation tensor Q into elastic and plastic parts [2]. This leads to the formulation of an appropriate logarithmic strain tensor A = | l n Q , and the strain rate tensor H = ^Q^^Q can then be additively decomposed into an elastic and plastic part W^ and H^'. The work-conjugate Kirchhoff stress tensor T is related to the elastic logarithmic strain tensor A^^ via a hyperelastic constitutive law based on a free energy potential function, and the corresponding generalization of Hooke's law for large strain analyses has been shown to be in good agreement with experimental results. In order to characterize the hydrostatic stress-dependent plastic material behavior, a phenomenological I1-J2-J3 flow theory including isotropic work-hardening is used [6] and the yield condition is written in the nonlocal form fP\h,

Ji, /3, c) = « /i + y j 2 + b ^ , - c(y,

y)=0 (1)

* Corresponding author. Tel: -h49 (231) 755-2536; Fax: -H49 (231) 755-2532; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

where /i = trT denotes the first invariant of the stress tensor, J2 = \ dev T • devT and J3 = det(devT) the second and third invariants of its deviator, a and b the hydrostatic stress coefficients, and c represents the strength coefficient

H. Obrecht et al. /First MIT Conference on Computational Fluid and Solid Mechanics

379

1.2 1

0

fiiiiMiiMiiiiniiiiniiiiniiiiiiiiiiiiiiiiTiiiiiiiiiiiiiiniiiMiiiiiiiiiiiiniuiniiiiiiiniiiiiiiiiiiiiiiiiiiiiiiriiiiiiiiiiiHiiiiMMii^

0

0,01

0,02

0,03

0,04

0,05

0,06

u/loH

Fig. 1. Load-deflection curves. which depends on the equivalent strain y and its nonlocal counterpart y, where the latter accounts for the inhomogeneously evolving microstructure. The inelastic part of the strain tensor is governed by the plastic potential function g^^ = ^2 which leads to the non-associated flow law W^ =X-

ax

= x

1 2^72

devT

(2)

where A is a non-negative scalar value.

3. Numerical solution procedure The computation of the local stress and strain histories is performed via an accurate and stable explicit scalar integration procedure which is derived from the well-known plastic predictor method. In addition, due to the nonlocal yield criterion, the consistency condition f = 0 leads to an elliptic partial differential equation. While Sluys et al. [3] and others satisfy the consistency condition in a weak sense, e.g. via mixed-type finite elements, the respective partial differential equation is solved explicitly using the finite difference method at each iteration of a loading step. As a result, no additional boundary conditions are needed, the displacement-based finite element procedure is gov-

erned by the standard principle of virtual work, and the associated linearized variational equations are obtained in the usual manner from a consistent linearization algorithm.

4. Numerical example The influence of geometry and boundary conditions as well as of various model parameters on the deformation behavior and failure modes of hydrostatic-stress-dependent metals is studied. For a uniaxially loaded rectangular specimen under plane strain conditions, for example, corresponding load deflection curves are shown in Fig. 1. Numerical calculations which neglect the effect of hydrostatic stress on the flow characteristics of metals predict the onset of localization at an elongation of U/IQ = 1.77% and the corresponding shear band inclination is 43° with respect to the loading axis. As can be seen from Fig. 2, relatively wide shear bands with a superposed diffuse neck are formed in the center region as the axial elongation is increased. The numerical calculations based on the generalized I1-J2 yield condition, on the other hand, predict initial localization at a noticeably smaller elongation of u/lo = 1.33% (lower curve in Fig. 1) together with a smaller load maximum. Localization also occurs in the center region of the specimen and the shear band direction

380

H. Obrecht et al /First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 2. Deformed configuration (a = a/c = 0).

I I I I I I I I I I I I I 111 rrr

Fig. 3. Deformed configuration (a = a/c = 20 TPa ^).

is again 43° (see Fig. 3), but the shear bands developing across the specimen are now somewhat narrower. [2] 5. Conclusions An efficient nonlocal numerical model for the large strain elastic-plastic deformation and failure behavior of metals which are sensitive to hydrostatic stresses is presented. The numerical analyses show that nonlocal parameters lead to mesh-independent deformation modes and that hydrostatic components may have a significant effect on the onset of localization and the associated failure modes. In particular, they may lead to a noticeable decrease in ductility.

References [1] Spitzig WA, Sober RJ, Richmond O. The effect of hydrostatic pressure on the deformation behavior of maraging and

[3]

[4]

[5]

[6]

HY-80 steels and its implications for plasticity theory. Metall Trans 1976;7A: 1703-1710. Briinig M. Numerical simulation of the large elastic-plastic deformation behavior of hydrostatic stress sensitive solids. Int J Plast 1999;15:1237-1264. Sluys LJ, de Borst R, Miihlhaus HB. Wave propagation, localization and dispersion in a gradient dependent medium. Int J Solids Struct 1993;30:1153-1171. Li X, Cescotto S. A mixed element method in gradient plasticity for pressure dependent materials and modelling of strain localization. Comput Methods Appl Mech Eng 1997;144:287-305. Briinig M, Ricci S, Obrecht H. Nonlocal large deformation and localization behavior of metals. Comput Struct, submitted for publication. Briinig M, Berger S, Obrecht H. Numerical simulation of the localization behavior of hydrostatic-stress-sensitive solids. Int J Mech Sci 2000;42:2147-2166.

381

Estimation of tool/chip interface temperatures for on-line tool monitoring: an inverse problem approach Lorraine Olson *, Robert Throne Department of Mechanical Engineering, University of Nebraska, Lincoln, NE 68588-0656, USA

Abstract We examine a steady inverse heat transfer problem that arises in on-line machine tool monitoring: identifying tool/chip interface temperatures from remote sensor measurements. We develop a new set of inverse approaches specifically for this problem, and inverse solutions are computed with all methods for two temperature profiles and various noise levels. Two of the new methods are robust and give reasonable accuracy. Combining data from temperature and flux sensors (data fusion) is far more effective than using temperature sensors alone, and with data fusion, the inverse can be computed robustly with information from only four sensor locations. Keywords: Inverse problem; Heat transfer; Data fusion; On-line monitoring

1. Introduction On-line machine tool monitoring, or tool condition monitoring, has been of interest for more than two decades because it promises to improve throughput, reduce downtime, and increase part quality. Ulsoy and Koren [1] have estimated that a $30K controller/monitor added to a $300K machine tool would produce a savings of $3 million over the life of the machine tool, which results in a return of 1000 times the original investment. Although significant progress has been made in the field, more work still remains to be done before the promises are fully realized. With recent advances in computational power, sensor fusion approaches which employ numerous types of sensors (cutting force, acoustic emission, motor torque, etc.) as inputs to a data synthesis program have been widely investigated. In this study, we will focus our attention on a steady inverse heat transfer problem that arises in on-line machine tool monitoring and data synthesis — identifying tool/chip interface temperatures from remote sensor measurements in turning. For turning processes that are not yet optimized in terms of feeds, speeds, or depth of cut, information about the tool/chip interface would help guide adjustments of * Corresponding author. Tel.: +1 (402) 472-5082; Fax: +1 (402) 472-1465; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

operating parameters. For processes in which the preferred cutting conditions have already been established, continuous monitoring of tool temperatures and stresses would allow the tool condition to be estimated. Identification of tool/chip interface temperatures from remote measurements in turning has been attempted in the past, but with only limited success. Most of the existing approaches have focused on the forward problems, in which the entire system is simulated and all material properties and boundary conditions must be specified precisely. Accurate models of the frictional properties, extent of the tool/chip contact region, and the heat generated by friction and material deformation are essential. Although the models are gradually becoming detailed enough to give quantitatively accurate results, the computational effort required for each simulation is enormous, and measured data on temperatures and stresses is not directly factored into the solution. Only a few researchers have considered inverse simulation of the tool itself in order to infer tool/chip interface conditions from remotely measured data. In inverse techniques, we need only accurate geometry and heat transfer properties for the tool, the boundary conditions (temperatures and surface stresses) are not required at all boundary locations, and the relationship between the remotely measured data and the variables we wish to monitor is directly included in the formulation.

382

L. Olson, R. Throne/First MIT Conference on Computational Fluid and Solid Mechanics

Tool/chip Interface (Prediction Surface)

Toolholder

(a)

Ambient Temperature Edges

Fig. 1. Schematic of simplified physical system. (All linear dimensions in mm.) 2. Modeling For this study, we will model the physical system with the approximate two-dimensional geometry shown in Fig. 1. We assume that the governing equation is that of steady heat conduction, with known (constant and isotropic) conductivity k. We replace the tool-chip interface with a specified temperature distribution over a known distance of 2 mm. Convection from other surfaces is neglected and replaced with an insulated boundary condition. The conduction of heat through the toolholder to the lathe is modeled by taking two of the sides of the toolholder to be at an ambient temperature of 30 C. The conductivities of the tool holder and shim were both taken to be 50 W/mC, while the conductivity of the tool was taken as 35 W/mC. Fig. 2 shows the finite element mesh employed. With this geometry, we studied two tool temperature profiles previously reported by other researchers from experimental measurements. The first temperature profile is that reported from experiments by Ostafiev et al. [2] for machining of mild steel (BHN 120) with a carbide tool (72.7% WC, 17.3% (Ti, Ta, Nb)C, 10% Cobalt). The second temperature profile is for orthogonal machining of 12L14 steel with an M34 tool, as reported by Stevenson et al. [3]. We use standard finite element methods [4] to create the numerical model. For computational purposes, it is useful to recognize that the prediction surface temperatures completely determine the remaining temperatures, so that a transfer matrix may be created to directly relate the sensor

Fig. 2. Finite element model for tool, (a) Overall view, (b) View near tool tip. (Circles indicate sensor locations for 45 sensors. Filled circles indicate sensor locations for four sensors.) temperatures T^ and the prediction surface temperatures (1)

T. = Z T T Z .

Similarly the heat fluxes at the sensors q due to these temperature distributions may be calculated Q =

(2)

^T,Tp

3. Inverse algorithms In the vector projection inverse methods, we minimize n with respect to an estimate of the prediction surface temperatures Xp-

Zr,r^

(3)

The first term on the right hand side of this equation is the error between the measured temperatures on the sensor surface and the values associated with the forward projection of our estimated prediction surface temperatures. The second term is the error between the measured fluxes on the sensor surface and the values associated with the forward projection of the estimated prediction surface temperatures. P represents the weighting of the two terms and is based on

L. Olson, R. Throne/First MIT Conference on Computational Fluid and Solid Mechanics

383

1600r

Distance from Tool Tip (mm) Fig. 3. Typical inverse solutions on the prediction surface for profile one, a = 5, two modes. (Solid line, true solution; dashed line, trigonometric basis vectors; dotted line, polynomial basis vectors.) the relative sizes of the measured values,

in the middle third, and the third term is unity over the last third of the range, and so forth. (4)

Alternatively, either the temperature information or the flux information can be used alone. For our new vector projection inverse methods, we assume a set of basis vectors for the prediction surface temperatures X^ = X^a, and then insert directly into Eq. (3). By minimizing n with respect to a, we can identify a linear algebraic system of equations for the expansion coefficients g^. The problem is regularized by limiting the number of basis vectors chosen for the expansion. Because we will solve for only a few coefficients, the choice of the vectors for these vector projection methods is crucial. We will examine three choices: a trigonometric series, a polynomial series, and a series of 'on-off' functions. Each of these functions is only defined on the prediction surface. If L^ is the length of the prediction surface and d is the distance from the tool tip, the trigonometric series consists of a constant, cos27td/Lp, cosAnd/Lp, etc. The polynomial series also begins with a constant, but then includes the linear function d/Lp, the quadratic function {d/LpY, etc. The 'on-off' functions are slightly different: if one term is used the function is a constant, if two terms are used we have one term which is unity over the first half of the range and a second term which is unity over the second half of the range. If three terms are used, the first is unity over the first third of the range, the second is unity

4. Testing and results For each of the two temperature profiles, we first computed the forward solution for the temperatures and fluxes at the sensors. Then we added random Gaussian noise with a specified noise level (variances of cr = 5°C and 20°C in the temperature and o/p W/m^ in the flux) to create simulated measurements. Each inverse method was then used to predict the tool tip (prediction surface) temperatures, and a relative error between the estimated and predicted temperatures. This process was repeated 100 times for each inverse method and for each temperature profile, and a mean and standard deviation for each error measure was computed. Initially, we examined results using 45 sensor locations, and studied the effects of varying the number of modes used for each method. Regardless of the basis vectors chosen, only two coefficients could be chosen with reasonable accuracy. This is a result of the extremely poor condition number of the coefficient matrix XTTSince we can identify only two coefficients, we also examined inverse results for four sensor locations. We saw very little degradation in the inverse solution as the number of sensor locations was reduced from 45 to 4. In fact, we found inverse solutions using four sensor locations with both temperature and flux data (fusion) were substantially

384

L. Olson, R. Throne /First MIT Conference on Computational Fluid and Solid Mechanics 800 r

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distance from Tool Tip (mm) Fig. 4. Typical inverse solutions on the prediction surface for profile two, a = 5, two modes. (Solid line, true solution; dashed line, trigonometric basis vectors; dotted line, polynomial basis vectors.) better than inverse solutions using 45 temperature sensors. Clearly the fusion of the temperature and flux data which occurs in Eq. (3) is important. Two of the best methods over all of the noise levels, error measures, and temperature profiles were the trigonometric and polynomial basis vectors. Figs. 3 and 4 show typical inverse solutions for the temperature along the prediction (tool/chip interface) surface. These pictures show the true temperature on the prediction surface, as well as inverse solutions from four sensors with a noise level of (7 = 5°C. The same noise realization was used in calculating the inverse for both techniques on a given plot.

noise levels with two different tool/work interface (prediction surface) temperature profiles. Because of the severe ill-conditioning of the system, each method could identify only two coefficients. However, good performance of the vector projection algorithms could be obtained with only four sensors. Of the vector expansion methods examined, a polynomial series and a trigonometric series gave the best results, with good stability and low errors. In addition, we found that employing temperature measurements alone was not nearly as effective as a fusion of the temperature and flux data.

References 5. Concluding remarks A difficult inverse problem arises in the context of online machine tool monitoring: identifying the tool/work interface temperatures from measurements made at more remote locations. In this study, we examined a two-dimensional model problem in which the geometry, material properties, and boundary conditions were closely related to the fully three-dimensional physical problem. We introduced a new class of vector projection inverse methods for this problem, with several different basis functions. Each of the inverse methods was tested for various

[1] Ulsoy A, Koren Y, Control of machining processes. J Dyn Syst Meas Control 1993;115:301-308. [2] Ostafiev V, Kharkevich A, Weinert K, Ostafiev S. Tool heat transfer in orthogonal metal cutting. J Manuf Sci Eng 1999;121:541-549. [3] Stevenson MG, Wright PK, Chow JG. Further developments in applying the finite element method to the calculation of temperature distributions in machining and comparisons with experiment. J Eng Ind 149-154, 1983. [4] Bathe KJ. Finite Element Procedures in Engineering Analysis. Englewood Cliff's, NJ: Prenfice-Hall, 1982.

385

Instability problems in shell structures: some computational aspects Costin Pacoste *, Anders Eriksson Structural Mechanics group, KTH, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Abstract The paper discusses certain numerical aspects related to the stability analysis of shells or shell like structures. A general multi-parametric description is used from the onset. The numerical solution procedures are based on systematic manifold evaluations, as described in Eriksson et al. [Comput Methods Appl Mech Eng 1999;179:265-305]. Efficient implementation of these procedures requires however a non-linear finite element which is accurate, but also inexpensive. In the context of element formulation, the emphasis is placed on the parameterization of finite 3D rotations. Keywords: Shell structures; Instabilities; Numerical methods; FEM; Parameter dependence; finite rotations

1. Introduction

1.1. The multi-parametric setting

The development of an accurate and efficient analysis strategy for instability problems in shells or shell like structures, should necessarily be based on a multi-parametric formulation with higher-dimensional solution sets as the natural outcome. The present paper discusses certain aspects related to the numerical treatment of such parameterized solution sets. The solution methods are based on generalized path-following procedures for augmented equilibrium problems. A brief review of this topic is given in this paper, with further details in Eriksson et al. [1,2]. The efficient implementation of the above mentioned procedures requires, however, a non-linear finite element which is not only accurate but also inexpensive. The formulation of a co-rotational flat facet triangular element, specially designed to fit the algorithm for multi-parametric analyses, defines the second aim of the paper. In this context special emphasis is given to the parameterization of finite 3D rotations which is an issue of relevance for both the element definition and solution algorithms. Further details on the element formulation are given in Pacoste et al. [3].

The general problem setting is that of a discrete model of a quasi-static elastic structural problem, with N degrees of freedom. The governing equations can then be written:

* Corresponding author. Tel.: +46 (8) 790-8044; Fax: +46 (8) 216-949; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

G(z).

0

(1)

where F is the set of N equilibrium equations, and g is a set of r auxihary conditions, which defines a subset of equilibrium states sharing a certain common property. A specially interesting case of augmenting equations are criticality indicators. Several augmentations have been proposed for this condition, as described by Eriksson [4]. 1.2. Solution manifolds The solution to the problem gives a set of combinations z = [d^, A^]^ of N state variables d and p control variables A, fulfilling both sets of equations. This represents a set of (p — r)-dimensional manifolds, with curves as special cases, i.e. p = r -\- I. The possibiHties to solve for one-dimensional generalized equilibrium paths are extensively discussed in Eriksson et al. [5]. Both basic equilibrium paths and fold lines are described. In addition, Eriksson et al. [2] tentatively discusses the possibilities to generalize and extend the

C Pacoste, A. Eriksson /First MIT Conference on Computational Fluid and Solid Mechanics

386

one-dimensional numerical procedure, in order to investigate two-dimensional equilibrium manifolds, i.e. solution surfaces. The solution methods are viewed as extensions to common incremental-iterative algorithms. Standard choices for the constraining equation and continuation step length can be adopted. The modifications which are required in a multi-parametric context, are related mainly to the evaluation of the tangent space, scaling of variables as well as equations and use of Lagrange multiplier relations. One additional aspect refers to the detection and isolation of so-called 'special solution sets'. The term is introduced as a generalization of the critical points occurring on a loaddisplacement curve. In the multi-dimensional setting, the special solutions will become manifolds, rather than points. Thus, for a two-dimensional solution surface the special solution states will be similar to contour lines of a variable or a stability coefficient, cf. Eriksson et al. [2].

2. Element formulation The objective of this section is to discuss the formulation of a co-rotational flat facet triangular shell element. The central idea in this context, is to introduce a local coordinate system which continuously rotates and translates with the element. The local displacements q] are then obtained from the global 'q] as q^j = q^iiqj), and used in the strain energy function 0 = 0 (q^j). The internal force vector and tangent stiffness matrix in global coordinates are then obtained through successive differentiation, according to:

k =

d^0

dq'i dq]

dq^drj dq[ dq^

d0

d

dq'j dq^

(2)

2.1. Parameterization of finite 3D rotations For the definition of the element kinematics, an orthogonal matrix R is used to represent large 3D rotations. The admissible variations 8R required in Eq. (2), are evaluated as: 8R = o)R

(3)

with (o denoting spatial angular variations. This alternative, which is adopted in most papers dealing with the co-rotational formulation, leads to an updating of the rotation variables via: R'^\=cxp(Aa>i^,)Ri^,

(4)

for each iterative correction Acol. In contrast with the above mentioned procedure, two different alternative are explored in the present paper. The

first one is based on the rotation vector ^ while the second is based on the spatial form of the incremental rotation vector W and its iterative increment, as advocated by Ibrahimbegovic [5]. Corresponding to these two choices, additional changes of variables from w to «f^ or alternatively from (o to W, must be introduced, as described in Pacoste et al. [3]. 2.2. Change of variables: (o —> ^ The key equation for constructing such a change of variables is given by: (5)

(o = Tsms^

relating the spatial angular variations o) to the variations 8^ of the rotation vector. The particular expression of the linear operator Ts can be found in Ibrahimbegovic [5] or Pacoste et al. [3]. Now, let uj and ^ / denote the translation and rotational vector at node /. In terms of these variables, the internal forces at node / can be obtained through differentiation. Thus, using the chain rule.

F,=

duj

d0

03

ra^i

d(Oi

d0

T

' d0 " z=z

03

.dWi_

a^/J

dUi

= H]f,

(6)

_9(«>/_

Using Eq. (5), the transformation matrix Hj corresponding to node /, can further be written as: H,=

h

O3

O3

Ts{^i)

(V)

The corresponding transformation equation for the tangent stiffness is obtained from Eq. (6) through an additional differentiation. The main advantage of this parameterization is that the rotational variables become additive and the necessity of a special updating procedure is avoided. This is of special importance for an efficient implementation of a multi-parametric solution algorithm. 2.3. Change of variables: (o

W

The drawback of the previous parameterization lies in the fact that the magnitude of the rotation vector must be restricted to values less than 2;r. In order to remove this drawback, a parameterization based on the incremental rotation vector W and its iterative increment 8W is also explored. Note that additive updates will still apply, but this time only within an increment at the level of the iterative corrections. The key equation in this case is given by: 00 = Ts{W)8W

(8)

387

C. Pacoste, A. Eriksson /First MIT Conference on Computational Fluid and Solid Mechanics which provides the connection between (o and 8W. Note that since Eqs. (5) and (8) are formally identical, the expressions of the transformation matrices are very similar to the preceding case.

3. Numerical examples 3.1. Deploy able ring

z,w

The first example refers to a ring with geometrical and material characteristics as shown in Fig. 1. The ring is fully clamped at one end and loaded at the other end by a twisting moment, through a rigid plate. At the loaded end, the center point of the section (i.e. (240, 0, 0)) is restricted to move along and rotate around the X axis. The results obtained for this problem are shown in Fig. 2. Both types of rotation parameterization were con-

E = 200000; i/=0.3

E=2100000 -

v=0.3

25

y.v

75 Fig. 3. Simply supported compressed C-beam.

sidered in the analysis. As expected, the first formulation based on the total rotational vector, fails for values of &x{A) > 27r, whereas the second is unaffected and the computations can be continued up to any value of @xiA)- It should also be mentioned that for the loading and boundary conditions of this problem, the two formulations produce identical results. 3.2. Compressed C profile

MwA^;0y,

Fig. 1. The deploy able ring problem.

The second example refers to a simply supported beam of symmetric C section. The material and geometrical properties of the beam are defined in Fig. 3. A uniform compressive force P = 125000 A, is applied at the two ends. For the reference length Lo = 900 a cluster of buckling modes is present on the fundamental path. Twelve bifurcation points were thus isolated for values of the load factor 1.41 < X < 1.65. These include 10 local modes with different numbers of half-waves plus two global modes, i.e. bending and torsional modes. The fold-lines obtained for the first five critical states for variable length are shown in Fig. 4. Identical results were obtained for both type of rotation parameters.

4. Conclusions

Rotation at A Fig. 2. The deployable ring problem: load-displacement curves, 2 X 80 mesh.

The paper has discussed certain aspects related to the numerical analysis of instability phenomena in shell structures. A multi-parametric problem definition is used, in order to reflect the parameter dependence in the critical response of the structure. Certain aspects related to the definition of a shell element were also addressed. These aspects refer to the parameterization of large 3D rotations. It has been shown that in the context of a multi-parametric description the use of rotational variables which are addi-

388

C. Pacoste, A. Eriksson /First MIT Conference on Computational Fluid and Solid Mechanics References

Fig. 4. Fold lines for the simply supported compressed C-beam. live, at least at the level of iterative corrections is essential for the algorithm.

[1] Eriksson A, Pacoste C, Zdunek A. Numerical analysis of complex post-buckling behaviour using incremental iterative strategies. Comput Methods Appl Mech Eng 1999; 179:265305. [2] Eriksson A, Pacoste C. Two-dimensional solution sets in non-linear quasi-static structural mechanics. Technical report, Dept Struct Engrg, Royal Institute of Techn, Stockholm, 2000 (manuscript). [3] Pacoste C, Eriksson A. Element formulations and numerical techniques for stability problems in shells. Comput Methods Appl Mech Eng (submitted). [4] Eriksson A. Structural instability analyses based on generalised path-following. Comput Methods Appl Mech Eng 1998;156:45-74. [5] Ibrahimbegovic A. On the choice of finite rotation parameters. Comput Methods Appl Mech Eng 1997;149:49-71.

389

Genetic algorithm for crack detection in beams Magdalena Palacz *, Marek Krawczuk Institute of Fluid Flow Machinery, Polish Academy of Sciences, Ul. Fiszera 14, skr. poczt. 621, 80-952 Gdansk, Poland

Abstract The aim of this paper is to test five of the most popular damage indicators with a genetic algorithm. The influence of measurement errors is also analysed in all cases. For the analysis are chosen those damage indicators which use changes in such modal parameters as natural frequencies and mode shapes and also differences between the curvatures of the damaged and undamaged structure in a given frequency range. All numerical calculations are based on a mathematical model of a cracked beam shortly presented in the paper (see details in [3]). Keywords: Damage detection; Genetic algorithm; Modal parameters

1. Introduction Last decades show a great interest in non-destructive damage detection techniques for a large complex structures. In most procedures the comparison of the baseline and subsequent vibration surveys have utilized such modal parameters as structural natural frequencies, mode shapes, and also direct frequency response functions. In this paper we try to elaborate a new efficient and time-saving method of estimating the crack parameters. For this purpose genetic algorithm is introduced with fitness functions built on the base of the most often used damage indicators.

2. Genetic algorithm Genetic algorithms (Fig. 1) are stochastic search methods that mimic the metaphor of natural biological evolution. Genetic algorithms operate on a population of potential solutions applying the principle of survival of the fittest to produce better and better approximations to a solution. At each generation, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics. This process leads to the evolution of populations of indi* Corresponding author. Tel.: +48 (58) 346-0881, ext. 109; Fax: +48 (58) 341-6144; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

viduals that are better suited to their environment than the individuals that they were created from, just as in natural adaptation. Genetic algorithms differ substantially from more traditional search and optimisation methods. The most significant differences are: • genetic algorithms search a population of points in parallel, not a single point, • genetic algorithms do not require derivative information or other auxiliary knowledge; only the objective function and corresponding fitness levels influence the directions of search, • genetic algorithms use probabilistic transition rules, not deterministic ones, • genetic algorithms are generally more straightforward to apply, • genetic algorithms can provide a number of potential solutions to a given problem.

3. Fitness functions In the genetic algorithm used there is a minimum of analysed functions searched. Fitness functions have been constructed on the base of the most known five damage indicators, which use the changes in dynamic characteristics of the structure. Changes in the natural frequencies have been utilized in the fitness function built on the base of the Cawley-Adams [1] and DLAC [5] criterion, changes in the

M. Palacz, M. Krawczuk / First MIT Conference on Computational Fluid and Solid Mechanics

390

Evaluate objective function

Generate initial population

START

(

Generate a new population

j

yes

Are optimisation criteria met?

^ /

Best individuals

Selection

i

Recombination

I

— Mutation

Fig. 1. Structure of a simple genetic algorithm.

mode shapes have been utilized in the fitness function built on the base of the MAC [2] and COMAC [6] criterion, and changes in the differences between curvatures in a given frequency range have been utilized in the fitness function built on the base of the FRCM [4] criterion.

4. Mathematical model of the cracked beam In this section the mathematical model of the cracked beam used in numerical tests will be shortly presented (Fig. 2). The beam is divided to two segments connected by elastic element, which stiffness is calculated according to fracture mechanic's law — see [3]. The equation of natural vibration for a Bemoulli-Euler beam can be presented as follows:

a'^U, t)

a ' j U , t) -{-pF 0 (1) 3^4 ' "' ar2 where p is the material density, F denotes the cross-sectional area of the beam, y{x,t) the deflection of the beam, / the geometrical moment of inertia of the beam cross section and E is Young's modulus. The solution of Eq. (1) is sought in the form: >'(L, r) = y(L) sincot. Substituting this solution into Eq. (2), after simple algebraic transformation, one has EI

y^L)

k'^y(L) = 0

where k*

• w^pFIL* EI

(2)

Taking the function y(L) in the form of a sum of two functions. yi(L) = A^coihik

• L) + Bt smh{k • Z.) + Ci cos(/t • L)

+ D, sin(fc-L),

Le[0,lp)

(3)

Fig. 2. The model of a cracked beam, with a crack at location Ip.

y2{L) = Ai cosh(k •L)-\-B2sinh(k -\-D2sm{k-L),

Le{lp,l]

• L) + C2C0s(k • L) (4)

The boundary conditions in terms of the non-dimensional beam length Ip = x/L, can be expressed as follows: is zero displacement of the beam at the Ji(0) = 0 restraint point, is zero angle of rotation of the beam at the y\{0) = 0 restraint point, y\(ip) = y'li^P) is compatibility of the displacement of the beam at the location of the crack, y'jilp) - y'\{lp) = 0y2{lp) is total change of the rotation angle of the beam at the location of the crack, y'lQp) = y2{lp) is compatibility of the bending moments at the location of the crack, y'{'(lp) = y'jilp) is compatibility of the shearing forces at the location of the crack, is zero bending moment at the end of the y'lW -- 0 beam, =0 is zero shearing force at the end of the beam. Taking into account the boundary conditions one obtains the characteristic equation, which can be solved to determine the natural frequencies as a function of a location and the depth of the crack. With the natural frequencies found adequate mode shapes can be determined.

5. Exemplary results All numerical experiments has been led for a 1 [m] long steel cantilever beam with a cross-section equal 0.0001 [m^]. In the numerical tests without taking into consideration the influence of the measurement error used algorithm allowed to access the parameters of the damage correctly for almost every criterion tested in over 85% of tests done. It was possible to obtain proper parameters of the damage for any configuration of the location and the depth of crack. Fig. 3 presents the exemplary results obtained for a DLAC criterion for a crack located in 0.3 of the relative beam

M. Palacz, M. Krawczuk /First MIT Conference on Computational Fluid and Solid Mechanics 14

• Best value = |.0727e-010

"

1

1

12

0.8

8

i

max depth = 0.050337 min loc = 0.13749 max loc = 0.69845

1

'^-

min depth = 0.043599

o

co.6 Q

391

O0.4 0.2

20

40 generation

60

80

0.2

1 0.8

t

0.6

0.4 depth

0

Best depth = 0.050136 Best loc = 0.3

^0.6

u

2 0.4

(/>

WT'

>'•vv..^.vV"^v'-..,v.x»..-'-y-v'^-A,/

d>

u.

0.2

20

40 generation

60

80

0

20

40 generation

60

80

Fig. 3. The change of thefitnessfunction based on the DLAC criterion and of the crack parameters for a 5% depth. 1 0.8

0.6

min depht = 0.050081 max depth = 0.50936 min loc = 0.46876 max loc = 0.50002

> 0.4 0.2 20

1 0.8

I 0.6

A

40 generation

60

0.2

1

\ \ o o

\

0.8

best depth = 0. best loc = 0.

•«0.6 x: •3.

:o.4

a> •°0.4 "5)

( "0.2

0.6

0.4 depth

\^t\—A>\^^A^ 20

q>

-...^_.v-/v..A,y^--"

40 generation

60

80

i3

0.2 20

40 generation

60

80

Fig. 4. The change of thefitnessfunction based on the MAC criterion and of the crack parameters for a 10% depth. length with a 5% depth of the crack. Fig. 4 presents results obtained for a MAC criterion for a crack located in the middle of the beam for a 10% depth of the crack. Table 1 presents only the results for numerical calculations for a 10% depth of the crack in the middle of the beam with the influence of growing measurement error.

6. Conclusions After analysing the results of all experiments, one can conclude: • using the genetic algorithm gives a very fast estimation of the parameters of a damage,

392

M. Palacz, M. Krawczuk /First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 The percentages of properly estimated task parameters Criterion

CA

DLAC

MAC

COMAC

FRCM

No error Max. error 0.1% Max. error 0.5% Max. error 1% Max. error 2% Max. error 3% Max. error 4% Max. error 5% Max. error 10%

90%

100% 100% 40% 10% 0% 0% 0% 0%

100% 100% 100% 100% 100% 100% 100% 100% 90%

30% 30% 30% 10% 0% 0% 0%

80% 50% 40% 40% 40% 40% 30% 20%

60% 10% 0% 0%

growing measurement error causes smaller number of properly estimated task parameters for every criterion tested, the genetic algorithm used allows to obtain perfect damage estimation not for every damage indicator, for those criteria which use changes in natural frequencies high percentage of properly estimated task parameters rapidly falls down when the measurement error is bigger than 0.1%, direct comparison of mode shapes seems to be the most resistant criterion to the measurement error, identifying coordinates is completely not useful (COMAC criterion).

tests of the criterion based on the changes in amplitudes of forced vibration show that 3% measurement error allows to obtain still a relatively high number of properly estimated task parameters, proper control of the genetic algorithm parameters makes the searching process much more effective.

References [1] Adams RD, Cawley P. The localisation of defects in structures from measurements of natural frequencies. J Strain Anal 1979;14(2):49-57. [2] Kim J-H, Jeon H-S, Lee C-W. Application of the modal assurance criteria for detecting and locating structural faults. IMAC 1992;10:536-540. [3] Ostachowicz W, Krawczuk M. Analysis of the effect of cracks on the natural frequencies of a cantilever beam. J Sound Vibrat 1991;150(2):191-201. [4] Maia NMM, Silva JMM, Ribeiro AMR, Sampaio RPC. On the use of frequency-response functions for damage detection. Identification in Engineering Systems, Proceedings of the Second International Conference held in Swansea, March 1999, pp. 460-471. [5] Messina A, Jones I A, Williams EJ. Damage detection and localisation using natural frequency changes. Identification in Engineering Systems, Proceedings of the Conference held at Swansea, March 1996, pp. 67-76. [6] Rytter A. Vibrational Based Inspection of Civil Engineering Structures. Ph.D. Thesis, University of Aalborg, 1993.

393

A geometric-algebraic method for semi-definite problems in structural mechanics M. Papadrakakis *, Y. Fragakis National Technical University of Athens, Institute of Structural Analysis and Seismic Research, Zografou Campus, Athens 15780, Greece

Abstract A method for the general solution of structural semi-definite problems is described in this paper. The proposed method does not possess the restrictions of existing methods in this field. It is robust, cost-effective and can be combined with any, open or closed, serial or parallel, solver for symmetric positive definite problems. Keywords: Semi-definite problems; Singular problems; Structural mechanics; Domain decomposition; Rigid body modes; Zero energy modes

1. Introduction

and the general solution of the system (1) is given by

In structural mechanics, semi-definite problems are usually associated with floating structures, namely structures which do not possess enough external constraints to restrain all possible rigid body modes or internal mechanisms which constitute the so-called zero energy modes. A general solution of these problems is obtained by the computation of a particular displacement field, which ensures the equilibrium of the structure, and of its zero energy modes. Semi-definite problems are encountered, among other, in static or vibration analysis of floating structures (partially constrained or totally unconstrained structures), such as satellites, airplanes or multi-body structures, as well as in high performance domain decomposition methods where floating substructure problems need to be solved [1-3]. In structural semi-definite problems it is usually required to find the general solution of the linear system of equations Ku = f

(1)

where A^ 6 R"""" is a symmetric semi-definite positive stiffness matrix, and u and / are the displacement and the external load vectors, respectively. \f E e W^^ is a matrix whose columns constitute a basis of the null-space of K, then this singular system of equations has a solution only if E^f = Q

(2)

* Corresponding author. Tel: +30 (1) 772-1694; Fax: +30 (1) 772-1693; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

u = K+f + Ea

(3)

where ^ + is a generalized inverse of K and a e R^ is a. vector of e arbitrary entries. The vector ^ + / is a particular solution of the linear problem (1). Typical methods for handling semi-definite problems, such as the Singular Value Decomposition (SVD) or eigensolution schemes, entail substantial computational cost when appUed to relatively large problems. A basic costeffective method for semi-definite problems is a straightforward modification of the standard LDL^ Choleski factorization method [1]. This method, however, lacks robustness when applied to ill-conditioned semi-definite problems [4]. Another computationally efficient method combines geometrical concepts with the factorization method [4]. This method proved to be robust in handling structures with rigid body modes only and can be used with direct factorization methods after a modification of their algorithm. Moreover, the FETI domain decomposition solver for Symmetric Positive Definite (SPD) problems, has been recently extended to treat semi-definite problems [3]. In this paper, a new method is presented for the efficient solution of semi-definite problems. This method combines geometric and algebraic concepts and goes beyond the restrictions of existing methods in this field. Its main features are the following. It is fully robust, cost-effective and accounts for all rigid body and mechanism modes, in either floating structures, or semi-definite subdomain problems.

394

M. Papadrakakis, Y. Fragakis / First MIT Conference on Computational Fluid and Solid Mechanics

encountered in domain decomposition methods. It can be combined with any open or closed, serial or parallel solver for SPD problems, at a very low cost. The proposed method consists of three distinct tasks. First the rigid parts of the structure are detected, then the zero energy modes are computed, followed by the computation of a particular solution of the singular system of equations (1).

2. Detection of rigid parts The first task deals with the identification of the rigid parts of the given structure. By the term 'rigid parts', we simply designate deformable parts of the structure that do not possess any internal mechanism. For example, the model in Fig. 1 with one mechanism consists of two 'rigid parts' without internal mechanisms. In order to build a rigid parts detection algorithm, one has to determine the cases in which no mechanism can exist between two or more adjacent finite elements. For example, no mechanism can exist between any two structural type elements that are rigidly connected in at least one common node. In the case of continuum type elements, with no rotational d.o.f., two adjacent surface (volume) elements can be glued together, without inducing any mechanism, if they share at least two (three) common nodes. In this case, no relative rotation is possible between the two adjacent elements and the two elements are firmly connected. On the other hand, in pin-jointed structures, the basic mechanism-free set of elements is a triangle of bar elements in 2-D analysis and a tetrahedron of bar elements in 3-D analysis. Based on these remarks, the simple rigid parts detection algorithm proceeds as follows. Choose an element of the structure and start gluing other elements to it without inducing mechanisms. When no more elements can be added in this rigid part without inducing mechanisms, define this rigid set as the first rigid part of the structure. Then, choose an element that has not been glued to the first

rigid part and start gluing other elements to it in order to form the second rigid part. After the identification of the second part, the procedure is repeated with the next rigid part, until all rigid parts of the structure are identified.

3. Computation of the zero energy modes After the identification of the rigid parts, the computation of the zero energy modes is based on the following remarks, i. Every zero energy mode is also a rigid body mode of each rigid part, ii. Every rigid body mode (of a rigid part) is a specific linear combination of the basic rigid body modes of this part (six in a three-dimensional space and three in a two-dimensional space), iii. The rigid body modes of the rigid parts must fulfil the displacement compatibility conditions in the externally constrained d.o.f. and in the interface d.o.f. of the rigid parts of the structure. The enforcement of the displacement compatibility conditions requires the computation of the null-space of a typically small rij xrib matrix, where rih is the total number of the basic rigid body modes of all parts (rib = 6 x p in 3-D analysis and n^ = 3 x p in 2-D analysis, where p is the number of rigid parts) and rij = «c + ^i, in which He denotes the total number of external constraints and rii denotes the total number of interface d.o.f. between the rigid parts. The null-space of the rij x fit matrix is computed with the Singular Value Decomposition Method. The numerical stability of the computation of the zero energy modes is further enhanced with a simple scaling strategy.

4. Computation of a particular solution The computation of a particular solution of (1) adopts the following reasoning. The natural equilibrium conditions related to the zero energy modes of the structure can be expressed by the equation (4)

E^f = 0

where / are the external loads. Suppose a permutation of the columns of E^, such that Ej is a non-singular square matrix. Eq. (4) becomes [EJ

E]]

:0

(5)

Then, Eq. (5) is equivalent to the equation

/, = -EfE]ff Fig. 1. A 2-D system with two parts pinched together in a common node.

(6)

Eq. (6) implies that for any applied loads / / , there is a set of reactions fs, which ensures equilibrium, in other

M. Papadmkakis, Y. Fragakis / First MIT Conference on Computational Fluid and Solid Mechanics words restrains all possible zero energy motions. Therefore, if e artificial constraints, which correspond to the reactions fs, are added to the original constraints of the structure, then the problem becomes positive definite. Since the e artificial constraints correspond to e redundant equations of the singular system of equations (1), the particular solution of (1) can be computed by the solution of the SPD problem which is derived after the addition of the artificial constraints. The reactions fs correspond to the d.o.f. which are related to the columns of E^. Thus, the problem of finding e suitable artificial constraints reduces to the detection of e columns of E^, which form a square non-singular matrix £ j , or equivalently, to the detection of e linearly independent columns of E^. One of all possible sets of linearly independent columns can be very efficiently computed with a simple Gaussian elimination of ^"^ with full pivoting.

5. Conclusions The proposed method goes beyond the restrictions of existent methods in the field of fast solution strategies for structural semi-definite problems. In particular, it is robust, cost-effective and accounts for all rigid body and mechanism modes, in either floating structures, or semi-definite subdomain problems encountered in domain decomposition methods. The computational overhead associated with the implementation of the method to any serial or paral-

395

lel solver for symmetric positive definite problems is in general insignificant. Furthermore, the method can be used with any solver for symmetric positive definite problems, even if its source code is not available, thus extending the capability of the solver to the general solution of structural semi-definite problems.

Acknowledgements This work is partially supported by the research project 'Archimedes' of the National Technical University of Athens.

References [1] Farhat C, Roux F-X. A method of finite element and interconnecting and its parallel solution algorithm. Int J Numer MethEng 1991;32:1205-1227. [2] Bitzarakis S, Papadrakakis M, Kotsopoulos A. Parallel solution techniques in computational structural mechanics. Comput Methods Appl Mech Eng 1997;148:75-104. [3] Rixen D. Dual Schur complement method for semi-definite problems. Contemp Math 1998;18:341-348. [4] Farhat C, Geradin M. On the general solution by a direct method of a large-scale singular system of equations: application to the analysis of floating structures. Int J Numer Meth Eng 1998;41:675-696.

396

Parallel algorithm for explicit dynamics with support for nonlocal constitutive models B. Patzak^'*, D. Rypl'', Z. Bittnar'' ^ Department of Civil Engineering, Laboratory of Structural and Continuum Mechanics, Swiss Federal Institute of Technology at Lausanne, 1015 Lausanne, VD, Switzerland ^ Department of Structural Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thdkurova 7, 166 29 Prague, Czech Republic

Abstract The present paper describes the parallel algorithm for explicit time integration with efficient nonlocal material model support, within the framework of finite element method. A central difference method is used to discretize equation in time and the application of both dual partitioning techniques (node-cut and element-cut) is discussed. The efficiency of the proposed algorithm is demonstrated by 3D analysis on a PC cluster. Keywords: Parallel computation; Explicit dynamics; Nonlocal material model

1. Introduction The description of material failure is one of the actual problems in structural mechanics. Realistic analyses of failure processes require the use of complex FE discretizations and advanced constitutive models. The parallel processing is a tool, which makes such complex analyses feasible from the point of view of both time and available resources. Parallelization of the problem reduces the computational time and, for some cases, it allows large analyses to be at least performed. The architectures of parallel computers can be classified into three basic classes: shared memory systems, distributed memory computers and virtual shared memory computers. From the software development point of view, portability and efficiency of a parallel code can be obtained if so-called parallel programming models are used. There are three most common models: the message passing model, shared memory programming model, and the data parallel model. Among these, only the message passing is available on all platforms. This broad portability has been the main reason, why message passing programming model was selected. Explicit integration schemes are very popular for solving time dependent problems. Their application leads to * Corresponding author. Tel.: +41 (21) 693-2418; Fax: +41 (21) 693-6340; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

effective numerical algorithm, which can be parallelized in a straightforward way. In this study, the central difference time-stepping algorithm, assuming lumped mass matrix and damping expressed in Rayleigh form, is adopted. Domain decomposition, based on the node-cut or element-cut approaches (see Krysl and Bittnar [2] and Patzak et al. [3]) represents a tool for formulating an efficient parallel algorithm. Nonlocal material models are recognized as a powerful localization limiter, which is necessary to capture the localized character of a solution, for example in tension regime of quasibritle materials. Nonlocal approach is based on the averaging of certain suitable local quantity over characteristic volume, which is considered to be a material property. The nonlocal quantity is then substituted into local constitutive equation. Due to the nonlocal character (local response depends on material state in the neighborhood), these models require special data exchange algorithms to be developed in order to efficiently handle the nonlocal dependency between partitions.

2. Parallel algorithm The first design step of parallel code is to decompose the computation into a number of concurrent tasks. There are generally two basic partitioning techniques, based on

B. Patzdk et ah /First MIT Conference on Computational Fluid and Solid Mechanics domain or functional decomposition. For the problems with large regular data sets the domain decomposition is recognized as the better approach. In this approach, one firstly partitions the data and then identifies the operations associated with these data. In general, one can distinguish two dual domain decomposition approaches: the node-cut and element-cut concepts. Node-cut approach partitions the mesh into a set of non-overlapping groups of elements. The nodes at interpartition boundaries are marked as shared nodes. Typically, they are assigned with local degrees of freedom at each partition. At local partition nodes, the equilibrium equations can be solved using standard serial algorithm. However, at shared nodes, one has to exchange data between neighboring partitions to guarantee the correctness of the overall algorithm. In our case, the internal nodal force contributions are exchanged for shared nodes. This process of mutual data contribution exchange has to be performed at each time step. Similar process has to be invoked at the very beginning, when the mass matrix is assembled (in this case, the mass exchange for shared nodes is necessary).Element-cut approach partitions the mesh into a set of non-overlapping groups of nodes. The elements divided by cut are dupHcated on neighboring partitions. To describe the geometry of these shared elements, the local copies of remote nodes are introduced. Such nodes are called remotecopy nodes. The equilibrium equations at each partition are solved for both local and remote-copy nodes — thus all nodes are assigned with local degrees of freedom. In each solution step, a mutual data exchange between partitions is necessary. For example, internal nodal forces can be exchanged between adjacent partitions to guarantee the correctness. The internal nodal forces are computed at each partition for its local nodes, those for remote-copy nodes are then received from partition maintaining its local counterpart. The assembly process of the mass matrix is similar to the internal force exchange. The node-cut concept leads to duplication of finite element nodes on neighboring partitions. On the other hand, the element-cut approach leads to duplication of elements. Since the computational demands associated with element computations are typically superior to that for nodes, the node-cut approach is computationally more efficient. Due to this fact, only the node-cut approach will be considered further in this paper. When the nonlocal constitutive model is considered, some additional issues have to be taken into account. Due to the nonlocal dependency of material model, the parallel algorithm has to perform additional inter-partition communication to compute the nonlocal contributions for points near the inter-partition boundary, where the nonlocal quantity consists of local as well as remote contributions. To avoid redundant requests for same remote values from different local integration points (leading to an extremely fine communication pattern, which must be avoided), the band

397

Table 1 Central difference node-cut stepping algorithm with nonlocal extension Mass contribution exchange for shared nodes; while not finished loop (i) Assemble load vector Rt; (ii) Exchange local values to be averaged for all integration points of remote-copy elements; (iii) Compute local real nodal forces Ft; (iv) Exchange real nodal force contributions for shared nodes', (v) Solve displacement increment 6rt; (vi) Compute acceleration and velocity fields; (vii) Update displacement vector r^+Ar = ^t -\- 8rt; (viii) Increment time t = t + At; endloop

of remote-copy elements is introduced at each partition. A remote-copy element is established for each element, which belongs to neighboring partition, and values of any local integration point depend on it. After local quantities, which undergo nonlocal averaging, are computed at every local element, their exchange to the corresponding remote element values is done. The remote-copy elements are intended only to store copies of relevant quantities undergoing the nonlocal averaging. By using the local copies, the nonlocal values can be easily computed, instead of invoking cost communication. No computation is associated with these remote-copy elements, since their contributions are taken into account on partitions owning their local counterparts. Typical central difference algorithm, extended by two communication schemes — first due to node cut approach (exchange of shared node contributions) and the second due to remote-element data exchange, is presented in Table 1.

3. Example A 3D notched specimen has been analyzed in threepoint-bending using the direct explicit integration. The employed constitutive model is a nonlocal variant of rotating crack model with transition to scalar damage (see Jirasek and Zimmermann [1]). The mesh contains 1964 nodes and 9324 linear tetrahedral elements. The total number of time steps analyzed was 7500. The partitions have been generated prior the analysis and have been kept constant throughout the whole analysis (static load balancing). The analysis has been performed on workstation cluster running under Windows NT and Linux operating systems. The cluster consists of six workstations DELL 610, each equipped with two processors. The workstations contain dual PII Xeon processors at 400 and 450 MHz with 512 MB of shared system memory and PHI processors at 450 MHz with same amount of memory.

398

B. Patzdk et al. /First MIT Conference on Computational Fluid and Solid Mechanics Real time [sec* 1000]

Speedup

nproc

4. Conclusions

nproc

Fig. 1. Real times consumed and obtained speedups for Windows NT cluster. User time [sec] 1

'

1

1

Speedup

'

1

. 32648

30000 20000

-

1« 460 •S4OO2O

10000

6766

1

1

1

5419

-

'

'

1

1

6.02 -

-

4^2/ 3^26/

^

\Tl/'

1

1

.

1

The efficient algorithm based on node-cut strategy for explicit time integration schemes has been formulated. Particularly, general support for nonlocal constitutive models was considered. The implementation is based on message passing concept. The attention has been focused on the design and implementation of inter-partition data exchange. The described strategy can be easily implemented into any existing explicit finite element code. The described approach has been demonstrated on structural analysis of 3D specimen using heterogenous parallel computers. A significant reduction of the computational time and reasonable parallel efficiency have been evidenced.

1

Acknowledgements This work has been supported by the Ministry of Education of Czech Republic under Contract 104/98:210000003.

-

References Fig. 2. User times consumed and obtained speed-ups for Linux cluster. The workstations were connected by Fast Ethernet 100 Mb network using 3Com Superstack II switch, model 3300. This cluster represents a heterogenous parallel computing platform with the combination of shared and distributed memory. The MPI based message passing libraries used were IVIPI/Pro for Windows NT (IVIPI Software technology, Inc. ^) that supports both the distributed and shared memory communication, and for Linux MPICH ^ (a freely available, portable implementation of MPI). The computational times and obtained speed-ups for both platforms are presented in Figs. 1 and 2.

1 www.mpi-softech.com ^ http://www-unix.mcs.anl.gov/mpi/mpich/index.html

[1] Jirasek M, Zimmermann T. Rotating crack model with transition to scalar damage. J Eng Mech ASCE 1998;124:277284. [2] Krysl P, Bittnar Z. Parallelization of finite explicit dynamics with domain decomposition and message passing. Int J Numer Methods Eng, submitted for publication. [3] Patzak B, Rypl D, Bittnar Z. Explicit Parallel Dynamics with Nonlocal Constitutive Models. In: Topping BHV, Kumar B (Eds), Developments in Analysis and Design using Finite Elements Methods. Edinburgh: Civil-Comp Press, 1999. [4] Patzak B, Rypl D, Bittnar Z. Parallel Explicit Finite Element Dynamics with Nonlocal Constitutive Models. Comput Struct, submitted for publication. [5] Message Passing Interface Forum. MPI: A Message-Passing Interface Standard. University of Tennessee, 1995.

399

Rheological effects and bone remodelling phenomenon in the hip joint implantation Marek Pawlikowski *, Konstanty Skalski, Maciej Bossak, Szczepan Piszczatowski Warsaw University of Technology, Institute of Mechanics and Design, ul. Narbutta 85, 02-524 Warsaw, Poland

Abstract In the paper the description of bone adaptation (bone remodelHng) phenomenon for the case of hip joint implantation is presented from the biomedical point of view as well as the kinetics equations of the phenomenon. Their mathematical forms in terms of femur-bone density change in time due to the application of various mechanical stimuli (i.e. strain, stress, energy density) are discussed. The initial-boundary value problem for the femur-implant system is formulated. In this formulation the rheological (visco-elastic) properties of femur and its adaptation are taken into account. A numerical example of strength analysis performed by means of finite element method (FEM) is also given. Keywords: Visco-elasticity; Remodelling and kinetics equations; Femur-implant system; Numerical strength analysis; Strain-stress fields

1. Introduction

2. Materials and method

The realisation of the bone functional adaptation processes depends on many biomechanical factors including the stress and strain distribution in the bone-implant system [1]. The press-fit and exact-fit fixation of an endoprosthesis into the bone during the hip joint arthroplasty causes the developing of stress and strain fields. That leads to the bone density decrement in the vicinity of the implant [2]. Such an alteration of the stress-strain status quo in the femur-implant system leads to the stress-shielding region forming which, in accordance with Wolff's law, causes bone resorption, loss of bone-implant connection (aseptic loosening) and often the necessity of reimplantation. It is of paramount importance, then, to know exactly how the processes mentioned above pass in the bone tissue especially in the vicinity of the implant. The aim of the paper is to include the remodelling phenomenon into the strength analysis performed by means of FEM system, which is very useful in the design process of the hip joint endoprosthesis [3].

Living bone tissue undergoes continuously the remodelling process. It consists of bone resorption (internal remodelling), bone apposition (external remodelling) and alteration of bone structure. These alterations may be evoked by mechanical factors as well as by hormonal, genetic and metabolic ones. As the mechanical stimuli influence the remodelling phenomenon most, only these particular stimuli will be considered in the strength analysis of the implanted femur. The results [4] have shown that the implantation of the endoprosthesis into the femur causes the bone to atrophy in the proximal part of the femur and along the stem of the implant. The aseptic loosening of the implant and/or bone fracture are the consequences of such an activity of the bone tissue. That entails the reimplantation. It is obvious that one cannot completely avoid this phenomenon. However, one can limit its range to some extent by designing an endoprosthesis of the stem exactly fitting the medullary canal of the femur of the given patient. The application of such an endoprosthesis would prevent the formation of undesirable stress concentrations and non-strain regions in the bone-implant system. One can make a hip joint endoprosthesis even more stable and durable by taking into account the bone functional adaptation phenomenon.

* Corresponding author. Tel.: -^48 (22) 660-8444; Fax: +44 (22) 848-4280; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

400

M. Pawlikowski et al. /First MIT Conference on Computational Fluid and Solid Mechanics

b) SETINrnAL INTEGRATION INCREMENT SETINniAL VALUE OFDENSnV

PRINT CALCULATED STRESSES AND OTHER DESIRED VARIABLES DURING STRESS PEONT-OUT

END Fig. 1. Flow chart of: modified user-supplied module (a), remodelling module (b).

The kinetics equations of the internal remodelling phenomenon may be formulated in three different forms dependently on the mechanical stimulus. The most common stimuli used by researches to describe the remodelling phenomenon are the fields of stress, strain and strain energy density. However, the most popular formulation of the kinetics equation for the internal remodelling is the strain formulation. Eq. (1) represents its the most general form [5]. dt

Bi(^(a,pJ-^o)

(1)

where Pa = apparent bone density; a = stress field; ^ , ^Q = present and reference stress functions, respectively; t = time; ^i = material constant. The structural and strength analysis of the bone-implant system is stated by the initial-boundary value problem to which the solution may be obtained by the use of the finite element method. The complete initial-boundary value problem consists of the constitutive equation for the visco-elastic body, equilibrium equation, complementarity equation, internal and external remodelling equations and the continuity, displacement and load boundary conditions.

However, it is very difficult to solve this complex system of differential and integral equations determining the initial-boundary value problem. Therefore, it has been proposed to simplify it by eliminating the remodelling equations and solve it simulating, in the preliminary approach, only the visco-elastic properties of the bone tissue. The FEM-ADINA software, which has been utilised to perform the strength analysis of the bone-implant system, makes it possible to assign the visco-elastic properties to the bone by the usage of the, so called, user-supplied module. As remodelling plays an important role in the alloplasty we intend in the future to modify the module so we can perform the strength analysis of the bone-implant system with the remodelling phenomenon taken into account. The flow chart of modified user-supplied module is shown in Fig. la. The modification consists in adding the remodelling module (Fig. lb) into the original flow chart.

3. Results The model of the implanted femur utilised in the numerical analysis is given in Fig. 2. To calculate at the first

M. Pawlikowski et al /First MIT Conference on Computational Fluid and Solid Mechanics

R

401

E«1B0H

m^irmm

rftoiffj

CWftfCM

0 6M MM 1,011 l^ll

.1 m tm 1000 iom>o 4230 HJLTl 6.D42 S 3 ^ : 5,446 mi7S mm^ i%m$ Wjffi TiiM l,OC^ I 0S^ Qsm 0»8tl ijm I Q^n cws 0 ^ 1

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WMl m:3m 16*1S0 15 J ^ mr^^ 0,S34 mvr Ov^T T w l iSPcIo) 0^17 0 ^ ^ l ^ J \^^m d ^ p i m tinii »l i^i^ii ^^tm&^^m m mtml^^0,^4^m&^d hmt wM4

r's«t|si 0 myom 1,0 1 wmmi IP

1

2

3

4

5

6

7

10

m

100

m

04>4O 0^10 0,SS4 0,$75 %U$ 0^57 p,S52 0 ^ 4 0 3 0,73 0 3 0 ^ 0 0^10 0,$S4 0,8B o,ass 0,857 %^^1 0,834 0 3 [ 0 3 0 3 !

Fig. 2. Model of the bone-implant system, applied load and the visco-elastic properties of the two bone tissues.

/

^f?^

A

Fig. 3. Distributions of dimensionless effective strains (left), dimensionless effective stresses (middle) and dimensionless strain energy density (right) in the bone-implant system after time f = 160 s (initial state — elastic t = 0). approach the visco-elastic effects in the above system the material properties given in the tables are assigned to the trabecular and cortical tissues at two different time scales (Fig. 2). In Fig. 3 and Fig. 4 the results of the strength

analysis are presented in non-dimensional strain, stress and strain energy density (SED) distributions. In Fig. 3 the values of strain, stress and SED at the time 160 s are compared to the initial (elastic) state of the system (^ = 0 s). As

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M. Pawlikowski et al /First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 4. Distributions of dimensionless effective strains (left), dimensionless effective stresses (middle) and dimensionless strain energy density (right) in the bone-implant system after time t = 10,000 s (initial state — visco-elastic r = 160 s).

one can see, at the beginning the creeping process of the trabecular bone dominates and at the further period of the analysis (160-10^ s) the creeping process of the cortical bone tissue is clearly visible (Fig. 4).

4. Discussion The behaviour of bone tissue can be simulated with high accuracy as a visco-elastic one. As the density of trabecular bone as well as its architecture (porous structure) are different from those of cortical bone, the rheological changes occur in a different way in these two tissues. The results of the strength analysis show that the rheological changes occur sooner in trabecular bone than in cortical one. It is caused by the anatomical structure of the trabecular bone which is a highly porous structure. It seems to be essential to take into consideration the visco-elastic behaviour of the bone tissue in the strength analysis of the bone-implant system. However, we intend to include into the analysis assumptions one more factor which plays an important role in the human joint alloplasty, namely the remodelling phenomenon. We are going to do it by modifying the user-supplied module which is, as it was stated above, a computer program written in FORTRAN language so it is very easy to modify.

Acknowledgements The authors would like to thank Dr. J. Walczak (ADINA R&D, Inc., USA) for his precious suggestions regarding the simulation of the remodelling phenomenon and visco-elastic properties of the implanted femur. This research has been partially supported by Warsaw University of Technology Bioengineering Program under grant no. 503|R| 1101130011200.

References [1] Weinans H, Huiskes R, Grootenboer HJ. The behaviour of adaptive bone-remodeling simulation models, J Biomech 1992;25(12): 1425-1441. [2] Cowin SC, Arramon YP, Luo GM, Sadegh AM. Chaos in the discrete-time algorithm for bone-density remodeling rate equations, J Biomech 1993;26(9): 1077-1089. [3] Dietrich M, Kedzior K, Skalski K. Design and manufacturing process of the human bone endoprosthesis using computer aided systems. J Theor Appl Mech 1999;3(37):481-503. [4] Allain J, Le Mouel S, Goutallier D, Voisin MC. Poor eightyear survival of cemented zirconia-polyethylene total hip replacement. J Bone Joint Surg 1999;81B(5):835-842. [5] Fyhrie DP, Schaffer MB. The adaptation of bone apparent density to applied load. J Biomech 1995;28(2): 135-146.

403

Computational synthesis on vehicle rollover protection Xiao Pei Lu General Motors Corporation, Warren, MI 48090, USA

Abstract Vehicle rollover crashes are serious and complex events [1]. In order to identify ways to better protect occupants from rollover crashes, the synthesis approach can be utilized to meet specified requirements in whole designing iterations. Synthesis capability elements include: requirement specifications establishment; modeling and simulation techniques; and computational codes and computing performance. The rapid growth of the combination of high computing performance and advanced math-based technologies have already made the synthesis approach available as an analytical designing tool for the auto industry. This paper is a brief status survey. Keywords: Computational synthesis; Rollover protection

1. Introduction

2.2. Requirements

The complexity of rollover crashes results from the interaction of driver, vehicle and environment. Significant enhancements in the capability of analysis/simulation relating to rollover resistance and occupant protection in rollovers for different driver/environment scenarios were accomplished. The key elements of synthesis with their capability status are briefly discussed.

The following have been used in the automotive industry to assGSS'.Rollover resistance: stability margin, static stability factor, tilt table angle, side pull ratio, and critical shde spQcd.Occupant protection regulations relating to occupant protection in rollovers are: - FMVSS 201: Occupant protection in interior impact - FMVSS 208: Occupant crash protection (30 mph dolly rollover test) - FMVSS 216: Roof crush protection (quasi-static roof crush test to define the requirements of roof strength) - FMVSS 206: Door locks and door retention components (specifies door-latch strength) - FMVSS 220: School bus rollover protection

2. Discussion 2.1. Synthesis A requirement-driven analytical process to be used throughout the design of a vehicle for meeting performance targets. For many years, vehicle development has relied on real prototype test iterations to validate and assure vehicle quality because of limitations in math-based tools. Recently, math-based technologies have been developing rapidly, including high performance computing power, advanced modeling, simulation techniques, and computational codes development. These progresses make the apphcation of synthesis on vehicle development more feasible. The status of the basic capabiUty regarding the synthesis approach as applied to vehicle rollover will be discussed in the following paragraphs.

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

2.3. Modeling and simulation techniques 2.3.1. Rollover resistance Among many available existing modeling/simulation techniques, ADAM, developed by MDI, is one sample which can be used to assess and potentially improve simulation of rollover events [2]. Full vehicle model, virtual prototype, assembled by each mechanical part (axle and body), subsystem (suspension, steering), components (spring, shock and tire/wheel) and their interconnections even tire/ground friction. Component information includes geometric data, inertial data and force-deflection characteristics. For example, tire

404

X. Pei Lu/First MIT Conference on Computational Fluid and Solid Mechanics

force-deflection characteristics are important. Very often, the saturation of rear outside tire lateral force would make vehicle spin out and result in either on/off road rollover or frontal/side crash. The tire lateral/longitudinal forces, align moment and overturn moment are influenced by normal loads, slip angle/ratio and camber. All of these characteristics can be measured and made part of model elements. The initial condition and driver maneuver can be selected and included in the simulation and the vehicle performance response can be simulated and evaluated in either normal or rollover condition. The modeling and simulation techniques show the capability to predict and potentially improve rollover resistance through engineering design. The character of the inputs and the validation of the model limit the usefulness of simulations, such as ADAMS. It is essential that the data used to describe vehicle characteristics be accurate in order for simulation results to be meaningful. 2.3.2. Occupant protection Once the vehicle exceeds rollover onset, occupant protection becomes a serious issue. In order to determine potential for occupant injury, the impact forces applied on the occupant and the relative motion between occupant and vehicle are investigated. One sample showed the capability of modeling and simulation during a vehicle rollover event [3]. MADYMO, developed by TNO of Delft, the Netherlands, is used in this study: the vehicle model, also a virtual prototype, is assembled by exteriors, interiors, suspension, tire, restrain systems, and dummy. Exteriors and interiors are modeled as a series of ellipsoids, planes and elliptical cylinders. The vehicle suspension may also be included in the vehicle model. The tires in this scenario would have the same stiffness regardless of the direction of loading. The restraint systems include seat, seat belt and air bag. The seat can be modeled as a combination of cushion, and seat back with cushion stiffness characteristics. The seat back to seat bottom torsion/bending stiffness characteristics may be included. The airbag can be modeled as a finite membrane element. Various techniques are being used and developed to actually fold the bag into its container. Also, the gas generator can be modeled with the rate and the total volume of the gas. The occupant is modeled as a 50th percentile part 572 or Hybrid II dummy. The Bronco II dolly begins with the vehicle on a dolly canted at 30° along the roll axis in the direction of travel. The dolly had a velocity of 13.3 m/s in the lateral direction. The simulation produced comparable results with real vehicle test results on the acceleration pulse experienced by

the occupant. This study concludes that this is an effective tool in studying occupant motions, impacts and the function of restraint systems in rollover crashes. Parallel to the application of MADYMO, the DYNA3D (Lawrence Livermore) has been used to simulate the vehicle rollover event [4]. The DYNA3D occupant models were 50th percentile male Hybrid III dummies, restrained using both lap and three-point seat belts. The simulation investigated the roof strength and performance of seat and seat belts in a rollover crash. It is claimed that the use of simulation can dramatically reduce the lead-time and cost to develop weight-effective and cost-effective improvement in vehicle structures. 2.4. Computational codes MADYMO is one of the many codes used in the occupant protection analysis [5]. It is flexible in the early design stage to be used for synthesizing rollover occupant protection. This code combines in one simulation program, the capabilities offered by multi-body and MADYMO 3D finite element techniques. MADYMO includes multi-body systems (flexible body, kinematics joints, dynamic joint, restraint, body surfaces), control module (sensor, signal, controller, actuator) and finite element models (material, elements, acceleration field, contact interaction, belt, airbag, tire, road, injury parameters). The simulation could provide information useful in determining how specific occupant protection features can be modified or added to enhance protection of occupants in a rollover crash. DYNA3D is an explicit 3D finite element code for analyzing the large deformation dynamic response of inelastic solids [6]. It is used for the same purpose. If the synthesis, design-evaluation-redesign, process is done manually, it means the engineers manually manipulate the simulation tools changing and evaluating the designs. This is inefficient for both quality and timing of the design. Some codes like iSIGHT that is developed by Engineous Software, Inc. can make the synthesis process more effective by its Design Exploration Methods along with process automation capabilities.

3. Conclusion Synthesis is an analytical designing tool, which can be used to assess and potentially improve vehicle rollover resistance and occupant protection. Synthesis capability growth relies on requirement/specification establishment, math-based technology advancement and high computing performance.

X Pei Lu/First MIT Conference on Computational Fluid and Solid Mechanics References [1] DOT, NHTSA 49 CFR Part 575 [Docket No NHTSA-20006859 RIN 2127-AC64]. [2] Chace MA, Wielenga TJ. A test and simulation process to improve rollover resistance. SAE 1999-01-0125. [3] Renfroe DA, Partain J, Lafferty J. Modeling of vehicle

405

rollover and evaluation of occupant injury potential using MADYMO. SAE 980021. [4] Randell N, Kecman D. Dynamic Simulation in the Safety Research, Development and Type Approval of Minibuses and Coaches. SAE 982770. [5] MADYMO v5.4 Theory Manual. [6] Theoretical Manual for DYNA3D.

406

Sensitivity study on material characterization of textile composites Xiongqi Peng, Jian Cao * Department of Mechanical Engineering, Northwestern University, Technological Institute, 2145 Sheridan Road, Evanston, IL 60208-3111, USA

Abstract Material characterization is one of the key elements in analyzing the forming process of textile composites. The various length scales, from micrometers of each glass fiber to meters of final products, have placed a challenge issue on how to accurately and effectively model the material behavior. A novel finite element approach for predicting the effective nonlinear elastic moduli of textile composites was proposed by the authors in [7], where a unit cell was first built to enclose the characteristic periodic pattern in the composite and numerical tests were performed on the unit cell to extract the effective nonlinear mechanical stiffness tensor as functions of elemental strains. In this paper, parametric studies on the friction coefficient and fiber yam width are conducted for the objective of future simulation and optimization of textile composite stamping. Keywords: Material characterization; Textile composites; Finite element method; Sensitivity study

1. Introduction Stamping of textile composite materials has recently received considerable attention, due to its potentiality for the mass production of composites at a reasonable cost. Though textile composites have many structural advantages as high specific strength, high specific stiffness and improved resistance to impact, the high heterogeneity of the material properties makes it difficult to fully understand, from the micro level, the mechanical behavior of textile composites during stamping. To simulate and optimize the stamping process numerically, it is essential to obtain the effective homogeneous material properties of textile composites from the known material properties of the constituent phases. Many efforts have been given to the estimation of effective material properties of composite materials [1-7]. The approaches developed include the homogenization method, the finite element method, analytical model and experimental approach. Due to the immense variety of available composite materials, possible fabric construction geometry, and the change of composite material properties with processing temperatures, it is impractical and very time consuming * Corresponding author. Tel.: +1 (847) 467-1032; Fax: +1 (847) 491-3915; E-mail: [email protected] © 2001 Elsevier Science Ltd. Allrightsreserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to obtain material characterizations of various composites by an experimental approach. Analytical methods, on the other hand, cannot deal with complex fabric construction geometries. The huge computational cost limits the application of the homogenization method in simulating the forming of complex structures. The authors proposed a novel procedure for predicting the effective nonlinear elastic moduli of textile composites in [7]. In this procedure, a unit cell is first built to enclose the characteristic periodic pattern in the textile composites. Using the unit cell, various numerical tests can be performed. By correlating the force versus displacement curves of the unit cell and a four-node shell element with the same outer size as the unit cell, the effective nonlinear mechanical stiffness tensor can be obtained numerically as functions of strain tensor. The entire approach is illustrated in Fig. 1. The procedure is here applied in the parametric studies of the friction coefficient and fiber yarn width for the objective of future simulation and optimization of textile composite stamping. 2. Unit cell for plain weave composites A plain weave E-glass/PP composite is used in this paper to illustrate the procedure of material characteriza-

X Peng, J. Cao/First MIT Conference on Computational Fluid and Solid Mechanics

407

Equiv, Shell Element 5.14mm

-^^^s^>^.,s,Jo.;.-,,^^;.^.

MacroscopicStructure

ou

UnitCell

Uii2J

^l;:,^•^•^;<>o^^%p.;|;1••r

u OIOIOI

c\c\c\ iololoi Meso-microscopic Structure

Fig. 1. Multi-scale material characterization.

tion. The geometric description for plain weave composites presented by McBride and Chen [8] was used to model a unit cell, as shown in Fig. 2. The characteristic values of w, s and h at room temperature were w = 3.72 mm, s = 5 . 1 4 mm, h = 0.39 mm. The pin-jointed net ideal-

ization was assumed. Contact conditions were prescribed between the possible interlacing surfaces of the fiber yams. Unless specified, the friction coefficient was assumed to be 0.05 at room temperature as from [9]. The material properties of the constituent phases can be found in literature [7]. The volume fraction of the E-glass was 70%. The predicted elastic constants for the fiber yams by the homogenization method were [7]: El = 51.92 GPa, vtt = 0.2143,

Et = 21.97 GPa, Git = 8.856 GPa,

v^ = 0.2489 G^ = 6.250 GPa

where 1 represents the longitudinal direction and t for the transverse direction of the fiber yarn.

3. Effective mechanical stiffness tensor for textile composites

Fig. 2. Unit cell for plain weave composites.

A four-node shell element with the same outer size of the unit cell was built to obtain the effective mechanical stiffness tensor of the plain weave composite. Large deformation and geometric non-linearity are considered in the FEM analysis. The material constitutive equation is given

X. Peng, J. Cao/First MIT Conference on Computational Fluid and Solid Mechanics

408

Trellising Test

in Fig. 3 with circles and can be represented as: Gi2(MPa) =

^ 1.7444)/i2 + 4.96 x lO'^

yu < 0.207

[ Csrh + <^2Ki'2 + ^"1X12 + Co

712 > 0.207 (4)

where C3 = 11.3103,

C2 = 22.3030,

Ci = -8.9429,

Co = 1.2059

Finally, the above representations of the effective elastic moduli and the constitutive equation were implemented in a user material subroutine. 0.4 0.6 Shear Strain y 12

0.8

4. Validation of material model

Fig. 3. Equivalent shear modulus Gn for shell element.

t^l2^2 1 -

yi2V2i

1 -

E2

G2

1 - vnV2\

an

0

1

0

V,2V21

£\

0

< £2

y,2y2i

0

(1)

Yn]

Gn_

Numerical tests were carried on the unit cell for the determination of the equivalent elastic constants. Trial mechanical stiffness tensor was iteratively modified and imposed on the shell element in each increment of the FEM analysis to match the force-displacement curves of the unit cell under different numerical tests, respectively. Details can be found in [7]. The obtained equivalent Poisson's ratio v^ for the shell element was vn

0.02

£, < 0

[ 346.634£2 _ 25.9684ei + 0.54416

(2)

5. Parametric study It is observed from the trellising that the characteristic behavior of the textile composites varies at elevated temperatures. The textile composite will experience various temperatures during the entire stamping process. The resin will melt down from the textile composite at elevated temperatures and hence result in the decrease of the friction

0 < Si < 0.04 5000

The equivalent tensile elastic modulus was: 10/22 + 0.1

To validate the obtained material model, experimental trellising tests were conducted on composite patches with different area: 0.012 m^, 0.025 m^ and 0.044 m^ [9]. The normalized load-shear angle curves were shown in Fig. 4. The FEM simulation was then implemented on a composite patch of 24 X 24 cm. The user material subroutine was imposed on each shell elements. The patch was then undergoing trellising test numerically. The resulting normalized load-shear angle curve was shown in Fig. 4. As can be seen from Fig. 4, very good agreement was obtained.

£i < 0

£,(MPa) = \A3sf -\-A2£J-^Ai£i-\-Ao + 1^1 - S2\ QXp{Bi£i -\- Bo)

4000

(3) 0 < £i < 0.04

FEM simulation Experiment 1 (0.012m^) Experiment 2 (0.025m^) Experiment 3 (0.044m^)

-E 3000

where i = 1, 2 and A3 = 7.60033 X 10^ A2 = -9.81521 x 10^ Ai = 4.35522 x 10^ AQ = 4.64523 x 10\ 5i = -87.34587, BQ = 13.02737 The equivalent shear modulus Gn was numerically determined from the trellising test by matching the forcedisplacement curve of the unit cell to those of the corresponding single shell element. The obtained Gn is shown

2000

1000

15 20 25 Shear Angle (deg) Fig. 4. Comparison with experiment results.

40

X. Peng, J. Cao/First MIT Conference on Computational Fluid and Solid Mechanics (s=5.14mm, w=3.72mm)

409

modulus G12 increases at the same strain level with the increment of the friction coefficient. Fig. 6 shows the effect of the fiber yam width on the effective shear modulus G12. The friction coefficient JJL and the yarn space s are kept unchanged SLS fi = 0.05, s = 5.14 mm, respectively. The fiber yam width is chosen as 3 mm, 3.4 mm and 3.72 mm for the parameter study. As shown in Fig. 6, the effective shear modulus G12 increases at the same strain level with the increase of the fiber yam width.

6. Conclusions

0.4

0.6

0.8

1

Shear Strain fn Fig. 5. Effect of friction coefficient on Gi2. coefficient and the fiber yam width. The effect of temperature on the material properties can be investigated through the parametric study on the friction coefficient and fiber yam width. Since trellising is the main deformation form in the stamping, here, we will only consider the effective shear modulus Gn. Parametric studies are conducted on the numerical trellising. First we investigate the effect of the friction coefficient II on the effective shear modulus Gu- The geometric parameters, yam space s, yam width w and fabric thickness h, are kept unchanged as s = 5.14 mm, w = 3.72 mm, h = 039 mm, respectively. Three different friction coefficients will be used: 0.03, 0.05 and 0.08. The corresponding effective shear moduli Gu for the shell element are shown in Fig. 5. As can be seen from Fig. 5, the effective shear (s=5.14mm,^=0.05)

I M

M

M

Shear Strain Y12 Fig. 6. Effect offiberyam width on G12.

With the combination of the homogenization method and the conventional finite element method, it is feasible to obtain the effective material properties of textile composites. By adjusting the friction coefficient and the geometric parameter values of the unit cell, the effective material properties of textile composites at various temperatures can be easily obtained. Hence, it is possible to simulate and optimize the stamping process of composites by using common shell elements with the effective homogenous material properties.

References [1] Dasgupta A, Agarwal RK. Orthotropic thermal conductivity of plain-weave fabric composite using a homogenization technique. J Compos Mat 1992;26(18):2736-2758. [2] Boisse P, Cherouat A, Gelin JC, Sabhi H. Experimental study and finite element simulation of a glass fiber fabric shaping process. Polym Compos 1995;16(l):83-95. [3] Gowayed Y, Yi L. Mechanical behavior of textile composite materials using a hybrid finite element approach. Polym Compos 1997;18(3):313-319. [4] Weissenbach G, Limmer L, Brown D. Representation of local stiffness variation in textile composites. Polym Polym Compos 1997;5(2):95-101. [5] Wang C, Sun CT. Experimental characterization of constitutive models for PEEK thermoplastic composite at heating stage during forming. J Compos Mater 1997;31:1480-1506. [6] Takano N, Uetsuji Y, Kashiwagi Y, Zako M. Hierarchical modeling of textile composite materials and structures by the homogenization method. Model Simul Mater Sci Eng 1999;7:207-231. [7] Peng XQ, Cao J. A dual homogenization and finite element approach for material characterization of textile composites. Submitted to Composites Part B, 2000. [8] McBride TM, Chen J. Unit-cell geometry in plain-weave fabrics during shear deformation. Compos Sci Technol 1997;57:345-351. [9] Prodromou AG, Chen J. On the relationship between shear angle and wrinkling of textile composite performs. Composites Part A 1997;28:491-503.

410

Uncertainty analysis of large-scale structures using high fidelity models Ravi C. Penmetsa*, Ramana V. Grandhi Wright State University, 209RC 3640 Col. Glenn Hwy, Dayton, OH 45435, USA

Abstract The objective of this paper is to develop a robust uncertainty analysis method to handle large-scale structural problems. As the complexity of the structure increases, the cost of manufacturing the prototype to validate the designs and determine the level of safety would increase alarmingly and thus the need for analytical certification is compelling. The analytical certification is not to replace the prototyping and actual testing, but to reduce the number of actual tests to a minimum. When dealing with complex structures, the use of high fidelity models (FEM, CFD) to predict the response is inevitable in order to increase the accuracy of the prediction methods. In this research, a methodology for estimating the system reliabiUty of complex structures subject to multi-disciplinary failure criteria is discussed. The accuracy and efficiency in estimation process were achieved by the use of high fidelity models such as multi-point approximations and two-point adaptive nonlinear approximations. A turbine blade example was used to demonstrate the proposed methodology. Keywords: Reliability; Failure probability; Approximation; Uncertainty analysis; Non-linear approximations; Monte Carlo

1. Introduction A real structure typically consists of many components, of which, each has the potential to fail, and the individual component failure might lead to a structural failure. Even in simple structures composed of just one element, various failure modes such as bending action, buckling, axial stress, temperature, frequency, etc., may exist and be relevant in the solution. A 'structural system' is one that is composed of many and this system may be subject to many forms of loads, either single or various combinations. Therefore, the reliability analysis of structural systems will involve consideration of multiple, perhaps, correlated limit-states which can be defined in any discipline. The system failure probability estimation involves large computational effort and methods are currently being developed to reduce this computational time. In structural system reliability analysis, the bound methods and numerical integration methods have practical significance. In this, the failure probability of the individual components has to be estimated and First-Order Reliability Method (FORM) is considered in this paper. If the com* Corresponding author. Tel. +1 (937) 775-5040; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ponents of the system are assumed independent, then the system failure can be obtained easily. However, in practical problems, the failure conditions depend on the same random variables and therefore the components are correlated. Cornell [1] has developed bounds on the system failure probability for systems subjected to multiple failure modes. The upper bound on the system failure was obtained by assuming perfectly correlated components and the lower bound is obtained by assuming statistically independent components. If all the components are perfectly independent then the failure probability of the component can be determined by using the approximation techniques developed by the authors in their earlier work [2]. The method of narrow bounds presented by Ditlevsen [3] for the system failure probability had wider applicability due to its accuracy. These bounds considered the correlation between each of the two failure modes, making more physically reasonable. Using this method, the system failure probability can be expressed from the bounds of first-order or second-order joint probabilities. However, these bounds are accurate only when the limit-states are of linear form. In situations where this assumption is not valid, the alternate procedures have to be developed to estimate the failure probability. When dealing with highly nonlinear problems with a

R.C. Penmetsa, R.V. Grandhi/First MIT Conference on Computational Fluid and Solid Mechanics large number of non-normal random variables and implicit limit-state functions, both the FORM and SORM approximations fail to give accurate results. Therefore, better approximations such as two-point adaptive non-linear approximations (TANA2 [4] or TANA3 [5]) have to be used to approximate the limit-state functions. The approximations capture the information of the limit-state accurately in the vicinity of the MPR When dealing with multiple limit-states, information about the Most Probable failure Point (MPP) of each limit-state is vital for the accurate estimation of the system failure probability. Therefore, the two-point approximation is used as a local approximation at each of the MPPs of every limit-state and then the multi-point approximations (MPA) are constructed. This multi-point approximation retains the information of each of the failure surfaces and constructs a joint failure domain. Since this joint failure domain is constructed using more accurate approximations of the individual failure domains, it can be integrated using the Monte Carlo simulation technique to obtain the system failure probability. The reduction in computational cost of the system reliability prediction greatly helps in the preliminary design of multi-functional structures.

411

values and gradients have been obtained. In this research, data points that are obtained in the process of searching for the MPP and the points obtained while estimating the intersection point of the limit-states closest to the origin in the normalized domain are used to construct the local approximations. Once the local approximations are obtained, an MPA is constructed that contains the information of all the local approximations. The MPA adaptively adjusts itself to behave as a local approximation when a design point is close to one of the data points. Function and gradient values of this MPA correspond directly with their exact counterparts at the points where the local approximations were generated. Monte Carlo simulation is performed on this MPA to obtain the system failure probability.

3. Examples Various examples have been studied to prove the efficiency and accuracy of this method. The system failure probability of the system estimated using MPA is compared with the results obtained directly on the exact limit-state using the Monte Carlo simulation. Each Umit-state is approximated using an MPA and the Monte Carlo simulation is performed on the approximate limit-state functions.

2. Proposed methodology 3.1. Turbine blade Reliability analysis typically begins with the identification of the most probable failure point. The MPP of each limit-state function can be efficiently estimated using the algorithm presented by Wang and Grandhi [6]. This algorithm uses a two-point approximation, TANA2, of the actual limit-state in the search procedure in-order to reduce the computational expense of searching the MPP, which is formulated as an optimization problem. In the process of searching for the MPP of each Hmit-state function, a series of data points information including the function

In this example the system failure probability of a turbine blade shown in Fig. 1, with a 45° twist angle, is studied. The blade is modeled with 80 quadrilateral plate-bending elements with 99 nodes. All the degrees of freedom along the hub are fixed. The thicknesses of the plate elements are considered as the random variables and using physical linking only 10 random variables are considered. All the chord wise elements are assumed to have the same thickness. All the element thicknesses have

Fig. 1. Turbine blade.

R.C. Penmetsa, R.V Grandhi/First MIT Conference on Computational Fluid and Solid Mechanics

412

a mean value of 0.35 with a coefficient of variation of 0.10. Three different failure criteria are considered: (1) Displacement in Z direction: Ai, 1.0 < 0.0 Safe, 0.015 (2) First natural frequency: 1.0-

^1

< 0.0 Safe, 2500 (3) Stress in element 77 (Root): V 65000/ V 65000/ 0^0^ / ^xy y 65000x65000 V9000/

has produced a converged result in 25,000 actual simulations, whereas the MPA based method required only 16 actual simulations to predict the failure probability. In the case of MPA simulations, both the gradient and function information are generated in each simulation whereas in Monte Carlo approach, only function value is considered.

4. Summary

1.0 < 0.0

Safe

The MPP is estimated for all the limit-states and then using the intermediate points in the MPP search algorithm local approximations are constructed. Additional points obtained from the search for the closest intersection point of the limit-states were also added to increase the accuracy of the approximation. The MPP search using stress constraint required six iterations and these intermediate points were used to construct the MPA for the stress constraint. The displacement constraint converged in three iterations resulting in only three design points. Therefore, only two TANA2 approximations could be used to construct the MPA for the displacement limit-state. For the frequency constraint, four TANA2 approximations were constructed at various intermediate design points and these approximations were blended into a third MPA. Only three additional simulations were required to obtain the closest intersection point because the local TANA2 approximations were used in this process. The information at these three additional points is also incorporated into each of the MPAs to improve their accuracy in the failure domain. Once these limit-states were available as closed form MPAs, the Monte Carlo simulation was used to obtain the system failure probability. The results are presented in Table 1. The results obtained from MPA are compared with the actual Monte Carlo results and FORM results. The Monte Carlo simulation on the actual limit-state functions Table 1 MPA and Monte Carlo results for turbine blade Method

Probability of failure

Monte Carlo simulation MPA (m = 2.0) First-order series bounds

0.0121 0.0132 0.00590 to 0.0125

! Difference

8.59 -51.45 to 3.55

The use of MPA has enabled modeling of the «-dimensional joint failure domain for using the Monte Carlo simulation. This approximation reduces a considerable amount of computational effort (MPA of each limitstate function is explicit) without sacrificing much accuracy. Because information at more points is used to construct MPA of each limit-state function, MPA is accurate over a larger region. It is possible to give a good prediction of the intersection point of different limit-state functions. MPA has a tremendous potential for problems where the limit-states are not unimodal and exhibit high nonlinearity.

Acknowledgements This research work has been sponsored by the U.S. Air Force under Contract F33615-98-C-2895. The support for the Graduate Research Assistant was provided by the Dayton Area Graduate Studies Institute (DAGSI).

References [1] Cornell CA. Bounds on the reliability of structural systems. J Struct ASCE 1967;93(1): 171-200. [2] Penmetsa RC, Zhou L, Grandhi RV. Adaptation of fast fourier transformations to estimate the structural failure probability 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, GA, April 3-6, 2000. [3] Ditilevsen O. Narrow reliability bounds for structural system. J Struct Mech 1979;7:453-472. [4] Wang LP, Grandhi RV. Improved two-point function approximations for design optimization. AIAA J 1995;33(9):17201727. [5] Xu Suqiang Grandhi RV. Multi-point approximation for reducing the response surface model development cost in optimization. Proceedings of the 1st ASMO UK/ISSMO Conference on Engineering Design Optimization, Ilkley, West Yorkshire, UK, July 8-9, 1999, pp. 381-388. [6] Wang LP, Grandhi RV. Safety index calculation using intervening variables for structural reliability analysis. Comput Struct 1996;59(6): 1139-1148.

413

A note on symmetric Galerkin BEM for multi-connected bodies J J . Perez-Gavilan, M.H. Aliabadi * Department of Engineering, Queen Mary, University of London, London El4 NS, UK

Abstract The non-uniqueness of the symmetric Galerkin boundary element method in its standard form when applied to multiple connected bodies was highlighted in [1]. In this paper, the causes of the problem are reviewed and two methods to overcome the singular behavior of the system of equations are presented. A numerical example is solved with both methods and shown to produce good results. Further comments on the relative strengths of the methods are given. Keywords: Boundary element method; Elasticity; Symmetric Galerkin

1. Introduction

2.7. Symmetric Galerkin

The symmetric Galerkin boundary element method was brought to the attention of the researchers more than 20 years ago [2] and in the last 10 years the development of the method has been substantial [3]. However all this research, no remarks can be found in the literature on the fact that for multiple connected bodies the scheme is unable to produce a unique solution for displacements, and consequently, the resulting system of equations is singular. Recently, Perez-Gavilan and Aliabadi [1] demonstrated the non-uniqueness properties of the symmetric Galerkin and proposed a modified formulation to overcome the difficulty. In this paper, the problem is briefly reviewed and two methods are presented that produce unique solutions; one which, partially destroys the symmetry and another that avoids the multiple connectivity altogether.

In the symmetric Galerkin method, the displacement and traction equations are used as follows

j UkfdTy= f f I Uiktidr.dF dT, dTy Tu

• \ t

In this section, a brief description of the non-uniqueness of the symmetric Galerkin boundary element method when applied to multiple connected bodies will be given. Full details can be found in [1].

* Corresponding author. Tel.: +44 020 7882-5301; Fax: +44 020 8983-1007; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

\ TikUi dV, dVy

(1)

T;

Tf/

and / tkf dTy= Tj

2. The uniqueness problem

Tn-Tj

j ir j Dikti dP, dP, Tj

r

- \ f

\

SikUidT^dTy

i dP^c dP^

(2)

where Uik and 7^^, denote the displacement and traction fundamental solutions and Dtk and Sik their corresponding gradients; Ut and ti are the unknown displacements and tractions and V^ is a weighting function. The equations are written for an internal point y' which, is taken to the boundary by a limiting process. The first equation above is

/ / Perez-Gavildn, M.H. Aliabadi/First MIT Conference on Computational Fluid and Solid Mechanics

414

used for the part of the boundary where the displacements are constrained i.e. Tu and the second equation or traction equation is used for the Neuman part of the boundary Tj. Boundary Tj is a closed unrestrained boundary, over which the integration is shown explicitly. The use of the equations in this way produces a symmetric system of equations. 2.2. Non-uniqueness The non-uniqueness of symmetric Galerkin BEM is best demonstrated by adding a rigid body motion: rotation and translation, to the displacements of an unrestrained boundary, here Tj, and observing that the equations are still satisfied. A rigid body rotation and translation can be written as 1

I = J

(3) ' -Iji i + J Adding these displacements to the ones in the last integrals of Eqs. (1) and (2) gives Ui(x)

= lirXr - Xi + C/

ir

Tik (lirXr

X € Tj

hj —

x, -\-Ci -\-Ui) dr^ dr,.

(4)

+r u^{x) = u^{x) xeT, t,{x) = t,(x) xeFr Fig. 1. Modified symmetric Galerkin scheme for mulfi-connected bodies. Detail showing an unrestrained boundary segment i.e. three degrees of freedom taken from two consecutive nodes of an element, for which the displacement equation is applied instead of the tracfion equafion.

and \l/

Sik {lirXr - Xi + Ci + Ui) dT_, dTy

(5)

The non-uniqueness follows from the observation that j Tik{y\ x) {l,rXr - Xi + a) d r , = 0

(6)

This will violate the symmetric Galerkin scheme and the symmetry of the system will break, although only a limited part of the system is affected leaving the main part of the system symmetric. The resulting block structure of the matrix can still be exploited using standard matrix algebra techniques.

and

/

Sik(y\

X) QirXr - Xi + Ci) d f , = 0

(7)

from the properties of the double layer potential and its normal derivative [5]. Consequently, numerical implementation will lead to a coefficient matrix that is singular. The above result was possible because the displacement equation is not written over Tj, as in that case, adding to the displacement on Tj also alters the free term displacements which, would not vanish after integration. This also explains why when the displacement equation is used over the entire boundary the problem does not arise and the solution is unique. The free term of the gradient equation is a traction that is not affected when altering the displacements over Tj.

4. Solution by subregions In this method the idea is simply to avoid multiple connectivity altogether, by creating sub-regions each of which should be simply connected. See for example Fig. 2.

3. Modified symmetric Galerkin (MSG) [1] One way to deal with the problem is to include sufficient displacement equations over the unrestrained boundary to constrain the rigid body motion (see Fig. 1).

Fig. 2. Multi-zone model with simple-connected regions.

J.J. Perez-Gavildn, M.H. Aliabadi/First MIT Conference on Computational Fluid and Solid Mechanics !

{

H-HK—H—y- X X X X

—^—~j^—^

,^ ^

*Sr—

>

415

H ' •' •

^ X i

I

T

I

t

u—^—%

1

;

1

n a)

• f—^—^

1

Fig. 3. Shear wall, (a) Model, (b) Deformed model using standard symmetric Galerkin showing characteristic singular behavior, (c) Deformed model using modified symmetric Galerkin scheme and with dashed line the solution calculated with standard collocation method. A fully symmetric scheme to deal with multiple regions is now available and is reported in [4]. Eliminating multiple connectivity readily eliminates the non-uniqueness of the displacement solution.

~ ¥ • • - • > ? • •• $

i

1

a) Fig. 4. Wall with simple-connected sub-regions, (a) Model, (b) Displacement solution using multi-region algorithm showing good agreement with collocation solution (dashed line).

breaking the symmetry of the system can be implemented to be transparent to the system user. The method using sub-regions preserves the symmetry of the system but the user is asked to introduce a new boundary with no physical interpretation, and for 3D model could require a significant amount of work.

5. Example Acknowledgements To demonstrate the effectiveness of above methods a structure consisting of a wall 10.8 m wide by 19.2 m tall, with four openings 3 by 3 m, was solved with each method. Only three of the openings form internal boundaries and their axis are 0.9 m to the left of the wall axis. The wall is subjected to uniform shear of 1000 Mpa on its top and clamped on its base. The internal boundaries are traction free and unrestrained. The material properties are Young's modulus E = 2.2 X 10^ MPa and Poisson's ratio v = 0.15. The results obtained with modified and sub-region methods are presented in Figs. 3 and 4, respectively.

6. Conclusions The symmetric Galerkin boundary element method cannot be used in its standard from for multiple connected bodies with closed unrestrained boundaries. Two methods, which produce unique solutions and accurate results, were presented. The modified Galerkin scheme while partially

The first author would like to acknowledge the support of CONACYT (The National Council for Science and Technology of Mexico) for this work.

References [1] Perez-Gavilan JJ, Aliabadi MH, Symmetric Galerkin BEM for multi connected bodies. Electr Bound Elem J, submitted for pubhcation. [2] Sirtori S, General stress analysis method by means of integral equations and boundary elements. Meccanuca, 14, 210218 (1979). [3] Bonnet M, Maier G, Pohzzotto, Symmetric Galerkin boundary element methods. Appl Mech Rev 1998;51(ll):669-704. [4] Gray LJ, Paulino GH, Symmetric Galerkin Boundary Integral Formulation for Interface and Multi-Zone Problems, Int J Numer Methods Eng 1997;40:3085-3101. [5] Chen G, Zhou J, Boundary Element Methods. Academic Press, London, 1992, Chapter 9.

416

Vibration suppression of laminated composite plates using magnetostrictive inserts S.C. Pradhan^'*, K.Y. Lam% T.Y. Ng\ J.N. Reddy'' " Institute of High Performance Computing, 89C Science Park Drive, 02-11/12 The Rutherford, Singapore 118261, Singapore ^Department of Mechanical Engineering, Texas A &M University, College Station, TX 77843-3123,USA

Abstract In the present work, first-order shear deformation theory (FSDT) is used to study vibration suppression of laminated composite plates. The magnetostrictive layers are used to control and enhance the vibration suppression via velocity feedback with a constant gain distributed control. Analytical solutions of the equations governing laminated plates with embedded magnetostrictive layers are obtained for simply supported boundary conditions. Effects of material properties, lamination scheme, and placement of magnetostrictive layers on vibration suppression are studied in detail. Keywords: Composite laminate; Magnetostrictive material; Velocity feedback control; Analytical solution

1. Introduction Under the influence of an external magnetic field the grains of certain materials such as Terfenol-D, can rotate and align according to the applied magnetic field. These Terfenol-D layers are easily embedded into laminates made of modem composite materials, without significantly effecting the structural integrity. Much work has been previously done to study the interaction between magnetostrictive layers and the composite laminates and the feasibility of using these magnetostrictive materials for active vibration suppression [1-3]. Some researchers [4,5] have also carried out nonlinear interaction of magnetostrictive layer and the composite laminates. Although there have been important research efforts devoted to characterizing the properties of Terfelon-D material, fundamental information about variation in elasto-magnetic material properties is not available. A few studies [6,7] report experimental evidence for a significant variation in material properties such as Young's modulus and magneto-mechanical coupling coefficients. In the present work, a plate formulation based on first-order shear deformation theory is presented to bring out the effects of the material properties of the lamina, lamination scheme and placement of the magnetostrictive layers on vi-

bration suppression characteristics. This paper extends the work of Reddy and Barbosa [3] on beams to composite plates and brings out the importance of the location of the magnetostrictive layers and lamination schemes in controlling the vibration by suppression of fundamental as well as higher modes.

2. Theoretical formulation Consider a symmetric laminated composite plate composed of A^ layers with the Mth and (A^ — M + l)th layers being made of magnetostrictive material. The remaining layers are made of a fiber-reinforced material, that are symmetrically disposed about the midplane of the plate (Fig. 1). The displacement field of the first-order shear deformation theory (FSDT) and the associated linear strains are given in [8,9]. The constitutive relations of the /:th lamina are [9]

1 ^yy

I ^^>'

•

(*)

1

f ^ ik) IcTxx

Cii

Qn 2 l 6

— I 2l2 222 G26 =

Gi6 Qie e66_

"^31 ^yy

Yxy

' -

Z ' ^32 1

m

H,m

"^36]

-\ik)

* Corresponding author. Tel.: +65 770-9493; Fax: +65 7709902; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

QAA

Q45

Yyz I

Q45

Q55

Yxz

(2)

411

S.C. Pradhan et al. /First MIT Conference on Computational Fluid and Solid Mechanics where Stj, Ctj and Mij are defined by

Zf

533 = K {2A,s^i^2 + AuPl + As5^l), 534 = K (A55P1 +

A45h),

535 = K (A45^i + A44P2) C33 = C34 = C35 = 0, M33 = /o, M34 = M35 = 0 i>^r

i^^r

i^
"1^^

1 ^ ^

D^3:

~^^r

1 ^ ^

N-M+1 layer M layer

543 = K {A45P2 + 844 = DnPl

AssPi),

+ DeePl + K Ass,

S45 = {Dn + De6)Pih

Fig. 1. Laminated composite plate with embedded magnetostrictive layers.

+ KA45

C43 = ^i£:3i, C44 = C45 = 0, M44 = /2, M43 = M45 = 0 553 = /^(A44ft + A45A),

where Qij and "^y represent transformed stress induced stiffness coefficients and transformed magnetostrictive moduli, respectively. The magnetic field intensity H^ of the velocity proportional closed-loop feedback control is expressed in terms of coil current as coil constants as y/b^, + 4r2

dy

+ q = h-

dt

(4)

3f2

3M

(5) 3My,

(6)

dx

The equations of motion (6) can be expressed in terms of the generalized displacements (wo,(px and 0^), material coefficients (A/y, Btj, Dtj), shear correction factor (K), and magnetostrictive coefficients 8ij. For simply supported rectangular plates, the generalized displacements (ifo,0x,0y) are expressed in double Fourier series (see [8,9]) with undetermined coefficients. The mechanical loads and magnetostrictive moments are also expanded in double Fourier series. Substituting the expansions into Eq. (6) yields the result S33

S34 S35

S43

S44 S45

^53

'^54 '^ss

+

0

0

w

C43

0

0

-^mn j

C53

0

0

J-mn

w

0

0

M44 0

•^mn

0

y

0

0

0

M33

M55

C53 = ^^2^32, C54 = C55 = 0, M55 = /2, M53 = M54 = 0 (8) where y^i = m7t/a and ^^2 = WTT/Z?. For vibration control, we assume that the mechanical load is zero, q = 0, and seek solution of Eq. (7) in the form

where be re and ric are coil width, coil radius, and the number of turns in the coil, respectively. The governing equations of the first-order theory are (see Reddy [8]) dx

Ss5 = DeePi + D22P2 + ^ ^ 4 4

y^mn

\lmn •

=

•

0

0 J

(7)

Wmn(t)

= Woe'\

Xmn{t):

Y^n{t) = Y^e^' (9)

Substituting Eq. (9) into Eq. (8), for a non-trivial solution we obtain the result ^33

534 535

^43

^44 ^45

S53

»^54 '^'55

= 0

(10)

where 'Sij = Stj + XCtj + X^Mij. This equation gives three sets of eigenvalues. The lowest one corresponds to the transverse motion. The eigenvalue can be written as A = —a + i(Od, so that the damped motion is given by sm codt sm COd

mnx . nny sm —— a b

(11)

3. Results and discussion Numerical studies are carried out to obtain the natural frequencies, magnetostrictive damping coefficients and the vibration suppression time. Various lamination schemes are used to study the influence of the position of the pair of magnetostrictive layers from the neutral axis on the vibration suppression time. The material properties used for carbon fiber-reinforced polymeric (CFRP), Graphite epoxy (Gr-Ep(AS)), Glass epoxy (Gl-Ep) and Boron epoxy (Br-Ep) composite laminates are same as those in [3]. The plate is assumed to be square and of dimensions Im X Im. Magnetostrictive material properties are taken to be Em = 26.5 GPa, Vm = 0.0, Pm = 9250 k g m - ^ 4 = 1.67"^ mA~^ and c(t)rc = 10"^. The numerical values

418

S.C. Pradhan et al /First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Inertial and material stiffness coefficients and the parameters a and codr Material CFRP

Gr-Ep(AS) Gl-Ep Br-Ep

Laminate [±45/m/0/90]5 [45/m/-45/0/90]5 [m/±45/0/90]5 [m//904]5 [m/04]s [±45/m/0/90]5 [±45/m/0/90]5 [±45/m/0/90]5

h

h (10-4) (kg/m)

-[^31,^32]

—a ±(jOdr (rad s ^)

(kg/m) 33.09 33.09 33.09 33.09 33.09 30.1 33.7 34.1

2.461 3.352 4.54 4.54 4.54 2.196 2.514 2.55

22.13 30.98 39.83 39.83 39.83 22.13 22.13 22.13

6.588 9.224 11.861 11.866 11.866 7.244 6.474 6.388

of moment of inertia, magnetostrictive material constants, magnetostrictive damping coefficients and natural frequencies of various lamination schemes are listed in Table 1. A comparison of uncontrolled and controlled motion at the mid point of a plate is shown in Fig. 2. Table 2 shows the influence of the position of the magnetostrictive layer (in the z-direction) from the neutral axis, and the influence of the lamination scheme on the damping of the vibration responses. The value of a [see Eq. (11)] increases when

± 254.823 ± 240.724 ±221.418 lb 187.435 ± 187.435 ± 264.443 db 187.440 ± 309.992

the magnetostrictive layer is located farther away from the neutral axis, indicating faster vibration suppression. This is due to the larger bending moment created by actuating force in the magnetostrictive layers. Present results also show that the vibration suppression time decreases very rapidly as the mode number increases. From Fig. 3, it can be observed that attenuation favours the higher modes. In this study, the vibration suppression time is defined as the time required to reduce the uncontrolled vibration

0.2

0.3 Time (s)

Fig. 2. Comparison of uncontrolled (

) and controlled (

) modon at the midpoint of the plate for lay-ups [m/45/-45/0/90]5.

Table 2 Suppression time ratio for different CFRP laminates, h = 5 mm and hm =2 mm. Laminate

±0}dr

(m) [±45/0/90/m]5 [±45/0/m/90]5 [±45/m/0/90]5 [45/m/-45/90]s [m/±45/0/90]s

0.001 0.006 0.011 0.016 0.021

1.527 12.308 25.987 39.767 53.669

1287.271 1086.117 1073.565 1007.686 906.182

t at Wmax/10

Ts

1.426 0.192 0.095 0.064 0.050

1.000 0.135 0.067 0.045 0.035

(mm) 0.764 0.818 0.798 0.827 0.962

S.C. Pradhan et al. /First MIT Conference on Computational Fluid and Solid Mechanics

419

0.005 0.004 III III

moclen=1 - mode n=2

>l^\j'^\J^^'^^'-^KJ<-^'~>"

0.4

0.6

0.8

Time(s)

Fig. 3. Comparison of original and controlled motion at the midpoint of the plate for modes n = \,2 (lay-up [45/-45/in/0/90]5). amplitude to one-tenth of its initial amplitude. Further, numerical simulations are carried out to estimate the vibration suppression time ratio (suppression time divided by the maximum suppression time) as the distance between the magnetostrictive layers and the neutral axis are varied. The suppression time is longest when the magnetostrictive layer is closest to the neutral axis and this is simply because the neutral axis generally undergoes the lowest overall displacement. The vibration suppression characteristics are improved as the magnetostrictive layers are moved away from the neutral axis with the suppression time ratio decreasing as a result. It is also observed that the suppression time ratio does not change with the intensity of control gain of the magnetic field. Finally, it is observed that a relatively thinner magnetostrictive layer leads to better attenuation characteristics.

4. Conclusions A theoretical formulation for laminated plates with embedded magnetostrictive layers is presented, analytical solution for simply supported boundary conditions is developed, and numerical results are discussed. The formulation is based on the first-order shear deformation plate theory (FSDT), and the analytical solution for simply supported plate is based on the Navier solution approach. The effects of the material properties of the lamina, lamination scheme, and placement of the magnetostrictive layers on the vibration suppression time have been examined in detail. It was

found that attenuation effects were better when the magnetostrictive layers were relatively thinner and placed as far away from the neutral axis as possible.

References [1] Anjanappa M, Bi J. Modelling, design and control of embedded Terfenol-D actuator, Proc. SPIE-Smart Structures and Intelligent Systems 1917, 1993, pp. 908-918. [2] Krishna Murty AV, Anjanappa M, Wu Y-F. The use of magnetostrictive particle actuators for vibration attenuation of flexible beams. J Sound Vibr 1997;206(2): 133-149. [3] Reddy JN, Barbosa JA. Vibration suppression of laminated composite beams. Smart Mater Struct 2000;9:49-58. [4] Pratt JR, Oueioni SS, Nayfeh AH. Terfenol-D nonlinear vibration absorber. J Intell Mater Syst Struct 1999;10:29-35. [5] Armstrong WD. Nonlinear behaviour of magnetostrictive particle actuated composite materials. J Appl Phys 2000;87:3027-3031. [6] Hudson J, Busbridge SC, Piercy AR. Dynamic magnetomechanical properties of epoxy-bonded Terfenol-D composites. Sensors Actual A-Phys 2000;81:294-296. [7] Busbridge SC, Meng LQ, Wu GH, Wang BW, Li YX, Gao SX, Cai C, Zhan WS. Magnetomechanical properties of single crystal Terfenol-D. IEEE Trans Magnet 1999;35:38233825. [8] Reddy JN. Mechanics of Laminated Composite Plates: Theory and Analysis. Boca Raton, FL: CRC Press, 1997. [9] Reddy JN. On laminated composite plates with integrated sensors and actuators. Eng Struct 1999;21:568-593.

420

PDFs of the stochastic non-Unear response of MDOF-systems by local statistical linearization H.J. Pradlwarter*, G.I. Schueller Institute of Engineering Mechanics, L.-F. University, Technikerstr 13, A-6020 Innsbruck, Austria

Abstract A novel computational approach called 'Local Statistical Linearization' (LSL) based on statistical equivalent linearization and Gaussian superposition has been introduced recently [4]. The methodology allows to extend the concept of statistical equivalent linearization to proceed from estimates of the second moment properties of the stochastic response to estimates for the non-Gaussian probability distribution. Locally linearized non-linear systems do not have the properties of a linear system but approximate closely the non-linear characteristic of the system. The suggested approach employs the well developed equivalent linearization procedure or Gaussian closure to compute the non-Gaussian distribution of the response of a non-linear system. Since equivalent linearization is applicable for higher dimensions and FE-models, the suggested approach lends itself to providing numerical solutions for higher dimensional cases. In this paper, the computations of probability density functions of non-linear MDOF-systems are discussed. Moreover, the basic methodology is extended to increase the efficiency of the numerical procedure when dealing with non-linear MDOF-systems. Keywords: Statistical linearization; MDOF-systems; Probability density function; Cumulative plastic deformation

1. Introduction The procedure of equivalent linearization is well known and widely used for the estimation of the stochastic response of non-linear systems [5]. This procedure allows to estimate the mean as well as the covariance matrix of the stochastic response. The strength and weakness might be summarized as follows: It is beside Monte Carlo simulation the only general applicable tool to analyze the stochastic non-linear response of many-degrees-of-freedom systems. Statistical equivalent linearization could be developed further to be usable in practice for the analysis of multidegrees-of-freedom systems, non-stationary excitations and of hysteretic non-linear systems. The introduction of complex modal analysis for the evaluation of the linearized stochastic response contributed further considerably to the computational efficiency of the linearization procedure [7]. Hence, it is justified to state that statistical equivalent linearization matured already to a stage where the procedure is suitable to be employed in engineering design practice. * Corresponding author. Tel.: -h43 (512) 517-6846; Fax: -H43 (512) 517-2905; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

All the above benefits of statistical equivalent linearization go hand in hand with several crucial drawbacks. First of all, one should keep in mind that the replaced linearized system can never represent the typical characteristics of a nonlinear system, irrespective of the sophistication of the criteria used for the equivalent linearization. One of the most discussed issues is certainly the fact that only the first two moments of the response can be determined which define uniquely only a normal distributed response. Since it is well known that the response is generally non-Gaussian distributed, the first two moments of the response are insufficient. Moreover, it can be proved that the accuracy of estimating the moments depends on the difference between the assumed and true probability distribution [2]. Hence, not even the first two moments can be estimated with confidence because the true distribution is unknown. Therefore, statistical equivalent linearization is a methodology which allows only quite approximate estimates of the distribution of the non-linear response. In a recent paper [4], it is shown that a computational approach can be used to remove to a large extent the above mentioned serious limitations. The key is the local linearization in phase space which allows to approximate the

H.J. Pradlwarter, G.I. Schueller/First MIT Conference on Computational Fluid and Solid Mechanics actual nonlinear behavior. It is a computational approach where the achieved accuracy depends mainly on the level of discretization. Local statistical linearization (LSL) employs the well developed Gaussian closure technique or standard equivalent linearization which can be employed for MDOF-systems. Hence, LSL procedure has the potential to estimate the probability density function of the non-linear response. In this paper, first steps toward the computation of PDFs of MDOF-systems are shown. For this purpose the concept of SLS is extended by an adaptive algorithm for controlling the accuracy. The extended procedure is then applied to a hysteretic MDOF structural system.

All

where Ej{h{x,t)) means the expectation with respect to the local density pj(x,t). The local linearization in Eq. (4) differs from the global linearization fundamentally. Most crucial, the locally linearized non-Unear function geq,L(x,t) is not a linear function anymore, but a non-linear function which closely resembles the original non-linear function g(x,t). Therefore, the locally linearized system does not have the characteristics of a linear system but represents closely the properties of the original nonlinear system. The LSL procedure is based on the Gaussian superposition representation, where the non-Gaussian PDF is represented as sum of Gaussian densities. p(x, t) = Y^Aj pj(x, t),

(6)

Y^Aj = \

2. Review of local statistical linearization In this paper only some basic features of LSL will be discussed briefly in context with the analysis of MDOFsystems. The reader may find [4] quite useful for the following. The LSL-procedure replaces the non-linear system not by a single global linear system, but by many local Hnear systems. The local system properties are determined by local Gaussian densities exactly in the same manner as in the standard (global) statistical linearization procedure. Performing statistical linearization, the non-linear function is replaced by a globally linear substitute geq(x,t). In particular, ge,{x, t) = E[g{x, 0} + E j ^-^j~^ ] • (^ - E{x}).

(1)

Evaluating the stochastic response on the basis of the linearized system introduces local deviations S{x,t) from the non-linear system behavior which might be measured globally by the positive quantity: ^g{x, t) dx

D{t)

^g{x, t) dx

' p(x, t)dx

(2)

The deviation D(t) in Eq. (2) depends on the non-Unear function g(x,t) and there is no possibility to lower this value due to the global linearization. For this purpose, the PDF p(x,t) will be represented by a sum of densities,

p(x,t) = Yl^jPj^''^^^^

^^^

Yl'^j = ^

j

j

where the local densities pj(x,t) have a considerable smaller standard deviation (aj < a) than the standard deviation a of the original density p(x,t) and the amplitudes Ai (meaning of probability) sum up to one. The representation in Eq. (3) allows to define a local linearization where the nonlinear function is replaced by the sum: Aj pj(x,t) qj(^^t) (4) ••-E p(x, t) with geqjix, t) := Ej{g(x, t)]-\rEj

dg(x, t) dx

'{x-E{x])

(5)

Moreover, it is assumed that all coefficients Aj are nonnegative Aj > 0 to ensure non-negative densities everywhere in phase space. Gaussian densities are chosen as basis distributions.

Pj{x,t) = ^ ^ cxp\-Ux-fijfCjHx-,ij)\ y(2;r)«det(Cy) "^ I 2 ' ^

M

(7)

After the establishment of the representation of the density at a fixed time f, it is necessary to consider the evolution of the PDF p(x,t) with respect to time t. For this purpose the PDF at two subsequent instants t and t -\- At is considered. The corresponding PDF's are denoted by p(x, t) and p(x, t + AO, respectively. Suppose, the total density at instant t is specified by a sum of local densities according to Eq. (6) using normal distributions. Let the number of local densities be denoted as A^. Hence, the total density is described by the data set. pixj):

{Aj{t),tij{t)Xj{t),

j =

h....N}

(8)

comprising all amplitudes, mean vectors and covariance matrices of the local Gaussian densities. The evolution of the local densities with respect to time will be represented by the map. {Ajit), pj(x, 0 ) ^ (Aj(t + AO, Pj(x, t + AO)

(9)

where two properties are fundamental. First, the amplitudes Aj remain constant with respect to time t, i.e.. Aj{t) = Ajit -^ At)

(10)

and secondly, the shape of the density will in general change due to the non-linear dynamics and the stochastic excitation. This evolution is approximated by considering only the evolution of the properties of the first two moments and assuming that the densities remain Gaussian distributed. Therefore, the evolution of the PDF p{x, t + AO [Aj{t), Hj(t), Cj(t)} ^ {Aj(t), fijit + AO, Cjit + AO} (11)

H.J. Pradlwarter, G.I. Schueller/First MIT Conference on Computational Fluid and Solid Mechanics

422

is well defined according to Eq. (8). This approach corresponds exactly to Gaussian closure and statistical equivalent linearization. Hence, for general non-linear systems, these mappings are performed by using the well developed procedure of statistical equivalent linearization or Gaussian closure, with the only significant difference that the linearization coefficients are now determined on the basis of the local densities Pj{x, t). This implies, however, that the linearization and integration must be carried out N times.

3. Control of local statistical linearization The LSL procedure determines the non-Gaussian response distribution at discrete instants tt^ = k • At. The computation involves three distinct steps: • Each gaussian density gj(x, r^) is checked if its spatial extent (covariance matrix Cj) is sufficiently small to represent local properties of the non-linear system. If this is not the case, the density is decomposed into many gaussian densities with a considerably smaller extent. (Aj, fij, Cj) i-> {(Ajk, fijk, Cjk), k = - m , . . . , m} (12) The corresponding procedure is documented in [4]. • To avoid an exponential growth of local Gaussian densities with respect to time, 'clumping' of neighboring densities is introduced. A further development which is especially suitable for MDOF-system is described in this section. • The evolution of the local Gaussian density is computed for the next time step, {Ajit,),iij{t,),Cj{t,)} ^ {Aj(tk), fijitk + AO, Cj(tk + At)}

(13)

using the standard equivalent linearization for general non-linearities like hysteresis or Gaussian closure for polynomial non-linearities. The first and third item, namely the decomposition and the computation of the evolution of the Gaussian densities is straight forward. This is not so for the second item, i.e. the clumping procedure, especially when MDOF-system should be analyzed. For making the LSL-procedure manageable for MDOF systems, it is therefore necessary to improve on the clumping algorithm. Hence, the following focuses on this item. For the purpose of clumping, the state vector x of the n-dimensional system is normalized to the vector y using a reference vector Ax = {Ax], Axj, . . . , AJC,,}. X \^ y,

yk=

Xk/Axk

(14)

The transformation x \-^ y serves two objectives. First, the components of the vector y can be made independent from

the used units (km or m, s or h) they are dimensionless quantities. Secondly, this transformation can be used to control the resolution of the Gaussian superposition as it is discussed later on. After establishing the reference vector AJC, all mean vectors of the local densities fij are transformed into normalized vectors Vj. flj ^

Vj

Vkj =

(15)

likjl^^k

It is suggested to clump those pairs of local Gaussian densities (g/, gj), of which the normalized mean vectors (v/, Vj) have a relative short distance ||v/ — Vj\\ from each other. Basically, any norm can be used, but for computational efficiency the norm using the sum of absolute differences dij = X^l^-^'

(16)

%-|

seems to be most appropriate. A pair of local Gaussian densities (g/, gj) is then reduced to a single local density gk, preserving the second moment properties: Ak = Ai + Aj /t^

={fiiAi-\-fijAj)/Ak

(17)

Ck = {Aild + (fii - iij,){iii - /^,)^] + Aj[Cj + iiij - fik)(fij -

tikV]}/Ak

Of course, the accuracy is affected adversely by clumping. Although the first two moments are maintained, the sum of both densities differs slightly from the clumped density. The deviations depend mainly on the distance \\fii - fij\\ and on the difference ||C/ — Cj\\ of the covariance matrices. For this reason, the reference vector Ax controls the accuracy of which the densities of the components are represented. Relative small components A, result in a higher accuracy for the component x, to the disadvantage of less accuracy of components Xj with a larger Aj. Hence the reference vector Ax can be also used to shift accuracy from some components to others. Clumping close pairs {giix,t), gj(x,t)) based on the distance dij has the following benefits: • The accuracy is distributed approximately equal in phase space, since the mean vectors of the local Gaussian densities gj(x,t) tend to be equal dense, i.e. the nearest neighboring mean vectors (/i,^, fij) tend to have equal distance. • The criterion for clumping based on the distance dij is independent of the dimension n of the state x. • The method is self-adjusting. Generally, all components are correlated and the response occupies, roughly speaking, just a curved hyper-plane of the fz-dimensional phase space. Using the distance as criterion for clumping, the local Gaussian densities are distributed equally within these hyper-planes. The required total number N of local Gaussian densities depends mainly on the complexity of the non-linear

H.J. Pradlwarter, G.I. Schueller/First MIT Conference on Computational Fluid and Solid Mechanics

423

+n (d)

system and the desired accuracy. The complexity increases with increasing non-Unearities. Fortunately, the complexity increases for most non-linear structures only moderately with increasing dimension n because the components of the state vector x are then in general much stronger correlated. For determining accurately the PDF q(x,t) of a complex structure, the required total number A^ might be of the order 10,000 and larger. It is therefore important to have algorithms available to compute efficiently close pairs in a n-dimensional space. Such algorithm have been developed in context of 'Distance Controlled Monte Carlo Simulation' [3] where a similar problem arises. It should be mentioned that the effort to determine these close pairs is proportional to A^ \n(N) and nearly independent of the dimension n.

A

4. Numerical example The PDF of the stochastic non-linear response of MDOF-system is computed. The structural system is shown in Fig. 1, representing a hysteretic six story building subjected to instationary random earthquake loading. This simplified structural model has been investigated previously [6] in context of a reliability estimation using Monte Carlo simulation and non-Gaussian equivalent linearization. In this paper, we attempt to estimate accurately the PDF of the non-linear stochastic response of a MDOFsystem with a state space dimension considerable larger than three or four, where alternative solution based on the Fokker-Planck equation might still be feasible [1]. Each floor is represented by a rigid mass, connected by flexible hysteretic columns. This system requires 26 components in phase space representation, namely for each mass

Fig. 2. Hysteresis.

m/, / = 1 , . . . , 6, the displacements w/, its velocities w/, positive plastic deformations Vf and negative plastic deformations Wi. Moreover, the non-stationary horizontal ground motion is presented as time modulated filtered white noise using a linear second order filter. The filter augments the state space dimension by two, leading to a total of 26. The restoring force n between the floors (/ — 1, /) is governed by the inter-story displacement dt = ui — Ui-\, the linear stiffness kt, and two further parameters vty and Tip. As indicated in Fig. 2, the restoring force is linear within the range -vty > vt > -i-ny, where r/^ denotes the limit restoring force where the plastification of the cross section starts. For a restoring force absolutely larger than rty, part of the cross section plastifies leading to the positive plastic displacements Vj and negative plastifications Wj. The minimum and maximum restoring force is bounded by the limits ±rip which denotes restoring force where plastification occurs in the whole cross section. These hysteretic characteristics of the inter-story force can be modeled analytically by the relations, r,(i) = kMiit) - vt(t) - Wi(t)) -h diit) = Ui{t)

-Ui- i ( 0 ,

h

-^diit), (18)

uo(t) = 0,

Fig. 1. Six-story building.

where the relative displacements at the basement is defined as wo(0 = 0 and considering P-A effect due to the normal force Ni in the columns which is here assumed negative for compression. The symbol h denotes here the height of

424

H.J. Pradlwarter, G.I. Schueller/First MIT Conference on Computational Fluid and Solid Mechanics

mm%m%^%m%mHU rJdJ

rJd^)

rJd^)

2'"2^

'31-3/

rjdj

'41-4

rJdJ

rJdJ

Fig. 3. Shear beam model. the columns. Assuming further a linear transition of the tangential stiffness from the start of yielding r/y until total plastification r/^, the plastic deformations can be expressed by the differential equation: Vi{i)

=di

0 n(t)-

0 Wi(i) = di \ -njt)

for Zi(t) < r , v f o r z / ( 0 > rriy forz/(r) > -r,v -ny

(19)

forz/(r) < - r „ v

Acknowledgements

Then, structure can be modeled by a shear beam model as shown in Fig. 3 for which the non-linear equation of motion reads for all six masses m, rriiiii + c,ii, + r/(r) - r/+i(0 = / = ! , . . . , 6,

-mia{t), (20)

rn{t) = 0

LSL allows to compute probability density functions of the non-stationary non-linear stochastic response of MDOF-systems. To the authors knowledge, there are no alternatives besides Monte Carlo simulation available to determine the PDFs of hysteretic MDOF-systems. At the same level of accuracy, the LSL procedure is several orders of magnitudes more efficient than Monte Carlo simulation.

The research is partially supported by the Austrian Research Council (FWF) under contract No. MAT 11498, which is gratefully acknowledged by the authors.

References

where the horizontal acceleration a{t) is specified by, (21)

a{t) = x{t) x{t) + 2i;f(jOfx{t) + cjolx{t) =

w(t)e(t)

(22)

in which f/, cof reflect soil properties, e(t) is a deterministic envelope function, and w{t) denotes unit white noise with autocorrelation function RXVW(T) = S(T).

5. Conclusions Based on the results obtained so far (see also [4]), the following conclusions are drawn: • The computational Local Statistical Linearization (LSL) procedure can be applied to MDOF-systems as it is typically used in context of the conventional equivalent linearization procedure. • The original procedure has been extended for the treatment of MDOF-system by introducing a novel clumping procedure. • Similar as in FE-analysis, the accuracy of the computed PDFs can be controlled by the level of discretization expressed by the number of N of local Gaussian densities. Hence any desired accuracy can be reached by the LSL procedure.

[1] lASSAR Subcommittee. A state-of-the-art report on Computational stochastic mechanics. Schueller GI (Ed). Probab Eng Mech 1997; 12(4): 197-321. [2] Kozin F. The method of statistical linearization for nonlinear stochastic vibrations. In: Ziegler F, Schueller GI (Eds), Proc lUTAM symposium, Austria. Berlin: Springer Verlag, pp. 44-56, 1988. [3] Pradlwarter HJ, Schueller GI. Assessment of low probability events of dynamical systems by controlled Monte Carlo simulation. Probab Eng Mech 1999;14:213-227. [4] Pradlwarter HJ. Non-linear stochastic response distributions by local statistical linearization. Int J Non-Linear Mech, in press. [5] Roberts JB, Spanos PD. Random Vibration and Statistical Linearization. Chichester: Wiley, 1990. [6] Schueller GI, Pradlwarter HJ, Bucher CG. Efficient computational procedures for reliability estimates of MDOF-systems. Int J Non-Linear Mech 1991;26(6):961-974. [7] Simulescu I, Mochio T, Shinozuka M. Equivalent linearization method in nonlinear FEM. J Eng Mech 115;1989:475492.

425

Effects of uncertainties on lifetime prediction of aircraft components C. Proppe*, G.I. Schueller Institute of Engineering Mechanics, Leopold-Fmnzens University, Technikerstr 13, A-6020 Innsbruck, Austria

Abstract Today's highly competitive environment in the commercial air transportation business instigates airline companies to continue operating their aircrafts beyond the design life, as long as the maintenance ascertaining their safe operation is more cost-effective than the investment into a new aircraft. In the context of damage tolerance of aging aircrafts, the residual strength of the fuselage skin damaged by multiple fatigue cracks received considerable attention. In the stress field of the fuselage skin, the rivet holes of the skin sphce joints constitute a number of stress concentrations, where multi-site damage (MSD) may develop: several small cracks may coalesce to form a large crack, after which damage may progress fast to ultimate failure. The small cracks may also lead to widespread fatigue damage (WFD), where unstable crack growth of a lead crack becomes possible due to strength reduction. As experimental work revealed clearly the stochastic nature of MSD and WFD failure, a realistic analysis of the uncertainties is necessary. It is the aim of this paper to develop a probabilistic framework for lifetime predictions of aircraft fuselages with MSD or WFD which incorporates an efficient method frequently employed in deterministic analyses, the finite element alternating method (FEAM). The uncertainties are characterised by random variables and importance sampling is applied in order to obtain robust and efficient estimations of the failure probability as a function of the cycle number. Preliminary results are given in this paper and important aspects of future research addressed. Keywords: Probabilistic lifetime prediction; Multi-site damage; Widespread fatigue damage; Aging aircraft

1. Introduction MSD and WFD have obtained great interest in the literature, notably after the accident of Aloha Airlines Boeing 737 in 1988, where MSD played an important role. As has been pointed out by several authors, the analysis of the fuselage with local damage can be carried out by a hierarchical model including a global, intermediate and a local analysis. As the first two steps can be done by standard FE programs, we focus our attention on the last step, the local analysis. Several deterministic investigations were devoted to the influence of certain parameters and the degree of sophistication necessary to model MSD and WFD locally in an adequate manner. Fracture mechanics has been most frequently employed, where the stress intensity factor has been calculated by analytical methods based e.g. on the strip yield model [2], the boundary [11] and * Corresponding author. Tel: -^43 (512) 507-6843; Fax: -h43 (512) 507-2905; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

dual boundary element method [10], and the finite element alternating method (FEAM) [14]. A detailed discussion of these methods is contained in [16]. From the results one can conclude that (1) the analysis has to take the redistribution of stresses due to MSD into account [16]; (2) linear elastic analysis using plastic zone linkup might lead to errors when compared with elasto-plastic analyses using the crack tip opening angle or the T*-integral as linkup criterion [6]; (3) a two-dimensional analysis considering straight cracks might be sufficient [15]. Experimental work reported in [8,12,16] revealed clearly a large scatter in the results, notably in the number of cycles until ultimate failure. Thus, a realistic analysis of the uncertainties that contribute to this scatter is necessary. To this end, several authors proposed probabilistic analyses. However, most of them, e.g. [1,19], deal with simpfified models that neglect most of the efforts achieved in deterministic analyses. This might be attributed to the additional simula-

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C. Proppe, G.I. Schueller/First MIT Conference on Computational Fluid and Solid Mechanics

tion loop a probabilistic analysis requires. A more advanced probabilistic investigation has been published recently [24].

2. Modeling aspects The problem of multiple cracks occurring in the row of the rivet holes in aircraft fuselage skin is rather complex. It is assumed that the MSD cracks emanate from the rivet holes. In the developments of the present paper, they are idealized as straight, collinear cracks. The hole with cracks is approximated as one straight crack of length equal to the sum of the hole diameter and the lengths of the two cracks, cf. [25]. Alternatively, it is possible to include the rivet holes in the modeling and then remove the residual stresses at the rivet hole surfaces through the FEAM algorithm [14]. For the structural component under investigation, the dominant part of the load is due to the fuselage pressurizing, and thus, tension perpendicular to the line of the cracks will prevail. Hence, only mode I fracture is considered here, although an extension of the approach to mixed-mode fracture is possible [14]. Experimental results [5,23] show that fracture of materials is a highly uncertain phenomenon. Several important uncertainties in crack initiation, crack propagation, and failure are identified here. The initial lengths of the cracks have a pronounced influence on the remaining fatigue life. The uncertainties in distribution and detection of MSD cracks in aging aircrafts necessitate modeling the initial crack length stochastically. Various approaches, including pitting corrosion [7] and fretting fatigue [21] can be found in the literature. Probabilities of crack initiation can be obtained from experimental results [20], where usually the number of cycles to reach a measured crack length is determined. On the other hand, the distribution of the initial crack lengths can be related to the probability of detection. In this context, the crack length distribution is often modeled by the exponential distribution. The lengths of the cracks emanating from the same rivet hole are assumed to be correlated. This is also to the benefit of approximating the hole with cracks as a single straight crack, which is not a good approximation when one of the cracks is very short, while the other one is relatively long. Several approaches may be used in modeling the crack growth phase, including the random variable approach and the stochastic differential equations approach (see e.g. [13, 22]). Using the random variable approach, the variance of the estimated crack length at a certain number of load cycles will be somewhat underestimated as compared to the stochastic differential equations approach, but on the other hand, a straight forward use of the importance sampling technique is possible [17]. Here, the material uncertainty is captured by representing the material coefficient of the Forman law used herein as a random variable.

The physical failure of the structure, i.e. the limit state, is as a combination of brittle and plastic failure. The actual failure mode is interpolated using the R6 interpolation function [9] between the two extreme failure cases. Fracture toughness and yield stress are also modeled as random variables, as they may vary considerably from specimen to specimen. In addition to the R6 curve criterion described above, the plastic zones linkup criterion is implemented.

3. Simulation procedure The proposed physical model and stress analysis methods underlying the reliability analysis require a rather involved computational algorithm entailing definition of the problem, generation of the random variables samples with importance sampling, FEAM solution, integration of differential equations, statistics, and efficient data handling. Owing to its modular concept, advanced macro language, and a wide range of available tools, the COSSAN™ software [3] appears to be suitable for implementing the analysis. As a first step, the samples of random numbers realizations are generated according to the respective distribution models and the correlation structure. Applying the importance sampling technique (see e.g. [18]) computational efficiency is most effectively influenced by the sampling distribution of the initial crack lengths. Thus, importance sampling is used only for the initial crack lengths, while the sampling distributions of the remaining variables were the same as the original distributions. By a careful choice of the sampling densities, more samples fall into the failure domain, while at the same time the initial crack growth phase is effectively captured by the probabilistic model, thus saving in each simulation run computational time in the crack growth integration. Fatigue crack growth integration is performed for each of the simulation samples. Fracture parameters for the current crack lengths are obtained by a call to the FEAM routine. The convergence of FEAM is evaluated in terms of the SIF increment from the current iteration. After each load block, the multiple failure criteria of the R6 curve and the plastic zones connecting, are evaluated. If the current sample is still in a state belonging to the safe domain, the crack growth analysis proceeds. Otherwise, the sample is marked as failed and a crack growth integration of the next sample is started. Having completed all simulation runs, failure probability is evaluated by means of the weights of the failed samples.

4. Results As a simple example, the method described above is applied to a reduced problem of a plate with four rivet holes, from which emanate the cracks cf. Fig. 1.

C. Proppe, G.I. Schueller /First MIT Conference on Computational Fluid and Solid Mechanics

o-
-^.

ao

ao

ao

if^

-A

ao

ao

ao

7^—^

TF

TF

All

ao

^

Fig. 1. An example of a plate with four rivet holes.

Table 1 Distribution parameters of variables Parameter

Units

Hole distances Initial length Law exponent Law multiplier Toughness Yield stress Young's modulus Poisson's ratio Stress amplitude

m m

Distribution

deterministic exponential deterministic normal Pa-v/m normal Pa normal Pa deterministic deterministic Pa constant

Mean

St.dev.

0.0254 0.002 2.7 4.5 . 10-19 1.1-10^ 3.5 • 10^ 7.0-1010 0.3 1.0-10^

_

0)

0.002

'co *o

-

^

4.5 • 10-20 1.1 10^ 3.5 • 10^

-

(0

o

0.01

0.001

0

20000

40000 60000 number of cycles

80000

Fig. 2. ProbabiUty of failure for various numbers of load cycles. Table 1 shows the distribution parameters of the variables used in the analysis. The original exponential distribution of the initial crack lengths has the mean value and standard deviation 0.002 m (single-parametric distribution), which was for the time being not obtained using inspection data, but rather corresponds to experience-based values used in the MSD research [1]. The length of the initial crack is measured form the rivet hole edge. The initial crack length sampling distribution is likewise exponential with mean and standard deviation equal to 0.005 m. The initial lengths of the two cracks emanating from the same hole are correlated with a correlation coefficient 0.58. The 10% coefficient of variation of the fracture toughness and yield stress correspond to the scatter indicated for the 2024 T3 Alclad aluminium alloy in [4]. The mean values of the parameters of the Forman law were adopted from [1]. Fig. 2 displays the estimated failure probabilities from 1000 samples for various numbers of load cycles.

5. Conclusions, future work The results presented in Fig. 2 show the expected range and trend of values. They illustrate the applicability of

the proposed method to the problem of MSD in aircraft fuselages. In the next steps of this research project the reliability analysis will be performed considering realistic load histories, relating the parameters of initial crack lengths distribution to in-service inspection data, and obtaining the distribution parameters of the Forman law material coefficient from statistical evaluation of the equation fits to test data. The computing time (1000 Monte Carlo simulations of crack growth over 25,000 load cycles were completed in a few hours for the example problem), which is mainly influenced by the size of the FE problem, allows for a more detailed analysis, incorporating elasto-plastic material properties and curved crack growth in the FEAM. Having determined the probability of failure as a function of the cycle number in a robust and efficient way, the expected total costs, including production, maintenance and repair costs can be calculated. In the future, non-destructive inspection and crack repair will also be included in the analysis with the ultimate goal of giving recommendations for inspection interval length with respect to an optimized balance between economy and structural reliability.

428

C Proppe, G.L Schueller /First MIT Conference on Computational Fluid and Solid Mechanics

Acknowledgements This work was supported by DaimlerChrysler Aerospace Airbus, which is gratefully acknowledged by the authors. The continued interest of Dr. G. Bertrand, DCA Airbus, Hamburg, Germany, in this research is also deeply appreciated. Thanks are also due to Mr. L. Nespurak for his help in carrying out the numerical analysis.

[12]

[13] [14]

References [15] [1] Bertrand G. Risk assessment of damage-tolerant metallic structures. In: Schueller GI, Kafka P (Ed.), Safety and Reliability: Proceedings of ESREL '99 — The Tenth European Conference on Safety and Reliability, Vol 1, MunchenGarching, Sept. 13-17, 1999, Balkema, Rotterdam, pp. 207-212. [2] Collins RA, Cartwright DA. On the development of the strip yield model for the assessment of multiple site damage. Theor Appl Fract Mechan 1996;22:167-178. [3] COSSAN. COSSAN™ — Computational Stochastic Structural Analysis — Stand-Alone Toolbox, User's Manual: IfM-Nr: A, Institute of Engineering and Mechanics, Leopold-Franzens University, Innsbruck, Austria, 1996. [4] Gallagher J. Damage tolerant design handbook. Metals and Ceramics Information Center, 1983, Columbus, OH, USA. [5] Ghonem H, Dore S. Experimental study of the constantprobability crack growth curves under constant amplitude loading. Eng Fract Mech 1987:27(l):l-25. [6] Gruber ML, Wilkins KE, Worden RE. Investigation of fuselage structure subject to widespread fatigue damage. In: Bigelow CA, Hughes WJ (Eds), Proceedings of the FAA-NASA Symposium on the Continued Airworthiness of Aircraft Structures, Atlanta, GA, August 28-30, 1997, pp. 439-459. [7] Harlow DG, Wei RP. Probabilities of occurrence and detection of damage in airframe materials. Fatig Fract Eng Mater Struct 1999;22:427-436. [8] Harris CE, Newman JC Jr, Piascik RS, Stames JH Jr, Analytical methodology for predicting the onset of widespread fatigue damage in fuselage structure. In: Bigelow CA, Hughes WJ. Proceedings of the FAA-NASA Symposium on the Continued Airworthiness of Aircraft Structures, Atlanta, GA, August 28-30, 1997, pp. 63-88. [9] Kanninen F, Popelar H, Advanced Fracture Mechanics, New York: Oxford University Press, 1985. [10] Kebir H, Roelandt JM, Gaudin J. Computation of life expectancy of mechanical structures. Proc. European Conference on Computational Methods in Applied Sciences and Engineering, 2000, Barcelona. [11] KontouUs PG, Kermanidis ThB. The LTSM-MSD boundary element code for fracture mechanics problems. In: Papadrakakis M, Samartin A, Onate E (Eds), Proc. Inter-

[16] [17]

[18]

[19]

[20]

[21]

[22]

[23]

[24] [25]

national Conference on Computational Methods for Shell and Spatial Structures, 2000, Athens, Greece, ISASR. Nesterenko GI. Fatigue and damage tolerance of aging aircraft structures. In: Bigelow CA, Hughes WJ. Proceedings of the FAA-NASA Symposium on the Continued Airworthiness of Aircraft Structures, Atlanta, GA, August 28-30, 1997, pp. 279-299. Ortiz K, Kiremidjian AS. Stochastic modelling of fatigue crack growth. Eng Fract Mech 1988;29(3):317-334. Park JH, Atluri SN. Mixed mode fatigue growth of curved cracks emanating from fastener holes in aircraft lap joints. Comput Mech 1998;21:477-482. Park JH, Singh R, Pyo CR, Atluri SN. Integrity of aircraft structural elements with multi-site fatigue damage. Eng Fract Mech 1995;51(3);361-380. Pitt S, Jones R. Multiple-site and widespread fatigue damage in aging aircraft. Eng Fract Mech 1997;4(4):237-257. Proppe C, Schueller GI. Statistical processing of fatigue crack growth data. In: Kareem A (Ed), Proc. 8th ASCE EMD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, Notre Dame, USA, July 2426, 2000, pp. 307-311. Schueller GI, Stix R. A critical appraisal of methods to determine failure probabilities. Struct Saf 1987;4(4):293309. Shoji H, Shinozuka M. An evaluation of reliabiUty on fuselage rivet splice with multiple-site fatigue cracks. In: Shiraishi N, Shinozuka M, Wen YK (Eds), Structural Safety and Reliability: Proceedings of the International Conference on Structural Safety and Reliability, ICOSSAR '97, Kyoto, Japan, Nov. 24-27, 1997, Balkema, Rotterdam, 1998, pp. 1189-1194. Silva LFM, Gon9alvs JPM, Oliveira FMF, de Castro, PMST Multiple-site damage in riveted lap-joints: experimental simulation and finite element prediction. Int J Fat 2000;22:319-338. Szolwinski MP, Harish G, McVeigh PA, Farris TN. The role of fretting crack nucleation in the onset of widespread fatigue damage: analysis and experiments. In: Bigelow CA, Hughes WJ (Eds), Proceedings of the FAA-NASA Symposium on the Continued Airworthiness of Aircraft Structures, Atlanta, GA, August 28-30, 1997, pp. 585-596. Tang J, Spencer BF, Jr. Reliability Solution for the Stochastic fatigue Crack Growth Problem. Eng Fract Mech 1989;34(2)419-433. Virkler DA, Hillberry BM, Goel PK. The statistical nature of fatigue crack propagation. Trans ASME J Eng Mater Technol 1979;101:148-153. Wang GS. Analysing the onset of multiple site damage at mechanical joints. Int J Fract 2000;105:209-241. Wang L, Brust FW, Atluri SN. The elastic-plastic finite element alternating method (EPraAM) and the prediction of fracture under WFD conditions in aircraft structures. Comput Mech 1997;19:275-277.

429

Computational and physical modelling of penetration resistance M.F.Randolph* Centre for Offshore Foundation Systems, The University of Western Australia, Nedlands, WA 6907, Australia

Abstract This paper summarises theoretical results for penetration analysis in soft clays. This is an area of increasing importance in offshore engineering as developments move into deeper water, where there is greater reliance on in situ tools to obtain strength profiles. Four different penetrometers are discussed, namely cone, T-bar, ball and plate, and results are also shown from physical model tests, comparing the penetrometers among themselves, and also with the theoretical solutions. Keywords: Computational modelling; Physical modelling; Penetration resistance; Cone penetration test; Plate loading test; Undrained shear strength

1. Introduction The design of an offshore foundation or anchoring system relies on first establishing shear strength profiles, generally acknowledging differences in strengths mobilised in triaxial compression or extension, or in simple shear, and then using those profiles to estimate the capacity of the foundation. In deep water, there is increasing reliance on in situ testing methods, with the main tools being cone penetration and vane testing. An inconsistency now arises. Typically, the cone penetration test is interpreted by correlating the net cone resistance with either laboratory strengths (usually that measured in triaxial compression) or field vane strengths, using an empirical cone factor, Nc. The foundation or anchor capacity is then evaluated using computational methods (limit analysis or finite element) - effectively using a theoretical bearing capacity factor. Why should the theoretical bearing capacity factor be appropriate for the predictive stage of design, but inappropriate for the interpretive stage? 2. Cone penetration resistance Empirical values of cone factor adopted for soft sediments lie in the range 11-17 [1]. Computationally, the cone test is difficult to analyse because of the large deformations *Tel.: +61 (8) 9380-3075; Fax: +61 (8) 9380-1044; E-mail: randolph @ civil.uwa.edu.au © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

needed to reach a steady state. The strain path approach has proved the most successful [2], but even that has a degree of uncertainty owing to approximations in satisfying global equilibrium. Large strain finite element analysis is more rigorous [3], and may be used to assess the effects of (a) interface friction ratio, oc\ (b) the rigidity index, /^ = G/Su (where G is the shear modulus and Su the undrained shear strength), and (c) the in situ stress ratio. A, defined as A =

(1) 2su where a'^ and a^ are the vertical and horizontal effective stresses. Teh and Houlsby [2] have proposed the folllowing expression for the cone factor: V

2000/

+ 2 . 2 a - 1.8A

= I [1 + ln(/,)] (^1.25 + ^ ^

+ 2.2a - 1.8A

(2)

where A^^ is the normalised limit pressure for spherical cavity expansion. It is intriguing that the correlation with rigidity index shows a stronger variation than the classical spherical expansion limit pressure, first proposed as a basis for deep bearing capacity in the 1950s. Teh and Houlsby [2] noted that the above expression for A^^ leads to a plausible range of 6-18, or a more likely range of 9-17. While this is consistent with the empirical range, it is disconcertingly broad, and with little guidance on what value should be adopted a priori on any given site.

430

M.F. Randolph / First MIT Conference on Computational Fluid and Solid Mechanics

3. Alternative penetrometer shapes

50

The wide range of cone factor suggests a need for alternative forms of penetrometer that measure the bearing resistance to local flow only, avoiding the cavity expansion mode (where dependence on A and /;- are introduced). Three alternative forms of penetrometer are considered here, namely a cylindrical (T-bar [4]), spherical (or ball) and a circular flat plate. Closely bracketed plasticity solutions are available for all three shapes, with the only dependence being on the interface friction ratio. Fig. 1 summarises theoretical results, and includes two curves for the cone. It should be noted that the extreme smooth and rough results for a plate (12.42 and 13.11) are exact [5], but the intermediate curve has been estimated from upper bound calculations only. The curves for the T-bar and ball are averages of upper and lower bound solutions, with a maximum error of less than 2% (and noting that the fully rough T-bar solution of 11.94 is exact). The variation with interface roughness is very similar for cone, T-bar and ball, although the cone resistance factor has wide variability due to other factors. For the T-bar and ball, those factors will have very little effect, provided the shaft area is maintained at or below about 10% of the projected penetrometer area. The plate penetrometer appears the best choice, with a bearing factor that lies in the range 12.8 ib 0.3, with the upper limit likely for plates of practical roughness (a > 0.5). The bearing factor will increase marginally for plates of finite thickness (e.g. a thickness of 10% of the plate diameter would increase the bearing resistance by 5-7%, depending on the side roughness). There is one caveat in the above discussion, illustrated in Fig. 2, which is that physical modelling has indicated

16 Cone dr = 300, A = -0.5)

2

14

h-

1 Exact

•""lian Plate (estimated)

,^

gg 12 T-bar^

*10 Cone (If = 100,A = ().5)

0.2

1

1

0.4

0.6

0.8

Interface friction ratio, a Fig. 1. Theoretical bearing resistance.

Exact

1

Net bearing pressure, qnet (kPa) 100 150 200 250 300 J_.

0H 5 \

1

NormaDy consolklated ^ kaolin

I 10^

11

\

350

\

i^Overconsolidated ^ kaolin

CL.

& 1520 -

Cone

1

T-bar Ban

25 -

^^K^cv^

Fig. 2. Measured bearing resistance [6]. almost identical bearing resistance of all the penetrometers! It has been customary to adopt a bearing factor of A/^pbar = 10.5, and such a value gives reasonable agreement with average strengths measured in different laboratory devices [6]. Although no data for plate penetration are shown in Fig. 2, Watson [6] has shown that factors of N^piate between 9.5 (for clay soils) and 12 for a calcareous silt) fit data from other penetrometers, where a factor of 10.5 has been used. Thus, on average, the plate has shown a similar bearing resistance to the other penetrometers, but with some differences in particular materials. The similarity in measured penetration resistance of axisymmetric (ball, plate) and 'plane strain' (T-bar) penetrometers contrasts with the theoretical difference of 20%. This difference needs resolution, and may possibly be attributed to strength anisotropy.

References [1] Lunne T, Robertson PK, Powell JJM. Cone penetration testing in geotechnical pracfice. Blackie Academic and Professional, London, 1997. [2] Teh CI, Houlsby GT. An analytical study of cone penetration test in clay. Geotechnique 1991;41(l):17-34. [3] Hu Y, Randolph MF, Watson PG. Circular skirted foundafions on non-homogeneous soil. ASCE J Geot Eng 1999; 125(11):924-935. [4] Stewart DP, Randolph MF. T-bar penetration testing in soft clay. ASCE J Geot GeoEnv Eng 1994; 120(12): 2230-2235. [5] Martin CM, Randolph MF. Applications of the lower and upper bound theorems of plasticity to collapse of circular foundations 2001. Proc 10th lACMAG, Tucson, 2:1417-1428. [6] Watson PG. Performance of skirted foundations for offshore structures. PhD Thesis, The University of Western AustraHa, 1999.

431

h- versus p-version finite element analysis for J2flowtheory E. Rank*,A. Diister Technische Universitdt Munchen, Fakultdt Bauwesen, Lehrstuhl fiir Bauinformatik, Arcisstrasse 21, D-80290 Munchen, Germany

Abstract In this paper a comparison of h- and p-extensions of the Finite Element method for the J2 flow theory is presented. Based on higher-order quadrilateral element formulations, a global Newton Raphson iteration scheme combined with a local radial return algorithm is applied to find approximate solutions for the underlying physically nonlinear model problem. Curved boundaries are taken care of with the blending function method, allowing an accurate representation of geometry with only a few /^-elements. Numerical examples demonstrate that the p-version on a moderately refined mesh suppUes efficient and accurate approximations to this class of physically nonlinear problems. Keywords: p-version; Finite element method; Plasticity

1. Introduction The p-version and the /1/7-version of the finite element method have now been widely accepted as efficient, accurate and flexible tools for analysing many linear problems in computational mechanics [2,8]. It is yet still unclear if and how this behaviour can also be observed for nonlinear problems. Only few publications have addressed problems of elasto-plasticity until now [1,4,5,9], and theoretical results on the expected rate of convergence or an optimal choice of the element size and /?-degree are even more rare [6,10]. Considering e.g. the J2 flow theory of elastoplasticity, a loss of regularity has to be expected along the boundary of the plastic zone. Following the classification of Szabo et al. [8], this problem is of Class C, i.e. has a line (in 2D) or a surface (in 3D) of singular behaviour in the interior of the domain. Therefore, only algebraic rate of convergence can be expected. Yet, this asymptotic rate does not give information on the pre-asymptotic behaviour, i.e. on the accuracy of a p-extension for a finite number of degrees of freedom, and especially on the question of computational cost for a desired accuracy of quantities of engineering interest. To shed a little light upon this question, we will investigate a model problem with respect to h- and p-extensions.

* Corresponding author. Tel.: +49 (89) 2892-3047; Fax: +49 (89) 2892-5051; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

This paper, giving a short summary of a more detailed investigation to be published elsewhere [3], will briefly define the model problem of J2 flow plasticity as well as our benchmark example and show selected numerical results.

2. /7-version FEM for J2 flow theory Our ;7-version code is based on hierarchical shape-functions on quadrilaterals, using the blending function technique for exact geometrical mapping of boundary curves. For details of the formulation see Szabo et al. [8]. As Ansatz spaces we use in this investigation the 'trunk space' <Sts'^(^st) being spanned by the set of all monomials • ^'r]J /,7-0,l,...,/7;/+7=0,l,...,p •

^Y]

fOYp=l

• ^Pr],^r]P for/7 > 2 The indices /, j denote the polynomial degrees in the local directions ^,r]. We assume that strains are small € = i(Vu+(Vu)^)

(1)

and can be decomposed into an elastic and a plastic part e = e'-\-eP

(2)

Following Simo et al. [7] the differential-algebraic equations of the J2 flow theory for perfect plastic behaviour are summarised in Box 1.

432

E. Rank, A. Duster/First MIT Conference on Computational Fluid and Solid Mechanics

1. Linear isotropic elastic stress-strain relationship: 2. Elastic domain in stress space:

Ea = [a\ f{G) < 0} .p dev[a] e^ = y dev[or]

3. Flow rule: 4. Kuhn-Tucker loading/ unloading conditions:

y > 0, f{a) < 0, yfi(T) = 0 yficr) = o

5. Consistency condition:

Box 1. Classical J2 flow theory Admissible stress states are defined by the von Mises yield criterion / ( a ) = ||dev[a] 1 - ^ 1 ( 7 0

<0

(3)

where •) is the Euclidean norm of a tensor. d e v [ ] : = ( • ) - ^tr[-] 1 is the deviatoric part and t r [ ] denotes the trace operator. The differential-algebraic equations of the J2 flow theory are treated with a radial return algorithm, based on an implicit backward-Euler difference scheme [7]. Due to the nonlinear stress-strain relationship depending on the strain history, the weak formulation of equilibrium is a nonlinear

functional which is solved incrementally. In each load step a Newton Raphson scheme is applied in order to linearise the weak formulation. The numerical example to be considered is a perforated square plate under plane strain condition with monotonous loading (see Fig. 1). This problem was defined by Stein [11] as a benchmark of the German research project 'Adaptive finite element methods in applied mechanics'. A quarter of the plate is investigated, being loaded by a traction p = 100.0 MPa which is scaled with a factor 1 < A. < 4.5 in 10 load steps as plotted in Fig. 1. At the lower and right side of the plate symmetry conditions are imposed. The bulk modulus is taken to ^ = 164206.0 MPa, the shear modulus is /x = 80193.8 MPa and the yield stress is ao = 450.0 MPa. The plate is meshed with 4 (mesh A), 48 (mesh B) and 192 (mesh C) quadrilaterals as depicted in Fig. 2. Considering a p-extension for a linear elastic problem, mesh A would lead to an efficient discretisation while mesh B and C would be by far too fine.

3. Numerical examples We will restrict our investigations to the accuracy of the von Mises stress for a load factor of A = 4.5, where the 'ex-

1

2

3

4

5

6

7

load step

8

9

Fig. 1. Perforated square plate under plane strain condition with monotonous loading.

mesh A

mesh B

mesh 0

Fig. 2. Perforated square plate meshed with 4, 48 and 192 quadrilaterals.

10

E. Rank, A. Duster/First MIT Conference on Computational Fluid and Solid Mechanics

433

Fig. 3. Reference solution: mesh, von Mises stress and pi istic zone act' reference solution is obtained with a mesh consisting of 5568 quadrilaterals based on the tensor product space, Szabo et al. [8], with p = 1 resulting in a total number of 546,756 unknowns. The corresponding von Mises stress as well as the plastic zone are plotted in Fig. 3. Fig. 4 shows the distribution of the error

mesh A

-100[%]

(4)

^0

relative to the yield stress (TQ = 450.0 MPa on meshes A, B and C for polynomial degrees p = 3, 5 and 10. For comparison, in Fig. 5 the error is plotted for an /;-extension on meshes with 64, 512 and 4000 elements using p = 1.

mesh B

mesh C

Xmesh 290 100

er < 70.9%

er < 23.9%

er < 3.2%

52 dof

552 dof

2016 dof

2

^

1-8 1.6 1.4 1.2

1

p=5 tr < 32.9%

er < 7.5%

er < 1.2%

116 dof

1272 dof

4768 dof

•••-

0.8 0.6

te

0.4

0.2

•H0 relative error [%]

p = 10 er<24.6%

e/<1.8%

e^ < 0.5%

416 dof

4752 dof

18368 dof

Fig. 4. Error in von Mises stress: p-extension.

434

E. Rank, A. Duster/First MIT Conference on Computational Fluid and Solid Mechanics

Xmesh 64 elements

512 elements

4000 elements

2.90 100

•1 M-

• ^^M

1-8 1-6 1-4 1.2

B 1°" 1°' 1 . 4 1

p=l Cr < 43.3% 144 dof

Cr < 16.4% 1088 dof

er < 5.7% 8160 dof

• o^ |

o

relative error [%]

Fig. 5. Error in von Mises stress: /z-extension. It can be concluded from these results that an /i-extension with p = 1 based on a sequence of meshes being refined toward the hole — where the most critical stress concentration occurs — shows very poor accuracy compared to higher order approximations. An accurate approximate solution can yet be easily achieved when a moderately refined mesh (compared to the very coarse meshes usually used for ^-approximations to linear problems) is combined with a /7-extension. Considering e.g. mesh B with /? = 10 or mesh C with p = 5 a. relative error of nowhere above a level 1.8 % or 1.2 %, respectively, is achieved with less than 5000 degrees of freedom.

References [1] Duster A, Rank E. The p-version of the finite element method compared to an adaptive /z-version for the deformation theory of plasticity. Comput Methods Appl Mech Eng. Accepted for publication. [2] Duster A, Broker H, Rank E. The /7-version of the finite element method for three-dimensional curved thin walled structures. Int J Numer Methods Eng 2000. Submitted. [3] Duster A, Rank E. A p-version finite element approach for two- and three-dimensional problems of the Ji flow theory with non-linear isotropic hardening. Int J Numer Methods Eng 2000. Submitted.

[4] Holzer S, Yosibash Z. The /?-version of the finite element method in incremental elasto-plasdc analysis. Int J Numer Methods Eng 1996;39:1859-187. [51 Jeremic B, Xenophontos C. Application of the p-version of the finite element method to elasto-plasticity with localization of deformafion. Commun Numer Meth Eng 1999;15(12):867-876. [61 Li Y, Babuska I. A convergence analysis of a p-version finite element method for one-dimensional elastoplasticity problem with constitutive laws based on the Gauge function method. SIAM J Numer Anal 1996;33(2):809-842. [7] Simo JC, Hughes TJR. Computational Inelasticity. New York: Springer, 1998. [8] Szabo B, Babuska I. Finite Element Analysis. New York: Wiley, 1991. [9] Szabo B, Actis R, Holzer S. Solution of elastic-plastic stress analysis problems by the p-version of the finite element method. In: Babuska I, Flaherty J et al. (Eds), Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations. IMA Volumes in Mathematics and its Applications, Vol. 75, New York: Springer, 1995, pp. 395-416. [101 Wieners C. Theorie und Numerik der Prandtl-ReuB-Plastizitat. Habilitationsschrift, Universitat Heidelberg, submitted 1999. [Ill http://www.ibnm.uni-hannover.de/Forschung/Paketantrag/ Benchmarks/benchmark.html

435

Simulation of interface fatigue crack growth via a fracture process zone model K. Roe, T. Siegmund * School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-1288, USA

Abstract Fatigue crack growth along interfaces is studied via the use of a fracture process zone model. The present approach departs from the description of fatigue crack growth via the Paris equation and instead aims to describes the actual process of material separation by an appropriate constitutive equation accounting for the creation of new surface under subcritical loading conditions. The model definition, its implementation and an application to fatigue crack growth in a double cantilever beam are reported. Keywords: Fatigue crack growth; Fracture process zone; Interface fracture

1. Introduction Fracture mechanics has been widely used to study failure at interfaces under monotonic loading. In contrast, only a rather small number of significant investigations were concerned with fatigue crack growth (FCG) along interfaces. There, the description of interface FCG is based on the Paris equation [1], relating the appUed strain energy release rate range, AG = Gmax - G^min to the crack growth rate, da/dN, by:

sessment procedures, FCG resistance has to be determined experimentally for each new material combination and specimen geometry. To obtain a more fundamental view of FCG, a different concept is adopted here in which the competing actions of: (1) the material separation processes in the fracture process zone (FPZ) at and near the crack front; and (2) the deformation of material elements surrounding the fracture process zone determine the FCG.

2. Fracture process zone models ^C(AGr

(1)

with C and m as experimentally determined values. The fact that the Paris equation is empirical and provides a data correlation scheme rather than a predictive capability is especially important for interface FCG since experimentally determined AG-da/dN curves for interfaces depend on factors not of concern in homogeneous materials. FCG rates and threshold values are dependent on the mode-mixity. Crack paths observed for fatigue cracks in multi-layer structures are different from those under monotonic loading, which subsequently influences da/dN. Also, crack growth rates do not follow either of the constituents in the multi-layer structure. Thus, following the current as* Corresponding author. Tel.: +1 (765) 494-9766; Fax: +1 (765) 496-7536; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Recently, FPZ models have been used in a series of studies of crack growth under monotonic or dynamic loading situations, for an overview see [2]. Crack propagation in homogeneous materials, in composites and at interfaces was investigated. For FCG simulations the FPZ models have to be extended to account for irreversible deformation, incorporate loading-unloading conditions [3], crack surface friction [4], and effects of accumulation of damage [5]. Here, the FPZ approach of [6] is extended to include account for these effects. The governing equations used for the cohesive elements representing the fatigue FPZ (FFPZ) are embedded in a correlation between the tractions, Tn, and displacement jumps, w„. For pure mode I loading, as considered here, the correlation between Tn and Un for

436

K. Roe, T. Siegmund / First MIT Conference on Computational Fluid and Solid Mechanics

loading, and unloading/reloading are: Loading

Unloading/Reloading

Tn = ^max^ (^ — j CXp (^- — j

Tn = G^^^e —

1.4 w

CQ

K^)

if iin < 0 or w„ < max(w„) Here, o^^^ is the current cohesive strength of the FFPZ and 8 is the cohesive length, respectively. For cyclic loading, the evolution of the cohesive strength of the FFPZ is accounted for by the use of a damage factor, D:

<7max.o is the initial cohesive strength of the FPZ. The evolution of the damage variable, D, depends on the amount of the total accumulated displacement jump, w„.tot, [4]: \Un\ r ^ n - ^ f ] ^,

withaf = C-o-max, and M„.tot = / \un t\dt

u S 1 ^ 1 0.8

/

S t? 0.6 0.4 ^

0.2 n

/ /

_r _r

/

f

/

/

J^

/

/ jJ"^

_r

AG/(|)o = 0.056

/

AG/(|)o = 0.036

- //X , 0

10 20 30 40 50 60 70 80 90 100 110 A^ - Number of Cycles

(3)

o(l-^)

A

AG/o = 0.0814

1.2

Fig. 1. Crack extension in dependence of the number of applied cycles, CTf = 0.25<7niax-

_^

C < 1.0

(4)

if r„ > 0.0

where af is a fatigue limit stress and H denotes the Heaviside function. The FFPZ model is implemented into the finite element code ABAQUS [7] via the 'user defined element' capabilities of this code.

0

10 20 30 40 50 60 70 80 A^ - Number of Cycles

Fig. 2. The evolution of the FPZ traction in dependence of the number of applied cycles at jc = A«o, (Aflo + 20/), (Aao + 40/), for AG/00 = 0.056 and df = 0.25ormax.

3. Results The present fatigue fracture process zone model was applied to a study of FCG in a double cantilever beam specimen. The DCB specimen was composed of two aluminum substrates (£• = 65 GPa, y = 0.334) bonded by an adhesive. The FFPZ model represents the mechanical properties of the adhesive. The DCB specimen possesses an aspect ration L/h = 10, an initial crack length QQ = L/2 and unit specimen width. Eighty cohesive elements of length 1/8 = 16.6 were placed in front of the initial crack. The surrounding continuum elements all possess the dimensions (/ X / ) . The properties of the FFPZ model were taken to be h/8 = 266.7 and (Jma\.o/E = 1.07 x 10"^. Three values of the fatigue strength, af, were considered, i.e. o-f = 0.35, 0.25, 0.0 o-^ax , respectively. Cyclic loading under load control was simulated for Gmin = 0 0 and the ratios of the applied A G relative to the cohesive energy in the undamaged state, 0o = crmax,o^^. were taken to be A G / 0 o = 0.036, 0.056, 0.0814. Typical results for the crack growth as a function of the number of applied load cycles are depicted in Fig. 1 for the three applied load levels. The simulations predict that an initial number of applied cycles are necessary to start the growth of the fatigue crack, this initial number of cycles being dependent on the applied A G . Subsequently, crack

extension depends rather linearly on the number of applied load cycles. This predicted behavior is in good agreement with the general trends obtained in experimental investigations. Fig. 2 depicts the development of the FPZ tractions and the damage variable at three locations, x = ao, ao-\- 20/ and ao + 40/ for A G / 0 o = 0.056 and df = O.25o-n,ax. This plot demonstrates that the material separation process at the initial crack tip is different from the process at locations away from the initial crack tip due to the impact of the first load cycle. For locations remote from the initial crack tip, the maximum traction value obtained is approximately 0.6 c^max.o and represents the steady-state delamination process for locations where the initial load does not impact the separation process. For the location x = GQ -\- 20/ the traction-separation behavior is depicted in Fig. 3 for A G / 0 o = 0.056 and CTf = 0.25o-max- Duriug the early cycles of loading little or no accumulation of damage occurs and the traction separation curve follows the slope of (crjnax,oe/8). As damage is accumulated, a^ax drops and unloading/reloading occurs with decreased stiffness. The stress carrying capacity of the FPZ drops to zero if the maximum of the FFPZ displacement jump during loading reaches approximately u„ = 0.85.

K. Roe, T. Siegmund/First MIT Conference on Computational Fluid and Solid Mechanics

437

4. Conclusion

Fig. 3. The traction displacement jump correlation at x = A(2o + 20/, for AG/00 = 0.056 and a^ = 0.25ormax. 0.1

The present paper represents an attempt to develop numerical models for fatigue crack growth by extending FPZ models to include effects of irreversible deformation. Based on the concepts of damage mechanics, a dependence of the FPZ model parameters on the accumulated amount of deformation is introduced. One additional material parameter, the fatigue strength cxf, is required for the proposed model. A parametric study on the influence of this quantity on the crack growth rate is performed, demonstrating a significant sensitivity of the crack growth rates on the fatigue strength. Further extensions of the model to include the effects of crack closure, crack surface friction as well as a crack initiation criterion are under development to improve the capabilities of the present model.

da/dN =0.611 AG/o° Of = 0.25 a ^

^0.01

t

References

da/dN = 2.16AG/(t)o^'^'

da/dN = 3.35 AG/(/>o

0.001 0.01

AG/(f>o

0.1

Fig. 4. Crack growth rates in dependence of the normalized applied energy release rate range, AG/cpo, for three levels of fatigue strength. Finally, Fig. 4 summarizes the numerically obtained crack growth rates in dependence on the normalized applied A G for three levels of fatigue strength, af. The numerically obtained crack growth rates were determined by a linear fit to the curves in Fig. 1 at crack growth initiation. The numerically obtained da/dN values can be described well by the use of the Paris relation, Eq. (1). Predicted values of the exponent m in Eq. (1) are in the range of 0.9-2.0, depending on the choice of af.

[1] Paris PC, Gomez MP, Anderson WP. A rational analytic theory of fatigue. Trend Eng 1961;13:9-14. [2] Hutchinson JW, Evans AG, Mechanics of materials: topdown approaches to fracture. Acta Mater 2000;8:125-135. [3] de-Andres A, Perez JL, Ortiz M. Elastoplastic finite element analysis of three-dimensional fatigue crack growth in aluminum shafts subjected to axial loading. Int J Solids Struct 1999;36(15):2231-2258. [4] Chaboche LJ, Girard R, Levasseur P. On the interface debonding models. Int J Damage Mech; 1997;6:220-257. [5] Lemaitre J. A Course on Damage Mechanics. Berlin: Springer, 1996. [6] Needleman A. An analysis of decohesion along an imperfect interface. Int J Fract 1990;42:21-40. [7] ABAQUS — A General Purpose Finite Element Code, Pawtucket, RI: HKS.

438

Improved direct time integration method for impact analysis B.T. Rosson*, C.W. Keierleber University of Nebraska — Lincoln, Civil Engineering Department, Lincoln, NE 68516, USA

Abstract A general procedure is presented for the use of higher-order Lagrange polynomials to represent the variation of acceleration in an implicit direct time integration scheme. The procedure is especially effective for impact problems. Amplitude decay and period elongation are used to compare the accuracy of the solutions. The relative merits and stability aspects of the equations are described. Keywords: Direct time integration; Accuracy; Stability; Lagrange polynomials

1. Introduction Most commonly used implicit integration methods use as their basis for development the assumption that the acceleration varies linearly from time r, to time ti+\. If higher-order polynomials are used instead, the results can be much more accurate provided At/T is below the stability limit of approximately 0.35.

(2) yield very accurate results and can be used simply as they are without further complication, although they are not self-starting beyond the linear acceleration method (n = 1) and are only conditionally stable. Eqs. (3) through (6) were developed in a manner similar to that used in the Wilson-^ method. The velocity and displacement at time tt+e are explicitly determined from previous time steps in Eqs. (3) and (4). The acceleration at time ti+i from Eq. (6) is used to update the velocity and displacement in Eqs. (1) and (2), respectively.

2. Temporal integration using polynomials Xi+e = — Y^m^jXi-rn+j-i The velocity and displacement at time fz+i are given in Eqs. (1) and (2). They were developed using Lagrange polynomials of degree n to represent the variation of acceleration over the time step. Refer to the top of Table 1 for the constants B„ ^Pj D^ and „5y for n = 0, 1, 2, 3. •

_

•

^t"^

-"P

At

Fi+0 - cXi+0 - kXi+0

-^/+i —

Lfn

. .

(4) (5)

m

(-i)"i„

(1)

2 "+1

= Xi + AtXi + -—-

^^^^^ p = m-\-l

(3)

7=1

.,

Error r,+i = E„ (Atf Xi +1

'i+9 = y - X ! P^J^i-P+J-^

whcrc m > n + 1

Xi+e — —

Ln

^„8jXi-„

y=i

Error T/+I = E„ (At)'«+3

|„(«+3)|

* Corresponding author. Tel.: +\ (402) 472-8773; Fax -hi (402) 472-8934; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

n^inlJ^j)

Xi-n+j-

7=2

(6)

(2)

The constants have also been evaluated for n = 4 and 5, although they are not listed in Table 1. Eqs. (1) and

Z-^ v"^^' ~ ( ~ 1 )

3. Conclusions The methods for w > 1 are conditionally stable for all 0 values and are not self-starting. However, as shown in Fig. 1, the period elongations and amplitude decays for

439

B.T. Rosson, C.W. Keierleber/First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Time integration coefficients

B„

n 0 1 2 3

1 2 12 24

n)^l

n^2

1 1 -1 1

1 8 -5

n

/)«

n^l

n<52

0 1 2 3

2 6 24 360

1 2 -1 7

1 10 -36

n

Ln

0 1 2 3 4 5

1 1 2 6 24 120

n 0 1 2 3 4 5

n-^l

n-^l

nft

nP4

En

5 19

9

1/2 1/12 1/24 19/720

nh

n^A

En

38

1/3 1/24 7/360 17/1440

3 171

n^3

n^4

n^5

n^6

(2 + ^)1^2 (3/2)(3 4-^)2^2 (6/3)(4 + ^)3X2 (10/4)(5+^)4A.2

(3 + ^)2A3 (4/3)(4 + ^)3A3 (10/6)(5+^)4>.3

(4 + ^)3;^4 (5/4)(5+^)4A4

(5+^)4^5

1

-e

(1+^)

2(2 + 0)iXi 3(3 + ^)2M 4(4 + ^)3^1 5(5 + 6>)4Ai

O1X2

-O2X3 ^3^4 -^4^5 nMl

nf^2

n/X3

nM4

nM5

-1

2 -3 4 -5 6

3 -6 10 -15

4 -10 20

5 -15

-1

(a)

M/T

n 0 1 1.4 2

1.5

- - - 3 1.53 1 1.0 2 0.75 3 0.85

Fig. 1. (a) Spectral radii, (b) percentage period elongations, and (c) percentage amplitude decays.

nM6

6

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B.T. Rosson, C.W. Keierleber / First MIT Conference on Computational Fluid and Solid Mechanics

n > 1 and ^ < 1 are much less than those obtained from the commonly used implicit methods for the exact solution condition of;: = cos cot as described in Bathe and Wilson [1]. Higher-order methods may be useful for problems involving impact where significant 'overshoot' occurs in the first few steps when the Wilson-^ method {n = \, ^ = 1.4) is used with large time steps [2].

References [1] Bathe KJ, Wilson EL, Stability and accuracy analysis of direct integration methods. Earthquake Eng Struct Dyn 1973;1:283-291. [2] Goudreau GL, Taylor RL. Evaluation of numerical integrafion methods in elasto-dynamics. Comput Methods Appl Mech Eng 1972;2:69-97.

441

The /7-version FEA: high performance with and without paralleHzation Martin Riicker ^, Ernst Rank b * ^ AutoForm Engineering Deutschland GmbH, Dortmund, Germany ^ Lehrstuhl fUr Bauinformatik, Fakultdt Bauwesen, TU MUnchen, MUnchen, Germany

Abstract The p-version of the finite element method is now commonly considered to be a very accurate discretization method for linear elliptic partial differential equations, but many researchers still doubt the efficiency of this method, when compared to the classical /i-version and applied to more complex problems. In this paper the high performance capabilities of the /?-version finite element analysis are outlined (considering high performance as to obtain most accurate results in a reasonably short time of computation). It will be shown that there are some special techniques being appHcable to the /7-version, yielding a highly efficient method. This is demonstrated for a Reissner-Mindhn plate problem for both a sequential implementation and a parallel implementation on a workstation cluster. Keywords: p-Version; Finite element method; Parallel computing

1. The /;-version While in the conventional /?-version of the finite element method convergence is achieved by refining the mesh, the polynomial degree of the shape functions remains unchanged. Usually, low order approximation of degree p in the range of 1 or 2 is chosen. The p-version instead is performed on the same mesh with a locally or globally increased polynomial degree of the shape functions. In most implementations a hierarchical set of shape functions is applied, providing a simple and consistent facility of implementation in 1-, 2- or 3-dimensional analysis. Guidelines to construct suitable meshes a priori can often be given much easier for the /7-version than for the /z-version [2,3]. For linear elliptic problems it was also proven that a sequence of meshes can be constructed so that the approximation error only depends on the polynomial degree p and not on the order of singularities in the exact solution. Following Szabo and Babuska [2], our p-version implementation uses hierarchical basis functions, which can easily be implemented up to any desired polynomial degree. The ansatz space SP'P{Q!lt) appHed in this report is * Corresponding author. Tel.: +49 (89) 2892-3047; Fax: +49 (89) 2892-5051; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

the space of polynomials on Q% = [ ( - 1 . 1) x ( - 1 . 1)1 being spanned by the set of all monomials ^'r]\ i, j = 0 , 1 , . . . , p. Considering e.g. a finite element for plane elasticity problems based on this ansatz space with p = S, the size of the n xn-element matrix is n = 162. The amount of degrees of freedom corresponding to the bubble modes is 98, being about 60% of the total number of degrees of freedom on element level. As the bubble degrees of freedom are purely local to the element, they can be condensed using a modified Cholesky decomposition for the element stiffness matrices (see the following section). This results in further increase of computation time on element level but drastic decrease of solution time because the condition number of the global stiffness matrix is strongly reduced. Several authors [1,4] have investigated these observations in detail, interpreting the bubble mode condensation as a preconditioning procedure for iterative solvers.

2. Parallelization of the /7-version FEA The implemented domain decomposition is a variant of the primal subdomain implementation (PSI algorithm, see e.g. [4]) which is based on a two-level domain decomposition. On the first level, the finite element mesh

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M. Riicker, E. Rank/First MIT Conference on Computational Fluid and Solid Mechanics

Q, is decomposed into n non-overlapping subdomains {Q^k, k = l,...,n), each consisting of a set of elements being identified with non-overlapping subdomains of level two {Q^^\ j = 1 , . . . ,^). Each /^-element has nodal-, edge- and bubble-modes. A numbering scheme for all degrees of freedom is applied so that all bubble modes for all elements get the lowest numbers, whereas all nodaland edge-modes are numbered last. Note that these modes correspond to 'interface nodes' in a domain decomposition for the /z-version, whereas they do not necessarily correspond to interfaces of the level-one subdomains Qi, • Now, the equation of equilibrium Ku = f for the p-version finite element method can be written as K (1)

0

0

0

K;

0

^(2) i

= 0

0

(1)

K

(1)

'^ib

(2)

Uz,

^bb

with ^bb

'^j) and f, = jy,

=

(2)

where ^j^ _ K•bb ^bi

-

R<^')'K'0")i>(y) ^ ' ^ ^ ^ b '^b

'^bi '

^U)

K•ib

K;^'B<^\

V=^'f^'-

(3)

Aj) K// correspond to the bubble modes and K^^^ to the nodal and edge modes of Q^-^\ while K|^\ K^f correspond to the U) r(J) ^ 0 ) interaction of these mode groups. K^^, K Ki^.'andr/' are the element matrices and vectors, respectively, and B), is the connectivity matrix of element j mapping local nodal and edge mode numbers to their global numbering. The concept behind this p-version PSI is exactly the same as that of the corresponding /z-version implementation, i.e. to reduce the initial system of equations to a system comprising only the edge and nodal degrees of freedom. Applying a static condensation to the subdomain internal unknowns, Eq. (1) can be transformed into the following problem for the nodal and edge unknowns:

(K,,-x:KifKrK;f)u.=f.-EKifKrf;^'(4) 7=1

7=1

or Su^ = ib

(5)

where

(6)

S<^'=Kli'-Kl;;"'K
(7)

(8) 7=1

Matrix S is the Schur complement of K^^ in K and each subdomain matrix S*^^^ corresponds to a local static condensation operator or a local Schur complement. Note now that all element Schur complements can be formed in parallel without any communication, so that for this reduction operation s processors (s is the number of elements) could be used. Yet, usually s is greater than the number of available processors n, so elements are grouped together to subdomains of level-one and all Schur complements for elements in one of these subdomains computed sequentially by one processor. As for high p-degrees bubble modes take the major part of all degrees of freedom, the size reduction of this element condensation is very significant. Furthermore, it was proven e.g. in [1], that this Schur complement is an excellent preconditioning for the remaining equation system, being solved either by a sequential or a parallel CG-solver in our implementation [5].

3. Numerical example The accuracy and the efficiency of the sequential and the parallel p-version approach are demonstrated by the application to a complex floor plate of a multi-storey office building. The plate is supported by 45 columns with quadratic cross-sections and the walls of the three stairwells. The columns are arranged on inclined grids with a grid space of ~6.25 m. The plate is uniformly loaded by its dead weight. For the p-version finite element analysis of the plate the structure is discretized by 419 quadrilateral elements (Fig. lb). On all free and soft simple supported boundaries a thin element layer of about twice the plate thickness is generated. Furthermore, the mesh is refined towards the reentrant comers of the structures. The columns are modelled by mesh-independent elastic foundations with a constant distributed column stiffness (see [5]). For the evaluation of the accuracy an /?-version analysis is performed on meshes with different refinement levels. The column's geometry is recognized in the mesh generation process, and therefore it is possible to model the columns by elastic foundation of full elements, to get comparable results from h- and p-version. While the mesh in Fig. la represents a practical mesh, the refinement level of the 41184 element mesh shown in Fig. 2 is much higher than in common engineering practice. The values of main interest in engineering practice are the stress resultants. As an example, the bending moment M;^: is considered in the following. The relative error in the bending moment depicted in Fig. 2 represents the error

M. RUcker, E. Rank/First MIT Conference on Computational Fluid and Solid Mechanics

443

Fig. 1. Floor plate of a multi-storey office building —finiteelement meshes for /z-version and />-version.

Fig. 2. Floor plate of a multi-storey office building — percentage error in bending Moment M^ at a re-entrant corner (top) and at a column support (bottom): /z-version with 41184 MITC4-elements, 126387 dofs (left), p-version with 419 elements, /? = 8, 81968 dofs (right). in relation to the yield stress and is thus of high practical significance. The accuracy of the /?-version approximation with a moderately high polynomial degree of p = 8 (81968 dofs) is much higher than of the finest /z-version (126387 dofs). The largest errors are concentrated to the small vicinity of the singularities in the columns interior and at the re-entrant comers where the exact solution is infinite. The increasing error at the corner singularity is essentially limited to the refined elements. For the p-version the error in the interior of the columns becomes very small towards the columnplate interface, so that the accuracy at the points of interest,

the values at the columns boundary, is clearly within the practical requirements. On the other hand, in the classical /z-version analysis the maximum error appears right next to the columns boundary. The efficiency tests of the parallel p-version implementation are performed on a workstation cluster consisting of seven COMPAQ XPIOOO alpha, each with 512 MB RAM, connected via a 100 MBit network. The sequential reference computations are made on the same workstation type with 1 GB RAM. Note that the efficiency for the implemented parallel algorithm is machine independent, because using a more powerful machine leads to a speed-up in the

M. Rucker, E. Rank/First MIT Conference on Computational Fluid and Solid Mechanics

444

Table 1 Floor plate of a multi-storey office building — sequential computation times for /z-version analysis with iterative solvers from the experimental code with MITC-elements Elements

dof

Memory (MB)

t (s)

8346 9874 11747 41184

25955 30695 36613 126387

100.5 119.0 141.6 494.5

23.6 30.8 43.3 268.1

sequential part as well as in the parallel part. Due to the minimized communication and parallel overhead the difference in speed between the sequential and the algorithm running on one single processor is negligibly small. The computational times of the /z-version analyses are listed in Table 1 to be able to value the numbers from the parallel computations. A comparison with a commercial code (see [5]) shows a high efficiency of the experimental code already for the /z-version implementation. Considering the computational times of a parallel approach with a parallel eg-solver in Table 2 it can be noticed that the absolute speed-up becomes best for the highest polynomial degree. This is due to the fact that the computational effort shifts more and more from the solver to the element based part and that the positive preconditioning effect increases with the polynomial degree. The superlinear speed-up resulting in an efficiency of more than 100% for the highest polyno-

mial degrees and a small number of subdomains is due to the deteriorated sequential times as a consequence from the memory (RAM) limitation. The efficiency of the parallel implementation applied to a practical example is very close the theoretically expectations. The speed-up in an environment of a few workstations being available e.g. in larger construction offices justifies the parallelization approach. The present results are obtained in a 'pollution-free zone' (in a computational sense); therefore, it should yet also be noted that experiences from numerous efficiency tests on different examples show the importance to observe the overall communication load in the local network.

References [1] Ainsworth M. A preconditioner based on domain decomposition for hp-FE approximation on quasi-uniform meshes. SIAM J Numer Anal 1996;33(4):1358-1376. [2] Szabo B, Babuska I. Finite Element Analysis. Chichester: Wiley, 1991. [3] Szabo B. Mesh design for the p-version of thefiniteelement method. Comp Meth Appl Mech Eng 1986;55:181-197. [4] Bitzarakis S, Papadrakakis M, Charmpis D. Parallel solvers — performance assessment. Esprit Project INSIDE, Report D51a, 1997. [5] Rucker M. A parallel /7-version finite element approach for civil engineering and structural mechanics problems. Dissertation, TU Munchen, 2000.

Table 2 Floor plate of a multi-storey office building — parallel computation times p 4 5 6 7 8 9 10 11 12 13 14

dof 20868 32372 46390 62922 81968 103528 127602 154190 183292 214908 249038

Memory^ (MB)

ts

ti

t7>

tA

^5

t6

ti

(s)

(s)

(s)

(s)

(s)

(s)

(s)

36.972 51.958 74.126 105.718 149.731 212.450 293.932 397.796 528.017 688.907 885.139

14.5 24.3 40.1 66.8 107.5 172.7 292.0 421.9 666.6 1052.8 1898.8

13.6 18.5 26.7 40.2 61.0 93.1 146.6 224.3 330.5 520.3 963.5

11.3 16.2 21.4 31.1 45.1 66.5 101.8 153.3 224.5 356.0 591.1

10.3 13.4 19.1 26.0 37.6 54.3 82.2 128.9 178.8 304.0 484.3

10.6 13.2 18.4 24.5 33.9 46.9 68.1 103.8 147.4 241.2 389.3

10.6 14.3 18.7 23.7 32.0 42.7 63.6 91.1 133.0 217.7 350.2

9.2 11.5 14.6 19.6 26.7 35.8 50.3 76.5 105.7 168.2 289.7

445

Finite-element simulation of complex dynamic fracture processes in concrete G. Ruiz^'*, A. Pandolfi'', M. Ortiz'^ " ETSI Caminos C. y P., Universidad de Castilla-La Mancha, Paseo de la Universidad 4, 13071 Ciudad Real, Spain ^ Dipartimento di Ingegneria Strutturale, Politecnico di Milano, 20133 Milano, Italy ^ Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA

Abstract We describe complex fracture processes in concrete specimens subjected to dynamic loading by using cohesive theories of fracture combined with the direct simulation of fracture and fragmentation. We account explicitly for microcracking, the development of macroscopic cracks and inertia, and the effective dynamic behavior of the material is predicted as an outcome of the calculations. The model is vahdated against tests performed under two very different experimental configurations: the brazilian test and the three-point bend test with a precrack displaced from the central cross-section. The model is able to capture closely the fracture patterns and displacement fields for both tests, even those in which several microcracked zones develop simultaneously under mixed-mode loading conditions. The simulations also give accurate histories of crack extension and velocity, and predict the load histories and other far-field experimental measurements over a range of strain rates. Keywords: Concrete; Fracture; Cohesive elements; Dynamic strength; Mixed mode fracture; Size effect; Strain rate effect

1. Introduction We use the direct simulation of fracture and fragmentation in conjunction with cohesive theories of fracture, in order to describe processes of tensile and shear damage in concrete specimens subjected to dynamic loading. By explicitly simulating the fragmentation of the solid, individual cracks are tracked as they nucleate, propagate, branch and possibly link up to form fragments, being incumbent upon the mesh to provide a rich enough set of possible fracture paths. The ensuing granular flow of the comminuted material is also simulated explicitly. The coupling existing in the cracks between shear and tensile damage is partly accounted for by an effective cohesive law based on static material parameters such as the fracture energy, the tensile strength and the ratio between mode II and mode I toughness. Mode coupling also comes up explicitly as a result of accounting for contact and friction upon closure of the microcracks forming the process zone, since the different slopes of the crack planes * Corresponding author. Tel.: -f-ll (34) 926-295-398; Fax: +11 (34) 926-295-391; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

would result in normal and tangent stress components even if coupling was not taken into account in the constitutive description. In addition, the effective dynamic behavior of the material is predicted as an outcome of the calculations rather than it be buried within the cohesive law description. Indeed, the cohesive properties of the material are assumed to be rate-independent. This is so because cohesive theories endow the material with an intrinsic time scale [1], as well as they build a characteristic length into the material description. This intrinsic time scale permits the material to discriminate between slow and fast loading rates and ultimately allows for the accurate prediction of the dynamic effects. The model is validated against two very different experimental configurations: the diametral compression test on cylinders performed by means of a Hopkinson bar [2], and the three-point bend test of an asymmetrically notched beam performed in a drop weight tester device [3]. In both cases the simulations are fed with static material parameters obtained by experimentation. A brief description of the simulation results for each case is described next.

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G. Ruiz et al. /First MIT Conference on Computational Fluid and Solid Mechanics

2. Simulation of dynamic brazilian tests on concrete cylinders Dynamic brazilian tests performed using a split Hopkinson pressure bar (SHPB) have been proposed as a convenient method to determine the tensile strength of cohesive materials. The SHPB consists of an incident bar and a transmitter bar, with a short specimen placed between them, and a striker bar that impacts the incident bar to produce a longitudinal compressive pulse that is partially reflected in the border of the incident bar, and partially transmitted through the specimen. The subsequent diametral loading generates tension perpendicular to the load plane which eventually causes the specimen to split. Our numerical tests refer to experiments reported by Hughes et al. [4], Tedesco et al. [5] and Ross et al. [6]. The specimens are concrete cylinders of 50.8 mm diameter and height obtained by coring from a concrete block. The model gives detailed information on the threedimensional growing of the crack which keeps very well the available experimental information. Fig. 1 depicts a sequence of snapshots recording the evolution of the crack pattern. Symmetric lens-shaped cracks initiate from the free surfaces and grow towards the interior of the specimen as the load increases, which eventually coalesce into cracks passing through the load plane. There is also a secondary multiple-cracking that takes place in the surroundings of the supports. The full development of the central crack zone and its opening process give rise to a peak in the

transmitted load followed by some softening. The specimen then turns into two half cylinders stuck together by a severely damaged zone, and this new geometry is also able to withstand some load, which gives rise to a secondary peak. The specimen finally collapses due to the comminution of the material close to the supports, as can be seen in the last shot in Fig. 1. The effective dynamic behavior of the material is predicted as an outcome of the calculations. In particular, our simulations capture closely the experimentally observed rate-sensitivity of the dynamic strength of concrete, i.e., the nearly linear increase in dynamic strength with strain-rate. More generally, our simulations give accurate transmitted loads over a range of strain rates, which attests to the fidelity of the model where rate effects are concerned.

3. Simulation of mixed-mode tests on concrete beams Another interesting configuration to validate the model is the three-point bend beam with a precrack shifted from the central cross-section, so that symmetry is broken and a mixed-mode process zone is obliged to develop. The specimens are tested dynamically by means of a drop weight device [7], or of a modified Charpy tester [8,9]. Fig. 2 shows the resulting crack pattern in one of the numerical tests in [8,9], corresponding to a 76.2 mm depth beam. Specifically, the figure shows the intersection of the cohesive elements with the surfaces of the specimen. As

Fig. 1. Development of fracture patterns in the diametric compression test.

G. Ruiz et al /First MIT Conference on Computational Fluid and Solid Mechanics

447

Fig. 2. Crack patterns obtained in the mixed-mode TPB tests. may be seen from the figure, the dominant fracture mode for this geometry consists of the extension of the pre-notch towards the point of application of the load. This structural crack is accompanied by a certain amount of distributed microcracking and profuse branching, as expected in concrete. However, it should be carefully noted that some of the microcracks and crack branches are partially failed and should not be construed as completely formed free surfaces. Simultaneously with the growth of the main crack, a diffuse microcracking zone forms around the tensile fiber of the central cross-section. There is also evidence of distributed damage at the point of contact with the impactor. These distributed damage zones act as additional energy sinks. The tensile microcrack zone remains diffuse throughout the simulation and fails to develop into a structural crack. Thus, Fig. 2 suggests that two failure modes are in competition: the extension of the pre-notch and the nucleation of a new crack within the loading plane. For the particular geometry shown in Fig. 2, the pre-notch extension mode clearly wins out at the expense of the nucleation mode. The histories of crack extension and external load that come out from the calculations also fit very well their experimental counterparts, in spite that the constitutive equations do not account for rate-dependency at all. This can be explained, on the one hand, by the direct resolution of microinertia on the scale of the cohesive zone, which captures most of the rate-dependency of the material. On the other hand, cohesive theories discriminate between slow and fast rates of loading in the same way they do between small and big specimen sizes. By accounting for both effects, the model accurately predicts the dynamic behavior of concrete.

4. Concluding remarks We have taken a cohesive formulation of fracture as a basis for the simulation of processes of combined tension-

shear damage and mixed-mode fracture in solids subjected to dynamic loading. The cohesive model accounts for key fracture properties such as the cohesive strength and the fracture energy, and, in conjunction with inertia, introduces characteristic length and time scales into the material description. The particular model proposed by Camacho and Ortiz [1], and subsequent extensions thereof [2,3,10-12], also accounts for tension-shear coupling and permits according the shear and tensile strengths independent values bearing arbitrary ratios. The cohesive law is assumed to be rate-insensitive and, therefore, all rate effects predicted by the theory are a consequence of the interplay between inertia and fracture. In calculations, the fracture surface is confined to interelement boundaries and, consequently, the structural cracks predicted by the analysis are necessarily rough. However, in simulations of concrete this numerical roughness can be made to correspond to the physical roughness by the simple device of choosing the element size to be comparable to the aggregate size. This choice of element size also serves to resolve the cohesive zone size, which in concrete is a small multiple of the aggregate size. The resulting simulations thus have a multiscale character, in as much as the discretization is charged with resolving both a micromechanical scale — the cohesive length scale — and the structural dimension.

Acknowledgements Gonzalo Ruiz gratefully acknowledges the financial support for his stay at the California Institute of Technology provided by the Universidad de Castilla-La Mancha, Spain. We are indebted to Santiago Lombeyda, Research Scientist at the CACR, Caltech, for his artistic interpretation of our simulation results.

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G. Ruiz et al. /First MIT Conference on Computational Fluid and Solid Mechanics

References [1] Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. Int J Solids Struct 1996;33(2022):2899-2938. [2] Ruiz G, Ortiz M, Pandolfi A. Three-dimensional finite-element simulation of the dynamic brazilian tests on concrete cylinders. Int J Num Methods Eng 2000;48:963-994. [3] Ruiz G, Pandolfi A, Ortiz M. Three-dimensional cohesive modeling of dynamic mixed-mode fracture. Int J Num Methods Eng In press. [4] Hughes ML, Tedesco JW, Ross A. Numerical analysis of high strain rate splitting-tensile tests. Comput Struct 1993;47:653-671. [5] Tedesco JW, Ross CA, Kuennen ST. Experimental and numerical-analysis of high-strain rate splitting tensile tests. ACIMaterJ 1993,90(2): 162-169. [6] Ross A, Tedesco JW, Kuennen ST. Effects of strain rate on concrete strength. ACI Mater J 1995;92:37-47

[7] Guo ZK, Kobayashi AS, Hawkins NM. Dynamic mixed mode fracture of concrete. Int J Solids Struct 1995;32(17/18):2591-2607. [8] John R. Mixed Mode Fracture of Concrete Subjected to Impact Loading. Evanston: PhD thesis. Northwestern University, 1988. [9] John R, Shah SR Mixed-mode fracture of concrete subjected to impact loading. J Struct Eng 1990;116(3):585602. [10] Ortiz M, Pandolfi A. Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer. Methods Eng 1999;44(9): 1267-1282. [11] Pandolfi A, Ortiz M. Solid modeling aspects of threedimensional fragmentation. Eng Comput 1998; 14(4):287308. [12] Pandolfi A, Krysl P, Ortiz M. Finite element simulation of ring expansion and fragmentation. Int J Fract 1999;95:279297.

449

General traction BE formulation and implementation for 2-D anisotropic media A. Saez*, J. Dominguez Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092-Sevilla, Spain

Abstract A general mixed boundary element approach based on the displacement and the traction integral equations for anisotropic media is presented. Integration of the hypersingular kernels along general curved quadratic line elements is carried out by transformation of the integrals into regular ones, which are numerically evaluated, plus simple singular integrals which can be integrated analytically. The generality of the method allows for the use of curved elements and quarter-point elements to represent Fracture Mechanics problems. Stress Intensity Factors can be accurately computed from the crack opening displacement at quarter point nodes. Keywords: Boundary element method; Hypersingular; Anisotropic elasticity; Fracture mechanics; Stress intensity factor

1. Introduction Boundary Element (BE) formulations which make use of the traction boundary integral equation have been studied rather extensively in the last 15 years. These formulations have appeared under the name of hypersingular integral equations [1,2], strongly singular integral equations [3], mixed [2] or dual boundary elements [4]. In all these cases, the hypersingular kernels appearing in the traction integral representation are regularized in order to evaluate the strongly singular integrals. In recent works, Sollero [5] and Sollero and Aliabadi [6] have presented the Dual Boundary Element Method for anisotropic media. They used a mixed formulation including the displacement integral equation and the traction integral equation, for the analysis of crack problems. The displacement integral equation formulation for anisotropic media had been presented by Cruse and Swedlow [7] in 1971. It can be followed in the book by Balas et al. [8]. Sollero and Aliabadi [6] started from this classical formulation to obtain the hypersingular traction representation in order to complete a Dual BE approach which follows the same scheme as that presented for isotropic elasticity by Portela et al. [4]. In their method, Sollero and Aliabadi [6] carry out a regularization of the hypersingular kernels which is limited * Corresponding author. Tel.: -^34 (95) 455-6606; Fax: +34 (95) 448-7295; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

to straight line discontinuous elements. They compute crack Stress Intensity Factors (SIF) using the J-integral. In the present work, a general regularization and integration approach of the hypersingular traction integral equation for anisotropic 2-D domains is presented. It follows the formulation presented by Saez et al. [2] for isotropic media. The actual expressions for regularization and analytical integration of the hypersingular kernels for general curved quadratic elements are shown. The elements may have any general quadratic geometry and therefore quarter point elements can be used without difficulty for crack problems. The SIF can be computed from the crack opening displacements at nodes of the quarter point elements.

2. Basic equations The mixed formulation for the BE solution of crack problems is based on both the displacement and the traction integral representation. In the case of zero body forces, the displacement integral representation for a point with coordinates §i, ?2 can be written as

r

= fu;j(zt,zi)pj(zl)dr(zt)

(1)

450

A. Sdez, J. Dominguez/First MIT Conference on Computational Fluid and Solid Mechanics

where i, j,k = 1,2; p* and u*j are the fundamental solution tractions and displacements, respectively; Cij(z^) = 8ij when § is an internal point and C/y(z^) = ^8ij for a smooth boundary point; and z^ and z"" are points on the complex plane defined by Zl = ^l -\-flk^2,

Zl = Xi -\-IXkX2,

where /x^ are the roots of the characteristic equation of the anisotropic material. The fundamental solution displacements and tractions are u*j(zl, zi) = 2Re{P,-i An ln(zj - z\) + PpA,^ Mz^ - zf)} (3)

-n2)-

Ai2

Z2 -

+ RkiQjiiiim,

Rjk = fik^ji

(4) Z\

where

- n2)—^^

(^2 - 4 ) ' J

(11)

(12)

+ ^ji

3. Evaluation of the singular integrals

p*0,cir

/.

(13)

where 0^ are boundary element shape functions, contain basic singular integral as (5) «12MA: H

-J

«26

Ilk Qjk = i^k^ji -

(6)

s72

\

1 - 1 1 - 1 -Ml

M2

-M2

^n

Pn

-Pii

Pi2

-Pi2

Aj2

-P21

P7'22

)

^'\

—&j\/2iii

0 0

The traction integral equation for a point ^ with normal N is given by Ci,{^)Pk{z^) + Nhjsl,,(z',z«)«,(z')dr(z^)

= N,jd*.,(z', z«)/7,(z^)dr(r')

1

-«2) ^k

drk djci

dr

(7)

{llkUx

0,cir

(14)

^k

which can be transformed into a regular integral plus a singular integral with known analytical solution. Calling rk='zl-zl = {xx - ?i) -\- fik(x2 - §2), it follows drk

'A,I

Ml

ypii

Zi)

Integrals of the kernels w* are weakly singular and can be integrated without difficulty. Integrals of the type

z\ - z\

+ Qjiil^inx

(^1

(2)

k=\,2

pUzt^zl) = 2Rc

where atj are the compliance coefficients and C/y are the elastic constants of the anisotropic media and

djci d r

drk dx2

djc2 d r

2 -r ^-/c i

where /ii, ^2 are the components of the external unit normal to the boundary (Fig. 1). Taking into account Eq. (15), Eq. (14) can be rewritten as

//

=

(16)

/ —(pqdrk J rk

which can be decomposed into the sum of a regular integral and a singular integral with known analytic solution (8)

h = J/ ' -rk( 0 ,^- l ) d r , - h J\ rk-drk Te

where s*^^ and £(*^ are obtained by differentiation of p* and M* , respectively, with the following expressions X2A

= -2Re(ene.i^ + a2Q.2-^ I

^1 -A

4i(z"'Z^)

C i i C12 C16

42(Z"'Z^)

C12 C22 C26

5*2(Z^Z^)

y C i 6 C26 ^ 6 6 ^

(9)

z\- z\

Pi2.2

(15)

(10)

x^

(dx2/dr,-dx/dr)

Fig. 1. Normal vector at boundary point.

(17)

A. Sdez, J. Dominguez/First MIT Conference on Computational Fluid and Solid Mechanics The integration of the d^jj^Nj = J*^ kernels can be done in a similar way as for the /?* kernels since they contain singularities of the same type when x - ^ ^ (=^ n^ -> Nj) (fikNi

A^2)-0,dr rk

-JH: + / -(0,-l)dr,+ I I. l^kNi - ^ 2 -

-^

(pa dr

re

'

rk

•drt

451

eral and allows for the use of curved or straight elements along the crack or any external boundary. In particular, the generality of the integration method allows for the use of straight line quarter-point quadratic elements and collocation points located extremely close to the crack tip (as in [2]). By doing so the ^ behavior of the Crack Opening Displacement (COD) is reproduced and the SIF can be evaluated from the COD at points where this behavior is highly dominant.

(18) Acknowledgements

The integration of the 5*^ kernels leads to hypersingular integrals that can be again decomposed into the sum of a regular integral plus singular integrals with known analytic solution by using Eq. (15) and the series expansion of the shape function 0^

The authors wish to acknowledge the financial support provided by the Comision Interministerial de Ciencia y Tecnologia of Spain under the Research Projects PB961380 and PB96-1322-C03-01.

(pqdrk Te

References r(0^ -(pqo - 0 ^ 0

•n)dn

+ 0^0 / -^ drk + 0^0 / - drk

(19)

The above approach permits the simple integration of all the kernels appearing in both displacement and traction integral equations for any general curved quadratic element. After discretization using a similar approach as Saez et al. [2] for isotropic media, one can easily obtain SIF values for general 2-D crack problems in anisotropic media.

4. Conclusions In this paper the mixed BE formulation using the displacement and the traction integral equations for 2-D anisotropic media is used and a general approach for the evaluation of the strongly singular and hypersingular integrals for quadratic curved boundary elements is presented. The integration approach is based on a regularization process and the analytic integration of the remaining basic singular integrals. The procedure is completely gen-

[1] Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. J Appl Mech 1992;59:604-614. [2] Saez A, Gallego R, Dominguez J. Hypersingular quarterpoint boundary elements for crack problems. Int J Numer Methods Eng 1995;38:1681-1701. [3] Rudolphi TJ, Krishnasamy G, Schmerr LW, Rizzo FJ. On the use of strongly singular integral equations for crack problems. In: Proc. Boundary Elements X. Southampton: Computational Mechanics Publications, 1988. [4] Portela A, Aliabadi MH, Rooke DR The dual boundary element method: effective implementation for crack problems. Int J Numer Methods Eng 1992;33:1269-1287. [5] SoUero P. Fracture mechanics analysis of anisotropic laminates by the boundary element method. Ph.D. Thesis, WIT, University of Portsmouth (UK), 1994. [6] Sollero P, AHabadi MH. Anisotropic analysis of cracks in composite laminates using the dual boundary element method. Compos Struct 1995;31:229-233. [7] Cruse TA, Swedlow JL. Interactive program for analysis and design problems in advanced composites technology. Carnegie-Mellon University (USA), Report AFML-TR-71268, 1971. [8] Balas J, Sladek J, Sladek V. Stress Analysis by Boundary Element Methods. Amsterdam: Elsevier, 1989.

452

Boundary and internal layers in thin elastic shells J. Sanchez-Hubert * Universite de Caen Basse-Normandie, boulevard Marechal Juin, 14032 Caen, France

Abstract We consider the boundary layer phenomena which appear in thin shell theory as the relative thickness s tends to zero. We deal with developable middle surface. Boundary layers along and across the characteristics have very different structures. It also appears that internal layers are associated with propagation of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small 6 have a very large intensity. Layers along the characteristics have a special structure involving subspaces, the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive meshes in these problems. Keywords: Thin elastic shell; Boundary and internal layers; Anisotropic mesh

This paper is devoted to the boundary and internal layers in thin elastic shells of thickness 2s. Let a(u,v) and £^b{u, v) be the two bilinear forms associated with the deformations of the intrinsic metrics and the variations of curvature (membrane and flexion form, respectively). The second one involves a factor £^, accounting for the small rigidity of a thin body to flexion. This fact entails very specific asymptotic properties for small e [1-3]. The corresponding system is of the form A -\- s^B, where B is elliptic, but A is of the same type as the points of the middle surface S. The order of derivation in B is higher than in A so that as s tends to zero there are a singular perturbation phenomenon. We shall focus on the case when Ais uniformly parabolic which corresponds to developable surfaces. The limit process ^ \ 0 is very singular, as it goes from a higher order elliptic system to a parabolic system. For £ > 0, the energy space V is chosen so that a + e^b be continuous and coercive on it, whereas the limit problem involves a new energy space V^ such that the form a is continuous and coercive on it. In fact, V^ is the completion of V with the norm ^a{- , •) . Obviously, the space Va contains functions which are less smooth than the functions of V. Consequently, the solutions u^ belong to V, but the limit as s tends to zero is a less smooth function. As a consequence, u^ for small s exhibits boundary layers. In fact, the most important reason for the * E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

presence of boundary layers is the following. As V C Va, the dual spaces verify Vj C V so that the admissible forces which are in V for £ > 0 may not be admissible for the limit problem. The corresponding solution of the limit problem is out of Va and exhibits distributional singularities so that u^ involves boundary layers of large intensity [4]. It should be mentioned that this situation is very common in shell theory where the space V^, which depends on the geometry, is often very large and consequently Vj is a very small space. For instance, in ruled surfaces with a free boundary along a generator, any loading not vanishing on that generator is out of Vj [5]. It should be emphasized that the non-smoothness of solutions of the limit problem has important consequences on the finite element computations of u^ for small s. It is not hard to prove [6] that when / ^ Vj the convergence of the finite element approximations ul to u^ cannot be uniform with respect to ^ G (0, ^o) with values in Va (and then also in any 'smaller' space!). In other words, the smaller s is the smaller h must be chosen in order to get a good approximation [7,8]. The scaling for the layers, either along or across the characteristics is obtained by a method based on asymptotic trends of exponential solutions. The specific equations and boundary conditions are obtained by formal asymptotic expansions directly from the variational formulation. For a free characteristic boundary, two cases appear according to the loading vanishes or not on the boundary. The intensity

J. Sanchez-Hubert / First MIT Conference on Computational Fluid and Solid Mechanics of the layer is then much larger than for a clamped characteristic boundary because f ^ V^. Internal layers along the characteristics are presented, which exhibit the phenomenon of propagation of the singularities. The structure of the characteristic layers is special, involving Lagrange multipliers. Moreover as a consequence of the structure of the layer we consider the numerical approximation using anisotropic meshes (i.e. with triangles elongated in the tangential direction) for the finite element approximation. We recall that error estimates for anisotropic meshes were obtained in [9] for problems with a singularity along an edge and in [10] for convection-diffusion problems depending on a parameter e with a limit solution exhibiting a discontinuity along a characteristic curve. We may compare [9] with [11] which used an isotropic adapted mesh to handle the same problem. Error estimates for anisotropic meshes in the model problem are considered. More precisely, in the layer region, we take advantage of the structure of M^ given by the formal asymptotic expansion procedure and we replace M^ by the leading term of the expansions in order to handle a simple, but essentially correct, description of u^. Knowing the description in the internal variables, we use classical error estimates for isotropic meshes (i.e. satisfying the classical non-flattening condition) [12] in these variables. Coming back to the outer variables, we get the corresponding estimates for an anisotropic mesh the dimensions of the triangles being H and 8(£)H (8 thickness of the layer) in the tangential and normal directions, respectively. It should be noticed that these estimates may also be obtained directly (and we checked them in certain cases) from error estimates for anisotropic interpolation theory [10], but our method shows more explicitly the influence of the asymptotic structure. The problem of the error estimates for the Galerkin approximation is studied. The advantage of an anisotropic mesh follows from the comparison with classical estimates for an isotropic (in outer variables) mesh: for a given error the number of triangles of the anisotropic mesh is 8 = s^^^ times the corresponding number for an isotropic mesh. Moreover, the local structure of the layer which involves a constraint evokes the possibility of improving the estimates by using special locking-free finite elements. Numerical experiments for the model problem may be found in [13]. These experiments include several cases

453

of loading and exhibit the corresponding patterns of layers (boundary and internal layers). Both isotropic and anisotropic meshes are used. The advantage of a refined anisotropic mesh appears clearly in certain cases. References [1] Chapelle D, Bathe KJ. Fundamental considerations for the finite element analysis of shell structures. Comput Struct 1998;66:18-36. [2] Goldenveizer AL, Theory of Elastic Thin Shells. Pergamon, New York, 1962. [3] Sanchez-Hubert J, Sanchez Palencia E, Pathological phenomena in computation of thin elastic shells. Trans Can Soc Mech Eng 1998;22:435-446. [4] Leguillon D, Sanchez-Hubert J, Sanchez Palencia E, Model problem of singular perturbation without limit in the space offiniteenergy and its computation. CR Acad Sci Paris Ser lib 1999;327:485-492. [5] Sanchez-Hubert J, Sanchez Palencia E, Coques Elastiques Minces. Proprietes Asymptotiques. Masson, Paris, 1997. [6] Gerard P, Sanchez Palencia E, Sensitivity phenomena for certain thin elastic shells with edges. Math Methods Appl Sci 2000;23:379-399. [7] Karamian P, Nouveaux resultats numeriques concernant les coques minces hyperboHques inhibees: cas du paraboloide hyperbolique, CR. Acad. Sci. Ser. lib 1998;326:755-760. [8] Karamian P, Coques elastiques minces hyperboHques inhibees: calcul du probleme limite par elements finis et non reflexion des singularites. These de I'Universite de Caen, 1999. [9] Apel T, Nicaise S, Elliptic problems in domains with edges : anisotropic regularity and anisotropic finite element meshes. In: J. Cea, D. Chesnais, G Geymonat, J.L. Lions (Eds.), Partial Differential Equations and Functional Analysis, in memory of P. Grisvard. Birkhauser, Boston, 1996, pp. 207-220. [10] Apel T, Lube G, Anisotropic mesh refinement in stabilized Galerkin methods. Num Math 1996;74:261-282. [11] Medina J, Picasso M, Rappaz J, Error estimates and adaptive finite elements for non-linear diffusion-convection problems. Math Methods Appl Sci 1996;6(5):689-712. [12] Ciarlet PG, Raviart PA, General Lagrange and Hermite interpolation in R" with applications tofiniteelement method. Arch Rational Anal 1972;46:177-199. [13] Karamian P, Sanchez-Hubert J, Sanchez Palencia E. A model problem for boundary layers of thin elastic shells. Model Math Anal Num 2000;34:1-30.

454

General properties of thin shell solutions, propagation of singularities and their numerical incidence E. Sanchez Palencia * Laboratoire de Modelisation en Mecanique, Universite Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France

Abstract Solutions of very thin shells enjoy propagation of singularities associated with the geometrical properties of the surface. These peculiarities imply local properties of numerical locking and usefulness of finite elements elongated along the asymptotic curves of the surface. Keywords: Thin shells; Local locking; Anisotropic finite elements

1. On the singularities We consider general properties of the membrane system (i.e. without flexion or shear terms) of tin shells. It is known that such a system describes the asymptotic behavior of shells which are geometrically rigid in very general situations, for instance, starting from three-dimensional elasticity, Kirchhoff-Love or Mindlin models (see for instance [1-3]). The variational formulation of the membrane problem is associated with the bilinear form of membrane energy and the corresponding finite energy space is a 'very large space', the properties of which are highly dependent of the geometric properties of the middle surface and on the boundary conditions. As a consequence, the dual space, which is the natural space for the loadings f, is 'very small'. The mathematical proof of the above mentioned convergence towards the membrane problem only holds for loadings such that the variational formulation of the membrane problem makes sense, that is, for f in the 'small space'. Nevertheless, there is numerical evidence that such convergence also holds in more general cases, namely for hyperbolic and parabolic surfaces with loadings in the dual space of the Kirchhoff-Love formulation. This includes, for instance, distributional normal loadings of the form 8{L) or 5'(L) (i.e. Dirac or its first derivative on a curve L of the surface). The membrane system for the unknowns u\,U2,ui (the components of the displacement vector) and the data ^ E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

f \ f^^ f^ (the components of the loading) has total order 4, with the indices 5i = 1, 52 = 1, 53 = 0 (i.e. the maximal order of the jih unknown in the /:th equation is Sj -\- Sk) ant the characteristics of the system are the asymptotic curves of the surface counted twice, so that in hyperbolic surfaces there are two families of double characteristics and in parabolic surfaces there is one family of characteristics with multiplicity 4. Regularity theory in the directions transversal to the characteristics enjoys properties analogous to those of elliptic systems (as the problem is microlocally elliptic in that directions). As the third index is 53 = 0, the resolvent does not improve regularity when passing from the data / ^ to the unknown M3. If /3 is piecewise smooth and has a step along a non-characteristic curve, W3 will also have a step along the curve; of course analogous properties hold in the case of Dirac's loadings. But much more singular solutions appear when the singularities of the loading are located along a characteristics. The corresponding study involves propagation of singularities. It appears, for instance, that, when f^ involves a step along a characteristic curve L, U3 involves a singularity in 8'{L) or 8"'{L) in the hyperboUc or parabolic cases, respectively. Moreover, these singularities have not a local character as in the microlocally elliptic case, but enjoy propagation properties [4]. The study of these propagation phenomena may be performed by methods analogous to those of [5] for discontinuous solutions of hyperbolic first order systems. The coefficients of intensity of such singularities satisfy transport differential systems along the

E. Sanchez Palencia / First MIT Conference on Computational Fluid and Solid Mechanics characteristic, of order 2 or 4 in the hyperboHc or parabolic cases, respectively. It is a very remarkable fact that step singularities along a characteristic (and their consequences!) often appear in cases where they are somewhat 'masked by other apparently more important singularities'. Let us consider, for instance, a parabolic surface in a parameterization {y^,y'^) such that the characteristics are the curves y^ = const, (this means that we have a developable surface with the generators y^ = const) with the loading f^ z= 8(y^)Y(y^) where Y denotes the Heaviside step function. For y^ > 0 we have a loading with a 8 singularity transversal to the characteristics, which implies an analogous singularity for u^. But, in addition, we have a singularity in Y along the characteristic y^ = 0; its factor is 8{y^), which is not a smooth function, but this fact is within the framework of propagation of singularities; as a result, there is a singularity in 8"'(y^) along the characteristic y^ = 0 issued from the 'extremity of the loading'. The order 4 differential system in y^ describing the propagation of that singularity involves the distribution 8(y^) in its right hand side. In the previous context of propagation of singularities along the characteristics, we have a differential system of differential equations along the characteristic, which exhibits qualitative properties of the propagation phenomena. Nevertheless, the complete integration of such a system involves the boundary conditions at the points where the characteristic intersects the boundary of the surface. Consequently, the determination of the constants of integration of the system implies a study of the reflections of the singularities at the boundaries. It appears that, for both fixed and free boundaries, the boundary conditions are not in the classical framework of reflection theory, and singularities do not reflect in a classical way. Instead of this, there are 'pseudo-reflections' [6]: the incident and reflected singularities are not of the same nature, there is a lowing of the order of the singularity; for instance an incident singularity in 8 implies a reflected one only in Y. As indicated above, the previous considerations are concerned with the membrane problem. When taking into

455

account the flexion terms, the singularities along the characteristics become thin layers of thickness of the order s^^^ or e^/"^ in the hyperbolic and parabolic cases, respectively, where s denotes the relative thickness of the shell. A study of that layers, using a formal asymptotic expansion procedure is in progress, and more or less done for a simplified model problem [7]. An analysis of the corresponding finite element approximation [8] shows two facts. First, the interest of using finite elements elongated in the direction of the layers [9] and second, the presence of local locking phenomena associated with the local structure of the layers, which involve constraints and Lagrange multipliers.

References [1] Sanchez Hubert J, Sanchez Palencia E. Coques elastiques minces, proprietes asymptotiques. Masson 1997. [2] Ciarlet PG. Mathematical Elasticity III Theory of Shells. North Holland, to appear. [3] Chapelle D, Bathe KG. The mathematical shell model underlying general shell elements. Int J Numer Methods Eng 2000;48:289-313. [4] Egorov YuV. Microlocal analysis. In: Encyclopaedia of Mathematical Sciences 1992;33:76-80. Springer. [5] Egorov YuV, Shubin MA. Linear partial differential equations, foundations of the classical theory. In: Encyclopaedia of Mathematical Sciences 1992:30:153-157. Springer. [6] Karamian Ph. Reflexion des singularites dans les coques hyperboliques inhibees. C R Acad Sci Paris II B 1998;326:609-614. [7] Karamian Ph., Sanchez Hubert J., Sanchez Palencia E. A model problem for boundary layers in thin elastic shells. Math Modell Numer Anal 2000;34:1-30. [8] Sanchez Hubert J, Sanchez Palencia E. Singular perturbations with non-smooth limit and finite element approximation of layers for model problems of shells. In: Ali Mehmeti F, von Below J, Nicaise S (Eds). Partial Differential Equations on Microstructures. Dekker, in press. [9] Aple T. Anisotropic finite elements: local estimates and applications. Teubner 1999.

456

Reliability analysis of structures against buckling according to fuzzy number theory M. Savoia * DISTART— Structural Engineering, Faculty of Engineering, University of Bologna, Bologna, V. le Risorginento 2, 40136 Bologna, Italy

Abstract Fuzzy number theory can be very effective to represent non deterministic quantities for which an uncertain or imprecise body of information only is available. In fact, a fuzzy number can be used to represent a possibility measure and, consequently, a wide class of probability measures. Fuzzy number theory is used here to perform a stability analysis against buckling where a fuzzy number defines the structural imperfection. The corresponding fuzzy number of the maximum load is obtained. Reliability of the structure is then briefly discussed. Keywords: Reliability; Fuzzy sets; Uncertainties; Buckling design

1. Introduction For very slender structures, safety against buckling may represent the fundamental design requirement, and structural or loading imperfections may significantly reduce reliability [1]. Due to the random nature of imperfections and the strongly non linear behavior of slender structures (especially for imperfection-sensitive schemes), deterministic or 2nd-level probabilistic analyses are often inadequate to estimate safe values of buckling limit loads, so that full probabilistic analyses or Monte Carlo algorithms are usually recommended. Moreover, the evaluation of small (from 10~^ to 10~^) fractiles of limit load requires the definition of very precise random distributions for imperfections. Probability theory has been considered for many years the only way to treat uncertain variables. Nevertheless, probability can be consistently used only when variables are random in nature, i.e., they are disperse but precise information on their fluctuation are available so that their pdf's can be defined. There are two cases at least where variables cannot be considered as random variables: when the available body of information is small and when their definitions come from subjective judgement. The first case

* Tel: -f-39 (51) 209-3254; Fax: -h39 (51) 209-3236; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

typically arises in structural design, where statistical information on loads and resistances may be even absent and normative provisions only are available. It has been recently shown that problems involving uncertain or subjective variables can be treated in the context of theories of possibility and fuzzy sets [2]. Possibility of an event is a weaker information than probability: the confidence on the occurrence of an event can be estimated when its possibility and the possibility of the contrary event are known. Differently from probability, possibility theory is also able to point out those situations where the body of information is so scarce that no useful estimates of the occurrence of the event can be obtained. In the present paper, structural reliability will be discussed in the framework of possibility theory [2,3]. Structural imperfections will be modelled as fuzzy numbers. In fact, when the body of information is insufficient to define the pdf of a random variable, a fuzzy number can be adopted to represent the equivalence class of probability distributions compatible with the few available data. Moreover, fuzzy number theory gives conservative bounds, with respect to probability, for characteristic values corresponding to prescribed occurrence expectations [3], requiring much simple operations than probability, especially when several variables are involved. Since fuzzy set theory allows for the use of subjective variables also, human errors can be also considered in the analysis.

M. Savoia/First MIT Conference on Computational Fluid and Solid Mechanics

457

A reliability analysis against buckling will be finally performed, using fuzzy numbers for structural imperfections and evaluating the corresponding maximum load possibility distribution. The imperfection-sensitivity of structures will be briefly discussed in terms of increase of the fuzziness of the number representing the maximum load.

where F%x) = P*((-oo,x]), F^(x) = P,((-oo,x]) Finally, a possibility distribution 7t(co) can be represented by means of a fuzzy number Q, whose membership function is IIQ^CO) = nico).

2. Fuzzy numbers to represent uncertain quantities

3. Operations with fuzzy numbers

When uncertain information only are available, probability of single elementary events (and, consequently, the pdf) cannot be defined in a unique way. Consider the case the probabiUty corresponding to subsets Et of the sure event Q (called the nested focal elements, with El c E2 C ... C EN C Q) can be assigned, but the probability distributions inside the subsets are unknown (see Fig. 1). Then, lower and upper bounds can be defined for the probability associated with a general event A, as depicted in Fig. 1. An equivalence class of probability measures P admissible for the event A compatible with available data can then be defined. It can be demonstrated that lower and upper bounds for probability are necessity and possibility measures, respectively [2]. The possibility distribution 7t(cjo) (see Fig. 1) can be used to represent the equivalence class of probability measures, since the following relations hold:

Consider two uncorrelated variables Xi,X2 which are represented through fuzzy numbers Xi selecting two classes Pi(Xi) (i = 1, 2), of probability measures compatible with the available data. Moreover, consider the output variable 7, expUcit function of the input variables, Y = Y(Xi, X2). A whole class P(Y) should be obtained, if all the probability measures obtained starting from probabihties Pi(Xi) are collected. It has been shown in Ferrari and Savoia [3] that, using extended fuzzy operations to obtain the fuzzy number Y, conservative bounds for characteristic values corresponding to prescribed occurrence expectations (i.e. small and large fractiles) may be obtained. The extension principle:

P^ (A) = 1 - sup TT (co) = inf [1 - TT (co)], coeA

P* (A) = sup TV (co)

^eA

(1)

coeA

For instance, for a problem where events are function of a single coordinate x, by setting A = (-00, x] lower and upper bounds for the CDF can be obtained from Eq. (1):

F,{x) < F(x) < F*W,

(2)

sup min(/xxi(^i), Mx2(^2))

l^Xi^X2(z) =

(3)

y=xi XX2

allows for the extension of a general operation between real numbers y = xi x ^2 to the fuzzy counterpart. Several techniques have been proposed to compute fuzzy numbers through extension principle. In general, if fuzzy numbers are defined by their a-cuts (intervals with membership greater than or equal to or), the usual methods of interval analysis [4] can be used. Moreover, if j = ;ti x ^2 is a monotonic function, the vertex method can be used [5], using the extreme values of of-cuts of input variables. Equivalence class of Probability Measures: P={P/\/A,PXA)
Lower and Upper Bounds:

P.(A)= ^ m , ,

P\A)=

EfQA

J^m, EinA:A0

Probability Masses: mi=Pi-Pi-i

Possibility Distribution: N

^4 CO 7i;(a))=5^m, = l - / ^ _ i

if coe £",,(0^ £",_!

Fig. 1. Focal elements, lower and upper bounds for probability, possibility distribution.

458

M. Savoia/First MIT Conference on Computational Fluid and Solid Mechanics

(b)

(a)

(c)

Fig. 2. (a) The imperfect structure; (b) moment-rotation relations for the elasto-plastic spring; (c) maximum load-imperfection relations (solid line: Spring I; dashed line: Spring II). A^ = (pe/L.

4. An example: reliability of structures against buckling In design of slender structural members subject to compressive loads, reliability analysis against buckling must be performed. In fact, structural and/or loading imperfections cannot be assigned in a deterministic way. Nevertheless, in civil engineering area, usually not statistical data but at most normative provisions are available during design i.u

-

a

stages, so that probabilistic methods usually require the introduction of arbitrary assumptions. In the present study, a simple structure is considered to describe the method of analysis (see Fig. 2a), where the rotational spring has the elasto-plastic behaviors described in Fig. 2b. The curves giving the maximum load as a function of imperfection value can be explicitly obtained and are reported in Fig. 2c. For more complex structures. 1.0 n

v\

"i 1 1

1 l\ (2)

0.5-

1 0.0-

-A1 ' 1 ' ^

\

\

' r ' r ' 1'1

I 0.0

'

0.2

I

•

0.4

I

'

0.6

I

'

0.8

I 1.0

(b)

(a) 1.0-

1.0 n

0.5 H

P/Pe

0.0 0.0

0.2

0.4

0.6 (C)

0.8

1.0

I

0.0

0.2

'

I

0.4

'

I

0.6

'

I

0.8

'

1.0

(d) Fig. 3. (a) Fuzzy number giving structural imperfections: Case (1): e„/Ae = 0, Cy/l^e = 1; Case (2): e^/Ae = 2, e^/Ae = 1; Case (3): Cn/Ae = 1, Cy/Ae = 3. (b-d) Fuzzy numbers of maximum load for imperfection cases (1) to (3), respectively (solid line: Spring I; dashed line: Spring II).

M. Savoia/First MIT Conference on Computational Fluid and Solid Mechanics asymptotic perturbation methods can be used to obtain the dependence of the Hmit load on structural imperfections in algebraic form (see for instance Casciaro et al. [6]). The structural imperfection e is the sum of nominal (i.e. deterministic) imperfection e„ and variable imperfection given by a fuzzy number. Variable imperfection is defined using the 5 percentile Cy (as usually prescribed by normative requirements). The criterium used to define the corresponding fuzzy number is described in Ferrari and Savoia [7]; it is the extension to continuous membership functions of the technique proposed by Dubois and Prade [2]. The result is reported in Fig. 3a for three different cases of imperfection values. The fuzzy numbers representing the non dimensional maximum loads are reported in Fig. 3b-d. Values of the membership function close to one denote loads with maximum possibility of occurrence. Of course, higher dispersions of variable imperfections give more disperse results. Nevertheless, as shown in Fig. 3c,d, an almost crisp value delimiting from below the interval of possible values of maximum load can be usually identified. That value is very significant from the design point of view, since it can be used to define a reliable value for maximum load having a low possibility of not being reached, starting

459

from a wide class of probability measures for structural imperfections compatible with normative provisions.

References [1] Elishakoff I. Probabilistic Methods in the Theory of Structures. New York: Wiley, 1983. [2] Dubois D, Prade H. Fuzzy Sets and Systems: Theory and Applications. Mathematics in Science and Engineering. San Diego: Academic Press, 1980. [3] Ferrari F, Savoia M. Fuzzy number theory to obtain conservative results with respect to probability. Comput Methods Appl Mech Eng 1998;160:205-222. [4] Moore RE. Interval Analysis. Englewood Cliffs, NJ: Prentice Hall, 1966. [5] Dong W, Shah HC. Vertex method for computing functions of fuzzy variables. Fuzzy Sets Syst 1991;42:87-101. [6] Casciaro R, Salerno G, Lanzo A. Finite element asymptotic analysis of slender elastic structures: a simple approach. Int J Numer Methods Eng 1992;35:1397-1426. [7] Ferrari F, Savoia M. Non probabilistic analysis of stability of structures. AIMETA Congress 1997;4:55-60 (in Itahan).

460

Simulation of cup-cone fracture in round bars using the cohesive zone model Ingo Scheider * GKSS, Max-Planck-Str. 21502 Geesthacht, Germany

Abstract The cohesive zone model for numerical crack propagation analysis has attracted considerable attention during the last decade. It is used here for the simulation of a ductile round bar under tensile loading. In those specimens, the crack initiates in the middle of the bar perpendicular to the loading direction and changes its direction to 45° as it propagates towards the outer surface (so called cup-cone fracture). Since the point of crack path deflection is not known in advance, the model must be able to change the crack path arbitrarily. In addition, the fracture mechanism in the 45°-regime constitutes a mixed mode problem. The present paper describes how shear fracture and coupling between normal and shear parameters are implemented into the model and shows the prediction of cup-cone fracture using FEM simulation. Keywords: Cohesive zone model; Cup-cone fracture; Crack propagation

1. Introduction The tensile test is one of the most wide-spread tests of all. It is the very foundation of characterization of ductile materials. Even though the specimen has a simple geometry and the loading is uniaxial, the mechanism of failure of that specimen is very complex due to the mixed fracture which starts with a pure normal crack opening in the center of the bar, propagates to the surface and deviates after some amount of crack growth in a 45° direction. The shape of the fracture surface leads to the given name cup-cone fracture. With an elastic-plastic FEM analysis, the local necking of the specimen can be simulated up to the point where damage becomes dominant. In most cases the specimen fails immediately after that point, so the load drops to zero very fast. Calculations of tensile specimens including damage and failure were first done by Tvergaard and Needleman [1] using the Gurson Model [2], but have received little attention since then. A recent publication [3] models the cup-cone fracture using the Gurson Model and the Rousselier model [4]. A simulation of cup-cone fracture of tensile specimens using the cohesive zone model has not been published up to now. *Tel.: -h49 (4152) 872599; Fax: -h49 (4152) 872534; E-mail: ingo. scheider @ gkss. de © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The basic idea of the cohesive zone model, first used for numerical ductile fracture analysis by Needleman [5] is to split the material's behaviour in deformation, which is modelled by continuum elements, and damage or separation, which is associated to the interface elements. In the latter so-called cohesive elements, the opening stress is controlled by a separation-dependent law, called the traction separation law (TSL). The separation, 5, can occur in normal (5^) or tangential direction (5^). In most implementations, we find two variables which are defined as material parameters. These are the maximum stress of the TSL, cTmax, and the separation where the cohesive zone element fails, 8Q. These two parameters can be different in normal and tangential direction. The function of the TSL is chosen in various ways by different authors. A third parameter, the total energy dissipation at fracture FQ, can be derived by FQ = j^^ a{b)d8. The value of Fo can be easily obtained by experiment, since it corresponds with the crack initiation 7/. The main goal is to show that the cohesive zone model is capable of simulating the fracture of round bars under tension. Two difficulties have to be overcome during the calculation. First, the point of deflection is not known before the simulation, so the mesh has to provide the opportunity for the crack to deflect when the status of cohesive zone elements indicates propagation in that direction. The

/. Scheider /First MIT Conference on Computational Fluid and Solid Mechanics

461

Traction Separation Law

second aspect is the coupling of normal and shear fracture, which is important for the point of deflection as is shown later. The cohesive elements normal to the loading direction fail under pure normal stress, whereas the diagonal break under competing shear and normal stress (that is the difference between this kind of fracture and pure shear fracture). Nevertheless, it will be shown that the main mechanism for failure is shear.

2. Cohesive zone model for crack propagation analysis 2.1. Traction separation law

0.5

The cohesive element is an interface element, i.e. it has zero volume in the case of 3D calculations and zero area in the case of 2D in the initial state. However, it has an upper and a lower face, with separate nodes which can move apart. With the ratio of actual to maximum separation a damage variable D is defined by D=j-

6

T(D) =

Fig. 1. Traction separation law with Di = 0.05 and D2 = 0.75. tangential separation is defined by

-m

= ^0 n 1 - ( ^ )

and

' respectively.

(4)

The damage variable D for Eq. (2) has now to be divided in a normal and a tangential part, which is defined by and

DT = -^j^, respectively

(5)

D < Di

DM =

Di < D2 (2)

The cohesive zone parameters for shear fracture are different from the parameters for normal separation. The maximum dissipated energy, P j , is lower than TQ and the ratio between tangential and normal cohesive traction is (as a rule of thumb) olJa^^^ ^ 1/3.

max

+ 1 I D2< D 1 or a decreasing function with Di = D2 ^ 0. With Di = D2 = 1/3 the function is quite close to the polynomial function defined by [5]. Fig. 1 shows the separation function of Eq. (2) with Di = 0.05 and D2 = 0.75. 2.2. Coupling of normal and shear fracture When normal and tangential separations occur in combination, the total damage is defined by

Hid^iM

Separation &8Q

(1)

The separation in the inface elements is defined by the displacement jump of the opposite faces. The traction is then given by a TSL. In this paper, two functions for the normal and the tangential traction are defined separately by the polynomial forms [6]:

62/80

(3)

Since the cohesive element fails at Z) = 1, the presence of tangential separation lowers the maximum normal separation and vice versa. For that, a maximum normal and

3. Simulation of round bar tensile tests A standard round tensile bar is simulated using the Finite Element code ABAQUS [7]. The material is a power law harding elastic-plastic solid with hardening n = 5. The N parameters for the interfaces are a^max 3.33ao, 8^ =0.1 and 8Q = 1.2af 0.15 mm. The ratio between mm, a^ normal and shear cohesive energy results in F ^ / F Q ^ 2. The cohesive zone elements are implemented by user defined elements [6]. The cross section at the center of the specimen where crack evolution is expected is modelled using triangular elements. Between each of these elements, cohesive interfaces are generated. The whole model in the undeformed state and the center region after failure of the specimen can be seen in Fig. 2. The crack starts in the center line. After about 2/3 of the way through the bar, it

462

/. Scheider / First MIT Conference on Computational Fluid and Solid Mechanics

:—1

1

1

1

1

1

r

I /^"" 12

z'

1

1•

^~^\

7

I - 1 —1

-

:

^ '-

^

"

CO

o

_i

-

4

Al (mm)

Fig. 2. Simulation of cup-cone fracture of a round bar. Left: FE-mesh of the specimen modelled including detail of the triangular mesh with cohesive interface elements between each of the continuum elements after failure of the specimen. Right: load-displacement curve of the simulation.

deviates and breaks with little normal separation in a line which is very close to the 45° line. An important aspect is the element length ratio used in the simulation. As can be seen in the figure, the pattern of four deformed triangular elements is nearly quadratic, so that the 45° direction is possible there. If a different aspect ratio is chosen, the crack path can be different. It can be shown, however, that within a range of aspect ratios the point of deflection is the same as in the figure. The point of deflection has a negligible influence on the load-displacement curve, since the crack propagates unstably and the curve drops to zero.

4. Conclusion The simulation of cup-cone fracture in a round bar under tensile loading using the cohesive zone model was quite successful. Even though some authors argue that the cohesive parameters might be dependent on the stress state (namely on triaxiality), see e.g. [8], the mechanism of crack path deviation due to the changing triaxiality can be predicted. For precracked solids of quasi brittle material, arbitrary crack propagation is modelled, e.g. by Xu and Needleman [9]. It is shown here that the cohesive zone model is a powerful tool for arbitrary crack path evaluation, even of ductile materials.

References [1] Tvergaard V, Needleman A. Analysis of cup-cone fracture in a round tensile bar. Acta Metall 1984;32. [2] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: Part I — yield criteria and flow rules for porous ductile media. J Eng Materials Technol 1977;99:2-15. [3] Besson J, Steglich D, Brocks W. Modeling of crack growth in round bars and plane strain specimens. Int J Solids Struct, submitted for publication. [4] Rousselier G. Ductile fracture models and their potential in local approach of fracture. Nucl Eng Des 1987;105:97-1 IL [5] Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl Mech 1987;54:525-531. [6] Scheider I. Bruchmechanische Bewertung von LaserschweiBverbindungen durch numerische Simulation mit dem Kohasivzonenmodell. PhD Thesis, Technische Universitat Hamburg-Harburg, 2000. [7] ABAQUS User's manual. Version 5.8. Hibbit, Karlson and Sorensen, Inc., 1080 Main Street, Pawtucket, RI, USA, 1999. [8] Siegmund T, Brocks W. The role of cohesive strength and separation energy for modelling of ductile failure. In: Paris PC, Jerina KL (Eds), Fatigue and Fracture Mechanics: 30th Volume, ASTM STP 1360. American Society for Testing and Materials, 2000, pp. 97-116. [9] Xu X-P, Needleman A. Numerical simulations of fast crack growth in britfle solids. J Mech Phys Solids 1994;42:13971434.

463

Response of a continuous system with stochastically varying surface roughness to a moving load C.A. Schenk'^'*, L.A. Bergman^ ^Institute of Engineering Mechanics, L.-F. University, Technikerstn 11, 6020 Innsbruck, Austria, EU ^ Aeronautical and Astronautical Engineering Department, University of Illinois, Urbana, IL, USA

Abstract The problem of calculating the second moment properties of the response of a general class of non-conservative linear distributed parameter systems excited by a moving concentrated load with stochastically varying surface roughness is investigated. Keywords: Moving load; Modal series expansion; Surface roughness; Colored noise; Runge-Kutta method; Second moment properties

1. Introduction Engineering structures such as highway and railway bridges on which vehicles move are subjected to loads that vary both in time and in space. The problem of calculating the dynamic response of such a combined system (see Fig. 1) has been of interest to engineers for many years; see, e.g., [1,2]. In this paper the effect of a non-conservative moving oscillator traversing a continuum with irregularities (i.e., surface roughness) is investigated. Therefore, we apply the method of solution for the moving oscillator problem as proposed in [3]. UtiUzing this approach, the response of the system is obtained by means of a series expansion in terms of complex eigenfunctions of the continuous system. Surface roughness is modeled as Gaussian, stationary colored noise, obtained by the response of a dynamic shape filter of second order excited by stationary white noise.

LM

The vibration of a spatially one-dimensional, linear nonself-adjoint distributed system is governed by the partial differential equation * Corresponding author. Tel.: -^43 (512) 507-6844; Fax: +43 (512) 507-2905; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

+ LD

^

+ LKW(X,

t) = f{x,

t),

(1)

ot

where x e [0, L], L is the length of the continuum; w{x,t) is the lateral displacement of the continuum; LM, LD and LK are real spatial differential operators defined on functions satisfying necessary smoothness requirements and given boundary conditions; and f{x,t) is an external force. In the present problem, LM and LK are positive definite and Lo is at least positive semi-definite. Boundary conditions are assumed to be given and the initial conditions are defined according to (2) w(x,0) = dw(x, 0) .0. dt

It can be shown, see [3], that the response w(x,t) due to a moving concentrated load can be obtained by a series expansion in terms of complex eigenfunctions of the continuum system according to (x,t)

2. Statement of the problem and previous results

iV-^ dt^

=

y^Rt[(l)n(x)qn{t)l

(3)

n=\

where the

.„(0 = ^ / . Mit-x) fn{^{x))y{z)dx

(4)

are the temporal coefficients and A„ = «„ + icOn and 0„(A:) are the (complex) w-th eigenvalue and eigenfunction of the distributed system, respectively, which are solutions to

CA. Schenk, LA. Bergman /First MIT Conference on Computational Fluid and Solid Mechanics

464

ql{t)=cOnq^{t) + anql,{t) - (al + coir' {(On^^{^{t)) + anfi{^{t)))

y{t)

qo

= (Poyit)

yit)

= k [{gt' - cpoqoit) - {0(?(O)r {^(0) + K^(0)]

-^d[gt-(poqo(t)-[HUt))}''{q(t)]

-[kat))]\q(t)}^r{^(t))]

(6)

and w{x,t) = {4>(x)}''{q{t)],

Fig. 1. A distributed elastic system carrying a moving linear non-conservative oscillator with stochastically varying surface roughness. the direct eigenvalue problem. In addition, IA„(JC) denotes the n-ih eigenfunction of the adjoint eigenvalue problem and <^(r) denotes the actual position of the load along the continuum axis. The overbar denotes complex conjugation. The problem of finding the coefficients ^,,(0 can be reduced, see [3], to that of solving a set of linear ordinary differential equations.

3. Method of analysis

(7)

where (po = 1 / V M is the normalized rigid body eigenfunction of the isolated mass, qo(t) is the corresponding temporal coefficient (initial conditions ^o(O) = —M^^'^g/k, ^Q(O) = 0), and curly brackets denote real vectors; e.g., {qit)} = {{q'it)V{q'(t)VV3.2. Mathematical model for surface roughness In (5) the surface roughness is introduced as a general implicit function of time. For further computations both a realistic and simplified mathematical model describing the random irregularities on the surface of the continuum are defined. In this paper the surface roughness r(^) is modeled as a Gaussian, stationary colored noise. Its one-sided auto spectral density Gr(y) and corresponding auto correlation function /?,(Af) (A^ = ^2 - fi) are (see [4]) y^ + a^-^-i

3.1. Solution to the non-conservative moving oscillator and implementation of stochastically varying surface roughness

Griy)

In the moving oscillator problem, the mass interacts w^ith the continuum through a spring (stiffness k) and a dashpot (damping constant d). The possibility of the onset of separation between the oscillator and continuum is not considered in this paper; i.e., the interaction force can take on both positive and negative values. Surface roughness of the continuum can be modeled as a general function r(f) describing the deviation of the ideal continuum surface to the distorted surface. The interaction force y{t) between continuum and attached mass is then given by

where a, ^ are shape parameters and a^ denotes the variance of the surface roughness. Instead of varying explicitly with time, the surface roughness r(f) is an explicit function of distance, implying that the auto spectral density is a function of the wavenumber y [rad/m]. The temporal spectrum of the surface roughness r(^(t)) can be obtained from the spectrum of the stationary response of a dynamic shape filter of second order excited by zero mean white noise W(t) (intensity qw) (see, e.g., [4]) as

y(t) =

k[z(t)-(w(i(t),t)-r(t))]

+ d i{t)-

dt

-r{t)

)]

(5)

If the only external force acting on the oscillator mass is its weight, it can be shown that, in this case, the set of equations governing a moving non-conservative oscillator including the effect of surface roughness for a linear non-self-adjoint distributed system is given by q^{t)--a,q^^{t)-oj,ql{t) + {al + 0?X'

{anfn(i(t))

- COntn{^{t)))

y{t),

K

(y2_a2_^2)2_^4c^2^2'

Rr{A^) = c r V -«|A^I cosySAf.

r{^{t)) = h V ) s(0 = Fs(0 + GW{t)

(8)

(9) (10)

with proper initial condition s(0) and i; = ^ is the translational velocity of the oscillator. 3.3. Application to conservative systems In order to apply the method discussed, the distributed system must be specified. This can be done in exactly the same way as proposed in [3], keeping in mind that, due to the surface roughness and the dashpot, the motion

C.A. Schenk, LA. Bergman /First MIT Conference on Computational Fluid and Solid Mechanics

465

xlO'

length x/L [m] time t [s] CJ

(x) at specific times

o

xlO""

0.25 s — 0.5 s ~ 1s

ffjando

(vt,t)

—

7

oscillator beam

6

t 2 1 0

0.2

0.4 0.6 length x/L [m]

0.8

1

0

0

0.2

0.4 0.6 time t [s]

0.8

1

Fig. 2. Variance function T?^^ix, t) of beam displacement. of the moving oscillator is now described by the set of equations given in (6). It is well known that a conservative system (i.e., L/) = 0) exhibits purely imaginary eigenvalues kn = io)n and real, orthogonal eigenfunctions, thus, the general set of equations given in (6) can be rewritten in the simple form 00

X(0 = A(OX(0 + GAt)fd(t) +

GMUt),

(12)

where X(t) denotes the state vector of the temporal coefficients qn(t) and the colored noise process s(t), A{t) is the system matrix, Gj(0 and Gs(t) are the distribution vectors for the deterministic and stochastic inputs, respectively, and fj(0 and fs(t) are the deterministic and stochastic input vectors, respectively.

m=0 00

m=0 00

+ dcp„(t;(t))J^
= (Pn(i(t)) {k (^gr2 + r(^(0)) +d(gt-^

linear ODE's given in (11), now truncated at order N, and the dynamic filter given in (9) and (10) can be combined to a system of ordinary differential equations according to (12)

r(?(0)))

(11)

3.4. Mean and variance of response 3.4.1. State space formulation In the following we restrict our consideration to conservative continua and assume constant velocity of the oscillator {v = 0). Using matrix notation, the system of

3.4.2. Mean and variance of temporal coefficients The differential equation for the mean response of the temporal coefficients, mx(0 = E{X(t)], can be obtained by applying the mean operator to (12), which is identically the response of the system without considering surface roughness (initial condition mx(0) = 0, except mo = E{qo(t = 0)} = -M'/^g/kl mx(0 = A(Omx(0 +

GAOUO-

(13)

466

C.A. Schenk, LA. Bergman / First MIT Conference on Computational Fluid and Solid Mechanics

1

"

length x/L [m]

time t [s] (S

(x) at specific times

a _ (t) and o 0.01 r

(vt,t)

oscillator —— beam |

0.008

•g 0.006 \

D 0.004

0

0.2

0.4 0.6 length x/L [m]

0.8

0.2

0.4 0.6 time t [s]

0.8

Fig. 3. Monte Carlo simulation (200 simulations): Variance function ll^^^{x, t) of beam displacement

In order to establish the differential Lyapunov equation for the variance function TxxCO, it is convenient to define the centered process X(r) X(0 = X ( 0 - m x ( r ) .

The auto covariance function for the scalar stationary white noise input W(t) is given by Twwit — s) = qw ^(t — s), from which the matrix D(r, ^) is found to be

(14)

Substituting (13) and (14) into (12), we get the differential equation for the centered process X

X = A(OX + G,(r)f,(0.

(15)

The covariance matrix T^^{t) = E{X{t)X^{t)} computed from (see, e.g., [5])

can be

D(r,5) =

0

t > s

\Gs{t)qwG]is):

t = s.

Thus, D ( 0 = \Gs(t)qwG](t), problem reduces to r x x ( 0 = Mt)T^x(t)

+ T^x(t)A\t)

(18)

and the solution of the + 2D(0.

(19)

where r x x ( 0 = ^ x x ( ^ 0-

f x x ( 0 = A ( O r x x ( 0 + r x x ( O A ^ ( 0 + D(r, t) + D^(r, 0 (16) with initial condition m,n = 0,l,2,...

,(2N-\-2)

(m ^ n)

rxx(O)

n = 0,l,2,... Tnn = S(0)

n = 2N -\-l,2N

(17) ,2N + 2

3.4.3. Mean and variance of response of continuous system The mean function of the continuous system can be obtained by applying the mean operator to (7), m^ix,

t) = E {w(x, t)} N

N

= ^ < / ^ . ( - ^ ) ^ [qniO] = ^ ( ^ „ ( x ) m 2 „ - i ( 0 . n=l

(20)

n=l

Similarly, the time varying covariance function describing

C.A. Schenk, LA. Bergman /First MIT Conference on Computational Fluid and Solid Mechanics the spatial correlation between two different locations xi and X2 along the continuous system is defined by A WW {XuX2,t)

= E[[w{xu t) - m^{xu t)][w{x2, t) - m ^ f e , 01} = E\W{X\,

t)w(x2, 0 }

N

N

n=l

m=l

N

N

n=l

m=\

m—l,2n—1 (0-

(21)

The variance function is obtained from (21) for jci = ;c2 = X, which is nothing but the main diagonal of ru;u;(-^i, -^2, 0 , ^L(-^' 0 = diag r^^(xi, JC2, 0 A^ =

N

Xmi^«W<^m(-^)r2«-l,2n-l(0-

467

damping constant are c = 2000 N/m and d = 44.72 Ns/m, respectively. The shape parameters for the one-sided auto spectral density Gr(y) are assumed to or = 0.5 rad/m and p = 0.2 rad/m; variance a^ = 0.005 m^. Since no closed form solution for the systems of ODE's (12), (13) and (19) is available, they have been integrated numerically using a Runge-Kutta method. The whole procedure has been programmed in MATLAB [7]. Although a slow convergence of the series for the variances has to be expected by definition, a study demonstrates reasonably convergence of the variance in 7 terms. The variance function ^IJ^(X, t) for the beam displacement obtained by the semi-analytical method proposed in this paper is presented in Fig. 2, which coincides with the corresponding Monte Carlo simulation (200 simulations), shown in Fig. 3, very well.

(22)

Acknowledgements 3.4.4. Mean and variance of moving oscillator The mean function of the moving oscillator can be calculated by m,{t) = E {z(t)} = E [\gt^ - (poqoit)} = \gt^ - (pomo(t).

(23)

The variance function is then given by a^^it) = E {[z(t) - m,(t)] [z(t) - m,(t)]} = E {z(trz(t)} = (Po E {qo(t)qo(t)] = (p^roo(t).

(24)

3.4.5. Monte Carlo simulation Another powerful tool for calculating the response statistics is Monte Carlo simulation. However, here it is used to verify the results obtained in Sections 3.4.2 and 3.4.3.

4. Numerical results and concluding remarks In this section we apply the proposed method of solution to a non-conservative oscillator traversing a conservative simply supported Euler-Bemoulli beam with constant velocity. To the authors' knowledge, nothing exists in the literature with which to quantitatively compare these results. The numerical values of the beam parameters are the same as those reported in [2,6] and are as follows: L = 6 m, El/m = 275.4408 mVs^ The ratio of the mass of the oscillator to the beam mass is M/ml = 0.2; the velocity of the oscillator is i; = 6 m/s, its spring stiffness and

The authors would like to thank Prof. G.I. Schueller and Assoc. Prof. H.J. Pradlwarter of the Institute of Engineering Mechanics, L.-F. University of Innsbruck, and C. Bilello and Y. Song from the Aeronautical and Astronautical Engineering Department, University of Illinois, for their support. The second author (Bergman) is particularly grateful to Prof. Schueller for providing him the opportunity to lecture on some of this material while serving as Distinguished A.M. Freudenthal Visiting Professor at the Institute of Engineering Mechanics during May, 2000, and to the National Science Foundation for partial support of this work through award no. CMS-9800136.

References [1] Fryba L. Vibration of Solids and Structures Under Moving Loads. Noordhoff International Publishing, 1972. [2] Sadiku S, Leipholz HHE. On the dynamics of elastic systems with moving concentrated masses. Ingenieur-Archiv 1987;57:223-242. [3] Pesterev A, Bergman L. Response of a nonconservative continuous system to a moving concentrated load. J Appl Mech 1998;65:436-444. [4] Miiller PC, Popp K, Schiehlen WO. Berechnungsverfahren fiir stochastische Fahrzeugschwingungen. Ingenieur-Archiv 1980;49:235-254. [5] Soong TT, Grigoriu M. Random Vibration of Mechanical and Structural Systems. Prentice Hall, 1993. [6] Pesterev A, Bergman L. Vibration of elastic continuum carrying moving linear oscillator. J Eng Mech 1997; 123:878884. [7] The MathWorks, Inc. Using MATLAB, 1999.

468

Elastic stability problems in micro-macro transitions J. Schroder^'*, C.Miehe*' " Institutfur Mechanik, Technische Universitdt Darmstadt, Hochschulstr. 1, 64289 Darmstadt, Germany ^ Institut fiir Mechanik (Bauwesen), Universitdt Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany

Abstract One problem in homogenization of nonlinear elastic micro-structures is the possibility of the occurrence of structural instabilities on the micro-structure. Due to this possible instabilities the size of the representative volume element is a priori unknown. In this presentation we investigate the connection between microscopic and macroscopic instability phenomena and present some representative numerical examples. Keywords: Micro-macro transition; Homogenization; Instability; Loss of rank-one convexity

1. Introduction In this paper, we discuss stability problems which may occur in the framework of micro-macro transitions of heterogeneous elastic materials at large strains. One of the main difficulties of the macroscopic description of nonlinear micro-heterogeneous materials is that the overall response can be completely different to the microscopic behavior of their constituents. The framework of the micro to macro transition is based on the notion of a homogenized macro-continuum with locally attached micro-structure (see Fig. 1). The basic assumption is that the macroscopic behavior of the heterogeneous structure can be described by the introduction of two different scales. On the macroscale, we consider a homogenized continuous medium and on the micro-scale we assume a periodic heterogeneous micro-structure. In the considered two-scale homogenization approach we have to replace the energy functional, which takes into account the heterogeneous micro-structure, with a homogenized one. The homogenization based on average theorems as outlined for example by Hill [2], Suquet [7], Miehe et al. [4], and others is well established in the literature. In the case of convex free energy functions x// = ^(F),itis sufficient to minimize the associated stored energy function on the micro-scale in one periodic cell. The assumption of convexity of the energy functional with respect to the deformation gradient is not suitable * Corresponding author. Tel.: -h49 (6151) 163174; Fax: -h49 (6151) 166117; E-mail:[email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Macro-Variable

B-A

iL

X J

Macro-Structure

Micro-Variable

/

1 1

\ \

\

HI X

s ^

U,

1

T^B

Micro-Scale

Fig. 1. The notion of a macro-continuum with micro-structure. Associated with a point X e B oi di homogenized macro-continuum is a micro-scale. On the micro-scale B d'R? denotes a unit cell with B ^ Be of the critical representative volume element Be C n^.

in nonlinear continuum mechanics, see e.g. Marsden and Hughes [3]. The nonconvexity of the integral functional can be related to nonuniqueness phenomena, e.g. to structural instabilities on the microscale, see Miiller [6]. For non-convex free energy functions a minimization of the associated stored energy function on the micro-scale has to be performed over a critical ensemble of periodic cells. Geymonat et al. [1] studied the connection between microscopic bifurcation and the loss of macroscopic strict rank-one convexity in the general framework of nonlinear functional analysis. One of the main difficulties in the two-scale homogenization approach for the description of nonlinear composites is that the size of a representative volume element is not a

/. Schroder, C. Miehe /First MIT Conference on Computational Fluid and Solid Mechanics priori known, for a detailed description we refer to Miehe and Schroder [5].

2. Micro-macro transitions at large strains The macroscopic deformation gradient F is related to the deformation of the microstructure x and the macroscopic first Piola-Kirchhoff stress tensor P to its microscopic counterpart P via the surface and volume integrals

'^h

^N&A

and

-vl

PdV

(1)

with DIV P = 0 in B, respectively. Here V := vol{B) denotes the volume of the micro-structure B and A^ the outward unit normal of the outer surface dB. The deformation of the micro-structure is assumed to be linked with the local values of the macro-continuum ai X e B via the ansatz X = FX + w

in B,

(2)

with the fluctuation field iu. In a deformation driven process of the macro-continuum, this coupling can be realized by three alternative constraints of the micro-structure deformation: (1) zero fluctuations on the domain; (2) zero fluctuations on the boundary; or (3) periodic fluctuations on the boundary. These three conditions satisfy the so-called averaging theorem, see Hill [2], which states that the average of the microscopic work is equal to the macroscopic work of the conjugate macroscopic quantities. P : F

-vf

P:FdV.

(3)

Due to the considered periodic micro-structures we assume periodic boundary conditions. If stability problems occur on the micro-scale we do not know a priori the size of the representative volume element which depends on the wavelengths of the bifurcation modes. Thus we have to take into account an ensemble of n cells Be := IJLi ^k with volume Vc. The macroscopic free energy function is then defined in terms of the microscopic free energy by the expression xl/(F) :=inf ^ inf

vj iA(F + vw#)dy

(4)

with the periodic fluctuation field w# over B^ see Muller [6]. Based on (4), it is possible to determine Be. The expression for the stored free energy function n^ associated with a fixed domain Be on the micro-scale is given by Ueim, Be) '=y

f fiF + Vu;#) dV Be

inf.

(5)

469

There exists a close connection between bifurcation on the microscopic level, characterized by the second variation of ric, and the possibility of shear band localization on the macroscopic level. Long wavelength instabilities for the linearized homogeneous problem on the microscale lead to a loss of rank-one convexity of the homogenized free energy function on the macro-scale. The possibility of macroscopic shear band localization is characterized by the loss of rank-one convexity of the homogenized (macroscopic) tangent moduli C := dj,p\lf(F). This condition can be checked by the investigation of the determinant of the so-called macroscopic acoustic tensor g , i.e. Min det Q(N)

> 0 before localization < 0

else.

(6)

The acoustic tensor is a function of the orientation A^ of the singular surface at X e B and the macroscopic tangent moduli, i.e. Qab := Ca^b^NANB3. Numerical example: Out-of-phase buckling of fiber reinforced composite In this example, we investigate the effect of the dimension of a fiber reinforced micro-structure on the overall behavior of the composite in case of buckling on the microscale. The considered micro-structure consists of a weak matrix material with horizontal aligned fibers with a volume fraction of the fibers of 20%. The bulk modulus of the matrix is KM = 49.98 N/mm^ and the shear modulus IJiM = 74.97 N/mm^ and the parameters of the fibers are set to Ki = 10"^/CM, III = 10"^/XM- The applied homogeneous macroscopic deformation mode is a horizontal compression governed by the macroscopic deformation gradient F with the components Fn = 1 — A and F22 = F33 = 1. All other components are set equal to zero. In compression, we distinguish between two characteristic failure modes which may occur on the micro-scale, an in-phase and an out-of-phase microbuckling of the fibers. Here we concentrate on the latter one and consider a micro-structure with two parallel aligned fibers. As pointed out above, the basic problem is the yet unknown size, the length, of the micro-structure. Thus, the stability analysis on the micro-scale requires the minimization of the macroscopic critical load parameter with respect to the length / of the micro-structure. We start the simulation with the length / = 3 and increase / by increments of 1 until a value of / = 12 is reached. Fig. 2a depicts the macroscopic Kirchhoff stress component fn versus the compression parameter k for some different lengths. As shown in the load-deflection diagram the characteristic of the curves changes significantly with increasing length. Fig. 2b summarizes the critical stress component fn,ertt at the onset of the out-of-phase microbuckling. A minimum for the critical macroscopic

470

J. Schroder, C. Miehe /First MIT Conference on Computational Fluid and Solid Mechanics —^W.crit

6000

1

i = 12 ^ /= 9 ^ ; = 7 ^

5000 4000

6000

/\ \ =3 -

3000 2000 1000

a.

:/7<

0
0

1

0.01

—

-5

1--= 4

/= 8

.J__

0.02

0.03

/ = 7

Fig. 2. Out-of-phase buckling mode, (a) Macroscopic Kirchhoff stress component T\\ versus the compression parameter X. (b) Critical macroscopic stresses fw^cm at the onset of microbuckling versus length /.

Fig. 3. Out-of-phase buckling of the fibers. Post-critical microscopic buckling modes for the length / = 12 at the compression values (a) -k = 0.011, (b) \ = 0.017 and (c) X = 0.035. Kirchhoff stresses inherit is obtained for the length / = 7, as visualized in Fig. 2b. A sequence of post-critical deformation stages for the out-of-phase mode is illustrated in Fig. 3a-c for / = 12 at the compression parameters X = 0.011, 0.017, 0.035, respectively. For the micro-structure with / = 12 a buckled microstate is observed after a moderate compression, where the micro-structure contains three characteristic microbuckling modes in the post critical region as depicted in Fig. 3c.

References [1] Geymonat G, MuUer S, Triantafyllidis N. Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch Rat Mech Anal 1993;122:231-290. [2] Hill R. On constitutive macro-variables for heterogeneous solids at finite strains. Proc R Soc Lond Ser A 1972;326: 131-147.

[3] Marsden JE, Hughes TJ. Mathematical Foundations of Elasticity. New Jersey: Prentice-Hall, 1983. [4] Miehe C, Schroder J, Schotte J. Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 1999;171:387-418. [5] Miehe C, Schroder J. Computational homogenization analysis in finite elasticity. Overall response and stability of composite materials. Comput Methods Appl Mech Eng, submitted for publication. [6] Mtiller S. Homogenization of Nonconvex Integral Functionals and Cellular Elastic Materials. Arch Rat Mech Anal 1987;99:189-212. [7] Suquet PM. Elements of Homogenization for Inelastic Sohds. In: Sanchez-Palenzia E, Zaoui A (Eds), Homogenization Techniques for Composite Materials, Lecture Notes in Physics 272. Springer, 1986.

471

Modeling of adaptive composite structures using a layerwise theory J.E. Semedo Gar9ao% CM. Mota Scares'^'*, C.A. Mota Scares^ J.N. Reddy'' ^ IDMEC — Instituto Superior Tecnico, Departamento de Engenharia Mecdnica, Av. Rovisco Pais, 1049 001 Lisbon, Portugal ^ Texas A & M University, Department of Mechanical Engineering, College Station, TX 77843-3123, USA

Abstract In this work a layerwise theory is used in modeling a composite laminated plate with embedded piezoelectric elements. The necessary formulation for the linear case is developed. The numerical solution of the structural problem using the finite element method is discussed. Keywords: Adaptive structures; Composite plates; Finite elements; Layerwise; Piezoelectric

1. Introduction The modeling capabilities of equivalent single layer theories depend on the laminate characteristics and on the degree of precision that is necessary on the results. When great precision is demanded a 3D modeling is necessary. Several kinds of layerwise theories have been proposed [1,2]. 2. Model description In this work the layerwise theory of Reddy [1] is considered. The model developed here for composite laminates with piezoelectric plies or patches is a linear elasticity model, valid only for small deformations. The materials are assumed homogeneous, general anisotropic, linear elastic, with some including linear piezoelectric effects. For the electrical problem it is assumed an electroquasistatic approximation, thus considering a negligible influence from possibly present small magnetic fields and magnetization. It is considered that the electric field and polarization, along with the mechanical variables, are the only important interactions when describing the motion and deformation of the material.

* Corresponding author. Tel.: +351 21 841 7455; Fax: +351 21 841 7915; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Consider a composite laminate with thickness h and arbitrary planar geometry Q. The whole volume may contain the structure and some free space necessary to obtain a more accurate solution for the electric field. The behavior of the structure is obtained solving the mechanical and electrical coupled problem, described by the elasticity and electrostatics equations and the respective boundary conditions. The set of equations is completed introducing the constitutive law for each ply k of the laminate. For a piezoelectric ply or patch, the constitutive law is given by: [a] = [Cf{s] - [e]'{E]

(1)

{D} =

(2)

[ef'{s}^[^f{E}

For a composite ply the constitutive law is: {a} = [Cf{s}

(3)

{D} =

(4)

[^cfiE]

In this case there is no coupUng between the deformation material and the electric field. The assumed displacement field is tridimensional with the components given by products of functions of (x, y,t) by functions of z (the upper indexes are not exponents). u(x, y, z,t) = Y^ U^(x, y, t)^^(z)

(5)

472

J.E. Semedo Gargdo et al /First MIT Conference on Computational Fluid and Solid Mechanics N

v(x,y,z,t)

= J2

'j{x,y,z,t)

(6)

V'(x,y,t)^\z)

= Y^

W'{x,y,t)^lf'{z)

(7)

The electrical problem is solved using the electrical potential function, related to the electric field as E = —V(p. For the electric potential, the same structure of functions is assumed. p

(8)

(p(x, y, z,t) = J2 ^'(^^ y^ t)@'(z)

v(x, y,z,t)

NV

= J2J2 1=1

K=l

N

NW

V^(t)b'^(x, y)
(12)

^K(tK(^^ >^)^'(^)

(13)

w(x, y,z,t) = J2Yl 7=1

P

K=l Ncp


Considering the test functions of the form of the approximation functions, with arbitrary time-independent coefficients, the final system of equations may be written as:

r[M„„ 1

[0]

[0]

[0]

lU}

3. Finite element formulation

[0]

[A/„„]

[0]

[0]

[V]

The finite element formulation is obtained from a variational formulation. The equations of the problem are demanded to be satisfied in an average sense:

[0]

[0]

[Muiw]

[0]

{W}

L [0]

[0]

[0]

[0]_

{^}

[Kuu]

///(Sf-.-S?)Q

\8pdzdxdy

(9)

=0

Applying the divergence theorem we arrive at the following variational formulation: d^Ui

/ / / PM -r-f Sai dzdxdy + n

Q

h

- 11 DiHiSp dS-

fff Q

95«/

dzdxdy

[Kucp]

[W] [K^u.

[K(pv]

I'^cpwi

L^^^l

{
[FuiO] lF„it)]

IF^iO)

d^cr

4. CoiM:lu sions

fj j BMi dz dx dy

=0

^i^W4(^' >')^'(^)

{U}

<{V}

h

where M, is the displacement field, a,y the stress tensor, Bj the body forces, PM is the material density, D, the electric displacement vector, pf the unpaired charge density. Hi is the unit normal to the boundary surface, dQa and dQo, respectively, the portions of the boundary where stresses and electric displacement are prescribed. The displacements and electric potential are assumed prescribed, respectively, in the portions 9Q„ and dQ^ of the boundary. The approximation functions are assumed to be:

u(x, y,z,t) = Y,Yl

[Kuw]

(15)

{FUt}]

(10) + Ijjpf8pdzdxdy

IKuA

+

h

(14)

VKiOfxi^' y)®'(^)

K=\

(11)

The layerwise finite element formulation has been developed and implemented in a computer code. The model is presently being tested. The influence over the numerical solution, of various possible interpolation schemes is being evaluated by comparison with analytical solutions and available experimental data.

References [1] Reddy JN. Mechanics of Laminated Composite Plates — Theory and Analysis. Boca Raton, FL: CRC Press, 1997. [2] Saravanos DA, Heyliger PR, Hopkins DA. Layerwise mechanics andfiniteelement for the dynamic analysis of piezoelectric composite plates. Int J SoHds Struct 1997;34(3):359378.

473

The local boundary integral equation and its meshless implementation for elastodynamic problems J. Sladek^'*, V. Sladek^ R. Van Keer'' ^Institute of Construction and Architecture, Slovak Academy of Sciences, 84220 Bratislava, Slovakia ^ University of Gent, Department of Mathematical Analysis, B-9000 Gent, Belgium

Abstract A new meshless method for solving elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in the present paper. This idea has been utilized in the development of a new formulation for the solution of a b.v.p. for the diffusion equation [1]. Applying the MLS approximation for spatially dependent terms the local boundary integral equations are transformed into the ordinary differential equations with spatially dependent matrix-coefficients. The system of ordinary differential equations is solved by the Newmark method. Keywords: Moving least squares approximation; Newmark method; Two-dimensional problem

1. Local boundary integral equations in elastodynamics Let us consider a linear elastodynamic problem on the domain Q, bounded by the boundary F. Then, the displacements are governed by the equation MM/,M + (A. + f^)Uk,ki + Xi = piii

integral in Eq. (2), we have / [fiulj^j^ (x) + (A + M)M* ^. (x)] Ui(x, 0 d^

+ f ti(x,t)u*(x)dr-

where ut and Xt are the components of the time dependent displacements and body force vector, respectively, and p is the mass density of the material. Dots indicate differentiations with respect to time and A, /i are Lame constants. The weak formulation of Eq. (1) can be written as

J

[piiiix, t) - Xi{x, 0]ulix) d^,

/ [l^Ui^kkix, 0 + (A. + lJi)uk,ki{x, t) + Xi{x, t) (2)

where u*(x) is a weight field. Applying the Gauss divergence theorem to the domain

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

+

f^Ukjrik

(4)

is the traction vector with the unit outward normal vector Hi to the boundary F. The weighted traction vector r* is defined by Eq. (4), where only the trial field ut is replaced by the weight field w*. ff the weight field is selected as the fundamental solution of the elastostatical governing equation l^Uij^kk (x,y)-\-(X-^

* Corresponding author. Tel.: -^421 (7) 5478-8662; Fax: -f-421 (7) 5477-2494; E-mail: [email protected]

(3)

where ti = jjiUi^knk + ^itk,kni

- piii(x, t)] u*(x) dQ = 0.

f t;(x)ui(x,t)dr

(1)

liWkjM (x,y) = -8ij8(x - y),

(5)

with M* = Uijix - y)ej(y) and t* = Tij{x, y)ej(y), one

474

J. Sladek et al. /First MIT Conference on Computational Fluid and Solid Mechanics

obtains the integral representation of the displacement field Ujiy, t) = j ti(x, t)Uij{x -y)dr~

f Tijix, y)Ui(x, t)dr

LBIE (7) we obtain the ordinary differential equation for the unknown Uai (0 /;(j)X!0«(y)i^a/(O=

•j[X,(x, t) - pUiix, t)] Uij(x -

y)d^.

/ T;j(x,y)^(l)a(y)Uai(t)

dr

(6) + I U*j{X - y)t{x, t)dr + j [/* (x - y%{x) AT

If, instead of the entire domain Q of the given problem, we consider a subdomain Qs. which is located entirely inside Q, the following equation (local boundary integral equation, LBIE) should also hold over the subdomain ^jiy^ 0=

ti(x, t)Uij(x -y)dr-

JTijix, y)Ui{x, t)dr

/

" pU*^{x - y)Y,4>.{x)ii^,{t)AU,

where n

i{x,t) = + / [Xi(x, t) - piiiix, t)] Uij(x - y)dQ

(7)

where 9^^ is the boundary of the subdomain Q^- Both boundary quantities (displacement and traction) are unknown on dQs- To eliminate the traction vector a 'companion solution', Uij, is introduced to the fundamental solution in such a way that the final modified fundamental solution, U*, is zero on the boundary dQ^ [2].

For the spatial approximation of displacements on dQs we have used the moving least square (MLS) formula [3]

J2Mx)u,,(t),

^NDB^{x)u^i{t), a=l

with the matrix N corresponding to the normal vector, D is the stress-strain matrix and 5„ involves gradients of the shape function. The matrix Ctj is the well-known free-term coefficient in the BEM analysis. There are many time integration procedures for the solution of the system of the ordinary differential equation (9). In the paper we have used the Newmark method.

References

2. Discretization and numerical implementation

u(x,t) =

(9)

(8)

where u«(r) is the time dependent fictitious nodal value and (pa(x) is the shape function. Substituting Eq. (8) into the

[1] Sladek V, Sladek J, Van Keer R. New integral equation approach to solution of diffusion equation. In: Kompis V (Ed), Numerical Methods in Continuum Mechanics. Zilina: 2000. [2] Atluri SN, Sladek J, Sladek V, Zhu T. The local boundary integral equation (LBIE) and its meshless implementation for linear elasticity. Comput Mech 2000;25:180-198. [3] Belytschko, Lu YY, Gu L. Element free Galerkin methods. Int J Numer Methods Engin 1994;37:229-256.

475

Structural analysis of composite lattice structures on the basis of smearing stiffness D. Slinchenko*, V.E. Verijenko Department of Mechanical Engineering, University of Natal, Durban 4041, South Africa

Abstract The numerical approach employed in the present study is necessitated by the computational inefficiency and conventional difficulties of linking together optimizer and FEM analysis package for calculating the stress resultants used in the optimization of such structures. This paper is aimed at verification of new mathematical model of the composite grid plates and shells subjected to a variety of loading conditions. The main objective of this research is to apply new homogenization approach and determine optimum values of geometric and other properties of the lattice and structures subjected to different loading/constraint conditions. Also, to improve the accuracy and efficiency of the tools used for design/optimization process. Keywords: Isogrid; Composite lattice structures; Smearing stiffness; FEM; Structural analysis; Homogenization; Optimization

1. Modelling on the basis of homogenization Composite grid structures (Fig. 1) represent a innovative concept that currently finds its wide spread application in different areas of modem engineering: aerospace (solar arrays, pay load shrouds, fuselages), civil engineering (concrete reinforced structures, bridge decks), transport (rail cars, trucks) to mention a few. Modeling of the lattice structures on the basis of FEM is quite laborious and leads to excessively complicated models. The further analysis of these models requires high computational power and in the case of design/optimization study requires recreation of the FEM model on each successive iteration. In order to avoid this the homogenized model with the stiffness equivalent to the original model can be used. This mathematical model is further referred as an Equivalent Stiffness Model (ESM). There is a need in homogenization principle, which aims at enhancing the scope and effectiveness of conventional tools for design/optimization of grid structures. The proposed approach deals with the elastic shell as a continuos system, i.e. external loads and the stress-strain state * Corresponding author. Tel: +27 (31) 260-1225; Fax: +27 (31) 260-3217; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

is described by the functions of constantly changing arguments. The governing system of six differential equations is derived (see [1] for details) and presented in the following form:

y(z«) = p(z«)y(z«) + /(z«),

(1)

where P is a square matrix with known coefficients that are calculated from the stiffness properties and geometric characteristics of the given shell [1] and y is the vector of unknown displacements (w°, w^, yi), forces (A^f, g ? ) and moment (Mf) acting in homogenize model given as: y^ = w^,

y2 = W^,

j3 = yu

y , = Nl

ys = Ml

ye = g?.

(2)

Also in Eq. (1) we have / , which is a known vector describing loading conditions of a shell: / l - /2 = /3 = /5 - 0,

/4 = -AX,

fe

-AZ,

(3)

Solution of the system of differential equations (1) must satisfy the following boundary conditions:

Biy{z'')=bi

for zO = z?.

(4)

476

D. Slinchenko, V.E. Verijenko / First MIT Conference on Computational Fluid and Solid Mechanics

1

.
I Ooss-sectionofarlb

1,2- Diagonal Ribs 3 - Vertical Ribs Fig. 1. Isogrid cell pattern. The given system of differential equations (1) represents a boundary value problem that can be solved numerically reducing it to the solution of Cauchy problem using method of Runge-Kutta.

2. Structural analysis and results Mathematica symbolic computation system was used for derivation of analytical expressions for the stress resultants with further calculation of the numerical values. In order to validate homogenization approach used the numerical results obtained for equivalent homogeneous model of the isogrid cylindrical structure are compared to those obtained from commercial FEM code MSC/NASTRAN. It is found that developed homogenized mathematical model accurately predicts stress resultants in the members of the structure with the discrepancy of the results less than 10%. However, it is observed that higher discrepancy of the results occurs in the regions subjected to the boundary effects.

3. Optimum design The governing system of partial differential equations (1) is derived for composite isogrid cylinders in such a way that allows analysis to be performed for the variety of loading conditions and applied constraints. The solution can also be used for a number of grid types: orthogrid, anglegrid and isogrid. Based on the proposed homogenized model, an approach for the optimal design of grid shells is formulated. A sample cylindrical grid structure is optimized taking cell configuration {(p) and geometric parameters of ribs (b, h: width and height respectively) as design variables to maximize the applied load (q). The optimization problem is stated as: max((^, h, b)
(5)

subjected to the following constraints: (6) The maximum (q) represents a load that the structure can sustain before the failure. It is computed on the basis of three dimensional interactive Tsai-Wu failure criterion which takes into account the influence of all stress components. The failure of the isogrid cylinder is characterized by the failure coefficient r: S^a^iX^, -Xl)-^S

(5^2(^3 -

^3? 1/2

(7) Failure coefficients are calculated for the vertical and diagonal families of ribs and the structure considered failed when either of them is equal to one. Special purpose computation routines are developed using symbolic computation package Mathematica for the calculation of equivalent stiffness of a structure, the analytical derivation of the objective function, failure analysis and calculation of optimum design parameters. An optimum yields to a maximum (critical) load that can be applied to a structure just before its failure, having the optimal geometric parameters h, b and cp. Computation of the optimum values of design variables is based on the use of robust multidimensional search methods which give fast convergence. A Sequential Quadratic Method (SQM) was used for calculating the optimum. In this method, a quadratic programming sub-problem is solved at each iteration. The isogrid cylindrical structure with material properties given in Table 2 and geometric parameters given in Table 3 is analysed. The numerical results are presented for the case of the structure subjected to tension and torque loads. The optimum sequences of the cross-sectional parameters are presented in Table 1. The values for critical load during the optimisation runs are plotted in Figure 2.

D. Slinchenko, VE. Verijenko /First MIT Conference on Computational Fluid and Solid Mechanics

Aril

Table 1 Optimum values iovb, h, cp Ws

Vertical family

b(m) 1 2 0.5

h (m)

0.014 0.033 0.024

0.004 0.0063 0.0135

Diagonal family

^O 0 0 0

^ten (N/m)

qtov (N/m)

1.38 X 10^ 5.43 X 10^ 4.9 X 10^

1.38 X 10^ 1.1 X 10"^ 2.45 X 10^

b{m) 0.021 0.027 0.052

h(m) 0.014 0.005 0.026

(PO

^ten (N/m)

qtor (N/m)

61.3 73.3 40.5

1.4 X 10^ 6 X 10^ 8.6 X 10^

1.4 X 10^ 1.2 X 10^ 4.3 X 10^

The range of design variables 35 dcg < cp < 75 deg, 0.002
IxlO^T'^"---.

Fig. 2. Critical load for the vertical family of ribs (left) and diagonal family of ribs (right). Load scahng factor: w;^ = 1Table 2 Material properties of the ribs Material

El (Pa)

E2 (Pa)

vn

Gu (Pa)

T300/5208

1.81 X 10^1

1.03 X lO^o

0.28

7.17 X 109

Table 3 Dimensions of the structure and parameters of the unit cell Height (m)

Diameter (m)

(PO

h (m)

b(m)

7.56

5.44

60

0.02

0.00667

on the basis of failure analysis shows that the difference in the value of maximum load applied to the optimal and non-optimal isogrid structure can be quite substantial, emphasizing the importance of optimization for the composite isogrid structures. Several isogrid cylinders are optimized on the basis of the proposed homogenization approach, such that both width and height of the ribs comprising vertical and diagonal families and the angle of cell configuration are determined optimally. These structures are analyzed for different loading conditions such as tension and torque, and their combination. The numerical results obtained during several optimization runs can be used for the design of isogrid structures subjected to the arbitrary loading combination.

4. Conclusions The optimization study shows the maximum values of the combination of the loads correspond to the values of optimum design parameters. Results presented here are given for the model that neglects the influence of the boundary effects on the general state of stress. The computational efficiency of optimization algorithm in the design optimization of cylindrical isogrids is improved and good accuracy of the results has been achieved. The investigation

References [1] Slinchenko D, Verijenko VE. Optimum design of grid cyhndrical structures using homogenised method. Proceedings of the Twelfth International Conference on Composite Materials ICCM/12, July 1999, Paris, France. [2] H-J Chen, Tsai S. Analysis and optimum design of composite grid structures. J Composite Mater 1996;30(4):503532.

478

Computer techniques for simulation of nonisothermal elastoplastic shell responses J. Soric *, Z. Tonkovic Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, 10000 Zagreb, Croatia

Abstract In the present paper, an efficient numerical simulation technique for nonisothermal elastoplastic responses of shell structures will be proposed. A realistic highly nonlinear and temperature-dependent hardening model will be appUed. The closest point projection integration algorithm applied in conjunction with a consistent elastoplastic tangent modulus ensures accuracy and robustness of the computational procedure. The derived algorithm has been embedded into a layered assumed strain isoparametric finite element which is capable of geometrical nonlinearities. Under the assumption of an adiabatic process, the increase of temperature will be computed during the elastoplastic deformation of a spherical shell. Keywords: Shell structure; Finite element analysis; Elastoplasticity; Nonisothermal hardening response; Integration algorithm; Tensor formulation

1. Introduction Inelastic phenomena and their numerical simulation have gained increasing attention in the research of shells and other fight weight structures [1,2]. A more realistic material modelling demands for consideration of nonisothermal hardening responses. This paper is concerned with the numerical simulation of nonisothermal elastoplastic deformation processes of shell structures. The applied material model proposed by Lehmann [3] adopts highly nonlinear hardening behavior with temperature dependent material functions obtained experimentally for mild steel by Szepan [4]. Small strains and an associative flow rule are assumed, and an adiabatic process is considered. The closest point projection algorithm according to Soric et al. [5], together with the consistent tangent operator technique which ensures high convergence rate in the global iteration procedure are derived. The computational algorithm, based on the simulation strategy presented in Kratzig [6], has been implemented into a four-noded isoparametric, assumed strain layered finite sheU element as presented in Basar et al. [7]. The finite element formulation employs the Reissner-Mindlin type shell * Corresponding author. Tel.: +385 (1) 616-8103; Fax: +385 (1) 616-8187; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

theory, and material nonlinearity will be combined with geometrically nonlinear analyses assuming finite rotations. The numerical example in Section 3 demonstrates robustness and numerical stability of the proposed computational approach. Spreading of plastic zones and changes of temperature are analysed during the elastoplastic deformation process modelled.

2. Constitutive model and numerical formulation The basic constitutive equations of the material model applied are compiled in Table 1. Herein, Latin indices take the values 1, 2 and 3. S'^ and p"^ are the deviatoric components of the stress tensor a'^ and back stress tensor p'^, respectively. X denotes the plastic multiplier, while 8ij represents the Kronecker delta. The temperature-dependent material parameters ^, x» ?. b\-b^, Cp and ar, which have been obtained experimentally for German mild steel St 37.12, are taken from Szepan [4]. Cp and p are the specific heat capacity and the mass density, respectively, while ar describes the thermal expansion coefficient. The temperature rate as given by Eq. (7) is obtained from the balance equation for the remaining energy as described in Lehmann [3]. Herein the terms depicting the coupling of the non-dissipative external and internal pro-

479

J. Soric, Z. Tonkovic /First MIT Conference on Computational Fluid and Solid Mechanics Table 1 Basic constitutive equations

3. Numerical example

Decomposition of total strain rate into elastic, plastic and thermal parts: Yii = Y!j+Y!; + Y^-

(1)

Associative flow rule: .p .dF(a^J,p^J,a,T) Yij = ^ ^—y .

(2)

The von Mises-type yield condition: F(a^j,p'j,a, T) = (S^J - p''J)(Sij - p[^) - k\a,

T) < 0.

(3)

Nonlinear kinematic hardening rule: p^^J=gyP^J-XP''^yfi^, whcrQg = g{p'^J,T),x

(4)

= x(T).

Nonlinear isotropic hardening rule: k^(a, T) = bi+ b2a + /73(1 - e^n, where bi .. .b4 = f(T).

(5)

Internal variable describing isotropic hardening: d = (S^J -p'^J)yP.

(6)

Temperature changes during the elastoplastic deformation process: t=—(S^^-p''J)y' CpP where ^ = ^a^J, p'J, T), Cp = Cp{T), p = p{T).

(7)

Thermal strain rate: y j = artSij, where ar =

(8)

ariT).

Consistency condition in Kuhn-Tucker form: F(a'J,p'J,a,T) <0, i > 0, XF(a^J, p'J,a,T)

= 0.

The integration algorithm has been implemented into the computer code of an assumed-strain layered finite element as described in Basar et al. [7]. The numerical simulation has then been carried out within the finite element program FEMAS [9]. Thereby, material nonlinearity is combined with the modelling of geometrically nonlinear phenomena assuming finite rotations. For tracing of load-displacement curves, Newton-Raphson and Riks-Wempner-Wessels iteration schemes, both enhanced by line search procedures [10], are applied. As an example, a hemispherical shell subjected to axial compression then is analysed. The shell is clamped along the bottom end, and only axial displacements are allowed on the upper boundary. Along the top end, the hemispherical shell is compressed by an increasing line load with the reference value of ^o = 10-0 N / m m . Employing symmetry, one quarter of the shell is discretized by 24 x 24 finite elements. The loading, geometry and finite element mesh are shown in Fig. 1. Computations are carried out at initial temperature of radius: R = 100 mm thickness: h =5 mm load: q =10 N/mm

(9)

cess contributions are neglected. Additionally, the heat flux is also neglected, because only adiabatic processes are considered in the paper. In order to integrate the nonisothermal elastoplastic material model on material point level, a closest point projection scheme is applied where the rates of all measures are replaced by their incremental values. Since the ReissnerMindlin type shell kinematics is employed, the stress and strain measures are described by eight tensor components: a G R ^ y € R^ and /) G R ^ while all deviatoric components of the stress and back stress tensor, 5 G R^ and p' G R^, are explicitly included in the complete context, an advantage of the tensor formulation applied. Due to the initially isotropic material, the proposed integration method yields only one scalar nonlinear equation which has to be solved for the plastic multiplier X using the Newton iteration procedure. After determination of the plastic multiplier, the updated values of stresses as well as all internal variables can be calculated. In order to preserve numerical efficiency of the global iteration strategies, the consistent elastoplastic tangent modulus, derived by linearization of the updated algorithm, is applied. More details connected with the applied computational strategy can be found in Tonkovic et al. [8].

Fig. 1. Geometry and finite element mesh for axially compressed hemispherical shell.

-64

-56 -48 -40 -32 -24 vertical displacement, mm

-16

-8

Fig. 2. Load-axial displacement curve for the top end of hemispherical shell.

480

J. Soric, Z. Tonkovic / First MIT Conference on Computational Fluid and Solid Mechanics employing a realistic material model with highly nonlinear isotropic and kinematic hardening properties has been successfully applied. During the elastoplastic deformation processes of the hemispherical shell, the spread of plastic zones and the increase in temperature are computed. The numerical example demonstrates robustness and numerical stability of the presented computational algorithm.

Acknowledgements

Fig. 3. Deformed configuration and spread of plastic zones throughout shell thickness for the load level at point A. 51.3 C

68.2 C

50.1 C

24.9 C

Fig. 4. Temperature increase along outer shell generatrix for the load level at point A. 25°C, and the increase in temperature is considered under the assumption of an adiabatic process. The load factor versus axial displacement of the upper boundary is plotted in Fig. 2. The deformed configuration and the plastic zones throughout the deformed shell thickness for the load level at point A in the diagram are presented in Fig. 3. By the elastoplastic process, temperature has been increased considerably. The changes of temperature along the outer shell generatrix for the load level at point A in Fig. 2, are portrayed in Fig. 4. Temperature distributions are plotted on the undeformed shell configuration. As expected, the largest increase in temperature is produced in the plastic regions undergoing large plastic deformations.

4. Conclusion An efficient computational strategy for modelling of nonisothermal elastoplastic behavior of shell structures.

The authors would like to express their gratitude to the Volkswagen-Stiftung, German Science Foundation, for generous financial support. This work has also been supported by the Ministry of Science and Technology of the Republic of Croatia.

References [1] Biichter N, Ramm E, Roehl D. Three-dimensional extension of nonlinear shell formulation based on the enhanced assumed strain concept. Int J Numer Methods Eng 1994;37:2551-2568. [2] Wriggers P, Eberlein R, Reese S. A comparison of threedimensional continuum and shell elements for finite plasticity. Int J Solids Struct 1996;33:3309-3326. [3] Lehmann Th. On a generalized constitutive law for finite deformations in thermo-plasticity and thermo-viscoplasticity. In: Desai CS et al. (Ed), Constitutive Laws for Engineering Materials, Theory and Applications. New York: Elsevier Science, 1987, pp. 173-184. [4] Szepan F. Ein elastisch-viskoplastisches Stoffgesetz zur Beschreibung groBer Formanderungen unter Berucksichtigung der thermomechanischen Kopplung. Institute for Mechanics, Ruhr-University Bochum, 1989. [5] Soric J, Montag U, Kratzig WB. An efficient formulation of integration algorithms for elastoplastic shell analyses based on layered finite element approach. Comp Methods Appl Mech Eng 1997;148:315-328. [6] Kratzig WB. Multi-level modeUng techniques for elastoplastic structural responses. In: Owen DRJ et al. (Eds), Computational Plasticity, Part 1. CIMNE, Barcelona, 1997, pp. 457-468. [7] Basar Y, Montag U, Ding Y. On an isoparametric finite element for composite laminates with finite rotations. Comp Mech 1993;12:329-348. [8] Tonkovic Z, Soric J, Kratzig WB. On nonisothermal elastoplasfic analysis of shell-components employing realistic hardening responses. Int J. Solids Struct, in press. [9] Beem H, Konke C, Montag U, Zahlten W. FEMAS 2000-Finite Element Modulus of Arbitrary Structures, Users Manual. Institute for Statics and Dynamics, RuhrUniversity Bochum, 1996. [10] Montag U, Kratzig WB, Soric J. Increasing solution stability for finite-element modelling of elasto-plastic shell response. Adv Eng Software 1999;30:607-619.

481

The successive response surface method appHed to sheet-metal forming Nielen Stander * Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA 94550, USA

Abstract This paper focuses on a successive response surface method for the optimization of problems in nonlinear dynamics. The response surfaces are built using linear mid-range approximations constructed within a region of interest. To assure convergence, the method employs two dynamic parameters to adjust the move limits. These are determined by the proximity of successive optimal points and the degree of oscillation, respectively. An example in sheet-metal tool design is used to demonstrate the robustness of the method. Keywords: Optimization; Response surface methodology; Approximations; Sheet-metal forming; Process optimization; Experimental design

1. Introduction The Response Surface Method has become popular for addressing the 'step-size dilemma' (see e.g. [1], Section 7.2) especially as it may occur in nonlinear dynamic response as calculated by explicit dynamic methods. In this case, the degree of random error is difficult or impossible to determine analytically which complicates the a priori determination of the window size for a finite difference procedure. A 'gradient'-based method which has a better chance of addressing this problem is one which starts by constructing a linear subproblem in a large region of interest while reducing its size after each step. If designs are forced to remain within the bounds of the chosen region, such a method would allow the 'big picture' to be captured initially, while refining and guiding the solution to an improved design. Such a method may, of course, be prone to failure since premature convergence can occur if the region of interest shrinks too rapidly. Trials [2-5] have shown that the method allows significant improvement of the design, although some cases display premature convergence requiring a restart with a large window for further improvement of the design [3]. * Tel: +\ (449) 2500; Fax: +1 (449) 2507; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The method presented here is based on the principles outhned by Roux et al. who focused on response surfaces and experimental design in [6]. As in [6], the current approach uses the D-optimal experimental design based on a 50% over-samphng of the design space. Kok applied linear approximations only and automated the method to reduce user intervention [3]. This method was largely successful, but did not address the possibiHty of oscillation. In the present study, this feature is added to constitute a two-parameter method with the following steps: • The identification of oscillation vs. 'panning' (translation of the region of interest in the design space). The reversal behavior of each variable in the region of interest is used to determine a parameter for reducing the size of the region. • The use of move characteristics (the difference between the trial and optimum design variable in relation to the size of the region of interest). An example in sheet-metal forming is used to demonstrate the method.

2. Successive response surface method: algorithm The successive response surface method uses the region of interest, a subspace of the design space, as a trust region to determine an approximate optimum. A range is chosen

N. Slander / First MIT Conference on Computational Fluid and Solid Mechanics

482

A

A

pan

pan & zoom

zoom

(0)

r[/,0

subregion

range r^

1

r'°'-^

X

•

m. X,

_^(i)

f 1

(0)

^.

rL,0 ^(1) (1) range r^

X<'»=JC">

subreg ,on<'> X2

X2

X2

Fig. 1. Successive response surface methodology. for each variable to determine its initial size. A new region of interest centers on each successive optimum. Progress is made by moving the center of the region of interest as well as reducing its size. Fig. 1 shows the possible adaptation of the subregion. The starting point x^^^ will form the center point of the first region of interest. The lower and upper bounds (x[ ' , x[^'^) of the initial subregion are calculated using the specified initial range value rj^^ so that ri.O^

,0,_

/ = !,...,«

,0,

and

in iteration k as

c!*'=rf'V'-'* where

Axl''=xl''

Jk-\)

(3) The oscillation indicator (purposely omitting indices / and k) is normalized as c where

x;-^-o = x;^^+o.5r;^^

(4)

c = yicjsign(c). (1)

The contraction parameter y is then calculated as

where n is the number of design variables. The modification of the ranges on the variables for the next iteration depends on the oscillatory nature of the solution and the accuracy of the current optimum. 2.1.

(2)

y =

ypan(l + c ) + }/osc(l -

c)

(5)

(Fig. 2, left). The parameter /osc is typically 0.5-0.7 representing shrinkage to dampen oscillation, whereas /pan represents the pure panning case and therefore unity is typically chosen.

Oscillation

A contraction parameter y is firstly determined based on whether the current and previous designs jc^'^^ and jc^^~^^ are on the opposite or the same side of the region of interest. Thus an oscillation indicator c may be determined

2.2. Accuracy The accuracy is estimated using the proximity of the predicted optimum of the current iteration to the starting

Xk pan

'OSC

Fig. 2. Oscillation and proximity criteria.

A^. Stander /First MIT Conference on Computational Fluid and Solid Mechanics

483

Fig. 3. Finite element model of tools and blank. (previous) design. The smaller the distance between the starting and optimum designs, the more rapidly the region of interest will diminish in size. If the solution is on the bound of the region of interest, the optimal point is estimated to be beyond the region. Therefore a new subregion, which is centered on the current point, does not change its size. This is called panning (Fig. 1, left). If the optimum point coincides with the previous one, the subregion is stationary, but reduces its size {zooming) (Fig. 1, center). Both panning and zooming may occur if there is partial movement (Fig. 1, right). The range r/^"^^^ for the new subregion in the (k + l)-th iteration is then determined by: 'i

— ^i

^i

i = 1,

k = 0, ..., niter

(6)

where A/ represents the contraction rate for each design variable. To determine Xi, jf^ is incorporated by scaUng according to a zoom parameter r] which represents pure zooming and the contraction parameter y to yield the contraction rate h = r] + \di\{y - rj)

Fi

n T2

\

n

Fig. 4. Parametrization of cross-section.

(7)

for each variable (see Fig. 2, right).

3. Example: sheet-metal form design A sheet-metal forming problem (Fig. 3) is presented in which the maximum radius of the cross-sectional die geometry has to be minimized. The simulation program used is LS-DYNA, an explicit dynamic analysis code. Fig. 4 shows

the three radius variables ri, r2 and rs, of the die cross-section. The constraints are the forming limit criterion (FLD) [7] (zero is the upper bounding value) and the maximum thinning of 20%. Mesh adaptivity is used during analysis to improve the curvature of the deformed model (shown with a uniform coarse mesh in its undeformed state in Fig, 3). The parameters are /osc = 0.5; }/pan = l.O\ri = 0.5. The history of Fig. 5 shows that the thinning and FLD responses converge in about two iterations. Each solid point

484

A^. Stander / First MIT Conference on Computational Fluid and Solid Mechanics

Q

3 4 5 Iteration Number

3 4 5 Iteration Number Fig. 5. Metal forming: optimization history of responses.

0

1 2

3 4 5 Iteration Number

6

7

Fig. 6. Metal forming: optimization history of variables.

represents a verification run of the predicted optimum for that iteration. Two or three further iterations are required to minimize the maximum of the three radii (Fig. 6). The dotted lines in the figure represent the upper and lower bounds of the region of interest. Oscillatory behavior can be observed for r2 and panning behavior for variable ri.

A violation of the bounds of the region of interest occurs in the first iteration because a feasible design could not be found and therefore the bounds are compromised by the core optimization solver. Fig. 7 shows the baseline and optimal flow limit diagrams with the degree of violation clearly visible for the baseline case.

A^. Stander /First MIT Conference on Computational Fluid and Solid Mechanics FLD-diagram (baseline)

485

size and response noise prevalent in nonlinear dynamic analysis. The method, which employs the move and oscillation properties of the solution, is shown to provide a high degree of accuracy, robustness and efficiency for the optimization of a sheet-metal forming problem. Starting from an infeasible initial design, an optimal design of reasonable engineering accuracy was obtained rapidly. Linear approximations make the approach viable for a large number of design variables at a computational expense similar to methods based on numerical sensitivity analysis.

References Minor Strain FLD-diagram (Optimal)

Minor Strain Fig. 7. Baseline and optimalflowlimit diagrams. 4. Conclusion An adaptive successive response surface method has been devised to circumvent difficulties related to the step-

[1] Haftka RT, Giirdal Z. Elements of Structural Optimization. Kluwer, Dordrecht, 1990. [2] Etman P. Optimization of Multibody Systems using Approximation Concepts. PhD thesis, Technical University Eindhoven, The Netherlands, 1997. [3] Kok S, Stander N. Optimization of a sheet metal forming process using successive multipoint approximations. Structural Optimization 1999; 18(4): 277-295. [4] Stander N, Reichert R, Frank T. Optimization of nonlinear dynamic problems using successive linear approximations. Proceedings of the 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, 2000. [5] Stander N, Burger M. Shape optimization for crashworthiness featuring adaptive mesh topology. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [6] Roux WJ, Stander N, Haftka RT. Response surface approximations for structural optimization. Int J Num Methods Eng 1998;42:517-534. [7] Stander N. LS-OPT User's Manual, Livermore Software Technology Corporation, September, 1999.

486

Hierarchic modeling strategies for the control of the errors of idealization in FEA Bama A. Szabo^*, Ricardo L. Actis^ " Center for Computational Mechanics, Washington University Campus, P.O. Box 1129, St. Louis, MO 63130-4899, USA ^ Engineering Software Research and Development, Inc., St. Louis, MO 63141-7760, USA

Abstract The key challenge in the research and development of software tools for the numerical simulation of physical systems and processes is the improvement of the reliability of information generated by mathematical models. Aspects of reliabihty pertaining to model selection in solid mechanics are discussed. Keywords: Finite element method; Adaptivity; A posteriori error estimation; Hierarchic model; Solid mechanics

1. Introduction There have been three paradigm shifts in finite element analysis (FEA) since its inception in the 1960s. Each was related to the improvement of the reliability in FEA computations. The first two were concerned with the problem of verification, that is, the development of methods for ensuring that the desired information is computed with sufficient accuracy for a particular mathematical model. The third is concerned with the problem of validation, that is, the development of methods for ensuring that the mathematical model chosen for a particular purpose is a sufficiently complete mathematical description of the physical system it is supposed to represent so that correct engineering decisions can be based on it. Following is a brief summary: (1) The development of h-adaptive methods for linear elasticity was completed by the late 1970s. It was proven theoretically and demonstrated through numerical experiments that it is possible to construct sequences of finite element meshes automatically, such that the corresponding finite element solutions converge to the exact solution in (energy norm) at the optimal algebraic rate. See, for example [1]. (2) The development of p- and hp-adaptive methods was completed by the mid-1980s. It was proven and demonstrated that it is possible to achieve exponential rates of convergence when p-extensions are used on properly de* Corresponding author. Tel.: +1 (314) 935-6352; Fax: +1 (314) 935-4014; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

signed meshes. In addition, superconvergent extraction procedures for the computation of the data of interest were developed in this period [2]. The importance of estimating and controlling the errors of discretization in terms of the data of interest in practical engineering appUcations was quickly recognized. (3) In engineering practice, the proper choice of model is even more important than control of the errors of discretization. To provide analysts with means for systematic selection of the proper mathematical model capable of delivering the information needed for arriving at a correct engineering decision, a hierarchic approach to modeling is required. This is analogous to, but much more complicated, than the hierarchic approach to controlling the errors of discretization in p-extensions. The development of hierarchic modeling procedures for the control of the errors of idealization begun in the late 1980s and is now sufficiently mature for practical applications in certain areas [3]. The concepts and techniques involved in the construction of hierarchic models are discussed in the following. The discussion covers: (a) the conceptual framework for controlling errors of idealization; (b) design requirements for software systems capable of controlling both the discretization and idealization errors; and (c) the problem of complexity.

2. Hierarchic models: conceptual basis In attempting to describe a physical system by mathematical methods it is necessary to introduce a set of

B.A. Szabo, R.L. Actis/First MIT Conference on Computational Fluid and Solid Mechanics

simplifying assumptions. There are two reasons for this: first, without the simphfying assumptions, the mathematical model would be very complicated and hence expensive to solve; second, detailed mathematical models typically require input data which are not readily available, or are not known with sufficient accuracy to justify the use of a detailed model. In mechanical and structural engineering practice, for example, the theory of elasticity is typically used as the basis for simplified models, often further simplified by dimensional reduction (plane stress, plane strain, axial symmetry, various plate and shell theories, etc.), briefly discussed in Section 2.1. The generalized form of a mathematical model is: Find UEX ^ E{Q) such that B{UEX. V) = F{v)

for all v e EQ{Q)

(1)

where UEX is the exact solution; Q represents the solution domain; E{Q) is the space of admissible functions, EQ{Q) is the space of test functions; B (respectively, F) is a bivariate (respectively, univariate) form. The principle of virtual work is a well known example of generalized formulations where Eq. (1) is given the physical interpretation that B(UEX, V) is the virtual work of internal stresses and F(v) is the virtual work of external forces. A generalized formulation is completely defined by the forms B, F and the spaces E{Q), EQ{Q). In general, B is nonlinear in its first argument but linear in the second. Each mathematical model has some restrictive assumptions incorporated. For example, models based on the theory of linear elasticity incorporate the assumptions that the strains are small; the deformation is negligible; the stress is proportional to the strain and independent of the magnitude of stress. Furthermore, the tractions on the boundaries are either given, or are proportional to the displacements. The hierarchic concept envisions a systematic relaxation of the restrictive assumptions incorporated in mathematical models. That is, the problem statement is: Find u^^^ e ^^'H-^^'^) such that Bi{ul'^, v) = Fi{v)

for aU v e E^'\Q^'^)

(2)

where the index / represents a model in a hierarchy in the sense that the ith model is a special case of the / -f 1 model. For example, the first model may be based on the assumptions of linear elasticity, the second model may account for inelastic deformation through the deformation theory of plasticity, but retains all other assumptions of the first model, etc. At present, the selection of a mathematical model is largely left to the experience and judgment of analysts. The underlying considerations are based on: (a) the goals of computation (typically design, design certification, interpretation of experiments, forensic analysis); (b) the data of interest (stresses, stress intensity factors, ultimate load, margins of safety, etc.); and (c) the accuracy and reUabiUty

487

of the data that characterize the physical system being modeled (material properties, constraints, loading, etc.). The data of interest are functional (^kiu^Ex)^ ^ ~ 1,2,... ,N that depend on the choice of the model and hence influenced by the restrictive assumptions incorporated in the model. In a properly chosen model, 0k(^^Ex^ would not change significantly if it were replaced by a more comprehensive model. In general, it is necessary to perform an investigation to estimate the effects of the restrictive assumptions on 0k(u^^\). This usually involves the computation of a reasonably close approximation to ^yt(^^x )• In many engineering apphcations, the objective is to find bounding values for the data of interest. The computation of bounds is usually motivated by uncertainties in the data. 2.1. Dimensionally reduced models Dimensionally reduced models are widely used in numerical simulation. A dimensionally reduced model is a semi-discretization that can be viewed as a particular member of a hierarchic system of models: ^{x, y,z) = Yl ^k(x,

y)fk(z)

(3)

where x, y, and z are the independent space variables; Wk(x,y) are the dimensionally reduced unknown functions and fk(z) are fixed functions, called director functions. As the model index / increases, the solutions w^^ converge to the solution of the fully three-dimensional model. The problem of hierarchic model definition for plate and shell structures is an important and complex problem which cannot be addressed here in detail. We refer to [4] and the references cited therein. For dimensionally reduced models in linear elasticity the model index / has a simple and natural meaning. In the general case geometric, material and other nonlinearities must be considered.

3. Implementation To provide analysts with a means for choosing a mathematical model capable of delivering information needed for arriving at correct engineering decisions, it is necessary to create a software infrastructure capable of supporting model-adaptive processes. As noted in Section 2, model / is completely characterized by the forms Bi, Ft, and the trial and test spaces E^\Q^^), E^\Q^^). The bivariate form has two key components; the differential operator matrix Di and the matrix of material properties Mi: Bi(u,v)=

f(DiufMi(DiV)d^

(4)

In the early papers on FEA, it was customary to combine model definition, discretization and element topology. This

488

B.A. Szabo, R.L. Actis/First MIT Conference on Computational Fluid and Solid Mechanics

led to the implementation of a great variety of elements that can be found in the element libraries of conventional FEA software products. Unfortunately, such element libraries make systematic control of the errors of idealization impractical. It is necessary to create an FEA software infrastructure that differs from conventional ones in that the differential operators /),, the topological description of elements and the element-level basis functions are treated separately. The software infrastructure of StressCheck' was designed with the objective of supporting hierarchic modeling procedures in mind.

the requisite expertise in FEA, it is necessary to introduce safeguards against the occurrence of those errors. 4.2. Conceptual errors and errors of interpretation Conceptual errors occur when data that are inconsistent with the formulation are used. An example of inconsistent data is the use of point constraints in linear elasticity for purposes other than rigid body constraints subject to the requirement that the external loads are in equilibrium. An example of errors of interpretation is when the functional 0kif^%) is infinity, yet the corresponding 0kiu%) is reported, such as the maximum stress or strain in linear elasticity at sharp re-entrant comers.

4. Control of errors: the problem of complexity 4.3. Complexity Errors in practical applications of FEA technology can be grouped into two categories: controllable (benign) errors and conceptual (malignant) errors. A brief discussion follows. 4.1. Controllable errors: the errors of discretization and idealization There are two types of controllable errors: the errors of discretization and the errors of idealization. The errors of discretization are controlled by the finite element mesh (h-refinement); the polynomial degree of elements (p-refinement); a combination of the two (hp-refinement); space enrichment techniques and, in some special cases, mapping procedures. The objective is to ensure that

mK\c) - ^k{u%)\ < T miu%)\

k = \,2,.

(5)

where up^ is the finite element approximation to u^^\ and r is a tolerance value. The errors of idealization are controlled by proper selection of the mathematical model, as discussed in Section 2. The conventional (h-version) FEA software products employed in professional practice today do not provide means for systematic and reliable control of the errors of idealization or discretization. Their proper use requires a high degree of expertise and a considerable amount of time. Various limitations often force analysts to forgo checking for errors of idealization and discretization altogether. Consequently, erroneous and misleading results are produced. A well-documented example of the dangers associated with basing engineering decisions on erroneous information is the sinking of the Sleipner A offshore platform in 1991 ^. The economic loss was estimated to be 700 million dollars. Undoubtedly, there are many undocumented instances of substantial economic loss attributable to errors in applications of FEA technology. For persons who do not possess ^ StressCheck is a trademark of Engineering Software Research and Development, Inc., St. Louis, MO, USA.

It is possible to conceive of a 'general purpose' FEA product that would incorporate procedures for systematic control of the errors of idealization and the errors of discretization for a large and diverse field, such as solid mechanics. Such a software product would be highly complicated, however, requiring an extraordinarily large investment and highly qualified expert operators. The complexity of the software would impose severe limitations on its usefulness. In order to reduce complexity to a manageable size, we must consider the alternative approach of producing FEA software tools for specific problem classes corresponding to the various engineering sub-disciplines. Recognizing the fact that a very large percentage of mechanical and structural design is routine design, i.e. the goal is to modify an existing part so as to satisfy some new criterion or requirement in an optimal way, it is possible to produce efficient and highly reliable software tools for large and important classes of problems. Modifications typically involve changes in dimensions and/or material properties and usually several load cases are of interest. This kind of problem can be treated very efficiently through parametric finite element models. Parameterized models can be created for each problem type and the finite element mesh can be designed so that the integrity of the mesh is preserved when the parameters are varied within pre-specified ranges. In this way, the user is freed from the burden of having to design meshes or to check the adequacy of the meshes. Automated error indicators can be provided. Parametric analyses can be performed to obtain design curves. Families of parametric problems can be developed and organized in an FEA-based handbook framework. ^ http://www.math.psu.edu/dna/disasters/sleipner.html. The sinking was caused by a failure in a cell wall: "The wall failed as a result of a combination of a serious error in the finite element analysis and insufficient anchorage of the reinforcement in a critical zone."

BA. Szabo, R.L. Actis/First MIT Conference on Computational Fluid and Solid Mechanics 5. FEA-based engineering handbooks Engineering handbooks have served to accumulate, preserve and distribute expert knowledge. In the present computational environment, there are new opportunities for achieving the same objectives on a much larger scale and with greater reliability. The means for obtaining solutions for parameterized models have been greatly enlarged by the availability of reliable numerical solution methods and powerful computers. The extension and adaptation of handbooks to the present computational environment offers new opportunities in the exploitation of the continuously increasing power of digital computers. The handbook framework of a finite element analysis program needs to provide the following capabilities: (1) Parametric definition of the topology, material properties, loading and constraints. Means for enforcing relational restrictions among the parameters. (2) Associativity between the topological data and the finite element mesh: when a parameter is changed then the mesh is updated automatically. (3) Error estimation and means of error control in terms of the engineering data of interest. (4) Hierarchic modeling capabilities: it must be possible to examine whether the data of interest are sensitive to the choice of modeling decisions. (5) Flexible post-processing: the user should be able to access in graphical or tabular form any engineering information once the solution has been obtained. It should be convenient to examine the effects of various load combinations in post-processing operations.

6. Closing remarks In order to realize the full potential of computers in engineering decision-making processes and take advantage

489

of research accomplishments in the past 15 years, it will be necessary to extend the notion of adaptivity to include model selection, in addition to the control of the approximation error. Importantly, the implementation must allow users who do not have expertise in analysis to be able to extract the data of interest with a high degree of reliability and with estimated error bounds. These requirements are strong motivators for thoroughly re-thinking the design of simulation software products taking into consideration the new paradigms summarized in Section 1 and the supporting research accomphshments.

Acknowledgements Support of this work by the U.S. Air Force Office of Scientific Research through Grant F49620-98-1-0408 is gratefully acknowledged. References [1] Babuska I, Rheinboldt WC. Adaptive approaches and reliability estimates in finite element analysis. Comput Methods Appl Mech Engrg 1979;(17/18):519-540. [2] Szabo BA, Babuska I. Finite Element Analysis. New York: John Wiley and Sons, 1991. [3] Szabo BA. Quality assurance in the numerical simulation of mechanical systems. In: Topping BHW (Ed), Computational Mechanics for the Twenty-First Century. Edinburgh: Saxe-Coburg, 2000, pp. 51-69. [4] Actis RL, Szabo BA, Schwab C. Hierarchic models for laminated plates and shells. Comput Methods Appl Mech Engrg 1999;(172):79-107.

490

Techniques to ensure convergence of the closest point projection method in pressure dependent elasto-plasticity models Benabdellah Tahar, Roger S. Crouch * Department of Civil and Structural Engineering, University of Sheffield, Sheffield SI 3JD, UK

Abstract This paper considers certain aspects of elasto-plasticity models for concrete. It starts by identifying some difficulties associated with the lack of C1/C2 continuity and hence the unwanted presence of vertices in some existing yield surfaces. A form of smoothing function is offered to avoid these difficulties. Two techniques are added to an implicit BackwardEuler scheme to integrate the rate expressions. The modified Closest Point Projection algorithm with automatic scaling and sub-incrementation is shown to enable convergence to be met for all trial states examined. Keywords: Stress return algorithm; Closest point projection; Plasticity; Constitutive model; Concrete

1. Introduction The development of a unified elasto-plasticity constitutive model for concrete owes much to an original paper by Willam and Wamke in 1974 [1]. Here the basic shape of the strength envelope was defined by single polynomial expression in the meridional planes and an ellipse in the deviatoric planes. Since that time a number of modifications to extend this model have been made (for example [2-7]). Collectively, these features offer a neat, transparent (albeit phenomenological) approach to the continuum modelling of concrete, in both the brittle (near tensile states) and ductile (high confinement) regimes. In the extensions, effort has concentrated either on increasing the simulation capabilities of the model or on providing a stable computational procedure for efficient inclusion in a general-purpose finite-element code. All the various versions of the model share the common approach of linking some scalar measure of plastic strain during softening to the material fracture energy. Furthermore, all models referred to above have been constructed as isotropic. A number operate with a non-associated flow rule, rendering the elasto-plastic tangent matrix non-symmetric. Despite these modifications, it appears that it is only the recent * Corresponding author. Tel.: +44 (1142) 225716; Fax: +U (1142) 225700; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

models of Tahar [8] and Kang [7] which have overcome the difficulties associated with non-smooth yield surfaces in the compression region. When considering the earlier models, the following observations may be made. (1) There is a lack of continuity in the yield surfaces at the intersection points on the hydrostatic axis both in the tension and compression regions. (2) There is no continuity in the derivative of the yield surface at the transition point (defining the level of confinement beyond which no softening takes place). This lack of continuity implies non-unique surface normals at this location. (3) It is difficult to control the geometric form of the yield surface functions. In particular, existing formulations cannot provide a close bunching-together of yield surfaces near the peak strength envelope in the tension while allowing them to be spread further apart in the compression zone. In the following three sections, a C2 continuous model overcoming all these difficulties is presented. In Section 5 emphasis is placed on creating a stable stress return strategy to handle highly curved regions of the yield surface and zones where the yield function is not defined. Note that this formulation [8] differs in a number of ways from the approach adopted by Kang [7].

491

B. Tahar, R.S. Crouch/First MIT Conference on Computational Fluid and Solid Mechanics 2. Some existing yield surface functions for concrete

Table 1 PNS surface constants for various criteria

The peak nominal stress (PNS) achievable by an isotropic concrete specimen under multi-axial loading may be expressed in terms of the three principal stress components. Fig. la shows the typical shape of the PNS envelope. It is open-ended in the hydrostatic compression zone but closed in the hydrostatic tension zone. Note that in this example, discontinuities appear in the gradient of the surface at the compression and extension meridians. The PNS envelope may also be defined in terms of the normalised (cylindrical co-ordinates) stress invariants. 7=

fc^

P

=

0

Huber-von Mises

0

Leon

where fc is the uniaxial compressive strength, a/j is the Cauchy stress tensor and stj are the deviatoric stress components (Sij = Oij — ^8ij). 8ij is the Kronecker delta and the summation index is implied for repeated subscripts. The Lode angle, 0, equals — f, 0 and + | on the extension, shear and compression meridians respectively. Many concrete PNS criteria can be expressed in a common form, encompassing the Rankine, Mohr-Coulomb, Huber-von Mises, Leon, Drucker-Prager, Hoek-Brown, Etse-Willam, Menetry-Willam and the Bicanic-Pearce criteria. For example, we could write F = ao {prf + Qfipr + y^op + y^if - 1 = 0

(2)

where r is a deviatoric shape function, UQ, ai, PQ and P\ are material constants as defined in Table 1 by Tahar and Crouch [9]. All the pressure-dependent criteria listed above intersect the hydrostatic axis in the tension region exhibiting a vertex at this point. In the contest of an elasto-plasticity model, the surface normal at this point will be non-unique. Furthermore, the surface will be undefined for tensile zones

ftVe

Vi

3

0

Willam-Warnke

0

Hoek-Brown

3 2

Etse-Willam

3 2

Menetry-Willam

3 2

Bicanic-Pearce

17^

2 ft + I

ft-I

ftVe

2^

(1) 3/2

1

0

Mohr-Coulomb

Drucker-Prager

fc (^ijSji)

Rankine

bo

Po

Oil

QfO

fjft ^'

+^ ft 1

ftVe i-f? ftV6 ^'

ft(l+e)

h

Pi 1

0

IT!

0

1-/.

0

0

0 0

ftV3

2(1 - / , )

ftV3 V3(l - ft) 2ft

b2

bi_

3bo

V3bo

0 0 0

ftV3 ftV3 eV3{l - ff)

0

ft(l+e) 1

bo

outside this region. Both cases lead to problems in the stress return algorithm. 3. Smoothing functions: Meridian plane To remove the discontinuity at the intersection points on the hydrostatic axis, a modification to the PNS expressions is proposed. In all that follows, the Hoek-Brown criterion has been adopted as the basic criterion to illustrate

Fig. 1. Typical yield surfaces (a) PNS envelope (b) cross-section through hardening and softening yield surfaces.

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B. Tahar, R.S. Crouch/First MIT Conference on Computational Fluid and Solid Mechanics

the process as it offers a good balance between physical relevance, ease of calibration and accuracy. However, it should be pointed out that the same procedures could be applied to many of the other PNS criteria. Inserting the appropriate values from Table 1 into (2) and considering the compression meridian we obtain V6

(3)

V3

where mHB = cii/y/6 Re-arranging (3) one obtains

One may ensure that the meridian always intersects the hydrostatic axis normally in the tension quadrant by defining (5) where typically 0.9 < y < 1. The corresponding generalised hardening and softening surface may be written mky

V3

-

-2

(6)

where

:fe<''M

-(I)

In order to describe the shape of the PNS surface for any Lode angle 0, the Bhowmik-Long [10] deviatoric shape function r is used to define smooth convex traces which ensure a stable material behaviour according to Drucker's stability postulate, having unique gradients to define the direction of the plastic strain rate. Unlike the original elliptic function of Willam and Wamke [1], this formulation introduces an addition control point on the shear meridian.

(4)

^^ = IV I f ~^HB + V'^HB^ - 12\/3mHB? + 36 j

^=(yi^O'^?(^^^0'

4. Smoothing functions: deviatoric plane

(7)

and c is a softening variable (Fig. lb), k defines a measure of plastic strain and |/, identifies the position on the compressive hydrostatic axis where the yield surface forms an intersection. This ensures Cj continuity for all values of | from l/j to I = ftt (ftt gives the tensile hydrostatic intersection point). Note that in this model, softening is achieved by translating the yield surface along the hydrostatic axis away from the tensile region.

5. Stress return algorithm aspects This section provides the strategy used to integrate the rate expressions for the new concrete model. A modified Closest Point Projection (CPP) algorithm [11] with automatic (bisector) scaling and sub-incrementation is used. This approach overcomes the problems associated with accidentally jumping between different principal stress sectors when returning the stress (for example, see [6]) and handles regions of high curvature plus zones where the yield surface is not defined. The four-stage algorithm is presented in Boxes 1-4. Note that zone II refers to the region beyond (that is, more tensile than) the hydrostatic cut-off in the tension region (Fig. lb). Tensile stress states are treated as positively valued in all that follows. A full description of the method is given by Tahar [8]. Fig. 2 shows typical contour plots of the number of iterations needed to satisfy the tight tolerances imposed by the consistency condition. Two different deviatoric planes are considered; ^ = /^ in the upper two plots (a and b) and ^ = — l.lf/i in the lower two plots (c and d). In each case, 6400 trials stress states are defined over a square grid extending up to 4fc from the hydrostatic axis.

Step 1. Compute the trial stresses

r+^(jM = ro-} + [D]r+»A6} Step 2. Store the current variables Step 3. Determine the hydrostatic components | and ftt If I > / r go to Box 4 Step 4. Check the current yield condition "+^F({"+^cr}, "k) If {"+^F < tot) then [a] = {"+^a'} and EXIT If not then go to Box 2 Step 5. If convergence has not been met within a predefined number of iterations then go to Box 3 Box 1. Master stress return algorithm.

B. Tahar, R.S. Crouch/First MIT Conference on Computational Fluid and Solid Mechanics

493

Step la. Initialise local iteration counter i = 0 and bisector scaling ratio y = I Initialise the current variables

Step 2a. Establish the ordering of the principal stresses Step 3a. Check the current yield condition and evaluate the residuals n+lfH)

^ n+l^(0/|n+l^(/)|^n+l

j^m

If n+1 f(i) < tol and {"+1R^'^} < tol, then EXIT If (/ > 100), then EXIT and indicate that an error has occurred Step 4a. Calculate the change in the plastic multiplier and evaluate the change in stress

n+l^^ii)

^ gn+l pii)

H

—^^— Step 5a. Store the current variables r(stored)^ _ | « + 1 ^ 0 ) |

^^(stored) _ n+l ^^(i)

Step 6a. Update the current variables

Step 7a. Check the ordering of the principal stresses If the ordering is preserved, then update the internal variables n+lj^ii+l) ^ n+lj^(i) _^ n+l^j^ii)

^^^ get y = 1

If the ordering is not preserved, then determine y by using the bisector scaling scheme and set the current variables equal to the stored ones Step 8a. Increment / = ^-f 1 and go to Step 3a Box 2. Bisector-scaled CPP method. Two different curvatures of the yield surface are examined; a = 0.6 in the left-most plots (a and c) and a = 0.1 in the right-most plots (b and d). The lower value of a represents a very severe test for the algorithm as it describes a highly curved surface close to the compression meridian. The red zones indicate regions where the stress returns with very few iterations, the yellow indicates a moderate number of iterations and the dark green zone show where a large number of iterations are needed. One important observation is that convergence is obtained in all cases (Tahar [8] shows how the standard CPP method fails to achieve this). The deviatoric plane beyond the tensile-cut off represents the most severe test (with a = 0.1) as can be seen from the contour plot. During the simulations it has been observed

that trial stress points located outside the yield surface are strongly attracted to the regions of high curvature. It is well established that such regions cause great difficulty in standard return algorithms. Therefore special attention (in the form of the automatic sub-incrementation scheme Box 3) has been paid to ensuring a proper convergence in these regions.

References [1] Willam KJ, Warnke EP Constitutive Model for the Triaxial Behaviour of Concrete. lABSE Seminar on Concrete Structures Subjected to Triaxial Stresses III-l, 1974.

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B. Tahar, R.S. Crouch / First MIT Conference on Computational Fluid and Solid Mechanics 4T-

Fig. 2. Iso-iteration maps for the modified CPP algorithm using the hardening/softening plasticity model with different yield surface curvatures, in different deviatoric planes: (a) a — 0.6 and ^ = ftt; (b) a = 0.1 and | = ///-, (c) a = 0.6 and f = -l.l^h', (d) a = 0.1

andf = -l.lf/,.

Step lb. Initialise / = 0, y = 1, /? = 1 and m =2 Define the strain increment {A6'"^} = {"^'Ae} and also define the residual strain increment as {A^''} = {"^^Af} Define the previously converged stress state as {"+V^} = {"a} Step 2b. Apply the CPP plus bisector scaling to the incremental strain jAf'"^} acting on{"+V'} (go to Box 2) If convergence has not been met within a predefined number of iterations then go to Step 4b Step 3b. Increment / = / + 1 Define the newly converged stress state as {"'^^a'} Determine the remaining incremental strain as {A€''} =

{A€''}-{A€'"'}

and update j , j = p - i then go to Step 5b Step 4b. Divide {A6'') by (m x j) to create the new increment {Ae" Set i = 0, p = (m X j), j = j -\- \ and go to Step 2b Step 5b. If j is greater than zero, then return to Step 2b Otherwise, EXIT to the next strain increment Box 3. Sub-incrementation technique.

(m X j)

B. Tahar, R.S. Crouch /First MIT Conference on Computational Fluid and Solid Mechanics

495

Step Ic. Compute the trial stresses. (2) where r + ^ A a } = [£)]{"+! Ac} Step 2c. Initiahse / = 0 and determine the hydrostatic components of |A, ftt (using ftt = — ) , I B , and (IB - ftt). Set i = i + l m Step 3c. If (^B —ftt) > tol then the trial stresses are considered in zone II. If this is the case, initialise rj = 0 and 5 = 0 Step 4c. Determine {ft — |/) where 1/ corresponds to the previous return stress state Step 5c. If (1^ - IA) < tol, then EXIT from the routine If it is not, then Srii = r^^—J—.

(h-U)

Here, r = 0.95

If (I - 7]) < 8r}i then 8r]i = (I - yj) and 5 = 1

If Sr]i < tol, then set 5 = 2 and EXIT Step 6c. Find the new trial stress state

r+Vc} = r^a^}+8^i (r+^a^ - r+1^^}) Step 7c. Up-date the scaHng factor r) = r] -\-8r]i and call the CPP plus bisector scaling routine, that is go to Box 2 Step 8c. If 5 = 0 then update i = i -\- I and return to Step 4c If 5 = 1, then EXIT from this routine. Box 4. Algorithm to return trial stresses from zone II.

[2] Smith S. On Fundamental Aspects of Concrete Behaviour. MSc Thesis, University of Colorado, 1987. [3] Pramono E, Willam KP. Fracture energy based plasticity formulation for plain concrete. J Eng Mech 1989; 106(9): 1013-1203. [4] Etse G, Willam KJ. Fracture energy formulation for inelastic behavior of plain concrete. J Eng Mech 1994;120(9):1983-2011. [5] Menetrey P. Numerical Analysis of Punching Failure in Reinforced Concrete Structures. PhD Thesis, Ecole Polytechnique Federale de Laussane, 1994. [6] Pearce CJ. Computational Plasticity in Concrete Failure Mechanics. PhD Thesis, University of Wales, 1996. [7] Kang HD. Triaxial Constitutive Model for Plain and Reinforced Concrete Behaviour. PhD Thesis, University of Colorado, 1997.

[8] Tahar B. C2 Continuous Hardening/Softening Elasto-Plasticity Model for Concrete. PhD Thesis, Department of Civil and Structural Engineering, University of Sheffield, UK, 2000. [9] Tahar B, Crouch RS. Hardening/softening in plasticity models for concrete. Complas V, Barcelona, pp. 15741580, 1997. [10] Bhowmik SK, Long JH. A general formulation for the cross-sections of yield surfaces in octahedral planes. In: Pande and Middleton (Eds), NUMENTA 90. Amsterdam: Elsevier, 1990, pp. 795-803. [11] Simo JC, Hughes TJR. Computational Inelasticity. Berlin: Springer.

496

Molecular dynamics calculation of 2 billion atoms on massively parallel processors Akiyuki Takahashi *, Genki Yagawa University of Tokyo, Department of Quantum Engineering and Systems Science, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Abstract The molecular dynamics method is one of the powerful methods to analyze the nonlinear problem near a crack tip, in the sense that the calculation is conducted from the microscopic view point. This paper describes a molecular dynamics method to analyze the crack problem which consists of billions of atoms. In order to employ a massively parallel computer, the whole domain is decomposed into subdomains and the parallel performance is evaluated with example cases. Keywords: Molecular dynamics method; Spatial decomposition; Large scale analysis; Parallel computing

1. Introduction In recent years, molecular dynamics methods have been applied to some analyses of crack initiation and propagation. Due to the recent developments in computer technology, especially in massively parallel processors (MPPs) and PC clusters, the number of atoms which can be calculated has become very large [1], but it is not yet sufficient. In this study, a molecular dynamics method to analyze the crack problem has been developed. In order to employ a massively parallel computer, the whole domain is decomposed into subdomains. Its parallel performance on MPPs is evaluated and the calculation of 2 billion atoms is finally shown.

2. Molecular dynamics method and its parallelization With the current molecular dynamics method, a-iron is employed as the material. Many accurate potentials for a-iron have been proposed by several authors [2]. In order to calculate a large-scale atomic system, the potential must be in a simple form. The Johnson potential [3] is then employed. The potential is a simple pair potential as

* Corresponding author. Tel/Fax: +S\ (3) 5841-6994; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

follows: O'^

Co(r'^-Ci) + C2r'^+C3

(1)

where, <J>'^ is the potential which works between / and j atom, r'^ is the distance between / and j atom, Q , Ci, C2, C3 are constants which depend on r'J. As a time integral method, the velocity Verlet method [4], which is very stable and gives accurate solution, is used. The molecular dynamics method is parallelized by spatial decomposition method using message passing interface (MPI). When a simple pair potential, for example the Johnson potential, is used, the communication is only carried out once in each time step. Therefore, the communication time becomes very small.

3. Benchmark of parallel molecular dynamics calculation on MPPs Parallel performance of the parallel molecular dynamics method is evaluated on MPPs, HITACHI SR2201. The analysis model is a plate with an edge crack and consists of 1,153,545 atoms. The thickness of model is only five lattices, and periodic boundary condition is applied to the thickness direction to assume plain strain condition. Fig. 1 shows the speedup by this parallelization method. It can clearly be observed that the speedup line is close to the ideal speedup line, and the curvature is nearly linear.

A. Takahashi, G. Yagawa /First MIT Conference on Computational Fluid and Solid Mechanics

497

Fig. 2 shows the result of the calculation. In the figure, the crack propagates in a straight line at first, and the created crack surface is very flat. After that, the crack propagates in two directions and the created surface is very rough, like a dimple fracture.

5. Molecular dynamics calculation of 2 billion atoms

30 40 Number of PEs

Fig. 1. Speedup (1,153,545 atoms). Then, it can be estimated that good scalability is obtained by this parallelization method.

4. Numerical example of molecular dynamics calculation

Large scale molecular dynamics calculations are conducted on MPPs HITACHI SR2201 with 1024 PEs. The maximum number of atoms is 2,097,034,205. The size of the analysis domain of 2 billion atoms is about 4.14 x 4.14 |xm, and the calculation takes about 187 s for each time step. Fig. 3 shows the time required for each time step. It can be clearly observed that as the number of atoms is increased, the required time is increased linearly. In addition. Fig. 4 shows the communication time via the computation time. In the figure, as the number of atoms is increased, the compared time is largely decreased. Therefore, it can be considered that high parallel efficiency is obtained in these large scale molecular dynamics calculations.

In this section, a simple numerical example of molecular dynamics analysis of the crack problem is shown. The atomic model is a plate with an edge crack, and consists of 2,050,065 atoms. The size of model is about 0.129 x 0.129 |xm and 1.43 nm thickness, and the crack length is about 58.4 nm. The model is divided into 4 x 4 subdomains and 16 PEs are used. In this analysis, Kj is set to 8.114 M P a v ^ , and temperature of the system is controlled at 300 K by the velocity scaling method.

0.8 1 1.2 1.4 Number of Atoms [G]

Fig. 3. Computation time for each timestep.

0.8 1 1.2 1.4 Number of Atoms [G]

Fig. 2. Crack propagation.

1.6

Fig. 4. Communication time via computation time.

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A. Takahashi, G. Yagawa / First MIT Conference on Computational Fluid and Solid Mechanics

6. Concluding remarks

References

A molecular dynamics method for large scale analyses of cracks has been developed. It was parallelized by the spatial decomposition method. A high parallel performance was obtained on MPPs. In the numerical example, fracture mechanism, like a brittle-ductile transition could be observed. Using the parallel molecular dynamics method, 2 billion atoms could be calculated on MPPs. The calculation of each time step took about 187 s and a high parallel efficiency was obtained.

[1] Abraham FF, Schneider D, Land B, Lifka D, Skovira J, Gemer J, Rosenkrantz M, Instability dynamics in threedimensional fracture: an atomistic simulation. J Mech Phys Solids 1997;45:1461-1471. [2] Finnis MW, Sinclair JE, A simple empirical N-body potential for transition metals. Phil Mag A, 1984;50:45-55. [3] Johnson RA, Interstitials and vacancies in a Iron. Phys Rev A, 1964;134:1329-1336. [4] Verlet L, Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys Rev Lett 1967;55:98-103.

499

Impact stresses in A-Jacks concrete armor units J.W. Tedesco'^'*, D. Bloomquist^ T.E. Latta^ ^ University of Florida, Department of Civil an Coastal Engineering, Gainesville, FL 32611, USA ^ Structural Affiliates International, Inc., Nashville, TN 37212, USA

Abstract Concrete armor units play a key role in providing stable protection for shorelines, groins, breakwaters and jetties in hostile wave environments. Because of its large hydrodynamic stability coefficient, the A-Jacks armor unit has emerged as one of the most popular armor unit designs. However, due to its long and slender appendages, it is susceptible to structural failure. The structural stabihty of A-Jacks armor units subject to placement-induced impact stresses is investigated through a series of finite element method FEM analyses. Keywords: Concrete; Armor unit; Impact; Breakwater

1. Introduction Providing stable protection for shorelines, groins, jetties, and breakwaters from wave-induced forces is a major area of concern in ocean engineering. Large stones have historically been employed to provide this protection as armor. Over the ensuing centuries it was observed that randomly placed stone dissipated more wave energy than placed stone blocks. Unfortunately, the randomly placed stones were more susceptible to displacement due to wave loads. Thus, larger stones were required. As the development of rubble structures extended into increasingly more hostile wave environments, the weight of the hydraulically stable stone became unreasonable, both in terms of availability and handling. As an alternative, a variety of concrete armor units have been developed (Fig. 1). The primary design consideration for these units has been to obtain high porosity and interlock among units to increase hydrodynamic stability. This approach has resulted in the use of complex geometries for armor units which have very high stability coefficients. Although these units exhibit excellent hydraulic stability, it is often at the expense of structural integrity. The A-Jacks has received considerable attention because it has a very high stability coefficient, but has also been susceptible to structural failure. It's long, slender appendages may fracture and the high degree of interlock is lost. * Corresponding author. Tel.: +1 (352) 392-9537; Fax: +1 (352) 392-3394; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(a) Accropode

(b) HARO

(c) Core-loc

(d) A-Jacks

Fig. 1. Common armor unit shapes.

Concrete armor units are essentially subjected to three types of loading conditions [1-4]: (1) static loads; (2) hydrodynamic loads; and (3) impact loads. The static loads are those associated with the unit's self-weight and the weight of other units in the structure which may bear upon it. The hydrodynamic loads are those imposed upon

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J.W. Tedesco et al. /First MIT Conference on Computational Fluid and Solid Mechanics

(a)

i I

-Strain Gages

(b)

Fig. 3. Impact conditions for drop test FEM simulation.

Hinge Support

Fig. 2. Schematic of A-Jacks impact simulations.

the unit directly from wave action on the structure. The impact loads result from the collision of units due either to wave-induced rolling and rocking motions, or from unit placement on the structure during construction. Only the placement-induced impact loads are considered in this study. Two types of impact conditions were investigated in the impact analysis: (1) the drop test simulation; and (2) the rollover test simulation. A schematic representation of the drop test is illustrated in Fig. 2a. The drop test simulation is representative of a drop condition that may occur during initial placement of the unit on the breakwater. The rollover test simulation is also representative of a condition that may be encountered during initial placement of the unit on the breakwater. A schematic of the rollover test simulation is presented in Fig. 2b.

2. FEM simulations The dynamic response of A-Jacks to specified drop heights in the impact test simulations was effected through

Fig. 4. Impact conditions for rollover test FEM simulation. a comprehensive series of finite element method (FEM) analyses. [5] Figs. 3 and 4 illustrate the orientation and impact conditions for the FEM simulations of the drop test and rollover test, respectively. The simulations were

J.W. Tedesco et al. /First MIT Conference on Computational Fluid and Solid Mechanics initiated by specification of initial nodal point velocities in the FEM model corresponding to the instant of impact. For the drop test, these initial nodal point velocities are linear (or vertical), and for the rollover test they are angular (or rotational). In all impact simulations, the target body was modeled as an elastic structure with a contact surface. The use of an elastic structure for the target body allows for a portion of the impact energy to be dissipated in the deformation of the elastic structure. Both a rigid and a semirigid condition for the target body were investigated. The rigid case is representative of A-Jacks impact with a concrete surface, such as a cap on a breakwater structure, or unit-tounit impacts. The semirigid impact case, in which the modulus of elasticity of the target body (elastic structure) is reduced to 50% of that for the rigid case, is representative of unit impact on a rubble structure, or surface conditions that might be encountered during transportation of the units.

501

6.0

5.0 fc=6000 psi •=. 4.0 fc=5000 ps

a

3.0

I"

c=3000 psi fc=4000 psi

1.0

0.0 ^ 24

48

72

96

120

144

Unit Length (in)

3. Results and conclusions

Fig. 6. Critical rollover height versus unit length. The results of the analyses are presented in the form of design nomographs to aid the engineer during the construction phase of the structure. Typical nomographs are presented in Figs. 5 and 6. The critical drop height, h, as a function of unit length, L, and concrete compressive

strength, is presented in Fig. 5 for the rigid base condition for the vertical drop simulation. The critical drop height increases with increasing unit length and concrete compressive strength. It is also noted that as the unit length increases, the rate of increase of the drop height (i.e. slope of the curve) increases. The critical rollover drop height as a function of unit length and concrete compressive strength is illustrated in Fig. 6 for the rigid base condition. As expected, the critical height increases with increasing unit length and concrete compressive strength. It is noted that as the unit length increases, the rate of increase of the drop height (i.e. slope of the curve) decreases.

References

72

96

120

Unit Length. L (in)

Fig. 5. Critical drop height versus unit length.

[1] Tedesco JW, McDougal WG. Nonlinear dynamic analysis of concrete armor units. Comput Struct 1985;21(2):189-201. [2] Tedesco JW, Rosson BT, Melby JA. Static Stresses in Dolos Concrete Armor Units. Comput Struct 1991;45(4):733-743. [3] Tedesco JW, McDougal WG, Melby JA, McGill PB, Dynamic response of Dolos armor units. Comput Struct 1987;26(l):67-77. [4] Tedesco JW, McGill PB, McDougal WG. Dynamic Response of prestressed concrete armor units to pulsating loads. Ocean Eng 1991;18(3):175-189. [5] ADINA, Automatic Dynamic Incremental NonLinear Analysis. System 7.2 Release Notes 1998. ADINA R&D, Watertown, MA.

502

A stabilized MITC finite element for accurate wave response in Reissner-Mindlin plates Lonny L. Thompson *, Sri Ramkumar Thangavelu Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA

Abstract Residual based finite element methods are developed for accurate time-harmonic wave response of the Reissner-Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variational form for the finite element approximation. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used to develop a new mixed least-squares (MLS4) 4-node quadrilateral plate element with improved wave accuracy. Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle, match both propagating and evanescent analytical wave numbers for Reissner-Mindlin plates. Numerical results demonstrates the significantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4 element. Keywords: Finite element method; Reissner-Mindlin plate; Mixed interpolation

1. Introduction When modeling the time-harmonic response of elastic structures, accurate plate and shell elements are needed to resolve both propagating and evanescent waves over a wide range of frequencies and scales. The propagating waves are characterized by sinusoidal components with phase speed determined by the material properties and thickness of the plate, while the evanescent waves are characterized by exponential decay with effects localized near drivers and discontinuities, e.g. near boundary layers. The accuracy improvement for intermediate to high frequencies plays an important role in modeling control-structure interactions, dynamic localizations, acoustic fluid-structure interaction, scattering from inhomogeneities, and other applications requiring precise modeling of dynamic characteristics. The numerical solution of the Reissner-Mindlin plate model for static analysis has been discussed by many authors. The primary focus has been various remedies to the well-known shear locking problem for very thin plates. Of the low order elements, the popular bilinear MITC4 ele* Corresponding author. Tel.: +1 (864) 656-5631; Fax: +1 (864) 656-4435; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ment [1] based on mixed interpolation of shear strains is one of the most attractive. The error analysis [2,3] performed on this element showed that it is optimally convergent for deflections and rotations on regular meshes. However, for the 4-node quadrilateral MITC4 element, it is not clear what is the optimal definition of the loading and mass which is consistent with the assumed strain field for dynamic analysis. While eliminating shear locking problems for thin plates, what is often overlooked is the large dispersion error exhibited in these elements leading to inaccurate resolution of propagating and evanescent wave behavior in dynamic analysis at intermediate to high frequencies. To address this problem, a residual-based modification of assumed strain mixed methods for Reissner-Mindlin plates is proposed. New plate elements are developed based on a generalized least-squares modification to the total energy for the time-harmonic Reissner-Mindlin plate model. Any of several existing mixed finite element interpolation fields which yield plate elements which are free from shear locking and pass the static patch test may be used. Here we start from the firm mathematical foundation inherent in the shear projection technique of the MITC4 element. A similar generalized least-squares approach was used in [4,5] to improved the accuracy of quadrilateral plate elements based

L.L. Thompson, S.R. Thangavelu /First MIT Conference on Computational Fluid and Solid Mechanics on assumed stress fields in a modified HelHnger-Reissner variational principle.

2. Wavenumber-frequency dispersion relation for Reissner-Mindlin plates We consider the Reissner-Mindlin plate bending model with thickness t. The deformation is defined by u — -zO(x, y) + w(x, y)ez,

(1)

where 0 = [Ox,Oy]^ e [H^{A)f denotes the two-dimensional vector of rotations, such that 0 J-e^, and w e HQ (A) is the vertical deflection of the midsurface. The curvatures K, are defined through the symmetric part of the rotation gradient, K(0) := V^^. The transverse shear strains are defined by the angle between the slope of the midsurface after deformation and the section angle, y = Vw — 0. The inclusion of nonzero shear deformation in the ReissnerMindlin model allows for a more accurate representation of high-frequency behavior. In the following, we assume time-harmonic motion with assumed time-dependence e~^^\ where co is the circular frequency measured in rad/s. In the absence of an applied load q, the plate equations of motion admit solutions of the form Wo c^''"'-'^,

e = Oov e^^^*'-^^,

(2) In the above, k is the wave number, v = [cos(p, sincp] defines a unit vector in the direction of wave propagation, with wave vector k = kv = k[cos(p,sin(p]. Conditions for the allowed waves are obtained by substituting the assumed exponentials (2) into the homogeneous equations of motion. The result is the dispersion equation relating frequency co to wave number k, V(k) := 0 - {k]

+ kl)k^^{ky^ -kt) = o.

(3)

Here, kp = co/Cp, ks = oo/Cs, k^ = {pto?/Dbyi^, where kb is the classical plate bending wave number for in vacuo flexural waves in the Kirchoff theory, and

-[

11/2 P(l-V2)

(?)

f G \ ^'^

Cs

3. Mixed least squares finite element formulation To develop a residual-based mixed formulation, we start with the total energy functional for Reissner-Mindlin plates and then add weighted differential operators acting on the governing steady-state equations of motion written in least-squares form. This approach may be considered an extension of Galerkin Least Squares (GLS) methods to mixed or assumed strain methods. Recall the discrete total energy for the Mindlin plate model: FM(0\w'):=nM

\l

-\-co-

(4)

In the above, Dt = EI/{I - y^), / = t^/U, with Young's modulus E, Poisson's ratio v, shear modulus G, and /c is a shear correction factor, Gs = KG, and pt is the mass density per unit area. Wave number solutions to the plate dispersion relation (3) occur in pairs: ±ki and ±k2. At frequencies below a cut-off frequency, the wave number pair ±ki occurs as purely real, while the pair ±k2 is purely imaginary. The real wave number pair corresponds to propagating waves while the imaginary pair corresponds to evanescent waves characterized by exponential decay.

f-

Qh\2-i + pl(0'y] dA - I w'q dA,

[pt(w'f

(5)

A

nM = ^B(e\0')

+

¥f'r'

fdA

(6)

EI

B ( r , r ) : = (1 + v) X j f ic(0') : K(0') + ( Y 3 ^ ) (diye'fdA

i ,

(7)

where y^ is the assumed strain. Our Mixed Least Squares (MLS) method is then based on the functional: FMLS(0\W')

div ^ = ikOo(ikv-x) e

503

(8)

= FM + FJ^S,

where TdVRifdA

niRlfdA

+ Ae

(9)

is the least-squares modification. In the above, R\ := D, div y^ + (ptco^)w^ + q

(10)

R^ := {DbV^ + ploj^) div 0^ + A div y^

(11)

are residuals for the finite element approximation to the governing equations for Mindlin plates. Here, A = Gst, V^ = div V, and ri{co) and Xiico) are frequency dependent local mesh parameters determined from dispersion analysis and designed to improve the accuracy of the finite element solution. Any of several existing mixed finite element approximation fields which produce elements which are free from shear locking and pass the static patch test may be used. In this paper, we use the field- and edge-consistent interpolations of the MITC4 plate bending element proposed by Bathe and Dvorkin [1]. The finite element interpolation of the element domain A^, together with the displacement field if^, and 0^, follows the standard isoparametric procedure. The displacement and rotation interpolation are constructed using the standard bilinear functions: 4

4

i=i

i=i

(12)

504

L.L. Thompson, S.R. Thangavelu /First MIT Conference on Computational Fluid and Solid Mechanics

We let [/] be the Jacobian transformation matrix of the mapping x: A -> A^, i.e. V = [/]^V, where [J] := [x,^], J = det[J], and V stands for the gradient operator with respect to ^ and r]. For the MITC4 mixed interpolation [2,3], the assumed strain is defined by a reduction operator Rh: [H\Ae)f -^ r/,(A^), which maps the shear strain interpolants to an auxiliary space r^, y^ = RniViv'-e') = {Vw'-R„0') = Vw'-[J]-"^ R^UfO'. The reduction operator R^: [H^(A)f -^ S'\A) = [e \ €i = ai-\- b\ Y], ^2= a2-\- ^2?, «i, ^1, «2, ^2 € M), is used to simplify the residuals appearing in the MLS functional. For square element geometries, the divergence of the MITC4 interpolated shear strains vanishes within the element div y' = jW-y'

= ~(y^^^ + >/,,,) = 0.

(13)

Furthermore, since 0^ e 2^(A), then div^^ e P^(A) = {0 I 0 = C] + C2? + c^T], Ci e R). With this result, it is clear that the Laplacian operator acting on the divergence of the rotations also vanishes for 4-node square elements, i.e.

v'(div^') = -V' (-^v. e''^ = 0.

(14)

Using (13) and (14) in (9), the generalized least-squares functional FLS reduces to

== E \

FTS

fnV(w'-^f)-V{w'^f)dA

+ - / r2(V.r)'dA,

(15)

where rx = Zi(ptco^y, r2 = T2{plct?f, and / = qjipto?). Substituting the bilinear interpolations for w^ and ^'', together with the assumed strain y^ defined by the MITC4 interpolation, into Eq. (15) and imposing stationary conditions with respect to w^ and (9^, results in the following system of linear algebraic equations for each element A^,

vs'\d' = r

(16)

where d^ is the 12 x 1 vector of element nodal displacements {{wi, Oi), / = 1, . . . , 4}, / ^ is the force vector resulting from the transverse loading, and 5'' is the 12 x 12 symmetric dynamic stiffness matrix for each element. S'(a)) =

Z'(co)-hKls(co),

(17)

where (18) Here, K^ and M^ are the stiffness and matrices for the plate, and Kl^{co) is a stabilization matrix resulting from FLS:

Kl^{co)=n(co)Klsi+r2{co)Kl^^

(19)

with frequency independent matrices, (20)

Kls2 = AiVe,.. + Ne^,yf(Ne,,,

+ iVe,,,) dA,

(21)

Ae

where N^, A^^^ and A^^^ are row vectors of bilinear basis functions defined by the interpolations (12) written in vector form.

4. Evaluating element mesh parameters Finite element difference relations associated with a typical node location (xm, jn) in a uniform mesh are obtained by assembling a patch of four elements. The result is a coupled system of three, 27-term difference stencils associated with the 3 nodal degrees-of-freedom at node (m,n). The effect of this stencil on the discrete solution dm^n is written in matrix-vector form as

J2 Y.^Dp,]E^EP{dU, = m,

(22)

-\q=

where E^ and E^. are directional shift operators. To obtain the finite element dispersion relation associated with this stencil, a plane wave solution is assumed for the nodal displacements, similar in form to the analytical solution:

**m.n — i

I

Go cos (p

(i/t{/zm)

(ik'l.hn)

(23)

9o sin (p where k'^ = k^cos(p. A:J = k^ sincp are components of the wave vector k^ = k^v, and h is the element length. Substitution of (23) into the stencil equations (22) results in the conditions for allowed waves in the finite element mesh: The resulting finite element dispersion relation for the plate is, D := H\\Hr

H'=0,

(24)

where Hjj are functions of matrix coefficients ztj of the element dynamic stiffness matrix [Z^] defined in (18), wave angle cp, and ri, r2. The finite element dispersion equation D = D(a),k^h,(p,kij,mij,ri,r2) defined in (24) relates frequency co, to the numerical wave number k^h and cp, and depends on the stiffness and mass coefficients kij = [K^]ij, and ruij = [M^]/y, and mesh parameters ri, r2. Similar to the analytic dispersion relation, there are two pairs of numeric wave numbers ib/:f and ±k2 that satisfy (24) which correspond to propagating and evanescent waves, respectively.

L.L. Thompson, S.R. Thangavelu /First MIT Conference on Computational Fluid and Solid Mechanics 1.08

1.24 1.22^ 1.2 1.18 1.16^ 1.14i 1.12 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94

1.06 H 1.04 1.02 1t'*'Jt**. \

0.98

v.-

•• *

^^::

******

• ^ • • > - -

^^

0.96 0.94 0.92 \ j

0.9 j

MITC(45)++++ MITC(30) MITC(15)oooo MITC(O)

0.88

'

^

^

^

•

•

•

•

•

•

•

°^^>^ •>j>1

1

r - — 1

•-

•

— 1

1

2 3 frequency(1 OMrad/s)

•

>

•

1

•

•

1

'

1

1.04^ 1.02 1

..>''^'

2 3 frequency(1 OMrad/s)

MLS(O) MLS 15) 0000 MLS(30) MLS(45)

l.ll

0.96 0.94

0.9

>^-

1.08 j 1.06 j 1.04 i 1.02 1 0.98^ 0.96 0.94

0.98

0.92

MITC(O) MITC(15)oooo MITC 30). MITC(45) H

0.92 1.24 1.22 1.2 1.18 1.16| 1.14-j 1.12^

505

MLS(45) H MLS(30). MLS(15 0000 MLS 0)

0.88

1

0.92

. 2 3 f requency(1 OMrad/s)

2 3 frequency(1 OMrad/s)

5

Fig. 1. Relative error k^/k at angles (^ = 0, 15,30,45 degrees. Top: MITC4. Bottom: MLS4 with (p = 30° in definition of mesh parameters ri and r2. Left: real propagating wave number ki. Right: imaginary evanescent wave number k2. We determine design parameters ri and r2 such that the finite element wave number pairs match the analytical wave number pairs ±ki and ±k2 for a given orientation (p = (pQ. Here, optimal values for ri and r2 are computed by setting k^ = kiico) and k^ = kiico) in the finite element dispersion relation (24). The result is two equations which may be solved for the mesh parameters ri((p,co,h) and r2((p,co,h): cn + cnn

+ Ci3r2 + Curir2 = 0

C2l + ^22^1 + C23r2 + C24rir2 = 0

(25) (26)

with coefficients Cu = Ci(ki, cp), and C2i = Ci(k2, cp). Eliminating r2 from (25) and (26), allows the design parameter ri to be obtained in closed-form by solving the quadratic equation, eir^-{-e2ri-\-€3

= 0,

(27)

where ei = ei(Cij). The solution of (27) results in two real negative roots. We select the largest root to determine ri.

as this root matches the analytical dispersion relations. The other design parameter can then be written in terms of the first, r2 =

• . C31 + C4iri

(28)

Hence, the design parameters r/ = ri(kij, mij,(o,h,(p), I = 1,2 are obtained in terms of the stiffness and mass coefficients in the underlying MITC4 element, the frequency dependent wave numbers satisfying the analytical dispersion relation, and cp. Using our definitions for ri and r2, for a fixed angle cp, the least-squares modification enables the finite element wave numbers to exactly match the analytical dispersion conditions, rendering a zero dispersion error solution. In general, the direction of wave propagation (p is not known a priori. However, similar to [6], we can select a (^ in the definitions for ri and r2 which minimizes dispersion error over the entire range of possible angles defined by the periodic interval 0 < (p < 7t/4.

506

L.L. Thompson, S.R. Thangavelu /First MIT Conference on Computational Fluid and Solid Mechanics

In implementing our mixed least squares method on nonuniform meshes, the element length h is defined by either a local size determined by the square root of the element area, he = -/A^, or by an average element length /zave computed over a local patch of similarly sized elements. While the optimal definition for the mesh parameters ri and r2 were derived from a dispersion relation on a uniform mesh, with constant element length /z, accurate solutions on nonuniform meshes are shown to be relatively insensitive to the precise definitions used for h.

5. Dispersion accuracy For a range of frequencies co, and wave angles (p, relative to uniform mesh lines, the wave number accuracy for our residual-based MLS method is compared with the underlying MITC method [1]. Results are presented for a steel plate with properties: E = 210 x 10'^ dynes/cm^, V =: 0.29, p = 7.8 g/cm^, plate thickness r = 0.15 cm, and shear correction factor K = 5/6. The relative error of the numerical wave number divided by the analytic wave number, k''/k is shown in Fig. 1. The frequency range is plotted over the range up to coh = 5 X 10^ cm/s corresponding to approximately four elements per wavelength. At low frequencies, the MITC4 element replicates the character of the analytic dispersion curves marginally well with error in the real propagating wave number less than 3% for discretizations finer than 10 elements per wavelength, i.e. 10/z = X. However, above this level, the error in both the real and imaginary wave number increases rapidly. To achieve a 1.5% level of accuracy would require more than 20 MITC4 plate elements per wavelength. The bottom two plots show the improved dispersion accuracy achieved for both the real and imaginary wave numbers by our residual-based MLS4 element. The MLS4 element replicates the character of the analytical dispersion curves well with significant reduction in numerical wave number error compared to the underlying MITC4 interpolation. Results for the MLS4 method give a maximum error in the real wave number less than 1% at a frequency of coh = I X 10"^ cm/s, corresponding to approximately 10 el-

ements per wavelength. This represents a nearly three-fold reduction in phase accuracy compared to the base MITC4 element. At the level of 10 elements per wavelength, the maximum error in the imaginary wave number is reduced from 3% for MITC4 to less than 2% for MLS4.

6. Numerical example Results are presented for forced vibration of a simply supported steel plate with a uniform distributed time-harmonic pressure loading q = 2 dynes/cm^. Fig. 2 shows the L2 convergence rates for the vertical deflection with uniform mesh refinement. Both MITC4 and MLS4 achieve the same rate of convergence at approximately A^ = 100 elements. However, as a result of improved dispersion accuracy, the MLS4 element decreases the L2 error for the same number of elements. We next study the performance of the MLS4 element for quasi-uniform meshes (parametric mesh grading). Here, the MLS4 element is computed with mesh parameters ri and r2 determined from an average element size /Zave, 10'

:

•

• ••

•

•

r— 1

•—r-i'

a -0-

MITC 1: MLS4 1 :

a.

R \

D

\ \

•^

\B

0. ^ V

"D

X

° X

1

•

•

'—

10'

Fig. 2. Simply supported steel plate example. Frequency / = 500 Hz. Convergence with mesh refinement. Relative discrete L2 error of vertical deflecfion versus A^, for a uniform mesh of N X N elements over one-quarter of the plate.

Table 1 Discrete L2 error for square plate example with quasi-uniform meshes at / = 500 Hz Mesh

Element type SRI4

MITC4

MLS4-ave

MLS4-local

QMeshl QMesh2 QMeshS Uniform

0.29952E 0 0.38511E0 0.33728E 0 0.18351E0

0.29952E 0 0.38514E0 0.33731E0 0.18352E0

0.55688E-1 0.12876E0 0.32663E-1 0.22147E-1

0.38146E-1 0.52325E-1 0.33728E-1 0.22147E-1

Results for a uniform mesh with equally spaced nodes shown for reference.

L.L. Thompson, S.R. Thangavelu /First MIT Conference on Computational Fluid and Solid Mechanics

507

Fig. 3. Quasi-uniform meshes with A^ = 50 elements per edge and 5:1 bias. Average element size /zave = 'sfAjW = 1-0. (Left) QMeshl, (Right) QMesh2, (Bottom) QMeshS. computed over the total mesh, denoted MLS4-ave, and from a local element size he = V^T, denoted MLS4-locaL Table 1 shows results obtained using the three different quasi-uniform meshes shown in Fig. 3. We observe that the large improvement in accuracy using the MLS4 element compared to the MITC4 element for uniform meshes is not drastically affected by the element distortions or higher aspect ratios. Showing the robustness of the MLS method, the discrete L2 error for the MLS4-local solution remains an order of magnitude lower than the underlying MITC4 element.

7. Conclusions A residual-based method for improving the dispersion accuracy of the 4-node MITC plate bending elements is developed. The property of field consistency in the MITC transverse shear strain interpolation is used to simplify the residuals appearing in the generalized least-squares operators, and leads to a simple modification of the element dynamic stiffness matrix with a frequency-dependent leastsquares matrix. Using complex wave number dispersion analysis, optimal values for the mesh parameters appearing in the least-squares matrix are determined such that finite element propagating and evanescent wave number pairs

match the analytical wave number pairs for a given wave orientation angle cp relative to a uniform mesh. Both dispersion analysis and numerical results show that the new mixed least-squares (MLS4) plate element significantly improves wave accuracy compared to the underlying MITC4 element.

Acknowledgements Support for this work was provided by the National Science Foundation under Grant CMS-9702082 in conjunction with a Presidential Early Career Award for Scientists and Engineers (PECASE), and is gratefully acknowledged.

References [1] Bathe KJ, Dvorkin E. A four node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. Int J Num Methods Eng 1985;21:367-383. [2] Bathe KJ, Brezzi F. On the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation. In: Whiteman JR (Ed.), Proc Conf Math Finite Elements Appl V, Academic Press, New York, 1985, pp. 491-503.

508

L.L. Thompson, S.R. Thangavelu /First MIT Conference on Computational Fluid and Solid Mechanics

[3] Bathe KJ, Brezzi R A simplified analysis of two plate bending elements — the MITC4 and MITC9 elements. In: GN Pande, J Middleton (Eds.), Proc Int Conf Num Methods Eng, (NUMETA 87), Martinus Nijhoff, Dordrecht, 1987. [4] Thompson LL, Tong Y. Hybrid Least Squares Finite Element Methods for Reissner-Mindlin Plates. Comput Methods Appl Mech Eng, accepted for publication. [5] Thompson LL, Tong Y. Hybrid least squares finite element

methods for Reissner-Mindlin plates. In: Proceedings of the ASME Noise Control and Acoustics Division - 1999, 1999 ASME International Mechanical Engineering Congress and Exposition, ASME, NCA Vol. 26, 1999, pp. 77-89. [6] Thompson LL, Pinsky PM. A Galerkin least squares finite element method for the two-dimensional Helmholtz equation. Int J Num Methods Eng 1995;38:371-397.

509

Modeling quasi-static fracture of heterogeneous materials with the cohesive surface methodology M.G.A. Tijssens ^'*, E. van der Giessen^, L.J. Sluys ^ "" Delft University of Technology, Koiter Institute Delft, Stevinweg 1, 2628 CN Delft, The Netherlands ^ Delft University of Technology, Koiter Institute Delft, Mekelweg 2, 2628 CD Delft, The Netherlands

Abstract The micromechanical fracture processes in cementitious composites are captured in a constitutive model for a cohesive surface. Using multiple cohesive surfaces in the numerical simulation of fracture, it is shown that the cohesive surface methodology is able to simulate the discontinuous crack growth process occurring in cementitious composites. Keywords: Cohesive surface; Finite element; Concrete; Numerical simulation; Discontinuous fracture

1. Introduction Several methods to numerically simulate fracture have emerged over the past years, one of which is the cohesive surface methodology. Based on the original ideas of Dugdale [1] and Barenblatt [2] to describe fracture as the separation between two surfaces bridged by tractions (see Fig. 1), Xu and Needleman [3] proposed a methodology in which cohesive surfaces are embedded in the continuum. Instead of specifying a single crack path by a cohesive surface, they used multiple cohesive surfaces uniformly distributed throughout the continuum. Continuum and cohesive surfaces are discretized separately and they both have their own constitutive laws. An illustration of this is given in Fig. 2, where part of a finite element discretization of a heterogeneous material is shown. The white lines are the cohesive surfaces made visible by shrinking the surrounding continuum elements. There are no cohesive surfaces in the inclusions. Using multiple cohesive surfaces, fracture evolves as a result of the competition between bulk deformation and loss of cohesion at multiple sites in the material. The crack path is an outcome of the computation. The strength of the methodology lies in the fact that no other criteria for crack initiation or crack front propagation are used other than the constitutive law of the cohesive * Corresponding author. Tel.: +31 (15) 278-6602; Fax: +31 (15) 278-6383; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

surfaces. Often, relatively simple relations for the cohesive surface traction T as a function of the separation A are used, see for example [3-6]. However, the cohesive surface methodology provides the opportunity to extend the description of r as a function of A and include other effects such as the separation rate A and the stress field surrounding the cohesive surface, see for example [7,8]. This paper demonstrates this for a cementitious composite.

2. Example: cohesive surface model for concrete As a specific example, consider the simulation of fracture of concrete. Concrete is a highly heterogeneous material and the details of the micromechanical processes preceding fracture are still not clear. Examining the microstructure of concrete, one observes that the heterogeneity finds its origin in the presence of voids, sand and aggregates, partly and fully hydrated cement grains and Fictitious crack

Fig. 1. Illustration of the fictitious extension of a crack by a cohesive surface with tractions J as a function of the separation

510

M.G.A. Tijssens et al. /First MIT Conference on Computational Fluid and Solid Mechanics

bulk material

cohesive surface

Fig. 3. Illustration of the model system that is used for the uniaxial tensile test to obtain the local softening curve shown in Fig. 4.

Fig. 2. Representative part of a finite element mesh with embedded cohesive surfaces. The white lines are the cohesive surfaces made visible by shrinking the continuum elements. microcracks. On a microscale, one finds the needle-like structure of the CSH (calcium-silicate-hydrates) and CH (calcium-hydrate) crystals. It can be expected that even for very small macroscopic loads, stress concentrations occur on the microlevel that cause microcracks to initiate and grow. We recently proposed [9] that the complex micromechanical fracture processes occurring in concrete can be captured in a damage formulation for the cohesive tractions. Motivated by analytical studies regarding planar microcracks [10,11] and the rate-dependent character of concrete deformation [12], the rate of evolution of damage is assumed to be a function of the current state of damage and stress. The proposed constitutive model for the cohesive surfaces reads r = (l

-oj)DA,

(tr-

sinh (1) (1 -o^Y in which T and A are the traction and separation vector of the cohesive surface and a; € [0, 1] the damage variable. Furthermore, co^, To,n and m are material parameters governing the damage evolution process in dependence of the traction component T„ = T n normal to the cohesive surface. The elastic stiffnesses in normal and tangential direction are assumed to be equal and uncoupled, i.e. D = kl, I being the unit matrix. Note that in contrast to traditional cohesive surface models, the traction is not a predetermined function of the separation between the cohesive surfaces. To get an idea of the local softening characteristics as described by Dugdale (1), consider a one-dimensional tensile test as shown in Fig. 3 in which a single cohesive surface and a single continuum element are subjected to a prescribed strain rate of ^ = 10~^ s~^ The material parameters used are representative for cement paste and are given in Table 1. Taking the

Table 1 Bulk and cohesive surface parameters used in all calculations Cement E (GPa) y

(bo (s-^) 7b (MPa) m n ^max ( M P a )

Gf (N/m)

20 0.2 10-35 2.1 5 5 5.0 19

Aggregate

Interface

55 0.2

-

-

10-34 0.9 5 5 2.1 3

Peak load amax and fracture energy Gf are results for a strain

accelerated evolution of damage as described by Dugdale (1) and numerically integrating with respect to time, one obtains a softening curve as given in Fig. 4. Note that the initial shape of the softening curve is similar to the softening curves that result from models describing the behavior

0

2

4 6 An (10"^ mm) Fig. 4. Traction-separation relation in a one-dimensional tensile test obtained by numerical integration of (1) with the parameters for cement paste given in Table 1.

M.G.A. Tijssens et ah /First MIT Conference on Computational Fluid and Solid Mechanics of planar microcracks (cf. [13,14]), i.e. a strong initial softening followed by a more gradual decrease of the cohesive surface traction. In our model, this is followed by breakdown of the cohesive surface in an accelerated manner.

3. Application: fracture of concrete The cohesive surface model is used in the numerical simulation of fracture of concrete in which the presence of aggregates is explicitly accounted for. The boundary conditions on the square specimens are such that the vertical faces remain traction free and the bottom face is restrained only in vertical direction. The cement paste and the interfacial transition zone, separating cement paste and aggregates, can fracture according to the cohesive surface model given in (1) with the parameters specified in Table 1. The maximum aggregate diameter is 2 mm. Aggregates are assumed not to fracture. Upon application of a vertical displacement at the top face of the specimen, damage develops in all of the cohesive surfaces. Due to the assumption that there is no initial damage in the specimen, the rate at which damage evolves

1

i

1

'

!

3

"

(MPa)

2

511

is very low at first, which results in a nearly linear elastic prepeak response shown in Fig. 5a. As the stresses become higher, more and more material in the specimen becomes damaged. Damage evolves faster on the poles of the aggregates, because the interfacial transition zone is specified to be weaker than its surroundings and because of the stress concentrating effect of the stiffer aggregates. Before the peak load is reached, damage development is diffuse which results in a deviation from linearity at approximately 60% of the peak load, see the inset in Fig. 5a. Note that often the local softening behavior of a material is obtained from macroscopic tensile tests. Therefore, the local softening behavior of cementitious composites is often taken as exponentially decaying. Here, no such assumption is made. Comparing Figs. 4 and 5a one sees that the shape of the local and global softening curves are totally different. A representative result of the final fracture pattern is given in Fig. 5b. Multiple cracks grow simultaneously and large crack face bridges are formed which extend the load carrying capacity of the specimen. These results indicate that the cohesive surface model is able to describe the discontinuous crack growth process occurring in cementitious composites. Note that the cohesive surface methodology as used here facilitates the description of multiple fracture without the necessity to keep track of all cracks that are nucleated. This is an advantage with respect to the node-enrichment techniques as proposed by Belytschko and Black [15] and by Wells and Sluys [16].

2H

\

1

\

0

X

!

()

0.05

0. 15

0.1

em^')

^~~"^^---~—

(a) £(10-^)

<^h (MPa) 3.3

0

Fig. 5. Stress-strain relation (a) and fracture pattern and hydrostatic stress distribution (b) for a typical concrete specimen. The black fringes indicate where damage has evolved.

4. Conclusions Many authors have shown that the cohesive surface methodology is a powerful method to describe crack growth. However, although most constitutive models for cohesive surfaces are physically motivated, often the underlying physical processes are captured in relatively simple traction-separation relations which are almost always independent of the loading rate. A side effect is that it is very difficult to obtain solutions to fracture problems in a quasi-static formulation of the problem. Here the cohesive surface methodology is exploited further and a cohesive surface model for the description of fracture of cementitious composites is proposed in which the traction is not a predetermined function of the separation. The description of the degradation of material integrity is irreversible and depends on the loading rate. This facilitates a quasi-static formulation of the fracture problem. The model is used in a numerical simulation of fracture of concrete. The outcome of the numerical simulations is a result of the competition between bulk deformation and fracture at multiple sites in the material. The cohesive surface methodology predicts the discontinuous crack growth occurring in cementitious composites.

512

M.G.A. Tijssens et al. /First MIT Conference on Computational Fluid and Solid Mechanics

References [1] Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Sol 1960;8:100-104. [2] Barenblatt GI. The formulation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axiallysymmetric cracks. J Appl Math Mech 1959;23:622-636. [3] Xu XP, Needleman A. Numerical simulation of fast crack growth in brittle solids. J Mech Phys Sohds 1994;42:13971434. [4] Tvergaard V, Hutchinson JW. The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J Mech Phys Solids 1992;40:13771397. [5] Repetto EA, Radovitzky R, Ortiz M. Finite element simulation of dynamic fracture and fragmentation of glass rods. Comput Methods Appl Mech Eng 2000;183:3-14. [6] Pandolfi A, Krysl P, Ortiz M. Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture. Int J Fract 1999;95:279-297. [7] Onck PR, Van der Giessen E. Microstructurally-based modelling of intergranular creep fracture using grain elements. Mech Mater 1997,26:109:126. [8] Tijssens MGA, Van der Giessen E, Sluys LJ. Modeling of

[9]

[10] [11]

[12] [13]

[14]

[15]

[16]

crazing using a cohesive surface methodology. Mech Mater 2000;32:19-35. Tijssens MGA, Sluys LJ, Van der Giessen E. Simulation of fracture of cementitious composites with explicit modelling of microstructural features. J Mech Phys Solids, submitted. Koiter WT. An infinite row of coUinear cracks in an infinite elastic sheet. Ing Archiv 1959;28:168-172. Westmann RA. Asymmetric mixed boundary-value problems of the elastic half-space. J Appl Mech 1965;32:411417. Malvar LJ, Ross CA. Review of strain rate effects for concrete in tension. ACI Mater J 1998;95:735-739. Ortiz M. Microcrack coalescence and macroscopic crack growth initiation in brittle solids. Int J Solids Struct 1988;24:231-250. Huang X, Karihaloo BL. Tension softening of quasi-brittle materials modelled by single and doubly periodic arrays of coplanar penny-shaped cracks. Mech Mater 1992;13:257275. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Num Methods Eng 1999;45:601-620. Wells GN, Sluys LJ. A new method for modelling cohesive cracks using finite elements. Int J Num Methods Eng, in press.

513

Application of numerical conformal mapping to micromechanical modeling of elastic solids with holes of irregular shapes I. Tsukrov *, J. Novak Department of Mechanical Engineering, Kingsbury Hall, University of New Hampshire, Durham, NH 03824-3591, USA

Abstract Pores and defects in real materials often have irregular (strongly non-circular) shapes. Thus, micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable. This paper presents a computational procedure to calculate the contribution of the irregularly shaped defects into the effective moduli of two-dimensional elastic solids. In this procedure, the hole compliance tensor of an individual defect is constructed using the numerical conformal mapping (NCM) technique. The effective elastic properties of a porous solid are predicted in the non-interacting approximation using the elastic potential-based approach. Keywords: Micromechanics; Effective property; Porous solid; Conformal mapping

1. Introduction Analytical predictions of the effective elastic properties of porous solids are limited by existing analytical solutions of elasticity, which are available for regular pore shapes only — spherical or ellipsoidal in the 3-D analysis, and circular, elliptical or right polygons in 2-D. Numerical simulations can be used for all kinds of shapes, but they require substantial computational power and are not universal. Our analysis combines numerical and analytical techniques: the elasticity problem for each type of defect is solved numerically, and this solution is used in the potential-based analytical procedure of micromechanical modeling. We do not consider non-linear effects caused by closing of holes by compressive stresses (stiffness increasing with compression). The effective moduli are derived in the approximation of non-interacting defects — it is rigorous at small defect densities and can be used as a basic building block for various approximate schemes (self-consistent, differential, Mori-Tanaka, etc.) in the micromechanical modehng of solids with interacting defects. The approach is based on the results for one hole. We represent the total strain in the reference area A subjected to a remotely applied stress a and containing a hole as a * Corresponding author. Tel.: +1 (603) 862-2086; Fax: -hi (603) 862-1865; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

sum 8 = S° : a + Ae

(1)

where S^ is the compliance tensor of the matrix, and a colon denotes contraction over two indices. Hole's contribution A 8 is given by the integral A8 = - ^

/'(un + n u ) d r

(2)

where n is the unit normal to hole boundary F (inwards the hole), u denotes displacements of points of F, and un, nu are dyadic products of two vectors. Due to Unear elasticity, A^ is a linear function of <x and hence can be written as A8 = H : a

(3)

where fourth rank tensor H is the hole compliance tensor (possessing the usual symmetries of the elastic compliance tensor). H-tensors were found for elliptical, polygonal and rectangular holes in 2-D and ellipsoidal cavities in 3-D by Tsukrov and Kachanov [1] and Kachanov et al. [2] (see, also, relevant results of Nemat-Nasser and Hori [3], and Jasiuk et al. [4]). In this paper, we present the derivation procedure and numerical results for 2-D holes of arbitrary irregular shapes. To obtain the effective elastic properties, we formulate the problem in terms of the elastic potential (rather than

514

/. Tsukrov, J. Novak/First MIT Conference on Computational Fluid and Solid Mechanics

compliancies): its structure implies the proper parameters of defect density and establishes the overall anisotropy. The elastic potential in stresses / ( a ) = ( l / 2 ) a : e(a) of a soUd with a hole is / ( a ) = ( l / 2 ) a : S^ : a + ( l / 2 ) a : H : a = /o + A /

(4)

where /o is the elastic potential of the matrix material, A / is the change in the potential due to the presence of a hole. In the non-interaction approximation, the stress field around each hole is not disturbed by the presence of other defects. Thus, the potential change due to holes is A / = ( l / 2 ) a : ^ H ^ ^ ^ : a = (l/2)<j : H*

(5)

where tensor H* = JZ^^^^ (summation may be replaced by integration over orientations, if computationally convenient) takes the individual cavity contributions with proper 'relative weights' and, thus, constitutes the proper parameter of defect density. The effective elastic compliancies Sijki are obtained from

a(/o + A/)

(6)

~ SijkiCJki

2. Calculation of the hole compliance tensor using numerical conformal mapping For any 2-D defect, tensor H can be represented in terms of the hole shape factors h\-h(,: H

PS

ASr,

2EQA

PS ASi2

=

2E^

s

X /zideieiei + /22e2e2e2e2 -h /i4(eieie2e2 + e2e2eiei) + ^5(eie2eiei 4- e2eieiei -f- eieieie2 + eieie2ei) + e2eie2e2 + e2e2eie2 + e2e2e2ei)

^3 1 + y (eie2eie2 + eie2e2ei + e2eieie2 + e2eie2ei) (7) where S is the area of the defect, Eo is the Young's modulus of the matrix material and Ci, Ci are the unit vectors of the local coordinate system JCIJC2 (Fig. 1). This representation yields the following expressions for additional stresses in a plate subjected to the uniaxial tension P remotely

[(h2 + h4) - (/i2 - h4) COS 20 + Ihe sinlO] [(/Z5+/l6) + (/l5

/z6)cos2^ + /23sin2^] (8)

The values of the shape factors hi-he for particular hole geometries are determined by comparing these expressions with the corresponding elasticity predictions obtained by numerical conformal mapping (NCM). In the NCM approach, we use the complex variable technique of 2-D elasticity [5] in combination with the conformal mapping method. We approximate the boundary of the hole by N-sided polygon and construct the singlevalued analytical function CL>(^) that maps the interior of the unit circle in the image plane f onto the exterior of the hole in the z = xi -\~ ix2 plane (Fig. 1). This conformal mapping function a)(^) is found numerically by evaluating the Schwarz-Christoffel integral [6]. Based on CL>{^), we calculate the stress functions (p(z) and \j/(z) using the traction-free boundary condition on the contour of a hole:

cpii) +

^(O

:Cp\i)-hifi^)=0

(9)

The displacements are expressed in terms of stress functions: Ml + iU2 =

EQA

+ hei^i^i^i^i

applied at an angle 0 with respect to xi-axis: PS Afn = zrzr^ [(hi +/z4) + (/?i - / 1 4 ) cos26> + 2/z5 sin2^] IEQA

Vo

Eo

(P(^)' ^^

Eo

[_co'(^) l^7(^

(P'(i)'hf(^)

(10) where Vo is the Poisson's ratio of the matrix material. These displacements are used in numerical evaluation of the integral (2) to obtain the additional strain tensor Ae. Table 1 presents the hole shape factors of the elliptical hole (of eccentricity 1 : 3 ) and irregular hole shown in Fig. 2. The elliptical hole has been approximated by a polygon with 81 vertices. For the hole of irregular shape, 101 vertices (marked by '*' in Fig. 2) have been used. The NCM results for the elliptical hole are compared with the analytical prediction of Kachanov et al. [2]. (Note that coefficient /14 in their paper corresponds to coefficient (—/i2/2) in the current publication). The maximum difference between non-zero shape factors is less than 2.5%.

Fig. 1. Hole of irregular shape: numerical conformal mapping.

515

/. Tsukrov, J. Novak/First MIT Conference on Computational Fluid and Solid Mechanics Table 1 Shape factors of elliptical and irregular holes hi

h4

hs

Elliptical hole Analytical prediction NCM calculation Percentage difference

1.666 1.658 0.5%

7.000 6.928 1.0%

5.334 5.396 -1.2%

-1.000 -0.976 2.4%

0.000

0.000

Irregular hole NCM calculation

3.920

6.900

7.440

-0.960

1.540

1.160

highly irregular hole shapes (characterized by six independent constants hi-he)- The effective Young's modulus and Poisson's ratio are: Eo E =

i + p(i(/^i + /^2) + K^3 + M '

Vo-p{l{hi+h2)-hlh4-\h3)

(12)

i + p(i(/^i + /^2) + |(^3 + M

-12

-10

-^

^

Fig. 2. Boundary of the irregular hole approximated by polygon (N = 100). 3. Effective elastic moduli of solids with holes of irregular shape

4. Conclusions

We present the formulae for effective elastic properties of 2-D solids containing defects of identical shapes. It is assumed that the location of holes is random and uncorrelated with their size and orientation. The approximation of non-interacting defects is adopted. Applying the procedure described in Section 1, the effective compliance tensor of solids with holes of irregular shape can be expressed in terms of the mechanical properties of matrix material, hole shape factors hi-he and porosity (relative cavity volume) p. For example, in the case of parallel holes. Sun

=

(l +

To

phi),

^2212 =

The computational procedure to evaluate the effective elastic properties of solids with holes of highly irregular shapes is presented. The contribution of each defect into effective moduli is defined in terms of the shape factors.

(l+p/i2), 1 /1+U„

Sn22 = T r ( - ^ o + ph4),

As an example, consider the elastic solid with randomly oriented irregular holes of the shape shown in Fig. 2. The effective Young's modulus of this solid is obtained using the values of /z-factors given in Table 1. Fig. 3 presents the dependence of E/Eo on porosity. The results for circular and highly elongated elliptical holes (a/b = 10) are also shown for comparison. It can be seen that elongated objects produce greater contribution into the effective compliance. This result is consistent with similar observations of Kachanov et al. [2].

hi\

to

= -rphs

5'l222 =

PK

(11)

where it is assumed that the local coordinate axes of each hole coincide with the global coordinate axes. Materials with randomly oriented non-interacting holes will exhibit the isotropic behavior even in the case of

Fig. 3. Effective Young's modulus of a solid with randomly oriented holes.

516

/. Tsukrov, J. Novak/First MIT Conference on Computational Fluid and Solid Mechanics

It is shown that these shape factors can be found for any (regular and irregular) defect shape using the numerical conformal mapping technique. Application of this technique to the regular hole shapes produces results that are in good correspondence with analytical predictions.

References [1] Tsukrov I, Kachanov M. Solids with holes of irregular shapes: effective moduli and anisotropy. Int J Fract 1993;64(1):R9-12.

[2] Kachanov M, Tsukrov I, Shafiro B. Effective properties of solids with cavities of various shapes. Appl Mech Rev 1994;47(1):S151-S174. [3] Nemat-Nasser S, Hori M. Mircomechanics: Overall Properties of Heterogeneous Materials. Amsterdam: Elsevier Science, 1993. [4] Jasiuk I, Chen J, Thorpe MF. Elastic moduli of two dimensional materials with polygonal and elliptical holes. Appl Mech Rev 1994;47(1):S18-S28. [5] Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Groningen: Noordhoff, 1963. [6] Driscoll TA. A MATLAB Toolbox for Schwarz-Christoffel mapping. ACM Trans Math Software 1996;22:168-186.

517

Impact simulation of structural adhesive joints Mark Tyler-Street ^'*, Nigel Francis ^, Roger Davis ^, Jeff Kapp ^ "" ESI North America, 13399 West Star, Shelby Township, MI 48315, USA ^ Ford Motor Company, Dunton, UK

Abstract A wide range of joining methods are employed for the assembly of vehicle structures and components; thermal (welding, brazing and soldering), mechanical (screws, nuts and bolts, clips and clamps, pins, rivets and staples) and chemical (adhesive) technologies. The appHcation of joining methods for vehicles may be broadly classified into two categories: use for the major load bearing joints of the BIW structure and use for minor load bearing and component applications. Typically, adhesives are used as a secondary bond to compliment existing joining methods for the major load bearing joints, whilst they are used as a primary joining method for the minor load bearing joints. The objective of the work described in this paper is to develop improved simulation methods for the prediction of structural adhesive impact performance applied to the major load bearing joints of a vehicle BIW. The methodology includes the numerical development of interface elements, boundary contact algorithms and material models to represent the adhesive bondline. Keywords: Adhesive bonding; Finite element simulation; Vehicle crashworthiness

1. Introduction The major load bearing joints of vehicle BIW structures typically use welding methods and mechanical fasteners as the primary joining method, adhesives may be introduced as a secondary joining method to enhance the joint properties. Adhesive bonding is employed in applications precluding reliance on welding or mechanical methods, for example; the joining of two different materials or aluminium panels. In these cases the adhesive bond is typically used in conjunction with mechanical fasteners Although structural adhesives are already employed in production to enhance the static properties of the vehicle BIW, the effect of adhesives on the impact performance of the structure is not fully understood. Previous work undertaken to evaluate high impact performance adhesives compared bonded and weldbonded (adhesives combined with spotwelds) joints with spotwelds in impact tests on beam structures and simplified vehicle front ends [1]. The results showed that the behaviour of the high toughness epoxy adhesives was a major improvement over conventional toughened epoxies. High impact epoxies maintained * Corresponding author. Tel.: +1 (810) 323-4610; Fax: -\-l (810) 323-4611; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

the integrity of the joints, so that regular folding of the metal occurred in the crumple zone, promoting effective absorption of impact energy. The test results showed that the weldbonded beams employing the high impact performance epoxies were effective at absorbing the impact energy, there was less post-impact deformation in comparison to the spotwelded beams (Fig. 1). Factors inhibiting the reliance of these adhesives in

100

ISO

Fig. 1. Force versus displacement results for box beams with and without adhesive — test.

518

M. Tyler-Street et al /First MIT Conference on Computational Fluid and Solid Mechanics

vehicle applications include the lack of understanding of material properties and failure behaviour of the adhesives and the limitations of current simulation methods for predicting the impact performance of the adhesive bondline. Despite these limitations, there has still been a dramatic increase in the numbers of weldbonded applications on high volume production vehicles [2,3]-

2. Existing simulation methods To date most of the analyses of adhesive joints in vehicle structures have been undertaken to investigate the effects of adhesives on the overall static stiffness, durability and NVH properties of vehicle structures. In these models, methods of representing the geometry of joints have improved correspondingly with increases in computer power. For example in early CAE models, bonded joints were either modelled as a continuous panel or with no connection between the flanges of the joint. Adhesives have been represented successfully by spring elements connected between nodes, similar to the method for modelling spotwelds [4]. Solid brick elements have also been used to represent the adhesive, however there are ongoing issues regarding their attachment to shell elements [5] because they have usually been connected directly to the mid-plane nodes of the shells. This results in unrealistically thick bonds as the thickness of the shell is included in the adhesive bond thickness. Many previous studies, investigating failure stresses in adhesives have only modelled the joints in isolation [6]. Multiple elements have been used through the thickness of

the adhesive to refine the mesh because of the high stress gradient that result in the loaded adhesive joint. Volkersen first reported the effects of this phenomenon for joints analysed in two dimensions only [7]. This showed that the adhesive at the ends of a lap joint had higher shear stresses than in the centre. Further analysis has shown that this also coincides with the highest peeling stresses [8] and this is where failure usually initiates. For the work described in this paper, numerical methods have been developed for the new adhesive bondline methodology. The developments include material models, contact algorithms and element formulations to represent the adhesive bondline material, boundary and geometric non-linearity.

3. Adhesive bondline representation and properties The properties of a high impact performance epoxy and conventional toughened epoxies under different stress states have been extensively investigated and measured by the National Physical Laboratory (NPL) in the United Kingdom under the DTI Materials Measurement Programme [9-11]. The adhesive yield stress has been found to be temperature and strain rate dependent. NPL have measured these properties for the adhesives at temperatures from —40°C to +50°C and at strain rates from low speed quasi-static to high speed impact. Measurements have been made for shear, tensile and hydrostatic stress states. The adhesive is represented with eight node brick elements to maintain the true geometry of the bondline. The thickness of the adhesive is geometrically represented in

c

J

T-peel joint illustrating adhesive and adherend thickness

i

shell element upper outer surface shell elements

Shell finite element model T-peel joint illustrating adhesive and centreline representation of shell thickness

4

nodes

Contact algorithm constrains adhesive interface element to outer surface of shell elements (offset equal to Vz material thickness)

/

•

»

X »

shell element lower outer surface

shell element centreline

Fig. 2. Representation of adhesive in Finite element model of t-peel joint.

M. Tyler-Street et al /First MIT Conference on Computational Fluid and Solid Mechanics

the finite element model, even if it may be only 0.1 to 0.2 mm thickness. A typical t-peel joint and the finite element representation are shown in Fig. 2. This illustrates the representation of the adhesive bondline thickness and the offset from the shell element centre line. Several different integration formulations have been developed for the thin adhesive bondline elements to maintain numerical efficiency and robustness. A material model has been developed to provide the essential characteristics of the adhesives being studied, using the experimental data available to describe the adhesive properties. The adhesive material model is non-linear and incorporates strain hardening and a pressure sensitive damage criteria. For the adhesives being studied, the yield stress is dependent upon strain rate, temperature and a function of both the deviatoric and hydrostatic stresses and accumulated damage.

519

O4.0,

Displacement - (mm) Fig. 4. Force versus displacement results for box beams with and without adhesive — simulation.

4. Simulation results The axial impact of two configurations of a box beam, with and without adhesive bonding have been simulated. A high toughness epoxy adhesive was used for the weldbonded box beam. The deformed geometry results of the impact simulation for the weldbonded beam are shown in Fig. 3. The simulation predicts the regular folding and crumpling of the metal. The adhesive bondline representation has remained intact for the simulation. The resulting force versus displacement relation for the two beam configurations is shown in Fig. 4. The magni-

tudes of the initial peak force and the average collapse loads during stable collapse of the beam are similar to the experimental test results (Fig. 1). The results predict the increases in collapse load and corresponding reductions in deformation achieved for the weldbonded in comparison to the spot welded beams.

5. Conclusions A new methodology has been successfully developed to simulate the impact response of adhesively bonded structures. It has been applied to the simulation of a box beam impact with and without adhesive bonding. The preliminary results compare well with test results and predict the higher energy absorption of the weldbonded beam in comparison to the spotwelded beam. The methodology is compatible with current crashworthiness simulation methods. It is simple to implement into finite element models if the joint flanges are already defined. It is also being apphed to joint test specimens, structural components and full vehicle models to assess the adhesive bond performance and further develop the adhesive bonding methodology. The methodology requires further validation, particularly for the adhesive material model. Further investigation will improve the overall characterisation of the structural adhesive and procedures for measuring the necessary data to describe the material properties.

Acknowledgements

Fig. 3. Deformed geometry of a weldbonded beam.

We gratefully acknowledge the support of Ford Motor Company and ESI for the resources for undertaking this

520

M. Tyler-Street et al /First MIT Conference on Computational Fluid and Solid Mechanics

research programme and for permission to publish the results, and the support from NPL for measuring adhesive properties.

[6]

References

[7] [8]

[1] Davis RE, Powell JH. High impact performance weldbonded steel bodyshells. In: Materials for Lean Weight Vehicles conference Institute of Materials, 1995. [2] Automotive Engineer 1997;(Dec):30-31. [3] Baker A. Automotive Engineer 1996;(April/May):57. [4] Wagner DA. FEA (finite element analysis) modelling for body-in-white adhesives. SAE Paper No. 960784, 1996. [5] Steidler S, Durodola J, Beevers A. Modelling of adhesive

[9]

[10] [11]

bonded joints. In: TWI symposium on Design and Manufacture of Lightweight Steel Vehicles, 1998. Adams RD, Peppiatt NA. Stress analysis of adhesivebonded lap joints. J Strain Anal 1974;9(3): 185-196. Volkersen O. Luftfahrtforsch 1938;15:41. Goland M, Reissner E. Stresses in cemented joints. J Appl Mech 1944;2:A17-A27. Dean G, Hu F, Duncan B. The application of finite element methods to the design of adhesive joints. 1998 October, pp. 7-9 BENCHmark Dean G, Duncan B, Lord G. Design for impact of adhesive joints. In: Adhesion '99 Conference. Inst, of Materials, 1999. Dean G, Duncan B, Read B, et al. NPL Reports 2, 4 to 11. Performance of Adhesive Joints Project PAJ2, DTI Materials Measurement Programme, 1997-1999.

521

On the stability of the tunnel excavation front p. A. Vermeer, N. Ruse * Institute of Geotechnical Engineering, Stuttgart University, Pfajfenwaldring 35, Stuttgart 70569, Germany

Abstract One of the major problems in shield as well as NATM tunneling is the stability of the excavation front. In shield tunneling the pressure has to be chosen such that there is neither a cave-in nor a blow-out. In NATM tunneling front stability considerations determine the precise way of staged excavation. If necessary a supporting pressure is created by means of temporary anchors. This paper addresses the question whether or not a particular tunnel heading needs a supporting pressure. Secondly the question on the required magnitude of the pressure is considered. A simple design rule is derived using FE-data from large series of 3D-analyses. Keywords: TunneUng; Excavation front; Cave-in; Tunnel heading; Tunnel face; Face stability; Finite element analyses

1. Introduction During boring of tunnels the face stability is one of the major problems to the geotechnical engineer. Both in shield tunnelling and NATM tunneling the 'tube' is continuously secured by a lining and this will seldom lead to a loss of stability. On the other hand, such tunneling projects often suffer from instabilities of the tunnel heading. For undrained soil conditions, the most important stability parameter would seem to be the cohesion number c^/yD, where y is the unit soil weight and c^ the undrained soil strength. For circular tunnels, D represents the diameter. Atkinson and Mair [2] have shown that the minimum front pressure Pf/yD depends both on Cu/yZ) and H/D, where H is the height of the soil cover on top of the tunnel. Their work will be confirmed by data from 3D FE-analyses. The problem of drained face stability has been addressed by several authors, e.g. Anagnostou and Kovari [1]. Here soil strength is defined by an effective cohesion c' and a friction angle (p. Again this paper concentrates on two questions. The first question is whether or not a face pressure is needed to prevent a cave-in of a particular tunnel heading. Secondly, considering situations with small cohesion numbers, the question arises which minimum face pressure is needed to maintain stability. For answering these

* Corresponding author. Tel: +49 (711) 685-2071; Fax: +49 (711) 685-2439; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

questions, we will present results of non-hnear elastoplastic finite-element analyses. Subsequently these results are used to derive simple design rules. Restriction is made to elementary situations of homogeneous ground.

2. Undrained FE-analyses For reason of symmetry, we consider half a circular tunnel as shown in Fig. 1. The dark tunnel surface presents a lining with EI = 2.25 GNm^/m. This tunnel lining is modelled by linear-elastic 8-noded shell elements. In front of the lining a relatively narrow ring with a width of only 10 cm is left unsupported. The soil is divided into 15-noded prismatic elements with a 6-point Gaussian integration rule. The analyses use a linear-elastic perfectly-plastic stressstrain law with a Tresca failure criterion. A typical mesh is shown in Fig. lb consisting of 1898 prismatic elements. The theoretical basis of the method is fully described by Smith and Griffith [4]. Besides the free surface all outer boundaries are roller boundaries. In the initial situation no tunnel exists and a geostatic stress field with a^o = cr^o = KQ • a^o is assumed. We used different values for ^o in the range between 0.5 and 1.0 in order to seek for its influence on the limiting face pressure. However, computational results showed no difference. The first computational step is to remove the tunnel elements. Here equilibrium is not disturbed as an equivalent pressure is put both on the tunnel lining and on

522

P.A. Vermeer, N. Ruse/First MIT Conference on Computational Fluid and Solid Mechanics

(a)

MBii

(b)

11.0m

Fig. 1. (a) Geometry of tunnel, (b) FE-mesh with 15-noded prismatic elements. 6 j

1

^

—•

1 »

•

5

^

_Pf_^

Pn-P 0.2

FEM Kovari'96 Atkinson + Mair '81

-^=0.1

2 1 0.2

0.8

u[mm]

Fig. 2. Typical load-displacement curve. The displacement u refers to a point in the middle of the tunnel.

the face of the tunnel. Hereafter this pressure is stepwise decreased down to failure. A typical load-displacement curve is shown in Fig. 2. This figure shows that the load reduction finally leads to failure, i.e. continuing displacement at a constant load. The computed failure mechanism for this situation is shown in Fig. lb by means of shadings for the intensity of incremental displacements. A series of analyses for different depths were performed to investigate the dependence of the failure pressure pf on the depth-factor H/D as indicated in Fig. 3. The numerical data show excellent agreement with a semi empirical formula by Atkinson and Mair [2].

3. Drained FE-analyses Similar to the undrained situation, finite element analyses have been performed for the more general situation of a fully drained tunnel face. The most general situation would include effects of ground water flow, i.e. seepage forces, but as yet this has not been included in the FEanalyses. In the following we thus consider a tunnel above the groundwater table. Attention is first of all focused on homogeneous ground with y = 20 k N / m \ c = 5 kPa and (f = 27.5°. Similar to the previous section, different depths of the tunnel will be considered. FE-meshes, boundary conditions and computational procedures are exactly as described before. This means that

0

3^^' 0

1

2

3

4

5

H/D

Fig. 3. Normalized failure pressure pf for undrained soil conditions. the face pressure is stepwise removed until failures occur. Similar to the undrained analyses, an elastic-perfectly plastic stress-strain law is used, but this time we used a Mohr-Coulomb failure criterion instead of a Tresca criterion. The ground is taken to be non-dilatant so that the analyses involve a non-associated flow rule. For details of this well-known model the reader is referred to Brinkgreve and Vermeer [3]. In contrast to the undrained FE-analyses, the drained situation is analysed by using a so-called tension cut-off. Some results of the analyses are shown in Fig. 4. In order to view the failure mechanisms shadings are shown for displacement increments of the very last calculation steps. The shallow tunnel shows a clear chimney failure, whereas the deeper one tends to a local failure mode in front of the tunnel face. Failure pressures for 8 different values of H/D are plotted in Fig. 5. These values appear to be virtually independent of relative depth. The FE-data compares well with results from the model by Anagnostou and Kovari [1], but significant deviations occur for very small relative depth. In fact, FE-results suggest that there is no influence of depth at all. On using different friction angles in the wide ranges between (p = 20° and cp = 40°, we confirmed the negligible role of H / D . All data could be well approximated by the equation Pf = c-Nc-\-y with A^c =

-

D'ND -1

tan<^

and

ND

=

1 9 • tan (^

- 0.05

(1)

PA. Vermeer, N. Ruse/First MIT Conference on Computational Fluid and Solid Mechanics

H/D=0.5 1824elements

523

H/D=2 2403elements

Fig. 4. Computed failure mechanism for drained soil conditions and different depths.

^.- '

• • - '

•

_PL

yD

' 0

/

/"'

FEM

- - Kovari (p = 27.5°

yD

0-5

1

1,5

2

2,5

3

0.05°

H/D

Fig. 5. Failure pressure as a function of the relative depth for drained soil conditions. at least for the range of 20° < y? < 40°. It would seem that frictional soil behavior deviates from non-frictional soil behavior in the sense that there is strong arching around the tunnel face such that the depth of the tunnel is unimportant. In particular horizontal arching explains the unimportance of relative depth.

4. Closing remarks Tunneling tends to be dominated by experience rather than by model analyses. We tried to improve this situation by developing an extremely simple formula for assessing face stability. In contrast to the existing line of thinking, the depth of the tunnel appears to be unimportant.

For non-shielded tunnel boring machines (TBM's), support is often conveniently postponed and a large part of the tunnel roof and tunnel walls are left unsupported. In such a case the basic stability problem is different and therefor not addressed on this paper. However, also for such cases, 3D FE-analyses might be used to asses design rules. In this study the analytical model has been applied to soils rather than to rock, but its range of application extends to homogeneous rock mass. For jointed rock, a homogenization makes sense as long as the joint spacing is small compared to the diameter of the underground opening. For faulted rock, homogenization is seldom possible and one will have to consider the implications of individual faults for tunnel stability.

References [1] Anagnostou G, Kovari K. Face stability conditions with earth-pressure-balanced shields. Tunnel Underground Space Technology 1996;11(2):165-173. [2] Atkinson JH, Mair RJ. Soil mechanics aspects of soft ground tunneling. Ground Eng 1981;July. [3] Brinkgreve R, Vermeer R Plaxis — manual, Version 7. Rotterdam: AA Balkema, 1998. [4] Smith IM, Griffiths DV Programming the Finite Element Method 2nd edition. Chichester: John Wiley and Sons, 1998.

524

Numerical aspects of analytical solutions of elastodynamic problems A. Vermijt* Delft University of Technology, Department of Civil Engineering, 2628 CN Delft, The Netherlands

Abstract The analytical solution of problems of the elastodynamic response of a loaded half space can sometimes be expressed in the form of a Fourier or Hankel integral. The accuracy of the numerical evaluation of such integrals may be improved by representing the integrand as the sum of two parts, one to be evaluated analytically, and the remaining part to be evaluated numerically. The analytical part incorporates some of the main characteristics of the original function, such as the behaviour for small or large values of the integration parameter. The numerical part of the integral then may be easier and more accurate to calculate. Keywords: Elastodynamics; Hysteretic damping; Numerical integration; Analytical approximation; Hankel transform

1. Introduction

2. The mathematical problem

The development of high speed railways in Europe has created a renewed demand for predictions of the behaviour of soil masses under dynamic loads. The dynamics of an elastic half space z > 0 may be studied using analytical or numerical solution methods. The basic equations are the equations of elastodynamics, with appropriate boundary conditions. The material is supposed to be linear elastic, characterized by a shear modulus /i and Poisson's ratio y. Because soils under cyclic or dynamic loading may exhibit a certain amount of hysteretic damping, this effect is introduced into the model by assuming a visco-elastic behaviour such that the relaxation time tr is inversely proportional to the frequency co of the loading. This means that the damping ratio may be defined as f = cotr, with f being independent of the frequency. If the problem is analysed in the frequency domain, using Fourier analysis, the damping ratio ^ can be considered as a constant. For such a material problems of elastodynamics can be solved relatively easy, for instance for a load moving with constant velocity (Verruijt [1]).

The problem considered refers to a uniform vibrating load pocos{(jot) over a circular area of radius a on the surface z = 0 of an elastic half space z > 0. Using the Hankel transform method (Sneddon [2]), the solution for the displacement of the origin, i.e. the point r = 0, z = 0, is found to be

*Tel.: +31 (15) 278-5280; Fax: +31 (15) 278-3328; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

oo

w = —Re

[po^exp(^ r

]

(1)

where u = coa/Cs, the dimensionless frequency of the load, y = r/a, the dimensionless radial coordinate, and the function F(x) is defined by F(x) =

(x^ + a2)2 _ 4x^apas

(2)

The parameters ap and as are given by al=x'-

(3)

1 + 2/f

1 1 + 2/^ where m is an elastic parameter, -2

" M

A. + 2/x

1 - 2V

2(1 - v)

(4)

(5)

525

A. Verruijt / First MIT Conference on Computational Fluid and Solid Mechanics and f is the damping ratio, which is considered as a material constant, denoting the relative loss of energy due to friction in a complete deformation cycle. It may be noted that the denominator of the function F{x) has a minimum for a value of x slightly larger than 1. If there is no damping (^ = 0) this minimum is zero, and corresponds to the Rayleigh wave singularity. When the damping ratio is not zero, the denominator will not be zero in the integration interval, but it may become very small if the damping ratio is small, thus causing very large values of the integrand in Eq. (1). It seems unlikely that a complete analytic solution of the Hankel integral (1) can be found. Therefore a numerical integration method will be used.

3. Numerical integration The integral in Eq. (1) can be evaluated numerically by separating the infinite interval from ;c = 0 to x = oo into a number of subintervals of gradually increasing length, separated at the points x = 1, 2, 5, 10, 100, 1000, 10000, and perhaps one or two more when necessary for convergence at infinity. In each subinterval the integral is calculated using Simpson's method. Because near .x = 1 the integrand may become very large the number of steps in the first few subintervals should be taken larger than in the other subintervals. In the numerical procedures used here the number of steps is taken as lOn, 25n, I0n,2n,n,n,n, respectively, where n is some basic value, say n = 20. The total number of subdivisions then is TV = 50n = 1000. The results of the numerical integration are shown in Fig. 1. The parameter I/I, plotted in the figure as a function of the dimensionless frequency u, is the ratio of the amplitude of the dynamic response w to the static response Wg, the limiting value for The vertical displacement of the origin shown in Fig. 1 seems to include a random numerical inaccuracy, which makes all results unreliable. The accuracy can, of course, be improved by taking smaller integration steps. The results for a numerical integration using A^ = 10000, with 10 times

1.0

^ 1 1 1 1 1 1 1 1 1 1 1 1 1

N

NN

M

\J M M

0.0

KM

0

J

|\Lxfn 1 T^J J^

5

10 u

15

20

Fig. 2. Numerical integration, 10000 steps. more integration steps in each interval, are shown in Fig. 2. At a cost of ten times more computation time, the errors have indeed been diminished, but there still seem to be some inaccuracies.

4. Partly analytical integration In order to improve the accuracy of the Hankel integral the function F{x) may be separated into two parts, with one of these parts allowing for analytical integration. For this purpose the function is written as , ,

F(x) =

VJC2 + b^

+

F\x),

(6)

where

sin p + / cos fi 1 + 2/^ (7) 2(1 - m)' 2(1 - m ) v ^ ( l + 4^^)1/4' where fi is defined by tan2y^ = 2^. The constants d and b have been chosen such that the behaviour of F(x) for X ^ 0 and for jc -> oo is approximated by the first part of Eq. (6). This means that the function F'(x) and its derivative will be zero foYx = 0 and will approach zero very rapidly as JC ^^ oo, so that it lends itself better for numerical integration than the original function. Substitution of Eq. (6) into Eq. (1) gives, using a standard Hankel transform from the tables by Bateman [3], oxp(i(jL>t)

W =

Wn —

/x

Re (1 + 2/^2

)Ji(xu)dx\,

(8)

where Wa is the analytical part of the integral,

1.0

Poa ^ \ / v^exp(-/y^) exp(icot) -[1 — &xp(—ub)] Rei (1+4^2)1/4

0.0

KLf R i 10

(9)

^

•/yHn^ 15

Fig. 1. Numerical integration, 1000 steps.

20

The results of this partly analytical, partly numerical evaluation of the Hankel integral is shown in Fig. 3, using only 100 intervals for the numerical integration. In this case using more intervals (e.g. A^ = 1000 or N = 10000) has no discemable effect on the results. Actually, the analytical part of the integral alone already shows most of the features of the complete solution.

526

A. Verruijt/First MIT Conference on Computational Fluid and Solid Mechanics

1.0

6. Conclusions

Fig. 3. Partly analytical integration, 100 steps. 5. Example of application The integration techniques outUned above have been used to solve the problem of a vibrating circular load on an elastic half space with hysteretic damping. As an example the surface displacements at a certain instant of time are shown in Fig. 4. The soil data for this case are y = 0, f = 0.05 and coa/Cs = 1, where co is the frequency of the load, a is the radius of the loaded area, and Cs = V M / P is the velocity of shear waves in the medium. The figure indicates that an axially symmetric wave is emanating from the loaded area, as expected. By performing the computations for various values of the soil parameters, it is found that the radial attenuation of the amplitude of the waves mainly depends upon the damping ratio f.

It has been shown that analytical solutions of elastodynamic problems may be solved by a combination of an integral transform method and numerical integration. If a fully analytical integration is impossible it may be worthwhile to at least using a partially analytical integration of an approximation of the integrand. In the present case the behaviour for small and large values of the integration parameter has been incorporated in the analytical integration. A further improvement might be possible by including the Rayleigh singularity, but this has not yet been accomplished. The integration technique can be used for the analysis of wave propagation in an elastic half space with hysteretic damping.

References [1] Verruijt A. Dynamics of soils with hysteretic damping. Proc 12th Eur Conf Soil Mechanics and Geotechnical Engineering. Rotterdam: Balkema, 1999, pp. 1-14. [2] Sneddon IN. Fourier Transforms. New York: McGraw-Hill, 1951. [3] Bateman H. Tables of Integral Transforms. New York: McGraw-Hill, 1954.

Fig. 4. Surface displacements for vibrating load.

527

Finite element modeling for surgery simulation Marina Vidrascu^'*, Herve Delingette^, Nicholas Ay ache ^ " INRIA Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France ^ INRIA Sophia Antipolis, BP 93, 06902 Sophia-Antipolis, France

Abstract In this paper, we discuss an approach using finite element models to design a surgery simulator. The main problem is that the desired realism of the mechanical model must be balanced against the need for speed of a real-time computation. For the human liver, a highly non-linear incompressible material, an accurate mathematical model based on non-linear elasticity cannot be used in practice for real-time simulations. Our approach considers two models: a linear coarse model which respects the real-time requirements; and a reference non-linear model for which accuracy is the main concern. The numerical predictions of these two models are compared and allow to define a real-time simulator which is both accurate and robust. Keywords: Biomechanics; Surgery simulation; Finite element; Linear and non-linear elasticity; Domain decomposition

1. Introduction

2. The real-time model

Laparoscopic techniques in liver surgery reduce operating time and morbidity. This novel technique has several advantages for the patients, but it is more complex and, in particular, it requires a perfect hand-eye coordination. As the mechanical properties of the tissues drastically change after death and, since, for obvious ethical reasons, the use of living animals is limited, surgery simulation remains the only appropriate training tool. Two components are needed to build a simulator: graphics to give realistic views of the surgery scene and provide surgeons with a visual illusion; and haptic interface obtained by force-feedback computation which gives the illusion of sensing. An accurate modeling of the deformation of the human liver remains a challenge. A realistic model is very complex as it should take into account the characteristics of the real material, i.e. at least the anisotropy due to the presence of the parenchyma and of the GUsson capsule. Such a model is clearly not affordable for real-time computations at present. The originality of our approach is that the comparison between the two models emphasizes the importance of the various components in the model.

A Surgery Simulator was developed at INRIA in collaboration with the IRCAD institute. In order to allow for real-time simulations, the liver is modeled by a simplified linear model [2]. The liver is treated as a linearly elastic nearly incompressible isotropic body. The problem to solve is then a standard linear elasticity problem. Real-time simulation can then be achieved by precomputing the response of the structure to a set of imposed displacements and constructing the corresponding Schur complement (transfer) matrix. This precomputed model is well-suited for computing the deformed mesh under the displacement constraints imposed by a virtual surgical tool. However, it does not allow any topology changes since the matrix inversion cannot be performed in real-time. Therefore, a dynamic finite element mesh using an explicit scheme of integration is also used at parts where the user is supposed to perform the cutting [4]. The precomputed and dynamic meshes can be combined into a hybrid mesh such that deformations and reactions forces are continuous across the two different meshes. A specific data structure allows to efficiently update the rigidity matrix when tetrahedra are removed (see Fig. 1).

* Corresponding author. Tel.: -h33 (1) 3963-5420; Fax: +33 (1) 3963-5882; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

528

M. Vidrascu et al /First MIT Conference on Computational Fluid and Solid Mechanics

Fig, 1. Real-time simulation of liver resection based on a linear elastic model.

domain is split into non-overlapping subdomains, local solutions are computed by direct solvers and an efficient iterative algebraic solver is used to properly interface these solutions. The finite element discretization uses P2 — PO elements. In this case, the displacement field is approximated by second order P2 Lagrange finite elements and piecewise constant PO elements are used to impose the incompressibility constraint and approximate the hydrostatic pressure field. Finally, the anisotropic membrane which envelops the liver (the Glisson capsule) has been treated as a hyper-elastic thin shell in large displacements, and its displacements are approximated by non-linear DKT shell triangular finite elements [1]. For kinematic compatibility reasons, we have identified the shell nodes with the three-dimensional nodes. This means that each boundary face of a volumic secondorder finite element corresponds to four DKT triangles.

3. The reference model

4. Comparison between models

A more realistic non-linear biomechanical model was developed. The liver is supposed to be a hyper-elastic incompressible material. In the reference configuration, the displacement field u{x) is the solution of the following variational problem:

The same finite element mesh is used for both linear and non-linear model. This mesh respects the real geometry of the liver. For this purpose, the outer surface of the liver under study is first reconstructed automatically from a three-dimensional medical image, using image reconstruction techniques as described in [2]. This step produces a smooth definition of the external surface endowed with a triangular surface grid. An unstructured tetrahedral mesh of the interior is obtained by using GHS3D, an automatic three-dimensional mesh generator [5]. Note that the use of domain decomposition allows to significantly improve computational time (for one Newton iteration we need about 10 h on a workstation and 3 min for our domain decomposition method), but we are still far away from real-time. The first comparison shows that from a visual point of view the difference between the linear and non-linear deformed configuration is irrelevant, on the other hand, as

/

aw

Vudjc

at

f^ 'Vdx+

j f

-vda,

Wve (Q)

where F = Id -\-Wu is the deformation gradient, / " and / ^ are the body forces and surface tractions. The MooneyRivlin hyper-elastic constitutive energy (which depends on (/i, h, h) the invariants of the right Cauchy Green tensor F'^ F) is given by: W(F) = C i ( / i - 3 ) + C2(/2-3) + a(J^ - 1) - (2Ci + 4C2 + 2a) log J The problem to solve is highly non-linear due to the specific form of the local energy density as a function of VM. Up to now, Newton type algorithms with arc length continuation [6] appear as the only robust technique for such problems. Their drawback is their cost, both in CPU time and in memory. In particular, for incompressible materials the tangent matrix obtained at each Newton iteration is very ill conditioned and it is then mandatory to use a direct solution method. The algorithms must therefore be adapted to modem powerful parallel computers. A good alternative is to use a domain decomposition method to solve each tangent problem. The domain decomposition algorithm used here is based on a generalized Neumann-Neumann preconditioner [7]. The original

Fig. 2. Deformations for the 3D model.

M. Vidrascu et al. /First MIT Conference on Computational Fluid and Solid Mechanics

529

in order to make the necessary trade-off between computational efficiency and biomechanical realism.

References [1] Carrive M, Le Tallec P, Mouro J. Approximation par elements finis d'un modele de coques minces geometriquement exact. Revue Europeenne des Elements Finis 1995, pp. 633662. [2] Cotin S, Delingette H, Ay ache N. Real-time elastic deformations of soft tissues for surgery simulation. IEEE Trans Visual Comput Graph 1999;5(l):62-73. [3] Delingette H. Toward realistic soft-tissue modeling in medical simulation. Proceedings of the IEEE, Special issue on virtual and augmented reality in medicine 1998;86(3):512-523. [4] Delingette H, Cotin S, Ay ache N. A hybrid elastic model allowing real-time cutting deformations and force feedback for surgery training and simulation. In: Thalmann N, Thalmann D (Eds), Computer Animation (Computer Animation '99). IEEE Computer Society, May 1999, pp. 70-81. [5] George PL, Hecht F, Saltel E. Automatic mesh generator with specified boundary. Comput Methods Appl Mechan Eng 1991;92:269-288. [6] Le Tallec R In: Ciarlet PG, Lions JL (Eds), Numerical Methods for Nonlinear Three-Dimensional Elasticity, Vol 3. 1994, pp. 465-622. [7] Le Tallec P, Vidrascu M. Solving large scale structural problems on parallel computers using domain decomposition. In: Solving Large Scale Problems in Mechanics, John Wiley and Sons, 1997, pp. 49-82. [8] Le Tallec P, Vidrascu M. Efficient solution of mechanical and biomechanical problems by domain decomposition. Numer Linear Algebra Appl 1999;6:599-616.

Fig. 3. Deformation for the 3D model with external shells. expected, the model with external shells is smoother that the non-linear one (see Figs. 2 and 3). The more spectacular difference is between reactions in a configuration with or without shells (Fig. 4). Nevertheless, given that the surgeon's reaction cannot be meaningfully quantified, the computation accuracy of the reactions is non-significant.

5. Conclusion The linear elastic model is well-suited for fast computation of deformable volumetric soft tissue models. However, the development of reference non-linear elastic models is necessary in order to quantify the errors induced by these simplified models. Our objective is to increase the biomechanical complexity of both real-time and reference models

178.7-

73

r e 0 c i i

03/06/99 reac_lnl.data NOMBRE DE COURSES : 2 EXTREMA EN X

119.2-

/

:

.OOE+00

2.2

.OOE+00

.18E+03

,H—*.—#

: 3D+SHELL

59.6-

_...-'"

TRACE DE COURSES

0.0-1 00

' '

'' 1

0.7

' '

22

1.5 d i s p 1 Q c 9 mg

ni

Fig. 4. Comparison between the 3D and 3D with shells models: reactions.

530

Distributed memory parallel computing for crash and stamp simulations S. Vlachoutsis*, J. Clinckemaillie Engineering Systems International Group, 20, Rue Saarinen, Silic 270, 94578 Rungis Cedex, France

Abstract The parallelization of the PAM-CRASH and PAM-STAMP codes for the distributed memory model is described briefly. Subsequently attention is focused on the parallelization of the adaptive meshing technique used primarily for metal stamping simulations. Numerical applications for a full car crash simulation as well as for stamping problems with adaptive meshing show the efficiency of the implemented model. Keywords: Parallel computing; Distributed memory; Message-passing program; Crashworthiness; Stamping; Adaptive meshing

1. Introduction Numerical simulation of crash and stamping problems in the design and production phases is now well established and commonly incorporated in industrial practice. The increasing needs of intensive computing is a serious challenge for engineering and computer communities. Distributed memory parallel computing (known also as message-passing or multiple-address-spaces parallel computing [1]) is an efficient solution which can be applied to a wide range of hardware platforms: dedicated parallel computer systems, clusters of engineering workstations etc. The PAM-CRASH and PAM-STAMP codes are explicit timemarching finite element codes used for the numerical simulation of the highly nonlinear dynamic phenomena arising in short-duration, contact-impact problems (see [2,3] for more information). In order to run the PAM-CRASH and PAM-STAMP codes in distributed memory parallel computing environments, new algorithms and methods were designed, implemented and used (see [4-7]). The principle is to split the structure into pieces (domains) using a domain decomposition method [8] and then apply the sequential code to each domain. To achieve a consistent solution, communications of forces and velocities at the interface nodes between processors are carried out accord* Corresponding author. Tel.: +33 1 49 78 28 37; Fax: +33 1 46 87 72 02; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ing to pre-established mailing lists, describing the domain connectivity. After each time step, the results of all processors are assembled in order to obtain the response of the total structure. Section 2 describes the parallelization of the adaptive meshing technique which was implemented in the PAM-STAMP code. Sections 3 and 4 present results for car crash and stamping simulations respectively. The conclusions and the references close the paper.

2. Adaptive meshing parallelization Adaptive meshing consists in subdividing the initial finite elements into smaller elements (h-type) or in using higher order shape functions for the initial mesh (p-type) when a re-meshing criterion is satisfied. The adaptive meshing technique of PAM-STAMP is of the h-type and the criterion used is an upper limit of the angle between adjacent shell elements. A typical stamping test case includes the blank sheet, which is deformed, and the tools (die, punch, blank holder, draw beads), which are rigid bodies. Adaptive meshing is only applied to the blank sheet. According to the domain decomposition partitioning used for stamping, a piece of blank sheet is computed by each processor. The variation of the number of elements and nodes results in additional difficulties for the parallel version with the distributed memory model. The principle of parallelization consists in the application of the adaptive meshing

S. Vlachoutsis, J. Clinckemaillie /First MIT Conference on Computational Fluid and Solid Mechanics algorithm to each sub-domain. The total response of the new mesh remains consistent by dynamically updating the mailing lists of interfaces between processors. For the contact-impact treatment, the blank sheet is distributed over processors according to the initial decomposition, while a copy of the tool geometry resides on every processor. Contact mailing lists for forces and coordinates/velocities are dynamically updated for the blank sheet. In the case of a force-driven press, the tool geometry is updated using the send/receive mechanism, while the update is local (inside each sub-domain) if tools are displacement-driven. The parallel efficiency may be adversely affected by load unbalance, caused by the non uniform refinement of the mesh. Load balance may be maintained by the dynamic redistribution of the elements of the refined mesh [9], but this was not applied to the test cases presented in the sequel. In the following some notations are used: A^ is the number of processors T^ is the elapsed time for a run with N processors SM is the speed-up for A'^ processors; by definition: SN =

40

60

80

100

531

120 140

Number of processors

Fig. 2. Results for car crash (CI) on 128-processor computer SGI Origin 3000 (400 MHz).

4. Stamping test cases (SI and S2) Adaptive meshing is appUed to both cases. The domain decomposition used is the Linear Coordinate Bisection method [8] and the parallel platform is PVM [11]. 4.1. Stamping test case 1 (SI)

TI/TM

3. Car crash test case (CI) A complete car is modeled with 111 852 nodal points, 119 284 shell elements, 504 beam elements, 303 rigid bodies and 11 contact interfaces (90% of the shell elements). The test is a frontal crash to a rigid wall and the total simulation time is 0.070 seconds. This test is run using two different platforms: (a) Vclass 16-processor Hewlett Packard (HP) computer and (b) 128-processor SGI Origin 3000 (400 MHz) computer. For both cases the domain decomposition used is the Multilevel Spectral Bisection method [8], and the parallel platform is MPI [10]. The total number of time steps is 58437. Figs. 1 and 2 illustrate results for (a) and (b) respectively. The sequential time is for (a) Ti = 126400 s and for (b) Ti = 60584 s. Elapsed time decreases for (a) from 35 h (A^ = 1) to 3 h (A^ = 15) and for (b) from 17 h (A^ = 1) to 23 min (A^ = 126). For (b) speed-up reaches values of 45 for 126-processor test. Both tests confirmed good speed-up and scalability.

The model used has 20 836 nodal points, 20 945 shell elements, 668 beam elements (representing the draw beads), 5 contact interfaces. The total simulation time is 0.031 seconds. Due to adaptive meshing, the number of shell elements of the blank sheet increases from 3036 initially to 30 078 at the end of the simulation. The machine used is the Vclass 16-processor HP computer and the sequential time is Ti = 19 810 s. Fig. 3 resumes speed-up results. 4.2. Stamping test case 2 (S2) This model has 31737 nodal points, 33 388 shell elements, 631 beam elements (representing the draw beads), 4 contact interfaces. The total simulation time is 0.072 seconds. Because of adaptive meshing the number of shell elements of the blank sheet increases from 5632 initially to 53 475 at the end of the simulation. The machine used is a SUN SPARC 10-processor computer and the sequential time is Ti = 5 8 020 s. Fig. 4 resumes speed-up results.

10 a ?

•o

.—•

8 6

I 4

(A

2 0

2

4 6 8 10 12 Number of processors

14

16

Fig. 1. Results for car crash (CI) on Vclass 16-processor HP computer.

0

L>

t

4 6 8 10 12 Number of processors

14

16

Fig. 3. Results for stamping test case 1 (SI) on Vclass 16-processor HP computer.

S. Vlachoutsis, J. Clinckemaillie / First MIT Conference on Computational Fluid and Solid Mechanics

532 o -

Q.4

I•6

00

2

4

6

8

10

Number of processors

Fig. 4. Results of stamping test case 2 (S2) on lO-processor SUN computer. 5. Conclusions For all test cases the speed-up obtained are satisfactory for industrial use. Scalability remains good even for a high number of processors. Slightly lower values of speed-up were obtained for the stamping cases with respect to the crash test case but this would be expected because adaptive re-meshing applied only to the former. More improvements for increasing speed-up are under investigation. In conclusion, numerical results confirm that parallel computing with distributed memory model (multiple-address-spaces model) is a powerful tool for crash and stamp simulations.

Acknowledgements The authors thank Dr. H. Chevanne from SGI Paris for running the SGI Origin 3000 test case.

References [1] Hennessy JL, Patterson DA. Computer Organization and Design. San Francisco, CA: Morgan Kaufmann, 1998.

[2] Zienkiewicz OC, Taylor RL. The Finite Elements Method, 4th edition, Volume 2. London: McGraw-Hill, 1991. [3] PAM-CRASH, PAM-SAFE Theory Notes Manual, Version 2000, ESI Group. [4] Lonsdale G, Clinckemaillie J, Vlachoutsis S, Dubois J. Communication requirements in parallel crashworthiness simulation. In: Gentzsch W, Harms U (Eds), Lectures Notes in Computer Science 796. Proceedings of HPCN Europe 1994, New York: Springer-Verlag, pp. 55-61, 1994. [5] Lonsdale G, Eisner B, Clinckemaillie J, Vlachoutsis S, de Bruyne F, Holzner M. Experiences with industrial crashworthiness simulation using portable, message-passing PAM-CRASH code. In: Herzberger B, Serazzi G (Eds), Lecture Notes in Computer Science 919: Proceedings of HPCN Europe 1995. New York: Springer-Verlag, pp. 856862, 1995. [6] Clinckemaillie J, Eisner B, Lonsdale G, Meliciani S, Vlachoutsis S, de Bruyne F, Holzner M. Performance issues of the parallel PAM-CRASH code. Int J Supercomput Appl High Perform Comput 1997;11(1):3-11. [7] Lonsdale G, Petitet A, Zimmermann F, Clinckemaillie J, Vlachoutsis S. Programming crashworthiness simulation for parallel platforms. Math Comput Modell 2000;31:61-76. [8] Floros N, Reeve JS, Clinckemaillie J, Vlachoutsis S, Lonsdale G. Comparative efficiencies of domain decompositions. Parallel Comput 21, 1823-1835. [9] DRAMA ESPRIT LTR Project N*" 249533, Dynamic reallocation of meshes for parallelfiniteelement applications. World-wide web document: http://www.cs.kuleuven.ac.be/ cwis/research/natw/DRAMA.html [10] MPI: A Message-Passing Interface Standard, Message Passing Interface Forum, June 12, 1995. See: http://www.mpiforum.org [11] Geist A, Beguelin A, Dongarra J, Jiang W, Manchek R, Sunderam V PVM: Parallel Virtual Machine. MIT, 1994.

533

The first-kind and the second-kind boundary integral equation systems for some kinds of contact problems with friction Roman Vodicka * Technical University ofKosice, Faculty of Mechanical Engineering, Department of Applied Mechanics, 041 87 Kosice, Slovak Republic

Abstract A boundary integral equation (BIE) solution of contact problem is presented. Two ways of defining BIE systems are shown, introducing the first-kind BIE and the second-kind BIE for solution by Galerkin boundary element method (BEM). The systems of BIEs are formed of both displacement and traction BIEs. The contact problem is assumed to obey the Coulomb friction law. Results of a numerical example are also presented. Keywords: Boundary integral equation; Second-kind boundary integral equation; Boundary element method; Galerkin boundary element method; Contact problem

1. Introduction The direct BEM is a numerical method for solution of BIE. However, the equation for boundary tractions is not usually under the consideration for contact problems. The idea of simultaneous application of both displacement BIE and traction BIE for multi-domain problems, as an application field of symmetric Galerkin BEM, see Gray and Paulino [2], helped the author to introduce systems of BIEs for frictionless contact problems presented in Vodicka [3]. The present paper discusses how similar systems of BIEs can be defined, when friction is taken into the consideration.

though in incremental notation, to obtain contact unknowns Aw* and AM;"^ in the slip and adhesion zones, respectively. Similarly, vectors Aiu^* and AM;^'' are introduced for given contact data. However, this is not exactly truth, as slip condition of coulombian friction is not included in contact functions, but a relation of the form Aw^i = ±/xAw;4,

(1)

should be satisfied for pertinent components of contact functions. The other components of the vectors with the superscript index '0' vanish according to contact relations. The sign of /x — friction coefficient — in Eq. (1) depends on the slip direction.

2. Contact problem 3. BIEs for contact problem Contact problem with Coulomb friction is taken into account. According to this requirement the contact zone can be divided into two parts: the adhesion zone and the shp zone. The relations between contact displacements and tractions are defined by the system of contact variables introduced in Vodicka [3]. The same set of functions is used.

*Tel.: +421 (95) 633-5312; Fax: +421 (95) 633-4738; E-mail: vodicka @ ccsun.tuke. sk © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

A formal representation of BIE system for the contact problem is founded on that defined in Vodicka [3]. The split of the contact zone, however, makes it possible to distinguish between the slip and adhesion zone. Nevertheless, the integral operator on the left-hand side of pertinent BIE system is either the first-kind or the second-kind integral operator, but the other side of the system contains an unknown function AUJ^J. Its remedy is treated in Section 3.2.

534

R. Vodicka /First MIT Conference on Computational Fluid and Solid Mechanics

3.1. A note about BEM discretisations The Galerkin method with element-wise Unear approximations is used for the first-kind BIE systems as well as for the other one. Such an approximation allows us to regularize the integrals in computation. The way of discretisation for frictionless case was shown in Vodicka [3], though standard collocations for the second-kind BIE system are considered there. On the other hand, in the present paper, the Galerkin method is also used for this BIE system in order to make a more compact comparison of both BIE systems. 3.2. An iterative method for numerical solution When a BIE system is to be solved numerically for given split of contact zone, it reduces to a discretised equation system of the form ,,(^-1)

(2)

3^^^=/^^^. The matrix is split into two parts such that the parts with either index '2' are created for items belonging to

contact functions Aii;4 on the left-hand side and /S.w^\ on the right-hand side. All other variables are gathered in the other blocks. The matrix jLt is diagonal, with i/x diagonal terms, depending on the slip direction.

4. Numerical results A simple numerical example will be considered. A rectangular punch is pressed against an elastic foundation. The punch is loaded by a uniformly distributed vertical pressure and by a lateral pressure, the total amount of which is balanced so that the punch tends to slide. The normal contact tractions r„ are singular at both comer points A and B of the punch, at least for small values of /x. The results of numerical tests and also the analytical singularity orders are given in Table 1, provided that the punch is sliding in the direction A -> B. A logarithmic scale plot is made in Fig. 1. The dashed lines at the pictures correspond to the analytical computations, the lines with open symbols belong to the results of the first-kind BIE, the lines with filled symbols to the other BIE. The symbol 5„ is used to indicate the distance along the segment AB from the point a = A ox B. Emphasizing that the analytical solution gives only an order of singularity, the graphs of this figure show actu-

Table 1 Singularity orders at points A and B compared with analytical results of Comninou [1] B

^l

0.00 0.10 0.20 0.25 l/TT

Ist-kindBIE

2nd-kind BIE

-0.21220 -0.14664 -0.04752 +0.03913 +0.36337

-0.21824 -0.15443 -0.06145 +0.01624 +0.28042

Analytical

Ist-kindBIE

2nd-kind BIE

Analytical

-0.22599 -0.17074 -0.10554 -0.06662 0.00000

-0.21220 -0.26460 -0.31013 -0.33121 -0.35865

-0.21824 -0.27036 -0.31633 -0.33781 -0.36597

-0.22599 -0.27512 -0.32008 -0.34137 -0.36943

Point A

s^(mm)

Point B

0.1

SgCmm)

Fig. 1. Normal contact tractions in logarithmic scale near the end points,

1

R. Vodicka /First MIT Conference on Computational Fluid and Solid Mechanics ally a good agreement between numerical and analytical solution.

5. Conclusions The simultaneous use of both displacement and traction BIEs for contact problems has been dealt with. An iterative way of solution for frictional contact is presented. The algorithm has been successfully tested by a simple example. Numerical experiments with more involved examples will be necessary in the future.

535

References [1] Comninou, M. Stress singularity at a sharp edge in contact problems with friction. J Appl Maths Phys (ZAMP) 1976;27:493-499. [2] Gray LJ, Paulino GH. Symmetric Galerkin boundary integral formulation for interface and multi-zone problems. Int J Num Methods Eng 1997;40:3085-3101. [3] Vodicka R. The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems. Eng Anal Bound Elem 2000;24:407-426.

536

On the computation of finite strain plasticity problems with a 3D-shell element W. Wagner^'*, S. KlinkeP, F. Gruttmann'' " Institutfur Baustatik, Universitdt Karlsruhe (TH), D-76131 Karlsruhe, Germany ^ Institut fur Statik, Techn. Universitdt Darmstadt, D-64283 Darmstadt, Germany

Abstract In this paper, we develop a finite element model for thin shell structures based on an eight-node brick element. An efficient and accurate element behaviour can be achieved using assumed and enhanced strain methods. Based on a hyperelastic orthotropic material model of St. Venant-Kirchhoff type and a nonlinear material law for finite strain /2-plasticity a wide variety of problems can be discussed. Keywords: Shell element; 3D formulation; Finite strain plasticity

1. Introduction The three dimensional efficient computation of thin structures in structural mechanics requires reliable and robust elements. In the past, several shell elements have been developed, where the normal stresses in thickness direction have been included. The basic associated variational formulation of the applied HAS methods has been developed by Simo and Rifai [5]. A geometrical nonlinear formulation in terms of the Green-Lagrangian strain tensor may be found in Betsch et al. [2] among others. However, for certain problems nodal degrees of freedom at shell surfaces are more advantageous. Examples are deformation processes with contact and friction or the delamination problem of layered shells.

2. Scope of the element formulation In this paper, a continuum based 3D-shell element for laminated structures is derived. The basis of the present element formulation is an eight-node brick element with tri-linear shape functions and displacement degrees of freedom. Thus, boundary conditions at the top or bottom surface of the brick-type shell element can be considered and standard CAD-data and mesh generators can be used. * Corresponding author. Tel.: -h49 (721) 608-2280; Fax: -1-49 (721) 608-6015; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The relative poor behaviour of the standard displacement element is improved using the assumed strain method and the enhanced assumed strain method. The associated variational formulation is written in a Lagrangian setting using the Green-Lagrangian strain tensor. This yields the same geometric stiffness matrix as in the pure displacement formulation and may be simpler than the corresponding formulation in terms of the deformation gradient. With restriction to physical linear behaviour, we implement a hyperelastic, orthotropic, three-dimensional constitutive equation of the St. Venant-Kirchhoff type. Hence the components of the constitutive tensor are given with respect to the convective coordinate system. In shell theory thickness, integration of stresses and linearized stresses yields the stress resultants and the shell stiffness. Stress resultants are not introduced in this paper. Here, virtual work expressions and associated linearizations are integrated in thickness direction. This simplifies the finite element formulation. A physical nonlinear material law for finite strain ^2-plasticity is implemented which bases on a multiplicative split of the deformation gradient in elastic and plastic part. The evolution law of the plastic strains and the internal variable are derived from the principle of maximum plastic dissipation. The numerical realization is proposed in several papers, see e.g. Simo [4]. Simo introduced an implicit exponential integration algorithm to integrate the evolution equation of the plastic strains. A logarithmic strain measure

W. Wagner et al /First MIT Conference on Computational Fluid and Solid Mechanics

537

OX

leads to an additive projection algorithm. Here we use a Lagrangian formulation of the flow rule, see also Ibrahimbegovic [3]. A radial return algorithm is developed with a spectral decomposition of the tensor fields.

3. Example Some numerical examples demonstrate the good performance of the developed element for anisotropic laminated materials and in the range of finite strain plasticity. As a representative example, we investigate an elastoplastic computation of a conical shell. The slightly modified geometry data are taken from Ba§ar and Itskov [1], who investigated the conical shell with an Ogden material. System and material data are given in Fig. 1. Considering symmetry, only a quarter of the shell is discretized with 8 X 8 X 1 elements and a nine-point Gauss integration. Fig. 2 shows the load deflection curve of the vertical displacement w of the upper outside edge.

References [1] Basar Y, Itskov M. Finite element formulation of the Ogden material model with application to rubber-like shells. Int J Num Methods Eng 1998;42:1279-1305. [2] Betsch P, Gruttmann F, Stein E. A 4-node finite shell element for the implementation of general hyperelastic 3Delasticity at finite strains. Comp Methods Appl Mech Eng 1996;130:57-79. [3] Ibrahimbegovic A. Finite elastoplastic deformations of space-curved membranes. Comp Methods Appl Mech Eng 1994;119:371-394.

Fig. 1. Finite element mesh with geometry and material data. [4] Simo JC. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comp Methods Appl Mech Eng 1992;99:61-112. [5] Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes. Int J Num Methods Eng 1990,29:1595-1638.

S

eq. pi. strains

1

1.5

Displacement w

Fig. 2. Load deflection curve and equivalent plastic strains.

0.45 0.36 0.27 0.18 0.09

538

Radial point interpolation method for no-yielding surface models J.G. Wang *, G.R. Liu Centre for Advanced Computations in Engineering Science, c/o Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore SI 19260, Singapore

Abstract This paper proposes a numerical algorithm for endochronic constitutive law using radial point interpolation method. First, a radial point interpolation method (radial PIM) is presented, which extends the compactly supported radial function to include polynomials terms. Then, a weak form based on reference constitutive law is developed in an incremental form. The accuracy of radial PIM is evaluated by a linear function and a complex function. As an example, an endochronic constitutive law for normally consoUdated soils is used to exploit the appUcabiUty of the proposed method in foundation problem. Keywords: Radial PIM method; Compact support; Polynomial reproduction; Reference constitutive law; Endochronic theory; Direct integration

1. Introduction Solution of elastoplastic problems using meshless method is an interesting topic in computational mechanics. Some publications on nonlinear or elastoplastic models are published within a framework of element-free methods or reproducing kernel approximation [1-3]. For example, Li and Liu [1] combined reproducing kernel method and multiple scale properties of constitutive laws to succeed in simulation of shear band formation. Their results overcome the mesh-size effect of finite element method during coalescence stage. For nonlinear/elastoplastic materials with volumetric compressibility, Wang and Liu [3] studied a foundation problem using Duncan EB and elastoplastic models. This paper will extend the radial PIM method proposed by Wang and Liu [4] to include compact support and polynomial basis, and the method is applied to an elastoplastic problem of foundation engineering. It is organized as follows: first, the radial PIM method is briefly proposed and its accuracy is evaluated through complex functions. Then, an elastoplastic boundary-value problem and its auto-corrector weak form are developed to form a novel algorithm through a reference constitutive law. * Corresponding author. Tel.: -^65 874-4796; Fax: -h65 8744795; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

An integration scheme is discussed for an elastoplastic constitutive law, endochronic model. Finally, a foundation problem is studied as an example.

2. Radial point interpolation method Give a set of arbitrarily distributed data points Pi(xi) (i = 1,2,... ,n) and the function value w/. Radial PIM method constructs a continuous surface u(x) from the neighborhood nodes of a point x using radial basis functions: n

m

uix) = V Biix)ai -h Y^ Pj(x)bj = B^(jc)a -f P^(;c)b .=1

;=i

(1)

This approximation includes two terms: one is radial function and the other is a polynomial. The polynomial term (m < n) is introduced to improve approximation accuracy. a"^ = [(3ifl2«3.--«n]

B'{x) = P''{x)

[B,{x)B2{x)...B,(x)]

= [pdx)p2(x)...Pm(x)]

(2)

The pre-assumed radial basis Bi(x) has following general form Bi(x) = Bi(ri) = Bi(x,y)

(3)

J.G. Wang, G.R. Liu/First MIT Conference on Computational Fluid and Solid Mechanics

539

In two-dimensional space, the Euclidean distance r^ is expressed as (4)

ri = [{x-Xif-\-{y-yify^

Let the interpolation pass through all n scattered points and the equation at the k\h point is expressed as n

m

^aiBiixk,

yk) + X l ^ ; ^ ; f e , yk) = Uk

-1

7=1

k=l,2,...,n (5)

The polynomial in Eq. 1 is an extra-requirement. Similar to the thin plate spline method (TPS) [5], a constraint is imposed as ^P,te,>;,M=0

; = l,2,...

(6)

i=l

The matrix form is expressed as Bo

Fig. 1. Compactly support function and its derivative.

Po

or

n 0

G

(7)

where U^ =

(8)

[U1U2U2 . . . Un]^

Bk(xi, yi) = Bi(xk, yk), thus the matrix BQ is symmetric. The solution is obtained as (9)

0

3. Reference constitutive laws and weak form A reference constitutive law as shown in Fig. 2 is used to join global computation: Here the Df-j^^ is a reference material matrix. Acr-^ is reference stress and Astj is incremental strain. The true stress increment Aatj is expressed as Aatj=Dfj,,AsM

And final interpolation is expressed as

0

:
(10)

where shape function (p(x) is defined as

j {8(As)f

[D^] [As] dv-

V

;=i

dx ^^dPj^

ds = - f {8(As)V {cr' + Aa'} dv

+ f {5(Aw)r {fy dv+ f {8(Aii)f {b']dv

(16)

/ Reference constitutive law

9^,

" 95.

{At}

(11)

where Gtk is the (/, k) element of matrix G~\ Its derivatives of shape function are

90, _ dy

I {S(AU)}^ {Ab} dv V

- f {8(Aa)f

(Pk(x) = J2 Bi{x)Gi,k + J2 PjMGn +j,k

j=i

(15)

+ Aa[j

Therefore, the weak form is

u{x) = [B'(x)F^(x)] G

dx

(14)

^<^.j = D^m^^u

True constitutive law (12)

A general form for any compactly supported basis function is

B(r) =

(l-rrS(r)

(13)

S(r) is a polynomial and (1 - r)^ is a compactly supported function. Fig. 1 shows the effect of q. When r approaches to one, function and its derivative all approach to zero. Schaback and his colleagues [6,7] discussed a particular class of positive define radial basis functions.

Local integration curve to get true stress numerically

•

Fig. 2. Schematic stress difference between two constitutive laws.

J.G. Wang, G.R. Liu/First MIT Conference on Computational Fluid and Solid Mechanics

540

The left hand side may be a linear system depending on the choice of reference constitutive law. Non-linearity is expressed by Aa'^ which is determined through iteration. Through a reference constitutive law, global iteration that is for whole structural system and local iteration that determines true stress can be separated formally.

4.2. Direct integration of endochronic model

4. Endochronic constitutive law and its integration

A direct integration scheme is obtained as

4.1. Endochronic constitutive law for soils

Zn

I'-"

,de'^. 1 z':!^^Z'^-AefXe~^^'"+e-^^'^-^) 2 '^^ dZ'

'J

Wang and Fan [8] proposed an endochronic model for normally consolidated soils. This model divides strain increment into elastic part (def) and plastic part (d^/") dSij = def. + de

Suppose information at Z = Z„_i is known and following approximation is true

(17)

(23)

(24)

B

where A,,=X^e-'''^2"4>(Z„_,) 3

B = i ^ C . ( l + .-^'-^")

4.1.1. Elastic part The elastic part follows incremental Hooke's law: d5,/ % + - ; f = D^.,;da,,

de!

(25) (18)

Where K and G are material parameters. d5/y = oij — Sijdakk/3 and Okk = ai + (T2 -f 0^3. 4.1.2. Plastic part Endochronic theory describes a no-yielding surface model with p(Z,-Z:)^dz:

F{a,e)j ZH

6) [
J

oz„

(19)

dz'„

d|.

d^s = lld^,''

dZ„ =

d|H

d^l=k^\def^\\'

A? = IIA^,^. II = CAZ, = {fAh) + i/,(?)/;(?)AZ„) AZ, (26) Combining Eqs. (24) and (26) can solve the stress increment if the strain path is given.

5.7. Evaluation of approximation accuracy Accuracy of approximation Eq. (1) is checked first. For simplicity, polynomials are taken linear terms (m = 3) and radial basis function is B(r) = (1 - r)'*(4r -hi) [7].

and scale and measure are defined as dZ,

A.^

5. Numerical examples

0

a = Gia,

4 k Z J = . - ^ ^ ^ ^ " 4 \ Z . _ 0 -f \Cr {l^e-^r^z.^

+ \de''

(20)

Kernel functions p(Z) and
(21)

where Cr, fir are material constants. F(a,d) and G(a,0) are associated with failure or hydrostatic hardening, /j and fn are hardening functions for shearing and hydrostatic stresses. They have following form f, = C-[C-

l]e-

h

J^''

The model constants are including C,a,

(22) p,k.

y - Axis

X - Axis

Fig. 3. Approximation error offittingsin(x) sin(};).

J.G. Wang, G.R. Liu/First MIT Conference on Computational Fluid and Solid Mechanics

541

algorithm is adopted through Eq. (24). Foundation soil is assumed to be endochronic model without yielding surface [8]. Fig. 4 gives a typical plastic strain contour, which has the same pattern as EFG results [3].

6. Conclusions Meshless method based on compactly supported radial basis functions is studied. This paper proposed a radial basis approximation that includes a compactly supported radial basis and polynomial. This polynomial makes it capable to reproduce linear polynomial, although it does not saliently improve approximation accuracy for complex functions. Compactly supported radial basis functions are split into two parts, compactly supported part that has the form (1 - ry and polynomial part such as 5(r). One form B(r) = (1 - ry(4r + 1) proposed by Wendland [7] is applied to a nonlinear foundation problem as an example. The proposed numerical algorithm is a feasible method for no-yielding surface model.

References Fig. 4. Plastic strain contour. Above: volumetric plastic strain. Below: generalized shear strain. Eq. (1) can reproduce linear function f{x,y) = x -\- y and its derivatives exactly. For complex function such as f{x,y) = sinxsin>7, it provides fair good approximation as show^n in Fig. 3. The linear polynomial slightly improves its approximation accuracy. 5.2. Application in nonlinear foundation problem A nonlinear foundation subjected to a strip loading is studied here (as shown in Fig. 4). This is a typical plane strain problem. Construction is composed of two stages: self-weight as first stage. At this stage, foundation soil is assumed to be linearly elastic. This stage obtains initial stress in foundation. The second stage applies a strip load of 20 kPa gradually. Ten (10) sub-steps are divided proportionally. Within each sub-loading step, an iterative

[1] Li S, Liu WK. Numerical simulations of strain localization in inelastic solids using mesh-free methods. Int J Num Methods Eng2000;48(9): 1285-1309. [2] Xu Y, Saigal S. An element free Galerkin analysis of steady dynamic growth of a mode I crack in elasto-plastic materials. Int J Solids Struct 1999;36:1045-1079. [3] Wang JG, Liu GR. A novel numerical algorithm for elastoplastic problems using element-free method, submitted. [4] Wang JG, Liu GR. A point interpolation meshless method based on radial basis functions, submitted. [5] Power MJD. Recent research at Cambridge on radial basis functions. 2nd Int Dortmund Meeting on Approximation Theory. Preprint, 1998. [6] Schaback R. Remarks on meshless local construction of surfaces. Preprint, University of Gottingen, Germany, 2000. [7] Wendland H. Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Adv Comput Math 1995;4:389-396. [8] Wang JG, Fan J. An endochronic model for normally consolidated soils. J Chongqing Univ 1991;14(4):l-7.

542

A stress integration algorithm for ^3-dependent elasto-plasticity models X. Wang^'*, KJ. Bathe \ J. Walczak^ ""ADINA R&D, Inc., 71 Elton Avenue, Watertown, MA 02472, USA ^ Massachusetts Institute of Technology, Mechanical Engineering Department, Cambridge, MA 02139, USA

Abstract In this paper, a stress integration algorithm is presented for a generalized elasto-plastic material model governed by the three stress invariants /i, J2 and J^. The methodology is successfully applied to the Mohr-Coulomb material model with a non-associated flow rule and implemented in ADINA. Keywords: Elasto-plasticity; Stress integration algorithm; Mohr-Coulomb yield criterion

1. Introduction

2.L The stress integration scheme

In the finite element method, the integration scheme of the inelastic constitutive behavior directly controls the accuracy and stability of the overall numerical solution [1]. So far, effective methodologies have been proposed for plasticity models whose yield criteria can be written as functions of /i (the first stress invariant) and J2 (the second deviatoric stress invariant) [1] [2]. However, no efficient algorithms are available for general material models of great engineering interests, in which not only /i and J2, but also the third deviatoric stress invariant JT, is used, such as in the Mohr-Coulomb material model. In this paper, an integration scheme is proposed for such models and specifically for the Mohr-Coulomb material description. The scheme is based on return mapping of the stresses. The objective of this paper is to briefly present the algorithm as implemented in ADINA and give some results obtained in the analysis of the excavation of a set of twin tunnels.

During the stress calculations, for any given strain increment Ae, the corresponding stress increment must be computed iteratively. The stresses at time t -\- At can be written in the following form ^+^V = C^^+^^e^ = C^ C+^^e - ^e^ - Ae^)

Aej; = Ae%

* Corresponding author. Tel.: +1 (617) 926-5199; Fax: -Hi (617) 926-0238; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

+ A^

(/, j = 1, 2, 3)

(2)

where Sjj is the Kronecker delta. Using the Euler backward method, the above mean plastic strain and deviatoric plastic strain increments based on the flow rule are, respectively t + At y

2. The algorithm An inelastic solution scheme requires two ingredients: the stress integration and the calculation of the consistent tangent stress-strain matrix [1].

(1)

where C^ is the elastic stress-strain matrix, ^"^^^e are the total strains at time t -\- At, ^+^^e^ are the total elastic strains at time r -f- Ar, ^e^ are the total plastic strains at time t, and Ae^ are the total plastic strain increments. For a general elasto-plastic material model with its plastic potential function written as g = g(Ii, J2, J3, Ha), where the Ha are the N state variables (a = 1,2, ...N), the plastic strain increments are given by

Ae'.'' = 'J

'(-

and the pressure and deviatoric stresses at time t -\- At are 3aM

\dpj

(5)

543

X. Wang et al. /First MIT Conference on Computational Fluid and Solid Mechanics ADINA

tion at time t -h At, consistent with the stress integration scheme, needs to b e calculated. The constitutive relation is defined as the variation in the stresses ^+^V as a consequence of a variation in the total strains ^+^^e _

t+AtQEP

o

t+At^

(9)

The components for the constitutive tensor take the form t+AtrEP

•^ijkl

_

^

d '+^'eki

t+Atf^EP _ ijkl

"

t+At (

'P,^3 9 t+^^eki

^'J

d t+At

^'7 Qt+At^^^

d ^+^^eki

a = j)

(10)

(

(11)

(i ^ J) 3'+^'Si

Fig. 1. 3-Dfiniteelement mesh for the twin tunnels.

\dSj

UE

(6) where pressure p is defined as — / i / 3 , p ^ and Sfj are the elastic predictors for p and Stj, respectively, a^ and GM are elastic constants, AX is called the plastic multiplier and the superscript t -\- At denotes the time of solution

(t) -d

(If).

[1]. It is noted that

f('^^'iy

a J

J2

a/ dH, = 0 dHa deki V

dCkl

1

t-\-At

J

t-\-At ]

'^"'h,

A e ^ Ae'P ij '

t+At

H^)

(a,fi = 1,2, ...TV)

ttE

of the consistent

tangential

2 d^g V dSijdp ~ 3 ^dJsdp '^

dh

AX

+ as

dSijdSm

3dJ3

dp dcki

3

(13)

deki

~ mn^ij ^mn^ii

-2,,

aE \dSij

ds^n

3aM dpdSmn deu

dSfj _ AX / dCki

d^g

dJsdS, \ aAA

dSm deki

3 'a/3

AX /

(8)

For a 3-dimensional problem, if A Ha, or ^^^^Ha, can be expressed explicitly in terms of the stress components and plastic strain increments, w e need to solve Eqs. (5), (6) and (7) for a total of 8 primary unknowns at time t + At: one pressure component ^+^^/7, six deviatoric stress components ^'^^^Sij, and one plastic multiplier AX. T h e solution is obtained using N e w t o n - R a p h s o n iterations. 2.2. Determination

AX

AX d^g dHa 3aM dpdHa deki

3flM dp deu

In general, the increments in the state variables can be written as functions of the stresses, plastic strain increments, and the state variables ^^^^ H^ themselves ^Ha = AhC+'^'Iu'^^'Ji,

dg dAX

(J)

t+At Tj \

(12)

dp 3aM a/?2 J 3eki

dp^

(t)

are functions of the stress components ^+^^/7, ^^^^Stj and the state variables '+^^/4. Therefore, '^^'p and '^^'Stj cannot be obtained explicitly from Eqs. (5) and (6), which represents some difficulties in the stress integration with the potential function g f+^7i, '+^'/2, '+^'/3, '^^'H^) and the yield function f =

are obtained b y the The expressions ^t+Atf and differentiation of the yield function, equations to calculate the stresses, and equations expressing the hardening relations with respect to the total strain components. Removing the left superscripts t -f At from all the variables for ease of writing, the differentiations are written as follows df dp df a / 2 dSrmn 9/ 9/3 dSm a / s dSmn deki dp dcki + 9 / 2 95, dCki

moduli

In a full N e w t o n - R a p h s o n scheme used to perform the global equilibrium iterations, the tangent constitutive rela-

d^g dHa (14) ?/2^^ deki ClE \dSijdHa 3 dJsdH^ In the above equations, i, j , k, I, m and n range from 1 to 3. The differentiation of each equation governing the strain hardening of the state variables, expressed as FH = 0, with respect to the strain components gives dFn dp dFn dS,^, ^ dFu aAA ^ dFn dHg 0 dAX deki dHa de^ dp deki dSmn deki (a =

l,2,...N)

(15)

Combining Eqs. (12), (13), (14) and (15), we have (8 + N) equations that are established using the stress integration algorithm and solved for the unknowns: dAX/de^, dp/deki, dSij/dekh and dHa/dCki-

544

X. Wang et al /First MIT Conference on Computational Fluid and Solid Mechanics ^

115. ~e- SeMement 81 time 2.0 (20 m advancement) -X- SetUement at time 2.0 {20 m advancement, elastic ans^sie)

20.

60.

70.

"ST

90.

100.

UMBnOO K I i0nyRUaHi8f CMBCillOii i m )

Fig. 2. Vertical displacement distribution along the tunnel crown at different excavation stages.

ADINA

z

Z-DISPLACEMENT TIME 5.000 f - -0.0080 1 ^ -0.0240 | - -0.0400 f - -0.0560 f - -0.0720 f - -0.0880 W- -0.1040

Fig. 3. Vertical displacement distribution at the sections X = 0, 30 and 60 m across the tunnel axis (m).

3. Application to the Mohr-Coulomb model The yield function and the potential function of the Mohr-Coulomb material model are [3]

/ = /i sin0 + -[3(1 — sin)sin^ -f \/3 (3 + sin0) cos ^ j V ^ — 3c cos 0

(16)

X. Wang et al /First MIT Conference on Computational Fluid and Solid Mechanics

545

ADINA

SMOOTHED STRESS-ZZ RST CALC TIME 5.000

,-;.„

i-^ ;\^?^i,'™"r '• ";.v~

'^^^tt

1 \C

-160. ,320.

I T "^^^• t-

-800.

h- -960. [--1120

Fig. 4. Vertical normal stress distribution at the sections X = 0, 30 and 60 m across the tunnel axis (kPa). g = /i sin i/r + - [3(1 — sin i/r) sin^ + A/3 (3 + sin -ij/) cos O^^fTi — 3c cos x/r

(17)

in which 0 is the internal friction angle, x// is the dilation angle, c denotes the material cohesion and 0 is the Lode angle. The proposed return mapping algorithm for the MohrCoulomb model has been implemented in ADINA, and we give here some results obtained in a simulation of the sequential excavation of a set of twin tunnels constructed in a soft rock layer, see Fig. 1. Each tunnel was 15.6 m in diameter. A soft rock domain of 100 x 100 x 50 m^ was taken for the analysis. The soft rock was assumed to correspond to the following material parameters: E = 240 MPa, y = 0.3, unit weight y = 21.5 kN/m^ 0 = 22.0°, xff = 10.0°, c = O.l MPa. The liner was modeled assuming a linear elastic material with E = 5000 MPa, v = 0.25, and unit weight y = 25.5 kN/m^. Five incremental excavation stages performed together for each tunnel were completed in the longitudinal direction, each incremental excavation comprising a 20 m advancement and containing 2 time steps to perform the radial excavation. The liner was installed right after each longitudinal excavation. To obtain three-digit accuracy in energy values, convergence was reached using a maximum of 3 iterations in each time step. Fig. 2 shows the vertical displacement distribution at different excavation steps. The displacements are plotted for the nodal points along the tunnel crown. The displacements are small before the tunnel face approaches the cross section, but increase immediately after the face has passed, and then increase further, as the face

progresses, until a level of about 11 cm. The displacement results assuming elastic conditions are also shown. These values are of course smaller. Similar conclusions can be drawn from Fig. 3, which shows the vertical displacement distribution at three sections at the time 5.0. Fig. 4 gives the corresponding vertical normal stress distributions.

4. Conclusions A stress integration procedure has been presented for a general elasto-plastic material model in which the stress invariants /i, J2 and J3 are relevant. The proposed formulation is capable of accommodating arbitrary yield criteria, flow rules and hardening laws provided, of course, the first and second derivatives of the yield and potential functions with respect to the stress components are available. The algorithm has been implemented for the Mohr-Coulomb material model with a non-associated flow rule. References [1] Bathe KJ. Finite Element Procedures. Englewood Cliffs, NJ: Prentice-Hall, 1996. [2] Borja RI. Cam-Clay plasticity, part II: impHcit integration of constitutive equation based on a nonlinear elastic stress predictor. Comput Methods Appl Mech Eng 1991;88:225240. [3] Desai CS. Siriwardane HJ. Constitutive Laws for Engineering Materials with Emphasis on Geological Materials. Englewood Cliffs, NJ: Prentice-Hall, 1984.

546

Numerical and analytical modeling of ground deformations due to shallow tunneling in soft soils A.J. Whittle*, Y.M. Hsieh, F. Pinto, Y. Chatzigiannelis Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, Cambridge, MA 02139 USA

Abstract This paper illustrates the role of analytical solutions in evaluating the settlements caused by tunnel excavation in soft soils. Analytical solutions have been obtained for shallow tunnels in isotropic and cross-anisotropic linear soil half-plane. These solutions provide a frame of reference for establishing numerical accuracy in FE analyses and for understanding the role of actual non-linear, inelastic soil behavior on far-field ground movements. Keywords: Tunnel; Ground deformation; Constitutive model; Elasticity; Anisotropy; Non-linearity

1. Introduction The prediction and mitigation of damage caused by construction-induced ground movements represents a major factor in the design of tunnels in congested urban environments. This is an especially important problem for shallow tunnels excavated in soft soils, where expensive remedial measures such as compensation grouting or structural underpinning must be considered prior to construction. Ground movements inevitably arise from changes in soil stresses around the tunnel face and overexcavation of the final tunnel cavity. Sources of movements are closely related to the method of tunnel construction ranging from (a) closed-face systems such as tunnel boring machines (with earth pressure or slurry shields), where overcutting occurs around the face and shield ('tail void'), but local ground loss is constrained by grout injected between the soil and precast lining system; to (b) open-face systems (such as the New Austrian Tunneling Method, NATM) where ground loss around the heading is minimized by expeditious installation of lining systems in contact with the soil (typically steel rib or lattice girder and shotcrete) with additional face support provided by a shield or other mechanical reinforcement (soil nails, sub-horizontal jet grouting etc.). In all cases, it is easy to appreciate the complexity of the mechanisms causing ground movement and their close * Corresponding author. Tel.: +1 (617) 253-7122; Fax: +1 (617) 253-6044; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

relationship with construction details, especially given the non-linear, time-dependent mechanical properties of soils and their linkage to groundwater flows. Indeed, this complexity has encouraged the widespread use of numerical analyses, particularly non-linear finite element methods, over a period of more than 30 years (see review by Gioda and Swoboda [1]). However, it is less clear how details of the construction methods or soil properties affect the magnitude and distribution of far field movements and hence, the potential damage to overlying structures. Although there have been some notable advances in numerical modeling of interactions between soil and overlying structures (e.g. Burd et al. [2]), nearly all FE analyses rely on approximate methods or empirical estimates of ground loss to simulate conditions occurring around the tunnel cavity (e.g. Schweiger et al. [3]). In practice, most predictions of ground settlements are still based on empirical methods [4]. Recent research by Sagaseta [5,6] and Verruijt and Booker [7] and strongly suggests that reasonable predictions of far field soil movements can be obtained using analytical solutions that simplify the tunnel geometry and assume linear elastic soil properties. The authors have investigated and extended analytical solutions for shallow tunnels in soil and have used these results to first establish the numerical accuracy in FE analyses and then to evaluate the role of soil behavior in controlling far field deformations. The following paragraphs summarize some results from this on-going study.

AJ. Whittle et ai /First MIT Conference on Computational Fluid and Solid Mechanics 2. Analytical solutions for shallow tunnels By assuming elastic properties for the soil, it is possible to derive closed-form expressions for the ground deformations as functions of displacements occurring at the tunnel cavity. Displacement boundary conditions can be expressed as the summation of three basic modes as shown in Fig. 1: (1) uniform ground loss (wj; (2) pure distortion (ovaUzation, U8, with no change in volume of the cavity); and (3) vertical translation (AM^). Following Sagaseta [5,6] and Verruijt and Booker [7], ground deformations can be obtained using fundamental solutions for a line sink in full-space (located at depth, H, corresponding to the axis of the tunnel). Boundary solutions for the stress-free ground surface are obtained by superimposing solutions for an image sink together with a distribution of corrective surface shear tractions. Solutions for the distortion mode are based on full-space solutions for deformations around a cylindrical cavity acted upon by uniform deviatoric stresses (after Kirsch, 1898). Both solutions produce vertical translations of the tunnel cavity. Pinto and Whittle [8] have evaluated the accuracy of these solutions by deriving solutions from a more complete formulation that includes the physical dimensions of the tunnel cavity with radius, R (using complex variables and conformal mapping after Verruijt [9]). The solutions for this 'exact geometry' are written in Laurent series format with coefficients that must be solved numerically in order to satisfy the far field boundary conditions. Fig. 2 shows that the approximate (point) solutions provide are in excellent agreement with the solutions obtained using the exact geometry for a tunnel with radius to embedment ratio, R/H = 0.2 (v = 0.25) in both the uniform convergence and ovalization modes of cavity deformation. Not surprisingly, the importance of the cavity geometry becomes more significant as R/H increases (i.e. for very shallow tunnels).

Sagaseta [5] and Pinto [10] have reported good agreement between analytical and measured settlement troughs from a number of instrumented tunnel projects. Other authors have reported that the shape of the settlement trough can be further improved by considering elastic anisotropy of the soil mass. Chatzigiannelis and Whittle [11] have extended the approximate superposition method for tunnels excavated in a cross-anisotropic soil using complex variable methods. Fig. 3 shows that the width of the surface settlement trough reduces significantly for soils with a low stiffness ratio, Gyh/Ey, as reported from FE analyses by Lee and Rowe [12].

3. Soil model and numerical analyses There is an extensive literature documenting the non-linear and inelastic constitutive behavior of soils. For example, Pestana and Whittle [13] have presented a generalized effective stress model, referred to as MIT-Sl, which is capable of predicting the rate independent, effective stressstrain-strength behavior of uncemented soils over a wide range of confining pressures and densities. Fig. 4 illustrates typical MIT-Sl predictions for a series of five standard, drained triaxial compression shear tests (with parameters derived for a reference cohesionless soil, Toyoura sand) performed at the same confining pressure but a range of initial void ratios (very dense, CQ = 0.6, to very loose, eo = 0.95). The predicted stress-strain behavior is consistent with measured data, with non-linear stiffness properties clearly seen at axial strains Sa < 0.01%, and dilation occurring ai Sa = 0.5% for dense samples at high mobilized friction angles. The role of soil modeling in predictions of tunnel-induced ground movements can now be appreciated by comparing numerical simulations using MIT-Sl, with analytical results for an isotropic elastic soil. The nu-

77777777777777777777777777777777777777777^7777^ -All,,

Uniform Convergence

Net Volume Change

547

Distortion Vertical Translation (Ovalization) (Downward Movement)

Final Shape

No Net Volume Change

Fig. 1. Deformation modes for a shallow tunnel cavity.

548

AJ. Whittle et al /First MIT Conference on Computational Fluid and Solid Mechanics A 0.2 0.0

+

-0.2

R/H = 0.2 v = 0.25 Exact geometry Point Sink FE

1.0

2.0

0.0

Lateral Location, x/H

C/2

-0.4 -0.6

b) Ovalization Mode 0.0

do

1.0 2.0 Lateral Location, x/H

Fig. 2. Comparison of analytical solutions for surface settlements for a shallow tunnel in a linear, isotropic soil. 0.2 0.0 B.

0.0

o

-0.2

R/H = 0.2 G^/E Line vh V 0.25 0.4 (Isotropic) 0.5 .

"2 -0.4 -0.6

b) Ovalization Mode

a) Uniform convergence mode .

0.0

1.0 2.0 Lateral Location, x/H

0.0

1.0

-0.2 o -0.4

j

-0.6

2.0

Lateral Location, x/H

Fig. 3. Analytical solutions for surface settlements in cross-anisotropic elastic soil. n^Q

001

0.02 ^0

0.1 .

1

.

,

0.L2 ^0 .

,

^

•

1

1—I

2

,—[-

/ /

"

3 I—1—I

1

0.7^

//^^^ //y\yy^^'S-

i^,.,j '

1 '

%=

-

0.6/^

MIT-Sl Toyour a Sand CI' =100kPa 0

0.01

0.02 • 0

0.95 -

0.1

0.2

0

.

1 .

,

1

a cohesionless, medium-dense Toyoura sand (eo = 0.75) in the uniform convergence and ovalization modes of deformation. The distribution of surface settlements predicted by MIT-Sl is very similar to the analytical solutions (linear, isotropic soil) for very small values of Ug/R (0.002, 0.02%, Fig. 5a). However, as the ground loss increases to Ue/R = 0.2%, there is a very substantial narrowing of the settlement trough associated with an incipient failure mechanism predicted by MIT-Sl. In contrast, Fig. 5b shows excellent agreement between the non-linear numerical solutions and analytical solutions for ovalization up to us/R = 0.2%. These results represent a first step towards a more comprehensive understanding linking soil properties to tunnel-induced ground movements.

.

Axial Strain, 8 (%) a

Fig. 4. Typical predictions of drained triaxial compression shear behavior for sand using MIT-Sl model. merical analyses have been carried out with prescribed deformation boundary conditions around the tunnel cavity. Fig. 5 compares results for a tunnel v^ith RjH = 0.45 in

4. Conclusions Reliable predictions of ground movements caused by shallow tunnel excavation represent are important for underground construction projects in urban areas. Recently developed analytical solutions provide a useful framework for estimating these ground movements, but are based on simplified assumptions of linear (isotropic or anisotropic)

549

A.J. Whittle et al. /First MIT Conference on Computational Fluid and Solid Mechanics 0.2 0.0 u^/R

Line „

_.

•••x Analytical (linear & isotropic) R/H = 0.4 MIT-Sl Model Toyoura Sand

a 1 — (

0)

o

t;

e„ = 0.75

1.0

2.0

Lateral Location, x/H

0.002% . 0.02% 0.2% Analytical R/H = 0.4 MIT-Sl Toyoura Sand

0.0

-0.2 I o

.

e^ = 0.75

a) Uniform convergence mode T 0.0

—

-0.4

-0.6 g -0.8 3 1

b) Ovalization Mode 1.0

^

-1.0 S -1.4

2.0

Lateral Location, x/H

Fig. 5. Role of soil model in predictions of surface settlements for shallow tunnel in sand.

material behavior. By comparing numerical predictions using realistic soil models with the analytical solutions, the results in this paper provide a first step towards understanding the role of non-linear and inelastic soil behavior on the distribution of ground movements.

References [1] Gioda G, Swoboda G. Developments and applications of the numerical analysis of tunnels in continuous media. Int J Numer Anal Methods Geomech 1999;23:1393-1405. [2] Burd HJ, Houlsby GT, Chow L, Augarde CE, Liu G. Analysis of setdement damage to masonry structures. Proc Numerical Methods in Geotechnical Engineering, 1, Balkema, 1994, pp. 203-209. [3] Schweiger HP, SchuUer H, Pettier R. Some remarks on 2-D models for numerical simulation of underground constructions with complex cross-sections. In: Yuan (Ed), Proc Computer Methods and Advances in Geomechanics, 2, Balkema, 1997, pp. 1303-1308. [4] Peck RB. Deep excavations and tunnels in soft ground. Proc. 7th Ind. Conf. on Soil Mechs. and Foundation Engrg. Mexico City 1969, SOA Paper, pp. 225-290. [5] Sagaseta C. Analysis of undrained soil deformation due to ground loss. Geotechnique 1987;37(3):301-320.

[6] Sagaseta C. On the role of analytical solutions for the evaluation of soil deformations around tunnels. In: Cividini A (Ed), Application of Numerical Methods to Geotechnical Problems, CISM Lecture and Course Notes 1998;397:3-24. [7] Verruijt A, Booker JR. Surface settiements due to deformation of a tunnel in an elastic half-plane. Geotechnique 1996;46(4):753-757. [8] Pinto F, Whittle AJ. Comparison of analytical solutions for ground movements caused by shallow tunneling in soil. ASCE J Eng Mech, submitted for publication. [9] Verruijt A. A complex variable solution for a deforming tunnel in an elastic half-plane. Int J Numer Anal Methods Geomech 1997;21(l):77-89. [10] Pinto F. Analytical methods to interpret ground deformations due to soft ground tunneling. SM Thesis 1999, MIT Dept of Civil and Env Engineering. [11] Chatzigiannelis Y, Whittle AJ. Deformations caused by shallow tunneling in cross-anisotropic soil. ASCE J Eng Mech, submitted for publication. [12] Lee KM, Rowe RK. Deformations caused by surface loading and tunneling: the role of elastic anisotropy. Geotechnique 1989;39(1): 125-140. [13] Pestana JM, Whittle AJ. Formulation of a unified constitutive model for clays and sands. Int J Numer Anal Methods Geomech 1999;23:1215-1243.

550

Identification of chaotic responses in a stable Duffing system by artificial neural network W. Witkowski*, L Lubowiecka Technical University of Gdansk, Faculty of Civil Engineering, Department of Structural Mechanics, ul. Narutowicza 11/12, 80-952 Gdansk, Poland

Abstract The paper deals with the problem of identification of chaotic responses in a forced stable Duffing system. The model analyzed is the von Karman type plate in the pre-buckling state, which in turn yields in positive linear stiffness. The system is studied for large amplitudes of the external load. The chaotic response for various values of amplitude and frequency is identified by the Lyapunov exponent. The neural network is used to estimate the further behavior of the system. Keywords: Nonlinear dynamics; Chaotic systems; Neural networks; von Karman plate

1. Introduction There has been a great deal of excitement about the problem of unpredictability in a dynamical system described by a set of differential equations since the pioneer work of Lorenz [1]. Subsequent studies have revealed that the phenomenon of chaotic motion is important in a vast class of problems. Perhaps the most extensively studied system is that described by Duffing's equation. Two different approaches can be distinguished here. The first one concerns the so-called unstable Duffing equation i.e. with negative linear part of the stiffness. This equation is known to exhibit chaotic solutions and to have three equilibria. It describes, for example, the snap through phenomena (see Tseng and Dugundji [2] or Moon [3]). Excellent reference in this field is Dowell and Pezehski [4] where the authors explained how the potential wells promote the chaos. The second approach deals with the stable Duffing equation i.e. with positive linear part of the stiffness. In this case, the equation has a single equilibrium point and it is of special interest to examine how the chaos occurs in this system. Based on numerical simulations Fang and Dowell [5], proposed a certain route to chaos involving period doubling scenario. Evidently, Duffing's equation received major attention in the literature. Yet, some steps must be taken towards the understanding this equation in the broader con* Corresponding author. Tel: -h48 (58) 347-2238; Fax: +48 (58) 347-1670; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

text of real structural engineering systems. In this paper, the stable forced Duffing equation is derived from the governing equations for the von Karman type plate. The response of the system is studied for different values of amplitude and frequency of excitation and identified using the Lyapunov exponents. In this way, a training set for an artificial neural network (ANN) is created. The network is then used in the practical problem of predictions the chaotic solutions.

2. Dynamic problem Consider an elastic thin rectangular plate presented in Fig. 1. The problem of motion of such a plate is described by the following set of partial differential equations (Fung [6]) DV^W

= ( F 2 2 W ; , 1 1 + ^,11^,22 - 2Fi2W^,12

-\-q{xi,X2,t)

V^F

-

phw)

(1)

= Eh[{w ^n)^ — w^iiw ^22)

where D is the flexural rigidity of the plate, q (xi,X2,t) is the external load, F — F(xi,X2,t) is the Airy stress function, E is the Young modulus and p is the material density. In the subsequent study, Eq. (1) is solved with the help of Galerkin's method using the simply supported boundary

W. Witkowski, I. Lubowiecka / First MIT Conference on Computational Fluid and Solid Mechanics

551

Fig. 1. Geometry of the plate. conditions. Assuming the first symmetric mode in the form sin a b and the external load in the form of point force

(2)

q{xi,X2,t)

(3)

w{xi,X2,t)

= y (0 sin

= Psm(cot)S

(xi~-\8l. 1 ^

2 - -

A.

we arrive at F + y y + y + aF^ = A sin (^T)

Table 1 Sample data obtained from 'Dynamics 2'

(4)

where Y{t) = y{t)/h, and y is a nondimensional linear damping coefficient. The remaining constants in Eq. (4) are defined in the Appendix. For the numerical analysis, the following assumptions are made: E = 205 GPa, v = 0.3, p = 7850 kg/m^ a = 3m, b = 2m, h = 0.01 m. Hence a = 1.44439. The value of y was fixed as y = 0.1.

16 16 16 16 16 16 16 16 16 16 16 17 17 17

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1 1.05 1.1

-0.05124 -0.05109 -0.05113 -0.04976 -0.049954 -0.050014 -0.050028 -0.049919 -0.050029 -0.050003 -0.050042 -0.0501 -0.050113 -0.050115

3. Numerical analysis Numerical simulations were carried out using 'Dynamics 2' software supplemented in [7]. The amplitudes A were taken as integer numbers from 16 to 28, whereas the frequencies P were chosen from the interval [1, 1.5] with the step 0.05. The first Lyapunov exponent A., was computed after 35,000 iterations to ensure that the transient phase dies out. The main information provided by this exponent is that the system has chaotic evolution if the exponent is positive, (cf. Wiggins [9] for the detailed information). A sample data is presented in the Table 1. The initial conditions were taken to be (7, Y) = (3,0).

4. Neural network simulations ANN simulations were carried out with the help of 'Neuronix' program [8]. The back propagation algorithm was used for learning (see Zurada et al. [10], Waszczyszyn [11] or Hajela [12] for the further details). A total of 138 patterns of form A, fi, as the input data and X as the output were used for training. The neural network was only supposed to identify the sign of the Lyapunov

exponent. All patterns were randomly presented during a single epoch. Initially, random weights were assigned to each synapse (Tadeusiewicz [13]). As the activation function, the sigmoid function was used. During the testing phase, each net was exposed to a set of 114 patterns never presented before.

5. Results The study involved testing of 15 different architectures with different parameters of learning [14]. The results reached were verified using the 'error ratio' defined as number of missed signs . 100%. number of patterns tested = 1 1 4

(5)

Among many networks tested, two with the lowest e ratio are presented below. The first one, presented in Fig. 2, consisted of two neurons in the input layer, 3 hidden layers having six neurons each and one neuron in the output layer. For the sake of brevity, this net is referred to as 2-6-6-6-1. In this example, the parameters were taken to be: the learning parameter r] = 0.37 and the momentum

552

W. Witkowski, I. Lubowiecka / First MIT Conference on Computational Fluid and Solid Mechanics output layer yf

term /x = 0. In Fig. 3 the results from 'Dynamics 2' (solid line) are compared with those obtained from ANN (dashed line). This net reached the error ratio e = 14.7%. Evidently, the results presented are in a good agreement. The second network is of the type 2-15-30-1. The learning parameters r] and /x were 0.4 and 0.7 respectively. In Fig. 4, a similar comparison is presented. As in the case of 2-6-66-1 net, the results obtained for this net are also in a good agreement. In this case, e = 12.8%.

6. Conclusions input layer A,/^ Fig. 2. Neural network architecture 2-6-6-6-1.

The authors have made an attempt to combine the two domains of computational structural mechanics: chaotic systems and artificial neural networks. The examples pre-

Fig. 3. Results from 'Dynamics 2' and ANN (2-6-6-6-1) for amplitude A = 30.

Fig. 4. Results from 'Dynamics 2' and ANN (2-15-30-1) for amplitude A = 30.

W. Witkowski, I. Lubowiecka /First MIT Conference on Computational Fluid and Solid Mechanics sented in this report show that the ANN approach gives qualitatively good results. By no means, it is due to the proposed weak criterion about the identified sign of the Lyapunov exponent. However, since we are interested whether there is a chaotic solution this criterion seems to be sufficient. It is also interesting to note that the trained ANN gives the picture of the system behavior for a wide class of the exciting parameters. A similar problem has been already treated by Ueda et al. [15] with the help of sophisticated numerical methods. The present authors are aware of the fact that ANN provides only a plausible range of the parameters responsible for the chaotic motion. The more detailed information about response should be reached by the numerical integration.

Acknowledgements This work was partially supported by the Polish State Committee for Scientific Research under grants No. 7 T07A 041 18.

Appendix A Eq. (4) is written with respect to the nondimensional time r = COQI, where CL>1 = [D7z'^{l -h m^f]la^ph is the square of the natural frequency of the plate and m = a/b. The constants in Eq. (4) are defined as: 3((3-i;2)(H-m4)-h4i;m2)

4(l+m2)2

co

and p = —. (JOQ

(3)

References [1] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20(March).

553

[2] Tseng WY, Dugundji J. Nonlinear vibrations of a buckled beam under harmonic excitation. ASME J Appl Mech 1971;38:467-476. [3] Moon FC. Experiments on chaotic motions of a forced Nonlinear Oscillator: strange attractors. ASME J Appl Mech 1980;47:638-644. [4] Dowell EH, Pezehski C. On understanding of chaos in Duffing's equation including a comparison with experiment. ASME J Appl Mech 1986;53:5-9. [5] Fang T, Dowell EH. Numerical simulations of periodic and chaotic responses in a stable Duffing system. Int J Non-linear Mech 1987;22(5):401-425. [6] Fung YC. Foundations of SoUd Mechanics. Prentice-Hall, 1965 [7] York JA, Nusse HE. Dynamics: Numerical Explorations, second edition. New York: Springer-Verlag, 1997. [8] AITECH Artificial InteUigence Laboratory: Sphinx 2.3. Cracow, Poland: Aitech. [9] Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer-Verlag, 1990. [10] Zurada I, Barski M, J^druch W. Artificial Neural Networks (in PoHsh). Warsaw: Scientific PubHsher PWN, 1996. [11] Waszczyszyn Z. Fundamentals of artificial neural networks. Materials from the course "Neural networks in mechanics of structures and materials" CISM Udine 1998. [12] Hajela P. Neural networks — appfications in modeling and design of structural systems. Materials from the course "Neural networks in mechanics of structures and materials" CISM Udine 1998. [13] Tadeusiewicz R. Neural Networks (in Pofish). Warsaw: Academic PubHsher RM, 1993. [14] Topping BHV (Ed). Developments in Neural Networks and Evolutionary Computing for Civil and Structural Engineering. Galashiels, Scotland: Civil-Comp Press, 1995 [15] Ueda Y, Thomsen JS, Rasmussen J, Moseklide E. Behaviour of the solution to Duffing's equation for large forcing amplitudes. In: Kreuzer E, Schmidt G (Eds), 1st European Nonlinear Oscillations Conference, Hamburg, August 16-20, 1993. Akademie Verlag.

554

Special membrane elements with internal defects Chunhui Yang *, Ai-Kah Soh Department of Mechanical Engineering, The University of Hong Kong, Hong Kong SAR, People's Republic of China

Abstract Special membrane elements with internal defects (holes/cracks/inclusions) have been developed using complex potentials and the conformal mapping technique. These elements can be easily combined with the conventional displacement elements, to analyze the complicated structures with different defects easier. Numerical examples have been employed to illustrate the potential accuracy and reliability of the proposed elements. Keywords: Defects; Membrane elements; Complex potentials; Conformal mapping technique; Interfacial boundaries

1. Introduction Almost all materials contain defects in different forms such as cracks, voids, inclusions or second phase particles. These defects are generally termed as inhomogeneities. Their existence in materials plays an important role and may even strongly influence the mechanical behavior of the whole structure. It is really crucial to determine the effects indicated by these defects. The finite element method (FEM) has been widely applied to solve such problems. However the conventional elements cannot assure high accuracy, especially in the vicinity of interfaces between the defects and the surrounding materials. A better way to solve such problems is to construct special elements containing internal defects shown in Fig. 1, explicitly implemented the interfacial boundary conditions at the interfaces between the different materials. Piltner [3] constructed some special finite elements with internal circular/elliptic holes or cracks based on two different variational formulae. He applied the Trefftz method for the use of trial functions, obtained by employing the complex potentials and the conformal mapping technique, developed by MuskheUshvih [1]. Meguid et al. [4] have developed an 8-node membrane element consisting of a circular inclusion using complex potentials to define stress and strain functions directly in the element. Soh and Long

[5] have also constructed some two-dimensional membrane elements containing a circular hole based on the same method. In the formulation of these elements, the boundary conditions at the free/interfacial boundary of the interfaces are implemented explicitly. It is worth noting that this approach leads to the use of only one element for the vicinity of the hole/inclusion, and high accuracy is achieved in the stress concentrated region. In the present paper, the 8-node rectangular membrane elements with different shaped internal defects will be developed using complex potentials and the conformal mapping technique. Numerical examples will be employed to demonstrate its potential accuracy and versatility by comparing the results obtained with those from plane elasticity theory.

•

•

I

m

(a) * Corresponding author. Tel.: +852 2859 2616; Fax: +852 2858 5415; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

•

y I

i (b)

Fig. 1. Eight-node rectangular membrane elements with an elliptical hole/inclusion.

C Yang, A.-K. Soh/First MIT Conference on Computational Fluid and Solid Mechanics 2. Basic theory and formulas

For holes or cracks,

To consider a two-dimensional plate with defects in plane elasticity, the fields of stress and displacement are expressed with the complex potentials [1] as follows:

2ix,y = 2 [zV{z)

G,-ay+

+ x\z)]

111 {ujc + iuy) = Kf{z) - zf\z)

- xG)

F,+iFy

x(z)^^

/ [ir{z) + zj\z) = -i [ir{zB) +

+

ZB^{ZB)

+

Op = Xpe =0

in the ^-plane

Fjc = Fy =0

in the z-plane

(10)

(1)

For inclusions with the assumption of perfect bonding between region M (p > I) and region / (p < 1),

(2)

(F,-i-iFy)M = (F,-\-iFy)l

(3)

in the z-plane

(u -\-iv)M = (M + iv)i

(4)

(11)

For the cases of the internal inclusions, the auxiliary stress function coig) is defined as follows:

co(g) = {7(l/g)/f(g)) n^) + x(^) X(ZB)-CA\

where z = x -^ iy; K = 3 — 4y for plane strain and /c = (3 — y)/(l + v) for plane stress. As the treatment of boundary conditions in the original domain usually fails, it is convenient and advantageous to make use of conformal mapping, for which the mapping function is given as follows: z = fig)

555

(12)

Obviously, the holomorphic functions \lr(g) and x(^) or co(g) can be approximated using the Laurent series of g with the limited terms in the different region where 1^1 > 1.0 or 1^1 < 1.0. For the cases of holes or cracks, JU

1^1 > 1-0

i^(g) = ifo + J2Ajg^+J2^^^~'

(13)

7=1

= c.(, + ^)

(5)

where g = ^ -\-iri and its inverse function g = f~^(z). By means of conformal mapping, the original domains with holes/cracks/inclusions are mapped onto the alternative boundaries as unit circles. Therefore, the stress and displacement fields are expressed in terms of f(g) and the functions xfrig) and x(^)» chosen in the ^-plane, as follows:

PU

x(g) = xo + X^ ^p^'+ E ^^^''

1^1 ^ 1-^

^14)

or in accordance with the expressions provided by Isida [2],

^;(^) = ^o + £A,-^^+£5,r^

(15)

;=i PU

—— + —

ap+ae=2\

c^e - < 7 p -\-2iTp0

(6)

X'(S) = c„

Co + ^

(16)

€,5'+ Y^ D,s=1

For the cases of inclusions.

=

(17)

2e'"'\f(s)\

V'^'\f'Hg) ^^^'f'Hs)) f'(s)\

2/x {u, + iue) = e-" ( Kf{g) - f{g)tj^

^' k\ > 1.0 (18)

- K ? ) | (8) and

F,+iFy f

(19)

(SB')

(9) where g = ^ -\-ir] = pe'*; a is the angle between axes x and p, and

^-la^i.JJil P

!/'(?)!

ande2'"= ^

f'is)

P' f'Cs) For the interfacial boundaries of the defects of unit circle in a plate, the boundary conditions to be satisfied at ^ = ^0 = e'^ or 1^1 = 1.0 are

J=i JM

-O'+l)

1/^
(20)

Once the complex coefficients in the above-mentioned series can be determined, all these potential functions can be given uniquely and hence the fields of stress and displacement can be calculated conveniently.

C. Yang, A.-K. Soh/First MIT Conference on Computational Fluid and Solid Mechanics

556

3. Construction of the typical rectangular elements with a central defect The nodal displacement vector {^Y for an 8-node element is defined as: {^r == |_(Wx)l

(Wv)l

{Uxh

{Uy)l

•••

where [A^Lxie = W W ' or [A^M]2X16 = [ ^ M ] [ A M ] and [iV/]2xi6 = [^/] [ A M ] ~ ^ are termed as the element shape functions in its different parts according to the conventional finite element method and A = /X///XM represents the shear moduli ratio of the inclusion with the surrounding matrix. Moreover,

( M ^ ) 8 (My)8 J

(21) To express V^(^) and x(^) or coig) by {5}^, their hmits are chosen in such a way that their number of independent terms is equal to the total number of its independently nodal values. Moreover, the displacement vector {5} of any point in the element can be expressed in terms of the above-mentioned independent coefficients as follows: for holes/cracks,

(^v)l (^.)2

2/x{5r

{r) = [A]{r}

(^v)2

(^v)8

1

\u,{p.O)\

(^.)i

(22)

for inclusions (^Mx)l

Ulx (P,0)

(3/}

2/x,

U,y(p,9)

[a,]{r)

(23) {^Mx)2

{SM}

=

\UMAP,0)

I 2fli

\uMy(p,0)

(24)

[it^

2MM W = I (i^A^y)2 I {r} = [A„] {D

where {T} is a matrix consisting of the unknown independent coefficients in those potential functions;

m= [ftM] =

n,(p,e)

.[ft/] =

^,Ap,d)

and

d)

Further, (r) = 2(1 [A]-'

{Sr

or ( D = 2MM [ A M ] " '

{SY

(^M;c)8 (^My)8

where (Qx)i. (^y)i or ( ^ M X ) / , (^My)i stand for the components of [ft] or [ ^ M ] at node / (/ = 1 , . . . , 8), respectively; [A] or [ A M ] is a real coefficient matrix, whose elements is the values of the functions of p and 6. Also, the strain vector {e} of any point in the element can be obtained as follows: For holes/cracks.

^ly{p,0)

[a„Ap,o) ^My(P,

(25)

r9

Substituting Eq. (25) into Eq. (22) or Eqs. (23) and (24),

m[Ar'{sr = [N]isr

[S} =

(26)

£x

— dx

£y

0

^xy

d

-a? {SM)

=

UMX

[^M][AMr'{Sr

= [NM]{SY

(27)

UMy

[Qf][AMr'{8r

=

[Nj]{8r

(28)

-1

0

a 9y 9 dx-

{S}

dQx

~a7 dQy

[S,

(29)

[A]-^

~W dQx

dy

_|_

9^v

— : dx

(30)

C. Yang, A.-K. Soh/First MIT Conference on Computational Fluid and Solid Mechanics or for inclusions,

or for inclusions,

r^ {SM}

= "

557

£MX

dx

^My

' := 0

() [S] ^

AM

Ih

-ay

(34)

+ / / [B,]UD,][B,]rdA

d dx

d

^Mxy

m= JJ\^Mf\l)M][^M]tdA

0

Ai

where [D] or [D^] and [D/] are the element elasticity matrixes; AM and A/ are the areas of its different parts respectively and A® is the total area; t is the plate thickness.

d^ Mx dx

[AMrM5r = [BM]{5r oi) aQMy dx

4. Numerical examples

r dx"^ {ei} =

3^

d

^Ixy

0 d

0

eiy

-1

w

d dx-

l^

da Ix dx dQ,iy

[AMrM5r = [B/]{5r dQ

02)

ly

dx where [B] or [BM] and [B/] are the element strain-displacement matrixes in its different parts. Therefore, the element stiffness matrix is deduced as follows: for holes/cracks. [K]' = / [B]^ [D] [B] t dA

(33)

To verify the accuracy of the above-mentioned method, some examples have been analyzed for which the theoretical solutions have been obtained in plane elasticity theory. To consider a square plate with a central circular hole, an internal crack and an elliptic inclusion subjected to different load cases, the results by using the coarse mesh with the proposed special elements with the different above-mentioned defects are provided, respectively. Table 1 presents the normalized hoop stresses, GQ/GQ, along the free boundary of the hole when the plate was subjected to uniform tensile stresses in x direction. Ten analyses were performed, by varying the ratio of the radius of the hole to half the side length of the proposed element, R/A. It is obvious that the numerical results obtained using the proposed element are in good agreement with the corresponding theoretical results and the maximum discrepancy is less than 5% when R/A < 0.7. From Table 2, the proposed special element also has been used to analyze the case of the single crack under uniform tensile and its maximum discrepancy is less than 5% at a/A < 0.5 and it shows the relatively large crack cause the calculation precision decreases very quickly. Fig. 2 provided the comparison between analytical (line) and numerical results (dot/circle)

Table 1 Normalized stress ae /CTQ at the hole in a square plate subjected to uniform tensile stress in jc direction

on 0.01

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Exact

R/A

0

15

30

-1.0244 (-2.4%) -1.0287 (-2.9%) -1.0291 (-1.9%) -1.0194 (-1.9%) -1.0089 (-0.9%) -0.9873 (1.3%) -0.9692 (3.0%) -0.9575 (4.3%) -0.9237 (7.0%) -0.8748 (-12.5%) -1.0

-0.7566 (-3.3%) -0.7602 (-3.8%) -0.7602 (-3.8%) -0.7521 (-2.7%) -0.7428 (-1.5%) -0.7249 (1.0%) -0.7120 (2.7%) -0.6991 (4.5%) -0.6931 (5.3%) -0.6843 (6.5%) -0.7321

0.0249 0.0206 0.0257 0.0216 0.0151 0.0635 0.0047 0.0091 0.0366 0.1034

0

(-) (-) (-) (-) (-) (-) (-) (-) (-) (-)

45

60

75

90

0.9747 (--2.5%) 0.9756 (--3.4%) 0.9778 (--2.2%) 0.9767 (--3.3%) 0.9802 (--2.0%) 0.9794 (--2.1%) 0.9724 (--2.7%) 0.9599 (--4.0%) 0.9227 (--7.8%) 0.8327 (--16.7%)

1.9743 (--1.3%) 1.9777 (--1.1%) 1.9813 (--0.9%) 1.9754 (--1.7%) 1.9772 (--1.1%) 1.9699 (--1.5%) 1.9619 (--2.0%) 1.9594 (--2.0%) 1.9539 (--2.3%) 1.9331 (--3.3%)

2.9738 (-0.8%) 2.9799 (-0.7%) 2.9849 (-0.5%) 2.9744 (-0.9%) 2.9752 (-0.8%) 2.9642 (-1.2%) 2.9624 (-1.3%) 2.9870 (-0.4%) 3.054 (1.8%) 3.1941 (6.5%)

1.0

2.0

2.7060 (-1.0%) 2.7113 (-0.6%) 2.7160 (-0.6%) 2.7067 (-0.9%) 2.7078 (-0.9%) 2.6977 (-1.3%) 2.6938 (-1.4%) 2.7097 (-0.8%) 2.7535 (0.8%) 2.8417 (4.0%) 2.7321

3.0

C Yang, A.-K. Soh/First MIT Conference on Computational Fluid and Solid Mechanics

558

(a)

3.0

(b) 5.0 c

>

'

4.5

2.5

1

''—

I

- —H

4.0 2.0

3.5 3.0

1.5

\

0.5 0.0

2.5

6=0''

1.0

^'

-•*^:

1.5 1.0

/

0.5

e=9o^

-0.5

0.0

•0.S -1.0

10

-

\

1 """'" " R

.

—

L:.^^

k P

r

- -\

J

j

0=0*^

J

^-^^"i W

9=90**

K > -"---Po -

i

».T=sr

__J

•

10

r

Fig. 2. Comparison between analytical and numerical results of the resulting normalized hoop stress at the interface between the inclusion and its surrounding material at ^ = 0° and 6 = 90° for elastic moduli ration X under uniform tension in the y direction. Table 2 Stress intensity factor Kj and Kji at the crack tip (z = ±a) of the internal crack with length 2a in a square plate subjected to uniform tensile stress in y direction a/A

0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Numerical solutions

Exact solutions

Error

Ki

Ki

KI

Kii

KI

0.1820 0.4071 0.5761 0.8167 1.0047 1.1684 1.3204 1.4685 1.6188 1.7757 1.9397

1.295e-5 4.258e-4 2.385e-3 1.285e-2 3.258e-2 5.993e-2 9.203e-2 1.250e-l 1.572e-l 1.860e-l 2.102e-l

0.1772 0.3963 0.5605 0.7927 0.9709 1.1212 1.2536 1.3734 1.4836 1.5862 1.6827

0 0 0 0 0 0 0 0 0 0 0

2.70% 2.73% 2.78% 3.03% 3.48% 4.21% 5.33% 6.92% 9.11% 11.95% 15.27%

KII

their high accuracy are achieved due to the exact definition of the stress and strain fields within the elements and the implementation of interfacial boundary conditions. These elements can be easily combined with the conventional elements, such as isoparametric elements, without any modifications, to obtain good results using simple finite element meshes. Thus, the finite element analyses of complicated plane problems, involving a number of holes or cracks, can be carried out easily. This is particularly useful to researchers who are interested in micro-mechanics.

References

-

of the resulting normalized hoop stress ag/ao at the interface between the typical circular/elliptic inclusion and the surrounding material at orientations 0 = 0° and 0 = 90° for different elastic moduli ratios X under uniform tension in y direction. They have good agreement.

5. Conclusions Special rectangular membrane elements with central defects (holes/cracks/inclusions) have been developed and

[1] Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity, 4th edition. Leyden: Noordhoff Intemafional Publishing, 1975. [2] Isida M. Methods of laurent series expansion for internal crack problems. In: Sih GC (Ed), Methods of Analysis and Solutions of Crack Problems. Leyden: Noordhoff International PubUshing, 1975. [3] Piltner R. Special finite elements with holes and internal cracks. Int J Numer Methods Eng 1985;21:1471-1485. [4] Meguid SA, Zhu ZH. A novel finite element for treating inhomogeneous solids. Int J Numer Methods Eng 1995;38:1579-1592 [5] Soh AK, Long ZF, A high precision element with a central circular hole. Int J Solids Struct 1999;36(35):5485-5497

559

Fatigue analysis during one-parametered loadings J. Zarka*, H. Karaouni Ecole Polytechnique, Laboratoire de Mecanique des Solides, 91128 Palaiseau cedex, France

Abstract We have developed a special approach for fatigue analysis of structures. At first, we introduce a global representation of any loading at the scale of the material. Based on this definition, we are able to give the equivalence rule between any two loadings at the scale of the material to allow to represent any random loading by an equivalent periodic loading. Then, we define also a procedure to represent the random loading by an equivalent periodic loading at the scale of the structure which is easier and faster to compute. Examples of application to some specimen/details/structures are given. Keywords: Random loading; Inelastic analysis; Damage

1. Introduction We are concerned by the fatigue life of a welded structure under one-parametered cyclic and random loadings. This structure can be made of several materials. During such a loading, the elastic response may be written in the form:

where G^^ is the elastic stress tensor, X{t) is any random or cyclic scalar function of time which takes the zero value at time / = 0, Acr^^ is the loading direction, and a^^ is the initial stress field. It means that the stress path in the space of the elastic stress field is along a straight line. During classical fatigue analysis of the structure, it is usually performed a linear or non-linear finite element simulation, and then eventually with some special methods to extract and to count the cycles from the random stress path (very often, the Rainflow method), it is needed to cumulate 'damage' based on S-N diagrams and the linear Miner-Palmgreen rule or some more elaborated non linear rules. The 'damage' factor (or tensor) is still the object of many researches as it may be seen in the proceedings of a recent Symposium on 'Continuous Damage and Fracture' (Cachan, October 2000). Everybody is speaking about such * Corresponding author. Tel.: +33 01-69-33-33-35; Fax: +33 01-69-33-30-86; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

a 'damage' factor but nobody has ever seen it; it is impossible to measure it during the whole loading path, even if, before failure, some slight changes on the elastic properties may be experimentally detected; different models and theories were created and do not communicate between them. Moreover, the representations of the loading and the counting methods are purely based on mathematics aspects and ignore the particular mechanical behavior of the present materials in the structure. The objective of our paper is not to take position relative to the various theories but to describe a possible approach which has (of course from our point of view) a physical meaning and which is very easy to perform. The principal idea is to find an equivalent simple loading suitable to represent the real random loading in terms of damage produced at the level of the material then at the level of the structure.

2. Local 'physical' quantification of the loading At the local scale, we choose as a 'measure' of any one-parametered random loading, the cumulated plastic strain which induces on a one-single-degree-of-freedom system representing the linear kinematic material (similarly to what it is done during a seismic analysis). It is intuitive that even if the cumulated plastic strain is not 'the damage factor', they are both going towards the same direction. Then, with this measure, two loadings are equivalent if they

J. Zarka, H. Karaouni / First MIT Conference on Computational Fluid and Solid Mechanics

560

induce the same cumulated plastic strain. Based on our approach of the modelling of inelastic materials, this one single degree of freedom system is schematically:

|J1IUNAAA._^CT Sn

^pc(0

— /

\6(u)\ du

=a

where Ke is the elastic modulus, h is the hardening modulus, S is the elastic limit or the threshold of the slider, a^ is the local stress at the level of the friction block, a is the internal parameter (here the glide of the slider), y^ is the transformed internal parameter(indeed, the opposite to the residual load at the level of the slider), Sp is the actual plastic strain, and Spc is the cumulated plastic strain. Let us assume that we have the time history of one particular random loading, we input it to the system and we compute the induced cumulated plastic strain. We have ACT > 25 (as there is accommodation)

s^^ = Aa

8l^2N\A8^\ =2S-\-hAs.

(3) based upon the one-dimensional system, which is specific to the material of the two selected points, we compute the cumulated plastic strains respectively for these 2 points (4) we deduce at the local scale, at each point, the equivalent cyclic loadings (A(T/,A^/) driving to the same cumulated plastic strain (/ = 1,2); two hyperbolae C/ (5) each such a C, hyperbola may be drawn in the global plane (AA.,A^) (6) the intersection of the two new hyperbolae gives the global equivalent amplitude or range and the number of cycles to apply to the structure (7) in order to maintain the same order of the maximum plastic strain, ^-max is obtained from only one a, max More precisely, as classically, the Von Mises stress is used to define the equivalent local stress a^^ within the structure:

where S^' is the deviatoric part of the elastic stress field tensor a^', we may write deg as a function of A., SQ and A5^^

^4 = J\ fe : ^0 + 2A^o - A^'^ + X^Ag'^ : A^^^^

or in the plane (Aa, A^) the hyperbola

Aa = 2S-\-

h(spc-s%) 2N

If we define a^^^ = S -\- hSp^^, for this Aa, we deduce

where a,, bi, c/ are known scalar function of the point. Given ACT, implies that we need to solve a simple quadratic equation to deduce the corresponding AX: AXlai -h AXkbi -h Ci - {Aa)] = 0

For this given cumulated plastic strain, we may built a family of 'equivalent' cyclic loadings with constant amplitude (or range) and a number of cycles (similar to the Whoehler curve) (ACT, A^) from the initial cumulated plastic strain. In order to insure that the physical phenomena will be the same, we also select amax corresponding to Spmax computed during the analysis.

3. Global 'physical' quantification of the loading We propose the following procedure to describe the one-parametered random loading: (1) we compute the elastic response of the structure to this random loading (as the loading is with one parameter, we perform, the elastic analysis for the value of X equal to 1, and then with the principle of superposition, we generate locally all the local stress paths) (2) selection of two points within the structure (in the sensitive zones) for which we compute the equivalent Von Mises stresses

This equation has always real solutions; only the positive root has to be kept. When there is no initial stress field {hi and c, = 0) the solution is obvious. We have shown how to represent any one-parametered random loading by a one-parametered cyclic loading [1]. Without taking position on the best endurance criterium, the real definition of the 'damage' factor, the cumulation rule, during cyclic loadings, we have described a procedure to conclude if the loading will imply failure and then, for such a case, the number of cycles which is necessary to apply to initiate this failure.

Acknowledgments This paper was written in part during the regular visits of the first author to UCSD's Center of Excellence for Advanced Materials, under ONR contract NOOO14-96-1-0631 (R. Barsoum, Coordinator) to the University of California at San Diego and for the other part by the second author under the sponsorship of Ligeron S.A (A. Azarian, Coordinator).

/. Zarka, H. Karaouni /First MIT Conference on Computational Fluid and Solid Mechanics References [1] Zarka J, Karaouni H, Nemat-Nasser S, Huang J. Fatigue analysis of welded structures: SMIRT 15-PCS 13. Intelligent software Systems in Inspection and Life Management of Power and Process Plants, August 1999.

561

[2] Dang-Van K. Sur la resistance a la fatigue des metaux: Sciences et Techniques de TArmement, 3eme fascicule. 1973, pp. 647-722. [3] Robert JL. Contribution a I'etude de la fatigue multiaxiale sous sollicitations periodiques ou aleatoires. Phd thesis, INS A Lyon, 1992.

562

Non-linear stability analysis of stiffened shells using solid elements and the /?-version FE-method Adam Zdunek * Swedish Defence Research Agency, Aeronautics Division, FFA, SE-172 90 Stockholm, Sweden

Abstract An efficient direct method to determine the hmits of elastic stabiUty for a sequence of problems by using a two-parameter formulation of critical mechanical equilibrium behavior is presented. The use and performance of higher-order solid p-version finite elements in small-strain large displacement type elastic nonlinear stability analysis of stiffened shells is discussed. Keywords: p-Version FE; Structural stability; Equilibrium paths; Fold lines; Parameter sensitivity; Stiffened shells

1. Introduction The use of the p-version FE-method at FFA started in the early '80s in the spirit taught in the book by Szabo and Babuska [1]. The sound mathematical foundation and the superior convergence properties using strongly graded meshes provided the motivations for the choice of the p-version FE-method for the reliable analysis of linear fracture mechanics problems, see for example [3] and references therein. Today the p-version FE-code STRIPE [2] is used as a platform also in a number of other research areas of engineering solid mechanics. Examples are contact mechanics, probabilistic stress analysis, small-strain elastoplastic analysis and nonlinear structural stability analysis, see for example [4,5]. This presentation elaborates on some recent achievements made in nonlinear static structural stability analysis [5] from the perspective of using the /?-version FE-method and 3D solid finite elements modeling the mechanical response of stiffened shell-like structures. The main motivation behind this strategy is the ability to efficiently and accurately predict a fully three-dimensional stress state where appropriate while avoiding shear and membrane locking problems [6] where a shell-like behavior is expected. It is tacitly assumed that the convergence

* Tel.: -h46 (8) 5550-4372; Fax: -h46 (8) 258-919; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

properties of the p-version FE method in classic linear elasto-statics carry over. An efficient direct prediction of the limits of elastic stability for a sequence of problems where the design is slightly varied, or where a spectrum of loadings is given, is presented. A /7-version FE based solution to this problem using a direct critical equilibrium path following technique [7] which avoids cumbersome re-analyses using basic equilibrium path following is given.

2. Problem formulation The mechanical equilibrium of a continuum is considered on the basis of p-version finite element discretizations of the virtual work principle in geometrically nonlinear quasi-static elasto-mechanics with conservative and proportional loadings. The resulting finite-dimensional equilibrium equations are studied on the form G(a, A,/x) = 0

(1)

where a e M" are generalized displacement degrees of freedom and (X, fji) e R^ are two control parameters. The first parameter X is the principal load proportionality factor. The second parameter /x is used to control a design variable, for example a geometric measure, a material property, or an auxiliary loading. A particular case /JL = /JLQ is denoted the reference problem. In Eq. (1) G is a continuously

A. Zdunek/First MIT Conference on Computational Fluid and Solid Mechanics differentiable operator mapping from R" x R^ into E". Due to the conservative loading the tangent stiffness (Jacobian) Ga{a, /x) e R"""" is symmetric and independent of X. Here, the shorthand notation Ga denotes the directional derivative in the a-direction. Moreover, for a properly restrained structure the tangent stiffness is positive definite in the neighborhood of the unloaded configuration on account of the restrictions on the material parameters E > 0 and — 1 < V < 1/2, respectively. 2.1. Basic equilibrium path following The fundamental path and intersecting secondary paths of the reference problem, have to be explored to a relevant extent. This basic equilibrium path following is done in the usual way by augmenting Eq. (1) with a suitable pseudo arc-length type constraint condition N(a,X,s) = 0 where s is used as control parameter. Exploration of the basic equilibrium paths for the reference case also leads to the problem of detecting and calculating a subset of its critical equilibrium points. 2.2. Direct calculation of simple critical points For a fixed problem /x = /XQ a critical point is a mechanical equilibrium state (^cr, ^cr) for which the tangent stiffness Ga(acr, Mo) is singular. The associated right null vectors of Ga(acr, Mo) are denoted 0 e R""""^ where a = dimN(Ga). At a simple critical state the tangent stiffness is rank-one deficient and a = I. The examples of complex instability behavior treated herein correspond to a number of closely spaced but distinct simple critical points. For a fixed /x = /xo a critical point is determined (isolated) using a direct method by augmenting Eq. (1) with the criticality constraint condition g(a,(l>,ix):=

'G,(fl,/x)0^ I "^ ' " I = 0 . kTu

V 0^0 - 1 /

(2)

as originally proposed in [7]. That is, for a fixed M = Mo the set of 2n + 1 equations H(a, A, 0,

M)

'•-

^G(fl,A, M) ^^(fl,0, M)

= 0

(3)

are solved for the simple critical point (flcr, -^cr) and the associated null vector 0. The direct calculation is started at a regular equilibrium state (ao, AQ) (obtained as described in Section 2.1 and a linearized version of Eq. (2) is used to supply a provisional null-vector 00- Confer [8,9] for a discussion on the risk of getting spurious eigenvalues in a linearized buckling analysis. Definition 1 (Stability - conservative systems). A sufficient condition for a solution (a, A, /x) of Eq. 1 to be

563

stable is that the corresponding tangent stiffness is positive definite. Thus, a critical point of Eq. 1 do not correspond to a stable equilibrium state. Definition 2 (Limit of elastic stability). For a fixed case /x = Mo the limit of elastic stability is defined as the solution of Eq. 3 with the smallest positive value k. Definition (2) is consistent with the fully nonlinear stability analysis approach presented, cf. [9,8] for the corresponding linear case. 2.3. Critical equilibrium path following -fold lines The determined critical point at M = Mo, («cr, -^cr) and the associated null vector 0 can now be used as a starting value for a continuation process that directly traces the limit of elastic stability as function of MIn order to do that Eq. (3) is augmented with a generalized pseudo arc-length constraint equation N(a,X, fi,Y]) = 0 where r) is used as control parameter. As long as the associated Jacobian of the augmented system is non-singular we can now follow a smooth path of critical points (flcr(^), Kr(^)^ Mcr(^)), forming a so-called fold-line.

3. A case study The nonlinear buckling of a sequence of T-shaped profiles is contained in this short note to illustrate the performance of the solid p-version finite element approach in combination with the direct method for determination of the elastic stability limit. The problem definition is given in Fig. 1. A model with 400 solid p-version finite elements sUghtly graded in the vicinity of the external load and towards the re-entrant comers is used. With a isotropic and uniform distribution of p = 4 this model has 24 thousand degrees of freedom. It is certainly a far from optimal discretization. It is however checked that it provides fairly converged overall mechanical behavior. The results obtained with a /z-version shell element model with 1536 elements and 4998 degrees of freedom are provided, [5]. A uniform traction over a small area is used to represent a vertical load of magnitude P = lOOOA, at point A. The shapes of the isolated null-vectors at the two first critical points, Xcr{i) = 2.715 and Acr(ii) = 3.282 for the reference problem are shown in Fig. 2. The linearly predicted critical loads are up to 6% and 20% lower, respectively [5, Table 2]. A direct determination of the limits of elastic stability using the described method in Section 2.3 for a sequence of T-profiles with varying width b of the flange is shown in Fig. 3. Finally, Fig. 4 shows the marked change in the primary post-critical

564

A. Zdunek /First MIT Conference on Computational Cross section I

b=38

Fluid and Solid

Mechanics

Boundary conditions

,

u(0, b/2, 0) = 0 v(0, b/2, z) = v(l, b/2, z) = 0 w(0, y, z) = w(l, y,z) = 0

Material E=70960 v=0.3321

A

f

A

B

Fig. 1. Transversally loaded T-profile. Definition of the reference problem. Mode 1: "Local mode" Mainly distortion of the flange under the load Mode 2: "Global mode" Twisting + bending of the ^veb^

Fig. 2. Transversally loaded T-profile. Isolated null-vectors at the two first critical points, ;.cr(i) = 2.715 and Acr(ii) = 3.282 for the reference problem. load-deflection behavior for a sequence of T-shaped profiles with varying lengths I. More details and other relevant examples will be included in the oral presentation.

4. Conclusions This contribution discusses the use of an efficient direct method to determine the limits of elastic stability for a se-

quence of problems by using a two-parameter formulation of critical mechanical equilibrium behavior. The importance of making parameter sensitivity analyses is evident. The use and the good performance of the higher-order solid /7-version finite elements in small-strain large displacement type elastic stability analysis is corroborated by numerical results. The ability to efficiently, accurately and robustly predict local three-dimensional stress states in a stiffened shell-like structure speaks in favor of the presented solid /7-version finite element approach.

A. Zdunek/First MIT Conference on Computational Fluid and Solid Mechanics

565

References

40 45 Width of the flange Fig. 3. Transversally loaded T-profile. Direct detemiination of the limits of elastic stability for a sequence of problems with varying width b of the flange. Fold lines for the second and third bifurcation point are also shown. The fold lines for the first and second bifurcation points do not cross. The shapes of the null-vectors are however exchanged, a so-called curve veering is observed. 1.1

\

1.051

•

'

\

\

—\i

\

_ — „ ^ 71=400 :

„„ ,

. 1

1 /if^^-S^^^-^50 ;

\^^^ 1 L^-<^ ^0.95 h--":;y'^

j>—i5 v.- 1=500 :

PH

0.9 L

\

.},.sbell.ro.o(lel...

£ ^ - ^

• - ^ ^ ^ " ^ ^ •*"vp.:. ^ • ^ - ^ ^

-. :

-• • :

-

^ 1

H "^ ••??••. : ^'-'^'i

vv,v.y.ai),p:-YersiQH

J

0.85K 0.8'

-

4 2 0 2 4 Lateral displacement at point C

Fig. 4. Transversally loaded T-profile. Post-critical paths for the first bifurcation point for a sequence of profiles with different lengths t.

[1] Szabo B, Babuska I. Finite Element Analysis. New York: Wiley, 1991. [2] STRIPE Users Manual. Bromma, Sweden: The Aeronautical Research Institute of Sweden (FFA), 1992. [3] Fawaz S, Andersson B. Accurate stress intensity factor solutions for unsymmetric comer cracks at a hole, FFA TN 2000-36. Bromma, Sweden: The Aeronautical Research Institute of Sweden (FFA), 2000. [4] Andersson B. Rehable analysis of 3D multiple-site fatigue crack growth, probalistic analysis of a butt joint, FFA TN 1999-20. Bromma, Sweden: The Aeronautical Research Institute of Sweden (FFA), 1999. [5] Eriksson A, Pacoste C, Zdunek A. Numerical analysis of complex instability behaviour using incremental-iterative strategies, Comput Methods Appl Mech Eng 1999; 179:265305. [6] Suri M. Analytical and computational assessment of locking in the hp finite element method. Comput Methods Appl Mech Eng, 1996;133:347-371. [7] Jepson A, Spence A. Folds in solutions of two parameter systems and their calculation. Part I. SIAM J Numer Anal 1985;22:347-367. [8] Szabo B, Kiralyfalvi G. Linear models of buckling and stress-stiffening. Comput Methods Appl Mech Eng 1999;171:43-59. [9] Suri M, Xenophontos C. Reliability of an hp algorithm for buckling analysis. lASS-IACM 2000. In: Fourth International Colloquium on Computation of Shell & Spatial Structures, June 5-7, 2000, Chania-Crete, Greece.

566

Random vibration of structures under multi-support seismic excitations Yahui Zhang *, Jiahao Lin Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116023, RR. China

Abstract For the seismic response analyses of multiply supported structures subjected to random excitations, a highly efficient algorithm, the Pseudo Excitation Method (PEM) is presented. The algorithm is an accurate CQC method because the cross-correlation terms between the participant modes and between the excitations have both been included. The algorithm is easy to implement on computers because any stationary random response analysis is exactly replaced by harmonic response analysis, while any non-stationary random response analysis is replaced by a step-by-step integration scheme. The Hong Kong Tsing-Ma suspension bridge is taken as an application to use the proposed method. Keywords: Structure; Earthquake; Multiple support; Random; Vibration

1. Introduction

2. Pseudo excitation method

Although the basic framework of linear random vibration theory has been well established, its applications to practical engineering have been seriously restricted mainly due to the huge amount of computational efforts required for complex engineering problems. Therefore, Ernesto and Vanmarcke [1] pointed out that the use of random vibration method by the earthquake engineering community is impractical, except for simple structures with a small number of degrees of freedom and supports. Kuireghian and Neuenhofer [2] held the similar view. They both selected the response spectrum method as approximate approach to the random vibration method. The present paper has completely resolved such difficulties by developing the Pseudo Excitation Method (PEM) [3,4]. It is an efficient and accurate CQC method. The variance analysis of 1800 displacements and 300 internal forces for a suspension bridge with 2300 degrees of freedom, 17 supports and 150 participant modes needs only 8 min in the case of stationary seismic excitations on a Pentium-2 personal computer, and 42 min in the case of non-stationary seismic excitations. The wave-passage effect, incoherence effect, local effect and the quasi-static displacements have all been accurately included. This method is now very popular in China.

The governing equations of motion for a discretized linear system are [2]

* Corresponding author. Tel.: +86 (411) 470-8402, E-mail: [email protected]

The pseudo responses [X'^] = {X^} exp(/ft;0 can be easily obtained. The corresponding PSD matrix can

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Mss Ml

\x„

Mm.

• +

c C^

0

Xs

Kss

c c

K„,

(1)

Jm

The response vector is usually decomposed into pseudostatic and dynamic components (2)

[Xs] = [Ys] + [Yr] They satisfy the equations

(3)

[Ys] = -[KssY\K,m\[Xm] [Mss][Yr]^[Css][Yr]

+ [Kss][Yr]

=

[Mss][KssY\K,m\{Xm]

(4)

For stationary and uniform seismic excitations, the pseudo ground acceleration is [3,4] { x ; (r, CO)} = ^S,^,^ {(D) exp {icot)

(5)

Y. Zhang, J. Lin/First MIT Conference on Computational Fluid and Solid Mechanics

567

KN 0 1 2 3 4

700 600

response spectrum method uniform ground motion wave passage effect Loll coherence model Harichandran coherence model

500 400 300 200 100

Fig. 2. Seismic shear force responses of the deck. Fig. 1. The Hong Kong Tsing-Ma Bridge. be accurately computed by [ 5 x . M ] = [X'XlX'V

=

For long-span structures, the wave passage effect, incoherence effect and sometimes the local effect should be included. This has long been considered as very difficult. For such cases, the ground excitation PSD matrix [^xmxm (<^)] can be decomposed into the sum of the PSD matrices of a limited number of single excitations, each corresponds to a pseudo harmonic excitation. Their linear superposition gives the accurate solution of such multiexcitation problems. The above method has be extended to non-stationary excitation problems [4].

3. Example The Hong Kong Tsing-Ma bridge (see Fig. 1) is analyzed by using the proposed method. The vertical shear force responses of the deck are illustrated in Fig. 2. It is seen that the wave passage effect is of particular importance, therefore, the widely used response spectrum method which neglects this effect is not so reliable.

4. Conclusions Accurate random seismic analyses for multi-support structures have become very simple by using the proposed method. Various complex engineering problems can be computed quickly and conveniently.

References [1] Ernesto HZ, Vanmarcke EH. Seismic random vibration analysis of multi-support structural systems. ASCE J Eng Mech 1994;120:1107-1128. [2] Kiureghian AD, Neuenhofer A. Response spectrum method for multi-support seismic excitations. Earthquake Eng Struct Dyn 1992;21:713-740. [3] Lin JH, Zhang WS, Li JJ. Structural responses to arbitrarily coherent stationary random excitations. Comput Struct 1994;50:629-633. [4] Zhang YH. Buckling and dynamic analysis of complex structures under multiple type of loading (in Chinese), Ph.D. Thesis 1999.

568

On simulation of a forming process to minimize springback Kunmin Zhao * General Motors Corporation, MC 480-400-111, 6600 E 12 Mile Rd, Warren, Ml 48092, USA

Abstract A forming process, known as SHAPESET, is simulated using the finite element code AutoForm. Simulation results agree with experimental observations. The SHAPESET process nearly eliminates springback of sidewall curl and is not sensitive to material properties. In comparison, the numerical simulation of the conventional draw forming is also performed. Keywords: Sheet metal forming; Stamping; Springback; SHAPESET

1. Introduction The automotive industry is becoming more interested in weight saving materials, such as high-strength steel (HSS) and aluminum alloy (AA), to improve fuel economy and upgrade vehicle performance. These materials are more difficult to stamp than conventional mild steel (MS) because they split at lower strains and exhibit more springback. Stamping a u-channel part results in two types of springback: angle open and sidewall curl, as illustrated in Fig. 1. Zhao and Lee [1] and Geng and Wagoner [2] have investigated the springback of aluminum sheets upon draw/bend angle open

loading. A stamping process, known as SHAPESET, was described by Ayres [3] to eliminate or reduce the sidewall curl springback. The main purpose of this study is to apply the finite element code AutoForm to simulate the SHAPESET forming process. For comparison, the conventional deep drawing process is also analyzed.

2. Simulation procedure and results Both SHAPESET forming process and conventional drawing process are simulated for a 75-mm deep and 70-mm wide plane strain u-channel. High-strength steel (HSS), aluminum alloy (AA) and mild steel (MS) are investigated. Their mechanical properties are listed in Table 1. The incremental finite element code AutoForm is used to perform the numerical analyses. 2.1. Simulation procedure

draw die shape

The SHAPESET (Ayres [3]) process consists two separate steps as shown in Fig. 2. In the first step, a preform is Table 1 Material properties

Fig. 1. Illustration of springback of a u-channel secfion. *Tel.: -hi (810) 578-4402; Fax: -\-l (810) 578-3578; E-mail: kunmin.zhao @ gm.com © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

AA HSS MS

E (GPa)

Y (MPa)

K (MPa)

n

R

70 210 210

143 360 159

514 111 520

0.26 0.16 0.22

0.11 1.0 1.5

K. Zhao/First MIT Conference on Computational Fluid and Solid Mechanics Step 1

569

— AA draw - ^ - - AA stretch —^

3.2nimShim

HSS draw

- X - - HSS stretch —A

MS draw

- - A - - MS stretch Step 2 Fig. 2. Schematic of the SHAPESET process. =^^=s

stamped with the drawbeads removed and with a 3.2-mm shim placed in the die cavity to limit punch travel. The partially formed stamping is then stretched to the final shape in the second step by firmly clamping the flanges and removing the shim. The SHAPESET process is modeled as a continuous forming process in the numerical simulation. The punch is positioned under the blank and remains stationary throughout the simulation. A pad is used to hold the blank against the punch. The upper binder contains locking drawbeads. The lower binder is positioned at a short distance to the punch bottom. That distance can be calculated by an inhouse code for an estimated wall strain. In the first step, the upper binder moves down while the lower binder remains stationary. The flanges are firmly clamped when the upper binder closes up with the lower binder. Then the upper binder presses the lower binder down to the final position. For convenience, the SHAPESET stamping process is named stretch forming in this paper, while the conventional deep drawing process is called draw forming. Various peak strains are achieved by applying different restraining forces in the draw forming operation and by setting up different positions of the lower binder in the stretch forming operation.

0

5

10 15 20 25 Peak strain along section (%)

^

30

Fig. 3. Flange angles developed by draw forming and stretch forming. (punch side) unloads elastically and reloads in compression. Similarly, the compressive stress in the outer surface (die side) unloads and then reloads in tension. Therefore, markedly different surface stresses are developed in the sidewall even though the strain is about the same on each side. For example, in the draw forming of the HSS channel, the strain on each side of the sidewall at the middle (point B in Fig. 1) is 9.8%, but a curvature of 5.52 (10"^) is created after springback because of the stress gradient. The first step of the stretch forming operation is identical to the draw forming, except that the stress gradient is much less severe. The second step of the stretch forming brings the through thickness stresses to the same level in tension. In the stretch forming of the HSS channel, the strain at point B is 5.1%, but a curvature of only 6.89 (10~^) is created due to the negligible stress gradient.

2.2. Simulation results and discussions

3. Conclusions

The finite element code is able to capture the angle open as well as the sidewall curl as shown in Fig. 1. The final flange angles developed by both stretch forming and draw forming are plotted in Fig. 3. Draw forming produces appreciable springback when the peak strain along the section is less then 15%. The material properties play a significant role. A larger restraining force reduces the amount of springback. The stretch forming is considerably effective in reducing the springback, even at low strain level of 5%. It is not quite sensitive to the material properties. The draw forming operation creates stress gradient through the sidewall to cause curl. As the sheet slides around the die radius, the tensile stress in the inner surface

(1) The finite element code AutoForm is able to simulate the stretch forming operation (SHAPESET), a two-step forming process that can reduce significantly the springback in straight, high-strength steel and aluminum u-channels. It also has the capability of capturing the springback of angle open and sidewall curl. (2) The stretch forming operation (SHAPESET) is not sensitive to material properties in reducing the sidewall curl springback. (3) Aluminum exhibits more springback than highstrength steel and mild steel. Mild steel develops lest springback because its (E/Y) ratio is the lowest and a greater wall load can be applied without splitting.

570

K. Zhao/First MIT Conference on Computational Fluid and Solid Mechanics

(4) The simulation technique developed in this paper is the major analytical tool in the formability group for early product design of high-strength steel structural parts.

Acknowledgements The author is grateful to Robert L. Frutiger and William W. Liu for their support and approval. Technical conversations with Chung-Yeh Sa have been invaluable throughout this study.

References [1] Zhao KM, Lee JK. Springback prediction using combined hardening model. In: SAE International Body Engineering Conference and Exposition, Detroit, Michigan, USA, Oct. 3-5, 2000. [2] Geng LM, Wagoner RH. Springback analysis with a modified hardening model. In: SAE World Congress, Detroit, Michigan, USA, March 6-9, 2000. [3] Ayres RA. SHAPESET: A process to reduce sidewall curl springback in high-strength steel rails. J Appl Metalwork 1984;3(2):127-134.

571

Linear multi-step and optimal dissipative single-step algorithms for structural dynamics Xiangmin Zhou^, Kumar K. Tamma^'*, Desong Sha^ ^ Department of Mechanical Engineering, 111 Church Street S.E., University of Minnesota, Minneapolis, MN 55455, USA ^Dalian University of Technology, Dalian 116024, People's Republic of China

Abstract The objectives of the present paper are: (i) to formulate the underlying spectral equivalence between the linear multi-step (LMS) methods and the so-called generalized single-step operators, and (ii) to design an optimal family of controllable numerically dissipation algorithms for linear structural dynamics. Keywords: Transient algorithms; Structural dynamics; Time discretized operators

1. Introduction For structural dynamics computations, emanating from a generalized time-weighted residual procedure, the resulting time-discretized operators have recently been characterized into three distinct classifications as Type 1, Type 2, and Type 3 [1]. Algorithms pertaining to a Type 1 classification are a direct result of selecting the tensor form of the homogeneous solution of the corresponding linear structural dynamic problem as the exact weighted time fields and with or without approximations introduced to the integration involving the load terms [2]. These representations for Type 1 classification of algorithms, even though they are explicit and unconditionally stable, they either involve an eigenvalue problem or the need to evaluate an exponential matrix. For nonlinear structural dynamic problems, the advantages and disadvantages of the Type 1 classification of algorithms are not fully explored to date; however, some limited work on the pros and cons appear in [1]. Algorithms pertaining to a Type 2 classification are a result from introducing approximations to the exact weighted time fields in the Type 1 classification of algorithms, which however retain the tensor form of the weighted time field, to reduce the computational effort. Type 2 classifications of algorithms preserve the unconditional stable algorithmic property and the capability to generate higher-order * Corresponding author. Tel.: +\ (612) 626-8102; Fax: +1 (612) 624-1398; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

accurate algorithms for linear situations as in Type 1 classifications of algorithms. The need for approximations or interpolations for the dependent field variable(s) are irrelevant for the Type 1 and Type 2 classifications of algorithms. Both Type 1 and Type 2 classifications do not pertain to linear multi-step (LMS) methods and hence circumvent the Dahlquist theorem [3]. On the other hand, algorithms pertaining to a special class of the Type 3 classification are a result from further introducing approximations to the weighted time fields in the vector or scalar function form in the weighted residual equation, thereby now necessitating further approximations or interpolations to the dependent field variable(s) and the associated updates. In all of the above classifications, for enacting the time-discretization process, (i) the burden of weight carried by the weighted time fields, and (ii) the conditions (if any) the weighted time fields impose (dictate) upon the corresponding approximations for the dependent field variables and the associated updates are uniquely characterized by the so-called discrete numerically assigned (DNA) algorithmic markers [2,4] which distinctly characterize time-discretized operators. Of attention here are the special class of the Type 3 classification of algorithms which are primarily the single-step representations of the classical hnear multi-step (LMS) algorithms wherein a wide variety of the so-called time-stepping algorithms which we are mostly familiar with pertain to this classification. To circumvent a certain drawback associated with the Newmark method, the first

X. Zhou et al. /First MIT Conference on Computational Fluid and Solid Mechanics

572

unconditionally stable and numerically dissipative algorithm with controllable numerical damping is the Wilson-^ method [5]. It is cited that the Wilson-^ method suffered from the drawback of second-order displacement overshoot and first-order velocity overshoot due to initial displacement [U2-V1]. In contrast, the Hoff-Pahl method [6] has only first-order displacement and velocity overshooting behavior [Ul-Vl]. Unlike these, the Hilber-Hughes-Taylor (HHT-a) method [7,8], the Wood-Bossak-Zienkiewicz (WBZ) method [9], and the Generalized-a method [10] belong to the same family of algorithms which inherently inherit zero-order displacement overshoot and first-order velocity overshoot [UO-Vl] due to initial displacement. Besides the overshoot properties, the HHT-a method, WBZ method, and the Hoff-Pahl method have similar algorithmic dissipation and dispersion properties for the same norm value (magnitude) of the principal roots in the highfrequency limit (poo)- The numerical dissipation and numerical dispersion properties among the currently available [UO-Vl] algorithms can be optimized by linear combination of a special case of the WBZ method and the HHT-a method such as appeared in the Generalized-a method, which is a result from setting the magnitude of the principal roots and the spurious root in the high-frequency limit to be equal. As a result, the Generalized-a method yields minimal numerical dissipation and numerical dispersion within the currently available [UO-Vl] algorithms for the same p^ and the same approximation for the weak form and the updates. In terms of the spectral radius curve, only one special case of the WBZ algorithm will give L-stability within the currently available [UO-Vl] algorithms. The range of Poo for the WBZ method is Poo ^ [O, l), the range of Poo for the Generalized-a method is Poc € (O, l), and the range of Poo for the HHT-a method and the Hoff-Pahl method is p^c G [5,1). In all of the above and existing to-date [UO-Vl], [Ul-Vl] and [U2-V1] controllable dissipative algorithms, an additional and major drawback is that they are only first-order approximations for the load except for the case of Newmark average acceleration or mid-point rule. The present paper establishes the underlying spectral equivalence between LMS methods and generalized singlestep operators with emphasis on subsequently designing an optimal family of controllable numerically dissipative algorithms for linear structural dynamics.

above two field forms, the characteristic equation for the homogeneous modal equation of motion can be shown to yield for the generalized LMS representation ^

[{a2)i + 2(^2)/?^ + ( A ) / ^ ' ] ' = 0

2.2. Generalized single-step methods Consider the single-field form of the equation of motion, Mu(0 -f CiiCO + Ku(r) = f(0, u(0) = uo,

(3)

u(0) = uo

The characteristic equation of the generalized singlestep methods is

^[a,-h2ft?^-h}//^']0'=O

(4)

As such, the generalized LMS representations and the generalized single-step representations for the second-order equation of motion are spectrally identical.

3. The optimal family of dissipative algorithms Employing the weighted residual method, and following the procedure shown in [1] to develop generalized single-step algorithms, and next imposing the conditions for second-order accuracy, unconditional stability, optimal dissipation and dispersion, and second-order approximation for the load term, the optimal family of dissipative algorithms is derived as: Algorithm 1. Given u„, u„, and u„, find u„+i, u„+i, and ii^+i from

/ u„Ar , u„A^' \ - K U„ -h — + 7— TT V

1+Poo

(l+Poo)V

u„A^^ u„+i = u„ + u„ Ar -h -— + 1+Poo •

2. From LMS methods to SS methods

(2)

/=o

u„+i =ii„ - f u „ A r + 2.1. Linear multi-step method

AaAr

AaA^^ (1+Poo)^

1+Po

u„+i = u„ -h Aa Consider the two-field form of the equation of motion, d - F + Ad = g(d,r),

d(0) = do

(1)

Following [11], and extending the classical LMS to the

where poo € [0, 1]. Fig. 1 shows the comparison of spectral radius, dissipation, and dispersion for the family of [UO-VO] algorithms

573

X. Zhou et al. /First MIT Conference on Computational Fluid and Solid Mechanics 0.2

^ "^ —^ —^

0.175 0.15 0.125 rjJLP 0.1

0.075 0.05 0.025

r

\ \

WBZ Hoff-Pahl Generalized- a O p t i m a l UO-VO

0.4 0.375 0.35 0.325 0.3 0.275 0.25 t l 0.225 EH 0.2 t^ 0.175 0.15 0.125 0.1 0.075 0.05 0.025

(b)

ii\

i _ 0

?L1

1 0 " 1 1 lO n n T - ^ 0.2 " 0.3 0.4 0.5

0 ^

0.

(c)

- HHT- a

- WBZ ^^- Hoff-Pahl ed- a ^- -- GO pe tniemr aall i zUO-VO

.-

0.1

•

•

•

•

L

0.2

I

.

.

.

r 0.

Fig. 1. Comparison of spectral radius, dissipation, and dispersion for [UO-Vl] and the optimal [UO-VO] algorithms with poo = 0.8: (a) spectral radius; (b) dissipation; (c) dispersion. n HHT- a A WBZ V Hoff-Pahl <5> Generalized- c O Optimal UO-VO

20 r F _F E /» t ^ O lofevy ^

D HHT- a A WBZ V Hoff-Pahl <5> Generalized- a O Optimal UO-VO

F ' *o

, J

.loE-W

ho''

D A V ^ O

HHT- a WBZ Hoff-Pahl Generalized- a O p t i m a l UO-VO

' „ "oon

o

(c)

-15 F-

•

I

I

I

Fig. 2. Comparison of overshooting behavior for [UO-Vl] and the optimal [UO-VO] algorithms with poo = 0.8. (a) displacement overshoot ^t/T = 10; (b) velocity overshoot Ar/T = 10; (c) velocity overshoot d.t/T = 100. and the [UO-Vl] algorithms. Fig. 2 shows the comparison of overshooting behavior to demonstrate the fundamental difference betw^een the [UO-VO] and the existing [UO-Vl] algorithms with p^o = 0.8 for a single-degree-of-freedom

—^

Optmal UO-VO — Generalized- oc HHT WBZ Hoff-Pahl

undamped modal problem for illustration. Fig. 3 shows the comparison of accuracy for the [Ul-Vl], [UO-Vl], and [UO-VO] algorithms of the singe-degree-of-freedom problem. An illustrative elastoplastic large deformation example is considered next with non-zero initial conditions (Fig. 4a). The time history of the vertical deflection at the midspan point of the plate for the optimal [UO-VO] family of methods with Poo = 0 . 8 is shown for illustrative in Fig. 4b for an instantaneous velocity VQ = —5000 in/s.

4. Concluding remarks The design and theory underlying LIMS and single-step methods were outiined first. Finally, an optimal family of algorithms was described for structural dynamics.

Acknowledgements

250

N

500

750

1000

Fig. 3. Comparison of accuracy for [Ul-Vl], [UO-Vl], and [UO-VO] algorithms with poo = 0.8.

The authors are very pleased to acknowledge support in part by Battelle/U.S. Army Research Office (ARO) Research Triangle Park, North Carolina, under grant number DAAH04-96-C-0086, and by the Army High Performance Computing Research Center (AHPCRC) un-

574

X. Zhou et ai /First MIT Conference on Computational Fluid and Solid Mechanics

0.00025

0.0005

0.00075

Time (sec) Fig. 4. Impulsively loaded clamped plate and comparative results for illustration purposes: (a) problem description and geometry; (b) results for a selected optimal UO-VO dissipative algorithm. der the auspices of the Department of the Army, Army Research Laboratory (ARL) cooperative agreement number DAAH04-95-2-0003/contract number DAAH04-95-C0008. The content does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. Support in part by Dr. Andrew Mark of the Integrated Modeling and Testing (IMT) Computational Technical Activity and the ARL/MSRC facilities is also gratefully acknowledged. Acknowledgement is also due to Dr. R. Namburu for other related discussions. Special thanks are due to the CIC Directorate at the U.S. Army Research Laboratory (ARL), Aberdeen Proving Ground, Maryland. Other related support in the form of computer grants from the Minnesota Supercomputer Institute (MSI), Minneapolis, Minnesota is also gratefully acknowledged.

[3] [4]

[5]

[6]

[7]

[8]

References [1] Tamma KK, Zhou X, Sha D. The time dimension: a theory of development/evolution, classification, characterization and design of computational algorithms for transient/dynamic applications. Arch Comput Mech 2000; 7(2):67-290. [2] Tamma KK, Zhou X, Valasutean R. Computational algorithms for transient analysis: the burden of weight and consequences towards formalizing discrete numerically as-

[9]

[10]

[11]

signed [DNA] algorithmic markers: Wp-family. Comput Methods Appl Mech Eng 1997; 147:153. Dahlquist G. A special stability problem for Hnear multistep methods. BIT 1963;3:27. Tamma KK, Zhou X, Sha D. A theory of development and design of generalized integration operators for computational structural dynamics. Int J Numer Methods Eng 2001;50:1619-1664. Wilson EL. A Computer Program for Dynamic Stress Analysis of Underground Structures. Berkeley: SESM, University of California, 1968. Hoff C, Pahl PJ. Development of an implicit method with numerical dissipation from a generalized single step algorithm for structural dynamics. Comput Methods Appl Mech Eng 1988;67:367. Krieg RD, Key SW. Transient shell responses by numerical time integration. Int J Numer Methods Eng 1973; 17:273286. Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Eng Struct Dyn 1977;5:283. Wood WL, Bossak M, Zienkiewicz OC. An alpha modification of Newmark's method. Int J Numer Methods Eng 1980;15:1562. Chung J, Hulbert G. A time integration method for structural dynamics with improved numerical dissipation: the generalized a-method. J Appl Mech 1993;30(371). Henrici PK. Discrete Variable Methods in Ordinary Differential Equations. New York: Wiley, 1962.

575

A 3D contact-friction model for pounding at bridges during earthquakes Ping Zhu *, Masato Abe, Yozo Fujino The University of Tokyo, Civil Engineering Department, 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan

Abstract A model is presented for three-dimensional pounding problems of bridge girders with friction. Contacting surfaces are assumed to be planes, and penetrations of contactor nodes to target surfaces are allowed and utilized to compute pounding reactions. On considering the pounding problems of bridge girders under earthquakes, the paper implements the model in a general-purpose dynamic analysis program for bridges. Experiments of pounding were conducted for verifying this model. The applicability of the model is illustrated by computations of pounding among a three-span steel bridge. Keywords: Pounding; Bridge; 3D modeling; Contact-friction model; Nonlinear analysis; Seismic evaluation

1. Introduction Unseating of bridge girders/decks during earthquakes is very harmful to the safety and serviceability of bridges. Evidence shows that in addition to damage along longitudinal direction, which is emphasized in current seismic design, lateral displacement and rotation of bridge girders caused by pounding can also lead to unseating. To analyze this effect, a 3D model of pounding is needed. Several methods are available to solve contact/pounding problems [1-5]. These methods can be categorized as: (1) capable of arbitrary contact of bodies/surfaces [1]; and (2) dealing with pounding of two prescribed points [2-5]. The advantage of first group is that it can be applied to a wide range of static and dynamic problems with material and geometric non-linearities. A relatively complicated algorithm is a disadvantage. Moreover, it is unsuitable to be used directly to a system mainly composed of bar elements, such as normal bridge structure models. For the second group of methods, clear physical meanings and simple algorithms are advantages. A ID fixed point-to-point contact element is adopted in analysis of pounding between bridge girders according to this group of methods. But a 3D arbitrary contact between girders cannot be simulated in this manner. The model developed in this paper is capable to ana* Corresponding author. Tel.: +81 (3) 5841-6099; Fax: +81 (3) 5841-7454; [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

lyze 3D pounding problem in bar based bridge structure systems. In addition, it is easy to be applied together with commonly used dynamic analysis methods.

2. A 3D contact-friction model for pounding The problem considered herein is a general case of pounding by two bridge girders (see Fig. 1). A 3D contact-friction model for the problem is illustrated in Fig. 2. The target surface, named as abed, is assumed as a rigid plane. (The surface does not have to be a rectangle.) During contact, the position and velocity at contact point p are girder!

glrden

tamtbod,

^^„^^]p^f„f contactorbody c

target sur^ce

outernormal VBCtor

con^ctor node

Fig. 1. Bridge girders in arbitrary contact.

576

P. Zhu et al. /First MIT Conference on Computational Fluid and Solid Mechanics and Slide: Kent

p

F.i.i-'T^^rF.u

Rk = Fk|, + Fe|„ + Ff|,.

(7b)

Components of damping forces at normal and tangent directions Fd^ and Fdt are given by Feln = - C - V k p L ,

(8a)

Fcl, = - G . V k p | , .

(8b)

Kinetic friction Ff|( is given by -Mk-|Fk|J

Fig. 2. Illustration of the 3D contact-friction model.

Vkpit

IVkpltl

(8c)

where /ik is the kinetic friction coefficient. During slide, the physical contact point p will be updated by the new position of node k.

functions of the target surface as: p = p(a, b , c, d)

(1)

Vp = Vp(Va,Vb,Vc,Vd)

(2)

2.3. Contact forces at target surface

The material overlaps at contactor node A;, A^, and the relative velocity of node k to point p, Vkp, are given by Ak = Xp — Xk,

(3)

Vkp = V k - V p .

(4)

Upon contact, a universal spring Kent between node k and point p is created to compute the force of contact. Two dashpots, C and C,, are also applied to node k for simulating energy loss during contact. The contact force at node k, Fk, is given by: Fk = Kent ' Ak.

(5)

The contact forces at four nodes of the target surface are given by

'target-Surface — i

*^target_surface — [ T ] ( - R k ) ,

(9)

(10)

where [T] is a distributed matrix by linear interpolation according to static equilibrium. 2.4. Governing equations of motion

2.1. Status of contact A contact happens when the contactor node k goes inside the target body and ends when node k moves out of the body. Contact status can be divided into stick and slide contact, which can be decided by Stick:

|Fk|J < M s | F k U ,

To apply the model to dynamic time-history analysis, incremental equilibrium equation of motion for iteration (/) at time f -h A/ is as follows:

(6a)

_ r+A/|^0-l) , r+ArpO-1)

and Slide:

|Fk|J > / X s | F k | J ,

(6a)

where Fkl^, Fklt are the normal and tangent components of Fk, respectively, and /Xs is the static friction coefficient. 2.2. Contact forces at contactor node

Rk = Fk + F e | „ + F e l t ,

where M, C, K

are mass, damping and stiffness matrices, respectively, AU^'^, AU^'^, AU^'^ are incremental vectors of accelerations, velocities and displacements at the /th, respectively,

t-\-Mj^{i-\)^t-^i^ty^{i-\) ^ g vectors of accelerations and ve-

Forces at node k are obtained separately for stick and slide conditions by Stick:

(11)

(7a)

^+^^R

locities after iteration (/ — 1), respectively, is the external load vector at time r -h A/,

r+Arp(/-i) jg i^j^g restoring force vector after iteration (/ — 1),

p. Zhu et al. /First MIT Conference on Computational Fluid and Solid Mechanics

511

r+Af^O^^i) jg ^YiQ vector of contact forces after iteration (/ — 1), can be obtained by -•V^nt —

/

(12)

^ ••V:i•ntlcnt-pair

cnt_pair

and

Rkl Rent Icnt^pair

Rk —

R,

Ra

. = , Rb

»et_surface 1

(13) time (sec)

Re

Fig. 3. Result comparison (ID case).

Rd] 2.5. Parameters of the model

3. Experimental verifications of the model

The axial stiffness of the contactor body can be used as the stiffness of the universal spring ^cnt (see Eq. (14)), where E, A and L are modulus of elasticity, cross-section area and length of the contactor element, respectively. ^cnt = —

(14)

Damping ratios C and Ct can be determined according to the restitution coefficient at normal and tangent directions by C =

?

2^IK,

M1M2

(15)

'M1+M2

-In^ y7r2 + (lne)2

(16)

where Mi,M2 are the masses of the two bodies in contact, is the restitution coefficient, range from 1 (elastic) to 0 (plastic), is the damping ratio according to restitution coefficient e, range from 0 to 1.

Experiments have been conducted to verify the model [6]. The experiments, with one model girder supported by rubber supports, and an abutment on a shaking table, were taken in ID and 2D cases according to excitation angles. Fig. 3 shows result comparison in ID case. Good agreements can be seen from experimental data and analytical results.

4. Computation of a three-span steel bridge with the pounding model A typical three-span steel bridge has been selected for analysis. As shown in Fig. 4, fiber model is adopted at the first segment of each pier from foundation. Base-isolation rubber bearings are applied to each pier [7]. For computing of pounding, a simple supported girder in each span is assumed. Computations were conducted in cases of with and without pounding under earthquake excitations of Takatori waves from the 1995 Kobe earthquake. Results of comparisons for mid-span are shown in Figs. 5 and 6. rigid bars

-^

X

Fig. 4. The three-span steel bridge.

P. Zhu et al /First MIT Conference on Computational Fluid and Solid Mechanics

578

girders, an algorithm for solution has been developed. Experimental verifications were also conducted. This model is suited to be combined with commonly used dynamic analysis methods. It is capable of analyzing poundings of bridges and also suitable for building structures. The applicability of the model can be seen from computations of a steel bridge.

References time (sec) Fig. 5. Longitudinal displacement of the mid span at node A. _J — w i t h o u t pounding

0.002

^—with pounding 1

0.001

L^.,-,wAr -^ ^'^ 1^ WrSli /K ' g -0.001

f

iwvr^

2 " ^

^Jg^^

1

I

I -0.002 -0.003 time (sec)

-0.004

Fig. 6. Rotating angle of the mid span at node A. The longitudinal displacement is reduced by pounding. A remarkable increase of rotating angle of the girder can be seen as a result of pounding.

5. Conclusions A 3D contact-friction model has been presented in this paper. On considering pounding problems between bridge

[1] Chaudhary AB, Bathe KJ. A solution method for static and dynamic analysis of three-dimensional contact problems with friction. Comput Struct 1986;24(6):855-873. [2] Papadrakakis M, Mouzakis H, Plevris N, Bitzarakis S. A lagrange multiplier solution method for pounding of buildings during earthquakes. Earthquake Eng Struct Dyn 1991;20:981-998. [3] Anagnostopoulos SA. Pounding of buildings in series during earthquakes. Earthquake Eng Struct Dyn 1988;16:443-456. [4] Ruangrassamee A, Kawashima K. Relative displacement response spectra with pounding effect. In: Proceedings of the First International Summer Symposium, International Activities Committee JSCE, Tokyo, Japan, 1999, pp. 9-12. [5] Jankowski R. Pounding of superstructure segments in elevated bridges during earthquakes and reduction of its effects. Doctoral dissertation, The University of Tokyo, 1997. [6] Yanagino K. Modeling of Pounding Behavior of Bridge Girders under Seismic Excitations (in Japanese). Master's thesis. The University of Tokyo, 2000. [7] Yoshida J, Takesada S, Abe M, Fujino Y. A bi-axial restoring force model on rubber bearings considering two-direction horizontal excitations (in Japanese). Proceedings of the 25th Conference of Research on Earthquake Engineering 1999;2:741-744.

579

Computational testing of microheterogeneous materials T.I. Zohdi*, P. Wriggers Institutfiir Baumechanik und Numerische Mechanik, Appelstrasse 9A, 30167 Hannover, Germany

Abstract In this paper, we investigate topics related to the numerical simulation of the testing of mechanical responses of samples of microheterogeneous solid material. Consistent with what is produced in dispersion manufacturing methods, the microstructures considered are generated by randomly distributing aggregates of particulate material throughout an otherwise homogeneous matrix material. Therefore, the resulting microstructures are irregular and nonperiodic. A primary problem in testing such materials is the fact that only finite sized samples can be tested, leading to no single response, but a distribution of responses. In this work, a technique employing potential energy principles is presented to interpret the results of testing groups of samples. Three-dimensional numerical examples employing the finite element method are given to illustrate the overall analysis and computational testing process. Keywords: Material testing; Random heterogeneous material

1. Introduction A primary research issue in the analysis of solid heterogeneous materials is the determination of 'effective' or 'homogenized' constitutive laws for use in macroscopic structural calculations (Fig. 1). The usual approach is to determine a relation between averages, IE*, defined through MACROSCOPIC STRUCTURE

NEW MATERIAL Fig. 1. Modifying a material with particulate additives.

((T)n = IE* : {€)Q. Here, {•)n = ]k\L'^^^ ^^^ ^ ^^^ € are the stress and strain fields within a statistically representative volume element (RVE) with volume |^|. If IE* is assumed isotropic one may write //ra^ 3^,def\ 3 IQ tr€\

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

def

2/x*'

{(T')Q

'

(€')^

: {€')n

{(^'}Q

'

(1)

(3 where tra and tr€ are the dilatational components of the stress and strain and a' and e' are the deviatoric stress and strain components. We note that even if the aggregate response is not purely isotropic, one can always interpret the above expressions as generalizations of isotropic responses. In theory, an effective response will be invariant for a sample that is infinitely large compared to the microstructure (Fig. 2). However, from a practical point of view, whether computationally or experimentally, we can only test finite sized samples of material. Therefore, the samples will produce 2E*'s that exhibit deviations from one another. Clearly, no single effective response appears. For example, in the isotropic case we have uncertainties, such as /x* ± A/x*

* Corresponding author. Tel.: -^49 (511) 762-19059; Fax: -^49 (511) 762-5496; E-mail: [email protected]

and

and

K* ± Aic*.

(2)

The determination of the magnitude of such uncertainties is the subject of this work.

580

T.I. Zohdi, P. Wriggers /First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 2. Left: A random microstructure consisting of 20 non-intersecting boron spheres, occupying approximately 22% of the volume in an aluminum matrix, as seen by the algorithm with a 24 x 24 x 24 trilinear hexahedra mesh density for a total of 46875 degrees of freedom (approximately 9 x 9 x 9 hexahedra or 2344 degrees of freedom per element). A '2/5' rule, i.e. a 2 x 2 x 2 Gauss rule if there is no discontinuity in the element, and a 5 x 5 x 5 rule if there is a discontinuity, was used. Right: A zoom on one particle.

2. Overall testing process: numerical examples

For the 100 sample tests, with 20 particles per sample, the results for the effective responses were

A typical example of a composite material combination is that of an Aluminum matrix (77.9, 24.9 GPa) embedded with (stiffening) Boron particles (230, 172 GPa). This is a widely used composite due to its light weight. We chose Al/Bo as a material combination which exhibits significant enough mismatch in the mechanical properties to be representative of a wide range of cases. All tests were run on a single IBM RISC 6000 workstation. Comparable hardware is available in most academic and industrial work places, therefore such simulations are easily reproducible elsewhere for other parameter selections. 2.1. Multiple sample tests We simulated 100 different samples, each time with a different random distribution of 20 nonintersecting particles occupying 22%. Consistent with the previous test's mesh densities per particle, we used a 24 x 24 x 24 mesh ( 9 x 9 x 9 trilinear hexahedra or 2344 d.o.f. per particle, 46875 d.o.f. per test sample), which provided mesh-insensitive results. The averages, standard deviations and maximum/minimum of these quantities are tabulated in Table 1, as well as a histogram in Fig. 3. Throughout the tests, we considered a single combined boundary loading satisfying Hill's condition, E/^ = 0.001, /, y = 1, 2, 3:

91.37 = {K-%^
30.76 = (At-i)^^ < A* = 42.35 < {/x)^ = 57.68, where ic* and jl* are the averaged effective responses from the 100 tests, and where the lower and upper bounds are, respectively, the classical Reuss and Voigt bounds. We also compared the computed results to the well-known Hashin-Shtrikman bounds which are, strictly speaking, only applicable to asymptotic cases of an infinite (sample length)/(particulate length) ratio and purely isotropic macroscopic responses. The 'bounds' were as follows: 94.32 = Ki-\- •

V2 1

^

K2 -ATI


3(1-i;2) 3/Ci + 4 / ^ 1

bulk modulus H/S lower bound
1 — 1'2

1 Ki -

31^2

= 102.38,

K2 + 2>K2 + 4/^2

bulk modulus H/S upper bound

35.43 = /ii +

6(1 -V2){ic\ /X2-A61

+2/xi)

5MI(3/CI+4/[ii)

shear modulus H/S lower bound Ell

E12

Ei3

Xx

Ei2

E22

E23

Xl

E31

E32

E33_

Xi^

< A* = 42.35 (3)


(1 - V2) 1

6u2(/<:2 + 2 / 1 6 2 ) • + /Xl - / i 2 ' 5 / X 2 ( 3 / C 2 + 4 A i 2 ) shear modulus H/S upper bound

= 45.64,

(5)

581

T.L Zohdi, P. Wriggers /First MIT Conference on Computational Fluid and Solid Mechanics Table 1 Results of 100 material tests for randomly distributed particulate microstructures (20, 40 and 60 spheres) Particulate

d/L

20 20 20 20

0.2763 0.2763 0.2763 0.2763

40 40 40 40 60 60 60 60

Max — min

Quantity

Average

Standard deviation

46,875 46,875 46,875 46,875

W (GPa) /c* (GPa) /x* (GPa) CG-ITER

0.001373 96.171 42.350 49.079

7.4 X 10-^ 0.2025 0.4798 1.5790

3.6 X 10"^ 0.950 2.250 9

0.2193 0.2193 0.2193 0.2193

98,304 98,304 98,304 98,304

W (GPa) /c* (GPa) /x* (GPa) CG-ITER

0.0013617 95.7900 41.6407 60.3200

5.0998 X 10-^ 0.1413 0.3245 1.6302

2.35 X 10-^ 0.6600 1.590 7

0.1916 0.1916 0.1916 0.1916

139,968 139,968 139,968 139,968

W (GPa) /c* (GPa) /x* (GPa) CG-ITER

0.0013586 95.6820 41.4621 66.500

4.3963 X 10-^ 0.1197 0.2801 1.9974

2.25 X 10-5 0.6214 1.503 10

d.o.f.

We note that for a zero starting guess for the iterative solver, the average number of CG-iterations was 55, therefore using the initial Voigt (M = E • jc) guess saved approximately 12% of the computational solution effort.

m 10

95.75

96

96.25

96.5

EFFECTIVE BULK MODULUS (GPa)

Fig. 3. The effective bulk modulus distribution for 100 samples. where KI, ixi and /C2, /X2 are the bulk and shear moduli for the matrix and particle phases. Despite the fact that the bounds are technically inapplicable for finite sized samples, the computed results did fall within them. Beyond a certain threshold, it is simply impossible to obtain any more information by testing samples of a certain size. The inability for obtaining any more information is the fact that the testing conditions are uniform on the 'subsamples'. This idealization is valid only for an infinitely large sample. However, suppose we increase the number of particles per subsample even further, from 20 to say 40 then 60, each time performing the 100 tests procedure. With this information, one could then possibly extrapolate to a (giant) sample limit. The results for the 40 and 60 particle cases are shown in Table 1 for 22% Boron volume fraction. Using these results, along with the 20 particle per sample tests, we have the following curve fits

W = 0 . 0 0 1 3 2 0 5 + 0.0001895

R^ = 0.9997,

d K* = 9 4 . 5 2 7 + 5 . 9 0 9 - ,

R^ = 0.986,

d M* = 3 9 . 3 4 5 + 1 0 . 7 7 5 - ,

R^ = 0.986,

(6)

where L is the sample size, d is the diameter of the particles. Thus as d/L -^ 0, we obtain an estimates of W = 0.0013205 GPa, 94.527 GPa and 39.345 GPa as the asymptotic energy, effective bulk modulus, and effective shear modulus, respectively. Indeed, judging from the degree of accuracy of the curve-fit, (R^ = 1.0 is perfect) the data appears to be reliable. The slightly less, although still quite accurate, reliability (regression values of i?^ = 0.98) of the effective responses, /c* and /i*, is attributed to the fact that absolute perfect isotropy is impossible to achieve with finite sized samples. In other words, the extrapolations using various samples exhibit sUght isotropic inconsistencies. However, the energy W, has no built-in assumptions whatsoever, thus leading to the nearly perfect curve-fit.

3. A minimum principle interpretation Consider the following process for a sample of material with u\dQ =E- x: 1. Step 1: Take a large sample, and cut it into N pieces, Q = U^^^QK' The pieces do not have to be the same size or shape, although for illustration purposes it is convenient to take a uniform (regular) partitioning (Fig. 4). 2. Step 2: Test each piece (solve the subdomain BVP) with the loading: M|a^^ =E x. The function UK is the solution to the BVP posed over subsample ^K-

582

T.L Zohdi, P. Wriggers /First MIT Conference on Computational Fluid and Solid Mechanics

LARGE PROBLEM STATISTICALLY REPRESENTATIVE SAMPLE

nnnnn nnnnn nnnnn nnnnn Einnnn

^^

PARTITIONED PROBLEM "SUBSAMPLES"

Fig. 4. The idea of partitioning a sample into smaller samples or equivalently combining smaller samples into a larger sample.

3. Step 3: One is guaranteed the following properties:

{ff>n

K —

u —"

K •

"^

£(Q)

=

(^> QK^ E : {m*

m''^ -m*)

~ def

< ^ ' < m*

u — Wife, + « i\n,

...UN

96.17-91.37 = 0.2141, 111.79-91.37 42.35 - 30.76 = 0.4305. 57.68 - 30.76

(8)

For traction tests, the testing of the smaller samples will bound the response of the very large sample from below. For more details on the presented results, see Zohdi and Wriggers [5].

N

1^1 '

References

:E|Q|

< E : ( ^ * - -{lE'^ >^'):E|<^|, {TE- ' ) ^ '

Therefore, for displacement tests, the averaged effective responses generated will always bound the response of the very large sample from above. Therefore, the average of the 100 sample tests provide us with tighter upper bounds on the response of a very large sample. Therefore, by the results in Box 7, the Reuss-Voigt bounds in Box 4 are tightened by the following factors

(7)

< ilE)^, \n^-

The same process can be done for the traction test loading case: t\dQ^ =Ln. The effective material ordering, line 3 in Box 7, has been derived by Huet [3]. The second line of Box 7, and generalizations to nonuniform loading, were developed in Zohdi and Wriggers [4]. The proofs result from a direct manipulation of classical energy minimization principles.

[1] Michaud V. Liquid state processing. In: Suresh S, Mortensen A, Needleman A (Eds.), Fundamentals of Metal Matrix Composites, 1992. [2] Mura T. Micromechanics of Defects in Solids, 2nd edn. Kluwer Academic, Dordrecht, 1993. [3] Huet C. Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 1990;38:813-841. [4] Zohdi TI, Wriggers P. A domain decomposition method for bodies with microstructure based upon material regularization. Int J Solids Struct 1999;36(17):2507-2526. [5] Zohdi TI, Wriggers P. Some aspects of the computational testing of the mechanical properties of microheterogeneous material samples. Int J Num Methods Eng, accepted for publication.

Optimization & Design

584

Shape optimization of frictional contact problems using genetic algorithm M. Al-Dojayli, S.A. Meguid* Engineering Mechanics and Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, ON, M5S 3G8, Canada

Abstract In this article, we develop and implement a new method for shape optimization of elastic solids in frictional contact using genetic algorithm. A continuous representation of contact surfaces was derived using C^-cubic splines. Consequently, a consistent linearization of kinematic contact constraints was derived and imposed exactly using Lagrange multipliers to solve the variational inequalities that define the contact problem. The resulting contact stresses were minimized using a genetic algorithm, which uses the control points of the splines as being the design variables. In this case, the potential energy was selected as the objective function. The best fit of the objective function was chosen using the tournament selection procedure. This approach avoids the difficulties associated with the use of scaling in the roulette wheel procedure. Finally, a numerical example is given to illustrate the improvement in the design resulting from the proposed technique. Keywords: Shape optimization; Contact; Cubic splines; Variational inequalities; Quadratic programming; Genetic algorithm

1. Introduction Shape optimization of solids in contact has gained special attention since the last two decades [1-3]. The focus was on the proof of uniqueness and existence of solution to such problems. The optimal shape design of contact problems is in general non-smooth and non-convex [2]. Consequently, either non-differentiable optimization or regularization techniques with gradient optimization algorithms were used. The regularization technique depends on the use of smoothing parameters, which might affect the accuracy of contact and optimization solutions. In addition, both techniques usually lead to local minima. Another problem encountered in shape optimization is the mesh mismatch introduced by shape modification during the optimization procedure. It was shown that using finite element linear discretization of contact surfaces could produce dramatic errors in contact stresses [4]. It is therefore the objectives of this paper to develop a new shape optimization algorithm, which overcomes the above drawbacks. The proposed approach is based on the use of * Corresponding author. Tel.: +1 (416) 978-7753; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Lagrange multipliers to impose the kinematic contact constraints exactly and the accurate representation of contact surfaces using C^-continuous cubic splines. In this case, control points that define the cubic spline segments were used as design variables to minimize the potential energy (objective function). In order to select the best fit in the genetic algorithm procedure, the tournament selection was used.

2. Contact formulation Contact between solids is generally governed by three constraints: (i) the magnitude of the normal contact pressure must be less than or equal to zero, (ii) the displacements of the contacting surfaces must satisfy the kinematic contact conditions, so as to avoid interpenetration, and (iii) the tangential forces and displacements along the contact surface are governed by a friction law. 2.1. Kinematic contact constraints The gap between the master and slave surfaces can be determined from the kinematics of deformation (Fig. 1), as

M. Al-Dojayli, S.A. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics

585

law, can be expressed as follows [5,6]: aCu, '\ - n) + y Cu, 'y) - jCu, 'u) > fCv - ^u), ^u, V V e K

(6)

a(u, v) represents the strain energy: a(u,\) = / aij(u)Sij (\)dQ

(7)

7(v) represents the virtual dissipation by the frictional forces: = j -fiaM(u)\yT\ d r (8) re f(v) represents the virtual work done by the external forces: j(u,\)

Fig. 1. Schematic of kinematic contact conditions.

f(v)= f f - v d ^ +

being x^ - X 5 = min Vxm - X5

(1)

Using linear discretization, the unit normal vector to both master and slave contact surfaces is generally not unique. Therefore, contact surfaces are represented using C^-piecewise cubic splines: x"(M) = ^5f(M)bf,

0<w< 1

(2)

where B and b represent the Bernstein polynomials and control point vector respectively. In order to preserve local support of cubic splines, Bessel tangents were used. In this case, the normal inward vector to a 2-D spline segment a can be defined uniquely using the first derivative:

where m"(M«) = £x"(i/„). Consequently, the gap function between two contact surfaces at unknown configuration t + At can be expressed as: ^+^^g(x) = p+^^x^ - ^^^^xs] . ^+^^N > 0 (4) In this case, the discretization of contact constraints for a general contact point p on the cubic spline can be represented as: ^Qa,p^^a,p < ^Goi,p

(5)

where Q«,^ represents the assembly of the nodal contribution to the kinematic constraint in the normal direction, b^^^ is the vector of nodal degrees of freedom associated with this constraint and G„,^ is the gap between two contacting surfaces.

Q.

M-vdr

There is no direct solution to Eq. (6) [5]. As a result, a reduced form of the VI is considered in which the normal contact stresses are assumed to be prescribed. In this case, the VI can be described [5] as: aCu, '\ - u) + JCY) ~ jCu) > f f V - ^u), 'u, V ' V G K

(10)

where j(y)=

f

-/x|aA,||vrl d r

(11)

t+Atrc

f

Start ^

Initialize population a (cubic spline vertices)

•

Fitness n(u(a), a) 1

"

GA Operation

r

H*

Fitness n(u(a'*'),a''')

^^;i;;;;?>ii <>.

0

v^onvergence^^,.,^-^^^

^f^s

2.2. Finite element solution tofrictional contact problems

f The variational inequality describing the solution u for the general contact problem, employing Coulomb's friction

(9)

Ft

End

^

Fig. 2. Schemal ic of shape optimization using genetic algorithm

M. Al-Dojayli, S.A. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics

586

L=10,H = 2 R=150 E = le6, V = 0.3 H = 0.1 P = 500

(a)

(b)

20

"•" 16

S

Initial Shape

—

Optimum Shape

4

k:• • • • 0.2* • • • • *0.4• "^

'

4

•

•

•—4

•

•

•

»

0

0.6 0.8 1 Distance along the contact surface (x/L) Fig. 3. Shape optimization of two elastic bodies in contact: (a) FE model, and (b) normal contact stress distribution for the initial and optimum designs. However, the above VI formulation (10) has the nondifferentiable frictional term ]{•). In order to overcome this difficulty, non-differentiable optimization is adopted [6]. In this work, a one-step algorithm has been developed, along with the use of incremental VI, in order to account for the non-linearity of the gap function. The kinematic contact constraints are updated globally and imposed in each iteration of the solution. In order to represent the plane of non-differentiability, additional constraint is imposed, which separates the regions of slip-stick conditions. The reduced VI formulation of (10) is equivalent to solving the following minimization problem: mm

^AFIJ + '+^'AUTS'+^'F'5

(12)

subject to: (13) ^S^r+^'AU > ^T C + ^ ' U ' - ' U )

(14)

where Eq. (13) represents the assembly of the kinematic contact conditions of the nodes on the candidate contact

surface Tc, and Eq. (14) represents the regions of slip-stick conditions. 3. Shape optimization using genetic algorithm The optimal shape design of contact problems can be described by the following inequality: n(a*,u(a*)) < n(a,u(a)) < n(a,v(a)), u, V V G K

Va*, V a e M

(15)

As indicated in [2], the solution of general shape optimization problems of solids in contact can be non-smooth and non-convex. Therefore, a global optimization solver using genetic algorithm (GA) was used. GA consists of search procedures that use random choice as a tool to guide a highly exploitative search through a coding of a parameter space. This approach is robust and can be used efficiently in shape optimization problems, since it does not deal with the derivatives of the objective function, as shown in Fig. 2. A simple genetic algorithm is composed of three operators: (i) selection, (ii) crossover and (iii) mutation. The tournament selection procedure is adopted in this work. This approach contains both random and deterministic features. Crossover and mutation operators are performed on

M. Al-Dojayli, S.A. Meguid/First MIT Conference on Computational Fluid and Solid Mechanics the binary coding of the design variables. The positions of crossover and mutation are selected randomly based on probability parameters. This algorithm is repeated until convergence is reached or the maximum number of iterations is exceeded. The efficiency of this approach can be best demonstrated by the Schema Theorem developed by Goldberg [7]. This theorem indicates that short, low-order, above-average schemata receive exponentially increasing trials in subsequent generations. This important property indicates that the proposed genetic algorithm converges within a finite number of iterations.

4. Numerical example The example shown in Fig. 3(a) was selected to show the ability of the proposed method to minimize contact stresses. In this example, an elastic clamped beam is in contact with an elastic foundation. The beam is subjected to a uniform distributed loading P at a distance / from the free end. The resulting initial and final normal contact stresses were normalized by the bending stiffness of the upper beam Go = E X {H/Lf. Fig. 3(b) shows the resulting uniform stress distribution along the contact surface of the optimum design. This stress was obtained as a result of the use of our optimization algorithms.

5. Conclusions In this work, a new method for shape optimization of contact problems was developed and implemented. The

587

method employs variational inequalities and genetic algorithms to minimize the potential energy, which resulted in a uniform stress distribution, as illustrated in the numerical example. This was achieved by manipulating the locations of the control points that define the contact surfaces. The work also reveals that the use of C^-cubic spHnes to model contact surfaces enhanced the accuracy of contact solution and ultimately the optimum shape.

References [1] Haslinger J, Neittaanmaki P. On the existence of optimal shapes in contact problems. Numer Funct Anal Optimiz 1984;7:107-124. [2] Haslinger J, Neittaanmaki P. Finite Element Approximation for Optimal Shape Design. Theory and Applications. New York: Wiley, 1988. [3] Sokolowski J. Sensitivity analysis of the Signorini variational inequality. Partial Differ Equat Banach Center Publ Warsaw, 1987;7. [4] El-Abbasi N, Meguid AS, Czekanski A. On the modeling of smooth contact surfaces using cubic splines. Int J Numer Methods Eng, in press. [5] Kikuchi N, Oden JT. Contact Problems in Elasticity: A study of Variational Inequalities and Finite Element Methods, SIAM. Philadelphia: Elsevier 1988. [6] Refaat MH, Meguid SA. A novel finite element approach to frictional contact problems. Int J Numer Methods Eng 1996;39:3889-3902. [7] Goldberg DE. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, 1989.

588

Simulation of non-gaussian wind pressures and estimation of design loads G. Bartoli, C. Borri*, L. Facchini, F. Paiar Universitd degli Studi di Firenze, Dipartimento di Ingegneria Civile, 50139 Firenze, Italy

Abstract In some previous research work, the authors dealt with the determination of the actual wind pressure distribution on the surfaces of 3-D irregular buildings of large size, (see Borri and Facchini [1-3]). Owing to the complex shape of the buildings, experimental campaigns in Boundary Layer Wind Tunnel (BLWT) are usually carried out, whose main results allow the determination of dynamic wind pressures, the evaluation of their time and space correlation structure, and the characterization of the stochastic properties of pressure fields. The present paper focuses on the computational aspects for determining the actual design loads leading to effects which are equivalent to the dynamic action caused by turbulent wind. Keywords: Wind engineering; Stochastic dynamics; Non-gaussian pressures; Neural networks; Numerical simulations; Design loads

1. Introduction

2. A neural network based approach

The present work is devoted to the numerical simulations for the reproduction of the actual complex aerodynamic circumstances arising, for instance, from the sharpedged shape of a 3-D bluff body. A neural network approach is currently successfully employed (Borri and Facchini [2]) to implement a nonlinear auto-regressive digital filter to simulate the pressure field on the whole external surface; such field is often evidently non-gaussian, as it appears in Figs. 1 and 2. Furthermore, a design load concept is introduced, whose main features take into account: • dynamic pressure fields (multi-correlated processes) as input data • two-dimensional influence coefficients relating dynamic pressures to internal forces • cross-correlation structure of the internal forces/stresses Finally, the effects of such non-gaussian properties are evaluated in terms of extreme values of the internal resultant stresses, investigating the reduction of non-gaussianity due to the integration process [4].

The simulation procedure is based on the decomposition of the pressure field in a series of Radial Basis Functions (RBF). In this way, the pressure field around the structure

* Corresponding author. Tel.: +39 (055) 479-6217; +39 (055) 479-6230; E-mail: [email protected]

Fig. 1. Time history for a pressure coefficient recorded on the windward side of a hip roof.

© 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

PRESSURE COEFFICIENT TIME HISTORY

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589

PEAK LOAD DISTRIBUTION FOR BENDING MOMENT MB

PRESSURE COEFFICIENT HISTOGRAM 350

200 150 100 50

static system: fixed-fixed

i . nn

-0.2

0

0.2

0.4

0.6

0.8

1

Hnnn^

1.2

1.4

(P)max = 0.68732 1.6

Fig. 2. Empirical histogram for the same pressure coefficient of Fig. 1. is described by a spatial RBF Artificial Neural Network with time-varying coefficients [3] p (x, t) = Y^Wk

^ 1

(P)min = -0.70451

Pressure {+)

^B

Depressure (-)

Fig. 3. Peak load distribution for bending moment MB identified using the LRC method. PEAK LOAD DISTRIBUTION FOR VERTICAL REACTION VB

it) (pk ( x ) ,

( _ ^ ^ )

(Pk (t) = exp

(1)

where p^ are the experimental pressure tap locations on the model and a a decay parameter. The coefficients Wk(t) are generated by means of auto-regressive nonlinear filters in the form v/(h8t) = f (w((/z - l)8t)..

.w((/z - M)8t)) + s{h8t)

A Static system: fixed-fixed

(2)

where the vector function f is modeled by means of an Artificial Neural Network whose coefficients are estimated according to the results of the wind tunnel tests [3]. The simulation tests will be finalized to the evaluation of the structural response and to the analysis of the extreme values of the internal stresses. The equivalent static design loads will then be evaluated to reproduce the same stress distribution as the dynamic ones.

3. Application examples The complete procedure is being tested and successfully applied to different structural typologies, like sharp-edged roofs, high-rise buildings and large industrial engineering facilities, such as interfering cooUng towers. The main steps consist in: (1) statistic analyses of experimental pressure fields; (2) simulation of non-gaussian pressure fields; (3) integration of (nonlinear) dynamic response with estimation of peaks and gust factors; (4) estimation of reduction of non-gaussianity in terms of internal stresses; and, finally.

^ B

(P)max = 0.32772 H i

Pressure (+)

(P)min = -0.71965 ^ "

Depressure (-)

Fig. 4. Peak load distribution for vertical reaction VB identified using the LRC method.

(5) the determination of the equivalent static design loads. The final step is usually carried out by means of quasistatic calculations, such as the definition of the peak factor [5], the load-response correlation (LRC) method [6] (see Figs. 3 and 4), and the POD method [7,8]. A comparison with the results given by such approaches is at present being carried out in order to define the proper dynamic amplification factor for the structures under examination.

References [1] Borri C, Facchini L. Wind induced loads on the monumental roof structure of the XII Century 'Palazzo delta Ragione' in Padova. In: Larsen A et al. (Eds), Proc. of the 10th Int Conf on Wind Engineering, lAWE, Balkema, 1995.

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G. Bartoli et al. /First MIT Conference on Computational Fluid and Solid Mechanics

[2] Borri C, Facchini L. Artificial generation on non-gaussian 3D wind pressure fields on structures or bluff body surfaces. 8th ASCE Spec Conf On Probabilistic Mechanics and Structural Reliability, 2000. [3] Borri C, Facchini L. Some recent developments in modeling turbulent wind loads and dynamic response of large structures. In: Fryba, Naprstek (Eds), Structural Dynamics, Vol. 1. Balkema, 1999, pp. 3-12. [4] Bartoli G, Borri C, Facchini L, Paiar F. Estimation of wind loads from wind tunnel experiments. 4th International Colloquium on Bluff Body Aerodynamics and Applications, 2000. [5] Davenport AG. The application of statistical concepts to the wind loading of structures. Proceedings of the Institution of Civil Engineers (UK) 1961;19:449-472.

[6] Kaspersky M, Niemann H-J. The LRC (load-response correlation) method. A general method to estimate unfavourable wind load distribution for linear and nonlinear structural behaviour. J Wind Eng Ind Aerodyn 1992,41-44:1753-1763. [7] Macdonald PA, Holmes JD, Kwok KCS. Wind loads on circular storage bins, silos and tanks. III. Fluctuating and peak pressure distributions. J Wind Eng Ind Aerodyn 1990;34:319-337. [8] Holmes JD. Analysis and synthesis of pressure fluctuations on bluff bodies using eigenvectors, J Wind Eng Ind Aerodyn 1990;33:219-230.

591

Optimization of helicopter subfloor components under crashworthiness requirements C. Bisagni * Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20158 Milan, Italy

Abstract This paper deals with the optimization of hehcopter subfloor components under crashworthiness requirements. The optimization procedure here proposed is based on the use of a neural network, that, after a proper training performed using an explicit finite element code, is able to reproduce the structure behavior. The optimum search is based on genetic algorithms and the objective function is represented by a combination between the specific absorbed energy and the ratio of the maximum force and the mean force during the crash. The obtained optimal configuration allows an increase of the crush force efficiency equal to 18%, together with a mass reduction equal to 8%. Keywords: Optimization; Crashworthiness; Finite element analysis; Neural networks; Helicopter; Subfloor

1. Introduction In the aerospace field, the crashworthiness requirements are imposed from the regulations and this is particularly true for the helicopter structures. Indeed, because of the flight conditions often at a low altitude and in difficult environmental conditions, the crashes are unfortunately frequent, but also potentially survivable in the case of low impact velocities. Nowadays regulations are established for military and civil helicopters, in which the characteristics for a crashworthy structure are defined. In particular, for helicopter crashes, the right design of the subfloor structure is extremely important, because the subfloor represents the structure that has to absorb the great part of energy during the crash [1,2]. Consequently, the subfloor structure has to be designed so as to limit the deceleration forces by structural deformation and to provide a post-crash structural integrity of the cabin floor. While until few years ago this kind of structure was designed just to satisfy the imposed requirements, the idea nowadays is to consider, already during the first phase of the project, the optimization also under crashworthiness requirements. Unfortunately, the definition of a structural optimization methodology under crashworthiness requirements is * Corresponding author. Tel.: -f-39 (02) 2399-8390, Fax: +39 (02) 2399-8334, E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

extremely complex and expensive from the computational point of view. Indeed, in spite of the performance increase in computers, there are a lot of difficulties due to the complexity of the considered dynamic behavior and due to the phenomenon length that is of a couple of milliseconds. For this reason, several examples of structural optimization are applied to aerospace structures, but very few examples of these methodologies are applied to optimal designs under crashworthiness requirements [3].

2. Helicopter subfloor structural components The energy absorption capabilities of a subfloor structure. Fig. 1, depend on the shape of each structural element, on the junctions between each part and on the topology of the whole structure. In particular, the design of the structural intersections among beams and bulkheads contributes essentially to the overall crash response of a subfloor assemblage, because, under vertical crash loads, these riveted intersections behave like 'hard-point' stiff columns, creating high deceleration peak loads at the cabin floor level and causing dangerous inputs to the seat/occupant system. Therefore, the first step for a global optimization of subfloor structures requires the capability to optimize the crash behavior of the intersection elements. The present research concerns the optimization under crashworthiness requirements of a riveted intersection of a

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C. Bisagni /First MIT Conference on Computational Fluid and Solid Mechanics

Keel beam

Fuselage skin Fig. 1. Typical subfloor structure.

typical helicopter subfloor structure. The starting configuration is characterized by a height of 195 mm and consists mainly of two vertical webs and four angular elements, that constitute a closed square section with the diagonal formed by the two webs. At the top and bottom there are two basis, connected by two vertical components, having the same height of the specimens, and by eight small L-shape components. The thickness is equal to 0.81 mm, except for the bases where is equal to 1.27 mm. The component is made of aluminum alloy 2024 T3 and is fastened using typical aeronautical rivets along the height and blind rivets on the basis.

strain characteristics according to the Cowper-Symonds law, while the riveted junctions are modeled by spot-welds and beam elements [5]. The model essentially requires also the definition of several contact surfaces to consider the interaction of the structure different parts. The finite element model is validated by means of experimental results. The crash tests are performed using a drop test machine by the Aerospace Engineering Department of Politecnico di Milano. During the tests, the displacement and the acceleration of the impact mass are recorded using respectively an incremental encoder and an accelerometer. The numerical model is able to well reproduce the loadshortening experimental curve and the real deformation, as shown in Fig. 2.

3. Finite element model The numerical analysis of the helicopter subfloor structural component is performed using the finite element method [4] and in particular the explicit code PAMCRASH. The component is modeled by four-nodes shell elements made of elastic-plastic material with rate-dependent stress-

4. Neural network Any optimization procedure requires carrying out analyses for several different configurations to compute both the structural responses and, when required, their sensitivities

Fig. 2. Comparison between the experimental and numerical deformations.

C. Bisagni /First MIT Conference on Computational Fluid and Solid Mechanics with respect to the design variables. This is the main difficulty in implementing an optimization procedure with a reasonable CPU time and above all with reasonable costs. The solution adopted in this study consists in substituting the expensive finite element analyses with a neural network able to simulate the structure behavior. The basic idea of the neural network consists in developing a structure that reproduces the thinking way of the human brain, so that after a training phase it can reproduce the behavior of the considered physical system. The training phase is based on a series of known inputs and the corresponding, and also known, outputs. The network optimizes the weight coefficients that are used by the different neuronal elements to interact, so to correctly combine known inputs with known outputs. The training is performed using a set of 36 PAMCRASH analyses. The considered inputs are the thickness of the webs, the thickness of the angular elements, the position of the angular elements and the number of the vertical rivets, while the outputs are the maximum force, the mean force and the force-time diagram. During the optimization, the neural network is used to evaluate the structural responses every time that the optimization algorithm requires them without the need to perform a complete finite element analysis. 5. Optimization The optimization method used in the present research is based on genetic algorithms that combine the advantage to find a global optimum to the advantage that they do not require the gradients computation. Nevertheless, these algorithms require performing a high number of analyses for the generation of an enough wide population, but these analyses are performed using the neural network. The objective function is represented by a combination between the specific absorbed energy and the ratio of the maximum force and the mean force obtained during the crash. The constraints consist of the maximum and mean force values and of the technological constraints representing the feasibility of the structural solution. The design variables include the shape and the size of the webs and of the angular components and the number of rivets. The objective function, together to the constraints and the design variables, is reported in Eq. (1). IV FMiax{x) ) Fjnean > F^

V Mass{x) ) \

X = {Web.Thickness, WebJShape, Angular.Thickness, Rivets jiumber) The optimal configuration is verified using the finite element code PAMCRASH. The load-time curves of the

593

.xlO •I

1 A

PAMCRASH analysis

4.5

/

4 3.5 Force [N|

3

Neuraf networks simulation

\1 l\

f

\^ \ \

/

2.5

/

2

/

^

X

\

" "^

/

^'^^ 1

1.5 1 0.5

^ 0.002

O.CK)4

0,0C6 Time [s)

0.008

0.01

0.012

Fig. 3. Load-time curves of the optimal configuration, obtained from neural network and from PAMCRASH. Table 1 Geometry and outputs of the optimal configuration compared to the values of the initial configuration

Web thickness (mm) Web shape (mm) Angular thickness (mm) Rivets number Maximum force (kN) Mean force (kN) Crush force efficiency

Initial configuration

Optimal configuration

0.81 40 0.81 10 51.45 24.29 0.472

0.72 37 0.82 8 46.29 25.79 0.557

final configuration, obtained by the optimization procedure and by PAMCRASH code, are both reported in Fig. 3. The difference on the maximum forces is equal to 7.5%, while that one on the mean forces is equal to 1%. The geometry and the outputs of the optimal configuration are reported in Table 1, where they are compared to the values of the initial one. The optimized structural component allows an increase of the crush force efficiency equal to 18%, together to a mass reduction equal to 8%. Considering that the webs, which thickness passes from 0.81 mm to 0.72 mm, are long the entire subfloor structure and that the studied component is present in a large number, the weight reduction of the whole subfloor structure would be even more considerable. The optimized configuration allows also a reduction of the maximum force equal to 10% and an increase of the mean force equal to 6%.

6. Conclusions The optimization procedure described in the paper appears as a reliable approach for the design of structural

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C Bisagni / First MIT Conference on Computational Fluid and Solid Mechanics

components under crashworthiness requirements. In particular, the use of neural networks seems a promising means to overcome the main difficulty typical of these optimization problems, represented by the high number of complete finite element simulations necessary to reach a suitable optimum. The next step of the research will be a global optimization of the subfloor structure, having as objective the optimal topology and size of the structural elements towards the crash energy absorption. Acknowledgements The author wishes to express her gratitude to Dr. Luca Lanzi for his contribution to the development of the optimization procedure.

References [1] Giavotto V, Caprile C, Sala G. The design of helicopter crashworthiness. AGARD, 66th Meeting of the structures and material panel, Energy absorption of aircraft structures as an aspect of crashworthiness 1988, pp. 6.1-6.9. [2] Kindervater CM, Kohlgruber D, Johnson A. Composite vehicle structural crashworthiness: a status of design methodology and numerical simulation techniques. Proceedings of Intemafional Crashworthiness Conference, Dearborn, Michigan, 1998, pp. 444-460. [3] Hajela P, Lee E. Topological optimization of rotorcraft subfloor structures for crashworthiness considerations. Comput Struct 1997;64(2):65-76. [4] Bathe KJ. Finite Element Procedures in Engineering Analysis. Englewood Cliffs, NJ: Prentice-Hall Inc., 1982. [5] Bisagni C. Energy absorption of riveted structures. Int J Crashworthiness 1999;4(2): 199-212.

595

Some results from the Self-Designing Structures research programme John W Bull* Engineering Design Centre, University of Newcastle upon Tyne, Newcastle upon Tyne, NEl 7RU, UK

Abstract An optimisation technique whereby where stresses are high, material is added and where stresses are low, material is taken away is described together with a series of example problems. Keywords: Self-designing structures; Optimisation; Stress concentrations; Finite elements

1. Introduction

3. Reverse Adaptivity (RA)

Fig. 1 shows the programme of work. The four significant ideas evolved during the project are described below.

The conventional concept of h adaptivity is that where stress gradients or other error indicators are high, the finite element mesh should be sub-divided into smaller elements to reduce errors on an element-by-element basis. The other way of carrying out adaptivity is p adaptivity, where extra polynomials are added to the element shape functions. In both cases the local accuracy of the solution is increased. Reverse adaptivity draws on the concepts of adaptivity but is a reversal of the original concept. In the optimization methods used in Self-Designing Structures, material that has low stress is removed. This has traditionally been done on an element-by-element basis and can lead to irregular edges. In reverse adaptivity where the stresses are low, the material is removed and the elements are sub-divided. Thus the removal of elements can more closely follow the stress contours. In this way the mesh is made finer in zones where the stress is low. This is the opposite of the conventional technique and enables the structure to become increasingly well defined in terms of its geometry [2,3].

2. Interactive Design Refinement (IDR) The first concept, developed, was Interactive Design Refinement [1]. It was a very simple, flexible idea. The designer starts with a domain of material, larger than that of the proposed component. A finite element mesh is developed, covering this domain, and then the boundary conditions and loads are applied. The structure is analysed and a graphical plot of the resulting stresses is produced. The designer then re-shapes the structure, using a mouse and clicking and pointing to change the structural shape to alter unacceptable stresses. The mesh generator then meshes the evolved structure. This process is continued until the designer has achieved the necessary structural improvements. Although this method is simple, it is very powerful allowing the building in of engineering intuition and feel. Material does not become disconnected and the boundary of the structure can be kept smooth, incorporating any production constraints.

*Tel.: +44 (191) 222-7924; Fax: +44 (191) 222-6059; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

4. Evolutionary Material Translation (EMT) The technique to add material is called Evolutionary Material Translation (EMT). Almost invariably, in practical engineering problems, the maximum stresses occur on the boundaries of the material. The EMT program searches for the higher stresses. Where these are above some limiting value, the boundary is moved outwards, perpendicular to

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J.W. Bull/First MIT Conference on Computational Fluid and Solid Mechanics 1996 July

1997 January

1997 July

1998 July

1999 January

1999 July

2000 January

[1] Development [2] Package Selection [3] Interfacing [4] Extension of Adaptivity [5] Case Study Selection [6] Component Redesign [7] 2D Development [8] 3D Development [9] Trajectory Development [10] 3D Redesign

Fig. 1. The Self-Designing Structures research programme. the original edge and new elements are constructed in the new zone of material. This movement of the boundary allows the stresses to be reduced in this area. The EMT method was later combined with reverse adaptivity to give a technique, which enables material in the structure to be translated from one point to another [4].

among numerical studies is a pin-jointed framework. This solution was achieved by all four methods. 6.3. Cantilever from a circle This is a point load at some distance from a fixed circle. The solution, a set of intersecting spirals was solved successfully by RA.

5. Approximated Contour Evolution (ACE) 6.4. Plate under uniform load In Approximated Contour Evolution, the structure is re-shaped along the stress contours within each element. RA removes entire elements, but ACE removes part of an element, so that the boundary of the structure corresponds to a given stress contour. ACE gives smooth shapes to the new structure and enables the retention of a moderately refined mesh without too many elements being generated. This is the most effective technique that was developed during the course of the research.

A simply supported plate was subjected to a uniform transverse load, and the optimum shape developed using RA. The resulting optimum was somewhat counter intuitive, consisted of comers of the plate and its centre remaining intact, while thin ligaments joined the two together. The research team was suspicious of this 'optimum' design, but similar results had been reported independently. 6.5. Optimal reinforcement of a circular hole in a plate

6. Theoretical problems considered 6.1. Michell strut and tie problem This classical problem has a point load, supported at some distance from a vertical wall. The theoretical solution, is a strut and tie, both at 45° to the wall. This solution was achieved by all four methods.

One topic of interest to the industrial partners, was the optimal layout of thickness around a hole so that the maximum stress in the plate did not increase. The so-called neutral hole. The result was a family of theoretical thickness distributions for a plate with a circular hole under equal bi-axial stress [5].

7. Industrial problems considered 6.2. Shallow cantilever 7.1. Extrusion problem A point load is supported as in the previous case, but the vertical extent of the structure is constrained. The theoretical optimal structure is not known, but the consensus

This problem was to optimise the shape of an aluminium extrusion containing two chambers, which are alternately

J.W. Bull/First MIT Conference on Computational Fluid and Solid Mechanics subjected to internal pressures. The problem was optimised using both the IDR and the RA algorithms. The best solution saved about \ of the total alumium, and $160,000 per annum for an industrial company.

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version of the EMT method. Although significant material reduction was obtained and the maximum stress did not increase, the resulting design was difficult to manufacture. A follow-on research project might well explicitly include production constraints.

7.2. Replenishment at sea support device 7.8. Welding This problem involved the design of a structure to fit on the deck of a vessel, to take the loading from a cable slung to another vessel. The structure was optimised using both the IDR and the RA algorithms, in both two and three dimensions. Various proposed shapes were generated, all based on beams.

The research considered the best shape for the weld between a cover plate and a base plate. Three optimal weld shapes were found depending upon the constraints [6].

8. Conclusions 7.3. Transition from square to circle at top of crane tower This problem concerned the selection of the best transition structure between a square column, and the luffing circle for a crane. The optimisation was attempted with IDR, RA and EMT. The best structural solution was a truss between the two ends of the model. 7.4. Bulkhead stiffening with penetration This problem involved a circular bulkhead in a submarine, subjected to uniform pressure on the circumferential exterior. The bulkhead had a circular penetration, of fixed specified size. The problem was to optimise the layout of stiffeners, on the plated bulkhead. The final design was based, perhaps surprisingly, on hexagons. Two solutions were attempted, one of which any pattern of stiffeners was allowed to evolve and another in which only one hole was allowed to develop. The latter turned out to be symmetrical 7.5. End of bilge keel transition This problem related to the best way to terminate a bilge keel running along the ship's hull to minimise the stress concentration. The three-dimensional RA method indicated that the existing solution could not be improved to any significant extent. 7.6. Bulkcarrier web frame layout This problem involved the best way of terminating the vertical side shell stiffeners, to minimise stress concentrations at the intersection of the stiffener and the bilges. The RA program suggested some small alterations to the layout of material at the intersection, but no great changes. 7.7. Tool box The question was whether a toolbox, which is also designed to be stood on, could be re-designed to reduce the volume of plastic used. This was attempted using the 3D

The research was most successful from the academic point of view. It trained six researchers in leading edge optimisation concepts. It led to the pubUcation of substantially more journal and conference papers than listed below. The research was also very successful in regard of the collaboration between industry and academia with genuine industrial problems being solved. It also involved a transfer of technology in that the latest developments in optimisation algorithms were transmitted from the research to the industrial sponsors. The only slightly disappointing feature of the project was the lack of transfer of the algorithms into a commercial finite element program. However the Self-Designing Structures research work will continue with commercializing the software being an over-riding priority.

Acknowledgements This paper is based on the Final Report to EPSRC, Self-Designing Structures (GR/79789) written by Prof. P Bettess and Dr John W Bull. The financial support of the Engineering, Physical and Science Research Council (EPSRC), Clarke-Chapman, (Rolls-Royce Materials Handling Group), dominick hunter, Rolls Royce Derby, Lloyd's Register, Kockums Computer Systems (KCS), the Ministry of Defence Procurement Executive, Black and Decker and Defence and Evaluation Research Agency (DERA) is gratefully acknowledged.

References [1] Christie C, Bettess P, Bull JW. Self-designing structures: a practical approach. Eng Comput 1988;15(l):35-48. [2] Reynolds D, McConnachie J, Bettess P, Christie WC, Bull JW. Reverse adaptivity — a new evolutionary tool for structural optimization. Int J Num Methods Eng 1999;45:529-552. [3] Neau E, Bettess P. Evolutionary material translation: an automatic tool for the design of low weight, low stress structures in 3D. Eng Comput, in press.

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[4] Reynolds D, Christie WC, Bettess P, McConnachie J, Bull JW. Evolutionary material translation: a tool for the automatic design of low weight, low stress structures. Int J Numer Methods Eng 2000. [5] Neau E, Bettess P. An axi-symetric reinforcement of a cir-

cular hole in a uniformly end loaded plate. Proc of IDMME 98, vol. 1, 1998, pp. 239-246. [6] Bull JW, Lim KH. The optimising of weldments using the self designing structures approach and the ANSYS optimisation module. Submitted to Comput Struct, 1999.

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On the use of 'meta-models' to account for multidisciplinarity and uncertainty in design analysis and optimization S. Butkewitsch * Federal University of Uberlandia, School of Mechanical Engineering, Uberlandia MG, 34800-089 Brazil

Abstract This paper presents and discusses techniques capable of broadening the scope of numerical analysis tools in the environment of integrated product development. The proposed approach consists of processing data generated by planned batches of numerical analysis in order to build symbolic models which are able to represent the physical behavior of a system. Such 'meta-models' (response surfaces, Bayesian models and neural networks, for instance) can then be used to integrate different analysis disciplines and techniques (automatic optimization, robust design, 'DFX' and others), besides accounting for uncertain/random operating conditions that span the overall product life-cycle. Based on the features and application possibilities of 'meta-modelling' techniques, an enhanced analysis software architecture is proposed. Keywords: Meta-modelling; Numerical analysis; Design optimization; Robust design; DFX

1. Introduction Based on the perspective that the 'art of design' can be represented by a systematic, scientific decision process, one can imagine engineering practice as the block diagram shown in Fig. 1. Numerical analysis tools, mainly those based on the Finite Element Method, have been responsible for a significant increase in the productivity of design activities. The addition of numerical optimizers, as in the context proposed by Schmidt [1] for automatic design synthesis, has further enhanced the efficacy of the scheme presented above. In the pursuit of more realistic and efficient designs, however, some additional steps can be considered when dealing with a design task. They are intended to address certain aspects that greatly affect the behavior of a product throughout its life cycle, and should be considered as fully integrated procedures pertaining to the design process: (1) Optimization formulations (Alexandrov et al. [2]) that consider multiphysics (multidisciplinary operating environments); (2) Consideration of variability in operating conditions. *Tel.: +55 (34) 239-4282; Fax: +55 (34) 239-4149; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

taking into account design factors that strongly influence the system behavior but cannot be deterministically controlled by the engineer (Dean et al. [3]); (3) Possibility to assure design reliability in the context stated by the ISO 8402 standard: Reliability is the projection of Quality in the time domain (ISO [4]); (4) Harmonization of conflicting requirements regarding assemblability, maintainability, usability, etc. The simultaneous fulfillment of requirements ' 1 ' (Multidisciplinary Design Optimization — MDO), '2' (Robust Design), '3' (ReUability), and '4' ('DFX'), although very beneficial to the overall quality and excellence of technical systems (Schulyak [5]), is not readily achieved by conventional means. The use of 'meta-modelling' techniques, based on data generated by numerical analysis technologies, can be a valuable resource on providing efficient design solutions from all these viewpoints.

2. Overview of 'meta-modelling' techniques As far as data generated by numerical models (i.e., FEM, BEM, etc.) are processed and used to create symbolic models of a physical reality, 'meta-modelling' techniques are being applied. Generally, 'meta-modelling' techniques

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COMPILATION

DATA

1

KNOWLEDGE (THEORY^

INFORMATION ORGANIZATION

OPTIMIZATION

IMPROVEMENT

PREDICTIVE CAPABILITY

SIMULATION

Fig. 1. Decision process flowchart for engineering design. whose overall quality and usefulness have to be evaluated by adequate sets of metrics. Each combination of design space sampling, model choice and fitting procedure leads to the use of specific verification procedures. A general overview of combination possibilities spanning the four major steps of 'meta-modelling' is presented in Fig. 2 (adapted from Simpson et al. [6]). Specific step combinations give rise to some 'meta-modelling' techniques which are very popular in a variety of applications. A few examples appear in Fig. 2 between the third (Model Fitting) and fourth (Quality Verification) columns. Response Surface Methods (RSM) are globalanalytical 'meta-models'. This means they are intended to represent physical relationships found in a design space by means of an unique equation whose coefficients have to

are developed in four steps: (1) Experimental design: a design space, including a range of design possibilities, is sampled in order to reveal its contents and tendencies; (2) Choice of a model: the nature of the 'meta-model' itself is determined, tacking into account that the relations contained in the data gathered in the previous step have to be symbolically represented, with the highest possible accuracy; (3) Model fitting: the model whose shape is defined in ' 2 ' is fitted to the data collected in T . Differences in fitting schemes may affect the efficacy of 'meta-modelling' techniques in the solution of a given problem; (4) Verification of model accuracy: the three precedent steps are sufficient to build a first tentative model,

EXPERIMENTAL DESIGN

MODEL CHOICE

(Fractional) Factorial

Polinomial

Central Composite

Splines

BoxBehnken Alphabet Optimal

Y N Frequency H Domain

I

Kernel Smoothing

Orthogonal PlackettBurman

Hybrid Latin Hypercube Enumerative Random

^

Least _ Squares Regression

Neural Networks

LogLikelihood

Rulebase or ^ Decision Tree

Bayesian Models

BackPropagation ^ Entropy (information' /theoretic)

•^Analysis of Residuals ANOVA

Best Linear Unbiased Predictor (BLUP) Best Linear Predictor (BLP)

v

Response Surfaces

Weighted Least Squares Regression

Radial Basis Functions

Hexagon

QUALITY VERIFICATION

MODEL FITTING

Neural _ Networks Inductive Learning

Fig. 2. 'Meta-modelling' techniques.

-^ Residual Error

S. Butkewitsch /First MIT Conference on Computational Fluid and Solid Mechanics be estimated through statistical techniques. The analytical form is a considerable advantage of RSM over other types of 'meta-models' in terms of physical insight and ease of use, but its global nature can be a handicap in the case of highly non-linear design spaces. If one searches for more symbolic/abstract 'meta-models', Bayesian or 'krigging' are of a kind that no longer offer analytical representation of the functional relationships pertaining to the design space. When compared to RSM, they are more difficult to implement and costly to run, but can cope better with non-linear design spaces due to their inherent structure intended to model local behaviors along design spaces. Increasing the level of abstraction, highly symbolic, heuristic models such as neural networks operate with transformation matrices that lead to the estimate of an output, given the corresponding input. Neural networks, in particular, exhibit a high degree of robustness (Rao et al. [7]) with respect to eventual noise collected during the 'Experimental Design' phase. From the brief comparison outlined in the latter paragraphs, it can be stated that each of the different 'meta-models' have its own advantages and drawbacks, and the choice for one of them will depend upon the particular problem to be solved and the resources available for the solution. On the other hand, all 'meta-modelling' techniques, regardless of abstraction level, offer two distinguished positive characteristics: • Low computational cost: if the 'meta-model' is a response surface, a low order polynomial equation has to be solved for a set of inputs. For the case of neural networks, a matrix multiplication operation has to be performed. Once they are constructed, 'meta-models' become more and more inexpensive to use in long term basis; • Superior numerical conditioning: this is a key characteristic in many fields of engineering. For example, if one intends to optimize a structure subject to crash loadings, it is virtually impossible to directly couple a numerical optimizer with a finite element solver due to the highly non-linear nature of the analysis. Instead, a response surface based on analysis results can be easily optimized (Yang et al. [8]). With the availability of low cost and well conditioned predictive tools, sophisticated design approaches can be adopted (that is, requirements ' 1 ' , '2', ' 3 ' and '4' mentioned above in this section can be considered as native parts of a broader design process): • If a certain design space is sampled by different types of analysis codes (fluid and structure, for instance), the resulting 'meta-models' can be combined and the problem solved through a multiphysics approach. For the example considered, optimization can be performed setting a structure feature as design objective and a fluid feature as design constraint, or vice-versa;

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• The experimental design procedure of 'meta-modelling' techniques can be used to sample the design spaces with combinations of both control and noise factors (Dean et al. [3]). An optimization procedure can then be used to robust design the control factors such that the system behavior is not significantly affected by to noise factors; • Extensive reliability studies can be performed by means of Monte Carlo simulations. This powerful technique usually requires a very large amount of simulations, which could be carried out at lower costs by using inexpensive 'meta-models'; • Since 'meta-models' are symbolic abstractions of physical systems, they can be designed to accommodate qualitative information, such as manufacturability, maintainability and usability metrics, enabling the use of DFX and KBE (Knowledge Based Engineering) techniques integrated to the design process.

3. Conclusions, perspectives and future research suggestions: an integrated numerical solution package based on 'meta-modelling' techniques A survey conducted on recent research related publications reveals an increase in the use of 'meta-modelling' techniques as an important approach to solve complex engineering problems (Simpson et al. [6]). Different methods involved in 'meta-modelling', however, tend to be employed in schemes with low levels of technological integration. This section suggests an architecture for future generations of engineering software, whose capabilities of dealing with multiphysics and uncertainty in engineering problems is standard, due to the use of 'meta-modelling' techniques as an intermediary step between numerical analysis and the integrated solution. Analysis results are no longer seen as 'final results'. They are data supplied for the realization of a more realistic, highly automated engineering design procedure. As in Fig. 3, data from separate analysis disciplines are transformed into an abstract dimension where they can receive unified treatment. Besides, aspects that are difficult or even impossible to be represented by numerical forms can be added to the symbolic representation of the problem. Examples are usability, reliability, maintainability, manufacturability and robustness requirements shown in Fig. 3.

References [1] Schmidt LA. Structural design by systematic synthesis. Proc 2nd Conference on Electronic Computation. New York: American Society of Civil Engineers, pp. 105-132, 1960.

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UNCERTAINTY (RANDOM OPERATING CONDITIONS)

DISCIPLINE

DISCIPLINE

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2

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"META-MODELLING"

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DISCIPLINE

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1

2

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• • •

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Fig. 3. Integrated development architecture based on 'meta-modelling' techniques.

[2] Alexandrov NM, Kodiyaiam S. Initial results of an MDO method evaluation study. AIAA-98-4884, 1998. [3] Dean E, Unal R, Taguchi. Approach to design optimization for quality and cost: an overview. In: Proc Annual Conference of International Society of Parametric Analysis, 1991. [4] ISO — International Organization for Standardization. Standard 8402, The Vocabulary of Quality. [5] Schulyak L. Introduction to TRIZ, http://www.triz.org/triz. htm.

[6] Simpson TW, Peplinsky JD, Koch PN, Allen JK. On the use of statistics in design and the implications for deterministic computer experiments. In: Proc of the ASME Design Engineering Technical Conferences, Sacramento, CA, 1997. [7] Rao VB, Rao HV. Neural Networks and Fuzzy Logic. New York: M and T Books, 1995. [8] Yang RJ, Tho, CH, Wu CC, Johnson D, Cheng J. A numerical study of crash optimization. In: Proc of the ASME Design Engineering Technical Conferences, Las Vegas, NE, 1999.

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Design of cams using a general purpose mechanism analysis program Alberto Cardona * Centro Internacional de Metodos Computacionales en Ingenieria (CIMEC-INTEC), CONICET-Universidad Nacional del Litoral, Giiemes 3450, 3000, Santa Fe, Argentina

Abstract We present a methodology to design cams for motor engine valve trains. We combine a multipurpose software for mechanism analysis with techniques of optimization trying to synthesize the cam profile that better suits the requirements. Maximum valve lift and timings are treated as optimization constraints, and the objective function to maximize is the time integral of the opened valve area to gas flow. The aspect of return spring dynamics is addressed in the paper. The spring model takes into account coil clash and spring surge, with distributed mass. Friction between inner and outer springs in dual assemblies is also modeled. Several proposed motion laws are analyzed and compared, both from the point of view of traditional approaches and with nonlinear dynamics simulation in a mechanism analysis program. Once an optimal lift profile is determined, the cam shape is computed by inverse kinematics analysis, taking into account all the geometric nonlinearities introduced by the kinematics chain. Finally, the whole mechanism is dynamically verified to check satisfaction of the design criteria. Keywords: Cam; Nonlinear dynamics; Synthesis of mechanism; Valve spring dynamics

1. Introduction Several factors should be considered in the design of motor engine valve trains and cams, which may be broadly classified into fluid dynamics and structural ones. The maximum valve lift and the valve timings are determined based on fluid dynamics considerations. Structural considerations are taken into account to satisfy these two factors while keeping integrity of the mechanism and optimize functioning. To this aim, efforts should be minimized to work within the allowable stress levels, and jumping between cam and follower should be avoided. At the same time, the gas flow through each valve should be maximized. Further complexity appears because of nonlinearities introduced by the kinematics chain usually interposed between cam and valve. Last, but not least, the feasible solution space is restricted to avoid mechanical interferences.

*TeL: +54 (342) 455-9175; Fax: +54 (342) 455-0944; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The spring dynamics plays a main role in the system behavior. In high speed applications, high harmonics in the spring are excited, causing jump between cam and follower and collision between coils. Several methods for cam motion synthesis have been proposed in the literature. Most of them analyzed a characteristic single degree of freedom equation [1,2]. Many authors reported that multiple degrees of freedom models [1,3,4], and even distributed models [5-7], of the return spring are needed. The Polydyne method was proposed to get tuned cam profiles for a given regime. However, away from the design speed, vibrations may be much greater than those coming from the non-corrected profile and special methods were developed to improve robustness [2]. Many authors pointed out that for high speed applications, the residual vibration spectrum does not necessarily give a indication of spring behavior, and proposed alternatives [3]. When including nonlinear phenomena as coil clash and friction, nonlinear dynamics simulation seems to be the only alternative to verify correct functioning. In a previous work, we presented a method to analyze the kinematics and dynamics of mechanisms with cams [8],

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using splines for describing cam geometry. In this work, we develop methods for obtaining optimal valve motion combined with the mechanism analysis program, allowing in this way a cam profile to be synthesized that takes into account the geometrical nonlinearities introduced by the kinematics chain.

2. Design factors

ment profiles in the form: u{6) =

f{oL,e)

(1)

where 0 is the crank angle and a a set of control parameters whose meaning depends on the specific choice. The design objective is to maximize the area below the lift curve, in order to maximize the net flow income. Let us define: Ova

Fluid dynamics factors usually impose both the maximum valve lift L and the valve timings Oyo (valve opening crank angle) and Oyc (valve closing crank angle). Practical experience recommends the maximum value of valve lift to be bounded at most at 40% of the valve diameter. Here, we assumed them as input data. The intake valve opening and exhaust valve closing are carried out in the proximity of the piston top dead centre. Since the distances between valves and piston and also between both valves are very small, it is necessary to detect eventual geometrical interferences and avoid them during the design of valve motion. This factor is very critical, especially in engines with large valves overlap. In order to reach the maximum valve lift L in the time interval where the valve remains open, we need to specify a motion profile that satisfies not only the interference constraints, but also the following dynamics restrictions [9]: (1) no jumping between cam and follower; (2) no impact in the valve seating; and (3) maximum stresses bounded for reliability and minimal wear. Spring dynamics greatly influence the system dynamic behavior. Dual spring assemblies are used to increase the spring eigenfrequencies and introduce damping by friction between inner and outer springs. Another solution to increase stiffness with minimum coil bind is to use wires with oval cross section. Some manufacturers are proposing springs in titanium for high speed applications. Varying pitch springs were also proposed to minimize resonance.

3. Constrained optimization strategy We may distinguish five zones in poppet motor valve motion: initial ramp, acceleration, spring-controlled, deceleration, and final ramp. The maximum values of positive acceleration are limited by the maximum efforts the system can sustain. On the other hand, during the spring-controlled zone, the negative acceleration imposed by the cam profile should be smaller than a given limit so that the inertia load is a fraction of the available spring force and jumping is avoided. Several proposals of valve motion are presented in the Uterature (i.e. cycloidal motion, polynomials [3], BersteinBezier harmonics [2], trigonometric splines [10]). We parameterized valve motion u{6) for several smooth displace-

A=

j u (6>)d6>

(2)

Ovo

integral in (2) is evaluated analytically for each motion profile. The definition of the optimization problem is completed with the set of constraints: (1) No interference between valve and piston. (2) No interference between valves. (3) Positive valve displacement. The objective function and restrictions are scaled so that the optimization problem is well defined. To this end, reference values of area, displacement and angular increments are defined. An optimization problem is then defined, whose solution a*

=argmaxA(a*)

(3)

was computed using standard routines for constrained optimization.

4. Cams and valves train design and analysis Fig. 1 shows schematic views of the mechanical system analyzed and a detail of the intake subsystem. The mechanism models were made with program Mecano [8,11,12] manual. The cam axis is centered at point O and is in contact with the rocker-arm roller centered at B. The follower is fixed in the pivot rocker-arm A. Note that there is a small cam at the end of the rocker-arm which is in contact with a thrust piece at the top of the valve stem. Fig. 2 plots the computed displacements for the intake and exhaust valves. We also display the piston displacement, and we can see that there is no interference between piston and valves. Using the valve displacement profile relative to the crank angle (obtained from the optimization stage) as input data, we got the cam profile necessary to produce the desired valve motion. A kinematics analysis was made by imposing adequately synchronized motion of valves and shafts, in a mechanism model without cams. As a result of the analysis, we calculated the distance between cam and roller centers, as a function of the cam's angular displacement. In order to account for the rocking motion of the rocker-arm, we also computed the angular relative position of the roller center with respect to the cam's center, in terms of the cam's

A. Cardona /First MIT Conference on Computational Fluid and Solid Mechanics

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goal of this computation is to check the operation condition of the whole mechanical system, specially to verify there is no separation between cam and follower, and efforts in each member remain lower than acceptable values.

5. Conclusions

spring

An optimization strategy for cam design is presented. The software obtained allows to compute cam profiles accounting for all dynamic effects present in a mechanism, including all geometrical nonlinearities of the kinematics chain. Once the cam profile is synthesized considering different motion constraints as avoidance of interferences and limitations on maximal and minimal acceleration, results are validated by a dynamic analysis of the full mechanism.

References

Fig. 1. Overview of the mechanism model. Left: global view. Right: detail of the intake subsystem.

0

45

90 135 180 225 270 315 360 405 450 495 540 585 630 675 720

Crankshaft angle [deg]

Fig. 2. Intake valve displacement (solid line) and exhaust valve displacement (dashed Une). Points show the piston displacement.

angular displacement. From these two measures, a simple geometrical analysis gave us the cam profiles. Finally, a dynamic analysis is performed, assembling the two cams in the whole mechanism shown in Fig. 1. The

[1] Wiederrich JL, Roth B. Dynamic synthesis of cams using finite trigonometric series. ASME J Eng Ind 1975;287-293. [2] Srinivasan LN, Jeffrey Q. Designing dynamically compensated and robust cam profiles with Bernstein-Bezier harmonic curves. J Mech Des 1998;120:40-45. [3] Yu Q, Lee HP, Influence of cam motions on the dynamic behavior of return springs. J Mech Des 1998;120:305-310. [4] Bagci C, Kumool S. Exact response analysis and dynamic design of cam-follower systems sing Laplace transforms. J Mech Des 1997;119:359-369. [5] Tiimer ST, Unliisoy YS. Nondimensional analysis of jump phenomenon in force-closed cam mechanisms. Mech Mach Theory 1991;26:421-432. [6] Unlusoy YS, Tiimer ST. Analytical dynamic response of elastic cam-follower systems with distributed parameter return spring. ASME J Mech Des 1993;115:612-620. [7] Lin Y, Pisano AP, General dynamic equations of helical springs with static solution and experimental verification. ASME J Appl Mech 1987;54:910-917. [8] Cardona A, Geradin, M. Kinematic and dynamic analysis of mechanisms with cams. Comput Methods Appl Mech Eng 1993;103:115-134. [9] Taylor C. The Internal Combustion Engine in Theory and Practice. Cambridge, MA: MIT Press, 1984. [10] Neamtu M, Pottmann H, Schumaker LL, Designing NURBS cam profiles using trigonometric splines. J Mech Des 1998;120:175-180. [11] Cardona A, Geradin M, Doan DB. Rigid and flexible joint modelling in multibody dynamics using finite elements. Comput Methods Appl Mech Eng 1991;89:395-418. [12] Samtech. Samcef-Mecano, user manual, 1996.

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On singular topologies and related optimization algorithm Gengdong Cheng *, Xu Guo State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, People's Republic of China

Abstract The present paper addresses a number of basis aspects and optimization algorithms in the area of structural topology optimization. Special emphasis is placed on the singular optimum design and related difficulties. New development of a proper problem formulation of truss structural topology optimization under buckling constraints is reported. Size effect on optimum topology is shown by numerical examples. Keywords: Topology optimization; Singular optimum; Buckling constraints

1. Background Singular optimum was first discovered by Sved and Ginos [1] (1968) when they applied the ground structure approach to truss topology optimization. Later it was understood that a wide range of topology optimization problems involve singular optimum, which causes numerical difficulties in the search process of mathematical programming approach. Indeed, singular optimum was one of the challenging problems in truss structural topology optimization subject to local bar stress and buckling constraints. In our previous studies [2-6] we demonstrated by simple example that for truss topology optimization under bar stress constraints singular optimum is connected to the entire feasible domain and locates at feasible sub-domain of low dimension. The correct picture of feasible domain provided non-trivial understanding of the essential cause of singular topology and hints for development of approach leading to a good algorithm. Further, we pointed out the different features of stress, displacement, and vibration frequency constraints in the context of topology optimization. There are essential differences among topology optimizations subject to different behavior constraints. The general condition of existence of singular optimum was obtained. To obtain the singular optimum topology within ground structure approach we presented a e-relaxation algorithm and its mathematical basis. The algorithm's performance * Corresponding author. Tel.: +86 (411) 4708-769; Fax: +86 (411) 4709-319; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

has been further improved by combination of continuation and relaxation. Numerical examples were given to show the fact that the new algorithm increases the probability of reaching global singular optimum. Because of its simplicity, the algorithm is easily incorporated in exiting structural size optimization algorithm and thus we unify structural topology and size optimization, or we can solve topology optimization by using efficient algorithm for size optimization.

2. Structural topology optimization under buckling constraints Structural topology optimization under buckling constraints is known to optimization community among the most difficult problems. Since allowable bar buckling stress becomes zero when bar area becomes zero, it is shown by simple example that bar local buckling constraint leads to disjoint feasible domain. Possible singular optimum is located at the tip of low dimensional feasible sub-domain, which is separated from the main feasible domain. Besides, phenomena such as bar buckling length jump and isolated bar, which are noticed by Rozvany and others in literature are characteristic for topology optimization under buckling constraints [7,8]. Since then, the problem is thrown into the shade for a quite long period. In the present paper, we present a detailed analysis of a simple four bar truss example and explain the reason why we may have isolated bar in the final optimum structure

G. Cheng, X. Guo/First MIT Conference on Computational Fluid and Solid Mechanics and how to avoid isolated bar by including the global buckling constraint. By the same example we demonstrate the possibility to include overlapping bars in the initial ground structure if structural compatibility is included in structural analysis. One common simplification made in structural topology optimization is to neglect compatibility constraints and linearize the optimization problem. Though it gives beautiful mathematical simplification, it does change the problem and gives true solution only in the case that the final optimum is statically determinate. For topology optimization under buckling constraints, it misleads the way and causes much more severe trouble. By neglecting the compatibility condition one cannot distribute the load between overlapping bars. And thus it rules out the possibility to include overlapping bars in structures and makes the node cancellation impossible. Without node cancellation, bar buckling length jump is not avoidable within the ground structure approach. Based on the above observation the present paper proposes a proper problem formulation of truss structural topology optimization under buckling constraints. By including compatibility conditions in analysis and global structural stability constraint in optimization formulation, isolated compressive bar could be avoided. In order to overcome the difficulty due to separate feasible sub-domains, we follow the idea of £-relaxation approach, which is shown to be effective for problem under stress constraints and propose a newly developed the second order smoothextended technique. By the technique, the gap between two separate feasible sub-domains is seamed and the shape of feasible domain is modified. Seamed irregular feasible domain is regularized by £-relaxation approach. Since the global buckling load is a continuous function of bar cross sectional area as long as the structure does not become mechanism, we do not need any special treatment for the global buckling load constraints. The above-mentioned formulation and tools enables us to explore the approach to deal with jumping buckling length. Numerical results show its effectiveness. A computer program is developed to implement the idea. To be general, we avoid any special treatment. Sensitivity of structural response with respect to bar cross sectional area is done by finite difference. A general mathematical programming technique is picked from optimization software package to perform optimization. 4 bar, 26 bar and 22 bar truss structures are optimized. And the results are compared with data in literature to show the effectiveness of the approach. 3. Size effect on topology optimum In nature, we often see the size effect on material properties. As nano-materials is concerned, size effect of the

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nano structure of material is extremely important. Most of topology optimization studies aim at minimizing structural compliance or maximizing structural stiffness or strength. Results from homogenization approach for effective material properties and inverse homogenization approach for determining material micro-structures show no sign of size effect. Very interesting, it is observed that structural size does affect optimum topology when buckling constraints are in problem formulation.

4. Conclusion To have realistic optimum design, buckling constraints must be taken into consideration. In order to overcome special difficulties caused by including buckling constraints into problem formulation, the present paper suggest a proper formulation and solution approach. A number of numerical examples are given to show the effectiveness of the formulation and the approach. Size effect on topology optimum design is reported.

Acknowledgements This project is supported by the National Natural Science Foundation of China(No.59895410) and National Development Program on Fundamental Researches of China, special grant G1999032805 (973 project)

References [1] Sved G, Ginos Z. Structural optimization under multiple loading. Int J Mech Sci 1968;10:803-805. [2] Cheng GD, Jiang Z. Study on topology optimization with stress constraint. Eng Optim 1992;20:129-148. [3] Cheng GD, Some aspects of truss topology optimization. Struct Optim 1995;10:173-179. [4] Cheng GD, Guo X. A note on jellyfish-like feasible domain in structural topology optimization. Eng Optim 1998;31:124. [5] Cheng GD, Guo X. e-Realxed approach in structural topology optimization. Struct Optim 1997;13(4):258-266. [6] Cheng GD. Some development in structural optimization. Sectional lecture at ICTAM, 19th International Conference of Theoretical and Applied Mechanics, Kyoto, 25-31 August, 1996. [7] Zhou M. Difficulties in truss topology optimization with stress and local buckling constraints. Struct Optim 1996;11(2):134-136. [8] Rozvany GIN. Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Optim 1996;11(3/4):213-217.

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Interactive design and investigation of physical bridges using virtual models Mike Connell ^'*, Odd TuUberg ^'\ Per Kettil ^ ^ Nils-Erik Wiberg ^ ^ Department of Structural Mechanics, Chalmers University of Technology, Goteborg, Sweden ^ AEC AB, Goteborg, Sweden ^ Skanska Teknik AB, Goteborg, Sweden

Abstract Use of interactive simulation and visualisation in the design of large-scale structures — in particular bridge structures is presented. We discuss the use of a software framework to provide Virtual Reality (VR) visualisation of results generated with the Finite Element Method (FEM) as part of an iterative interactive investigatory process. With this system, a user can enter a Virtual Environment (VE) and immediately observe the results of a FEM simulation upon a model as they make modifications to it. In order to achieve a response as close as possible to real-time, we have developed approximation methods that can generate a plausible result for immediate display to the user whilst the FEM process computes the correct model over a longer period of time. Keywords: VR; FEM; OOP; Design; Simulation; Integration

1. Introduction Prototypes of large scale structures such as bridges can not be built and physically tested — we must be sure of the correctness of the design before the structure is built. The use of Finite Element Analysis (FEA) to perform numerical simulation of the structure is a common tool in an iterative design process (i.e., Connell et al. [1]). Fig. 1 illustrates the process as a tetrahedron where the base represents the modelling, simulation and visualisation in an overall adaptive design process, and we place data communication at the top of the tetrahedron controlling data flow in the system under the overall control of the design team. A part of this process is the investigation of the properties of the current design by testing and simulation of the virtual model by numerical methods. For example, in Connell et al. [2], we presented a software system (named iFE) capable of allowing a user to interactively make small modifications to a bridge model, and observe the results of these changes whilst in a VE (Fig. 2). This system was designed to be a modular software framework where indi* Corresponding author. Tel.: -1-46 (31) 772-8572; Fax: +A6 (31) 772-1976; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

vidual parts could be replaced at will. Continuing work has resulting in an improved system being developed (named iFEM, Fig. 3) described in the next section. Brooks [3] describes the improved simulation of the VE as an important technology for VR in general, and that

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ing reusable software components. We have emphasised the stages of changing the FEM model, performing simulation, and the visualisation of result in the VE, but the system can be extended further; for example to allow the modification of the underlying CAD geometry. The design has changed substantially from that presented in Connell et al. [2] (Fig. 2). This change was motivated by a desire to enforce the separation between the framework itself, and the implementation modules that provide functionality. This is now performed by the use of Java Interfaces (Gosling et al. [8]) that a given implementation must provide. Additionally we have switched from traditional socket based communication (Stevens [9]) to a more abstract communication system utilising Java RMI (Remote Method Invocation) (Sun [10]). This allows the necessary movement of data between machines implementing modules of the framework to be hidden behind standard object method calls. Thus our framework appears to operate as a single program, whereas it is generally running on a number of machines with different modules executing on the machine that can provide the best functionality for the specific task.

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Fig. 2. The iFE framework. virtual objects should behave realistically. This is a more generalised case of what we attempt to achieve — the accurate behaviour of a specific object within the VE. The task of achieving a real-time interaction to complex simulation problems is found in almost every field where these simulations occur (for example, surgical training simulation in Rosen et al. [4]). Closer to our work is that of Taylor et al. [5,6] or Liverani et al. [7]. However, these systems work by the close coupling of the customised software components (visualisation, simulation, VE-interaction, etc.), and not within a dynamic reusable software system such as the one we present.

3. Interaction with the model We have been using the possibilities of interactive investigation to gain an intuitive understanding of the reaction of a bridge (Figs. 4 and 5) to multiple heavy loads. We use a collection of point loads to represent a vehicle, and this has a suitable graphic representation in the VE. Movement of this representation by the user triggers a new FEA with the modified load positions, and the results are displayed through modification of the VE using traditional visualisation methods such as surface stress contours or iso-surfaces.

2. Framework The framework shown in Fig. 3 is designed to allow the construction of certain interactive design systems by utilis-

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Fig. 4. The entire bridge structure in context.

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M. Connell et al. /First MIT Conference on Computational Fluid and Solid Mechanics 4. Approximation The framework described has a latency problem inherent in the execution of a complete FE simulation for every user modification. This simulation time is generally in the order of tens of seconds, or greater. In order to provide a faster response time, we provide the user with approximate data up until the point when the final results from the FEM are available. In Connell et al. [2], we used a results cache mechanism in order to record all previously known results. As part of the improvements to the iFEM framework we have generalised the concept of the results cache into a full Approximator module. This software module has the responsibility of determining the similarity between the current input data set (i.e. the input to the FEM module), and any previously computed cases. Once accomplished, the module outputs a suitable intermediate representation for visualisation in the VE. Our current implementation makes use of a representation of the FE mesh and the known nodal results for previous load cases: given a load at point P^, and previous loading positions of Pi, P2, ...Pn (which each provide a complete set of results Ri, R2, ...Rn) we can construct a new set of results by picking individual nodal results from each of the sets Ri, R2, ...Rn based upon calculating the distance of P, to each of Pi, P2, ...Pn, and then choosing and extracting a value from the appropriate result set Ry. This is based upon an assumption of proportionality between the nodal result and the position of the load: the influence of the result generated for a load is directly related to the proximity to that load's position to the current load position. In addition we have made modifications to a general purpose FEA program to offer features to minimise simulation time. For example, computation and LU-factorisation of the stiffness matrix may be unnecessary if we have a previously computed copy and the structure is unchanged with only the loading altered.

5. Conclusions Our work has illustrated several notable points. Firstly, by increasing the structure and formality of our framework

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we have not reduced its flexibihty or speed. Secondly, we have seen that the ability to interact with large and complex modules whilst being shown the results of numerical simulation leads to a better understanding of the model by the user. Finally, the use of approximation methods to provide 'close enough' results to the user whilst a more accurate FE simulation is in progress helps maintain the feeling of immersion necessary for using a VE.

References [1] Connell MA, Kettil P, Tagnfors H, Tullberg O, Wiberg N-E. Integrating modelling, simulation and visualisation in immersive environments — a tool in bridge design. In: Fourth International Colloquium on Computation of Shell and Spatial Structures (lASS-IACM 2000), Chania, Crete, Greece, 4-7 June 2000. [2] Connell MA, Tullberg O. A Framework for the Interactive investigation of finite element simulations within virtual environments. In: Topping BHV (Ed), Proceedings of Engineering Computational Technology, Leuven 2000: Developments in Engineering Computational Technology. Civil-Comp Press, pp. 23-28, 2000. [3] Brooks FP. What's real about virtual reaUty? IEEE Comp Graph Appl 19(6); 1999:16-27. [4] Rosen JM, Soltanian H, Redett RJ, Laub DR. Evolution of virtual reality. IEEE Eng Med Biol 15(2); 1996:16-22. [5] Taylor VE, Stevens R, Canfield T Performance models of interactive immersive visualization for scientific application. International Workshop on High Performance Computing for Computer Graphics and Visualisation, 3-4 July 1995, Swansea, UK. [6] Taylor VE, Chen J, Huang M, Canfield T, Stevens R. Identifying and Reducing Critical Lag in Finite Element Simulations. IEEE Comp Graph Appl 16(4); 1996:67-71. [7] Liverani A, Kuester F, Hamann B. Towards interactive finite element analysis of shell structures in virtual Reality. In: Proceedings of IEEE Information VisuaHsation, pp. 340346, 1999. [8] GosUng J, Joy B, Steele G, Bracha G. The Java Language Specification, Second Edition. New York: Addison-Wesley, 2000. [9] Stevens WR. Advanced Programming in the UNIX Environment. New York: Addison-Wesley, 1992. [10] Java Remote Method Invocation Specification. Sun Microsystems Inc., 1998.

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Design of an inertial safety barrier using explicit finite element simulation Gary R. Consolazio *, Jae H. Chung, Kurtis R. Gurley University of Florida, Department of Civil and Coastal Engineering, P.O. 116580, 345 Weil Hall, Gainesville, FL 32611-6580, USA

Abstract A new segmental concrete barrier system is presented that has been developed for roadside work zone environments. It is shown that by making use of nonlinear dynamic finite element simulation, several cycles of conceptual design refinement can be accomplished using simulation rather than expensive full scale crash testing. Full scale crash tests of the final design are presented and compared to simulation results. Keywords: Vehicle impact; Contact; Friction; Longitudinal barrier; Snagging; Energy dissipation

1. Introduction In the past, development of new roadside safety hardware systems such as railings and barriers typically required iterative cycles of conceptual design and full scale vehicle crash testing. Much of the conceptual design was done based on sound understanding of engineering principles, past development experience, and intuition. However, with only these tools available, several cycles of concept development and expensive crash testing were usually required to arrive at a successful design. Over the past decade, the design process has changed substantially as finite element analysis (FEA) has found increasing use. A substantial portion of design can now be performed computationally with subsequent experimental testing used primarily for simulation validation. In 1998, the Florida Department of Transportation (FDOT) awarded the University of Florida (UF) with a research project to develop a new safety barrier system for work zones. The new barrier is intended for temporary use in separating traffic from roadway construction crews. Desirable design characteristics were: • Portable and modular. System must be composed of easily movable units that can be assembled in the field and modularly replaced. • Low profile. Allow driver unhindered visibility of vehicles crossing perpendicular to traffic flow. • Minimal anchorage. Performance of the system should * Corresponding author. Tel.: -f-1 (352) 846-2220; Fax: +\ (352) 392-3394; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

not be highly dependent on anchorage of the barrier to existing pavement. In order to produce such a design, extensive nonlinear finite element simulation, using LS-DYNA3D, was conducted during the conceptual design phase of this project.

2. Finite element modeling of impacting vehicle For a longitudinal barrier design to be acceptable, it must be capable of smoothly redirecting errant vehicles of various sizes. NCHRP 350 requires crash tests be conducted using both a 2000P vehicle (a 2000 kg pickup) and an 820C vehicle (an 820 kg compact car). The 2000P vehicle delivers more kinetic energy on impact than an 820C and has a higher center of gravity making it more prone to rollover. As a result, primary focus was given to designing the barrier for impact by a 2000P vehicle. A modified version of the reduced resolution C2500 truck (Chevy C2500 pickup) model developed by NCAC [1] was used to simulate the 2000P vehicle. Modifications to the frontal suspension model in the C2500 had previously been made by Consolazio et al. [2] to correct problems arising when studying tire-curb interaction.

3. Inclusion of frictional effects in FEA impact simulation The design goals of smoothly redirecting a high center of gravity vehicle and maintaining a low barrier profile

G.R. Consolazio et al. /First MIT Conference on Computational Fluid and Solid Mechanics are at odds with each other. Low profile barriers, while providing good visibiUty, tend to allow vehicles to override the barrier system. The ability of a barrier to redirect an impacting vehicle depends in large part on the frictional contact forces developed between the vehicle and the barrier [3]. The primary parameters influencing friction forces are as follows. • Geometry of the roadside face of the barrier. • Contact algorithms employed in the finite element simulation. • Coefficients of friction at contact surfaces. • Finite element modeling techniques employed in representing tire, rim, and suspension assemblies. In LS-DYNA3D simulations, separate friction functions may be specified for each contact definition in the model. In the present research, individual contact definitions were specified for rubber to concrete, concrete to roadway, steel to steel, and steel to concrete contact. In general, friction /Xc at a contact surface between two objects moving relative to each other at velocity ^sliding can be represented by /Xc = Of + J

- y (^sliding)

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where a, p, and y are parameters chosen to produce the desired velocity dependence relation. Fig. 1 illustrates the upper and lower bound curves for tire to barrier contact that were used in this research (based on [3]). The authors have noted [3] that in a large number of roadside safety simulations reported in the literature.

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Friction model: //, = /«(v,,,^„J

Friction model: //^ = 0.1 Fig. 2. Curb impact simulation results (left to right) using different friction models (top to bottom).

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G.R. Consolazio et al. /First MIT Conference on Computational Fluid and Solid Mechanics

A of Fig. 1 and one in which a small and constant friction value of fjic = 0.1 is used. Fig. 2 illustrates the resulting vehicle behavior. When a realistic friction model is used, the vehicle overrides the barrier. However, when a low fixed friction value is used, the simulation predicts that the curb redirects the vehicle. Fig. 3 shows the kinetic and internal energies corresponding to these simulations. It is evident that more energy is dissipated in the case where a realistic friction model is used. The kinetic energy drops much faster than the internal energy rises, the difference being primarily attributable to the energy consumed by frictional forces and plastic deformation of vehicle components. Consideration of energy dissipation is important for segment-to-segment connection design.

Beason [4] was given careful consideration as it was reported to have performed well in crash tests. However, the authors of the present study made significant modifications to reduce the weight and improve the performance of the system. Initially the newly developed UF design called for large connection bolts installed through blocked out sections near the centerline of the barrier. However, after building formwork and a reinforcing cage for this initial barrier design, it was decided that there was excessive rebar congestion. A new connection detail in which external brackets were attached to the back faces of the segments was explored. In this new design, the bolt block outs were eliminated and the reinforcing cage was simplified. However, subsequent FEA simulation revealed a serious problem in this design — the bolts spanning from bracket to bracket transferred very little shear force across the joints. As a result, during an impact, barrier segments were able to transversely slide relative to one another (see Fig. 4). FEA simulations indicated serious vehicle snagging on the protruding portions of the downstream segments. Eventually a design evolved in which the connection bolts were still near the back face of the barrier but were embedded in the concrete cross section. A load transfer assembly was designed that takes the load from the connection bolts and transfers it to the concrete. This connection design is capable of carrying the substantial tensile loads in the bolts and is also able to transfer shear from one segment to the next during impact, thus eliminating the snagging problem. In addition, the connection bolts provide sufficient system stiffness during impact that the inertial resistance of the system is sufficient to redirect vehicles without the need for anchorage.

4. Barrier development using FEA impact simulation

5. Design validation by full scale crash testing

Using the friction models of Fig. 1, an iterative barrier design process was undertaken using FEA simulation. The segmental concrete barrier shape reported by Guidry and

Extensive FEA simulations were performed on the final design concept in an attempt to ensure success during full scale crash testing. A set of barrier segments were then

Fig. 3. Kinetic and internal system energies for curb impact simulations using frictional relationships /Xc = /«(fsliding) and lie =0.1.

tffl Fig. 4. Vehicle snagging on intermediate design that permitted excessive relative transverse sliding of segments.

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Fig. 5. Comparison of FEA simulation results with full scale crash testing of a 2000P vehicle. fabricated and full scale crash tests were conducted (by subcontract to E-Tech Testing Services, Inc.) using both a 2000P vehicle and an 820C vehicle. Fig. 5 shows a comparison of a priori FEA simulation results and full scale crash test results. The FEA simulation results shown were not modified or tuned using test results in any way.

6. Conclusions By making extensive use of finite element impact simulation, a new work zone barrier system was successfully developed and tested. Several cycles of design iteration were performed based purely on computational simulation thus substantially reducing the development cost for the system.

References [1] Zaouk AK, Bedewi NE, Kan CD, Marzougui D. Development and evaluation of a C-1500 pick-up truck model for roadside hardware impact simulation. FHWA/NHTSA National Crash Analysis Center, The George Washington University, 1997. [2] Consolazio GR, Chung JH. Vehicle impact simulation for curb and barrier design. Center for Advanced Infrastructure and Transportation (CAIT), Department of Civil and Environment Engineering, Rutgers University, 1998. [3] Consolazio GR, Chung JH. Simulation of vehicle impacts on curbs using explicitfiniteelement analysis. Submitted for publication, 2001. [4] Guidry TR, Beason WL. Development of a low-profile portable concrete barrier. Development and Evaluation of Roadside Safety Features, Transportation Research Record No. 1367, Transportation Research Board, 1992.

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An efficient thermomechanical modeling strategy for progressing cavity pumps and positive displacement motors D. DairAcqua^'*, A.W. Lipsett\ M.G. Faulkner\ T.M.V. Kaiser^ "" Noetic Engineering Inc., P.O. Box 49036, RPO Strathcona, Edmonton, Alberta T6E 6H4, Canada ^Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada

Abstract Progressing cavity pumps (PCPs) and positive displacement motors (PDMs) are used extensively in oilfield production and drilling operations. Both types of machines consist of a helical steel rotor rotating within a mating elastomeric stator. Cyclic loading of the stator elastomer generates heat that causes the temperature of the elastomer to climb v^ell above the environment temperature, sometimes resulting in failure or poor performance of the pump or motor unit. An efficient, pseudo-steady-state thermomechanical modeling strategy for determining the stabilized downhole elastomer temperature and associated structural response in PCP and PDM stators has been developed. Structural, heat generation, and thermal models form the basis for an iterative solution procedure that may be extended to any configuration of PCP or PDM. Preliminary testing using an instrumented stator shows that the modeling strategy provides a reasonable estimate of the stabilized operating temperature. Keywords: Thermomechanics; Elastomer; Hysteresis; Heat generation; Progressing cavity pump; Positive displacement motor; Cyclic loading

1. Introduction Progressing cavity pumps (PCPs) and positive displacement motors (PDMs) are used extensively in downhole oilfield production and drilling operations. Each consists of a rotating steel rotor and a stationary elastomeric stator which mesh helically to form a series of sealed cavities which move axially as the rotor is rotated. The primary goal of the research is to develop a coupled thermomechanical solution strategy for obtaining the temperature distribution in the stator in its stabilized downhole operating mode. The realization of this goal requires structural and steady-state thermal finite element modeling and quantification of the viscoelastic response of the stator elastomer to cyclic structural loads. The problem is somewhat unique because viscous energy dissipated within the elastomer causes significant changes in the geometry of the stator. This behavior and the temperature-sensitive nature of the elastomer properties necessitate the use of an iterative * Corresponding author. Tel.: +1 (780) 437-5919; Fax: +1 (780) 469-1250; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

solution strategy. It is expected that the modeling strategies developed through the study will prove valuable in PCP and PDM design optimization and failure prevention. 2. Solution requirements The following conditions have been met to make the solution strategy a feasible design tool: • The strategy is efficient. Each aspect of the thermomechanical solution strategy is computationally demanding; the use of this method as a design tool is conditional on reasonable solution times to enable parametric analysis studies to be conducted. • The strategy identifies the operating temperature distribution and structural loading state in the downhole operating environment under representative loads. Key design parameters include the peak elastomer temperature, distortional stress and/or strain energy distribution in the elastomer, and the distribution and magnitude of contact stress between the rotor and stator. • The strategy is adaptable to a wide range of pump and motor geometries.

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3. Thermomechanical solution strategy Fig. 1 shows a flow chart that outlines the iterative strategy for obtaining the stabilized temperature and loading distributions in a pump or motor. The iterative loop begins with a structural finite element analysis of the stator through a full rotor operating cycle under representative loads. The structural analysis provides a description of the loading history in the stator elastomer. This information may be used to calculate viscous energy losses within the elastomer using a viscoelastic heat generation model. The resulting viscous heat loss distribution is used in conjunction with a thermal finite element model to determine an average temperature distribution in the elastomer. This temperature distribution is then compared with the assumed temperature distribution as a convergence criterion for the procedure. If the temperature distributions differ, the newest temperature distribution is used as the applied temperature field for the structural analysis in the subsequent iteration. This iterative process continues until temperature distributions from subsequent iterations are within a specified convergence tolerance.

4. Structural modeling Static structural analysis is used to quantify the structural response history in the stator elastomer through the course of one rotor cycle. The ADINA 7.3 finite element package is used for all structural modeling work. Two-dimensional models are employed using a generalized plane strain approach to allow global axial expansion of the stator. Fig. 2 shows a sample effective (von Mises) stress distribution in a cross-section of the stator with the rotor at one position in its rotational cycle. Three-dimensional models have been created to understand the error associated with the planar approximation to the axial helical geometry of the components, but are not employed in the thermomechanical strategy because of the size of the finite element models.

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5. Heat generation modeling A custom routine is used for quantifying the heat generated in the stator elastomer from structural finite element results. Stress and strain histories at each integration point are characterized with Fourier sine and cosine series using

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dynamic elastomer testing procedures. Fourier series stress and strain descriptions also ensure an accurate depiction of the stress/strain hysteresis loop, especially as the loop can be highly non-elliptical at some points in the stator elastomer. Calculation of the area within the hysteresis loop is also simplified as the analytical functions describing the stress and strain histories are easily integrated. The average heat input density per rotational cycle of the rotor is calculated at each integration point and integrated over each element for subsequent use with the thermal finite element routines.

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D. Dall'Acqua et al /First MIT Conference on Computational Fluid and Solid Mechanics 7. Verification testing Preliminary verification testing shows encouraging agreement between theoretical results and the observed temperature distribution in the stator elastomer [3]. Fullscale progressing cavity pumps were run until a stabilized temperature distribution was obtained; this took on the order of 1 h for each loading scenario. Temperature measurement was achieved using embedded thermocouples at various locations in the stator elastomer. Corresponding results were generated using the iterative thermomechanical solution strategy described above. Fig. 4 shows a comparison of the predicted and observed peak temperature increases in the elastomer. While some discrepancy is apparent between modeling and testing results, much of this may be attributed to the uncertainty in controlling factors such as the component geometries and structural and thermal properties of the stator elastomer.

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thermal modeling components are required to make the strategy feasible as a design tool. The stabilized temperature distribution in the stator elastomer is governed by a number of factors. These include the initial interference between the rotor and stator, the thermal expansion coefficient and thermal conductivity of the elastomer, variations in dynamic elastomer properties with temperature, and applied loading and boundary conditions.

Acknowledgements The authors would like to acknowledge Weatherford Artificial Lift Systems for its support and cooperation through the course of this project.

References 8. Conclusions The investigation has shown that an iterative thermomechanical solution strategy is required in order to accurately depict the loading state in the stator elastomer in its downhole operating environment. Temperatures within the elastomer have been observed to exceed the environment temperature by more than 50''C in single-lobe progressing cavity pumps. Efficient structural, heat generation, and

[1] Bathe KJ, Finite Element Procedures. Upper Saddle River, NJ: Prentice Hall, 1996. [2] McAllen J, Cuitino AM, Sernas V. Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires, Part II — Thermal Modeling. Finite Elem Anal Des 1996;23(2-4):265-290. [3] Dall'Acqua, D. Thermo-mechanical Modelling of Progressing Cavity Pumps and Positive Displacement Motors, M.Sc. Thesis, Department of Mechanical Engineering, University of Alberta, 2000.

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Using Pro/MECHANICA for non-linear problems in engineering design L.E. Doxsee Jr. * PTC, 2590 N. First St, Suite 200, San Jose, CA 95131, USA

Abstract Non-linear analysis capabilities were recently added to a commercial /7-version finite element program, called Pro/MECHANICA. These capabilities include both geometrically non-linear large deformation analysis and frictionless contact analysis. As the program is intended for use by design engineers, the algorithms were developed to meet the twin goals of robustness and ease of use. This paper describes the implementation of geometrically non-linear analysis and contact analysis. Solutions of simple model problems are given that demonstrate how non-linear problems can be solved with minimal guidance from the user. Keywords: /?-Version; Finite elements; Nonlinear; Contact

1. Introduction Initial implementations of the p-version of the finite element method were restricted to linear analysis, in which displacements and strains are small and the assumed material laws defined a linear relationship between stress and strain, as described by Szabo and Babuska [1]. However, there is nothing inherent in the /7-version of the finite element method that limits its applicability to the linear regime. In fact, several researchers have developed /7-version finite element programs with geometric and material non-linear analysis capabilities [2,3]. Although it appears that contact analysis has not previously been implemented in a /7-version finite element code, the techniques are well established for the h-version of the finite element method [4]. It is standard practice for manufacturing companies to employ specialized analysts to validate proposed and nearly completed designs using commercial finite element programs. In addition, there are many benefits to using finite element analyses in the early phases of a design, where analysis results can greatly influence the ultimate design. These benefits include reduced cost and reduced time to market. However, to be used effectively in the early phases of a design, it is important that the finite element program *Tel.: 4-1 (408) 953-8556; Fax: 4-1 (408) 953-8700; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

be fast, robust and simple enough to use by the engineer that is responsible for the design. It is usually the case that this design engineer has had less formal training in finite element techniques than has a specialized analyst. This paper presents two recent enhancements to a commercial finite element program: geometrically non-linear large deformation analysis, and frictionless contact analysis. For both, the implementation will be described and solutions to simple model problems will be given. In both problems, accurate solutions were obtained with minimal guidance from the user of the program.

2. Contact analysis 2.1. Implementation Frictionless contact was implemented using an adaptive penalty method. The algorithm is described for the case of 3D models in which one or more surfaces may come into contact with one or more other surfaces, as the magnitude of a specified applied load is increased. Analogous algorithms were implemented for plane strain, plane stress and axisymmetric model types. Algorithms for contact between thin shells and beams have not been implemented. To define a contact region in the finite element pre-processor program, the user must select two surfaces on the geometric model that may come into contact with each

L.E. Doxsee Jr. /First MIT Conference on Computational Fluid and Solid Mechanics other. This step may occur before any elements have been created for the finite element model. During the finite element solution process, the program identifies faces of elements that lie on these previously selected contact surfaces. The element faces that lie on one of the contact surfaces are called dependent faces. For each of these dependent faces, the program identifies one or more faces, called independent faces, on the other contact surface that may come into contact with the dependent face. To determine the penalty terms, the program loops over several sampling points, xf, on the dependent face. The normal, nf, of the dependent face at xf is intersected with the independent face to find the potential point of contact, x|, on the independent face. Under loading, the points, xf and xj displace to points

x;^ = xf + uf

2 i ^ 5 8

400 350 T 300 \ 250 200 150 j 100

1 1 0

50

1

621

exact O Pro/MECHANICA f = 100 / unit area I E = 200o\

^ ^ R=8y/ ////////////////////// 1

» X

2.5 1.5 Position, X Fig. 1. An infinitely long half-circular cylinder contacting a rigid plane loaded by uniform pressure. The contact pressure is shown as a function of position. 0

0.5

where uf and u\ are the displacements of xf and x), respectively. Interpenetration occurs when 8i = (^/ - x[') . nf < 0, where 5, is the separation distance. If interpenetration occurs, then penalty terms, which are equivalent to stiff springs acting at the sampling points, are added to the global system of equations. An initial penalty parameter (or spring stiffness), is chosen automatically based on the material stiffness and element sizes. An incremental Newton-Raphson solution procedure is used to find a solution in which interpenetration is forced to be small via the penalty terms. The penalty terms contribute to both the tangent stiffness matrix and the residual force vector. The program monitors several error indicators and automatically improves the fidelity of the model by increasing the polynomial orders of the finite element basis functions and/or subdividing the elements (hp refinement). Also, the contact penalty parameter is adjusted automatically during the solution sequence to ensure that the interpenetration is small and that the solution time is not too long. 2.2. Sample problem The following simple problem from [4] demonstrates some of the features of the current implementation. An infinitely long half-circular cylinder contacting a rigid plane is loaded by uniform pressure, as shown in Fig. 1. Due to symmetry, half of the model was modeled with finite elements. Two different solutions were obtained to this problem. In the first, the problem was modeled with a single three-sided plane strain 2D solid element, with the polynomial order of the finite element basis functions set to degree 7. Fig. 1 shows good agreement between the finite element solution and an exact elasticity solution for contact pressure as a function of position.

Fig. 2. The 3D tetrahedral mesh after it was automatically refined during the contact analysis. This problem was also modeled with 3D solid elements in a model with width equal to 10. The initial automatically generated mesh contained two tetrahedral elements. During the solution, the program automatically subdivided the mesh and increased the polynomial orders of the elements. The final mesh is shown in Fig. 2. The program obtained a maximum contact pressure of 381, compared with an exact solution of 374. The program obtained a contact area of 27.18, compared with an exact solution of 27.23. Thus, without any input from the user other than defining the geometry, loads and constraints, the program automatically obtained an accurate solution for this contact problem.

3. Geometrically non-linear large deformation analysis 3.1. Implementation Geometrically non-linear analysis for solid elements was implemented by extending the well-known h-version finite element procedure [5] in a straightforward manner. At this time, only Neo-Hookean materials are supported. The

L.E. Doxsee Jr. /First MIT Conference on Computational Fluid and Solid Mechanics

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the ability of the automatic mesher to generate wedge and brick elements, so the second mesh contained 271 tetrahedral, 47 wedge and 67 brick elements. This second mesh also contained 72 interfaces between tetrahedral and wedge faces where multifunction constraints were automatically created to enforce displacement continuity between the dissimilar elements. Both meshes contained only one element through the thickness of the shell. Solutions at 20 different load levels were obtained in order to compare the current solution with previous results. A graph of the deflection of the center of the cap versus the applied total load is shown in Fig. 3. The current solution agreed well with the finite element and analytical results presented in [8].

Pro/MECHANICA analytical h-version FEM [8] 0.00

0.02

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0.10

0.12

0.14

0.16

Displacement of Center (in)

Fig. 3. A thin spherical cap loaded with a ring load. The total load as a function of displacement of the cap's center is shown.

Fig. 4. An automatically generated mesh for a thin circular cap.

quality of the solution is monitored using a procedure that is similar to one described by Zienkiewicz and Zhu [6,7] and the polynomial orders of the basis functions are updated automatically during the solution procedure. The non-linear equations are solved with an approximate Newton-Raphson method with automatic load step control, and BFGS updates to the tangent stiffness matrix. 3.2. Sample problem A thin spherical cap is loaded with a ring load, as shown in Fig. 3. Two separate load cases were considered. In the first, the load was applied at the center of the cap and in the second, the load was applied along a circle with radius r = w/4. Solutions to the problems were obtained by using the default settings of the program. The program automatically generated 3D solid meshes for each loading case. The first mesh contained 50 wedge and 72 brick elements, as shown in Fig. 4. The presence of the ring load influenced

4. Conclusion An overview of an implementation of contact and geometrically non-linear analyses for a /7-version finite element code was presented. The sample problems demonstrated that users can obtain accurate solutions without specifying solution procedure controls. Several key algorithms enabled accurate solutions to be obtained automatically. These include (i) an automatic p-element mesher, which can generate wedges and bricks, (ii) a p-element, geometrically non-linear version of the Zienkiewicz-Zhu error estimator, (iii) an algorithm that automatically adapts the basis function polynomial orders to maintain an acceptable level of error, and (iv) non-linear equations solvers, which automatically select the load step intervals and determine when to reform the tangent stiffness matrix.

Acknowledgements Pro/MECHANICA, the program discussed in this paper, is the result of years of effort by many dedicated PTC employees.

References [1] Szabo B, Babuska I. Finite Element Analysis. New York: John Wiley and Sons, 1991. [2] Krause R, Mucke R, Rank E. /ip-Version finite elements for geometrically non-linear problems. Commun Numer Methods Eng 1995;ll(ll):887-897. [3] Sorem RM, Surana KS. p-Version plate and curved shell element for geometrically non-linear analysis. Int J Numer Methods Eng 1992;33(8): 1683-1701. [4] Kikuchi N, Oden JT. Contact problems in elasticity: a study of variational inequalities and finite element methods. Philadelphia: SIAM, 1988. [5] Bathe K-J. Finite element procedures in engineering analysis. Englewood Cliffs, NJ: Prentice-Hall, 1982. [6] Zienkiewicz OC, Zhu JZ. The superconvergent patch re-

L.E. Doxsee Jr. /First MIT Conference on Computational Fluid and Solid Mechanics covery and a posteriori error estimates, part 1: the recovery technique. Int J Numer Methods Eng 1992;33(7):1331-1364. [7] Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates

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and adaptivity. Int J Numer Methods Eng 1992;33(7):13651382. [8] Zienkiewicz OC. The Finite Element Method. London: McGraw-Hill, 1977.

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Enhancing engineering design and analysis interoperability. Part 3: Steps toward multi-functional optimization Rodney L. Dreisbach^, Russell S. Peak^* " The Boeing Company, Computational Structures Technology ^ Mail Code 67-MW, 535 Garden Avenue North, Renton, WA 98055, USA ^ Georgia Institute of Technology, Engineering Information Systems Lab ^,813 Ferst Drive MARC 452, Atlanta, GA 30332-0560, USA

Abstract This is Part 3 in a series about knowledge representations that enable enhanced collaboration between computer-aided engineering design and analysis (CAD/CAE) processes. This paper describes how a new unified physical-behavior modeling representation, based on constrained objects (COBs), can be used to bridge the gaps associated with performing multi-functional digital optimization of a product that must satisfy diverse performance specifications. It exploits the underlying modeling concepts depicted in Parts 1 and 2. Keywords: Multi-functional optimization (MFO); Constrained objects (COBs); Multi-disciplinary optimization (MDO)

1. Motivation Current product optimization processes are largely performed by using idealized design variables that are not linked (associated) with either the actual product or the virtual product definition defined by its geometry, as specified by a CAD (computer-aided design) system. Typically, the so-called design variables are attributes of one or more disparate analysis models using CAE (computer-aided engineering) processes, methods and tools to simulate one or more different physical behaviors (functional performance requirements) of the product. This paper introduces the term multi-functional optimization (MFO), as coined at Boeing, to emphasize that, in general, a multitude of operational functional requirements of a product should be optimized concurrently during the product design process. Furthermore, the design variables used to perform the optimization steps should be associated directly with the product itself (and not the mathematical models of the product used in performing different analyses * Corresponding author. Tel.: +1 (404) 894-7572; Fax: +\ (404) 894-9342; E-mail: [email protected] ^ http://www.boeing.com/ ^ http://eislab.gatech.edu/ © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

associated with different technical disciplines). Several new techniques for achieving this associativity between product design, analysis and optimization strategies are described herein, along with a typical engineering appUcation. Wilson et al. [1] overview constrained objects (COBs) as an object-oriented representation of engineering concepts (Part 1), whereas Peak and Wilson [2] present a basic engineering example in Part 2 to show how COBs facilitate design-analysis integration for simulation-based design (SBD) (a step towards MFO). In this context, simulation and analysis refer to modeling physical behavior such as stress or temperature.

2. Multi-functional optimization In developing future aerospace vehicles, the mentality that 'high performance at any expense' has yielded to addressing affordable, safer and more durable solutions that enable new product functionalities. This new strategy presents challenges for more innovative technologies and products during the 21st century than ever before. As noted by Noor et al. [3], these demands for efficiency and multi-functionality of future aerospace systems will drive the need, for example, to integrate conformally the com-

R.L. Dreisbach, R.S. Peak/First MIT Conference on Computational Fluid and Solid Mechanics munications, thermal management, flight control, vibration and noise suppression, sensors, actuators and the associated cables, all within the load-bearing structure. New products having multi-functional operational specification envelopes are generally designed via sequential development processes, where much of the data associated with the product is re-interpreted as needed by the various technologies. In many cases, design choices spanning the multiple technologies must be made, whereby tradeoffs of conflicting design objectives must be decided somewhat arbitrarily. Obviously, there is a proclivity of introducing unnoticed violations of physics! In fact, the resulting product designs may well reflect numerical optimality of the design/analysis mathematical models, rather than have a direct bearing on the physical relevance of robust multi-functional operational requirements of the desired end product. The overall knowledge pertaining to a product matures throughout its lifecycle, as illustrated in Fig. 1. It is of particular interest not only to ensure rapid access to the product knowledge as needed by multiple technical disciplines during the development process, but also to utiHze that knowledge efficiently and effectively through any portion of the overall development cycle. These activities should be performed in a COMPOSE computing environment (Fig. 1) relying on a single-source PIM (Product Information Manager). This environment, however, is not sufficiently robust relative to the set of currently available computing tools. More advanced techniques for performing designanalysis-optimization-synthesis activities concurrently, in

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satisfying the multi-functional operational specifications of a product, are needed to attain a higher level of product functional prototyping in a virtual environment. Thus, for example, a new airframe might be developed by considering it as a multi-functional structure early in its conceptual design that is to be optimized across all of the associated technical disciplines. This is a step beyond what the industry currently touts as MDO (multi-disciplinary optimization), whereby concurrent exploitation of multiple diverse technologies can not be adequately achieved. Specifically, this new encompassing development process requires advances to be made in computational technology to virtually simulate the lifecycle of an aerospace system before physical prototyping. This leads us to multifunctional optimization (MFO) which encompasses smart (knowledge-based) techniques for handUng product definition data (involving CAx technologies), parametric product information representations (such as needed for analysis, simulation, support activities, etc.), mapping of multifidelity product information between different, but related, technologies (disciplines such as structures, aerodynamics, acoustics, thermal, manufacturing, etc.), and integration of the product lifecycle evolutionary process and support infrastructure. The system-level payoffs of integrating these techniques will be realized only through concurrent development across all of the technical disciplines involved in designing new products. Integration must begin early in the product development phase of the system development for these payoffs to materialize. An innovative technique that allows for close associa-

Product Design Requirements, Operational Specifications

Constrained Objects (COBs): Multi-Directional, Associative, Computer-Sensible, Knowledge-Based, Engineering Information Mapping

Fig. 1. Maturation of product lifecycle knowledge.

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R.L. Dreisbach, R.S. Peak/First MIT Conference on Computational Fluid and Solid Mechanics

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Fig. 2. Enhanced optimization model (EOM) for a multi-behavior flap link optimization problem [5]. tivity between design and analysis models, whereby automated transfer of product information can be imposed by one application onto another, is described in [1,2] as COBs (constrained objects). Techniques of this type are mandatory to provide MFO capabilities in a virtual environment that will allow multi-physical constraints on a product design to be cross-functionally optimized, with minimal compromising of the overall set of operational specifications.

3. Analysis and optimization integration 3.1. Using constrained

objects

Parts 1 and 2 overviewed COBs and their usage to enhance C A D - C A E interoperability via the multi-representation architecture (MRA) approach [1,2]. COBs and the MRA provide the following key steps toward MFO: (1) problem partitioning into semantically rich objects (including representing idealizations and other engineering knowledge), thus providing the foundation for further knowledge-based processing; (2) explicit associativity between a design model and its analysis models needed for different analytical technologies; (3) multi-behavior, multifidelity models that can co-exist in the same constraint

graph and be connected/disconnected as needed; (4) multidirectional ^ relations that enable information flow from one functional partition to another (and within a partition) in whatever direction is needed at a given point in the product lifecycle. We believe that in the future, the COB-based approach will allow multiple users to work concurrently on different parts of an engineering design problem and to synchronize their work periodically, thereby offering a necessary step toward practical MFO. With the macro-level constraint schematic in fig. 1 of Peak and Wilson [2], one can graphically imagine how this might work: one analyst could handle the extensional behavior portion of the graph, with another analyst managing the torsional portion, while a designer is controlling the APM details. The object-orientation provides natural divisions of labor, and the associativity relations enable control points for automatic synchronization. Furthermore, one can imagine the product lifecycle as a multi-layered, evolving constraint graph, where one team initiates the virtual product as a graph of functional require^ At the macro-level one may consider this to be bi-directional associativity between design and analysis. But at the object attribute level, one can have multiple information flow directions depending on the combination of selected inputs and outputs.

R.L. Dreisbach, R.S. Peak/First MIT Conference on Computational Fluid and Solid Mechanics ments and other teams progressively add higher-fidelity objects to the graph. The COB formalism enables this type of thinking above the fray of tool idiosyncrasies and data format details, as well as actual implementation that automates such details. Fig. 2 comes from emerging work [5] adding an automatic optimization layer onto the present MRA [2,4]. Parallels with the current MRA include: • rich optimization objects are added to the architecture that leverage existing analysis modules (CBAMs); • associativity is achieved between a product's optimization models, its analysis models (CBAMs) and its design model (APM); • optimization models of diverse fidelity are represented in a manner similar to current multi-fidelity analysis and design models; • SMMs provide automated interaction with traditional optimization methods and tools. 4. Summary Using constrained objects (COBs) to represent engineering concepts for enhanced collaboration between CAD and CAE processes has been introduced via a series of three presentations at this conference. The technique provides a unified physical-behavior modeling representation that captures reusable knowledge to satisfy the multi-functional operational requirements of typical products. The current methodology is necessary, but not sufficient, for performing

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practical MFO. Further work is required to develop compatible representations of high-level operational criteria [6], such as Product Design Requirements and Objectives, and their associativity with the detailed product structure. We befieve that COBs provide a basis for developing these additional capabilities, including a framework for leveraging existing MDO capabilities.

References [1] Wilson MW, Peak RS, Fulton RE. Enhancing engineering design and analysis interoperability. Part 1: Constrained objects. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [2] Peak RS, Wilson MW. Enhancing engineering design and analysis interoperability. Part 2: A high diversity example. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [3] Noor AK, Venneri SL, Paul DB, Hopkins MA. Structures technology for future aerospace systems. Comput Struct 2000;74(5):507-519. [4] Peak RS. X-Analysis Integration (XAI)^ Technology. Georgia Tech Report EL002-2000A, March 2000. [5] Cimtalay S. The Enhanced Optimization Model Representation: An Object-Oriented Approach to Optimization. Doctoral Thesis, Georgia Tech, 2000. [6] Deng YM, Britton GA, Tor SB. Constraint-based functional design verification for conceptual design. Comput-Aided Design 2000;32(14):889-899.

^ Some of these references are available at http://eislab.gatech.edu/ ^ X = design, manufacture, sustainment, etc.

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Stochastic process/field models for turbomachinery applications Dan M. Ghiocel * STI Technologies, Advanced Engineering, 1800 Brighton-Henrietta TL Road, Rochester, NY 14623-2572, USA

Abstract The paper discusses key aspects of stochastic modeling for aircraft jet engine appUcations. Specifically, the paper addresses issues related to engine rotating blade-disk assemblies (called simpler blisks). Stochastic process/field models are used to idealize space-time random variations of operational loading, material properties and blade geometry deviations due to manufacturing. Keywords: Blades; Turbomachinery; Stochastic; Manufacturing; Non-Gaussian models; Life

1. Introduction Turbomachinery applications, and especially those for aircraft jet engine turbines that operate in a continuously varying speed range, are difficult engineering modeling problems involving mathematical idealization of multiple, complex aero-structural dynamics interacting phenomena. Steady and unsteady pressures and temperatures on blisk surfaces are varying in time and space inducing a continuously transient multi-axial stress/strain state in blisk structure. Random aspects are always present due to complexity of the real phenomena. A key aspect of stochastic modeling is to incorporate significant macro- and microscale random space-time random variabilities; or in other words, to be intimately related with the physics of the phenomena involved in blisk design analysis. These spacetime random variabilities are always present in operational environment, unsteady forcing function, thermo-mechanical material behavior, aero-elastic instability, manufacturing geometric deviations, contact interface boundary conditions, multiple-site fatigue, progressive damage process, including different damage mechanism interactions, etc.

2. Stochastic modeling of space-time variabilities The paper addresses some of key aspects of stochastic modeling of space-time random variabilities for engine *Tel.: +1 (716) 424-2010; Fax: +1 (716) 272-7201; E-mail: dghiocel @ sti-tech.com © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

blisk applications, namely, (i) loading history and (ii) blade geometric deviations. Loading history or sequence plays an important role when the stress amplitude is highly variable in time that is the typical case for jet engine bhsks. Also, spatial deviations of blade geometry or properties can influence drastically blade vibration amplitude. This is due to the loss of cyclic symmetry geometry of the blisk (cyclic symmetry assumes equal nominal properties for all the blisk blades). A slight departure from the blisk cyclic symmetry geometric pattern may produce significant differences in blade forcing-functions, flutter stabilities and resonant stresses. Mistuning which manifests by inducing large vibration in blades is only one of the drastic consequences of stochastic spatial variability in blade dynamic properties. 2.7. Flight profiles A typical flight speed profiles is shown in Fig. L This figure indicates that speed mission profiles can be idealized using non-stationary, non-Gaussian stochastic processes. A typical flight profile can be modeled using a linear recursive pulse process model described by the sequence of realizations of two independent random variables Y and Z as follows [1]: Xk = Yi^Xii-\ + (1 — Yk)Zk

1,2,

(1)

where k is the step number in the random sequence. The Y variable is a binary random variable taking value 0 or 1 with probabilities \ — p and p, respectively. The Z variable is an arbitrarily distributed random variable with

D.M. Ghiocel /First MIT Conference on Computational Fluid and Solid Mechanics SEVERE FLIGHT OPERATING SPEED PROFILE

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(2)

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(4)

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assuming that the starting point is zero. An alternate approach is to relate the non-Gaussian mission profile, denoted as a continuous process X{t), to its Gaussian image, denoted by Z(r), by a nonlinear memoryless monotonic transformation, g(-), X(t) = g(Z(r)). This transformation can be further expressed by the relationship X(t) = F - i ( o ( Z ( 0 ) ) , where F'^-) is the inverse of marginal cumulative distribution function of X(0 and 0 ( Z ) is the standard Gaussian cumulative distribution function of Z(t). The pair correlation (an element of correlation matrix) of X(t), denoted E[X{t)X{t')], can be expressed in terms of the pair correlation of Z(t), denoted E[Z(t)Z(t')lby

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2.2. Blade geometry Manufacturing geometry deviations are idealized using stochastic field models. Fig. 2 illustrates two samples of simulated airfoil thickness variation using a non-homogeneous stochastic field model. Fig. 3 shows the drastic effect of airfoil thickness variation on the blade leading edge resonant stresses (computed for nominal airfoil geometry and airfoil perturbed geometry). From the mathematical modeling point of view, these stochastic fields are quite complex, being non-homogeneous, non-isotropic non-Gaussian vector fields. To handle these stochastic fields it is advantageous to factorize them in terms of an optimal orthogonal function basis. For general case, several techniques can be used for constructing factorable stochastic fields. For example, the use of the Pearson differential equation for defining different types of stochastic orthogonal polynomial series representations including Hermite, Legendre, Laguerre and Cebyshev polynomial series. One major application of theory of factorable stochastic fields is the spectral representation of stochastic fields [2,3]. The Karhunen-Loeve (KL) representation is an optimal spectral representation with respect to the secondorder statistics of the stochastic field. For typical continuum

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D.M. Ghiocel /First MIT Conference on Computational Fluid and Solid Mechanics

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mechanics problems the KL expansion is fast convergent series, i.e. it needs only few expansion terms. The use of the KL expansion assumes that the covariance function of the stochastic field is previously known which limits sometimes its area of applications. The KL expansion of a stochastic field X(p), where p is a generic parameter, is based on spectral representation of covariance function, Cov[Z(p), X(p')\ as follows:

The covariance function being symmetrical and positive definite has all its eigen-functions mutually orthogonal, and they form a complete set spanning the function space which contains the field X(p). It can be shown that if this deterministic set is used to represent the stochastic field, then the random coefficients used in the expansion are also orthogonal. The general from of the KL expansion is

X{p) = J2^'^'^P'^^^ C0v[X(p), X{p')] = Y,

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(5)

where X„ and 0„(/?) are the eigen-value and the eigenvector of the covariance kernel, respectively. They are computed by solving the integral equation

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(6)

(7)

where set {z,} represents a set of independent, standard Gaussian random variables. If the process being expanded is Gaussian, then the random variables [zt] form an orthonormal Gaussian vector and the KL expansion is meansquare convergent irrespective of the correlation structure. More generally, if the stochastic field X(p) is nonGaussian and/or covariance function is unknown, it can be formally expressed as a nonlinear functional of a set

D.M. Ghiocel /First MIT Conference on Computational Fluid and Solid Mechanics EFFECTIVE VS. RESONANT SPEED PROFILES

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TIME (seconds)

Fig. 4. Simulatedflightstress profile in a critical location (minimum predicted life), (a) Steady (LCF) and vibratory (HCF) stress profiles, (b) Excitation vs. mode resonant speed profiles. of standard Gaussian variables or in other words expanded in a set of random orthogonal random functions, denoted {\lfi]. Herein, for example, a stochastic polynomial expansion model of polynomial type series is suggested [3]. The polynomial expansion model often called 'polynomial chaos' is defined by the series: p

The polynomial expansion functions are orthogonal in the sense that their correlation, Eli^jx/fk], is zero. A given truncated series can be refined along the random dimension either by adding more random variables to the set {Zi} or by increasing the maximum order of polynomials included in the stochastic expansion. The first refinement takes into account higher frequency random fluctuations of the underlying stochastic process, while the second refinement captures strong nonlinear dependence of the solution process on this underlying process. Using the orthogonality property of polynomial chaoses, the coefficients of the stochastic expansion solution can be computed by E[xl/ku] fOYk= 1, Uk K (9)

Elfh

It should be noted that the polynomial chaos expansion can be used to represent, in addition to the solution process, any arbitrary stochastic input process. Polynomial chaoses are orthogonal with respect to the normal probability measure, dP = exp(|z^z)dz, which makes them mathematically identical with the multidimensional Hermite polynomials. The orthogonality relation between these random polynomials is expressed by the inner product in L2 sense with respect to Gaussian measure:

/

r „ ( Z / i , . . . , Zin)^m(Zn,

. . . , Zin)

X exp {-\z^z) dz = nlVlrtSm

(10)

2.3. Blade stress/strain profiles The computed flight stress profile in a critical blade location is shown in Fig. 4. The steady-state stress profile (LCF component) and the vibratory stress profile (HCF component) are comparatively plotted. The HCF stresses shown at the bottom of Fig. 4a) exhibit a highly skewed shaped, non-Gaussian probability density. The total stress variations were obtained by the superposition of a slow-varying/low-frequency stress component (defined by a pulse process with holding times), due to pilot maneuvers, and a fast-varying/high-frequency stress component (defined by intermittent narrow-band process/harmonic 'clusters'), due to unsteady aero-forcing when the excitation frequency is close to blade mode frequencies or is passing slowly through the blade mode resonances (Fig. 4b). The slow-varying stress cycles produce mostly low-cycle fatigue (LCF) damage and creep damage, while the fast, randomly occurring vibration stress cycles with low amplitude produce mostly high-cycle fatigue (HCF) damage.

3. Concluding remarks Stochastic modeling for jet engine applications represents a difficult engineering task involving multiple aerostructural dynamics uncertain phenomena. The paper indicates the need of using stochastic process/field models for capturing adequately the random aspects of the phenomena involved in engine blisk analysis. The paper proposes specific stochastic process/field models for flight speed profiles and blade geometry deviations.

632

DM. Ghiocel/First MIT Conference on Computational Fluid and Solid Mechanics

References [1] Grigoriu M. Applied Non-Gaussian Processes. Englewood Cliffs, NJ: Prentice Hall, 1995. [2] Ghanem R, Spanos P. Stochastic Finite Elements. New York: Springer, 1991. [3] Loeve J. Probability Theory. New York: Springer, 1977, 4th ed. [4] Ghiocel DM. Probabilistic Fatigue Life Prediction For Jet Engine Components: Stochastic Modeling Issues. In: ECOMASS 2000, Barcelona, September 11-14, 2000.

[5] Ghiocel DM. Factorable stochastic field models for jet engine vibration response. In: 13th ASCE Speciality Conference, Baltimore, June 13-16, 1999. [6] Ghiocel DM, Rieger NF. Specific probabilistic issues for gas turbine HCF life prediction. In: 3rd FAA/AFRL Workshop on Application of Probabilistic Design Methodologies to Gas Turbine Rotating Components, Phoenix, AZ, March 4-5, 1998.

633

Optimum design and sensitivity analysis of piezoelectric trusses Yuanxian Gu *, Guozhong Zhao, Yuhong Chen State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

Abstract The paper studies the optimum design for structural stiffness and free-vibration frequency of piezoelectric intelligent trusses. The computational formulations of sensitivities of structural displacement and free-vibration frequency with respect to size and shape design variables are proposed. The electric voltage is defined as a new kind of design variable, and the method to compute displacement sensitivity with respect to this kind of variable is proposed. A new method for structural displacement control by optimizing the voltages of piezoelectric active truss is implemented. Keywords: Piezoelectric intelligent truss; Mechanical-electric coupling; Design optimization; Sensitivity; Voltage; Stiffness; Frequency

1. Introduction To meet the requirements of the space engineering, especially aircraft, satellites and robot manufacturers, the intelligent structure has become an active research field in recent years [1]. The piezoelectric material, as an important part of intelligent structure materials, not only has the ability of carrying load, but also the mechanical-electric coupling property can serve as actuator and sensor. Because of mechanical-electric coupling effect and external electric load, the design of the piezoelectric intelligent structures is more complicated than conventional materials. By means of the design optimization methods, not only the structural weight and cost can be reduced, but also the structural behaviors, such as strength, stiffness, vibration, buckhng, and other performances can be improved efficiently. Therefore, research on design optimization methods for piezoelectric intelligent trusses is very useful. The numerical methods of design optimization and sensitivity analysis for piezoelectric intelligent trusses are studied in the paper. The design constraints considered are structural stiffness and free-vibration frequency. On the basis of finite element method and accounting the mechanical-electric coupling effect under electric and mechanical loads, the formula of sensitivities of displacement * Corresponding author. Tel/Fax: 86 (411) 470-8769; E-mail: guyx96 @ dlut.edu.cn © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

and free-vibration frequency is proposed with respect to the size, shape, and electric voltage variables. The voltage is a newly defined design variable by this paper to produce a new method controlling structural shape of piezoelectric active trusses. The numerical methods of design optimization and sensitivity analysis for piezoelectric truss structures are implemented in JIFEX software. Numerical examples are given to demonstrate the effectiveness of the methods of the paper.

2. Finite element model The linear mechanical-electric coupling effect can be expressed by the following equations. {a} = [c^m

- [e]{E}

{D} = [eV{8] + [E]{E}

(1)

where {a} is the stress vector, D the electric displacement vector, [e] the piezoelectric coefficient matrix, [s] the strain vector, [S] dielectric matrix, and {E} the electric field vector. The piezoelectric truss is made of piezoelectric thin pieces shown in Fig. 1. While exerting electrical voltage along the axial direction, the thin pieces will produce axial strain. Based on the Hamiltonian principle, the dynamical equation of piezoelectric truss can be derived from Eq.

Y. Gu et al /First MIT Conference on Computational Fluid and Solid Mechanics

634

N2

<

L

>

Nl

^t^^^

<

•

O

t

Fig. 1. Piezoelectric truss.

d{[K,Y[QY)

^

(1) with Guyan condensation to eliminate the electrical potent vector.

d{QY

=

M''[K2V[K,r'lK2]

[K,Y

=

m^[K2V[K,r

1

-1

-1

1

M

(3)

(4)

= {P}-[K,]IQ}

For free vibration situation, Eq. (2) can be written as {[K] + [KE] - co'[M]){ip} = 0,

IcpVlMUcp} = 1

(5)

where co is natural frequency, {(p] free-vibration mode vector.

3. Sensitivity analysis of piezoelectric truss Differentiating Eq. (4) with respect to the design variable at each element, we obtain axi

dxi

\

dXi

dxi

dXi

)

= M' (V

Eq. (6) is solved with the same procedure as that for Eq. (4) to get the sensitivities of displacement. Since the sensitivity derivative of the elastic stiffness matrix is the same as the usual materials, we discuss here only how to compute the derivatives of the mechanical-electric coupling stiffness matrix and the electric load vector. For the size design variable, the derivative of the mechanical-electric coupling stiffness matrix and electric load vector can be computed as

m

dXj

(7)

(8)

d[K2r dXi

[Ky (9)

dXj

It should be noted that the derivative of electric load vector to the size variable is not zero. This is different from the case of mechanical load. For the shape design variable, the semi-analytical scheme is used to simplify the calculations. The derivatives of the mechanical-electric coupling stiffness matrix and electric load vector in Eq. (6) are computed as below.

_ {[KEYY - i^^EYf

SIKEY

(10)

dXi

d{[KJlQr) _ {[K.YlQVy -

{[KJ{Qrf

dXi

(11)

where superscripts '0' and ' 1 ' denote quantities before and after the perturbation of design variable. After the perturbation of shape design variable, a shape updating technique [3] is performed to change the finite element model. Because the electrical voltages only change the electrical charges vector, the sensitivity of external load with respect to the electrical voltage is given as dXi

(12)

[KJ[CY

Differentiating Eq. (5) gives the derivatives of the freevibration frequency as the below d{[K] + [KE])

dco (6)

-1

dXi

dXi

(2)

\K,\{Q\

where [u] is the displacement vector, [M] the mass matrix, [K] the elastic matrix, [A^^] the mechanicalelectric coupling stiffness matrix, [P] the mechanical loads, [Q] = [C]{V} the electrical charges vector, [C] electrical capacitance matrix, [X] the coordinate translation matrix, and {V} is electrical voltage vector. Details of [K2] and [A^3] are described in [2]. For static strength case, we have {IK]^[KE]){U}

1

d[CY {V}

dXi

\KEY

-{KiMK^y

^[K,] dXi

dxi

[M]{M} + [\K\ + [/^d){«} = {P\ -

{Kir\K2Y

dXi

X ^[if3]-'[^2]^' dxi


O-

-C

[xY (2

dXi

dXi

2o)

29[M] . . .

dXj

(13)

4. Design model and optimization algorithms The general mathematical presentation of structural design optimization problems is min/(X) s.t. gj(X)<0 XiL < Xi < Xiu

; = ( l , 2 , . . . ,m) / = ( 1 , 2 , . . . , n)

(14)

635

Y. Gu et al. /First MIT Conference on Computational Fluid and Solid Mechanics where f{X) is the objective function, n the number of design variables Xi, m the number of constraint functions gj{X), and Xiu and xn the upper and the lower bounds of design variables. In our studied problem, the design model is that objective function and constraint functions are structural weight, the total piezoelectric voltage of electrical pieces, displacements and free-vibration frequencies. The design variables of the design model are of three categories of size variable (bar cross-section areas), shape variable (node coordinates) and the electrical voltage of the piezoelectric pieces. The optimization solution algorithm used here is the Sequential Linear Programming (SLP). In SLP algorithm, the objective and constraint functions are approximated with linear extensions at the current design point of the optimization iteration. Then the original problem Eq. (14) is transformed into the following linear programming problem. min/(Xo) + V V ( ^ o ) A X s.t. g,(Xo) + V ^ g , ( Z o ) A X < 0

(15)

all the same as L = 1.5 m, and the thickness of each piezoelectric piece is t = 0.005 m. The material is PZT-4 and parameters are C33 = 8.807E10 (N/m^), ^33 = 18.62 (C/m^), S33 = 5 . 9 2 E - 9 (C/Vm), and p = 7600 (kg/m^). The eight design variables are }7-coordinates of nodes 9-16, i.e. yg-yie- The low bound of constraint on the first order frequency is 36.0 Hz, and the initial and optimum values 36.78 and 36.0 Hz. The structural weights of initial and optimum designs are 164.16 and 144.28 kg, respectively, and the weight decreases 12.1%. Optimum design variables are 1.231, 1.231, 1.650, 1.872, 1.878, 1.645, 1.231 and 1.231. The optimum shape of truss is shown in Fig. 2, and the structural symmetry is kept. 5.2. Example 2 The 25-bar truss shown in Fig. 3 subjected to loads p^^ = P2z = - 7 0 0 0 0 . 0 N. The cross-section area of bars is A = 3.0 cm^, the material parameters are the same as in example 1. The electric voltage of each piezoelectric piece is 0 V. The maximum displacement occurs in the Z direction of node 1 or node 2, and the value is 0.172

where the V ^ / ( X o ) and V^gy(Xo) are derivative gradients of the objective function and constraint functions, respectively. The linear programming problem Eq. (15) is solved with the Lamke pivot algorithm to find a new design. This procedure is repeated until the convergence is reached. To overcome the error problem of the linear approximation of constraint functions and ensure iteration converge, the approaches of the approximate line search, the adaptive move limit, and the constraint relaxation have been employed [4], and computational experience has demonstrated that these approaches are effective.

0.75

5. Examples of piezoelectric trusses optimization 5.1. Example 1 The structure is shown in Fig. 2. The height of truss is h = 2.0 m, the distances between vertical bars are

Fig. 3. Twenty-five bar piezoelectric truss. 14

15

16

Fig. 2. Initial and optimum designs of frame structure.

636

K Gu et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 The design variables and optimum values of 25-bar truss Design variables Al

A2

A3

A4

As

Ae

Av

Node Nos. of bars

1-2

Optimum values (m^)

2.93E-4

2-5, 2-A 1-3, 1-6 4.00E-4

1-4, 2-3 1-5, 2-6 2.43E-4

3-6, 4-5 3-4, 5-6 1.50E-4

3-7, 6-10 5-9, 4-8 4.00E-4

4-9, 5-8 3-10, 6-7 1.79E-4

3-8, 4-7 6-9, 5-10 3.35E-4

cm. To optimize the truss with constraint on maximum displacement to 0.15 cm, the following two methods are used. (1) Modify the cross-sectional area of the bars. The cross-section areas of all the bars are divided into seven design variables A1-A7. The node numbers of bar elements and the bars of design variables are listed in Table 1. The initial values of all design variables are 3.0 cm^, upper bounds 4.0 cm^, and lower bounds 1.50 cm^. The objective function is weight, and constraint is the maximum displacement. In the optimum design, the structural weight reduces from 75.4 to 72.33 kg, design variable values are shown in Table 1, and the maximum displacement is 0.15 cm. (2) Increase the electric voltage of piezoelectric pieces. The electric voltage design variables are divided into seven groups V1-V7 as the same as the A1-A7, and the upper and lower bounds are 3500 and 0 V, respectively. Subjected to the displacement constraint, the total voltage of all the piezoelectric pieces is objective function. The optimization result is that the maximum displacement is 0.15 cm, the total voltage is 2.98 x 10^ V, and all of the electric voltage design variables are 0, 3500, 3500, 0, 3500, 955.7, and 0(V).

6. Conclusions The design optimization and sensitivity analysis methods for piezoelectric intelligent trusses with the displace ment and frequency constraints are proposed with respect to both size and shape variables, as well as the new defined

design variable of electric voltage. It can be concluded that the displacement of piezoelectric truss can be controlled by means of modifying the cross-section areas of bars or exerting the piezoelectric voltage to relevant bars. When the minimization of weight or energy cost is considered, how to arrange the cross-section areas or electric voltage is important to practical engineering. The numerical results presented here show that the optimization methods of the paper can settle the problem well.

Acknowledgements The project is supported by the NKBRSF of China (No. G1999032805), and the Foundation for University Key Teacher by the Ministry of Education of China.

References [1] Rao SS, Sunar M. Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey. Appl MechRev 1994;47(4): 113-123. [2] Nie RT, Shao CR. Adaptive truss structure Mech-Electric coupling dynamic equation. Mech Eng (in Chinese) 1997;10(2): 119-124. [3] Gu YX, Cheng GD. Structural modeling and sensitivity analysis of shape optimization. J Struct Optimiz 1993;6(1):2937. [4] Gu YX, Kang Z et al. Dynamic sensitivity analysis and optimum design of aerospace structures. Int J Struct Eng Mech 1998;6(l):31-40.

637

Vehicle crashworthiness design using a most probable optimal design method I. Hagiwara*, Q.Z.Shi Tokyo Institute of Technology, Department of Mechanical Engineering and Science, 2-12-1 0-okayama, Meguro Ku, Tokyo, 152-8552, Japan

Abstract Human safety is a very important factor to be taken into consideration in automobile design. The subject of this research is an optimal design of the vehicle side member. The research focuses on the optimization method of taking the comprehensive trade-off between the global approximation and computational cost. The optimization approach called most probable optimal design (MPOD) proposed by the authors is modified to be appHcable to the optimization of the vehicle side member with the mixed discrete and continuous design variables. Numerical simulation in crashworthiness analysis shows that the MPOD technique is effective in saving the computational cost. Keywords: Vehicle crashworthiness design; Energy absorption; Optimization; Response surface methodology; Most probable optimal design

1. Introduction Due to the recent developments in computer science and software technology, fluid analysis, nonlinear structural analysis and their combined analysis has become possible for large-scale models and will eventually be employed in manufacturing design [1]. The importance of being able to make rational and correct decisions during the early stages of design is currently being emphasized. However, crashworthiness analysis requires considerable time for one functional evaluation even using supercomputers. It is vital that rapid evaluation models of performance be used instead of computationally time-consuming simulation tools and key techniques in optimal design. For this purpose, response surface methodology (RSM) works effectively to achieve these design optimizations. Research has proven that RSM is a valuable tool for optimization problems in which the designer must work with functions with nonsmoothing response surfaces. A neural network is considered as the most feasible activation response function to RSM. Hagiwara et al. initially employed the holographic neural network for the activation function * Corresponding author. Tel.: -h81 (3) 5734-3555; Fax: +81 (3) 5734-3555; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

of the response surface and compared it to the multilayer feed-forward neural network to demonstrate that the former type of neural networks had a much higher approximation accuracy and training speed [2]. In this paper, a sequential approach using the HNN is introduced for the activation of the response surface to make the technique available for general application in structural optimization. We present the proposed approach, termed the most probable optimal design (MPOD) method to determine one local optimal design, which has the highest probability in the design space. A design example of crashworthiness for vehicular structure demonstrates the validity and utility of this method.

2. Theoretical background of holographic neural network The theory of the holographic neural network for the response surface is briefly introduced. In general, the form of stimulus of real data x = {^i, ^2, ^3, • • • , ?«} may be expressed by transforming the scalar values to complex space by the following nonlinear transformation.
(fc = 1,2, 3 , . . . , n )

(1)

638

/. Hagiwara, Q.Z. Shi/First MIT Conference on Computational Fluid and Solid Mechanics ITT

(y^ = 1 , 2 , 3 , . . . ,w)

( 1 4 - e~^ki^k-i^k)/crk)

(2)

where /x^t and a^ are the mean value and standard deviation of ^k, respectively, and X^ defines a vector magnitude bound within the unit circle (0-1), expressing a weighting or dominance for each element of the stimulus field (k = 1, 2, 3 , . . . ,n).lf no design variable dominates then, X^ = \(k= 1,2,... ,n). A similar transformation can be performed within the response field y. r = ye^

--My)

(3)

where fs and fr are transformation functions, which are always nonlinear functions. If p designs are trained, then (pk(^k) and r are summarized as the following matrix formulations.

xW^^

m=

X]e''"

[Y]

{y^^-

(4)

3. The most probable optimal design (MPOD) method The MPOD algorithm involves two steps sequential optimization illustrated in Fig. 1. The first step is to scan the whole design region to find where the global optimal exists most-likely in probability sense by conditioned random seeds. The second step is to improve the accuracy near the most-likely global optimal by adding information near the most-likely global optimal. The optimal problem is defined as.

,yV<}'

subject to: gi(x) < 0,

(/ = 1,2,.. . , m)

(13) (14)

where f{x) is the objective function, gi(x) < 0, (/ = 1,2, ...,m) are the constraints, m is the number of constraints.

XPpi^n

k'.e" y^e"''

(12)

{rj} = rnxj)]"" • {A]

max f(x)

Xy'2

Xfe'^

where eps is the threshold value to terminate the iteration. The predicted response of any design point X/ can easily be calculated,

(5)

Where ^/, (p^ and X[, yj represent the mapped phases and magnitudes of the stimulus and response, respectively. The training operation involves mapping the stimulus [O] and response {Y}. These mappings can be expressed by linear functions.

Step 1: To search the most-likely global optimal design over the whole design space. The number of Nd = {n -\- l){n + 2)/2 initial experimental designs with distribution conditioned random seed are used to construct the original approximations of both objective function and constraints,

r =

dm = min

^ak(pk(^)

(6)

where 0A(?) is the activation function decided by Eq. (1) and ak are the complex parameters of the model. Let the covariance matrix of experimental error be a unit matrix ([W]=[!]), then the estimators of parameters become {A} = ([fm)-'[ct>]"{Y}

(7)

To solve parameter {A} in Eq. (7), it is necessary to calculate the inverse matrix. When the number of activation functions is large, it will lead to considerable calculation time and may result in calculation error. The following iteration process is used to save calculation time. {ioi =

m"{Y}

{AAk+i} =

(8) m"{iY]-miAk])

{Ak+i) = {Ak} + {AAk+i}

(9) (10)

The iteration may be terminated when the error becomes smaller than the preset threshold value, eps. J({Ak+i]) = ([Y] - m{Ak+i})"(lY}

- [OKA,^,}) < eps (11)

Xi

Xj

(1 < /, ;• < A^, / 7^ 7, m = 1,2, . . . , p ) > Jm

Xc

(15)

where Xc is the scale parameter which can be selected to the size of design region or mean value of design Xk. drum is the threshold of distant which should be set as large as possible, p is the total combinations of designs. If no new design is found after thousands of random trials with the threshold distance, new threshold distance has to be decreased by ^min = P • ^min(0 < ^ < I). The corresponding objective function f(xk) and all constraints gi(xk) are trained using neural network to approximate the response surface f{x) and gi{x). The initial optimal design Xop{0) can be solved with approximation functions using optimization algorithm. Then creating w designs to improve the accuracy of approximation functions. The A:th trial approximation functions by neural network are used to obtain the new optimal design Xop(k). The convergent criteria of step 1 is, Ad =

„{k) -Xppjk-

1)

<Si

(16)

Xc

where s is the threshold of termination of step 1. The termination of step 1 is explained by Fig. 1. In the figure, the

/. Hagiwara, Q.Z. Shi/First MIT Conference on Computational Fluid and Solid Mechanics >

639

uniformly distribution randam designs

H

no

Step 1 search optimal region

acceptance ?

yes

response calculation

0

approximation with HNN

^ search optimal

acceptance ? I "~ add n points (Gauss distr.)

design

no

a = a*A^

M = ^^—^ n—f

Step 2 increase the accuracy near optimal design

^ ^1 Probability

add 1 point at optimal design

yes T end

design region

Fig. 1. Flowchart of the MPOD method. light dot line expresses the {k — l)th trial of approximation function and the dark dot line stands for that of ^th trial.

X) = 7.85E-3g/nim^ E = 210GPa Et = 2.5GPa cry = 220MPa 2/ =0.3 v= 54 km/h meshes= 2700 elements

Step 2: Step 2 is to verify and improve the accuracy of approximation of objective function and constraints at optimal design Xop. Accuracy check is based on, \\f{Xop{k))-f{Xo,{k))\\ \\f{Xop{k))\\ \gi{Xop{k)) -

gi{Xop{k))\

hi{xopik))\\

(17)

82

(/ - 1 , 2 , . . . , m )

(18)

(b)

where 82 and 6i are the accuracy criteria. Fig. 2. Analysis model of crashworthiness optimization. 4. Vehicle side member component optimal design The vehicle side member (component parallel to the central axis of vehicle) plays a role in energy absorption while the crashworthiness and energy absorption of the vehicle is determined by its size, shape and welding. In this study, the square cross-area, a perfectly straight side member with uniform thickness is investigated and reinforcement of the component is considered as a way to increase the energy absorption. The size and material property of the member is illustrated in Fig. 2a, and the form of reinforcement is depicted in Fig. 2b. All degrees of freedom at the bottom end are rigidly fixed, and at the top end, a rigid mass of 500 kg, and velocity of 54 k m / h are

used as a load to simulate a crash. The load-displacement behavior of the member while crashing is calculated by the finite-element method (FEM) solver LS-DYNA3D [3], in which 2700 shell elements and 2754 nodes are used. The design variables include the thickness of the baseplate (^1), the upperplate (^2) and the location of reinforcement (z). The total weight of the component is constant {w = 780 g), therefore only two design variables are independent. The objective function is the energy absorption of the component, 10 ms after crashing. The mathematical definition of the problem is max f{ti,

t2,z)

subject to: 0.5 < ^1, ^2 < 1.0, 1.5 < z < 250

/. Hagiwara, Q.Z. Shi/First MIT Conference on Computational Fluid and Solid Mechanics

640

FEM calculation HNN approximation 200 C3

150

5

100 0

50

100

150

200

250

Reinforced position z (mm) 11.141^

^

7604.377 L97.6I26

7897.6126—-

variable t1 Fig. 3. Approximation function by the proposed method.

'^ =

zb

The training procedure and stop thresholds are identical to the case of large deviation, that is, o? = 5, yS = 0.6, and Si = 0.15, 82 = 0.01. Step 1 is terminated after four iterations (corresponding to 12 functional evaluations) of random seed searches. Then, Eq. (17) is satisfied after two iterations and the total number of functional evaluations until convergence, is 14. The approximation optimal is (1.21, 1.39, 160), fixopik)) = 8484.11 kJ, which is 11.6% higher than that of the original design (1.00, 1.50, 0.0). The contour plot of the approximation function is depicted in Fig. 3. In the figure, the open circle indicates the optimal design using the proposed MPOD method and the plus sign expresses the designs used for approximation. The comparison between the approximation function and FEM values with ti = 1.21, and tj = 1.39 are fixed while the changing reinforcement location z is plotted in Fig. 4. It can be seen from the figure, and also mentioned in [4], that the FEM value varies, because of the heavy nonlinearity of the crashworthiness problem [4]. A smooth and robust approximation function is obtained by the holographic neural network approximation and an optimal design is realized successfully [5]. The MPOD approach based on the holographic neural network has a robust property against calculation noise. After this, crashworthiness analysis of vehicles characteristics and characteristics of passenger protecting devices will be investigated in future.

Fig. 4. Comparison of FEM and HNN when design variable z is changed.

5. Conclusions In this paper, the most probable optimal design (MPOD) method is proposed to search the design space containing the global optimal design using designs by conditioned random seeds to determine more accurate approximations. The MPOD method is a response surface methodology based on the holographic neural network, which uses the exponential function as an activation function. The proposed sequential optimal design method can be applied to design problems with regular and irregular boundaries of design space. Application to the vehicle crashworthiness design with multiple peaks reveals that the proposed method is a feasible and practical approach.

References [1] Hagiwara I. Review of optimization method and its tendency, Forum on Spring Conf., Society of Automotive Engineers of Japan, May, 1998 (in Japanese). [2] Hagiwara I. Innovative technique and future studies of optimal design from the viewpoint of automotive application example, Proc. of the 7th Design Engineering and Systems, JSME, No. 97-95, pp. 3-7, December, 1997 (in Japanese). [3] LS-DYNA, Livermore Software Technology Corporation, 1997. [4] Okamura Y. Application examples of optimum design tool PAM-OPT (in Japanese), Proc. of the 7th Design Engineering and System, JSME, 1997, pp. 451-454. [5] Shi Q, Hagiwara I, Azetsu S, Ichikawa T. Optimization of acoustics problem using holographic neural network (in Japanese), Trans JSAE 1998;29(3):93-98.

641

Computer simulations and crack-damage evaluation for the durability design of the world-largest cooling tower shell at Niederaussem power station R. Harte'''*,U. Montag'' ^ Institute for Statics and Dynamics of Structures, Bergische Universitdt-GH Wuppertal, Pauluskirchstrasse 7, 44285 Wuppertal, Germany ^ Krdtzig and Partner Civil Engineering Consultants Ltd. Bochum, Buscheyplatz 11-15, 44801 Bochum, Germany

Abstract The present paper will sketch the main aspects of the design and construction of the 200 m high cooling tower at the power plant Niederaussem. This tower definitely belongs worldwide to the largest and thinnest concrete buildings at present. Because of the combined action of wind, thermal and hygric effects, special care has to be taken on fatigue, cracking and corrosion to insure an adequate level of safety and durability. The present paper gives a short overview, providing additional Unks to more detailed references. Keywords: Cooling tower shell; High-performance concrete; Acid resistance; Crack damage; Nonlinear analysis; Durability

1. Introduction In 1995 the RWE Energie AG, the largest German energy supplier, initiated the conceptual design phase for the renewal of their power plants in the German lignite mining area west of Cologne. Since that time large efforts have been undertaken both in research and application to improve both the efficiency of the power unit and the durability of the structural components. This especially holds true for the 200 m high natural draft cooling tower. In the predesign phase, the first author has been especially concerned with the structural form-finding process, the treatment of turbulent wind action due to interference effects and the optimization of the reinforcement mesh. Later he verified the design of the cooling tower shell under the influence of a special high-performance concrete, which has been created under the leadership of RWE to meet the acid attack from the cleaned flue-gas vapor. In the phase of design and construction the second author was responsible for the proof of the complete structure — foundation, column framework and shell — and for the final drawings ready * Corresponding author. Tel.: +49 (202) 439-4080; Fax: +49 (202) 439-4078; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

for construction, with special emphasis on the development of the pre-stressing procedure of the upper ring beam. In addition his task was to supervise the reinforcement works on site.

2. Design specifications 2.7. Structural form-finding process Fig. 1 shows the main dimensions of the tower. In a wide range the shell has a thickness of 22 to 27 cm. The form has been the result of optimization calculations with respect to a best possible load-bearing behavior — lowest meridional stress and highest buckling safety — resulting in most economic concrete and steel masses. All in all it can be stated that the best possible load-bearing behavior can be reached when the curvature of the meridian increases continually from the base lintel to the throat and continues above throat without any drastic change. In addition, the shell's durability had to be increased. While a non-optimized shell may obtain crack damages already at moderate storm levels, which will cumulate during the cooling-tower's lifetime, an optimized shell will

642

R. Harte, U. Montag /First MIT Conference on Computational Fluid and Solid Mechanics 088.41 m

t^OO.OOr

10.30 m

Table 1 Material parameters of concrete

0.22 m

0.24 m

0.24 m

0.45 m

12.18 m

k

1.16m

^152^54 m

Fig. 1. Overall dimensions of cooling tower Niederaussem.

stay without cracks or at least poor of cracks for a longer period of time. Significant damages will not occur till storms of greater intensity [1]. The surrounding of the cooling tower is dominated by the boiler building of the new unit and the cooling towers and buildings of the existing power units. To consider interference effects most realistically, wind tunnel tests have been performed yielding non-axisymmetric wind distributions with different pressure intensities around the shell's circumference [2]. The soil characteristics deviate along the circumference. The columns are founded on a closed ring with partly deeper foundation on stiffer natural soil than the normal ring sections with soil replacement. Due to the hyperbolic character of the geometry these differences will influence the overall deformation and stress behavior of the shell [3].

Medium concrete compression strength /cm [N/mm^] Medium concrete tensile strength /ctm [N/mm^] Young's modulus [N/mm^]

B35

B85

SRB 85/35

35.0

85.0

82.03

2.675 34,000

4.773 43,000

2.880 40,400

have a diameter of 6.50 m, a total length of nearly 120 m, loading the shell with 2000 kN vertical and ± 4 0 0 kN horizontal. Further details are given in [4,5]. 2.3. Consideration of high-performance

concrete

Because of the inlet of the cleaned flue gases, which still contain residuals of sulfur, the inner surface of the shell will be attacked chemically by low concentrated acid vapor. To meet this, the shell's inner surface can be coated completely, but as each the outer and inner surface of the Niederaussem shell exceeds 60,000 m^, this procedure would be rather cost-intensive. In addition, the later repair of the coating would cause time-consuming maintenance works, thus resulting in expensive shut-downs of the complete unit. Thus a special acid-resistant high-performance concrete, named SRB 85/35, has been developed taking the ideal sieve-analysis curve by Fuller and Thompson [6]. The resulting compact packing of the aggregate combined with fly-ash and microsilica will protect the concrete against acid attack. Numerous mixtures have been tested at the Technical University Berlin and by the contractors HOCHTIEF and HEITKAMP [7]. A side effect of the new mixture was the increase of mechanical parameters, like Young's modulus, compression strength and tensile strength. Table 1 shows the difference of the special concrete SRB 85/35 to ordinary concrete B35 and to nominal values of high-performance concrete B85. These material specifications have to be considered within the structural proof and verification, especially in case of thermal and hygric action.

2.2. Flue-gas inflow in high position 3. Nonlinear verification Another interesting detail is the inflow of the cleaned flue-gas into the cooling tower at a height of nearly 50 m (Fig. 1). This is a speciality of German power plants, as — due to environmental requirements — the flue gas is cleaned from sulfur- and nitrogen-oxides. Since 1983 the cleaned flue-gas stream was injected into the thermodynamic uplift of the cooling tower via large pipes, supported on steel or concrete pipe bridges. In 1996 a new idea was added: the inflow at high position with support of the pipes on the shell's wall. In Niederaussem these GRP pipes

Besides the pure proof of sufficient safety against the ultimate limit state, the aspects of durability and lifetime increase strongly in importance. The accumulation of damages is a nonlinear process which cannot be covered by the usual static calculation techniques: nonlinear numerical simulation models are necessary. Taking account of the real load process as well as of realistic material models for reinforced concrete they reproduce the load-bearing behavior of the cooling tower more realistically.

R. Harte, U. Montag / First MIT Conference on Computational Fluid and Solid Mechanics

643

4

2 J 4 (8351

ii

T2.8 I

2,45 (SRB 85/35)

2,54 {B35)

2,45 (SRB 85/35)

•2.4

2.8 2.4

im

•2.0

^ S ^ ^ 174 (885)

fB8S)

2.0

• 1.6

g + i + X'^N V3[m]

\A

-1.6

g + X.W

•1.2 •0.8

-0.8

•0.4

0.4

0.0

-1.76 -1.60

-1.44

-1.28

-1,12

-0.96

-0.80

-0.64

-0.48

-0.32

-0.16

-1.2

0.00

0.16

V3[m]

^ *• ^

.^

,

,

,

-1,76 -1.60

,

r

-1.44 -1.28 -1.12

0.0

-0.96

-0.80

-0.64 -0.48

-0.32

-0.16

0.00

Fig. 2. Load-displacement curves {g = dead weight, t = thermal loads, w = wind load, k = amphfication factor).

B35

B85

Fig. 3. Crack patterns for X = 1.85. Comparison of normal concrete B35 and high-performance concrete B85.

Nonlinear solution techniques and material modeling still represent an actual research topic [8]. The previous calculations were achieved by means of the doubly-curved, stacked reinforced concrete shell element adapted from [9] with the nonlinear simulation models given in [10]. To minimize crack deterioration, the complete tower has to be modeled as a reinforced concrete structure considering • nonlinear stress-strain relations for concrete in compression, • tension cracking after exceeding the concrete tensile strength, • elasto-plastic stress-strain behavior of the reinforcement, • nonlinear bond between reinforcement and concrete. Details of the numerical modeling of these nonlinear components are given in [10,11]. Since all types of struc-

tural damage are nonlinear processes, the applied computer simulations are highly nonlinear. They follow the so-called multi-level simulation technique [12]. In case of the special concrete SRB 85/35, the material model has to be improved for the use of high-performance concrete. The Darwin/Pecknold-model [13] was extended in accordance with MC90 [14] and considering actual research results from TU Dresden [15]. Some results of the nonlinear computer simulations of the load combination g-\-Xxw (g = dead weight, w = wind load, A = amplification factor) are shown in Figs. 2 and 3. For comparison the cooling tower has been investigated for concrete B35, B85 and SRB 85/35. At the beginning of wind amplification the load-displacement path is linear, until the tensile strength is exceeded and initial cracks occur.

644

R. Harte, U. Montag /First MIT Conference on Computational Fluid and Solid Mechanics

The differences between the various concrete qualities are easily to recognize. The tower with B35 reaches the state of initial cracking early, continuing then with less stiffness, but finally reaching an ultimate load level of X = 2.54. Both SRB 85/35 and B85 show higher levels of initial cracking, but less ultimate safety against collapse. The higher cracking level naturally is caused by the higher tensile strength, freezing a higher energy potential within the uncracked structure. This energy is released abruptly in case of cracking, followed by drastic increase of deformation and decrease of load. The structure is not able to re-distribute the stresses to uncracked shell regions without local instabilities, before a stable equilibrium path is found again. This process leads to locally enhanced deteriorations and finally to a lower ultimate load level than for the ordinary concrete B35. A look at the crack evolution for both the B35- and B85-shells in Fig. 3 demonstrates this. The B35-shell exhibits a more regular crack distribution overall the shell than the B85-shell. This emphasizes that the stiffer shell with high-performance concrete provides less ductility and poorer capability to distribute stresses. The concrete mixture SRB 85/35 used in Niederaussem with less tensile strength shows a lower crack level with less deteriorating and finally an ultimate load level of A = 2.45. The cracks under serviceability loads g + t -\-\.0w {t = thermal loads) are definitely smaller than the admissible crack width of 0.2 mm. That means, the concrete mixture SRB 85/35 will not affect the safety and serviceability of the cooling tower considerably.

4. Conclusions The cooling tower shell at Niederaussem has been erected from autumn 1998 to spring 2000. It has withstood the heavy storm 'Lothar' at the end of 1999 without any damage, though the upper ring was not completed at that time. So the first real test for the shell's safety and reliability and through this for the design concept has run successfully. The effectiveness of the chemical resistance of the special concrete mixture has still to be proven under operation conditions next year.

Acknowledgements The presented research activities took place at the RuhrUniversitat Bochum and at the Bergische Universitat Wuppertal, partially supported by the Ministry of Science and

Research NW, the German Research Foundation, the European Union and the RWE Energie AG.

References [1] Harte R, Kratzig WB. Nonlinear analyses of reinforced concrete shells as preventive measure against damages. In: Astudillo R, Madrid AJ (Eds), Proc. lASS 40th Anniversary Congress, Vol. I. Madrid 1999, pp. A29-A40. [2] Busch D, Harte R, Niemann H-J. Study of a proposed 200-m-high natural draught cooling tower at power plant Frimmersdorf/Germany. J Eng Struct 1998;19:920-927. [3] Andres M, Harte R, Montag U. Computations for the innovative design and proof of a 200 m cooling tower. In: Topping BHV (Ed), Finite Elements: Techniques and Developments. Edinburgh: Civil Comp Press, 2000, pp. 367372. [4] Titze B, Harte R, Eckstein U, Schwickert M. Innovative flue gas injection into newly built cooling towers. VGB Kraftwerkstechnikl997;77(6):479-484. [5] Eckstein U, Harte R. Hochliegende Reingaseinleitung in Kuhlturmneubauten. Bau-technik 1996;73:485-491. [6] Busch D, Haselwander B, Hillemeier B, StrauB J. Innovative Betontechnologie fiir den Kiihlturmbau. Beton 1999;4:108-109. [7] Budnik J, Starkmann U. Der Naturzugkiihlturm NiederauBem. Beton 1999;10:548-533. [8] Harte R, Kratzig WB, Noh S-Y, Petryna YS. On progressive damage phenomena of structures. Comput Mech 2000;25:404-412. [9] Harte R, Eckstein U. Derivafion of geometrically nonlinear finite elements via tensor notafion. Int J Numer Methods Eng 1986;23:367-384. [10] Zahlten W. A contribufion to the physically and geometrically nonlinear computer analysis of general reinforced concrete shells. Technical Report 90-2. Ruhr-Universitat Bochum, 1990. [11] Kratzig WB, Mancevski D, Polling R. Modellierungsprinzipien von Beton. In: Meskouris K (Ed), BaustatikBaupraxis 7. Rotterdam: Balkema, 1999. [12] Kratzig WB. Mulfi-level modelling techniques for elastoplastic structural responses. In: Owen DRJ, Onate E, Hinton E (Eds), Computational Plasficity, Part 1. Barcelona: International Center for Numerical Methods in Engineering, 1997, pp. 457-468. [13] Darwin D, Pecknold DA. Nonlinear biaxial law for concrete. ASCE J Eng Mech Dev 1979;105:623-637. [14] CEB-FIP Model Code 1990. Bull Inf July 1991 ;203 and 204. [15] Hampel T, Curbach M. Behavior of high performance concrete under mulfiaxial loading. In: Proc. PCI/FHWA/FIB Int. Symp. on High Performance Concrete. Orlando, 2000.

645

Structural optimization in consideration of stochastic phenomena — a new wave in engineering D. Hartmann*, M. Baitsch, H. Weber Ruhr-University Bochum, Institute of Computational Engineering, Civil Engineering Department, D-44780 Bochum, Germany

Abstract In recent years, it has become obvious that deterministic optimization of structures no longer is appropriate. Increasing demands with respect to safety, reUabihty and quahty in structural engineering require the integration of probabilistic phenomena as well as the consideration of real-world stochastic processes induced in the structures. Within the present contribution two distinct research fields are discussed in which the embedding of stochastics and probability is absolutely mandatory. The first field is the reliability-based and lifetime-centered structural optimization, the second is the semi-probabilistic structural optimization regarding geometric imperfections inherent in structural systems. Keywords: Stochastic structural optimization; Structural rehability; Real-world modeling; Damage estimation

1. Introduction Safety, reliability and avoiding unnecessary risks are key issues in structural design of today and, therefore, in optimum design of structures as well. Failures, damages and accidents in buildings make us aware of the vulnerability of our structural systems. In particular, structural optimization often leads to vulnerable design solutions because, according to the attempts of an optimization logic to make the most of the constraints imposed, extremely sensitive structures may be created. In these cases, minor discrepancies of the applied structural mechanics model from the real-world behavior can easily evoke disastrous consequences. Due to these findings, the fundamental approach in structural optimization importantly has changed within the last years. A new wave of research in structural optimization has arisen that takes into account the 'full' reality of static and dynamic loadings and the 'true' nature of structures. It is obvious that, in this context, probabilistic phenomena and effects, inherent in many engineering problems, play a dominant role. As a consequence, conventional deterministic structural optimization alone is no longer acceptable and, therefore, has to be replaced by stochastic structural optimization models, or at least by models con* Corresponding author. Tel.: +49 (234) 322-3047; Fax: +49 (234) 321-4292; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

sidering probabilistic behavior of structures and their surroundings in an appropriate fashion. In the present paper two distinct problem fields are looked at to demonstrate the effectiveness of this new wave in structural optimization. First, the reliability-based and lifetime-centered optimum design of steel structures is introduced. Second, the imperfection-oriented structural optimization of arched structures are dealt with.

2. Lifetime-centered structural optimization Structural optimization with respect to the lifespan of a structure is a challenging task. The chief difficulty hes in the multi-level and multi-scale character of the optimization model that has to be built. Three levels determine the total optimization model (see Fig. 1): • the time-variant structural mechanics model in the kernel of the total system, allowing the estimation of potential damages along the lifetime; • the optimization level comprising the governing components of the optimization model (design variables, objective criterion and a set of constraints derived from the structural mechanics model); • the navigation of the structural optimization problem along with the structural model, based on an optimiza-

646

D. Hartmann et al /First MIT Conference on Computational Fluid and Solid Mechanics smeared constraint function. By way of example, Fig. 2 demonstrates the situation in the case of the optimization of a 2D-steel frame subject to a stochastic wind load process. Using only two optimization variables, the height x\ and the width xi of the cross-selection, and minimizing the weight with respect to the reliability constraint mentioned above and to side constraints for x\ and xi, three regions can be detected: (i) a definite failure area, (ii) a definite survival area, and (iii) a scattered failure/survival area in which the optimal solution is located.

Fig. 1. Multi-level problem structure. tion server, providing various powerful optimization strategies. The stochastic character of the optimization problem results from the loadings (e.g. wind, crane loads, sea waves, traffic loads, etc.) which — due to the demand of being as realistic as possible — are modeled as stochastic intermittent pulse processes within the time scale of several minutes. According to the governing differential equations of motion Mx(r) -f Dx(r) + Kx(r) = f(r)

(1)

where M, D and K are the well known system matrices of the structural system and x(r), x(r) and x(r) are describing the system kinematics, the loads that induce dynamic structural response in the system in a micro-time domain. By that, the structural response, in particular, the probability density functions of the displacements x(r) and the stress amplitudes, computed from the displacements, can be determined. Without going into details, it should be mentioned that the stochastic information needed is provided by a so-called covariance analysis making use of a form filter formulation and a singular value decomposition (see [1,2]). Once the density functions are known the relevant damages in the micro-time domain are accumulated based on a stochastic Palmgren-Miner-rule. Subsequently, the damages computed on the micro-time scale are transferred to the macro-time scale (measured in years) to form the stochastic structural mechanics constraints in the optimization model. Currently, primarily a single probabilistic reliability constraint is applied which leads to a randomly

width Xj [mm]

3. Imperfection-oriented structural optimization Imperfections in a structure can emerge due to a wide variety of causes. Focusing here solely on geometric imperfections, as a result of the fabrication or erection of a structural system, the concept of random fields [3] provides ways and means to describe the stochastic nature of geometric imperfections. In the case of structural optimization, however, computational efficiency is of extreme significance. Thus, the random field concept only serves as a starting point from which a numerically less expensive stochastic model can be deduced which, however, is sufficiently sophisticated in capturing the stochastic nature of the problem considered. Geometric imperfections of a 2D-curved girder, here serving as a test case, can be expressed by the elementary vector equation where the vectors Xjmp and Xper define the locations of the imperfect and perfect geometry, respectively. The vector function h(Xper) describes the offset of the perfect structure representing a random mapping function by virtue of the coincidences during construction. Based on the theory of random fields a spectral decomposition of the function h can be developed as follows h(Xper) = ^ } ^ / N , ( X p e r )

(3)

/ where y is a random vector of stochastically independent components j , and N, (Xper) are base functions. Hence, the

failure

height x^ [mm] 600 Probabilistic reliability constraint

700

800

900

1000

optimal area

Fig. 2. Optimization of a steel frame with respect to reliability.

D. Hartmann et al. /First MIT Conference on Computational Fluid and Solid Mechanics . I I 1 I I 1 1 i J 1 i 1^1 1 I

^

F = 20kN

loads

L_

system layout (initial solution)

647

width

1 = 84.0 m

elasticity mod ulus

E = 2.M0^N/mm^

density

p = 7850 kg/m^

max .imperfection

H<10cm

shape of imperfection

objective function

f-X||u.||

constra int s

G< 8400 kg h<8.05m b<2.10m

geometry

Fig. 3. Girder to be optimized. vector Ximp represents a random field. In harmony with this field, the geometry of the structure becomes random as well, resulting in a random structural response in terms of displacements u. Hence, we have the stochastic response u = u(y).

(4)

To streamline the numerical effort in the optimization model the random field problem is reduced to a semistochastic problem using the following approach: provided that imperfections are not excessively large an envelope is computed containing P* percentage (e.g. P* = 95%) of all potential shapes of imperfection. This can be achieved by choosing suitable base functions for the vector h(Xper). To secure the P*-quantile, the postulation P( y < £) = P*

(5)

has to be met such that P, the probability that the norm y being not greater than the quantity e, equals the given value P*. The quantity s determined by (6)

y <s

can thereby be interpreted as the envelope of the expected imperfection shapes.

Based on this idea, a specific two-level optimization problem is formulated which takes the following form min max {f(x, u(x, y)) | x e §, y < e} where in the feasible domain S is defined as \=\xe

I xi < X <Xu,g(x,u(x,y)) < 0 }

^ j L ; u u u u u k ^ ^

optimal girder 0.4

0.6

0.8

1.0

load-displacement-relation -0.1 o| -0.1 -42

^ 0

42

governing shape of imperfection

(8)

containing side constraints and the vector of constraints g. At the first level, the most disadvantageous shape of imperfection y (encompassed by the envelope) is computed for each current vector x. Based upon this envelope, the structural optimization is carried out at the second level. The applicability of this new approach is demonstrated for the optimization of a plane curved girder consisting of 129 truss bars. Applying non-uniform rational B-splines for the Ni-functions the structural optimization for 9 shape and 3 sizing variables is performed. The total stiffness (expressed as the sum over the nodal displacements) is maximized subjected to prescribed bounds to the girder's weight, height and width (see Figs. 3 and 4).

collapse load factor

0.0 0.2

(7)

structural response (>,=1.0

Fig. 4. Optimal girder.

displacement

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D. Hartmann et al. /First MIT Conference on Computational Fluid and Solid Mechanics

4. Conclusions

References

Taking into account the realistic, i.e. the stochastic, behavior of structural systems in structural optimization models ensures the reliability as well as the general acceptance of the optimum design, substantially. In particular, by the aid of two distinct application examples it has been verified that the results of a consequent embedding of stochastic phenomena in structural optimization leads to a new quality of results that demonstrably, and to a large extend, differs from conventional deterministic structural optimization.

[1] Weber H, Hartmann D, Faber O, Niemann H-J. Process analysis and reliability estimation for structural optimization. In: Kareem A. et al (Eds), Proceedings of the 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability. URL: http://www.nd.edu/'-pmc2000/ pmc2000/session/abstract/a081 .pdf [2] Faber O, Hartmann D, Niemann H-J, Weber H. Analysis of loading and damages processes and reliability estimation for lifespan-oriented structural optimization. In: Proc ESREL'99 — 10th Eur Conf Safety and Reliability, Munich-Garching, Germany, 13-17 Sept 1999, Vol 1. Rotterdam: Balkema, pp. 539-544. [3] Vanmarcke, E. Random Fields: Analysis and Structure. Boston, MA: The MIT Press, 1983.

649

NHTSA's supporting role in the partnership for a new generation of vehicles W.T. Hollowell *, S.M. Summers National Highway Traffic Safety Administration, US Department of Transportation, Washington, DC 20590, USA

Abstract On September 29, 1993, President Clinton, Vice President Gore, and the Chief Executive Officers of Chrysler, Ford, and General Motors announced the formation of a historic, new partnership aimed at strengthening U.S. competitiveness while protecting the environment by developing technologies for a new generation of vehicles. Tabbed the 'Partnership for a New Generation of Vehicles' (PNGV), the program's long-term objectives include developing a range of technologies to yield automobiles with a threefold improvement in fuel efficiency and reduced emissions. This is to be achieved without compromising other features, such as performance, safety, and utility. This also requires developing and introducing manufacturing technologies and practices that will reduce the time and cost associated with designing and mass producing this new vehicle. Within the Department of Transportation, NHTSA is the focal point for the PNGV program support. The agency's role is to ensure that the PNGV developed vehicles will meet existing and anticipated safety standards and that the overall crash and other safety attributes are not compromised by their light weight and the use of new advanced materials in production of the vehicles. This paper is written to outline the activities that NHTSA has initiated in support of its role in the program. Keywords: PNGV; Fleet evaluation; Finite element modeling; Computer modeling; Systems model; Crashworthiness; Optimization

1. Introduction On September 29, 1993, President Clinton, Vice President Gore, and the Chief Executive Officers of Chrysler, Ford, and General Motors announced the formation of a historic, new partnership aimed at strengthening US competitiveness while protecting the environment by developing technologies for a new generation of vehicles. Tabbed the 'Partnership for a New Generation of Vehicles' (PNGV), the program's long-term objectives include developing a range of technologies to yield automobiles with a threefold improvement in fuel efficiency and reduced emissions. This is to be achieved without compromising other features, such as performance, safety, and utility. This also requires developing and introducing manufacturing technologies and practices that will reduce the time and cost associated with designing and mass producing this new vehicle [1]. * Corresponding author. Tel.: +1 (202) 366-4726; Fax: -1 (202) 366-5930; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

To address the aforementioned objectives, the Federal Government and the United States Council for Automotive Research (USCAR), which represents Chrysler, Ford, and General Motors, have initiated activities to address the following three interrelated goals: Goal 1: Significantly improve national competitiveness in manufacturing. Goal 2: Implement commercially viable innovation from ongoing research on conventional vehicles. Goal 3: Develop a vehicle to achieve up to three times the fuel efficiency of today's comparable vehicle (i.e. the 1994 Chrysler Concorde, Ford Taurus, and Chevrolet Lumina). In developing a vehicle which achieves up to three times the fuel efficiency of today's comparable vehicles, the PNGV partners have determined a number of specific assumptions/requirements toward this venture. Of particular interest to the NHTSA is a requirement that the vehicles meet present and future Federal motor vehicle safety stan-

650

W.T. Hollowell, S.M. Summers/First MIT Conference on Computational Fluid and Solid Mechanics

dards, while also meeting equivalent in-use safety performance. Within the Department of Transportation, NHTSA is the focal point for the PNGV program support. Toward this, the agency's role is to ensure that the PNGV developed vehicles will meet existing and anticipated safety standards and that the overall crash and other safety attributes are not compromised by their light weight and the use of new advanced materials in production of the vehicles. The most recent projections indicate that a 40% reduction of the vehicle mass will be required to meet the fuel economy requirements of the PNGV program. This reduction, coupled with the potential use of materials other than the conventional steels used in automobile construction today and with possibly entirely unique power trains, requires that careful attention be given in determining the overall crash safety of the vehicles. Beyond the testing required by the Federal motor vehicle safety standards, the safety analysis must include evaluating the performance of the vehicles in crash modes that are representative of the real world crash environment. When considering the PNGV vehicles, interactions with the existing fleet, the mass reduction requires extra attention be given to crash energy absorption characteristics of the vehicle structure and to the performance of the occupant restraint systems. Furthermore, the potential of developing vehicles with mass distributions that vary significantly from today's vehicles may require careful scrutiny regarding how these vehicles will behave in their interactions with roadside safety hardware such as guard rails, breakaway luminaire supports, etc.

2. NHTSA research activities Toward meeting the aforementioned stated objectives, NHTSA has initiated efforts to develop advanced computer models and develop methods and techniques for evaluating the crashworthiness characteristics of alternate vehicle designs and any new lightweight materials. A large scale systems model is being developed to evaluate vehicle crashworthiness based on the safety performance of the vehicle when exposed to the entire traffic crash environment, i.e. across the full spectrum of expected collision partners, collision speeds, occupant heights, occupant ages, and occupant injury tolerance levels. The means of evaluating vehicle crash performance on a system-wide basis was first accomplished by the Safety Systems Optimization Model developed by Ford Motor Company and later enhanced by the University of Virginia [3,4]. Starting with SSOM as a foundation, the Vehicle Research Optimization Model (VROOM) computer model, as described below, takes full advantage of recent dramatic improvements in vehicle and occupant models, newly developed injury criteria, and a comprehensive projection of the crash environ-

Table 1 Vehicles selected for finite element models to represent the fleet Category

Vehicle

Subcompact car Compact car Midsize car Fullsize car Sport ufility vehicle Minivan Full size van Small pickup Full size pickup

1997 Geo Metro 1996 Chrysler Neon 1997 Honda Accord 1997 Ford Crown Victoria 1997 Ford Explorer 1998 Dodge Caravan 1998 Ford Econoline 1998 Chevrolet S10 1996 K2500 Pickup

ment for the years 2000-2005. Where possible, VROOM also will explore the feasibility of implementing promising algorithms from the Volkswagen ROSI system-wide optimization model [5]. Detailed finite element models are being developed for each of the PNGV baseline vehicles and for vehicles representing the fleet (see Table 1). This activity involves the tear down of the PNGV baseline vehicles and selected fleet vehicles for scanning the vehicles to develop geometric data to be used in prescribing the finite element mesh, and for measuring the inertial and other physical properties of the vehicles (see Fig. 1). Crash testing is being conducted to validate the models as well as provide for audits of simulations undertaken in support of the fleet analysis. Design concepts will be explored and evaluated for the various power trains under consideration for the PNGV vehicles. This includes exploring the use of advanced structural materials such as composites and aluminum. It is anticipated that research into improved material models will be required in the computer software to accommodate these studies. Finally, a system model is being developed for identifying optimal characteristics for the PNGV vehicles. The new vehicle FE models will be utilized in two ways. First, the models will be used to study specific crash configurations, with specific collision partners, and specific impact speeds (see Fig. 2). However, while FE models are potentially very accurate and geometrically fidelic, FE

Fig. 1. Vehicle tear down and measurement.

W.T. Hollowell, S.M. Summers/First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 2. Finite element crash simulation.

Fig. 3. MADYMO crash simulation. models are prohibitively expensive to execute for global design evaluation. A typical VROOM run requires over 10,000 simulations. The second application for the FE models will be to generate sophisticated, yet simpler and faster running, articulated mass models for the systems evaluation. Evaluations using the articulated mass models will provide broad design directions (e.g., double the aft frame stiffness) for improved crashworthiness (see Fig. 3). After optimization, these results can be used to design modified vehicle components and corresponding FE models for an optimized structure. The approach to be used in the system model is similar to that found in [2]. In particular, the approach to crashworthiness optimization may be stated formally as the following non-linear problem: Minimize Inj(x, u) = Y^PiSi{,x_, u) subject to W g t ( x ) < W g t m a x C 0 S t ( x , W{x))

(1)

< CoStmax

< X — < -^max X,

where x = vector of design variables; u = belt usage rate; Inj(x,w) = total injuries; Wgt(x) = incremental weight associated with design 'x'; Cost = incremental cost associated with x and Wgt(x); Wgl^ax = upper constraint on incremental weight; Costmax = upper constraint on incremental cost; pi = probability of event /; and st = injuries resulting from occurrence of event /. The objective expressed in Eq. (1) is to determine that vector of design variables which minimizes total injuries or

651

some measure of societal cost of total injuries [6]. The simulations will attempt to minimize normalized harm, defined as total harm in dollars normalized by the harm associated with an AIS 6 injury level. Total harm is computed by summing the harm incurred in each of crash encounters / weighted by pt, the annual expected probability of event /. The incremental weight penalty associated with any proposed design modifications w(x) is limited to the upper constraint Wgt^ax. Similarly, the incremental cost of the proposed design modifications is limited to an upper constraint of Costmax- The incremental cost in this context includes both the additional cost of design modifications and an estimate of the cost of material substitution to reduce weight. To ensure that design modifications lie within realistic ranges, the design variable vector is constrained by lower and upper limits on each design modification. The annual expected probability of a crash event /, sometimes referred to in the literature as exposure, is computed based on historical real world crash data. For the model, a crash event / is completely characterized by prescribing the crash speed, the impacting vehicle weight, the occupant seating location, the occupant height, the occupant gender, and the occupant restraint type.

References [1] Partnership for a New Generation of Vehicles Program Plan. United States Department of Commerce, Washington DC, 1994. [2] Gabler HC, Hollowell WT, Hitchcock RJ. Systems Optimization of Vehicle Crashworthiness. Fourteenth International Technical Conference on Enhanced Safety of Vehicles, Munich, Germany, May 1994. [3] Ford Motor Company, Safety Systems Optimization Model, Final Report, US Department of Transportation, Contract No. DOT HS-6-01446, November 1978. [4] White KP, Gabler HC, Pilkey WD, Hollowell WT. Simulation Optimization of the Crashworthiness of a Passenger Vehicle in Frontal Collisions Using Response Surface Methodology, SAE Paper No. 850512, March 1985. [5] Zobel R. Economically Justified Passenger Protection Results of Simulation. SAE Paper No. 850516, 1985. [6] The Economic Cost of Motor Vehicle Crashes. National Highway Traffic Safety Administration, NHTSA Report No. DOT HS-807-876, September 1992.

652

Towards a CAD design of cable-membrane structures on parallel platforms p. Ivanyi *, B.H.V. Topping, J. Muylle Heriot-Watt University, Edinburgh, Riccarton EH14 4AS, UK

Abstract The use of lightweight surface structures constructed in a free form from cables and membranes is now widely accepted. There is an increasing tendency to push both design concepts and material capabilities to their Hmits with the goal of covering increasingly greater areas. To achieve these ever increasingly complex designs parallel form-finding tools may be used. Unfortunately, the introduction of parallel computing into the design process requires extra tools and considerations on the part of the design engineer. The key to parallel form-finding is not only to develop accurate and fast algorithms, but to provide software which is easy to use. The ultimate goal is an integrated CAD system with an easy to use graphical interface. This CAD system would require several components, such as pre-processing, form-finding and/or analysis and post-processing tools. These tools should be modified and developed further for a parallel platform. In this paper, a general overview of the already developed tools is given which may be incorporated in a parallel form-finding CAD package. Keywords: Cable; Membrane; CAD; Parallel

1. Introduction

2. Mesh generation

In the design of cable-membrane structures the calculation of the geometry or form-finding of the structure is an essential part. One of the most widely used computer methods for form-finding is based on dynamic relaxation [1,2]. From the engineering point of view one of the main advantages of dynamic relaxation as a form-finding tool is that it simulates a physical process and the pseudo-dynamic trace of the deformed state helps to investigate the structural system in a similar way as model studies. Further more, because of its explicit formulation, using the natural stiffness matrix formulation [3], it can be easily parallelised [4,5] which helps not only to solve larger problems, but to provide form-finding results faster. The parallelised dynamic relaxation program may require additional pre- and post-processing tools.

Although the dynamic relaxation method does not require 'special' meshes the reason to use structured meshes is to assist the generation of cutting patterns, since it is easy to insert geodesic strings [6] into a regular patterned mesh. The input for the mesh generator is a geometric model. The construction of the initial geometric model follows some patterns [7] which have been identified in the design process. The available elements for geometric modelling are ID straight lines (cable, truss elements), 2D triangular or quadrilateral blocks (membrane surfaces). The final, structured finite element mesh is generated from the geometric model.

* Corresponding author. Tel: +44 (131) 449-5111, ext. 471^ Fax: +44(131)451-3593; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

3. Partitioning Even though structured, highly regular meshes are used in the form-finding and their partitioning may seem to be a trivial task for a CAD system, an easy to use module should be provided to carry out this task. It has been shown, that one of the best approaches is to offer a 'multiple choice solution' [8] for the engineer to decide which partitioning

p. Ivdnyi et al. /First MIT Conference on Computational Fluid and Solid Mechanics method is best suited for the particular problem. In the case of cable-membrane structures the difficulty is, that the meshes contain ID and 2D elements in the same finite element mesh. It has been shown, that the current methods of graph representation are not always applicable to cable-membrane structures which consist of both ID and 2D elements. A new method for generating graphs for the representation of these structures has been proposed [9]. The new graph representation is based on the usual, dual graph of finite element meshes. The adjacency of those finite elements which are left unconnected in the dual graph is determined using geometric information (distance in Euclidean space) of the mesh. The generated graph is called a 'bubble' graph. This graph partitioned with state-of-the-art graph partitioning tools [10-12] provides very good subdomains for mixed meshes.

4. Heterogeneous network of computers The availability of networks of workstations and personal computers is likely to increase over the next few years. Although these networks are generally provided for other reasons, engineers will increasingly seek to utilise this computing resource for distributed finite element analysis. The partitioning of finite element meshes for distributed analysis has to be able to account for the heterogeneous nature of these networks where each processor may be of differing capability. A CAD system for a parallel platform would require a module to provide the partitioning not only for supercomputers or homogeneous network of computers, but also for a heterogeneous network. The developed tool is based on the diffusion method which is combined with a modified Kemighan-Lin algorithm. The algorithm considers constant power of a computer and do not carry out dynamic load-balancing when the power of a computer has changed during calculation. The generated non-uniform partitions are comparable to partitions provided by modified graph partitioning tools.

5. Coupling of form-finding and cutting pattern generation Usually the form-finding stage results in a non-developable surface which means that the structure cannot be projected into plane impHcitly. Therefore, to determine the stress free side lengths of the membrane elements in a plane the structure should be 'unassembled' into pieces and then these pieces can be unstressed in the plane. To reduce the wastage of material the most often used form is a strip of cloth with edges as straight as possible. With the introduction of geodesic strings [6] this condition can

,653

be controlled more easily. The strips will be bounded by either the structural boundary itself or by a one dimensional element such as a cable or a geodesic string. Moncrieff and Topping [13] described a cutting pattern generation method, called flattening, which enables the use of strips consisting any kind of idealisation. The procedure is based on the dynamic relaxation method. A separated strip of cloth is projected into the plane, minimal boundary conditions are applied to satisfy the statically determinant condition then with the dynamic relaxation method the strip 'structure' is spread into an unstressed, equilibrium state in the plane providing the cutting pattern. In the case of a sequential program each strip is flattened one after another in sequence. This stage can be parallelised, since the cutting pattern generation for each strip of cloth can be viewed as an independent task. Moreover form-finding and cutting pattern generation together provide the detailed geometric design information therefore coupling the two problems within one program is a natural approach. In the case of parallel execution, further tasks such as preprocessing and partitioning must be undertaken. One solution to the problem is to use an initial partitioning for form-finding calculation and then do a repartitioning for the cutting-pattern generation. All decisions about the repartitioning are made using a software tool incorporating a genetic algorithm. Using this technique the required data movement can be considerably reduced compared to a processor farming approach for the cutting pattern generation. On the other hand, it can be noticed that by using repartitioning the mesh is processed twice. There is the initial partitioning for form-finding and before cutting pattern generation another partitioning is required to distribute only complete strips of cloth to a process, but the coupling of the two partitionings (initial and re-balancing) is also possible. In this case, the partitioning will not operate on an element level, but will consider only whole strips ensuring approximately equal load and minimum communication between the subdomains, but more importantly the boundaries of subdomains will coincide with some of the boundaries of strips at the same time. Experiments with state-of-the-artgraph partitioning tools [10-12] has shown that they are unable to handle small weighted graphs which are generated for the strips of a membrane structure. A genetic algorithm module has therefore been developed to provide the strip based partitioning. The tool uses the same technique as ESGM [14]. The method uses bisection, where the separation of partitions is determined by a straight line which is controlled by two variables, an initial position and an angle.

References [1] Day AS. An introduction to dynamic relaxation. The Engineer, 1965.

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P. Ivdnyi et al /First MIT Conference on Computational Fluid and Solid Mechanics

[2] Bames MR. Form-finding and analysis of tension space structures by dynamic relaxation. PhD thesis, City University, 1977. [3] Argyris JH. Recent Advances in Matrix Methods of Structural Analysis. Progress in Aeronautical Sciences, Vol. 4. Pergamon Press, London, 1964. [4] Topping BHV, Khan AI. Parallel Finite Element Computations. Saxe-Coburg Publications, Edinburgh, 1996. [5] Ivanyi P, Topping BHV. Parallel and distributed dynamic relaxation form-finding. In B.H.V. Topping and M.B. Leeming (Editors), Innovation in Computer Methods for Civil and Structural Engineering, Civil-Comp Press, Edinburgh, 1997, pp. 157-165. [6] Wakefield D. TENSYL' membrane analysis, implementation of geodesic strings. Technical report, Buro Happold, 1982. [7] Hudson P, Topping BHV. The design and construction of reinforced concrete 'tents' in the middle east. Struct Eng, November 1991;69(22):379-386. [8] Farhat C, Lanteri S, Simon HD. TOP/DOMDEC — a software tool for mesh partitioning and parallel processing. Comput Syst Eng 1995;6(l):13-26. [9] Ivanyi P, Topping BHV. Partitioning cable-membrane structures, submitted for publication.

[10] Hendrickson B, Leland R. The Chaco user's guide, version 2.0. Technical Report SAND95-2344, Sandia National Laboratories, Albuquerque, NM 87185-1110, July, 1995. [11] Walshaw C, Cross M, Everett M. Mesh partitioning and load-balancing for distributed memory parallel systems. In B.H.V. Topping (Ed.), Advances in Computational Mechanics with Parallel and Distributed Processing. Civil-Comp Press, Edinburgh, 1997, pp. 97-103. [12] Karypis G, Kumar V. METIS, a software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. Technical report. University of Minnesota, Department of Computer Science/Army HPC Research Centre, Minneapolis, MN 55455, November 1997. [13] Moncrieff E, Topping BHV. Computer methods for the generation of membrane cutting patterns. Comput Struct 1990;37(4):441-450. [14] Sziveri J, Scale CF, Topping BHV An enhanced parallel sub-domain generation method for mesh partitioning in parallel finite element analysis. Int J Num Methods Eng 2000;47:1773-1800.

655

The effect of hydrodynamic loading on the structural reliability of culvert valves in lock systems Randy J. James ^'*, Liping Zhang ^, David M. Schaaf ^, Gregory A. Wemcke^ "" ANATECH Corp., 5435 Oberlin Drive, San Diego, CA 92121, USA ^ U.S. Army Corps of Engineers, Louisville, KY 40201, USA

Abstract The hydrodynamic loads imposed on a culvert valve used in navigational locks on the Ohio River are evaluated using computational fluid dynamics. These loads, as a factor on the hydrostatic head across the valve, are needed as part of a structural reliability evaluation for fatigue cracking at welded connections. The modeling method is first benchmarked with test measurements taken at the Mc Nary Lock on the Columbia River in 1957 to diagnose a 'heavy pounding noise' that occurred as the valves were opening when that lock was put into operation. The method is then applied to the culvert valve design under investigation to determine the hydrodynamic loading effect for the structural reliability study. Keywords: Computational fluid dynamics; Hydrodynamic loads; Structural rehability; Culvert valves; Turbulence

1. Introduction A series of navigational locks and dams on the Ohio River provide the ability to efficiently transport large tonnages of materials and goods over long distances. A critical component of the locking operation is the culvert valves, which are opened and closed to fill and empty the lock chambers. These valves are large welded steel structures, typically constructed using skin plates on curved vertical ribs, which are connected by large horizontal load girders, as illustrated in Fig. 1. The valves pivot about a trunnion beam embedded into the lock wall with large strut arms supporting the gate. These valves must be opened under large head differentials with the water impinging on the concave side of the gate. For design purposes, a factor of 2 is typically used on the operating head differential to account for the hydrodynamic loads during the opening of the valve. This practice has been proven reliable by the many years of safe operation of the valves in the lock systems. However, many gates now have 40+ years in service, and some have failed recently due to cracking at welded connections, as illustrated in Fig. 1. As part of an overall reliability study of the navigational system * Corresponding author. Tel.: +1 (858) 455-6350, ext. 104; Fax: + 1 (858) 455-1094; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

on the Ohio River undertaken by the Corps of Engineers, a structural reliability evaluation for fatigue cracking at welded connections in the culvert valves was conducted. For this probabilistic based reliability evaluation, best estimates of loads are needed for establishing fatigue life and potential fatigue crack propagation rates, which could lead to failure of the valve. Thus, fluid flow modeling is used to evaluate the hydrodynamic loads imposed on the valve during a load cycle. The structural shape of the valve face is modeled as a rigid boundary in the fluid to monitor the pressure differential along the valve as the valve opens. The ADINA-F fluid flow program [1] is used in this investigation.

2. Modeling Culverts, typically with a rectangular cross-section, are embedded in the concrete walls of the lock, and two valves are used along each culvert to fill and empty the lock chambers. With the upper valve open, water is collected through inlet ports from the upper pool and discharged into the lock chamber. The lower valve is opened to discharge the water from the chamber to outlet ports in the lower pool to empty the chamber. The valves are opened under a head differential corresponding to the full lift height of

656

R.J. James et al. /First MIT Conference on Computational Fluid and Solid Mechanics

Fig. 1. Reverse tainter culvert valve and closeup of failure at welded connections. the lock, and are closed under 'equalized' pool conditions. The valves are referred to as reverse tainter gate type valves since the water impinges on the concave side of the valve. The valves are mounted in the culvert at a well or opening that extends from the top of the lock wall to the

culvert. This well provides access for repair or maintenance on the valve, and also provides a surge chamber for the water as the valves are closed. Upstream and downstream of each valve are slots in the culverts that also extend upward to the top of the lock wall. These slots are used

RJ. James et al /First MIT Conference

on Computational

2000

73.47 Ft/sec

1500

657

iiiiiiii lipliiiiiliiiftiliiliili

iili

gf 1000

55.00

I

| - 45.00

'^

ii

Mechanics

Benchmark for Culvert Discharge

VELOCITY TIME 20.00

t

Fluid and Solid

65.00

500

I h 35.00 | H 25.00 j j h 15.00

Surge Chamber

5.00

8

10

Fluid Pressure: — Measured 234' of Elevation Calculated 232' of Elevation

Valve Body

Rectangular Culvert

4 6 Time (sec)

\ El. 240'

Fluid Flow Model for Mc Nary Lock Culvert Valve Opened 3' Under 88' Head

Air Sucked in from Bulkhead Slot Caused Pounding Noise in Valve

Fig. 2. Benchmark of model against test data for Mc Nary Lock culvert valve.

to place bulkheads in the culverts to dewater the valves for inspection or maintenance. Fig. 2 illustrates the geometry and modeling of the culvert and culvert valve for the Mc Nary Lock, which is located on the Columbia River about 290 miles above the mouth, and is designed for a 92' maximum lift height. When this lock was first put into service, heavy pounding noises resembling thunder or cannon shots could be heard as the valves were opened. While opening the air vents that were installed at each valve eliminated the noise, a testing program was undertaken to determine the cause and effects of the pounding noise and resulting vibrations

in the valve and operating equipment and as an aid in future culvert valve designs. These test data on the Mc Nary Lock taken in 1957 [2] were used as a means of benchmarking the fluid flow models for use on this project. A 2-dimensional model of fluid elements is used with rigid boundaries for the shape of the culvert and surge chamber. The valve face is modeled as a rigid boundary within the fluid at various opening positions. A pressure distribution corresponding to the upper pool head is applied along the cut on the upstream boundary, and likewise a pressure distribution for the lower pool is applied on the downstream cut. A uniform pressure is applied across

658

R.J. James et al. /First MIT Conference on Computational Fluid and Solid Mechanics Velocity (ft/sec)

t

42.38 39.00 33.00

- 27.00 -21.00 - 15.00 -

9.00

lil

3-- 3.00

1 Pressure Difference Along Valve Face

Pressure (ft) Points Along Length

t• 1 ^' ° 45_Q

at Tip of Valve

5

10 15 Time (seconds)

-

36.0

L r

^^-^ 18.0

r

9.0

P-

0.0

20

Fig. 3. Fluid flow simulation for culvert valve under 30' head. the cut boundary of the surge chamber. The analysis is initiated with zero velocities. A stick boundary condition, which makes the tangential velocity zero at the boundary, is used along the culvert and surge chamber surfaces. However, during opening of the valve, the flow will be turbulent around the valve, and this turbulence factor must be calibrated or benchmarked for this application. Some of the turbulent flow models were evaluated, and some velocity slips along the valve and the culvert surfaces near the valve were also considered. These types of fluid flow modeling parameters were confirmed with the test data for fluid pressures and discharge rates from the Mc Nary Lock, as shown in Fig. 2. In the end, laminar flow conditions with stick type boundary conditions proved to be a good

simulation of the flow in the culvert. In addition, the flow characteristics downstream of the valve were in good qualitative agreement with the conclusions of the testing at the Mc Nary Lock. The conclusion from that study was that "the most probable cause of the pounding at the valve was the sudden relief of low pressures just downstream from the partially open valves by surges of air from the downstream bulkhead slot." For partially open conditions, the calculations showed negative pressures downstream of the valve and flow velocities moving toward the valve from the location of the bulkhead slot. This would confirm that air is drawn in through the downstream bulkhead slot, and that the air bubbles will travel toward the valve and burst.

RJ. James et al. /First MIT Conference on Computational Fluid and Solid Mechanics 3. Results The modeling philosophy and parameters used in the benchmark tests were then used to model the geometry and flow characteristics of the culvert valves for the lock systems along the Ohio River. Fig. 3 illustrates the fluid flow modeling and results. This lock system uses a slightly larger culvert size and has an operating head of 30' across the culvert valves. A valve is evaluated for several opening distances by positioning the valve surface within the fluid domain and initiating a transient analysis with zero initial velocities. While the valve is actually opened in a continuous process, the analysis considers the hydrodynamic loads that develop for a series of fixed opening distances. The calculated pressure differences across the valve are plotted in time for points along the valve. As shown in Fig. 3, the hydrodynamic factor on the head differential is found to be 1.3 for this culvert valve with a 30' operating head. This hydrodynamic factor is also seen to be relatively constant over the length of the valve face. For the probabilistic reliability study, this hydrodynamic factor is further characterized for variations in operating conditions, such as head differential.

4. Summary and conclusions Computational fluid dynamics is used to evaluate the hydrodynamic loads that develop on the structural compo-

659

nents of a culvert valve during opening of the valve under pressure heads. The loads are needed to determine the peak stress range in the load cycle of the valve for reliability evaluation of fatigue cracking at welded connections. The fluid modeling is first benchmarked with the results of test data that were taken at a lock site to diagnose heavy pounding noises at the culvert valve during opening. The modeling methods were then applied to the culvert valve geometry and operating conditions under investigation for fatigue cracking reliability. A factor of 1.3 over the head differential was identified for the hydrodynamic loads during opening of the valve.

References [1] ADINA User Interface, Volume III: ADINA-F, Report ADR 96-4, ADINA R&D, Inc., Watertown, MA, October 1996. [2] Hydraulic Prototype Tests of Tainter Valve, Mc Nary Lock, Columbia River, Washington. Technical Report 2-552, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, June 1960.

660

An efficiency of numerical algorithms for discrete reliability-based structural optimization K. Kolanek*, R. Stocki, S. Jendo, M. Kleiber Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Swietokrzyska 21 00-049 Warsaw, Poland

Abstract The paper presents two optimization techniques to approach the mixed (discrete-continuous) reliability-based optimization (RBO) problem, the enhanced controlled enumeration method and the new nonlinear mixed programming method. The mixed RBO problem is formulated as the minimization of the cost function subjected to the constraints imposed on the values of system and componential reliability indices. The transformed continuous RBO problem for mixed progranmiing method is formulated. For the controlled enumeration method, the improved algorithm for the cost optimization of truss structures is outlined. Keywords: Structural reliability; System reliability; Discrete optimization; Truss structure; Reliability-based optimization

1. Introduction The reliability-based optimization (RBO) problem is often formulated as a minimization of the initial structural cost under the constraints imposed on the values of structural and elemental reliability indices corresponding to various limit states, see e.g. [1]. Very often due to technological requirements, some design variables can take only discrete values from certain finite sets. In such a case, the RBO problem must be considered as a nonlinear mixed (discrete-continuous) programming problem. Due to high costs of reliability computations, it is justified to search for effective algorithms solving this problem. The efficiency of two discrete optimization methods when applied to the RBO problem is investigated. These are namely, the controlled enumeration method [2] enhanced by the use of constraints approximation technique and the new nonlinear mixed programming method, recently proposed by Wang et al. in [3].

minimize CI(JC'',X ),

(1)

subject to: A (JC^JC^) > Pf

i = 1,

, rrir,

ftys(:*:^x^)>iS,7

(2) (3)

c/(x^jc^)>0

i = l,...

,md,

(4)

xl

k = I,...

,n,

(5)

^k

—

^k

^k

e Zk = {zk,\,Zk,2^

k = l,...

,N,

'•'

^Zk,jk^

(6)

where Ci is the initial cost/weight of the structure, x^ and jc'' are the continuous and discrete design variables, respectively, ft, / = 1 , . . . , m^, are the chosen componential reliability indices, ^sys is the system reliability index, Cj, / = 1 , . . . ,m^, are deterministic constraints and (x^, "xj^, k = 1 , . . . , n, are the lower and upper bounds, respectively, imposed on the continuous variables. For each discrete variable, x^, k = I,... , N, Zj, is 3. discrete set of real numbers with J^ elements. 3. Transformation to continuous optimization problem

2. Mixed RBO problem The mixed RBO problem can be stated as: * Corresponding author. Tel: -h48 (22) 826-1281, ext. 331; Fax: +48 (22) 826-9815; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

One way of solving the above problem is to transform the mixed programming problem Eqs. (l)-(6) into the equivalent continuous optimization problem according to the method proposed in [3]. First, new discrete variables Vi^ = {vk,i, Vk,2^ . •. , ^kjk^, /: = 1, . . . , A^, are introduced

K. Kolanek et al. /First MIT Conference on Computational Fluid and Solid Mechanics by means of transformation

4

Jk

(7)

^Zk,jVk,j,

where Zkj e Zj, for j = I,..., straints are imposed on Vk'. ^kj

:0or 1,

j= h

Ji. The following con(8)

,Ji,

(9) 7=1

for any k = 1 , . . . , N. It was proven that the variables Vk can be made continuous by taking values in [0, 1] and that in the optimal point they will take values Vkj = 1 for only one j and Vkj = 0 for all other j if the additional constraints Q(Vk) = 0 are introduced, where

e(i^.) = EjV;-(E^MY. 7= 1

^ 7= 1

(10)

^

The original problem Eqs. (l)-(6) now can be expressed in the space of design variables x^ and i; as the following continuous optimization problem minimize Ci{x', x^(i;)) = Ci(x', v),

(11)

subject to: A•(x^ jc^(i;)) = ft (x", r) > pf"" 1=1,..., ^ys(x',x'{v))

(12)

Mr, =

'^,(x',v)>P^^

Ci(x',jc^cv)) = ^i(x', v)>o i = 'xl<xl<X k=l,... ,n,

(13)

i,. • • ,ma,

(14)

(15)

Jk

J2^kj - 1 = 0 k=l,...

,N,

(16)

7=1

Q(vt) = 0 0
k=

l,...,N,

j = 1,...

,Ji,k

(17) =

l,.. .

, N.

(18)

To solve the above optimization problem any nonlinear programming algorithm can be employed.

of the method to order the constraints checking process in a reasonable way. The strategy presented below takes advantage of the solution of the optimization problem obtained by relaxing discrete restrictions. This solution constitutes the lower bound for the cost produced by admissible discrete designs. Therefore, it is justified to exclude from the constraints checking process all combinations of design parameters giving lower cost value than assumed minimal one. Looking for the minimum it is reasonable to order the constraint checking process according to increasing cost function value of the designs. The time performance of the controlled enumeration method can be further improved by eliminating from exact constraints calculations all designs for which linearly approximated constraints are violated. Linear approximation of the constraints can be easily evaluated using the constraints sensitivities of the continuous optimal design. To present the controlled enumeration algorithm consider the problem of volume minimization of a truss structure. The discrete variables x"^ are cross-section areas of structural elements. Denoting them as Ak, k = 1 , . . . , N the cost function (structural volume) is expressed as

C(A) = J2lkAk

(19)

where k is the combined length of truss elements corresponding to the A:th discrete variable. The discrete sets Z^ (cf. Eq. (6)) are indexed according to increasing values of their entries. The tree graph (cf. Fig. 1) represents all possible cost function values. In the first layer, the vertices represent all possible volume values of the first structural element (or the group of elements with the same discrete cross-section). In the second layer, the vertices represent all possible combinations of the sum of volumes of the first two structural elements. Finally, the vertices in the A^th layer represent all possible combinations of the volume sums for the whole structure. A path connecting the root and the vertex in the A^th layer represents particular design (0.0)

4. Controlled enumeration method In the absence of the continuous design variables x'^ the mixed programming problem Eqs. (l)-(6) can be solved using the control enumeration techniques [2]. The main idea of this approach is to search for the set of all combinations of the values of design variables to find the optimum design in the subset of design combinations satisfying imposed constraints. The main advantage of this method is that it is free from the convergence problems observed with the previous algorithm for a large number of design variables. However, due to the huge number of combinations to check and very high costs of reliabihty constraints computation it is necessary for the effectiveness

661

Fig. 1. Tree graph of controlled enumeration.

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A^. Kolanek et al /First MIT Conference on Computational Fluid and Solid Mechanics

combination. If the path includes the vertex with index {k, j) it means that for the corresponding design the value of variable Ak is Zk,j • The main steps of the algorithm are as follows: 1. In the tree graph (cf. Fig. 1) generate the path representing discrete solution: 1.1 Consider subgraphs containing roots in the first layer with indices (1,7) where j = 1, . . . ,71. The maximum values of the cost function represented by the vertices in the layer A^ are: (20)

^TT = ^^^^j + X1^^^^.A--

For the first design variable assign the value which satisfies the condition: Al = min{zi,, I C* < Cj^f},

j = l,...,Ju

(21)

where C* is the structural volume corresponding to the optimal solution from continuous optimization. If the set minfzi^^ | C* < CJ"}^} is empty, it is necessary to redefine catalogues for design variables and restart the algorithm. 1.2 Consider the subgraphs with the vertices (2, y), located in the second layer, where j = I,... , J2, and with the vertex in the first layer (1, 7^,). The index 7^, corresponds to the first element volume assigned in Eq. (21). The values of maximum volume for the vertices located in the layer A^ of such subgraphs are:

C^f=hA,^l2Z2.j^J2^kZk.J,.

aft (A) I dA.

AA,

< £,

1,.. , rur, ^ 1,.

(24)

where AA^ = A^ — A^, and A* is the optimal continuous design. If the above conditions are satisfied, insert the design to the constraint checking list and order it by the increasing volume. 3. Compute the exact values of all the constraints for designs in the list starting from the one with the smallest volume. The first design satisfying the constraints is considered to be the problem solution. The efficiency of the two approaches will be analyzed on the numerical examples of the geometrically nonlinear truss structures with the cross-sections of the bars modeled as random variables and their mean values treated as discrete variables. Displacement, local and global stability type limit state functions will be used.

(22)

References

For the second design variable assign the value such that: A2 = rmn{z2,j I C* < C^f],

than the continuous one, so they can be excluded from further analysis. 2. Generate the ordered list of discrete designs to check the constraints. Starting from the path defined in the step 1, going to the right till the last path of the graph is reached, check the designs for the following conditions: • the discrete design volume is greater or equal to the optimal continuous one, • the variations of the values of ^ indices due to the perturbation of the discrete design variables are smaller or equal to the given prescribed value

7 == 1, . . . , 72- (23)

1.3 Continue the procedure listed above, until all the design parameters assume assigned values. Notice that all the designs represented by the paths located on the left-hand side of the one defined above (very often numerous) give smaller structural volumes

[1] Kleiber M, Siemaszko A, Stocki R. Interactive stability-oriented reliability-based design optimization. Comp Methods Appl Mech Eng 1999;(168):243-253. [2] Greenberg DE, Lee WH. Optimal synthesis of frameworks under elasfic and plastic performance constraints using discrete sections. J Struct Mech 1986;(14):401-430. [3] Wang S, Teo KL, Lee HWJ. A new approach to nonlinear mixed discrete programming problems. Eng Optim 1998;(30):249-262.

663

Genetic algorithms and high performance computing for engineering design optimization C.S. Krishnamoorthy * Department of Civil Engineering and Centre for Finite Element Analysis and Design, Indian Institute of Technology Madras, Chennai 600 036, India

Abstract With globalization of economy, optimization in design has assumed great significance in the engineering industry. Genetic Algorithms (GAs), as part of evolutionary computation, offer scope for size, configuration, topology/shape optimization. Genetic modelling, analysis and computational aspects for design optimization are discussed. Typical cases are demonstrated to focus attention on practical applications. Future research and development needs are highlighted. It is concluded that as a potential tool for multi-disciphnary optimization, GA-based methodologies, on high performance distributed computing environment, offer great scope for optimization in collaborative engineering design. Keywords: Engineering design; Optimization; Genetic algorithm; Space truss; Fuselage; High performance computing

1. Introduction Optimization plays a key role in engineering design. With the advent of computers and, development of finite element analysis (FEA) and numerical methods for mathematical programming techniques, intense research has been conducted in the area of minimization of cost or weight of structures used in aerospace and civil engineering [1]. Many of the practical problems in topology, shape and configuration optimization could not be addressed satisfactorily. Recent advances in cost-effective high performance computing, have encouraged researchers to turn towards realistic optimal design modelling and the following four methods are of significance in the area of topology optimization: optimality criteria [2]; homogenization method [3]; evolutionary structural optimization (ESO) [4]; and genetic algorithms (GAs) [5]. Genetic algorithms are search algorithms based on the principle of natural evolution. Research over the past 10 years has shown that it is a powerful optimization technique for complex problems and can be successfully applied for a wide range of problems of importance to engineering industry [6]. Also, engineering design is a multi-disciplinary * Corresponding author. Tel.: +91 (44) 445-8286; Fax: +91 (44) 235-2545/445-8281; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

activity and GAs have been found to be effective for optimization of real life problems that require satisfaction of multiple objectives. The aim of the paper is to focus attention on GA based methodologies for engineering design optimization and the potential it offers for implementation on high performance computing for practical application in industry.

2. Genetic algorithms and optimization modelling The most important step in GA-based optimization is the genetic modelhng of the problem, i.e. to develop a coding scheme that maps the variables into genospace to form an individual that adequately represents the problem. GAs offer scope to consider topology/shape and configuration variables and, constmctibility/manufacturability aspects which cannot be handled satisfactorily in other optimization methods. It is in this context that new coding schemes and strategies are needed. A variable genetic algorithm (VGA) has been developed to address the topology optimization [6,7]. Non-dominated Sorting Genetic Algorithm (NSGA) has been found to be effective in multiobjective optimization [6]. In addition to coding schemes, there are other problemdependent aspects such as, handling of constraints through

C.S. Krishnamoorthy /First MIT Conference on Computational Fluid and Solid Mechanics

664

outerDiaj ~|^outerDia{ ^ oiitcrPial -^outerDia -^•outerDiaj outerDial

1 2

3

thickness[0] [0] = 3.25 thickncss[0][I]==3.25 thickness(0] [2] - 4.05 thickness[0] [3] = 4.05 thickness[l][0]-3.25 .thickness[l][l] = 3.25 thickness[l][2] = 4.05 thickness[l][3] = 4.05

1=42.4; |«48.3 1-60.3 76.1 1=88.9: =114.3

4

1 2

1 2

©

0

u^

Fig. 1. Coding scheme for three part chromosome. penalty methods, schemes for selection, cross-over and mutation. To meet the requirements of an application environment, a core library, GALiLEO (Genetic Algorithms Library for Learning and Engineering Optimization) has been developed at IIT Madras [8]. GALiLEO is based on object oriented technology and allows a user to plug-in any optimization problem into the library and run the problem with different genetic algorithm parameters and coding schemes.

3. Applications Based on the recent work, typical practical problems are given here to illustrate the importance of genetic modelling and software tool mentioned in Section 2. 3.1. Optimization of space trusses with member grouping In the case of large space structures with around 2000 members or more, from constructibility point of view, grouping of members is needed. Hence the objective function is, NG

Mk

minimize W = Y^ A^ V " L,yo, A:=l

(7)

software GALiLEO has been used to solve a square-oversquare space truss with 611 nodes, the topology of which is shown in Fig. 2. The total number of members is 2304. A three-phase method is used to arrive at optimal number of member of groups [9]. The convergence plot is given in Fig. 3. 3.2. Optimization of middle fuselage of a transport aircraft Aluminium alloy and laminated composite models of the semi-monocoque fuselage structure are considered. The structure shown in Fig. 4 consists of a fuselage shell with cutouts for windows. For aluminium alloy structure, the variables considered are: fuselage shell 9; bulk heads 48 and stringers 3 (total 60). For laminated composite structure, the total number of design variables works out to 114. To handle mixed variables, adaptive movement of lower and upper bounds and a new genetic operator, cross-permutation are developed. MSC/NASTRAN is interfaced with GA solver. Distributed computing environment is used for solution [10]. 3.3. Application ofGAs in machine components and manufacturing industry

/= 1

where NG is the number of member groups and Mk is the number of members in member group k. A three-part chromosome is proposed for tubular space truss (Fig. 1). The

GA-based optimization of machine components using FEA packages, like ANSYS, for practical application is currently in progress.

C.S. Krishnamoorthy /First MIT Conference on Computational Fluid and Solid Mechanics

665

R

a eg)

2 18(®1.5m-27m

\AAA/\AAAAAAA/\A/\A7W ~J Fig. 2. Topology of 2304 member tubular space truss. 4. High performance computing in GA-based methodologies In design optimization problems, the fitness function evaluation is the most time-consuming part of the integrated GA because it involves a finite element analysis of the structure represented by each string in the population. Depending on the number of elements/members/components in the structure and the complexities of the structure, the fitness function evaluation usually consumes 85-90% of the total computation time. For a structure with m elements, the time for genetic operations (selection, crossover and mutation) is of the order of m (i.e. 0(m)) while that for finite element analysis is of m^ (i.e. 0(m^)) [11]. Another important question is how to choose the size of the population. In the case of large structural optimization problems, hundreds of strings are needed in order to avoid 'sub-space search problem'. These engineering problems cannot be solved easily in real time on most of the uni-processor machines. Hence high performance GAs with parallel processing capabilities are required. The search process can be accelerated by allocating either the

process of fitness evaluations or genetic operations or both to different processors of a parallel computer or to different workstations of a network. This idea leads to many models of HPC in genetic search algorithms [12]. Fig. 5 shows a schematic diagram of a master-slave implementation for single population band global parallel GA. There is another type of parallelization in which not only fitness, but also genetic operations are performed in parallel. There are two models, coarse- and fine-grained parallel GAs. Hybrid models have also been developed combining global and, coarse- or fine-grained GAs as shown in Fig. 6.

5. Conclusions Genetic algorithm-based methodologies for design optimization integrated with FEA packages offers great potential for application in engineering industry. There are several areas in which future research and developments are needed: special purpose optimization modules; efficient reanalysis techniques; multi-objective optimization to practical systems; reliability based optimization, hybrid

666

C.S. Krishnamoorthy / First MIT Conference on Computational Fluid and Solid Mechanics 180000

170000

160000 A c

a> 150000 4

~*~Popuiation Size = 40

5 >

- * - P o p u l a t i o n Size = 50 -•—Population Size = 60

140000 -{

to o CQ 130000

120000 H

110000

10

20

30

40

50

60

70

80

90

100

Number of Generations Fig. 3. Convergence plot for space truss optimization.

Acknowledgements The paper is based on research and development at IIT Madras. The support through Indo-US project on 'Optimal Design Processor for Structural Systems' with Dr. D.E. Goldberg at UIUC through NSF Grant INT-9421299 is acknowledged. Sincere thanks to a large number of research students for their contributions.

Fig. 4. FE model of the aircraft fuselage.

Fig. 5. Master-slave implementation of global parallel GA.

methods for topology optimization; HPC frame work for integration in a collaborative design environment. The potentials and diversity of high performance GA in parallel or distributed environments are still to be exploited in solving problems of design optimization in engineering and technology.

References [1] Vanderplats GN, Numerical Optimization Techniques for Engineering Design with Application. New York: McGrawHill, 1984. [2] Rozvany GIN (Ed), Shape and Layout Optimization of Structural Systems and Optimality Criteria Methods. Udine: Springer, CISM, 1992. [3] Hassani B, Hinton E, Homogenization and Structural Topology Optimization. Springer, 1999. [4] Xie YM, Steven GP, Evolutionary Structural Optimization. Springer, 1997. [5] Gen, Mitsuo and Cheng, Runwei, Genetic Algorithms and Engineering Design. New York: John Wiley and Sons, 1996. [6] Krishnamoorthy CS, Structural optimization in practice: potential applications of genetic algorithms. Struct Eng Mech, to appear. [7] Rajeev S, Krishnamoorthy CS, Genetic algorithms-based methodologies for design optimization of trusses. J Struct Eng ASCE 1997;123(3):350-358.

C.S. Krishnamoorthy /First MIT Conference on Computational Fluid and Solid Mechanics

667

T I T

•—•—•—•

•—•—• • •

Fig. 6. (a) Hybrid GA that combines coarse-grained upper with fine-grained at lower level, (b) Hybrid GA that combines coarse-grained with global at lower level.

[8] Prasanna VP, Sudarshan R, GALILEO — Genetic Algorithm Library in Learning and Engineering Optimization. Theory and Program Documentation. Civil Engineering, LLT., Madras, India: 2000. [9] Sudarshan R, Genetic Algorithms and application to the optimization of space trusses. Project Report, Civil Engineering, LLT., Madras, India, 2000. [10] Madhusudhan BS, Genetic Algorithm based optimum de-

sign of large scale isotropic and laminated composite structures. Ph.D. Thesis to be submitted, Civil Engineering, LLT., Madras, India, 2000. [11] Hojjat AdeH, Nai Tsang Cheng, Concurrent GA for optimization of large structures. J Aerospace Eng ASCE 1994;7(3). [12] Erick Cantu-Paz, Designing Efficient Master-Slave Parallel GAs, IIHGAL Report No. 97004, 1997.

668

Dynamics of wearing contact in groundwood manufacturing system S.S. Launis'''*, E.K. Keskinen^ M. Cotsaftis^ ^ Tampere University of Technology, Laboratory of Machine Dynamics, P.O. Box 589, FIN-33101 Tampere, Finland ^ Laboratoire des Techniques Mechatroniques et Electroniques, ECE, 53 Rue de Crenelle, 75007 Paris, France

Abstract A contact dynamics approach has been appUed for simulating wood grinding process in pulp and paper industry. The system level model combines different components of the system including mechanical parts of grinding chamber, grinding stone, shaft line, viscoelastically behaving wood and hydraulic loading circuit. These material elements are connected through different interfaces, of which the most important one is pulp producing wearing boundary between grinding stone and wood batch. Keywords: Multibody contact dynamics; Grinding

1. Introduction Mechanical pulp used in papermaking is traditionally produced from wood logs by groundwood process [1]. In grinding, coaxial logs are fed in grinding chamber between two guiding walls against rotating grinding stone by hydraulically actuated shoe bodies, as illustrated in Fig. 1. Although the feeding system has a feedback loop in order to limit certain quality indicators of the produced pulp within wanted bounds, major production quality

variation is typical in groundwood pulping. This is mainly due to the difficulties in applying the conventional actuator control loops to compensate the strong nonlinearities in the compression of logs and to maintain constant conditions in the wood-stone interface. This study introduces a contact dynamics approach for simulating the grinding process on the complete system level in order to study its dynamical behavior and further, to improve the process performance. The theory describes the properties of the log batch based on an averaged one dimensional viscoelastic continuum material model.

2. System modeling

Fig. 1. Typical PGW-grinder (Valmet Co.). * Corresponding author. Tel.: +358 (3) 365-2796, Fax: + 358 (3) 365-2307, E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The components of the groundwood system consists of: • The shaft line consisting of the stone, shaft and electrical drive • The wearing contact between wood and pulpstone • The shoes and the hydraulic system of pushing actuators • The propagating wood batch • The feedback control loops for pushing actuators and driving the pulpstone The models of the different components are presented in the following first as hand-derived and secondly by means of component oriented modeling.

S.S. Launis et al. /First MIT Conference on Computational Fluid and Solid Mechanics (a)

(b) k

T -4^

QR

<> t ^ /;

k

-m^

e

1 1

* h

T <4-

Fig. 2. (a) Stone-shaft configuration, (b) Mass-spring model of the shaft system. 2.1. Hand-derived models The studied grinder configuration consisting of a pulpstone loaded from one side and driven by an electrical motor is presented in Fig. 2a and the corresponding lumped parameter model of in Fig. 2b. The shaft between the motor rotor and the grinding stone is torsionally elastic and modeled as a spring. The state variables of the rotating system are the absolute angular position of the rotor 6 representing the propagating component of the motion and the small relative angular position 0 between the stone and the rotor due to torsional elasticity of the shaft. A straightforward derivation of dynamic equations of motion yields to vector system Mq + Kq = F(q,q,t)

(1)

where q = {0^}^. The inertia and stiffness matrices are M =

K =

h h

h h+h

k

0

0

0

(2)

(3)

and the loading vector has components F =

-RQ T-RQ

(4)

The chosen electrical motor model approximates a DCdrive. Running at nominal speed Q, the PID-controlled drive produces torque T = Kp{Qt -0) + Ko{Q - ^) + . , / {Qt - 0) dt

(5)

The interaction between stone and wood is described as a wearing process. Grinding stone surface is usually made of abrasive ceramic segments. It is assumed that the stone surface conditions remain constant during simulated grinding periods, which is realistic in modem machines where stone surface is washed and sharpened frequently with online high-pressure water showers. The momentum

669

transfer from stone to wood batch is governed by energy equation QT = CgAs

(6)

where Q is resulting tangential force in stone grinding zone, r = (0 -\- <j))R the sliding speed at stone-wood interaction surface, Ce specific energy consumption per unit volume, A the cross-sectional area of the wood batch and As the volumetric rate of produced pulp. The wood mass transfer out of the batch is governed by the wear equation [2] As = kuiNi

(7)

where N is the normal force and ky^ is the wearing factor. Once the sliding speed is computed from the stone motion, Eq. (7) may be used for solving the wear rate s of wood s = A-^k oNx

(8)

Further, by combining Eqs. (6) and (8) the relation between the normal force and the tangential force is Q = Cek^N

(9)

According to pilot machine observations Q is related to the sliding speed [3]. This means that the product Cekyj, which may be considered as effective friction coefficient in grinding contact, must be a function of f. The measurements scattered but suggested, that specific energy consumption Ce remained nearly constant whereas quantity indicated with wearing factor k^ which is also dependent on material and surface pair had a clear gradient inside the recorded rotating speed range. Due to this, Q is considered constant while k^ follows a second order polynomial curve fitted to the measurement points k^(0, 0) = a(0 + (pf + /7(6> + 0) + c

(10)

where a, b and c are constants that depend on grinding contact variables and sliding speed has been replaced by rotating speed. The logs are pushed against the grinding stone with hydraulically actuated shoe. The dynamic pressure p in the pushing cylinder is governed by differential equation -xApi + qpi Ppi = B-xApi + Vpi

(11)

where B is the bulk modulus of fluid, A pi the piston area and Vpi the dead fluid volume in the piping. The dynamic equation of shoe position is ,2lih-^L mx = ppi A pi — Ne'

(12)

The valve flow may be computed from turbulent flow model by qpi = cUy/ps - Ppi

(13)

1/2 where Ps is the line pressure and c = qnomU^l^Ap^ nom

670

S.S. Launis et al. /First MIT Conference on Computational Fluid and Solid Mechanics SLl

SL2

•

< Xpl

^

1>\<

M

X \Ppl

1

1

A ^ II P^

n

^A

h

H

X

i—^

J^shoe 4-

^

ir

qpi\\

X

•

r V-/ Y Vr

BASE VOLl

1 ^mi

nxMm<^

P^ Pt

1

ryifi>n\!ijr^^

VOLP

+

desired

- /

(14)

edit

where e is the servo error between the desired and measured process quantity (Fig. 3). When the batch end is fed with shoe velocity x and the other one is ground with wear velocity s depending on the grinding force itself, the normal force distribution in the wood batch follows an exponential curve because of the friction between the logs and the pocket walls [4]. The state equation of compression force A^ is written lAfjih ^l^lh-HLo-

X |£|i

-X) _

1

hL

- ( -

+ r)[x- A

K^

AE

AA-'k^Mr]

{N't + A^i'•)]

1

VOLT

Fig. 4. A schematic model of the grinding system divided into components.

is obtained by measuring volumetric flow qnom for fixed pressure difference AP^^m and full input voltage MmaxValve input is in case of classical PID-controller simply U = Kpe + Koe +

'

V0L2

VAL

Fig. 3. Shoe body governed by a hydraulic servo loop.

A^ =

\

J

1

•

CYLl

cycles of the grinding machine even if the configuration may vary. As an example, numerical simulations were accomplished using a system model composed of object oriented library models in Fig. 4. The system model itself is a list of calls to subroutines representing object oriented component models. Corresponding Fortran90 program code is presented in Fig. 5.

3. Numerical results Two different types of controlling the shoe motion were tested in computer simulations. The first one was simple constant speed pushing, so that desired shoe speed was constant i^. The second approach based on maintaining constant N = N^. This implied condition N = 0 had to be satisfied. Eq. (15) then gives required stationary pushing speed Xd at each shoe position Xd

(15)

where /x is the friction coefficient in wood-wall contact, h the height of the batch, Lo is the original length of the batch, AE is the axial elasticity and Ar] the axial viscosity of the batch. 2.2. Component oriented system descriptions A more flexible way to model multi-component technical systems is object oriented modeling, where existing library models constructed for previous problems can be reused in new systems. To be an object oriented model a component model should at least (a) have a compatible interface to other component models, (b) have routines for initializing and parametrizing itself and (c) to generate and solve its dynamical equations whether they are algebraic or differential equations. Component oriented modeling environment enables to carry out relatively rapid redesign

= ri_^,2..-..j

Sd

(16)

where Sd is required wearing rate. From Eq. (16) we are able to integrate the shoe position xj and use it as a control input in the feedback loop. The simulated volumetric production rates are presented in Fig. 6. The controllers have difficulties in reaching the desired value for input in the beginning of pushing when the hydraulic liquid and the wood batch are compressed heavily. After compression phase, the pressure losses in valves causes that the needed flow rate cannot enter the cylinder although the system is working at its maximum capacity. After desired shoe speed has been reached, the speed controller, due to its nature, continues full capacity driving until the shoe finally reaches the desired position. This causes overshooting in desired speed and in production rate. In the rule based control, also the grinding force A^ undergoes during that period overshooting. After desired position has been reached the production rate begins to drop.

S.S. Launis et al /First MIT Conference on Computational Fluid and Solid Mechanics

671

CALL MECBASE (M_NAME,"BASE",MEC,ARM(1)) CALL MECBODY (M_NAME,"A",MEC,ARM(2)) CALL WHEEL (M_NAME,"B",MEC,ARM(3)) CALL REVOLUTE (M_NAME,"JOl",MEC,ARM(1),ARM(3), & "NC2","NC3","NBl","NB2",JOINT(1)) CALL INTSLIDE (M_NAME,"SLl","SL2",MEC,ARM(1),ARM(2), & "NC1","NC2","NA3","NA2",J0INT(2),JOINT(3)) CALL GWOWEN (M_NAME,"WOOD",MEC,ARM(2),ARM(3),ARM(1), & "NA2","NBl","NC2",OWEN) CALL HYDCYL (M_NAME,"CYLl",MEC,ARM(1),ARM(2), & "NCI","NA1",V0L(1),V0L(2) ,CYL) TOR = l.E7*CTL(2).X IF (GET_DYN().EQ.1)CALL ADDLOAD (OWEN.NGRIND,0.DO,0.DO,TOR,ARM(3)) CALL SET_P (PRES(l),V0L{5)) CALL SET_P (PRES(2),V0L(6)) CALL HYDVOL (M_NAME,"VOLl",MEC,VOL(1)) CALL HYDVOL (M_NAME,"V0L2",MEC,VOL(2)) CALL HYDVOL (M_NAME,"V0L5",MEC,VOL(5)) CALL HYDVOL (M_NAME,"V0L6",MEC,VOL(6)) CALL SRV (M_NAME,"SRV1",MEC,V0L(5),V0L(6),V0L(1),V0L(2), & VAL) CONTROLSWITCH = 1 IN SPEED CONTROL, = 2 IN RULE BASED SPEED CONTROL SELECT CASE (CONTROLSWITCH) CASE(l) DXD=1.667E-3 El = DXD*GET_TIME()-(ARM(2).NODE(OWEN.NPUSH).UN-XINI) CASE(2) DXD=1.6E-3-5.2 92 9E-6*OWEN.EXPAL CALL INTTR(DXD,XD,WXD) El = XD-(ARM(2).NODE(OWEN.NPUSH).UN-XINI) END SELECT DEI = DXD-ARM(2).NODE(OWEN.NPUSH).DUN E2 = -31.1*GET_TIME()-ARM(3).Y(3) DE2 = -31.1-ARM(3).DY(3) CALL DCV_PID ("SRV1",E1,DE1,CTL(1),VAL) CALL CTRL_PID ("DRIVE",E2,DE2,CTL(2))

Fig. 5. System model of grinding machine presented in Fig. 4. model to minimize the variations of the grinding force for producing standard quality pulp. It is shown that the performance of both conventional and rule based speed controlled grinding machine is not optimal because of contradictory between PID-controller inputs and desired output. However, the present system model can help the design team to improve machine performance and increase pulp quality by means of more sophisticated model-based control approach.

References Fig. 6. Volumetric pulp production.

4. Conclusions A traditional grinding machine of pulp industry has been modeled using contact dynamics approach. The complete system model consists of rotating and feeding elements of the machine as well as the wood batch under wearing contact with the stone. Application of mass and momentum transfer equations yields first order differential equation for the normal force in grinding. Control algorithm governing the fluid powered wood feeding motion utilizes this wear

[1] Haikkala P, Liimatainen H, Tuovinen O. New grinder concept for continuos pressure grinding. Proceedings of the TAPPI International Mechanical Pulping Conference, Houston, TX, 1999. [2] Hailing J. Principles of Tribology. London: MacMillan Press, 1975. [3] Launis S, Keskinen E, Cotsaftis M, Raitaniemi M. Nonlinear dynamics and control of log pushing in groundwood systems. Proceedings of the ASME International Conference on Engineering Systems Design and Analysis, Montreux, Switzerland, 2000. [4] Launis S, Keskinen E, Cotsaftis M, Raitaniemi M. Mechanics of wood motion in groundwood manufacturing system. Proceedings of the lUTAM International Conference of Theoretical and Applied Mechanics, Chigaco, IL, 2000.

672

Design optimization of materials with microstmcture Shutian Liu *, Zhiqiang Lian, Xinguang Zheng State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, RR. China

Abstract In this paper, the design optimization on the microstmcture is studied for advanced materials, including functionally gradient materials (FGM) and materials with specific elastic properties. The homogenization theory and the optimization method are applied to determine the effective properties of materials, and to design the microstmcture of the materials with specific properties. The problem formulations, solving schemes, and numerical examples of designs on the FGM and materials with specific elastic properties have been presented. Keywords: Material design; Optimization; Homogenization theory; Microstmcture; Composite materials; Functionally gradient materials

1. Introduction The properties of Composite materials can be designed. This feature creates the opportunity to design specific physical properties into the individual regions of the stmcture requiring them. By changing the material microstmcture, composites can exhibit different properties. From the mechanical points of view, the aim of material design is to determine a distribution of microstmcture parameters (e.g. fractions) which represent microstmcture of composites, to realize a technical requirement. In this paper, the formulations and the solving schemes of the microstmcture design problems of materials, including functionally gradient materials (FGM) and materials with specific elastic properties, are given. The optimal distribution of materials to reduce an effective stress is obtained in a solid cylinder made from zirconium oxide (Zr02) and tungsten (W). In the end, the design of materials with specific Poisson's ratio is presented.

2. Design optimization of FGM FGM is a new idea for material design proposed by Waranabe [7] in 1987. As a new material, wide attention has been paid to its design methods e.g. [4,6]. For two-phase materials, let us consider a gradient ma* Corresponding author. Tel.: +86 (411) 470-7057; Fax: +86 (411) 470-8769; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

terial composed of two components and suppose that X3 is a gradient direction. On the macroscopic scale, the material can be supposed to be constmcted by periodic repetition of a unit cell, and the volume fraction of one component, / , can represent the microstmcture. The distribution of material is described by the distribution of volume fraction / , with / = 0 or 1, corresponding to single phase and 0 < / < 1 to composites composed of two or more components. The aim of the optimal design of FGM is to determine the optimal distribution of fraction / = /(X3) along gradient direction. Considering the practical process of material fabrication, the material can be laminated. Thus, the volume fraction / can be expressed as a step-wise function X°<X3<

/i

Xl

x\ <Xi < x\

/ =

(1)

< X3 < xj

fn

Choosing an appropriate number of layers, n, and taking the thickness, ti, and fraction, / , of each layer as design variables, the optimal design of FGM to reduce stress is formulated as Find tr.

fr r =

M i n M a x (Jij{u(x), s.t.

l,2,...,n Eijuifr),

«E(U,V) = L(V),

0
Olijifr),

V V G U^^

0 < /. < 1

tr)

(2)

S. Liu et al. /First MIT Conference on Computational Fluid and Solid Mechanics

673

Where, the biUnear form a^{\x, v) of the energy and load Unear form L(v) are defined as aE(u, v) = / Eijki -~—-—dx, J dxi dXj L(v) = / EijkiakiT-^

dx + / fiVi dx + i tiVt dP

(3)

The material properties are computed by use of homogenization theory (see [1,3])- Based on the homogenization theory, the thermoelastic constant can be determined by F^

- —-

/ (

^ijkl

Liijmn

d^[

i) ''• dy,

<

4 6 Number of layers Fig. 1. Variation of circumferential stress with respect to layer number.

(4)

l.Z

Where ^ f " ( j ) is the periodic solution of the following equations.

[(F

F ^^rv^'H

WveVY = {v(y) \yeY,

SI

1.0

=IE!J,,]-'P,

n

v(y) == v(y + Y)}

§ 0

c

.2

0.8 0.6

1 0.4 fa

(5)

The problem (2) is solved by the use of the sequential quadratic programming (SQP) method.

0.2 r\f\

•

r 1

1

1

1

1

1

^ 1

-3.0 -2.0 -1.0 0.0

1

1.0

1

1

1

1

2.0 3.0

Z /mm Fig. 2. The distribution of fraction of Zr02.

3. Design results and analysis Let us consider a composite solid circular cylinder composed of zirconium oxide (ZYO2) and tungsten (W) under a temperature reduction by 500°C, and determine the optimal distribution of the fractions of components to reduce the fabricating residual stresses. The cylinder is 6 mm in height and 8 mm in diameter. The reduction of stress is realized by choosing an appropriate combination of values of the fractions in layers between upper and lower ones. This problem is modeled by four-node axisymmetric isoparameter elements and the thermoelastic stresses are obtained by the use of finite element method. The results obtained are given in Figs. 1 and 2. If two half parts composed of pure phases of the cyHnder contact directly (n = 0), temperature variation of 500°C results in a maximum circumferential stress of 434 MPa. If we divided uniformly the middle part of the cylinder into 1, 2, 4, 6, 8 and 10 layers, respectively, the maximum stresses corresponding to each of the above divisions have great reduction. For example, the maximum circumferential stress for n = 10, is 85.9 MPa, this value is only about 20% of the value of the maximum circumferential stress for n = 0. Fig. 2 gives the distribution of fractions for eight layers divided.

Fig. 1 shows that, when the number of layers reached a certain number, the increase of the number of layers has very little effect on the reduction of maximum stress. Therefore, it is rational to divide the material into an appropriate number, instead of a very large one, during the design and fabrication of the FGM.

4. Microstructure design of materials with specific elastic property Composite properties are dependent on the microstructure of materials, which is depicted with a unit cell. The goal of material design optimization is to find these parameters to make the materials with desired properties. In this section, a shape optimization procedure is used to determine the distribution parameters on the bases of a given topology of a unit cell. The objective function /(E^Jj^^) can be any combination of the elastic properties Efji^^. The concerned example in this paper will be the case where we want to design a material microstructure with specific Poisson's ratio in a given direction. In this case, the objective function / will be the square of the difference between the value of

674

S. Liu et al /First MIT Conference on Computational Fluid and Solid Mechanics

WAJWW

-IL

DC X

JL

X

JL

X

:c :c :c DC : : DC : [ :c :c :c 1L :c :c :c :c :c :c

3Cn f X nr X n r X Fig. 3. Periodic structure (left) and a unit cell (right) of materials.

Fig. 4. Periodic structure (left) and unit cell (right) of materials with zero Poisson's ratio.

Poisson's ratio, y,^, and the given value, / = (yf^ - y|^2)^' where, v^^^ is the desired Poisson's ratio of the materials. Consider a two-phase material that is composed of a material phase and a void phase. If the topology of solid material phase is given, the parameter of the shape of the unit cell and the parameters which depict the shape of the solid structure in the unit cell domain will be the design variables. For a honeycombed skeleton structure shown in Fig. 3, design parameters are the thickness and the angle a,x = (r,a)'^. The thickness and the angle determine the shape of the solid structure and the shape of the unit cell. For the purpose of design materials with orthotropic, square symmetric or isotropic parameters, such constraints must be implemented in the optimization problem. These constraints are chosen as penalty functions added to the cost function, such as taken by Sigmund and Torquato [5]. In this paper, two symmetric lines are specified to obtain orthotropic properties, although only one symmetric line is needed to obtain this nature. The optimization problem including the above-mentioned features can now be formulated as

Table 1 Microstructure parameters and properties of materials with zero Poisson's ratio

Min fix) = (yf2 - ^?2)''

(6)

a < 90°,

-a < 0°

where the effective Poisson's ratio is determined by •^1122

E^

^1111

H

'

^1122

10 20 30 40

a

n

-0.3038 -0.9977 -2.4270 -5.1850

Fraction 0.1458 0.2848 0.2848 0.5696

El

El

G\2

(MPa)

(MPa)

(MPa)

7087.03 14462.00 22456.00 31842.00

103.81 645.08 1905.70 3955.00

41.3 275.96 934.38 2306.10

For a series of variable thickness values, the angles which make the materials exhibits zero effective Poisson's ratio are determined. When t = 20, the microstructure of materials with zero Poisson's ratio is expressed in Fig. 4. In this case, the angle is 0.9977°. The other parameters of properties of materials are listed in Table 1. For different thickness parameters, the angles are different.

5. Conclusion

X = (r, a)

s.t. t > 0

t

(7)

2222

The effective elastic properties ElJ,^i of porous materials are computed using above described numerical homogenization method. The optimization problem (6) will be solved using the Golden-division method for a series of thickness parameters t. The iteration procedure will end when the change in the angle or the objective function from step to step is lower enough, for example, less than 10""^. Now, a design example of materials with zero Poisson's ratio is given. The material is an aluminium material with uniformly distributed pores. The periodic microstructure and the unit cell are shown in Fig. 3. The Young's modulus and Poisson's ratio are 6.958 x 10^ MPa and 0.3148, respectively.

The numerical computation and optimization methods can be successfully used for the design of FGM and microstructures of materials with specific properties. From the example of the FGM design, one can see that it is rational to divide material into layers in design and fabrication of the FGM. The number of layers has an important effect on the optimization results only when the number is relatively small. When the layer number is relatively large, its increasing is of little importance to reduce stresses. Thus, the rational layers division should be made in design and fabrication. The design example of the materials with zero Poisson's ratio shows that the shape optimization of the microstructure of material phase can produce a specific overall behavior for material, e.g. zero Poisson's ratio.

Acknowledgements This research is supported by the National Natural Science Foundation of China (10072016) and by Foundation

S. Liu et al. /First MIT Conference on Computational Fluid and Solid Mechanics for University Key Teacher by the Ministry of Education of China.

References [1] Bendsoe MP, Kikuchi N. Generating optimal topologies in structural design using homogenization method. Comput Methods Appl Mech Eng 1988;71:47-58. [2] Lakes R. Foam structures with negative Poisson's ratio, Science 1987;235:1038-1040. [3] Liu ST, Cheng GD. Prediction of coefficients of thermal expansion for unidirectional composites using homogenization method. Acta Mater Comp Sin (in Chinese) 1997;14:76-82. [4] Sata N. Design and production of a functionally gradient

675

material by a self-propagating reaction process. Funct Mater (in Japanese) 1988;8(2):47-58. [5] Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Dannish Center for Applied Mathematics and Mechanics, Report No. 525, Technical University of Denmark, 1996. [6] Tanaka K. et al. Design of Thermoelastic Materials Using Direct Sensitivity and Optimization Methods: Reduction of Thermal Stresses in Functionally Gradient Material. Comput Methods Appl Mech Eng 1993;106:271-284. [7] Waranabe T. Fabrication of functionally gradient materials by powder technological process. Funct Mater (in Japanese) 1988;8(4):51-59.

676

Load lateral distribution for multigirder bridges Chunhua Liu ^'*, Ton-Lo Wang ^, Mohsen Shahawy ^ ^ Department of Civil and Environmental Engineering, Florida International University, Miami, FL 33199, USA ^ Florida Department of Transport, Structural Research Center, Tallahassee, FL 32310, USA

Abstract This paper presents the Uve load distribution for highway bridges based on dynamic moments and shears. The bridge structures are represented by grillage model. The trucks are selected from weigh-in-motion (WIM) measurements and mathematically modeled as spatial systems. Road surface roughness is generated as correlated processes. It is observed that: (1) the various truck types cause close lateral distribution of moment/shear; and (2) the calculated lateral distribution factors for interior girders based on loading of two lanes are lower than the values specified by AASHTO Specifications. Keywords: Bridges; Road surface roughness; Girders; Load distribution factor; Simulation; Correlation; Random processes

1. Introduction

2. Bridge and truck models

Live load lateral distribution is an important issue in the highway bridge design. This distribution can be calculated using numerical methods or classic plate theories. For the convenience of the design, the AASHTO Specifications adopted deliberately calibrated formulae to estimate the load lateral distribution factor. The AASHTO Standard [1] specifies the S/D approach. To improve the accuracy of the S/D approach, Zokaie [11] carried out a lot of research work and developed new formulae with higher accuracy and broader range of bridges. The new formulae were adopted in the AASHTO LRFD [2]. In most of the previous studies, static moments and shears generated by trucks have been used. Huang et al. [6] studied the relationship between the impact factors and the dynamic lateral distribution for prestressed concrete I-girder bridges. Kim and Nowak [7] validate the AASHTO formulae based on field measurements on two bridges using normal traffic. The objective of this study is to further investigate the lateral distribution based on dynamic moments and shears.

Four simply supported prestressed concrete highway bridges were designed according to AASHTO Specifications [1] and the Standard Plans for Highway Bridge Superstructures [9] from the U.S. Department of Transportation (DOT). All the bridges are of I-beam sections with a cast-in-place deck and the design truck is HS20-44. The bridges have a roadway width of 9.74 m (for two traffic lanes) and a concrete deck thickness of 0.19 m. All five girders have identical sections and are transversely connected to each other by diaphragms. The number of diaphragms is 0, 1, 2, and 2, respectively, for the shortest to the longest span length. The multigirder bridges are treated as grillage beam systems. Table 1 presents the mass and girder properties of these bridges. According to the traffic counts at two locations on the interstate highway L95 and 1-75, four typical trucks are selected in this study. Mathematical models of these trucks are established based on the data of nationwide-used trucks H20-44, HS20-44, type 3S2, and type 3-3. The gross vehicle weight (GVW) of these trucks is, respectively, 97, 151, 294, and 363 kN (21.8, 33.9, 66.1, and 81.6 kips). In this study, empty trucks are excluded by setting certain dividing line. These models simplify the trucks into several rigid bodies (tractor and trailer) connected with suspensions by springs and dampers. The masses of the bodies are derived according to their static equilibrium relationship with the measured axle weights. The total number of degrees of

* Corresponding author. Tel.: +1 (305) 348-3048; Fax: +1 (305) 348-2802; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

677

C. Liu et al. /First MIT Conference on Computational Fluid and Solid Mechanics Table 1 Mass and girder properties of bridges Span length L (m)

Girder / X 10-2 (m4) (2)

m (kN/m)

I X 10-2 (m^)

Jd X 10-2 (m^)

m (kN/m)

(6)

(7)

(3)

(4)

(5)

9.14

3.209

0.619

-

18.29

7.788

0.740

30.48

26.872

1.363

42.67

56.361

1.410

17.60^ 13.27^ 20.44^ 15.78 ^^ 27.07^ 22.43 ^^ 31.73^ 27.09^

(1)

Diaphragm at ends

Intermediate diaphragm Jd X 10-2 (m4)

2.060

6.972

5.777 14.87

9.103

I X 10-2 (m4)

Jd X 10(m4)

(kN/m)

(8)

(9)

(10)

0.428

2.094

1.646

3.029

0.478

3.530

1.664

4.676

0.718

5.182

2.014

6.830

1.142

6.801

2.522

/ = inertial moment; Jd = torsional inertial moment; and m = mass per unit length. ^ Exterior girders. ^ Interior girders.

freedom is, respectively, 7, 11, 16, and 18. The equations of motion of the vehicle systems were derived using Lagrange's formulation. Detailed derivation refers to Wang and Huang [10]. The equation of motion of a vehicle-bridge interaction

M A + C A + K,5, = F ,

(1)

where M^ = global mass matrix of bridge structure; K^ = global stiffness matrix of bridge structure; C^ = global damping matrix of bridge structure; 8b, 8b, 8b = global nodal displacement, velocity, and acceleration vectors; and F^ = global load vector due to the interaction between bridge and vehicle. The interaction force between the iih wheel of the vehicle and the bridge is given as follows: Kt = Ktzi Utzi + Ct,i Utz

(2)

where Kt, tire stiffness of the /th wheel; Ct, = tire damping coefficient of the /th wheel; Utzi = Zwi — {—Usri) — (—Zbi), the relative displacement between the /th wheel and bridge, and the superscript dot of Utzi denotes differential with respect to time; Zwi = vertical displacement of the /th wheel; Usri = road surface roughness under the /th wheel (positive upwards); and Zbt = bridge vertical displacement under the /th wheel (positive upwards), which can be determined by the nodal displacement 8^ and the displacement interpolation function of the element [3]. In the present study, the fourth-order Runge-Kutta integration algorithm is employed to solve the nonlinear equations of motion of vehicle. The dynamic equations of the bridge are solved by the modal superposition procedure based on the subspace iteration method.

3. Road surface roughness Honda et al. [5] presented the power spectral density (PSD) functions for the characteristics of road surface roughness along the longitudinal direction, as well as the calculated coherence function of the roughness at an interval of 1.5 m. To reflect the variation in the transverse direction, an approach suggested by Samaras et al. [8] is employed to simulate road roughness based on a given spatial correlation relationship:

Y, = J^^iXri

- ^A,-Y,_,- +BoX,

(3)

where Y^ = two-variate discrete time series (r = 1,2); Aj and Bj = 2 x 2 auto-regressive and moving average (ARMA) coefficient matrices, respectively, and these matrices can be derived from the prescribed (/? + !) x (/? + 1) correlation function matrix; X^ = two-variate Gaussian white noise series with mean zero and satisfying: £ [ X , X j ] - lm8r,

(4)

where I^ = 2 x 2 identity matrix; Srs — Kronecker's delta. In this study, the parameter p and q are chosen as 49 and 40, respectively. The spatial coherence function in the transverse direction can be derived from available measured data. Based on Honda's study [5], the value of correlation function Coh^{^, 0) in the range of 0 = 0.01 to 1.0 cycle/m is roughly 0.4. Thus, the coefficient of correlation can be obtained as c = Coh(^, 0 ) = 0.63. The midpoint in the good condition range is used for roughness coefficients a, i.e. a = 0.62 x 10"^ m V ( m - c y c - ^ ) . The simulated correlated road surface roughness and its autoand cross-correlation functions are illustrated in Fig. 1.

678

C. Liu et al. /First MIT Conference on Computational Fluid and Solid Mechanics

O) -1

o cc

10

15

20

25

Distance (m)

^

8e-6 -

£

4e-6 -

1

0 -

(b)

Target Simulated(Left Line) Simulated (Right Line) "v^=^s>--'^''^^^""^^^^^^^^^

20

40

60

80

100

Number of Intervals 6e-6

2

O

0

20

40

60

80

100

Number of Intervals Fig. 1. Simulated road surface roughness and correlation functions, (a) Road surface roughness, (b) Auto- and cross-correlation functions. 1.0 0.8

o

0.8

j

^^i^v

SPAN LENGTH L = 9 . 1 4 M

06

o

p ^ D O 0.4

n Q

0.6

1

au

-Q2 -j

— • — T Y P E 10

0.3

* TYPE 9 ^><—TYPE 8-1

0.2

^

2

HS20-44

—»—TYPE10

—^«—TYPES

1

4

2

0.6

SPAN LENGTH L » 30.48M

0.6

CD •|O0.3

4

0.7

0.7

z

3

(b)

NUMBB^ OF GIRDER

NUMBER OF GIRDER

o 0.5 ioo.4

^

0.0

f___

3

—^—

\

0.1

(a)

-0.4

SPAN LENGTH L =:18.29M

\

D O 0.5 CQ H 0.4

CD H "

0.2

0.7

o

— • — -HS20-44 -TYPE 10

0.3 0.2

-TYPES

0.1

(c)

0.0 1

2

3

4

SPAN LENGTH L = 4 2 . 6 7 M

0.5 0.4

.

CO ^ 0.2 -I O

V.

0.1 0.0

\

—^—

HS20-44

— • — T Y P E 10 —^<—TYPE 8-1 -^•e-TYPE 5

1

NUMBER OF GIRDB^ Fig. 2. Lateral distribution under various truck types.

(d)

^

___^ 2

3

4

NUMBB^OFGIRDBl

C Liu et al. /First MIT Conference on Computational Fluid and Solid Mechanics 4. Lateral distribution factor

Type 10 (dynamic) AASHTO Standard HS20-44 (Static) Type 9 (static)

The wheel load distribution factor is defined as Mi • n Mt

679

(5)

where Mt = the sum of maximum moment/shear of all girders at the specific section; n — number of wheel loads in the transverse direction; and Mi = maximum moment/shear of the /th girder at the section. In this study, the dynamic moment/shear for Mt and Mi (including impact effect) is taken into account. Fig. 2 shows the wheel load distribution factor of dynamic moment at midspan when a single truck travels along the center of lane 1. The results are obtained based on one simulation and a traveling speed of 88 km/h. From Fig. 2, it is observed that the five selected trucks cause similar lateral moment distribution among the five girders, regardless of the variation in their axle weights and configurations. To examine the distribution factors specified by AASHTO Standard [1] and AASHTO LRFD [2], the simulation is performed 20 times and an average is taken for each case. A two-lane loading (using the same truck) is considered in the analysis, which is achieved by the superposition of one-lane loading results. This assumes the symmetry of distribution factors for loading on each lane. Fig. 3 shows the maximum wheel load distribution factors of moment at midspan and shear at end along with the specified values for interior girders by AASHTO Specifications (transferred to wheel load case). Also, in Fig. 3, the distribution factors are calculated on the basis of static moments and shears. It can be seen that the maximum distribution factors are similar for different truck types. The computed maximum factors based on both static and dynamic moments/shears are similar. The calculated factors for interior girders are lower than the specified values. This is consistent with the measured results reported by Kim and Nowak [7]. However, it should be noted that in this study, a two-lane traffic is used, while the specified values are obtained based on the controlling static moment/shear caused by any number of trucks that fit the bridge transversely [11]. Thus, the specified values may lead to higher distribution factors.

9.14

18.29

30.48

42.67

Span Length L (m) 1.8 -rl - • — HS20-44 (dynamic) —ik—Type 9 (dynamic) 1.7 -AASHTO LRFD -Type 10 (static) 1.6 1.5 1.4 1.3 -I 1.2 j (0

O

Type 10 (dynamic) AASHTO Standard HS20-44 (static) Type 9(static)

1 . 1 -I

1 0.9 0.8 9.14

18.29

30.48

42.67

Span Length (m) Fig. 3. Comparison of wheel load distribution factor, (a) Moment, (b) Shear.

Acknowledgements This research project was sponsored by the Florida Department of Transportation (FDOT) State Project No. BC-379. This support is acknowledged and greatly appreciated. The presented results are those of the writers and do not necessarily reflect the opinions of the sponsor.

References 5. Conclusions (1) Despite the variation in axle weights and configurations, the five typical trucks cause close lateral distribution factors. (2) Calculated distribution factors based on both static and dynamic moments/shears are similar. (3) Calculated lateral distribution factors for interior girders based on loading of two lanes are lower than the values specified by AASHTO Specifications.

[1] AASHTO. Standard Specifications for Highway Bridges, 16th edn. American Association of State Highway and Transportation Officials, Washington, DC, 1996. [2] AASHTO. LRFD Bridge Design Specifications-Customary U.S. Units, 2nd edn. American Association of State Highway and Transportation Officials, Washington, DC, 1998. [3] Clough RW, Penzien J. Dynamics of Structures, 2nd edn. New York, NY: McGraw-Hill, 1996. [4] Florida Annual Average Daily Traffic Report. Florida Department of Transport, Tallahassee, FL, 1998.

680

C Liu et ai /First MIT Conference on Computational Fluid and Solid Mechanics

[5] Honda H, Kajikawa Y, Kobori T. Spectra of Road Surface Roughness on Bridges. J Struct Div ASCE 1982;108(9): 1956-1966. [6] Huang DZ, Wang T-L, Shahawy M. Impact Studies of Multigirder Concrete Bridges, J Struct Eng ASCE 1993; 119(8):2387-2402. [7] Kim S, Nowak AS. Load distribution and impact factors for I-girder bridges, J Bridge Eng ASCE 1997;2(3):97-104. [8] Samaras E, Shinozuka M, Tsurui A. ARMA Representation of Random Processes, J Engrg Mech ASCE 1985;111(3): 449-461.

[9] Standard Plans for Highway Bridge Superstructures. U.S. Department of Transport, Federal Highway Administration, Washington, DC, 1990. [10] Wang T-L, Huang DZ. Computer Modeling Analysis in Bridge Evaluation - Phase III, Final Report, Project No. FL/DOT/RMC/0542(3)-7851, Florida Department of Transport, 1993. [11] Zokaie T. AASHTO-LRFD Live Load Distribution Specifications. J Bridge Eng ASCE 2000;5(2):131-138.

681

Effects of diaphragms on seismic response of skewed bridges Shervin Maleki * Department of Civil Engineering and Construction, Bradley University, 1501 W. Bradley Ave., Peoria, IL 61625, USA

Abstract Steel cross-frames are on the load path for seismic forces moving from the superstructure to the substructure. This paper investigates the effects of this stiffness on the free vibration of skewed bridges. Slab-on-girder, single-span bridges with skew angles ranging from 15 to 60 degrees are considered. It is assumed that the bridges have elastomeric bearing at the ends. It is concluded that for the practical range of support stiffnesses the first period increases with increasing skew angle but the second period is almost a constant. It is also shown that in an actual earthquake, such as El Centro, the displacements are always perpendicular to the abutments and the demand goes up with increasing skew angle up to 45 degrees. Bridges with 60 degrees skew behave erratically. Keywords: Bridge; Skewed; Cross-frame; Diaphragm; Seismic; Elastomeric bearing

1. Introduction Skewed bridges occur routinely on highway projects. Their design for earthquake loads propose a challenge to structural engineers. This is due to the fact that their vibrational modes do not uncouple in orthogonal directions as non-skewed bridges do. In this study, the free vibration characteristics of skewed, simply supported slab-on-girder bridges with end-diaphragms in the form of cross-braces will be considered. In addition, the bridge bearings are assumed to be elastomeric with finite stiffness in the longitudinal direction. A typical plan view of a skewed slab-beam bridge is shown in Fig. 1. The partial end cross-sectional view of the same bridge is shown in Fig. 2. According to the American Association of State Highway Officials AASHTO-LRFD [1], every bridge has to be constructed with a diaphragm or cross-frame at its two ends. In addition, intermediate diaphragms shall be provided along the span based on a rational analysis. End-diaphragms provide an important load-path for the seismically induced loads. Seismic forces at the deck would have to pass through the cross-frame to arrive at the top of bearings. During recent earthquakes, many steel bridges have suffered end-diaphragm damage [2]. Diaphragm effects for *Tel.: -hi (309) 677-2713; Fax: +1 (309) 677-2867; E-mail: sherv @ bradley.edu © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

vertical loading has been a subject of study, e.g., [3]. However, recendy, Zahrai et.al. [4,5], studied the effect of diaphragms on seismic behavior of straight bridges. They have indicated that end-diaphragms in the form of single channel sections connecting the girders perform poorly in earthquakes. This is due to their flexibility and lack of connection to the lower part of the girder. They have also concluded that intermediate diaphragms do not affect the seismic behavior of bridges significantly. A typical elastomeric bearing detail for these bridges is flexible along the longitudinal direction, but restrained with a side-retainer in the transverse direction. The author has investigated the elastomer's effect for skewed bridges in more detail [6]. In this study the effect of skew angle and diaphragm stiffness, on seismic response of bridges is considered. It is assumed that the concrete deck is rigid in its plane. Turkington et al. [7], claim that this assumption is valid for decks with length-to-width ratios of less than eight. This is because the in-plane deck stiffness coupled with girder stiffnesses is very high in comparison with cross-frame and elastomer stiffnesses.

2. Free vibration formulation The finite element model of a typical skewed bridge is shown in Fig. 3. The degrees of freedom at the center of mass are also shown. Springs k^ represent the stiffness

682

S. Maleki/First MIT Conference on Computational Fluid and Solid Mechanics

-SKEWANGLE=a

EXP. JOINT ABUT. SEAT

Fig. 1. Bridge plan view.

- S (Typ.) •*'•'•

Brg. stiff. Typ^

•*• •

m:

rnwi2x40(American Std.f

L4x4xl/4 3/8'in.Bent Plate(Typ.) Fig. 2. Bridge cross-section.

Shell Element (Typ.) Bearing Stiflhessfryp.)

Fig. 3. Finite element model.

683

S. Maleki/First MIT Conference on Computational Fluid and Solid Mechanics of elastomeric bearings in the X direction, and springs ki represent the total lateral stiffness of the end-diaphragms in the T direction. Let M represent the total mass of the bridge superstructure. Then the undamped free vibration equation for our bridge subjected to ground acceleration in the X and Y directions can be written as follows,

M

0

1 Ux

0

M

yiiy

(1)

• +

Noting the summation of all spring stiffnesses as ^5 = X]^5 and Ki = ^ki, and transforming the local coordinates N and T of the Ki spring to the global X and Y coordinates and solving the eigenvalue problem of free vibration leads to the following equation for periods of vibration [6], ^TT^M

T =

[(K, + Ki) ± y/{K, +

Kif-4K,Kicos^a^

\ Introducing a non-dimensional parameter as p Eq. (2) can be rewritten as, STT^M

N

Ki [(1 + ^) ± V(l + «'-4iecos2Qf]

(2)

Ks/Ki, (3)

As shown above, for the case of a skewed bridge, the period of vibration is coupled and it depends on the skew angle a in addition to stiffnesses of springs Ks and Ki .

3. Cross-frame effects 3.1. Free vibration

the mass. Referring to Fig. 1, and assuming a spacing of girders S to be 2.134 meters, for spans of 9, 18 and 36 meters, one can obtain an estimate of the bridge mass to be 65, 130 and 260 tons respectively. These masses represent short, medium and long span slab-on-girder bridges. A typical value for a total of ten elastomers stiffnesses for this bridge is Ks = 32.7 M N / m and is used throughout this study. Based on the above values, and to demonstrate the effects of cross-frame stiffness on the vibration characteristics of a skewed bridge, a plot of Ki versus the first and second periods of vibration for a mass of 130 ton is shown in Fig. 4. Eq. (3) is used in all these plots. The author [6] has verified the accuracy of Eq. (3) by comparing the results with finite element model using SAP2000 [8] program. 3.2. Forced vibration To examine the effects of cross-frames on the response of a bridge in an earthquake, the bridge in Fig. 3 is subjected to El Centro earthquake in the X and Y directions. Program SAP2000 is used in the analysis and a 2% damping is assumed. Fig. 5 shows the results of this analysis for a mass of 130 ton. It is found that for ^ values (see Eq. (3)) higher than 0.02 the skewed bridge moves in the X and Y directions. However, for ^ values equal to and lower than 0.02, the bridge moves perpendicular to the abutments (A^-direction, Fig. 1) regardless of skew angle and for both loading cases. A practical maximum value of yS is 32.7/1500 = 0.022 for our bridge. Higher cross-frame stiffnesses result lower ^ values.

As shown in Eq. (3), the periods are proportional to square root of mass. Span and width of the bridge affect

a=0

M=130, Ks=32.7

1

Q

K

\

1\ \

3 2.5

LU Q_

2

CO

1.5

1

^i^""-!;"10

~. 100

. 1000

CROSS- FRAME STIFFNESS(MN/m) (a)

1

0 45

M=130,Ks=32.7

4.5 3.5

a=30

— - — - a-45 ————-a—60

5 4

O cc

— a=15

0.4

o

0.35

en

0.3

0)

Q O

0.25

QH LU CL Q Z

0.2 0.15

O

o

0.1

LLI

0.05

0.5 0

0 10

10000

100

1000

CROSS-FRAME STIFFNESS(MN/m) (b)

Fig. 4. Period vs. elastomer stiffness.

10000

684

S. Maleki/First MIT Conference on Computational Fluid and Solid Mechanics a=15

-a=30 -a=60

•a=45

a=15 a=45

-a=30 -a=60

80

80

70

70 60

60 _ : (0 Q. O

(0

1000

10000

50

E '^

50

40

_ : TO Q. O

40

30

30

20

20

10

10

100000

1000

10000

100000

CROSS-FRAME STIFFNESS(MN/m) (b)

CROSS-FRAME STIFFNESS(MN/m) (a)

Fig. 5. Displacement vs. cross-frame stiffness.

4. Conclusions

References

Referring to Fig. 4, it is observed that both translational periods decrease as cross-frame stiffness increases. Also, the first period increases as skew angle goes up. The reverse is true for the second period. Periods are sensitive to variation of cross-frame stiffness only in the low range. Note that a minimum requirement per reference [9] for end cross-frames of this bridge is shown in Fig. 2 and has a total stiffness of AT/ = 1500 M N / m . Hence, for practical applications, one can assume that the first period remains independent of cross-frame stiffness for all skew angles. For the second period, there is some sensitivity to cross-frame's stiffness but it is independent of skew angle. These observations are true regardless of bridge mass.

[1] AASHTO, LRFD bridge design specification, 2nd edition. American Assoc, of State Highway and Transportation Officials, Washington, D.C. 1998. [2] Astaneh-Asl A, Bolt B, McMullin KM, Donikian RR, Modjtahedi D, Cho SW. Seismic performance of steel bridges during the 1994 Northridge earthquake. UCB Report CE-STEEL94/01. Berkeley, CA, 1994. [3] Azizinamini A, Kathol S, Beachman M. Influence of cross frames on load resisting capacity of steel girder bridges. AISC Eng J 1995;32(3): 107-116. [4] Zahrai SM, Bruneau M. Impact of diaphragms on seismic response of straight slab-on-girder bridges. J Struct Eng ASCE 1998;124(8):938-947. [5] Zahrai SM, Bruneau M. Ductile end diaphragms for seismic retrofit of slab-on-girder steel bridges. J Struct Eng ASCE, 1999;125(l):71-80. [6] Maleki S. Effect of elastomeric bearings on seismic response of skewed bridges. In: Proceedings 5th International Conference on Computational Structures Technology. Civil-Comp Press, pp. 177-182, 2000. [7] Turkington AJ, Carr AJ, Cooke N, Moss PJ. Seismic design of bridges on lead-rubber bearings. J Struct Eng ASCE 1989;115(12):3000-3016. [8] Computers and Structures, Inc. SAP2000, Version 7.4, Integrated Structural Analysis and Design Software. Berkeley, CA, 2000. [9] Metric Bridge Manual. Illinois Department of Transportation. Springfield, Illinois, 1999.

Referring to Fig. 5, it can be concluded that the displacement demand is independent of stiffness for each skew angle. Furthermore, the demand for K-direction earthquake goes up with skew angles up to and including 45 degrees. Displacement demand for X-direction earthquake is the same, except that, for long span bridge it goes down with skew angle. Skew angles above 45 degree behave erratically and depend on frequency content of the input motion.

685

Applications of artificial-life techniques to reliability engineering Akinori S. Matsuho^*, DanM. Frangopol^ ^ Anan College of Technology, Department of Civil Engineering, Minobayashi, Anan 774-0017, Japan ^ University of Colorado at Boulder, Department of Civil Engineering, Boulder, CO 80309-0428, USA

Abstract This paper considers applications of artificial-life techniques, such as genetic algorithm and genetic programming, to reliability engineering. These techniques include reliability-based optimization, multiple-integration of joint-probability density function, and identification of a sample function of a random function. Several numerical examples are presented. Keywords: Artificial life; Genetic algorithm; Genetic programming; Reliability engineering; ReHability-based optimization; Multiple integration; Identification of sample function

1. Introduction In general, analysis and design of engineering systems based on reliability theory provide a consistent basis for combining the effects of different sources of uncertainties in the computational process. The process is not always efficient. Application of artificial-life techniques to such engineering problems may realize computational efficiency. This paper considers applications of artificial-life techniques, such as genetic algorithm and genetic programming, to reliability engineering. These techniques include reliabihty-based optimization, efficient multiple-integration, and identification of a sample function of a random function. Several numerical examples are presented.

2. Numerical examples This section presents several applications of artificiallife techniques to reliability engineering. 2.1. Reliability-based optimization Reliability-based optimization of a structural system can be formulated and solved based on the information integration method and the genetic algorithm. In this section, the method is reviewed according to Matsuho and Shiraki * Corresponding author. Tel./fax: +81 (884) 237-203; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

[1]. The information integration method, which is a rational evaluation method of a system [2], can be used to evaluate the alternatives with respect to many quality items including reliability of the relevant system. On the other hand, the genetic algorithm is an efficient optimization method which can be appHed to problems with continuous and/or discrete parameters. The proposed optimization method, in which a fitness of the genetic algorithm is evaluated by using the information integration method, is efficient and effective. In a numerical example, an optimal design problem of a highway bridge girder subjected to vehicle live loads is considered. This girder bridge [3] has four vehicle lanes, seven main girders I-beam simply supported and one cross beam. In the design, it is assumed that the design parameter is the web width t of girder, assumed equal to the thickness of flange. The numerical values used in the design are as follows: total height of I-beam section h = 2 m, total width of I-beam section B = 32 cm, allowable bending stress a« = 24,000 tf/m^, unit weight of steel p = 7.85 tf/m^ span length of girder L = 39.2 m, and interval between the main girders a = 2.85 m. The live load acting on each girder is modeled based on simulation using the observed traffic data [3]. The simulation shows that the annual maximum bending moment can be modeled by the type I extreme value distribution. This simulation also considers the effect of cross beams. The design considers the failure probability Pf and the weight of each girder per unit length w as system design parameters. Pf is assumed to be the probability that the maximum bending moment of each girder during lifecycle (50 years)

686

A.S. Matsuho, DM. Frangopol/First MIT Conference on Computational Fluid and Solid Mechanics

exceeds the allowable bending moment calculated according to the Japanese current design code [4]. As a result of the computation, the optimum solution is 26.2 mm and the associated failure probability is Pfopt = 2.12 x 10~^. Although the solution ^opt is associated with a cross-section larger than that of the actual bridge [3], a substantial reliability improvement was realized by using the proposed method. 2.2. Efficient multiple-integral

and its application

As indicated in Matsuho and Frangopol [5], multipleintegrals can be solved efficiently using quasi-random numbers and genetic algorithm. A /:-dimensional multiple integral / = f"-ff{x)dx = /..•//U,,...,x,)d;c, • • -dxi, can be approximated by using the following form [6]: ^(A^) = ECNnf{[na,l

[na2l • • • , [na,])

0

5

10 15 20 25 30 35 40 45 50 55

70 75 80 85 90 95

Fig. 1. Given data.

(1)

where ?„ = {[na\],[na2], - • • , [^Qf^t]) is a sequence of quasi-random numbers, and n = 1,2, The constants aj, where j = 1, 2 , . . . , ^, are real numbers. These numbers are computed to minimize the error between the strict solution / and the approximate solution S(N). In this study genetic algorithm is used to find the optimal constants Uj. Table 1 shows computational results of an integral / of a joint probability density function of two normal variables with various correlation coefficients p. Table 1 shows that the proposed method gives a good precision. The proposed method can be applied to various problems to evaluate the system reliability including a problem in which a first passage probabiHty of a stochastic process is evaluated [5]. 2.3. Identification of sample function of random

function

If a sample function xit) of the random function X{t) can be approximated by a chaotic function xdt) [7], the random function can be easily characterized [8]. When data on the magnitudes Xd, Xc2,. •., Xc, of x{t) at discretized values ti,t2,'--,tn are lacking a chaotic function model Table 1 Computational results of integral / of two-dimensional normal distribution Correlation coeffifient

SiN) using the proposed method

Strict solution

0.1 0.3 0.5 0.7 0.9 0.99

0.0002731 0.0007191 0.0015665 0.0030151 0.0053354 0.0062118

0.0002621 0.0007013 0.0015598 0.0030105 0.0053329 0.0062094

0

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Fig. 2. Identified function.

can be applied [8]. A numerical example identifies a given chaotic function by using genetic programming. In general, genetic programming can identify a function even if no information about shape of the given function exists. Let consider a logistic equation defined as:

>'/

: Xi+i = r X Xi X (l.O — Xi)

(2)

Fig. 1 shows the shape of the function associated with Eq. (2) under conditions that r = 3.95 and xi = 0.4. The identified function by genetic programming is shown in Fig. 2. By comparing Fig. 1 and Fig. 2, we could recognize the excellent approximation provided by genetic programming.

3. Conclusion Applications of the GA and the GP to reliability engineering were considered. Several numerical examples show the quality of approximations obtained by applying artificial-life techniques to various reliability engineering problems.

A.S. Matsuho, DM. Frangopol/First MIT Conference on Computational Fluid and Solid Mechanics References [1] Matsuho AS, Shiraki,W. Efficient method of reUabihty-based optimization design using information integration method and genetic algorithm. Proc. of ICASP-7 1995;1:371378. [2] Matsuho AS, Shiraki,W. Design method satisfying safety requirements for various Hmit-states based on information integration method. Proc. of ICOSSAR 1989;3(l):2243-2246. [3] HDL Committee. Report on Investigation of Design Load Systems on Hanshin Expressway Bridges. Osaka: Hanshin Expressway Pubhc Co., 1986, (in Japanese). [4] Japan Society of Highway. Japanese Steel Highway Bridge Specification 1, 1990, (in Japanese).

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[5] Matsuho AS, Frangopol DM. Development of efficient integral method using quasirandom number and genetic algorithm. In: Topping BHV, editor. Computational engineering using metaphors from nature. Proc. of CST 2000, pp. 85-89. [6] Tsuda T. Monte Carlo method and simulation. Tokyo: BaifuKan Co., 1987, (in Japanese). [7] Tanaka Y, Syohda K. ProbabiUstic consideration on behavior concerning random response of non-linear system. Proc. of the 15th symposium on Material and Structural Reliability 1997, pp. 127-132 (in Japanese). [8] Matsuho AS. Basic study on apphcation of artificial-life techniques to rehability engineering. Proc. of Symposium on ProbabiUstic and Statistical Methods of Decision Making 1998, pp. 63-68 (in Japanese).

688

HPC for the optimization of aeroelastic systems K. Maute *, M. Nikbay, C. Farhat Center for Aerospace Structures, Department for Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309-0429, USA

Abstract We consider the problem of optimizing for steady-state conditions a given aeroelastic system, by varying both aerodynamic and structural parameters such as the shape of the dry or wet surfaces, and the orientation of the composite fibers. The aeroelastic response of a given system is computed by coupling efficiendy a finite element solution method for the structure subsystem and a 3D Euler finite volume method for the fluid subsystem. The resulting large-scale systems of nonlinear/linear equations are solved by parallel computing based on a multi-level domain-decomposition approach. Keywords: Aeroelasticity; Optimization; Multi-level domain-decomposition

1. Introduction The design of complex aeronautical systems remains a challenging task because it must account for several nonlinear fluid, structure, and thermal coupling effects. While for such systems intuition, experience, and engineering skills still dominate the design process, numerical simulation offers an important potential for speeding up the design cycle. In general, a design problem is defined in terms of physical design parameters, objectives to be optimized, and constraints to be satisfied. Typically, the objective and constraint are functions of the system response which, in turn, is governed by a system of coupled partial differential equations (PDEs) that describe the various physical effects and their coupling. Because of complexity and computational cost issues, most often the coupling effects are neglected when developing an optimization method or applying it to a practical system. However, in many cases the overall performance of an aeronautical system can be governed by coupling effects and, therefore, multidisciplinary optimization methods in general, and aeroelastic optimization algorithms in particular, have flourished in recent years. Frequently, during an aeroelastic optimization, the aerodynamic loads are predicted by a linear theory (see, e.g. [8,6]). However, with the availability of faster computing platforms and the advent of parallel processing, nonlinear

flow theories and detailed finite element structural models have made their way into the aeroelastic optimization process (see, e.g. [7,9,11,12]). In this paper, we present a methodology for optimizing an aeroelastic system for steady-state conditions. The addressed aeroelastic problems are governed by a nonlinear flow theory and represented by a detailed structural finite element model. Our methodology is based on a modular framework for formulating and solving the target optimization problem, a three-field approach for describing an aeroelastic system, efficient staggered parallel procedures for evaluating an aeroelastic response, and for computing the analytically derived gradients of the optimization criteria. This methodology is applied to the optimization of the design of wing structures for steady-state conditions.

2. Aeroelastic optimization The aeroelastic optimization problem can be solved by combining three different numerical models, namely, the optimization, design, and analysis models (see, e.g. [14]). In the optimization model, the aeroelastic optimization problem is formulated as a generic abstract optimization problem: min z(s)

* Corresponding author. Tel.: +1 (303) 735-2103; Fax: -\-\ (303) 492-4990; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(1)

s

h(s) = 0

he

(2)

K. Maute et al. /First MIT Conference on Computational Fluid and Solid Mechanics g{s) > 0

ge

s = {s e .

SL

<s < Su)

(3)

3. Aeroelastic analysis and sensitivity analysis

(4)

In this work, we predict the aeroelastic response of the target system by coupling a 3D finite volume Euler flow code and a linear elastic finite element structural code. More specifically, we adopt the three-field formulation of Farhat et al. [5] for representing the coupled fluid/structure/moving-mesh problem, whose discrete form for a steady-state can be written as

where 5 is a set of ris abstract parameters restricted by lower and upper bounds SL and Su, z is a cost function of interest, h denotes a set of rih equality constraints, and g a set of rig inequality constraints. The abstract optimization problem (l)-(4) can be solved by numerical optimization algorithms (see, e.g. [17]). In this work, a Sequential Quadratic Programming (SQP) method is applied which has proven to be robust and efficient for a broad range of optimization problems [15]. In the design model, the physical design parameters are defined as functions of the abstract optimization variables s. For aeroelastic optimization problems, the structural and aerodynamic design parameters can cover, among others, the cross-sectional and thickness dimensions of the structural elements, the shape of the dry and wet surfaces, and the Mach number and angle of attack. In the simplest case, for example, the thickness of a structural element can be identified with an abstract optimization variable. However, in the case of shape optimization, for example, it is often necessary to define complex relations which link the shape variation to the abstract optimization variables. In this work, we follow the design element concept for describing the shape variations (see, e.g. [1]). In the analysis model, the optimization criteria are evaluated. The optimization criteria q — that is, the objective function z and the constraints h and g — can cover the aerodynamic performance factors such as the lift and drag, as well as the structural behavior descriptors such as the displacements, stresses, and strains. In general, the optimization criteria q depend on the aeroelastic response of the system characterized by the structural displacement vector u and the fluid state vector w, which in turn are functions of the physical design parameters, or rather the abstract optimization variables s. Therefore, q = q(s, u, w)

(5)

u = u(s)

(6)

w = w(s)

689

In this study, we follow the so-called Nested Analysis And Design (NAND) approach and assume that u and w always satisfy the aeroelastic state equations. For this reason, we do not include the aeroelastic state equations in the set of equality constraints (2), but determine the structural displacements u and fluid state variables w at each iteration of the optimization process. Also for evaluating the sensitivities of the optimization criteria, the state equations have to be taken into account and, therefore, the sensitivity analysis is assigned to the analysis model. Aeroelastic analysis and sensitivity analysis are discussed subsequently.

S(s,u,x,w)

(7)

=0

D(s,u,x)

= 0

(8)

F{s,x,w)

= 0

(9)

and where, in this work, S = Ku-

P(x, w)

(10)

D = Kx

with

(11)

X = u on F F/S

K is the finite element stiffness matrix associated with the structure, P is the external load vector that combines the aerodynamic load Pj transferred from the fluid to the structure and other specified structural loads such as gravity represented here by the vector PQ P = PO^PT

(12)

F is the vector of Arbitrary Lagrangian Eulerian (ALE) Roe fluxes resulting from a second-order finite volume discretization of the Euler flow equations, Fp/s is the interface between the fluid and structure meshes, and the motion of the fluid mesh x is determined by the improved spring analogy method described in Farhat et al. [3] and characterized by the fictitious stiffness matrix K. The coupled system of equations (7)-(9) can be solved efficiently by a staggered procedure which decomposes the coupled system into a fluid and a structure subdomain and allows the application of different solution methods to the different fluid and structure subproblems (see, e.g. [4]). In this work, we employ the specific staggered algorithm described in Maute et al. [11] which can be summarized as follows: • The Hnear elastic structural response (7) is computed for a given aerodynamic load by a direct solver. • The fluid domain is decomposed in multiple domains. Based on this decomposition the fluid mesh motion, which is described by a Dirichlet boundary condition problem (8), is computed by a parallel Preconditioned Conjugate Gradient (PCG) solver. • The fluid state variables are computed by applying a single Newton-Raphson subiteration combined with a homotopy approach to Eq. (9). The resulting linear system of equations is solved by a parallel Generalized Minimum Residual (GMRES) method preconditioned by a Restricted Additive Schwarz (RAS) method (see [2]).

690

K. Maute et al /First MIT Conference on Computational Fluid and Solid Mechanics

optimum shape

initial shape

• At > 0

^ At < 0

Fig. 1. Optimum shape and optimum thickness correction Ar. • The motion of the wet surface of the structure is transferred to the fluid system and the aerodynamic loads computed in the fluid mesh are projected onto the structure such that potentiaUy non-matching fluid and structure meshes are accounted for (see, e.g. [10]). The crucial point in the proposed methodology is an efficient and accurate sensitivity analysis. In this paper we apply the analytical method for computing the sensitivities of three-dimensional non-linear aeroelastic static problems which was presented in Maute et al. [11]. This approach follows the general framework for deriving the global sensitivity equations of coupled systems of SobieszczanskiSobieski [16], and is based on the above three-field formulation for describing an aeroelastic system. It includes the above multi-level domain decomposition and addresses the problem of nonconforming finite element-finite volume discretizations of the structure and the fluid. The gradients of the optimization criterion qj with respect to the optimization variable 5/ can be computed as follows dqj dqj ^ du dqj ^ dx dqj ^ dw dsi dsi du dsi dx dsi dw dsj where the derivatives du/dsf, dx/dsi, dw/dst of the aeroelastic response with respect to the abstract variable 5/ are given by differentiating the governing aeroelastic equations (7)-(9), which yields

- ds dSi

dD

+

dSi

dF -

dSi

-

- dS du

dS dx

dD 'du

dD 'dx

0

dF

ds -

- du

dw

0 dF dw -

dx dSi

du; - d^

(14)

where A denotes the Jacobian of the aeroelastic problem. Eqs. (13) and (14) can be evaluated following the direct approach or the adjoint approach (see, e.g. [12]). We note that whether the direct or adjoint approach is selected, a coupled system of linear equations has to be solved where, in the direct case, A and, in the adjoint case, A^ is the linear operator. In both cases, the coupled linear system can be solved efficiently by another staggered procedure which is similar to the one applied for computing the aeroelastic steady-state response. Again on a generic level,

the coupled system is decomposed into a fluid and a structure subdomain and different solution methods are applied to the different fluid and structure subproblems. Again, the fluid subdomain is decomposed into several subdomains. The differentiated governing equations for the fluid mesh motion are solved by a parallel PCG algorithm and the ones for the fluid state are solved by a parallel GMRES algorithm. The staggered procedures following the direct and the adjoint approaches are described in detail in Maute etal. [11,12].

4. Numerical examples The methodology presented above is applied to the optimization of the design of the ARW2 built at NASA Langley. We seek to optimize the drag/lift ratio under lift, stress and displacement constraints for cruise conditions in 12,000 m altitude by varying the wet shape and the thickness of the built-in stiffeners. The optimization problem is defined in detail in Maute et al. [13]. Here, we only summarize the optimization result. The initial and the optimum shapes of the wet surface as well as the optimal thickness corrections are shown in Fig. 1. The objective function, that is the drag-to-lift ratio is reduced by 19.54% by essentially decreasing more the drag than the lift. The backsweep is increased in order to reduce the wave and induced drag. The twist of the wing is adjusted such that the local angle of attack is slightly increased. The weight of the wing has been significantly reduced by reducing the thickness of the built-in stiffeners.

References [1] Bletzinger K-U, Kimmich S, Ramm E. Efficient modeling in shape optimal design. Comput Syst Eng 1991;2:483-495. [2] Cai X-C, Farhat C, Sarkis M. A minimum overlap restricted additive schwarz preconditioner and applications in 3dflowsimuladons. In: Mandel J, Farhat C, Cai X-C (Eds), Tenth International Conference on Domain Decomposition Methods for Partial Differential Equations, 1998. [3] Farhat C, Degand C, Koobus B, Lesoinne M. Torsional springs for two-dimensional dynamic unstructured fluid meshes. Comput Methods Appl Mech Eng 1998;163:231245.

K. Maute et al. /First MIT Conference on Computational Fluid and Solid Mechanics [4] Farhat C, Lesoinne M. Two efficient staggered procedures for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems. Comput Methods Appl Mech Eng 2000;182:499-515. [51 Farhat C, Lesoinne M, Maman N. Mixed exphcit/impHcit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution. Int J Numer Methods Fluids 1995;21:807-835. [6] Friedmann P. Helicopter vibration reduction using structural optimization with aeroelastic/multidisciplinary constraints — a survey. J Aircraft 1991;28:8-21. [7] Giunta A, Sobieszczanski-Sobieski J. Progress towards using sensitivity derivatives in a high-fidelity aeroelastic analysis of a supersonic transport. AIAA 98;4763. [8] Haftka R. Structural optimization with aeroelastic constraints — a survey of U.S. applications. Int J Vehicle Des 1986;7:381-392. [9] Hou G-W, Satyanarayana A. Analytical sensitivity analysis of a statical aeroelastic wing. AIAA 2000;4824. [10] Maman N, Farhat C. Matching fluid and structure meshes for aeroelastic computations: A parallel approach. Comput Struct 1995;54:779-785.

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[11] Maute K, Lesoinne M, Farhat C. Optimization of aeroelastic systems using coupled analytical sensitivities. AIAA 2000;560. [12] Maute K, Nikbay M, Farhat C. Analytically based sensitivity analysis and optimization of nonlinear aeroelastic systems. AIAA 2000;4825. [13] Maute K, Nikbay M, Farhat C. Coupled analytical sensitivity analysis for the optimization of 3-dimensional nonlinear aeroelastic systems, to be submitted. [14] Maute K, Schwarz S, Ramm E. Structural optimization — the interaction between form and mechanics. J Appl Math MechZAMM 1999;79:651-674. [15] Schittkowski K, Zillober C, Zotemantel R. Numerical comparison on nonlinear programming algorithms for structural optimization. Struct Optim 1994;7:1-28. [16] Sobieszczanski-Sobieski J. Sensitivity of complex, internally coupled systems. AIAA J 1990;28:153-160. [17] Vanderplaats G. Numerical optimization techniques for engineering design: with applications. New York: McGrawHill, 1984.

692

Updating of a plane frame using neural networks B. Miller*, L. Ziemianski Rzeszow University of Technology, Department of Structural Mechanics, ul. W. Pola 2, PL-35959 Rzeszow, Poland

Abstract This paper presents an application of artificial neural networks (ANN) for updating of a mathematical model of a structure. As input information to the network the frequency response function (FRF) is used. In order to reduce the dimension of input vector necessary for the model updating, the FRF is compressed even to one tenth of initial number of values by a dedicated neural network called characteristic replicator. Wide variety of network architectures and different input definitions have been tested. The sensitivity of this method in predicting selected model parameters has been checked and the optimal location of impulse excitation for dynamic measurements has been found. Keywords: Artificial neural networks; Updating; Vibrations; Simulation; Identification

1. Introduction An accurate and representative computer models are necessary to predict the dynamic characteristics of a structure under the study. A dynamic model of a structure (built with the finite element method or discrete modelling) is verified by testing physical model and comparison of responses from these two models. There are often discrepancies between these results so the dynamic model must be modified until a good agreement between responses of these models is achieved. The process of modifying the dynamic model is called updating [1]. In this paper a dynamic model updating technique using neural networks is demonstrated. To solve this problem multi-layer feed-forward (MLFF) neural networks are used [2,3]. FRFs are used as input data vectors for the networks updating the mathematical model. However, the FRF of even small models contains too many points to use them all as an input of a network. If the input vector has a high dimension, the number of learning patterns required for adequate network generalization is also very high [4]. Therefore it is necessary to reduce the number of inputs of the neural network. In this paper the input data have been reduced by the technique of compression by a neural network [5].

* Corresponding author. Tel.: +48 (17) 865-1482; Fax: +48 (17) 854-9027; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

In the presented updating algorithm two separate stages can be distinguished: • extraction of model dynamical features from FRF using the so-called characteristic replicator, • building of an ANN for the model updating based on a compressed FRF. This algorithm has been tested on a simple model of a 12-story, one bay frame on elastic foundation, consisting of concentrated masses coupled by springs (see Fig. 1) [6]. Two examples of updating are presented: updating of all stiffnesses of the model and updating of all masses of the model. In both cases the assumption that only one stiffness (or adequately one mass) at a time can vary has been made. All other parameters (including all masses in the first case or all stiffnesses in the second case) are invariable. The learning and testing patterns are taken from numerical simulations. They have been calculated by simulation of changes of parameters being updated in the range of 70130% of the initial value of the parameter in steps of 2%. The patterns concerning the change of the second stiffness of the model have been removed during stiffness updating because for this particular model they don't entail significant changes of the FRF [7]. In masses updating from the same reason all patters concerning the change of the first mass have also been removed.

B. Miller, L. Ziemianski /First MIT Conference on Computational Fluid and Solid Mechanics

a) -^0^ Point of measurement

^}S

— ^ ^ 1 3

^t2

—•qia

^ff

— • ^ 1 1

^10

---•%

Frequency Response 1 Function

—•qg

^8

—•^^s

^7

-—•<17

CL

—•qe

<

EJi

Impulse excitation

c)

>^9

m^ "^5

---•<^5

"^4

—•<^4

^3

--•^3

m^

\\y//y\\\//x\\\///>

•Qg -Qi

693

3

£

Mi l Frequency

Fig. 1. The model and the example of frequency response function. 2. Updating of the model

It is possible to compress a FRF calculated for the mathematical model even to one tenth of initial number of values without a loss of important information and a network used for model updating can base on condensed characteristics.

2.7. Characteristic replicator The calculated FRFs contain 205 points. In order to separate dynamic features from the FRF, a characteristic replicator has been built. Different networks of architecture 205-/Z-205 (where h is the number of neurons in a hidden layer) have been taught to replicate at the output the vector FRF data points given at the input. To obtain a network outputting a compressed FRF the output layer of the learned 205-/Z-205 network has been removed and its task has been taken over by the hidden layer. Network 205-/? give on the output FRF condensed to h values. Miller and Ziemianski [7] show that for the analysed problem of stiffnesses updating the optimal value of /z is 21. In masses updating the optimal value of h is also 21 (see Fig. 2). The replicator with 21 hidden neurons is able to decompress a FRF with accurate precision after the previous compression done in a hidden layer. Stiffnesses updating

10 12 14 16 18 20 21 22 24 Number of hidden neurons m 206- h-2QS replicator

2.2. Networks updating the model FRFs compressed by characteristic replicator are then used as input vectors for networks updating the model. The wide variety of different network architectures with one or two hidden layers have been tested. Furthermore two different definitions of the outputs have been tried out: one network with two linear outputs, trained to predict both the change of the stiffness (or mass) and the location of this change, or two separate networks each with one output, predicting the values of the parameters of the model independently. The results obtained from this calculations show that the separate networks predict the values of the parameters with Masses updating

10 12 14 16 18 20 21 22 24 Number of hidden neurons in 205- fr205 repHcator

Fig. 2. Optimal architecture of characteristic replicator for updating of (a) stiffnesses, (b) masses.

694

B. Miller, L. Ziemiahski /First MIT Conference a)

Stiffnesses updating: R^

on Computational

Fluid and Solid

b)

Masses updating: R^

• • • •

• Storey testing (1h) • Storey testing (2h) « Stiffness testing (1h) ^ Stiffness testing (2h) 200

300 400 500 600 700 Number of free parameters In the networks updating the model

250

Mechanics

Storey testing (1 hidden layer) Storey testing (2 hidden layers) Mass value testing (1 hidden layer) Mass value testing (2 hidden layers)

300 350 400 450 500 550 600 650 70 Number of free parameters in the networks updating the model

Fig. 3. Optimal architecture of the networks updating the model: (a) stiffnesses updating, (b) masses updating.

higher precision. The results of predicting the values of the change of the stiffness or the mass are more accurate then the results of predicting the location of this change. In order to improve the results of predicting the location of the change of the stiffness or the mass the networks with a bigger number of hidden neurons were applied. The best results for updating of stiffnesses are obtained from the networks with two hidden layers of the architecture 21-11-11-1 for the change of the stiffness prediction and 21-14-14-1 for the location of the change prediction (see Fig. 3a). In the case of updating the masses from among all tested network architectures the best results are obtained from the networks 21-27-1 for location of the change of the mass. In predicting the value of the change of the mass the network architecture hasn't a significant influence on the precision of obtained results, all tested networks with one hidden layer produce results of very similar precision, see Fig. 3b. a)

Change of stiffness: R^

2.3. The location of the impulse excitation The influence of the location of the impulse excitation on the efficiency of the stiffnesses updating has been checked. The excitation has been simulated on every degree of freedom except the 1st one. Calculated FRFs have been then compressed by characteristic replicator and passed to the separate networks determining the value and the location of the stiffness change. The most accurate results are obtained for both parameters being predicted when the impulse excitation is applied in the direction of the 5th degree of freedom. The value of the stiffness change is predicted with comparable precision also in the case of applying the excitation in the direction of the 11th degree of freedom, and the location of the stiffness change in the case of excitation in the direction of the 8th degree of freedom, see Fig. 4. Storey: R^

f • • • • •

3 4 5 6 7 8 9 10 11 12 Number of degree of freedom with applied impulse excitation

3 4 5 6 7 8 9 10 11 12 Number of degree of freedom with applied impulse excitation

Fig. 4. Optimal location of the impulse excitation for updating of stiffnesses: (a) change of stiffness, (b) location of stiffness change.

B. Miller, L Ziemianski /First MIT Conference on Computational Fluid and Solid Mechanics

695

Storey

Stiffness change

Relative error |%]

Fig. 5. Results of updating of stiffnesses: (a) change of stiffness (21-11-11-1 network), (b) location of the stiffness change (21-14-14-1) and updating of masses: (c) change of mass (21-13-1), (d) location of mass change (21-23-1). 2.4. Results of updating of the model

References

In the Fig. 5 there are shown examples of the obtained results. The precision of predicting stiffnesses and masses is very high, only in predicting the location of the change of the mass the results are not as good as in other parameters' predicting.

[1] Friswell MI, Mottershead JE. Finite element model updating in structural dynamics. Dordrecht: Kluwer Academic PubHshers, 1996. [2] Waszczyszyn Z, Ziemianski L. Neural networks in mechanics of structures and materials — new results and prospects of application. Proceedings of the ECCM'99 European Conference on Computational Mechanic, Miinchen 1999. [3] Levin RI, Lieven NAT. Dynamicfiniteelement model updating using neural networks. J Sound Vib 1998;210(5):593607. [4] Haykin S. Neural Networks. A comprehensive foundation, 2 ed. Upper Saddle River: Prentice Hall, 1999. [5] Ziemianski L, Miller B, Piatkowski G. Dynamic model updating by neural networks. Proceedings of 5th International Conference Engineering Applications of Neural Networks, Warszawa 1999, pp. 177-182. [6] Ziemianski L. Artificial neural networks in dynamic of structures — selected problems (in Polish). Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszow 1999. [7] Miller B, Ziemianski L. Neural networks in updating of a 12-storey frame model. In: Topping BHV (Ed), Computational Engineering Using Metaphors from Nature. Edinburgh: Civil-Comp Ltd, 2000, pp. 31-36.

3. Final remarks The neural network updating method presented in this paper has proved to be an effective tool to solve the model updating problem. The strong advantage of the suggested method is the capability of working with a limited number of measured degrees of freedom. In the presented examples only one FRF has been used to solve the problem of model updating. The presented method of extracting of dynamic features from FRF has proved to be very useful in reduction of dimension of the input vector.

696

Shape optimization problem based on optimal control theory by using speed method Y. Ogawa*, T. Ochiai, M. Kawahara Department of Civil Engineering, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo 112-8551, Japan

Abstract This paper presents the shape optimization problem of a body located in an incompressible viscous flow field. Shape optimization is to determine the domain that minimizes a performance function. Domain variation is formulated with a speed field. The formulation is based on the optimal control theory. The optimal state is defined by the fluid forces subjected to the body. The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint conditions by the Lagrangian multiplier or the adjoint equations using adjoint variables corresponding to the state variables of the state equations. Keywords: Domain variation; Material derivative; Speed method; Optimal control theory

1. Introduction What is the shape of the body which has minimum drag when moved at constant speed in viscous flow field? It may be suggested that such bodies are flat plates or streamline bodies and so on. But not much is known theoretically about this problem. Therefore it is difficult to design the shape of bodies in engineering fields. The purpose of this study is to apply a formulation of shape optimization to a numerical simulation of the body located in an incompressible viscous flow. Domain variation is formulated with a speed field. The shape optimization is based on the optimal control theory. In the optimal control theory, a control value which makes phenomenon an optimal state can be obtained. In this theory, a performance function should be introduced. When the performance function is minimized, it is assumed that the state is optimized, and then the control value is obtained. The performance function is defined by the fluid forces subjected to the body. The treated problem is to determine the domain that minimizes an performance function under the constrain of the state equation. This problem can be transformed into the minimization problem without constraint condition by the Lagrangian multiplier or the adjoint equations using adjoint variables corresponding to * Corresponding author. TeL: 03-3817-1814; Fax: 03-3817-1803; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

state variables of the state equations. The traction method is applied for minimizing the performance function.

2. Domain variation In this section, to formulate a domain optimization problem, a formula of domain variation by using speed method is introduced. 2.1. Material derivative Let F, : R^ ^ R^ fs{Xx,X2) = {Xx,X2)+sV{Xx,X2)

(1)

where V is a vector field and ^ c R^ a domain with a Lipschitz boundary. For (p e H\Q) and cp^ e H\Q) let us define (2)

(p' = (p^ o F.

The material derivative of cp for field V is given by ., , .. (p(x + sV)-(p(x) (p{x) = lim , {5) and the shape derivative of (p for field V is given by ,, , ,. (pMs -(p(x) w (x) = lim

^ ' '

5^0

s

= ^{x) - V{x) • Vcpix),

(4)

Y. Ogawa et al. /First MIT Conference on Computational Fluid and Solid Mechanics 2.2. Derivative of functions

where F\ and Fi are drag and lift forces, respectively. The fluid force F is obtained by integrating the traction hmt, which is written as follows,

If (p is given by a domain integration as follows

(5)

d dt

tdV.

I

-JA b=o

J=^-{F-FYQ{F-F) = / v + V-((^F)}dQ Q

=

(p'dQ-{-

(pV

-ndQ.

(6)

If (p is given by a Boundary integration as follows / , = / <^,dr,,

(7)

I

where Q is the weighting parameter, F is the objective value of fluid force which is pre-assigned value. The performance function should be minimized satisfying the state equation.

In this paper, some formulas of domain variation are introduced. These formulas are an attempt to apply to the shape optimization problem of a body located in an incompressible viscous flow field.

L_n

dt

=/(.' =

(10)

4. Conclusions

the derivative J has the following relationship

d

(9)

The fluid force is directly used in the performance function. The performance function / is defined by the square sum of the residual between values of computed fluid force and objective fluid force.

the derivative / has the following relationship J=

697

References -^(V(p-\-(pK)V

((p-\-(picV

'n)dr

-n}dr

(8)

where K is the curvature.

3. Formulation The shape optimization problem can be formulated by the optimal control theory. In case of optimal control problem with constraint conditions, the performance function should be minimized satisfying the state equations. This problem can be transformed into a minimization problem without constraint conditions by the Lagrangian multiplier method. 3.1. Performance

function

In this paper, a fluid force control problem is considered. The fluid force acting on the body B is denoted by F ,

[1] Maruoka A, Kawahara M. Optimal control in Navier-Stokes equations. IJCFD 1998;9;313-322. [2] Zolesio JP. The material derivative (or speed) method for shape optimization. In: Haung EJ, Cea J (Eds), Optimization of Distributed Parameter Structures, vol 2. Sijthoff and Noorhoff, Alphen aan den flijn, pp. 1089-1151, 1981. [3] Zolesio JP. Domain variational formulation for free boundary problems. In: Haung EJ, Cea J (Eds), Optimization of Distributed Parameter Structures, vol 2. Sijthoff and Noordhoff, Alphen aan den Rijn, pp. 1152-1194, 1981. [4] Leons JL. Some aspects of the optimal control if distributed parameter systems. Tegional Conference Series in Applied Mathematics, vol 6. [5] Pironneu O. On optimum profiles in Stokes flow. J Fluid Mechl973;59(l):117-128. [6] Pironneu, O. On optimum design in fluid mechanics. J Fluid Mech 1973;59(1):117-128.

698

Reliability based optimization using neural networks M. Papadrakakis *, N.D. Lagaros

National Technical University of Athens, Institute of Structural Analysis and Seismic Research, Zografou Campus, Athens 15780, Greec

Abstract In this paper a robust and efficient methodology is presented for treating large-scale reliability-based, structural optimization problems. The optimization part is performed with evolution strategies (ESs), while the reliability analysis is carried out with the Monte Carlo simulation (MCS) method incorporating the importance sampHng technique for the reduction of the sample size. The limit elasto-plastic analysis phase, required by the MCS, is replaced by a neural network prediction for the computation of the necessary data for the ESs optimization procedure. The use of NN is motivated by the approximate concepts inherent in reliability analysis and the time consuming repeated analyses required by MCS. A back propagation algorithm is implemented for training the NN utilizing available information generated from selected elasto-plastic analyses. Keywords: Structural optimization; Reliability analysis; Monte Carlo simulation; Evolution strategies; Neural networks

1. Introduction Reliability analysis methods have developed significantly over the last decades, Schueller [1], and have stimulated the interest for the probabilistic optimum design of structures. Despite the theoretical advancements in the field of reliability analysis serious computational obstacles arise when treating realistic problems. In particular, the reliability-based optimization (RBO) of large-scale structural systems is an extremely computationally intensive task, as shown by Papadrakakis et al. [2]. Despite the improvement in the efficiency of the reliability analysis techniques, they still require disproportionate computational effort for treating practical reliability problems. This is the reason why very few successful numerical investigations are known in the field of RBO and are mainly restricted to small-scale frames and trusses. In the present study the rehability-based sizing optimization of large-scale multi-storey 3-D frames is investigated. The objective function is the weight of the structure while the constraints are both deterministic (stress and displacement limitations) and probabilistic (the overall probability of failure of the structure). Randomness of loads, material properties, and structural geometry are taken into consid* Corresponding author: Tel. +30 (1) 772-1694; Fax: -h30 (1) 772-1693; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

eration in reliability analysis using Monte Carlo simulation (MCS). The probability of failure of the frame structures is determined via a limit elasto-plastic analysis. The optimization part is solved using evolution strategies (ESs) which in most cases are more robust and present a better global behaviour than mathematical programming methods, as shown by Papadrakakis et al. [3]. The limit elasto-plastic analyses required during the MCS are replaced by NN predictions. The use of NN is motivated by the approximate concepts inherent in reliability analysis and the time consuming repeated analyses required for MCS. The suitability of NN predictions is investigated in evaluating the probability of failure of real scale framed structures. An NN is trained first utilizing available information generated from selected conventional elasto-plastic analyses of different designs. The limit state analysis data is processed to obtain input and output pairs, which are used to produce a trained NN. The trained NN is then used to predict the critical load factor due to different sets of basic random variables. It appears that the use of a properly selected and trained NN can eliminate any limitation on the sample size used for MCS and on the dimensionality of the problem, due to the drastic reduction of the computing time required for the repeated analyses.

M. Papadrakakis, N.D. Lagaros /First MIT Conference on Computational Fluid and Solid Mechanics 2. Structural reliability analysis With the structural reliability analysis the design engineer can take into account all possible uncertainties during the design, construction and life of a structure, in order to calculate its probability of failure pf. Time invariant reliability analysis produces the following relationship (t)fs(t)dt

Pf = P[R < OO

/

F,(t)Mt)dt

(1)

in which R denotes the structure's bearing capacity and S is the external loads. The randomness of R and S can be described by known probability density functions //?(0 and fs(t), with FR(t) = p[R < t], Fsit) = p[S < t] being the cumulative probability density functions of R and S, respectively. Most often a hmit state function is defined SLS G(R, S) = S — R and the probability of structural failure is given by Pf = p[G{R, S) > 0]

-- f fR{R)fs(S)dRdS

(2)

G>0

It is practically impossible to evaluate R analytically for complex and/or large-scale structures. In such cases the integral of Eq. (2) can be calculated only approximately using either simulation methods, such as the Monte Carlo simulation, or approximation methods.

3. Reliability based structural optimization using ESs andNN During the last ten years various methodologies have evolved which deal with the reliability-based optimum design of structures. These attempts are restricted to relatively moderate size truss and frame structural problems using FORM and SORM reliability analysis methods. The probabilistic constraints enforce the condition that the probability of a local failure or the system failure is smaller than a certain value (i.e. 10~^-10~^). In this work the overall probability of failure of the structure, as a result of a limit elasto-plastic analysis, is taken as the global reliability constraint. The probabilistic design variables are chosen to be the cross-sectional dimensions of the structural members and the material properties (E,ay). After the selection of the suitable NN architecture the training procedure is performed using a number (M) of data sets, in order to obtain the I/O pairs needed for the NN training. Since the NN based structural analysis can only provide approximate results it is recommended that a correction on the output values should be performed in order to alleviate any inaccuracies entailed, especially when

699

the constraint value is near the limit which separates the feasible and the infeasible region. This is achieved by relaxing this limit during the NN testing phase before entering the optimization procedure. A "correction" of the allowable constraint values was therefore performed in proportion to the maximum testing error of the NN configuration. The maximum testing error is the largest average error of the output values among testing patterns. Whenever the predicted values were found smaller than those derived from a conventional structural analysis the allowable values of the constraints were decreased according to the maximum testing error of the NN configuration and vice versa. The proposed ES-NN methodology, implemented by Papadrakakis et al. [4,5], can be described with the following algorithms: 3.1. Algorithm 1 The combined ES-NN optimization procedure is performed in two phases. The first phase includes the training set selection, the limit elasto-plastic analyses required to obtain the necessary I/O data for the NN training, and finally the selection, training and testing of a suitable NN configuration. The second phase is the ESs optimization stage where the trained NN is used to predict the response of the structure, due to different sets of design variables, instead of the standard Umit elasto-plastic analysis computations. 3.2. Algorithm 2 The combined ES-NN optimization procedure is performed in three phases. The first phase is the ES optimization stage until a stationary point is obtained. The second phase includes the training set selection in the vicinity of the stationary point from the previous structural analyses during previous ES steps. The third phase is identical to the second phase of algorithm 1.

4. Test example The reliability based sizing optimization problem of a space frame is performed. The cross section of each member is assumed to be a W-shape and for each member two design variables are allocated. The probabilistic constraint is imposed on the probability of structural collapse due to successive formation of plastic nodes and is set to p^ =0.001. The probability of failure caused by uncertainties related to material properties, geometry and loads of the structures is estimated using MCS with the importance sampling technique. External loads, yield stresses, elastic moduli and the dimensions of the cross-sections of the structural members are considered to be random variables. The loads follow a log-normal probability density function.

M. Papadrakakis, N.D. Lagaros /First MIT Conference on Computational Fluid and Solid Mechanics

700

group 11 9

13

group 10 w h

3 O

m N.

^

a.

8

group 10 HH h^

group 11 \ H \H

i

12

CD

16

^ 1-

20

Plan view

a. 3 O O)

a. O

1 ^ D J b>

L

a. a o rn 1r

I

Q.

3

a. 3 O

7777'

7777'

24'

7777

24'

24*

7777

5. Conclusions The solution of realistic RBO problems in structural mechanics is an extremely computationally intensive task. In the test example considered the RBO procedure was found over forty times more expensive than the conventional deterministic optimization procedure. The aim of the proposed RBO procedure is to increase the safety margins of the optimized structures under various model uncertainties, while at the same time minimizing the weight of the structure and mainly the additional computational cost. This goal was achieved using NN predictions to perform the structural analyses involved in MCS.

r

9 1r u> J I

into eleven groups, as shown in Fig. 1, and the total number of design variables is eleven. For this example the additional optimum weight when probabilistic constraints are considered is approximately 10%. The results of the proposed method are given in Table 1.

'I

References

Front elevation

Fig. 1. The twenty-storey frame. Table 1 Performance of the methods for the twenty-storey frame Optimization procedure

ESs cycles

Pf

Optimum weight (tn)

Time (s)

DBO RBO

83 124

0.13 0.001

581.9 638.2

6,519 301,930

while random variables associated with material properties and cross-section dimensions follow a normal probability density function. The members of the frame are divided

[1] Schueller GI. Structural reliability — Recent advances, 7th International Conference on Structural Safety and Reliability (ICOSSAR '97), Kyoto, Japan, 1997. [2] Papadrakakis M, Papadopoulos V, Lagaros ND. Structural reliability analysis of elastic-plastic structures using Neural Networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 1996;136:145-163. [3] Papadrakakis M, Tsompanakis Y, Lagaros ND. Structural shape optimization using evolution strategies. Eng Optimization 1999;31:515-540. [4] Papadrakakis M, Lagaros ND, Tsompanakis Y. Structural optimization using evolution strategies and neural networks. Comput Methods Appl Mech Eng 1998;156:309-333. [5] Papadrakakis M, Lagaros ND, Tsompanakis Y. Optimization of large-scale 3D trusses using evolution strategies and neural networks. Int J Space Struct 1999;14(3):211-223 [Special Issue on Aircraft Hangars, OS Ramaswamy (Ed)].

701

Parallel computational strategies for structural optimization M. Papadrakakis *, N.D. Lagaros, Y. Fragakis

National Technical University of Athens, Institute of Structural Analysis and Seismic Research, Zografou Campus, Athens 15780, Greec

Abstract The objective of this paper is to investigate the efficiency of parallel computational strategies for large-scale optimization problems solved by evolution strategies (ESs). The basic solution method implemented in this work is the FETI method for solving the repeated structural analysis problems required by ESs. Furthermore, a two-level iterative method is proposed, particularly tailored to solve reanalysis type of problems, where the FETI method is incorporated in the preconditioning step of a parallel subdomain global PCG solver. Keywords: Structural optimization; Evolution strategies; Domain decomposition methods; Preconditioned conjugate gradient; Reanalysis problems

1. Introduction Optimization of large-scale structures, such as multistorey 3D frames, subjected to constraints imposed by design codes is a computationally intensive task. When evolution strategies are adopted to perform the optimization, the solution of the finite element equations is of paramount importance since more than 95% of the total computing time is spent for the solution of the finite element equilibrium equations. In the present study the FETI method of Farhat and Roux [1] is implemented for solving the repeated structural analysis problems required by ESs. Furthermore, a two-level iterative method, specially tailored for solving reanalysis type of problems, suggested by Papadrakakis and Tsompanakis [2], is also implemented in an effort to further increase the computational performance of the optimization procedure. In the two-level iterative solver the FETI method is incorporated in the preconditioning step of a global subdomain implementation of the PCG method.

2. Evolution strategies and reanalysis type methods The algebraic definition of an evolution strategy procedure applied into a structural system with the finite element equation Ku = f may be described as follows: * Corresponding author. Tel: +30 (1) 772-1694; Fax: +30 (1) 772-1693; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(1) Initialization: selection of Si {i = 1,2,... , /x) parent vectors of the design variables, set ^o = ^(^^i) and solve KQUI = f and (KQ + AK(si))ui = / , (/ = 2, 3 , . . . , /x) (2) Constraints check (3) Offspring generation: generate Sj, (j = 1,2,... ,A) offspring vectors of the design variables (4) Solution step: solve {KQ + AK(sj))uj = / , (/ = 2 , 3 , . . . ,A) (5) Constraints check: if satisfied continue, else change Sj and go to step 4 (6) Selection step: selection of the next generation parents according to (/x + A) or (/x, X) selection schemes (7) Convergence check: if satisfied stop, else go to step 3 Reanalysis solution schemes have been proposed by Papadrakakis et al. [3] for treating the repeated finite element equations encountered in ESs and proved to be very efficient, compared to the standard skyline solver, in sequential computing environment. Their parallel implementation, however, is hindered by the inherent scalability difficulties encountered during the preconditioning step, which incorporates forward and backward substitutions of a fully factorized stiffness matrix. In order to alleviate this deficiency a parallel global subdomain implementation (GSI) of a subdomain-by-subdomain PCG algorithm is implemented [4]. Furthermore, to exploit the parallelizable features of the GSI(PCG) method taking advantage of the efficiency of a fully factorized preconditioning matrix, the following

702

M. Papadrakakis et al. /First MIT Conference on Computational Fluid and Solid Mechanics

two-level methodology is proposed based on the combination of the global subdomain implementation and the FETI method. The GSI(PCG) method is employed, using a multi-element group partitioning of the entire finite element domain, in which the solution required during the preconditioning step is performed by the FETI method operating on the same mesh partitioning of the GSI(PCG) method. In the proposed methodology the preconditioning step of the GSI(PCG) method ^m + l — Cj^ rm + \

(1)

is performed by FETI. This two-level method is abbreviated as GSI(PCG)-FETI. 2.7. The two-level methodology In the GSI(PCG) method the iterations are performed on the global level with the GSI(PCG) method, using an incomplete Cholesky factorization of the stiffness matrix as preconditioned Thus, the incomplete factorization of the stiffness matrix KQ -\- /^K can be written as LDL^ = KQ ~\- AK — E, where E is an error matrix which does not have to be formed. Matrix E is usually defined by the computed positions of 'small' elements in L, which do not satisfy a specified magnitude criterion and therefore are discarded. For the typical reanalysis problem

(Ko + AK)u = f

Table 1 Initial parent vectors Test set #7 Parent 1 Parent 2 Parent 3 Parent 4 Parent 5

(24(s),16(s),16(s),10(t),10(t),10(t)) (24(s),16(s),16(s),10(t),10(t),10(t)) (24(s),16(s),16(s),10(t),10(t),10(t)) (24(s),16(s),16(s),10(t),10(t),10(t)) (24(s),16(s),16(s),10(t),10(t),10(t))

1169.19 tn 1169.19 tn 1169.19 tn 1169.19 tn 1169.19 tn

Test set #2 Parent 1 Parent 2 Parent 3 Parent 4 Parent 5

(20(s),13(s),12(s),6(t),5(t),5(t)) (21(s),8(s),15(s),9(t),5(t),10(t)) (24(s),14(s),14(s),7(t),5(t),5(t)) (22(s),15(s),10(s),5(t),6(t),6(t)) (21(s),13(s),15(s),4(t),9(t),7(t))

631.35 tn 815.03 tn 735.29 tn 639.55 tn 774.42 tn

Test set #3 Parent 1 Parent 2 Parent 3 Parent 4 Parent 5

(20(s),13(s),12(s),7(t),5(t),9(t)) (21(s),12(s),15(s),9(t),5(t),10(t)) (24(s),13(s),14(s),7(t),5(t),6(t)) (22(s),14(s),15(s),5(t),6(t),6(t)) (21(s),13(s),9(s),4(t),9(t),7(t))

681.25 tn 827.30 tn 737.45 tn 708.09 tn 691.32 tn

Test set M Parent 1 Parent 2 Parent 3 Parent 4 Parent 5

(23(s),13(s),13(s),6(t),5(t),5(t)) (24(s),13(s),15(s),9(t),5(t),10(t)) (24(s),14(s),14(s),7(t),5(t),5(t)) (22(s),15(s),10(s),5(t),6(t),6(t)) (21(s),13(s),15(s),4(t),9(t),7(t))

688.99 tn 878.63 tn 735.29 tn 639.55 tn 774.42 tn

(2)

matrix E is taken as AA:, SO that the preconditioning matrix becomes the complete factorized initial stiffness matrix Q = ^o- The solution of the preconditioning step of the GSI(PCG) algorithm, which has to be performed at each GSI(PCG) iteration, can therefore be effortlessly executed, once ^o is factorized, by a forward and backward substitution. With the parallel implementation of the two-level GSI(PCG)-FETI method the preconditioning step can be solved in parallel by the FETI method for treating the repeated solutions required in Eq. (1) using the same decomposition of the domain employed by the external GSI(PCG) method. The procedure continues this way for every reanalysis problem, while the direction vectors of FETI are being re-orthogonalized in order to further decrease the number of PCPG iterations of the interface problem within the preconditioning step. The solution of Eq. (1) is performed Hi xrir times via the FETI method, where «, and rir correspond to the number of GSI(PCG) iterations and the number of reanalysis steps, respectively.

bottom flanges. Group 2: Cross girders of the top and bottom flanges. Group 3: Bracing diagonals connecting top and bottom flanges to top and bottom chords of the space frame. Group 4: Top and bottom chords of the space frame. Group 5: Diagonal bracing members connecting top and bottom chords of space frame to middle chords. Group 6: Middle chords of the space frame. The hangar comprises 3614 nodes and 12,974 members. Members of Group 1 to 3 are to be selected from the structural sections and members of Groups 4 to 6 from the tube sizes. A constraint of 750 mm on the maximum deflection was imposed besides constraints on stresses. All computational results reported in this section were run on a SG Power Challenge XL shared memory machine with R8000 processors. Optimization was carried out by the ESs technique. In applying ESs, the (5-|-5)ESs scheme was adopted and four different initial population were used (see Table 1). The optimum weight achieved for all four different runs is of 862.78 tn for one half of the hangar and a maximum deflection of 637 mm (see Table 2).

3. Test example 4. Conclusions The optimum design of a long span aircraft hangar is investigated. The members of the space truss were grouped as follows. Group 1: Longitudinal members of the top and

An important characteristic of Evolution Strategies that makes them differ from other conventional optimization

703

M. Papadrakakis et al. /First MIT Conference on Computational Fluid and Solid Mechanics Table 2 Optimization results Test

Optimum weight (tn)

Optimum design

1 2 3 4

862.78 862.78 862.78 862.78

17s, 17s, 17s, 17s,

7s, 7s, 4t, 7s, 7s, 4t, 7s, 7s, 4t, 7s, 7s, 4t,

5t, 5t, 5t, 5t,

It It It It

algorithms is that in place of a single design point the ESs work simultaneously with a population of design points in the space of variables. This allows for a natural implementation of the evolution procedure in parallel computer environment. Since a number of finite element analyses of the structure can be performed independently and concurrently, a complete finite element analysis can be assigned to a processor without the need for inter-processor communication during the solution phase. The computational efficiency of the ES optimizer may be substantially enhanced with the implementation of parallel algorithms for the solution of each of the repeated finite element systems of equations encountered at each optimization step of the evolution strategy procedure.

No. of generations

No. of FE analyses

Seq. time (s)

Par. time (s)

65 32 41 56

213 100 123 159

20658 9750 11978 15454

7434 3370 4294 5706

References [1] Farhat C, Roux F-X. A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Methods Eng 1991;32:1205-1227. [2] Papadrakakis M, Tsompanakis Y. Domain decomposition methods for parallel solution of sensitivity analysis problems. Int J Numer Methods Eng 1999;44:281-303. [3] Papadrakakis M, Tsompanakis Y, Hinton E, Sienz H. Advanced solution methods in topology optimization and shape sensitivity analysis. Eng Comput J 1996;3(5):57-90. [4] Papadrakakis M. Domain decomposition techniques for computational structural mechanics. In: Papadrakakis M (Ed), Parallel Solution Methods in Computational Mechanics. Chichester: Wiley, 1997, pp. 80-147.

704

Enhancing engineering design and analysis interoperability. Part 2: A high diversity example Russell S. Peak*, Miyako W. Wilson Georgia Institute of Technology \ Engineering Information Systems Laboratory, 813 Ferst Drive, Atlanta, GA 30332-0560, USA

Abstract This is Part 2 in a series about knowledge representations that enable enhanced cooperation between engineering design and analysis models. A basic flap link example shows how the multi-representation architecture (MRA) analysis integration strategy supports computing environments that have a diversity of analysis fidelities, physical behaviors, and CAD/CAE tools. The constrained object (COB) technique from Part 1 provides the MRA with reusable, modular, multi-directional capabilities. Keywords: CAD-CAE interoperability; Multi-representation architecture; Simulation-based design; Multi-fidelity; Multidirectional

1. Introduction

2. Flap link tutorial example

At this conference, Wilson et al. [1] overviews constrained objects (COBs) as an object-oriented representation of engineering concepts (Part 1), and Dreisbach and Peak [2] discuss COB-based steps towards multi-functional optimization (MFO) (Part 3). This paper (Part 2) presents a basic example to show how COBs facilitate design-analysis integration for simulation-based design (SBD) as a step towards MFO. In this context, simulation and analysis refer to modeling physical behavior such as stress or temperature. Peak [3] overviews recent developments in X-analysis integration^ (XAI) technology including the COBbased multi-representation architecture (MRA). The MRA achieves advanced CAD-CAE interoperability in environments that have a diversity of tools and models. Interoperability can be defined as the ability for tools and models to communicate and share information in a seamless computer-based manner.

Fig. 1 overviews the main MRA concepts via an example. Traditional CAD tools (left side) are used to define the manufacturable description of this product. On the right are traditional CAE tools that solve discretized and symbolic mathematical problems. In between are the four main types of MRA objects: solution method models (SMMs), analysis building blocks (ABBs), analyzable product models (APMs), and context-based analysis models (CBAMs). These are stepping stones to help connect diverse tools and models in a flexible and modular manner.

* Corresponding author. Tel.: +1 (404) 894-7572; Fax: -f 1 (404) 894-9342; E-mail: [email protected] ^ http://eislab.gatech.edu/ ^ X = design, manufacture, sustainment, etc. © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

2.7. Analyzable product models Analyzable product models (APMs) [4] help coordinate and merge design-oriented details coming from possibly many design tools and libraries. The lower-middle portion of Fig. 1 shows the flap link APM constraint schematic, which has objects such as sleeves, shaft, cross-section, and ribs. The blue design-oriented relations show how design parameters like sleeve width and shaft width are related. APMs add idealizations (red) that may be used by multiple analysis models. For example, the ID torsion and extensional analysis models (Fig. 1) both use a parameter called effective length, Leff. This parameter is the distance between the edges of the sleeves; it is a geometric idealiza-

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tion of the main material region connected by the sleeves. While such a parameter is useful from an analysis point of view, it would not likely be included as a CAD parameter used to manufacture this part. Yet it is related to such parameters, so the APM provides a place to capture such relations.

ner (Fig. 1, far right). They support white box reuse of existing tools (e.g. FEA tools, math tools, and in-house codes) within a uniform constraint-based framework. ABBs generate SMMs based on solution technique-specific considerations such as symmetry and mesh density. 2.4. Context-based analysis models

2.2. Analysis building blocks Analysis building blocks (ABBs) represent analytical engineering concepts as semantically rich objects independent of solution method and product domain. The uppermiddle portion of Fig. 1 contains constraint schematics for a material model ABB and two continuum ABBs. The continuum ABBs are extensional and torsional rods, as covered in undergraduate mechanics courses. Applying object-oriented reasoning similar to that in [1], one recognizes that these and other continuum primitives are built from the same linear elastic stress-strain-temperature concepts. Thus the ID linear elastic model ABB captures this knowledge to reduce manual re-creation, provide modularity, and facilitate reusability. 2.3. Solution method models Whereas ABBs represent concepts at the analytical level, solution method models (SMMs) represent them at the detailed solution method level. SMMs can be viewed as object-oriented wrappers around CAE solution tools that obtain analysis results in a highly automated man-

Context-based analysis models (CBAMs) explicitly represent the fine-grained associativity between a design model and its possibly many analysis models (i.e. between an APM and its ABBs). CBAMs are also known as analysis modules and analysis templates. Fig. 1 depicts three flap link CBAMs and their macro-level connections to the APM. The right side of Fig. 2 is the flap link extensional model annotated with its key MRA and CBAM features. It captures explicit CAD-CAE associativity, i.e. how a subset of APM attributes like effective length, Leff, are connected to the extensional rod ABB it is using. Note that this same type of ABB can be used in other CBAMs for other types of products (e.g. circuit board solder joint analysis [3]). In addition to how the analysis is 'wired' to work, the CBAM shows why the analysis exists: to determine if the calculated stress exceeds the allowable stress. It uses a margin of safety ABB for this purpose as seen in the lower left corner of the constraint schematic. Similar to the spring examples in [1], Fig. 3 shows how a single CBAM can be run in several directions. In state 1, the APM details are inputs, and stress and margin of safety are outputs. In state 3 (the lower portion), the situation is

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R.S. Peak, M.W. Wilson/First MIT Conference on Computational Fluid and Solid Mechanics reversed in that margin of safety is now an input and APM cross-sectional area is an output. This capabihty allows one to directly compute the 'optimum' design variable (e.g. cross-section area) in subgraph cases where systems of relations analytically support directional changes. The same CBAM can be used to check the design again after its details have been developed. Considering the engineering semantics of the problem, one sees that state 1 typifies a simple design verification scenario, where the 'natural inputs' (physical design properties and a load) are indeed inputs and a 'natural output' (a physical response to the load) is the requested output. Hence, the design is being checked to ensure it gives the desired response. As a design synthesis (sizing) scenario, state 3 reverses the situation by making one natural output into an input and one natural input into the desired output. It effectively asks "what cross-sectional area (a design-oriented variable) do I need to achieve the desired margin of safety (which depends on the stress physical response)?" This COB capabihty to change input and output directions with the same object thus has important engineering utility. It is a multi-directional capability in that there are generally many possible input/output combinations for a given constraint graph. Fig. 1 also contains the flap link plane strain model, which simulates the same type of physical behavior (extension) as the flap link extensional model. It utilizes a finite element-based SMM to obtain more detailed stress and deformation answers (over a 2D field versus the ID field in flap link extensional model). Its constraint schematic graphically shows that its ABB connects with more APM geometric and material model idealizations than does the ID case. Thus, it is a higher fidelity CBAM and illustrates the multi-fidelity capabilities of the MRA. Typically engineers use quick lower fidelity models early in the lifecycle to size the design, and more costly higher fidelity models later to check the design more accurately. Finally, the flap link torsional model in Fig. 1 illustrates the multi-behavior capability of the MRA. This CBAM simulates a different type of physical behavior (torsion) versus the previous two extension CBAMs. Note that it uses the torsional rod ABB described before and connects to different idealized attributes in the APM (e.g. polar moment of inertia) as well as to some of the same ones (e.g. effective length). The analysis tool Mathematica again solves the formula-based relations as an example of CAE tool re-usage.

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3. Discussion The left side of Fig. 2 is a traditional documentation-oriented view of the ID extensional analysis. Shortcomings of this view are that it imposes a unidirectional sequence, it limits modularity and reusability, and it typically does not contain idealization relations like effective length. COBs overcome some of these problems today. In the future, such documentation views may be automatically derived from COBs using technologies like XML. In cases where relations cannot be inverted, at a minimum COBs can be used to try various inputs and attempt to achieve the desired result (a kind of manual optimization). Part 3 discusses steps towards automated optimization [2].

4. Summary This paper describes constrained objects (COBs) for a flap link analysis integration tutorial. It overviews concepts from the multi-representation architecture (MRA) that enable advanced CAD-CAE interoperability. Employing an object-oriented approach, the MRA defines natural partitions of engineering concepts that occur between traditional design and analysis models. The MRA is particularly aimed at capturing reusable analysis knowledge at the preliminary and detailed design stages. Other work [3] describes industrial applications including highly automated analysis module catalogs for chip packages that have reduced simulation cycle time by 75%.

References ^ [1] Wilson MW, Peak RS, Fulton RE. Enhancing engineering design and analysis interoperability. Part 1: Constrained objects. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [2] Dreisbach RL, Peak RS. Enhancing engineering design and analysis interoperability. Part 3: Steps toward multi-functional optimization. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [3] Peak RS. X-Analysis Integration (XAI) Technology. Georgia Tech Report EL002-2000A, March 2000. [4] Tamburini DR. The Analyzable Product Model Representation to Support Design-Analysis Integration. Doctoral Thesis, Georgia Tech., 1999.

^ Some of these references are available at http://eislab.gatech.edu/

708

Development of CFD-based design optimization architecture D. Peri, E.F. Campana*, A. Di Mascio INSEAN — National Ship Research Centre, via di Vallerano J39, 00128 Rome, Italy

Abstract In this paper, design optimization problems are faced using CFD solvers as analysis tools. The procedure is developed in the framework of the Multi-disciplinary Design Optimization (MDO), and is applicable in many multi-disciplinary analysis environment. We present a specific example of the design optimization of a surface ship, where a great effort is needed for the solution of the hydrodynamic problem, i.e. the free-surface flow past the ship hull. Objective functions are the total drag, the flow quality in the propeller region and the sea-keeping performances of a surface combatant of the US Navy. In the analysis, models of growing complexity have been used, ranging from potential to RANSE solver, to deal with the free surface flow past the ship. Keywords: Computational fluid dynamics; Numerical design optimization; Viscous resistance; Wave resistance; Ship-wave pattern; Seakeeping; Free surface viscous flow; Free surface potential flow; Finite volume; Boundary element method

1. Introduction

2. The optimization problem

Ship designers will be forced in the use of the CFD as an analysis tool in design optimization problem by the great amount of new design concepts that will be introduced in the next few years. In the design of a new ship, the clear definition of the requirements, setting the framework of the design, is of great relevance. Among these requirements, the sea-keeping performance, i.e. the behaviour of the ship in waves, plays a relevant role, and the designer usually has to establish criteria and limit values on ship motions, having assessed before their effects on the performances of the crew and on the operational capability of the ship. Drag reduction, beside the reduction in fuel consumption, has many positive side effects, such as the reduction of the produced wave pattern (which is associated with the wavy component of the drag) and the reduction of the wave breaking phenomenon, which is relevant to the ship detection by analysis of SAR images of the ocean surface. Finally, a reduction of the ship's wake is also desirable, hence enhancing the characteristics of the downstream flow, usually leading to better propulsion efficiency and reducing noise and vibration levels.

In the present design problem, all the aforementioned requirements have been taken into account. The sea-state and the advancing speed of the ship are input data as well as the initial shape to be optimized, and a numerical optimization scheme has been built to optimize the bow shape of a US Navy surface combatant (Fig. 1). The trade-offs between the requirements of the different disciplines have been addressed by using the MDO methodology (for a survey see Alexandrov and Hussaini [1]). A complete towing tank test programme has been performed to experimentally verify the success of the MDO procedure. Details about the formulation of the problem and the adopted numerical schemes maybe found in [2,3].

* Corresponding author. Tel: +39 (6) 5029-9296; Fax: +39 (6) 507-0619; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

3. Description of the numerical scheme A summary of the MDO procedure used to face the problem is the following. The Objective Functions (OF) to be minimized, under some linear and nonlinear constraints, are the total resistance (drag) of the ship, the height of the free surface wave pattern in the bow region, the wake produced by the sonar dome/hull junction, evaluated as the averaged axial vorticity of a certain control region, and the vertical motions (heave and pitch) of the ship advancing in

D. Peri et al /First MIT Conference on Computational Fluid and Solid Mechanics

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Fig. 1. Numerical grid for the panel solver. The USS DDG51 has been discretized with 2000 elements for the solution of the inviscid free surface flow problem. waves. Instead of trying to find a parametric representation of the geometry of the ship with some shape function, a perturbation approach is used. The above requirements have been obtained by using Bezier surfaces controlled by a desired number of control points, hence assuming the role of design variables. The use of the Parameter Space Investigation (PSI) (see Statnikov and Matusov [4]) allows for an approximate representation of the feasible solution and Pareto optimal sets, while the use of the Variable Complexity Modelling (VCM, see [1] for details) is used to reduce to CPU time of the overall procedure. VCM consists in the use of models of different complexity in the OFs evaluations: from a RANSE code (multigrid, multiblock, finite volume solver [5]) to full-3D potential flow solver [6], to a 2D-strip theory potential solver.

4. The free surface flow Both RANSE and the full-3D potential solvers deal with the solution of the free-surface flow past the ship. The so-

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Fig. 3. Eddy viscosity contours in a transversal section of the ship. The presence of the vortex is highlighted by the stream traces in the plane. Acknowledgements 1.06

This work was financially supported by the Ministero Trasporti e Navigazione, in the frame of the INSEAN research plan 2000-2002.

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[1] Alexandrov NM, Hussaini MY (Eds). Multidisciplinary Design Optimization. Proceedings of the ICASE/NASA Workshop on MDO, SIAM, USA, 1997. [2] Peri D, Rossetti M, Campana EF, Improving the hydrodynamic characteristics of a ship hull via numerical optimization techniques. In: 9th Conference of the International Maritime Association of Mediterranean, IMAM 2000, Ischia, Italy. [3] Peri D, Rossetti M, Campana EF. Design optimization of ship hulls via CFD techniques. J Ship Res, to be pubHshed. [4] Statnikov RB, Matusov IB. Multicriteria Optimization and Engineering. London: Chapman and Hall, 1995. [5] Di Mascio A, Broglia R, Favini B. A second-order Godunov-type scheme for Naval Hydrodynamics. In: Godunov (Ed), Methods: Theory and AppHcation. Singapore: Kluwer Academic/Plenum, 2000. [6] Bassanini P, Bulgarelli U, Campana EF, Lalli F. The wave resistance problem in a boundary integral formulation. Surv Math Ind 1994;4.

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The functional virtual prototype: an innovation framework for a zero prototype design process Douglas M. Peterson * Mechanical Dynamics, Inc., 2301 Commonwealth, Ann Arbor, MI 48105, USA

Abstract How do companies continue to design innovative products as the physical prototypes are eliminated, development times are shortened, and engineering teams are more globally distributed? A virtual prototyping framework based on open, object oriented, and web-based technology allows engineering teams to dramatically increase engineering insight and at the same time capture the important process knowledge that exists in each prototype. Using this framework, virtual prototypes of various systems are defined as modules that are then plugged together into platforms to test innovative design concepts, explore new design solutions, and drive towards design decisions. The virtual prototype then becomes a central technology that allows engineers to collaborate across design disciplines, more accurately predict real-world behavior, and in the end reduce the reliance on the expensive and time-consuming process of building, testing, and refining hardware prototypes. Keywords: Virtual prototype; Design process; Collaborate; Innovate

1. Introduction In his book, Schrage notes that a company's prototyping culture drives their ability to develop and deliver innovative products to the market [1]. Without a platform to test ideas, companies are limited in their ability to bring new and improved ideas together into the product design. A prototype is therefore the essence of engineering where teams validate customer satisfaction and product manufacturability, they check if the design will work or if it will break, they account for how much it will cost, and so on. It is the platform upon which the effort of the engineering organization is realized as a tangible, although ideally digital, representation of the collective product concept. An innovative prototyping process where ideas can efficiently be brought forth for the entire organization to conceptualize is the only way to consistently deliver superior products to market. Two aspects of a virtual prototyping process that help form this corporate prototyping culture are: 1) a framework where prototypes are developed once and shared enterprise wide, and (2) a framework where technology in one discipline is leveraged across all disciplines. These are two *Tel.: +1 (734) 913-2517; Fax: +1 (734) 994-6418; E-mail: dpete @ adams.com © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

similar yet distinct approaches that allow organizations to focus more on innovation in the design and less on computer simulation technology development.

2. Enterprise wide virtual prototyping A well-built virtual prototyping infrastructure is based on the foundation of a growing network where engineers can leverage information assets throughout the organization. There isn't time or resources to redevelop virtual prototypes or test procedures for each design program. They need to be developed once, bookshelved, and become a platform for collaboration between engineering teams so innovative ideas can be tested and evaluated. Traditionally, computer specialists, or analysts have focused on developing virtual prototypes for one design team. By redeploying these analysts towards developing standard templates that represent systems and test procedures, the technology will then be available for the platform team to test ideas. The bookshelved templates functioning together as an open, configurable, and extensible knowledge base will allow the analyst resources along with the virtual prototype to more strategically impact the development process.

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D.M. Peterson / First MIT Conference on Computational Fluid and Solid Mechanics

The technology to support this process is more evident by looking at comparisons to the general computer industry. The drive in the computer industry is to develop technology in an open, object oriented manner so components can be developed, reused, and formed into larger solutions. Software developers using these ubiquitous technologies can work faster, more efficiently, and globally. This technology approach is the foundation of the web. It follows that similar requirements in the product design process would profit from a prototyping framework built on similar technologies. Those analysts developing new virtual prototyping methods would define new classes, or templates, that are carefully correlated and refined. The platform teams would then quickly instance together various subsystems to form the basis for a virtual test. Since the methods had previously been validated, then the testing can occur early in the development process as teams quickly consider innovative design alternatives. A scenario of enterprise virtual prototyping is as follows: A template would be developed for a subsystem. Analysts and physical test engineers would then work together to validate the template. Once confidence is achieved in the underlying method, the template would be published to the engineering community on the web. Product designers working in packaging would use the template within the CAD package to verify packaging of the design. Development engineers working in a design discipline would also use the template to quickly perform parametric studies and validate customer satisfaction. A supplier working on another subsystem would also be able to integrate this template to understand interactions between the two different subsystems. One example of this technology is the Functional Digital Car. In this example, templates are built of suspension systems, steering systems, driveline systems, and engine systems, each by their respective experts or suppliers. Also, experts develop templates of vehicle test rigs where standard testing practices are defined. With these pieces in place, the platform teams are then able to focus on the product design. The templates of the various systems are pulled together and used as is required for the type of test being performed. For example, a shift quality test will pull together templates of the shift quality test rig along with detailed suspension and driveline systems. A steering feel test will use another set of templates including the steering feel test rig along with detailed steering and suspension systems. Suppliers and manufacturers can collaborate together to develop and publish various templates as new design concepts in a global development environment. The end result of this template approach is a digital process that leads the design, deals with the challenging issues of the design process, and captures the process knowledge in a form that can continuously expand.

3. Multi-discipline analysis Most engineering organizations today have a separation of design considerations along discipline lines. Those doing durability studies build their own virtual prototypes while those doing vibration studies build yet another. This difference in prototypes inevitably leads to turf wars and conflicts between design teams. It is increasingly important to avoid this separation so engineers can efficiently test design innovations and gain insight without struggUng across discipline lines. The prototype testing must be tightly coupled so the prototype data and methodology are leveraged rather than re-developed for each design discipline. Looking at an example of durability, vibration, and motion testing, there are three scenarios that demonstrate the need for cross discipline prototypes: The first scenario is when studying component durability in a system design; there is a need for finite element, motion, and fatigue prototypes to efficiently communicate. The component finite element provides flexibility information for the motion prototype, which provides loads to the component finite element, which provides stresses for the life prediction. Any drop in communication and the virtual prototype is of no consequence in supporting the design team. Another scenario is how virtual prototype correlation and method development for one discipline will improve the virtual prototype predictability for all disciplines. A new and improved template of a suspension subsystem for vibration testing will also improve the correlation of the motion test, as achieving improved frequency content improves the motion model. A final scenario is how the effort to collect and validate model data for one discipline is the same effort needed for another discipline. The hard point, bushing, and damper rate data for motion prototype testing is the same data for vibration testing. Collecting the data and conceptualizing it into a prototype twice doubles the effort to evaluate various design concepts. In the physical world, prototypes are built without consideration of the design discipline. The instrumentation, event, and environment are what define the discipline being studied, not the prototype itself. A virtual prototype built the same way will help bridge disciplinary boundaries and improve the efficiency of the teams. Teams co-investing in the core prototype will drive a prototyping culture that supports product innovation rather than disciplinary tugs-of-war.

4. Conclusion As the digital age continues to move forward, the reality of virtual prototyping is that it can no longer be an isolated development activity practiced by the experts. It is quickly becoming a global community that together forms a virtual

DM. Peterson /First MIT Conference on Computational Fluid and Solid Mechanics prototyping network. There are practical examples of this functioning in certain industries such as the Functional Digital Car. As this continues to move forward, there are real implications to all industries in terms of their ability to reduce reliance on physical prototypes, cut time to market, and improve the communication and innovation of global design teams.

References [1] Schrage M. Serious Play: how the world's best companies simulate to innovate. Boston: Harvard Business School Press, ^^^^•

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An HPC model for GA methodologies applied to reliability-based structural optimization C.K. Prasad Varma Thampan, C.S. Krishnamoorthy * Department of Civil Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Abstract In engineering design optimization, it is becoming increasingly important to consider requirements of reliability of the components or system or both. Reliability analysis is a computation-intensive task and is being incorporated in FEA packages. The paper presents a computational framework for HPC on NoWs for the reliability-based optimization of structures using genetic algorithm-based methodologies. A master-slave implementation with an adaptive load balancing algorithm is proposed. Industrial application is illustrated through an example of topology optimization of a transmission line tower. Keywords: Parallel genetic algorithm; Reliability analysis; Optimization; Transmission line tower; High performance computing

1. Introduction Genetic algorithms offer scope for configuration and shape/topology optimization of practical structures that are extremely difficult to solve by using conventional optimization algorithms. The fitness function evaluation is the most time-consuming part of the integrated genetic algorithm (GA) in the case of structural optimization problems, because it involves a finite element analysis of the structure represented by each string in the population. Depending on the number of elements/members/components in the structure and the complexities of the structure, the fitness function evaluation usually consumes 85-90% of the total computation time [1]. In the case of reliability-based structural optimization problems, the evaluation of fitness also includes a reliability analysis of the structure which needs additional time for computation. If the reliability assessment is performed at system level, the time requirement will increase considerably. Even though the time requirement for genetic operations are relatively small, when the nature of the problem being solved demands for large population sizes and longer strings, these operations also require more time in each generation. * Corresponding author. Tel.: +91 (44) 445-8286; Fax: 4-91 (44) 235-2545/445-8281; E-mail: [email protected] © 2001 Published by Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The GA-based structural optimization process can be made more fast and efficient with the help of parallel and distributed computing techniques. This paper presents the design and implementation details of a high performance GA/parallel GA model developed to perform reliabilitybased optimization of tower structures.

2. The parallel GA models Genetic algorithms have the inherent nature of implicit parallelism. They search simultaneously at multiple points in the search domain through the various distinct members of the population. Since each string represents an independent set of design variables pertaining to the problem being solved, the fitness function evaluations for all strings can be done simultaneously and parallelly in the same iteration. Similarly, mutation for each string and crossover for each pair of strings can also be done independently and simultaneously. Hence the search process can be accelerated by allocating either the process of fitness evaluations or genetic operations or both to different processors of a parallel computer or to different workstations of a network. This idea leads to many models of HPC in genetic search algorithms. Erick Cantu-Paz [2] has presented a survey of various parallel GAs.

C.K. Prasad Varma Thampan, C.S. Krishnamoorthy /First MIT Conference on Computational Fluid and Solid Mechanics Parallel GAs are basically classified into single population GAs (or global parallel GA) and multiple population GA. Global parallel GA has only a single population and only the task of evaluating the fitness of the population is divided among the various processors. Genetic operations (selection and mating) are performed considering the entire population. Multiple population-based parallel GAs were proposed to keep the diversity of the individuals of the population and thus force to search different and extensive domains to arrive at the optimum without premature convergence; in addition to considerations of speed-ups. The total population is divided into various sub-populations called demes, which are kept in isolation in various processors to evolve in semi-isolation with minimum intercommunications and transfers (migrations). Here, not only the fitness evaluations, but also the genetic operations are performed in parallel. Parallel genetic algorithm implementations with multiple populations can be classified into: (1) Coarse-grained parallel GAs; and (2) fine-grained parallel GAs, depending on their population density in the array of processors invoked.

Let A^^, tfm, tfi, Cm,

P be the size of the population number of slave machines average duration of one fitness evaluation by master average duration of one fitness evaluation by /th slave Optimal number of candidate solutions to be evaluated by master Csi, Optimal number of candidate solutions to be evaluated by /th slave tc, time for communication. The condition to be satisfied for best performance is Cmtfm + tcNs = Csitfi

(1)

such that Ns

:/xp,

/x< 1.0

(2)

From (1) and (2) ^1.0 lip - tcNs 2_^ — i=l ^f' Ns ^ ^

Cm

1.0 + . , . ^ 3. The master-slave approach The present work uses a model based on the concept of global parallel GAs. Global parallel GAs are usually implemented as master-slave programs and they require constant inter-processor communications. Fig. 1 shows a schematic diagram of the master-slave implementation. The masterslave model was implemented on a Network of Workstations (NoWs). The Parallel And Virtual Environment (PAVE) was chosen as the platform for parallel implementation of GA, since it can act as a layer over both MPI and PVM and is independent of the operating system(s).

foV / = 1, 2, . . . , A^,

i=l

(3)

— ^f'

Substituting c^ in Eq. (1), c,/ can be evaluated for / = 1,2,... ,Ns. During the first generation, the candidate solutions are equally shared by master and slaves and the average time for fitness evaluations tfm and tft are computed, based on which the master calculates c^ and Csi and re-allot the candidates from next generation onwards. The strategy is reviewed and modified at specified intervals of generations to keep up with the varying computational load at different nodes.

5. Example 4. Dynamic scheduHng algorithm

5.1. A 110-kV transmission line tower [3]

The load balancing in master-slave implementation of parallel GA may be either static or dynamic. Static type is sufficient, as long as the processors are fully dedicated and duration for fitness evaluation is a constant. Both the conditions are seldom true with topology optimization of practical structures on NoWs. In the case of RBSO with failure mode evaluations for system reliability assessment, the time for fitness evaluation is not a constant, even for member size optimization. The dynamic load balancing algorithms for master-slave implementation reported in the literature are based on a master dedicated for distribution of chromosomes to idle slaves on a 'busy-wait schedule'. The efficiency of such a scheme depends on the ratio of time for communication to computation and hence highly problem dependent. Hence an adaptive scheduling algorithm is proposed.

Reliability-based optimization is performed with variables pertaining to: (1) member sizes alone; (2) member sizes and configuration of the body; and (3) member size, configuration and topology of the body, in the global parallel computing environment. Configuration variables were the joint coordinates where as the topology variables consisted of different types of side and plan bracings. 5.2. Data Tower voltage = 110 kV, circuits = double; type of tower = tangent, normal span = 320 m; ground wire: diameter = 10.65 mm, weight per metre = 5.45 N; conductor: diameter = 15.81 mm, weight per metre = 4.94 N; annual wind speed is with Gumbel distribution, mean = 45

715

716

C.K. Prasad Varma Thampan,

C.S. Krishnamoorthy

/First MIT Conference

Master.

on Computational

Fluid and Solid

Mechanics

Ns Slaves Read problem details

Read problem details Read GA parameters Initialise first generation randomly and divide subpopulations equally to master & slaves and send them

Map to phenotypes and perform FEA and reliability analysis; evaluate objective function, constraints and fitness values of the alloted subpopulation Compute the average time of evaluating a member

Map to phenotypes and perform FEA and reliability analysis; evaluate objective function, constraints and fitness values for the self alloted subpopulation Compute the average time of evaluating a member

Send fitness, constraints and time of evaluation to master

Receive info on convergence from master

Receive results from all slaves

For first generation and generations at selected intervals perform load balancing and recompute subpopulation sizes.

T

Send notice to all slaves]

NO

/ Output results

Store the best individual Do genetic operations Send subpopulations of new generation to slaves

(

STOP

/

3

Fig. 1. Schematic diagram of master-slave implementation. m/s with COV = 0.20; strength in tension and compression are with normal distribution, mean = 150 N/mm^ and COV = 0.15; Young's modulus == 2.1 x 10^ N/mm^; target system reliability index = 2.08; number of members = 228; number of joints = 71; number of member groups = 11. 60 equal and unequal angles were in the database. The tower was optimized for the following load cases: (1) normal operating condition; (2) state of broken conductor; (3) state of broken earth wire; and each of them in combination with wind load. Fig. 2 shows the basic geometry of the structure. The optimization problem is based on target reliability approach and is stated as minimize the structural weight

W = Y^ A,L,p

subject to

^sys

_

Hsys-tgt

(4) (5)

where A, and L, are the cross sectional area and length of /th member, p is the material weight density, Psys and Psys.tgt are the actual and target system reliability indices. The details of genetic modelling are available in reference [4]. The system reliability was assessed by modeling the structure as a series system of parallel subsystems, using the branch and bound algorithm and bounding techniques of failure mode approach [5]. Variable length GA [6] was used to model the structure for topology optimization. The problem was solved with different population sizes and number of slaves. Fig. 3 shows the variation of the average time required for running one generation in member size, configuration and topology optimization processes. The process took total generations varying from 500 to 650. Table 1 shows the variation of optimum weights with respect to the three levels of optimization.

C.K. Prasad Varma Thampan, C.S. Krishnamoorthy /First MIT Conference on Computational Fluid and Solid Mechanics

111

L52 All dimensions are in m^res

Base Width

Fig. 2. A 110-kV transmission line tower: basic geometry. Table 1 Optimum structural weights of the tower (kN) RehabiUty-based optimization with respect to Optimal structural weight

Member sizes alone

+ Configuration

30.18

28.83

Topology 27.64

6. Conclusions The global parallel GA in the form of a master-slave implementation on a NoWs is cost-effective and efficient enough to perform the reliability-based configuration and topology optimization of practical structures within a reasonable time. The proposed adaptive scheduling algorithm in the Master-Slave Parallel implementation could solve the problems efficiently in a non-dedicated environment, with non-uniform duration for fitness evaluations. Higher speed-ups were noticed with higher population sizes when more slaves are employed. More similar studies will be required to exploit the full potential of other types of high performance GAs in solving design optimization problems.

References

Fig. 3. Average duration of one generation of parallel GA. Top panel: member size optimization. Middle panel: configuration optimization. Bottom panel: topology optimization.

[1] Adeli H, Cheng NT. Concurrent GA for optimization of large structures. J Aerospace Eng ASCE 1994;7(3):276-296. [2] Erick Cantu-Paz. Designing efficient master-slave parallel gas. ILLiGAL Report No. 97004, 1997. [3] Natarajan N, Santhakumar AR, Reliability-based optimization of transmission line towers. Comput Struct 1995;3:387403. [4] Thampan CKPV, Krishnamoorthy CS, Prasad M, Rajeev S. Reliability-based configuration optimization of trusses using genetic algorithms. Int J Evol Optim 2000;l(l):71-88. [5] Thoft-Christensen P, Murotsu Y. Application of Structural System Reliability Theory. Springer: 1986. [6] Rajeev S, Krishnamoorthy CS. Genetic algorithms based methodologies for design optimization of trusses. J Struct Eng ASCE 1997;123(3):350-358.

718

Reduced-basis output bound methods for heat transfer problems D.V. Rovas, T. Leurent, C. Prud'homme, A.T. Patera * Massachusetts Institute of Technology, Mechanical Engineering Department, Room 3-266, Cambridge, MA 02139, USA

Abstract We describe a technique for the rapid and reUable prediction of outputs of interest, of elliptic partial differential equations with affine parameter dependence. To achieve efficiency, the reduced-basis method is used; reliability is obtained by the development of relevant a posteriori error estimators. We apply this method to the problem of designing a thermal fin, to effectively remove heat from a surface. A number of design parameters/inputs are considered. Each possible configuration, corresponding to different choices of the design parameters, needs to be evaluated by solving the heat conduction equation and calculating certain outputs of interest like the average temperature on the fin base. Keywords: Reduced-basis method; A posteriori error estimation; Heat transfer

1. Introduction

2.1. Efficiency

In engineering and science, the use of numerical simulation is becoming increasingly important. The physical problems in consideration are often modeled by a set of partial differential equations and related boundary conditions; then, a discrete form of the mathematical problem is derived and a solution is obtained by numerical solution methods. As the physical problems become more complicated and the mathematical models more involved, current computational resources prove inadequate; the time required to perform the computation becomes unacceptably large. Especially in the field of optimization or design, where the evaluation of many different possible configurations is required — corresponding to different choices of the design parameters/inputs — reliable methods that reduce the complexity of the problem while at the same time preserve all relevant information, are becoming very important.

To achieve efficiency, we pursue the reduced-basis method; a weighted residual Galerkin-type method, where the solution is projected onto low-dimensional spaces with certain problem-specific approximation properties.The reduced-basis method has been proposed first by Nagy in [6], for the nonlinear analysis of structures. It has been further investigated and extended by Noor and Peters [7]. A priori theory has been developed by Fink and Rheinboldt [10], Porsching [11] and Barret and Redien [9].

2. Numerical method The method used in this paper is the reduced-basis output bound method developed in [1-4]; for details related to the implementation, see [2]. In designing new methods, certain qualities must be considered: efficiency, relevance and reliability. * Corresponding author. © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

2.2. Relevance Usually in a design or optimization procedure, we are not interested in the field solution or norms of it, but rather in certain design measures, such as the drag coefficient in the case of flow past a bluff body, or the average temperature on a surface in the case of heat conduction. The methods considered give accurate approximations to these outputs of interest, defined as functional outputs of the field solution. 2.3. Reliability To quantify the error introduced by the reduced-basis method, a posteriori error analysis techniques must be invoked. There has recently been much interest in methods for a posteriori error estimation, especially to estimate the

D. V. Rovas et al. /First MIT Conference on Computational Fluid and Solid Mechanics discretization error for the finite element method; a review can be found in [8]. Most error estimators developed give bounds for abstract norms of the error. A posteriori error estimators for outputs of interest have been developed in the reduced-basis context; for more details see [3]. Special affine parameter dependence of the differential operator is exploited to develop a two-stage offline/online blackbox computational method. In the online stage, for every new set of design parameters, an approximation to the output of interest and an associated error bound is calculated. The computational complexity of the online stage of the procedure scales only with the dimension of the reduced-basis space (which is usually 0(10)) and the parametric complexity of the partial differential operator; for more details see [2]. Solution of the governing equations in 'real-time', can thus be achieved.

3. Thermal fin problem In this example, we consider a three-dimensional thermal fin used to effectively remove heat from a surface. The three-dimensional fin, shown in Fig. 1, consists of a vertical central 'post' and four horizontal 'subfins'; the fin conducts heat from a prescribed uniform flux 'source' at the root, through the large-surface-area subfins to surrounding flowing air. The fin is characterized by a seven-component parameter vector, fi = (fi^,..., IJ7), where /x' = k\ i = 1, ,4; 5 _ Bi; /x^ = L; and JJL^ = t\ fx may take on any value M in a specified design space D C R^. Here k' is the thermal conductivity of the /-th subfin (normalized relative to the post conductivity Z:^ = 1); Bi is the Biot number, a nondimensional heat transfer coefficient reflecting convective transport to the air at the fin surfaces; and L and / are the length and thickness of the subfins (normalized relative to

the post width). The fin is one unit deep (the root is square) and four units tall. We consider several outputs of interest. The first output, Troot ^ I^. is taken to be the average temperature of the fin root normalized by the prescribed heat flux into the fin root. This output relates directly to the coohng efficiency of the fin — lower values of Troot imply better performance. Another output is the volume of the fin, which represents weight and material cost — thus lower values are preferred. In order to optimize the design, we must be able to rapidly evaluate T^ootifJ^) and the volume of the fin V for a large number of parameter values /x G D. The steady-state temperature distribution within the fin, w(x), is governed by the elliptic partial differential equation -k'V^u' = 0

"F"

^

OCR^

\3^

31 ^

^

31Root: Heat In Fig. 1. 3D thermal fin.

i =0,

,4,

(1)

where V^ is the Laplacian operator, and u^ refers to the restriction of u io Q,\ Here Q^ is the region of the fin with conductivity fc'; / = 0, ..., 4: Q^ is thus the central post, and Q^\ i = 1,..., 4, corresponds to the four subfins. We must also ensure continuity of temperature and heat flux at the conductivity-discontinuity interfaces F' = dQ^ (1 dQ\ i = 1,..., 4, where dQ' denotes the boundary of ^ ' :

-(Vw^-n =-t(Wu^

-ri)]

onr\

i = 1,...,4, (2)

here n' is the outward normal on 9 ^ ' . Finally, we introduce a Neumann flux boundary condition on the fin root (3)

-(Vi/°-n') = - l onFroot,

which models the heat source; and a Robin boundary condition -k'iVu'

dQ.\

- tip

fcOrrl

mO:

• n ) = Biw' on T[^,,

/ = 0 , . . . , 4,

(4)

which models the convective heat losses. Here F^^^ is that part of the boundary of Q' exposed to the fluid that is

Bi: heat transfer coefficient

W

719

Froot.

For every choice of the design parameter-vector /x — which determines the k\ Bi, and also the fin geometry through L and t — solution of the above system of equations yields the temperature distribution M(X, /x) in the fin. The output of interest, T^oot(l^), can be expressed as

C(^) = / V

(5)

(Froot is of area unity). As for the volume, it is given by the following formula V = 4 + 8Lr.

(6)

720

D.V. Rovas et al /First MIT Conference on Computational Fluid and Solid Mechanics

4. Results In multi-criterion optimization, we consider various (competing) outputs of interest, say volume, V, and root temperature, Troot- Changing the dimensions of the fin by selecting different L and t will (say) decrease the volume of the fin, and hence material requirements — but also (typically) increase the fin base temperature. It is thus of interest to determine all possible operating points, that is, to generate the map of the 'achievable set'. In general, this will be prohibitively expensive unless one has recourse to a very low-dimensional representation, such as the reducedbasis approximation. We consider this problem for constant conductivities hi - \, i z=z 0, . . . , 4 , and Biot number Bi = 0.1. We then select 100 points in the two dimensional design space [r, L] = [0.2, 0.4] X [2.0, 3.0] and evaluate our bounds for Troot with an error tolerance of 1%. Since in this design we wish to be sure that the actual temperature will be less than our prediction, we choose to construct our map based on the upper bound obtained by the error estimator. We are thus insured that at each design point, the actual temperature will be lower than that on our curve. Each evaluation produces a point on the Troot-V plane, thus generating the achievable set. Obvious optimality conditions require that we remain on the left or lower boundaries of the achievable set, known as the efficient frontier or trade-off curve in Pareto analysis. As we can see from Fig. 2, we can decrease the volume with relatively small fin3d 13.6320 p

12.0160 I-

o

>

10.4000 8.7840 7.1680 ^ • • •

1.17919

. i .

1.22078

1.26238

Troot Fig. 2. Achievable set.

I 1.30397

I 1.34557

increase in temperature up to the point were the left and lower boundaries cross; after that a small further possible volume reduction results in a steep rise in base temperature.

Acknowledgements This work was supported by the Singapore-MIT Alliance, by AFOSR Grant F49620-97-1-0052, and by NASA Grant NAGl-1978.

References [1] Machiels L, Maday Y, Oliveira IB, Patera AT, Rovas DV. Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C.R. Acad Sci Paris, Serie I, to appear. [2] Maday Y, Machiels L, Patera AT, Rovas DV. Blackbox reduced-basis output bound methods for shape optimization. In Proceedings 12th International Domain Decomposition Conference, Chiba, Japan, 2000. [3] Patera AT, Rovas DV, Machiels L. Reduced-Basis OutputBound Methods for Elliptic Partial Differential Equations. SIAG/OPT Newsletter Aug 2000;11(2):4—9. [4] Maday Y, Patera AT, Rovas DV. A blackbox reducedbasis output bound method for noncoercive linear problems. MIT-FML Report 00-2-1, 2000; also in the College de France Series, to appear. [5] Maday Y, Patera AT, Peraire J. A general formulation for a posteriori bounds for output functional of partial differential equations; application to the eigenvalue problem. C.R. Acad. Sci. Paris, Serie I, 1999;328:823—829. [6] Nagy DA. Model representation of geometrically nonlinear behavior by the finite element method. Comput Struct 1977;10:683-688. [7] Noor AK, Peters JM. Reduced basis technique for nonlinear analysis of structures. AIAA J 1980;18(4):455-462. [8] Ainsworth M, Oden IT. A Posteriori Error Estimation in Finite Element Analysis. John Wiley and Sons, January, 2000. [9] Barret A, Redien G. On the reduced basis method. Z Angew Math Mech 1995;75(7);543-549. [10] Fink JP, Rheinboldt WC. On the error behavior of the reduced basis technique in nonlinear finite element approximations. Z Angew Math Mech 1983;63:21-28. [11] Porsching TA. Estimation of the error in the reduced basis method solution of nonlinear equations. Math Comp 1985;45(172):487-496.

721

Multi-disciplinary optimization for NVH and crashworthiness Uwe Schramm * Altair Engineering Inc, 2070 Business Ctr Dr # 220, Irvine, CA 92612, USA

Abstract Designing an automobile is a multi-disciplinary task. Response surface based optimization methods and Design of Experiments approaches are used to combine frequency targets and crashworthiness into one design optimization problem. The example of a bumperbeam is given. Keywords: Multi-disciplinary optimization; Design of experiments; NVH; Crashworthiness; Response surface

1. Introduction Developing automobiles is a multi-disciplinary task. Noise, vibration and harshness, as well as the crashworthiness of the design are of interest. Also other concerns such as durability, driving and handling play an important role. Linear and non-linear finite element analysis is applied to predict the structural behavior. The evaluation of the results includes the decision about design changes to obtain a better product. Trade-off studies need to be conducted to account for the multi-discipUnarity of the design. Multi-disciplinary optimization and Design of Experiments (DOE) Studies based on computational methods are useful tools to support the process of finding the best design. The complexity of the layout can be described mathematically as an optimization problem. Using the results of a computational optimization, the decision process can be improved. Optimization of structural elements of an automobile can lead to significant cost reductions which are threefold: Firstly, the design cycle is shortened leading to reduced development costs; secondly, the manufacturing costs are reduced leading to higher profitability of the enterprise; and thirdly, the operating costs of the final product are reduced leading to a more competitive product. Industrial application of structural optimization techniques depends on the availability of software. For linear statics and dynamics such software is available and fairly well supported. Structural layout and shape optimization can be performed to design parts and assemblies. If crashworthiness needs to be considered currently no algorithms *TeL: +1 (949) 221-0936; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

are available to perform sensitivity analysis and optimization as known in Hnear statics. Alternative approaches use response surface techniques in conjunction with non-linear finite element software. There are different types of response surface method using different types of sampling the data. Some are based a fixed number of samples, some on the sequential creation of samples. Design of Experiments approaches can be used too.

2. Design optimization Objectives of the design process are structures and structural parts that fulfill certain expectation towards their economy, functionality and appearance. The design problem can, for example, be formulated as the minimization of costs, with constraints determined by the function of the product and its aesthetic design. In general, the design problem can be given in the form of an optimization problem such as Objective:

V^o(b) =^ min

Constraints:

i//"/ (b) < 0

Design space:

b' < b < b"

(1)

Objective and constraints are quantities determined by a computational analysis. Structural mass, stiffness and energies can be objective functions. Stresses, displacements, natural frequencies, forces and similar structural responses are constraints. The process to solve this problem goes through multiple iteration steps:

722

U. Schramm / First MIT Conference on Computational Fluid and Solid Mechanics

(1) Define an initial design h^'^^\ with the vector of design variables b = {^^}, ; = 1 , . . . , m. (2) Analyze the behavior of the structure. (3) Compare the results of the analysis with the objective and constraints. (4) If the requirements are not met, change the design such thatb^^+'^ =b^^^-f 5b. (5) Back to 2. The search direction in this process 6h can be determined directly from the results of computational or experimental analyses. Objective and constraints are considered as functions of the design variables V^, = V^,(b),

/ = i,

(2)

These functions can be linearized for each design b^""^ such that (3)

^-^ abi

The derivatives d^/idbj are the design sensitivities that can be obtained directly from the results of numerical analyses. They are used to built approximations of the design space. The solution of the optimization in the approximate design space then defines the search direction 8h. Other, higher order approximations that are called response surfaces can be built from numerical analyses too. 3. Parameter study Very common for design optimizations are parameter studies (Fig. 1). The selection of the sets of design variables used for a number of analyses is called the Design of Experiment. The intent of a parameter study is to understand how changes to the design variables of a model influence the performance of that model. In a parametric study, a model is being analyzed repeatedly through a simulation for various combinations of parameter settings. Effects and interactions of the parameters of the model can be studied.

I

A mathematical model is computed that describes the responses of the model as an algebraic function of its parameters. The algebraic expression that describes the response of a model as a function of the parameters is known as a response surface. Several Design of Experiment methods are described in the literature [2,3]. Once a set of response surfaces has been generated for a model, the response surfaces can act as a proxy for the model. New combinations of parameter settings that were not used in the original design can be plugged into the response surface equations to quickly estimate the response of the model without actually running the model through an entire analysis.

4. Optimization methods Above, it was established that the complex structural optimization problem is solved using sequential approximation. Except for the already mentioned local approximation there are also methods that allow the search for an optimum in a much larger range [4]. Both ways, local and global approximation will be discussed shortly. 4.1. Local approximation Local approximation methods determine the solution of the optimization problem using the following steps (Fig. 2): (1) (2) (3) (4)

Analysis of the physical problem using finite elements. Convergence test, if the solution is found. Design sensitivity analysis. Solution of an approximate optimization problem formulated using the sensitivity information. (5) Back to 1. This approach is based on the assumption that only small changes of the design occur in each optimization step. The result is a local minimum. It should be mentioned that the biggest changes occur in the first few optimization

Design-Analysis FEA

%

Design Analyses FEA

Design Variable Interaction

Respor^se Surface Computation

Fig. 1. Parameter study.

»^Ck)nv?

Optimum

Design Sensitivity Analysis (DSA)

Design Update Optimization

Fig. 2. Local approximation.

723

U. Schramm /First MIT Conference on Computational Fluid and Solid Mechanics steps. Therefore, very few system analyses are necessary in practical applications. The design sensitivity analysis of the structural responses with respect to the design variables is one of the most important ingredients to take the step from a simple design variation to a computational optimization. Further details about the local approximation approach for dynamic impact can be found in [5].

H JM

"

1

r-M

\

J

1

Design Analysis FEA

•

'

.1

Design Update Optimization

For many problems, the implementation of a design sensitivity analysis is not a simple task. This is the case, for example, for non-linear problems or if the space for the design search is quite large. In such cases it is convenient to introduce higher order analytical expressions called response surface to approximate the dependency between the objective or constraint functions and the design variables. An approximate analytical relationship between structural responses and design variables can be estabHshed with only few analyses. The solution to this problem can be determined using mathematical programming. It yields an approximate solution to the structural optimization problem. Problems of different physical content can be combined in one optimization problem easily. The solution of the optimization problem using response surfaces involves the following steps: (1) Finite element solutions of the problem. (2) Response surface computation for each response. (3) Solution of the approximate optimization problem. In this method, a predefined number of designs is analyzed followed by the computation of the response surfaces and the optimization solution (Fig. 3). After the evaluation of the complete response surface,the design domain can be redefined and the whole procedure is repeated until convergence. A different approach analyzes the designs as the optimization proceeds and is called sequential response surface method [1]. Using this method the response surface is up-

1

I

Response Surface Computation

4.2. Global approximation

J\ 1 nt 1 n

,

1

1 III r 1 U1 11

1

''

•

1

Design Analyses

FEA

^r Response Surface Computation

^r Design Update Optimization

Fig. 3. Ordinary response surface approximation.

Fig. 4. Sequential response surface method. dated in each optimization step (Fig. 4). This method leads to less design evaluations than an ordinary response surface method and is therefore much more efficient. Response surface methods are very useful for multidisciplinary design. Finite element solvers for different problem classes can be combined into a multi-disciplinary optimization tool.

5. Combining parameter studies and optimization Parameter study and optimization can be usefully combined to investigate and optimize the behavior of a structural model. Response surface methods can handle just a few design variables since otherwise the computational effort is too high. Especially in crash analysis, where a single analysis run requires from several hours to days of computer time, any effort to reduce the wait time needs to be made. The number of design variables for an optimization should be limited to about ten. If the number of design variables is very high, it is advisable to first run a so-called Screening Design of Experiment to determine design variables of large influence. Wizard type software implementations combine parameter studies using Design of Experiment approaches and sequential response surface optimization. The same parameterized model can be used in both parameter study and optimization. If the study wizard is integrated in the finite element pre- and post-processing tool, a powerful solution makes it easy for the engineer apply optimization to a design problem. This way all the interfaces to different solvers can be accessed and multi-disciplinary optimization can be performed.

6. Example problem The objective for this optimization analysis was to minimize the mass of a bumper beam while under a barrier intrusion (displacement) for a centerline barrier hit and un-

724

U. Schramm /First MIT Conference on Computational Fluid and Solid Mechanics 7. Conclusions

Fig. 5. Bumper beam-crash model. der a constraint on its first natural frequency. The Altair StudyWizard [6] was used to set up a multi-disciplinary optimization using LS-DYNA [7] for the non-linear crash analysis and Altair OptiStruct [8] for the linear frequency analysis. Gage and shape design variables are used to achieve an optimum design. The StudyWizard utilizes a sequential response surface method implemented in Altair HyperOpt [9]. It is integrated with the finite element pre-and post-processing software Altair HyperMesh and Altair HyperView [10,11]. The bumper beam was designed in two parts, front and rear, which were connected by a full seam weld across the top and bottom of the assembly. The front section was separated into two gage regions; upper and lower (Fig. 5). Due to symmetry, just half of the bumper is modeled. Further, two shape design variables have been defined to modify the inner shape of the bumper beam. Initial gage values for all three regions were set to 1.6 mm. A 5-mph centerline barrier hit analysis was made by constraining the rail bracket to ground and giving the barrier an initial velocity of 5 mph. In a linear normal modes analysis the first natural frequency of the bumper is determined. The design goal to minimize the mass of the bumper. The maximum barrier intrusion should not exceed 50 mm when 80% of the kinetic energy of the barrier is absorbed. The target frequency should be above 90 Hz. Hence, the optimization problem is: • Minimize the mass. • Upper bound constraint on the barrier displacement at 80% of the initial kinetic energy. • Lower bound on the first natural frequency. The baseline analysis showed that a barrier displacement of 43.7 mm, a frequency of 94.8 Hz and the half-bumper mass of 5.25 kg. The optimization ran through twelve iterations for convergence. The optimization was successful since it satisfies the barrier displacement (intrusion) design constraint of 50 mm and the frequency constraint while minimizing the mass of the bumper beam. The final design has a mass of 4.74 kg, the barrier intrusion is 50 mm, and the first frequency is 94.7 Hz.

The results from computational analyses can be employed effectively for the design of structural systems and parts if design of experiments and structural optimization methods are used. The manual effort for expensive design variations and comparisons is considerably reduced. The formulation of the design problem as an optimization problem allows an objective oriented search for the best design. Of course, the accurate analysis of the physics of the problem and the sufficient determination of the input data such as loading, boundary conditions, material data, is necessary. The application of structural optimization is not just limited to linear problems anymore. It can be applied to complex physical behavior too, such as analyzed in a multi-disciplinary structural analysis of automobiles. Optimization and methods of parameter study methods are of growing interest in the automotive industry. The examples in this paper show that structural optimization is a valuable tool in structural design even of structures with complex nonlinear dynamic behavior. To seemlessly integrate modem optimization technology into the CAE process, GUI and solver interfaces need to be provided at a high level. The integration of parameter study and multi-disciplinary optimization tools in a CAE environment helps to efficiently assess design targets. The data management is much easier to accomplish since interfaces, data structures and report capabilities are already available.

References [1] Schramm U, Thomas H, Schneider D. Crashworthiness design using structural optimizafion. Des Optim 1999;1:374387. [2] Grove DM, Davis TP. Engineering, Quality and Experimental Design. Longman, 1997. [3] Taguchi E. Introduction to Quality Engineering. White Plains, 1986. [4] Barthelemy J-FM, Haftka RT. Approximation concepts for optimum structural design — a review. Struct Optim 1993;5:129-144. [5] Schramm U, Pilkey WD. Review: optimal design of structures under impact loading. Shock Vib 1996;3:69-81. [6] Altair StudyWizard, Manual. Altair Engineering Inc., Troy, MI, 2000. [7] Hallquist JO. LS-DYNA, Theoretical Manual. Livermore Software Technology Corporation, Livermore, CA, 1997. [8] Altair OptiStruct, Manual. Altair Engineering Inc., Troy, MI, 2000. [9] Altair HyperOpt, Manual. Altair Engineering Inc., Troy, MI, 2000. [10] Altair HyperMesh, Manual. Altair Engineering Inc., Troy, MI, 2000. [11] Altair HyperView, Manual. Altair Engineering Inc., Troy, MI, 2000.

725

Optimum design of frame structures undergoing large deflections against system instability R. Sedaghati ^'*, B. Tabarrok% A. Suleman'' ^ Mechanical Engineering Department, University of Victoria, Victoria, BC V8W 3P6, Canada ^ IDMEC, Mechanical Engineering Department, University of Victoria, Victoria, BC V8W 3P6, Canada

Abstract An optimization algorithm for structural design against instability is developed for shallow beam structures undergoing large deflections. The algorithm is based on the maximization of the Hmit load under specified volume constraint. The analysis for obtaining the limit load involves coupling of axial and bending deformations, and is based on the nonlinear finite element analysis using the displacement control technique. The optimization is carried out using both the Sequential Quadratic Programming and optimality criterion techniques and the results are compared. An example, a shallow arch, illustrates the structural design optimization methodology and the results are compared with those in the literature. Keywords: Design optimization; Frame structure; Instability; Geometrical nonUnearity

1. Introduction Generally, in design optimization problems for system buckling, the design variables are selected so as to maximize the system-buckling load while keeping the volume constant. In several reported investigations [1-4], the system stability requirement has been posed as a linear buckling analysis. Such an analysis is restricted to small rotations and equilibrium in the initial state and may not be conservative enough for some flexible structures. For this reason, a nonlinear buckling analysis should be undertaken to find the more conservative buckling loads (limit point). Here, the nonlinear buckling analysis based on displacement control method due to Batoz and Dhatt [5] is used to capture the hmit load for shallow frame structures. Sedaghati and Tabarrok [6] have used the displacement control technique proposed by Batoz and Dhatt as an analyzer in the optimum design of truss structures undergoing large deflections subject to the system stability constraint. The method was found to be extremely efficient in optimizing the structures under snap-through buckling load. Two optimization methods based on optimality criterion * Corresponding author. Tel.: -Hi (250) 472-4214; Fax: -Hi (250) 721-6051; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(OC) and Sequential Quadratic Programming (SQP) are used. The OC is based on the uniform strain energy density distribution in the structure [6] and the SQP method is based on the work by Powell [9].

2. Nonlinear analysis In the presence of large deflections, geometrical nonlinearity becomes important. It is therefore necessary to write the joint equilibrium in terms of the final geometry of the structure. Using updated Lagrangian formulation, the incremental equilibrium equation can be expressed in matrix form as [7] (,KE + \KG)^\^

= ^^^^aPref - y

(1)

where \KE and \KG are the system linear elastic stiffness matrix and system geometric stiffness matrix, respectively. \V is the vector of the nodal resultant member forces at time step t, Pref is a given reference load, ^^^\a is a load factor parameter to denote the external load at time step t + ^t and AU is the vector of increments in nodal. The iterative solution of this equation is conventionally represented as an evolution in time t. The problem is, of course, static and t simply denotes incremental steps in the solution. To guarantee that both out-of-balance load vector

726

R. Sedaghati et al. /First MIT Conference on Computational Fluid and Solid Mechanics 3. Optimization The optimization problem can be defined as: find the vector of element cross-sectional areas, A (design variables) such that Maximize the limit load factor Subject to

C'^^^aPref - JP') and incremental displacements (AU) are small, the energy convergence criteria [7] has been used in this analysis. The element linear elastic stiffness matrix \kE does not depend on the geometry of the structure and can be found in [7]. For a beam element as shown in Fig. 1, the element geometric stiffness matrix J/TG in local coordinates can be shown as [8]: {AE/l}) SYM

3(U4-U,) -6(U2-U5)

L(U3+U6)

6(U4 - U i )

5

10

5

-L(U2-U5)

L2(4U3-U6)

L(U4 - U , )

10

30

10

15 L(U2 - U 5 ) 2,. — ^ ^ ^ + L ( 4 U 3 - -U6)

SYM 2L2(U4-UI)

6(U2 - U 5 )

L(U3 + U 6 )

6(U; - U 5 ) L(U3+U6) 5 10 —6(U4 - Ui

5

10

5

10

L2(U3 - 4 U 6 )

L(U4 - U i )

-L^(U4-Ui)

30

10

30

-3(U4 - U i )

L(U2 - U 5 )

10

(3)

where L is the vector of elemental length, V and V are the total volume and the specified total volume of the structure, Qfcr the limit load factor, and A/ is the lower bound of the design variable vector, A. In this study, the optimization methods based on the Optimality Criterion (OC) and Sequential Quadratic Programming (SQP) technique have been used and the results are compared. The employed OC is based on the uniform strain energy in optimum design and the detail of the algorithm can be found in [6]. The SQP method used in this study is based on the work by Powell [9], the required sensitivity of the limit load factor has been obtained efficiently using the adjoint method [10] based on the information obtained from the nonlinear buckling analysis.

Fig. 1. Beam element with positive displacements in local axis.

kG=k Din) =

acr(A) _ V = A'^L-V = 0 A/ - A < 0

)

-L(U4

-UI)

3(U4 - U i ) -6(U2-U5)

L(U3+U6)

5

10

L(U2 - U5) _ L ^ ( U 3 - 4 U 6 )

10

30

6(U4 - U i )

5 -L(U4 - U i )

2 L 2 ( U 4 -^ U i )

10

(2) where the matrix A;D is called displacement geometric stiffness matrix and depends on nodal displacement vector u. With a few assumptions, the conventional element stress stiffness matrix, ka (depending on the element forces) may be derived from the matrix kgBecause the displacement control method traces the post-buckling path, one can estimate the peak load by very small displacement increments. The true peak load is the largest load obtained as the displacement traces the equilibrium path. However, the solution obtained will be sensitive to the displacement increments employed and the cost of the solution would be prohibitive. The peak load in this study is obtained by performing a quadratic fit to the load-displacement curve near the critical load. The peak load determined through this procedure was found to be very accurate.

4. Illustrative example A simply supported sinusoidal arch and its finite element model are shown in Fig. 2. The arch is modeled using 10 plane beam elements, which have equal projections in the A:-axis. The moment of inertia I, of the beam elements is such that I = flA^, where A is the cross-sectional area, b = 1,2,3 and a is a specified constant. The Young's modulus is assumed to be E = 10^ psi. The downward vertical displacement at node 6 is taken as the controlling displacement. Two arches with different apex height (H) have been investigated. For apex height of H = 5 in, the limit load is maximized under specified volume of 40 in^ while for H == 10 in the specified volume of 35 in^ is considered. The main objective is to investigate how much increase in the limit load can be gained by redistributing the cross sectional area. The final results

R. Sedaghati et al. /First MIT Conference on Computational Fluid and Solid Mechanics

111

Table 1 Final design for area of cross-sections ^ (in^): shallow arch (OC) Element no.

h= 1 H = 5in h=\ H = 10 in h-1 H = 5in /7 = 2 H = 10 in ^7 = 3 H = 5in Z7 = 3 H = 10 in

Linear Nonlinear Linear Nonlinear Linear Nonlinear Linear Nonlinear Linear Nonlinear Linear Nonlinear

(l^cr)opt /v^crjinitial

1

2

3

4

5

0.2219 0.2936 0.1751 0.2570 0.2120 0.3102 0.1681 0.2685 0.2308 0.3296 0.1886 0.2853

0.2723 0.3167 0.2257 0.2775 0.2290 0.3103 0.1776 0.2687 0.2308 0.3297 0.1886 0.2854

0.2723 0.3165 0.2257 0.2771 0.2290 0.2892 0.1776 0.2466 0.2335 0.2980 0.2060 0.2552

0.3293 0.3746 0.2729 0.3140 0.4711 0.4499 0.4283 0.3881 0.5157 0.4546 0.4527 0.3913

0.8961 0.6900 0.8247 0.5922 0.8512 0.6311 0.7742 0.5451 0.7812 0.5781 0.6876 0.4980

0.8961 1.136 0.9607 1.136 0.6617 1.235 0.6587 1.236 0.5269 1.315 0.5596 1.314

^ The areas of the elements on other half of the arch can be found by symmetry 5000

Y= i

P

Y

-V-A-•-i^ -H- ^

4500

r

i

4000

H ^^^^^j

^3500

h=\, OC b=\, SQP A=2,0C h=l, SQP *=3, OC h=7>, SQP

5^3000 o

100 in

^ 2500

Finite Elenlent Model (Elements are circled)

^ ^•^^

3

_

\

-

1

ym p pi'iw l>

J 2000 10

Fig. 2. Sinusoidal arch and itsfiniteelement model. using the OC are given in Table 1. As expected, erroneous results are obtained using the linear buckling analysis. It is noted that when using the linear buckling analysis, the limit load in the final design was decreased. Next, the optimization results obtained by the OC were compared to those obtained using the SQP technique. This method was found to be computationally much more expensive than the OC. The iteration history for OC and SQP for the arch with H = 5 inch using nonlinear buckling analysis is shown in the Fig. 3. For arch with /? = 2 and /? = 3, the good agreement exists between the OC and SQP methods. However, for Z? = 1, the lower limit load ratio was obtained when using SQP. Considering that the shallow arch may exhibit no distinct limit load for some area distributions. This lower limit load ratio for Z? = 1 can be attributed to the premature termination during the optimization process in SQP, because of not finding the limit load. This problem has been addressed in [11]. The approach was based on the nonlinear analysis using the force control method. Both the gradient search techniques, and optimal-

1500 ^CTja0 p^iii ^M 1000

0

5

— _ — A A A A A

^ M M " -

10

15

6 6 6 6 6 6 20

Iteration for OC (multiply by 6 for SQP)

25

Fig. 3. The iteration history for the arch with H = 5 inch using OC and SQR ity criterion based on the maximum potential energy were used as optimizers. The problem was addressed for just H = 10 inch and it is mentioned that for H = 5, the optimization failed because no limit load was determined. For H = 10 inch, the limit load ratio for the arch with Z? = 1 was lower than that found in the present investigation (1.136). For b — 1 and b — 3, the limit load ratios of 1.064 and 1.092 were obtained, respectively, through maximization of the potential energy in [11] which is considerably lower than that of the current research. The lower limit load ratio obtained in [11] may be explained by not catching accurately the limit load during the course of the optimization.

5. Conclusions An optimization algorithm has been developed to maximize the limit load of frame structures under volume

728

R. Sedaghati et al /First MIT Conference on Computational Fluid and Solid Mechanics

constraint. The algorithm combines the nonlinear buckling analysis using the displacement control technique as analyzer, with optimality criterion technique based on the uniform strain energy, and mathematical programming technique based on the Sequential Programming method, as optimizers. It has been demonstrated that design based on the linear buckling can exhibit erroneous results. It is concluded that catching the limit load as accurately as possible is crucial in structural optimization of the frame structures against instability.

References [1] Kiusalaas J. Optimal design of structures with buckling constraints. Int J Solids Struct 1973;9:863-878. [2] Khot NS. Nonlinear analysis of optimized structure with constraints on system stability. AIAA J 1983;21(8):11811186. [3] Levy R, Pemg HS. Optimization for nonlinear stability. Comput Struct 1988;30(3):529-535.

[4] Canfield RA. Design of frames against buckling using a Raleigh quotient approximation. AIAA J 1993;31(6):11441149. [5] Batoz JL, Dhatt G. Incremental displacement algorithms for nonlinear problems. Int J Numer Methods Eng 1979;14:1262-1267. [6] Sedaghati R, Tabarrok B. Optimum design of truss structures undergoing large deflections subject to a system stability constraint. Int J Numer Methods Eng 2000;48(3):421434. [7] Bathe KJ. Finite Element Procedures. Englewood CUffs, NJ: Prentice-Hall, 1996. [8] Chang SC, Chen JJ. Effectiveness of linear bifurcation analysis for predicting the nonlinear stability limits of structures. Int J Numer Methods Eng 1986;23:831-846. [9] Powell MJD. A fast algorithm for nonlinearly constrained optimization calculations. Proceedings of the 1977 Dundee Conference on Numerical Analysis, Lecture Notes in Mathematics 1978;630:144-157. [10] Haftka RT, Gurdal Z. Elements of Structural Optimization. Dordrecht: Kluwer Academic, 1992. [11] Kamat, MP. Optimization of shallow arches against instability using sensitivity derivatives. Finite Elem Anal Des 1987;3:277-284.

729

CFD modeling applied to internal combustion engine optimization and design P.K. SenecaP, R.D. Reitz Engine Research Center, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI53706, USA

Abstract A methodology has been developed for internal combustion engine design that incorporates multi-dimensional CFD spray and combustion modeling and a global optimization scheme. This methodology, called KIVA-GA, performs full cycle engine simulations within the framework of a Genetic Algorithm (GA) search technique. Genetic Algorithms are artificial intelligence techniques that employ the processes of natural selection to drive a family of designs through a search space to an optimum. Design fitness is determined using a three-dimensional CFD code based on KIVA-3V for spray, combustion and emissions formation, coupled with a one-dimensional gas-dynamics code for calculation of the gas exchange process. The KIVA-GA methodology is apphed here to investigate the effects of engine input parameters on the emissions and performance of a heavy-duty diesel. The method allows parameters such as start of injection (SOI), injection pressure, amount of exhaust gas recirculation (EGR), boost pressure, split injection rate-shape, swirl, nozzle hole size, spray angle, and combustion chamber geometry to be included in the optimization. The predicted optima result in significantly lower soot and NO;^ emissions together with improved fuel consumption compared to baseline designs. Keywords: Multi-dimensional modeling; CFD; Optimization; Genetic algorithms; Diesel engines; IC engines

1. Introduction With the current status of computer CPU speed and model development, multi-dimensional modeling has become an increasingly important and sometimes necessary tool for engine designers and researchers seeking methods to reduce pollutant emissions without sacrificing performance. To this end, a number of investigators have computationally explored the effects of injection characteristics and exhaust gas recirculation on diesel engine performance and emissions. For instance, the work of Han et al. [1] explained how a split injection with a small percentage of fuel in the second pulse can reduce both soot and NO;^ simultaneously. In addition, Chan et al. [2] found that good agreement with experimental data was obtained when various EGR levels (from 0 to 10%) were combined with single, double and triple injection schemes. With increasing environmental concerns and legislated * Corresponding author. Tel.: +1 (608) 265-9469; Fax: +1 (608) 262-6707; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

emissions standards, current engine research is focused on simultaneous reduction of soot and NO^ while maintaining reasonable fuel economy. Factors including injection timing, injection pressure, injection rate-shape, combustion chamber design, turbo-charging and EGR have all been explored for this purpose [3]. With such a large number of engine parameters to investigate, it is evident that a computational methodology for engine design would significantly aid in the pursuit of cleaner and more efficient engines. The present study focuses on the development of such a methodology using multi-dimensional spray and combustion modeling through an improved version of the KIVA-3V code [4]. Physical submodels for turbulence, spray and combustion have been implemented in KIVA-3V and vaHdated against existing engine data [5].

2. Optimization methodology This section summarizes the key elements incorporated in the present design methodology including the baseline

730

RK. Senecal, R.D. Reitz/First MIT Conference on Computational Fluid and Solid Mechanics

design, the parameters of interest, constraints, the objective function and its evaluation, and vaUdation of the search technique. For brevity, a summary of the results for only one engine and operating condition will be presented here. Additional results are given in Refs. [6-9]. 2.1. Baseline design A single cylinder version of the Caterpillar 3400 Series diesel engine was chosen for the present study. The baseline engine specifications and operating conditions are presented in Table 1. For this study, a 57% load, 1737 rev/min operating point was investigated, as in the experimental study of Montgomery [10] which included a 68(10.5)32 split injection and 12% exhaust gas recirculated (EGR). 2.2. Parameters of interest The design factors and ranges considered in the present study are given in Table 2. The injection duration range specified in Table 2 corresponds to an injection pressure range of 100 to 200 MPa.

2.3. Constraints As described by Montgomery [10], physical constraints on this engine include a maximum exhaust temperature of 1023 K and a peak combustion pressure of approximately 15 MPa. The penalty method technique of Senecal [5] is used to inhibit convergence to a solution that violates the present constraints. 2.4. Objective function and its evaluation Since the goal of the present optimization process is to reduce emissions without sacrificing fuel economy, the objective, or merit, function should contain engine-out NO^c, hydrocarbon (HC) and soot emissions levels, as well as fuel consumption. In this study, the proposed merit function of Montgomery [10] is used and is given by 1000 /(X) = -R] . + Rl r . + Ri zwhere ^1

NO, + HC W, (NO,:+HCL PM

Table 1 Engine specificafions and operafing conditions for the baseline engine case Bore X Stroke Compression ratio Displacement Combusfion chamber Engine speed % of Maximum load Fuel rate Intake temperature Intake pressure Exhaust pressure Injection pressure Start of injection EGR level Split injecfion with 68% mass in first pulse. 32% in second and a 10.5 CA deg. dwell

137.2 X 165.1 mm 16.1 2.44 L Quiescent 1737 rev/min 57 6.8 kg/h 309 K 184kPaI 181 kPaI 150 MPa

+r atdc 12%

Table 2 Design factors and ranges considered Parameter

Range

Optimum

Boost pressure EGR level Start of injecfion Injecfion duration

165-284 kPa 0-50% -10to+10°atdc 20.5-29.0° (100-200 MPa:) 10-90% 5-15°

230 kPa 46% -l°atdc 21.2° (187 MPa) 80% 7.2°

Mass in first pulse Dwell between pulses

(1)

H'2PM.

R^ =

BSFC

(2)

(3)

(4)

BSFCo and the parameter vector X is defined in Table 2. (NO^^ + HC)^ and PM^ are EPA mandated emissions levels (3.35 and 0.13 g/kWh, respectively) and BSFCo is a baseline fuel consumption (215 g/kWh in the present work). W\ and Wi are weighting constants (safety factors) set to 0.8 for this study. The one-dimensional gas-dynamics code of Zhu and Reitz [11] was interfaced with KIVA-3V to allow simulations of the enUre engine cycle. The one-dimensional code not only provides initial conditions for KIVA-3V at the time of intake valve closure (IVC), but also provides an estimate of work during the intake and exhaust strokes for use in the BSFC calculation. The computational mesh used in the present simulations is a 60° sector of the combustion chamber due to the six-fold symmetry of the six-hole injector nozzle [5]. 2.5. Search technique The final, and perhaps most important, element of the KIVA-GA methodology is the micro-Genetic Algorithm (IJLGA) optimization technique. The KIVA-GA code is completely automated to simulate a |xGA generation (i.e., five designs) in parallel. Once the five simulations are completed, the genetic operators produce a new population and the process is repeated [6].

P.K. Senecal, R.D. Reitz/First MIT Conference on Computational Fluid and Solid Mechanics

(b)

50

100 Generation Number

731

150

Fig. 1. Two-dimensional Rastrigin's function (a) and convergence of |xGA to the global maximum (b). To test the convergence of the |xGA technique, the method was applied to seek maxima of several complex analytical functions [5]. For example, Fig. la shows Rastrigin's function

fovn = 2 parameters. As seen in Fig. lb the optimum {f = Xi = X2 = 0) is found in about 120 generations. Note that this is a severe test of the search algorithm due to the multitude of local maxima and the shallowness of the global optimum.

simulations are included in these figures, corresponding to about 12 days of continuous running on five CPUs of an SGI Origin 2000 system. It is clear from Fig. 2 that the present methodology has found an optimum design with significantly lower soot and NO;, emissions, and the design also had a 15% improvement in fuel consumption compared to the baseline case. The design factors for the optimum design are presented in Table 2. The optimum configuration has a higher boost pressure and EGR level, a slightly advanced SOI, a shorter injection duration (i.e., a higher injection pressure), more mass in the first injection pulse and a shorter dwell between injections.

3. Results

4. Conclusions

Fig. 2 presents the soot vs. NO;, points for the diesel engine optimization study. Data from approximately 250

This work demonstrates an efficient computational methodology for engine design using multi-dimensional CFD, spray and combustion modeling. The KIVA-GA code incorporates an improved KIVA-3V CFD model with a one-dimensional gas dynamics code for full cycle engine calculations. The |xGA optimization technique efficiently determined a set of engine input parameters resulting in significantly lower soot and NO;, emissions compared to the baseline case, together with improved fuel consumption. The present methodology provides a useful tool for engine designers investigating the effects of a large number of input parameters on emissions and performance. Current efforts include consideration of combustion chamber geometry parameters in the optimization search space [9].

f{Xi) = - j l O « + YX^^i - 10cos(27rZ,)]|

(5)

0.50

Acknowledgements Optimum

Fig. 2. Soot vs. NOx data from the present optimization study for all simulation cases including the baseline and optimum.

The authors would like to acknowledge the generous financial support of the Army Research Office, Caterpil-

732

P.K. Senecal, R.D. Reitz/First MIT Conference on Computational Fluid and Solid Mechanics

lar, Inc., TACOM and the Department of Energy/Sandia National Laboratory. [7] References [8] [1] Han Z, Uludogan A, Hampson GJ, Reitz RD. Mechanism of soot and NOj^ emissions reduction using muUiple-injection in a diesel engine. SAE Tech Pap 960633, 1996. [2] Chan M, Das S, Reitz RD. Modeling of multiple injection and EGR effects on diesel engine emissions. SAE Tech Pap 972864, 1997. [3] Hikosaka N. A view of the future of automotive diesel engines. SAE Tech Pap 972682, 1997. [4] Amsden AA. KIVA-3V: A block-structured KIVA program for engines with vertical or canted valves. Los Alamos National Laboratory Rep LA-13313-MS, 1997. [5] Senecal PK. Development of a Methodology for Internal Combustion Engine Design using Multi-Dimensional Modeling with Validation through Experiments. Ph.D. Thesis, University of Wisconsin-Madison, 2000. [6] Senecal PK, Montgomery DT, Reitz RD. A methodology for engine design using multi-dimensional modeling and

[9]

[10]

[11]

genetic algorithms with validation through experiments. Int J Engine Res 2000, accepted. Senecal PK, Reitz RD. Simultaneous reduction of diesel engine emissions and fuel consumption using genetic algorithms and multi-dimensional spray and combustion modeling. SAE Tech Pap 2000-01-1890, 2000. Senecal PK, Montgomery DT, Reitz RD. Diesel engine optimization using multi-dimensional modeling and genetic algorithms applied to a medium speed, high load operating condition. American Society of Mechanical Engineers, ICE Division, Peoria, IL, 2000, submitted to J Gas Turbines Power Wickman DD, Senecal PK, Reitz, RD. Diesel engine combustion chamber geometry optimization using genetic algorithms and multi-dimensional spray and combustion modeling. Accepted for SAE Congress 2001. Montgomery DT. An Investigation into Optimization of Heavy-Duty Diesel Engine Operating Parameters when Using Multiple Injections and EGR. Ph.D. Thesis, University of Wisconsin-Madison, 2000. Zhu Y, Reitz RD. A 1-D gas dynamics code for subsonic and supersonic flows applied to predict EGR levels in a heavy-duty diesel engine. Int J Vehicle Design 1999;22:227.

733

Difficulties and characteristics of structural topology optimization ChaiShan^'^'* ^ State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, P.R. China ^Shandong Institute of Technology, Academy of Science and Technology, Zibo 255012, Shandong, P.R. China

Abstract Two difficulties, 'limit stress' and 'singularity of optimum solution', are studied in this paper. After analyzing these two difficulties and their influencing factors, the concepts of two kinds of design variables, 'sectional design variable' and 'topological design variables', are introduced. Then, a mathematical model of topology optimization with two kinds of design variables is build up. Furthermore, it is pointed out that the difficulty of 'Hmit stress' can be avoided by using the proposed mathematical model, and the problem relative to 'singularity of optimum solution' can be solved by using the mixed discrete optimization or pure discrete optimization. Keywords: Structural optimization; Topology optimization; Singularity

1. Introduction As is well known, there are two difficulties in topology optimization: 'limit stress' and 'singularity of optimum solution'. When the size of one bar tends to zero, the stress expressed by the stress formulation does not become zero. In this case, the stress is called 'limit stress'. The 'singularity of optimum solution' means that the optimum solution of topology optimization is at the singular point of the feasible region. In recent years, many studies have been done to find the methods for solving the above two problems [1,3,7]. Why are there two difficult problems in topology optimization? What are the influencing factors for them? These two questions are investigated in this paper. The concepts of two kinds of design variables, 'sectional design variable' and 'topological design variables', are introduced, and then a mathematical model of topology optimization with two kinds of design variables is build. Furthermore, it is pointed out that the problem of 'limit stress' can be avoided by using the proposed mathematical model, and the problem of 'singularity of optimum solution' can be solved by using the mixed discrete optimization or pure discrete optimization. * Correspondence to: Shandong Institute of Technology, Academy of Science and Technology, Zibo 255012, Shandong, P.R. China. Tel: +86 (533) 2153259; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

2. The mathematical model of topology optimization of structures 2.1. The mathematical model with sectional variables only At present, the widely used mathematical model of topology optimization of truss structures is Pi find Ai

s.t.

M

(Til (A)
1 = 1,2, k = l,2,

,NL

(1)

,ND

Ai < 0 where M is the number of area design variables; NL, ND are the numbers of load cases and displacement constraints, respectively; A is the vector of area design variables; At is the area design variable and Ai = 0 means that the iih bar is cancelled from the structure. In mathematical model Pi, the design variables are only sectional variables and the change of topology of structure is realized by canceling some members with zero area and some other members according to some canceling or resuming rules after having obtained the optimum solution of Pi. For the static undetermined structures, the stress of a bar is not only dependent on the section of itself but

C Shan/First MIT Conference on Computational Fluid and Solid Mechanics

734

also dependent on the sections of other relative bars. The expression of stress is Oil

=

fii(P,A)

The mathematical model of topological optimization of trusses, including two kinds of variables and subjected to the stress and displacement constraints, is as follows.

(2)

P2

J2^.A,

findA/,a/ min W = y ^ piAiaiU

where / / ( P , A) is the internal force of the /th bar in the /th load case, and ki is a scale coefficient. It can be seen from formulation (2) that even if A, is equal to zero, the stress an may not be zero, and this stress is just the 'limit stress'. In some cases, the 'limit stress' will violate the corresponding constraints. On the other hand, the design variables have different feasible regions for a different topology. If the union set of all subsets of design variables in different topologies is taken as the feasible region for the whole topology optimization process, in some situations, the feasible region will show a strong 'non-convexity'. If the optimum solution is at the singular point, the 'singularity of optimum solution' will occur. 2.2. The mathematical model of topology optimization with sectional design variables and topology design variables Because the topology of structure and the sectional sizes must be designed in the topology optimization, two kinds of variables, topology variables a, and sectional size variables A,, should be included in the mathematics model of topology optimization. In order to express these two kinds of variables clearly, the definitions of topological variable and sectional variable are advanced here. Definition 1. The design variable that expresses the topological relation among the nodes of structure is defined as topological variable and is expressed by a,. Topological variable a, is a discrete variable, being 0 (expressing cancel) or 1 (expressing remain). Definition 2. The design variable that only changes the sectional size of structure is defined as sectional variable and is expressed by A/. Sectional variable A, is a continuous variable (for continuous variables optimization) or a discrete variable (for discrete variables optimization).

Fig. 1. Four bar truss.

s.t.

a/a,/(A, a) < a,-

/ = 1,2, . . . ,NL

5^/(A, a) < 5^

yt = 1,2,... ,ND

Ai > 0

for continuous variables

Ai e Si = {A/1, A / 2 , . . . ,Ai^Mi'.

(3)

for discrete variables

a, € {0, 1) where a is the set of all topology design variables o?/; stress cr//(A, a) and displacement 5^/(A, a) express the stress and displacement; and they are both functions of sectional design variables and topology design variables. The model P2 is a mixed discrete programming or a pure discrete programming including two kinds of variables. In P2, the stress constraint is a/a//(A, a) < a/

(4)

If the /th bar is cancelled from the structure, the corresponding topology design variable «/ = 0, stress a/cr//(A, a) = 0, and the stress constraint Qf/cr//(A, a) < a/ will be certainly satisfied, so the 'limit stress' will not disturb us. The feasible region Q of P2 includes topology design variables and sectional design variables. Let the feasible region of the /th topology of structure be ^ / , then Q = {^1,^2,

••

.

^N

(5)

where A^ is the total number of the structure's topology. It is obvious that P2 can be divided into N size optimization model PPi ~ PPA^ corresponding to A^ topologies, and there is no 'singularity of optimum solution' in every section optimization PP/, so there is no 'singularity of optimum solution' in P2. 3. Comparison and analysis of the two mathematical models The following topology optimization of four bar truss, shown in Fig. 1, is a well-known problem, and many scholars have studied it. The corresponding feasible region is shown in Fig. 3 and Fig. 2 shows the feasible region of model Pi. It can be seen from Figs. 2 and 3 that the feasible region in Fig. 2 is a union set of all feasible regions of all topologies in Fig. 3, and it is a strong 'non-convexity' feasible region. The optimum solution is at the singular

735

C Shan/First MIT Conference on Computational Fluid and Solid Mechanics .5

®

3

_ ^

Fig. 4. Plane truss with 10 bars. 2 4 6 8 10

.5

Fig. 2. Feasible region with one kind of design variables.

®

3

(2)

1

Fig. 5. Optimum topology of 12 bar truss.

2

4

optimum solution. In order to obtain the optimum solution, the above two kinds of design variables have to be treated simultaneously no matter what kind of algorithm is used. The following example illustrates the problem. A plane truss with 10 bars is shown in Fig. A. E = 68.96 GN/m^ p = 27150.68 N/m^ and the allowable stress is ±112A MN/m^ for all elements. There are two load cases. Load case I: P2y = -445 kN; load case 2: P^y = -445 kN. The allowable displacement in the y direction of each moveable node is 50.8 mm. The lower limit of each element's area is 0.645 cm^ and the initial value of each element's area is 64.5 cm^. The allowable discrete set is 5 = {6.45, 19.35, 32.26, 51.61, 64.51, 67.74, 77.42, 96.77, 109.68, 141.94, 154.84, 167.74, 180.64, 187.1, 200, 225.81} (cm^). The optimum solution is shown in Table 1. The computation converges through 6 iterations. Using the relative deference quotient algorithm, we obtain the structure shown in Fig. 5, which has a different topology from Wang and Sun [5], with an optimum value of objective function better than that of Wang and Sun [5]. Ringertz [6] had the same example with continuous size design variables and obtained a solution that is heavier than the solutions of discrete size design variables of Wang and Sun

6

Fig. 3. Feasible region with two kinds of design variables. point Ai = 0, A2 = 6, which can not be obtained by common algorithms. Now let us see a topology optimization with two kinds of design variables. For the topology T(3), the optimum solution is Ai = 0, A2 = 8; for T(2), it is Ai = 0, A2 = 6; and for T(l), it is Ai = 8, A2 = 0. The optimum solution of structure is Ai = 0, A2 = 6. Because there is no 'singularity of optimum solution' in all three size optimizations corresponding to the three topologies, so long as we solve the problem with the mixed discrete optimization algorithm, the puzzle of 'singularity of optimum solution' will not occur. In model P2, the topology design variables and size design variables are coupled and both have influence on the Table 1 Calculating results of 10 bar truss Reference

[6] [5] [4]

W{N)

21,792 20,477 20,122

Sectional area of each bar (cm^) Ax

Ai

A3

A4

As

Ae

Ai

As

A9

Aio

All

A12

194.19 167.74 167.74

0 0 19.35

142.58 109.68 109.68

96.77 96.77 77.42

0 51.61 0

0 0 19.35

39.23 32.26 32.26

137.42 96.77 109.68

137.42 141.94 109.68

0 0 32.26

0 0 0

0 0 0

736

C. Shan/First MIT Conference on Computational Fluid and Solid Mechanics

[5]. But it was pointed out in [5] that the same topology as in [6] would be obtained by canceling the 5th bar; however, the weight of the structure, which is heavier than the solution of [6], was W* = 23738 N. If the ground structure is 12 bars truss, the same optimum solution is obtained. Even for different original values of design variables, the same optimum solutions are obtained.

4. Conclusions The following conclusions can be summarized from the above analysis and discussion. (1) The origin of the problem with the 'singularity of optimum solution' is that the feasible region of topology optimization is strongly 'non-convex' because the union set of the feasible regions of section design variables in all topologies of structure is taken as the feasible region of topology optimization. (2) In nature, the topology optimization of structures is a mixed discrete optimization (sectional design variables are continuous variables) or a pure discrete optimization (sectional design variables are discrete variables) with topology design variables and section design variables. The puzzle of the 'singularity of optimum solution' can be avoided and the influence of 'limit stress' on the topology optimization can be canceled by constructing the mathematics model of topology optimization with topology design variables and section design variables and solving the model by using the algorithms of mixed discrete optimization or pure discrete optimization.

(3) Because the topology design variables and section design variables are coupled, some algorithms that treat the two kinds of design variables simultaneously must be used to solve the mathematics model. Acknowledgements This work is supported by the Visiting Scholar Foundation of State Key Laboratory and NSF of Shandong Province. References [1] Cheng G. Some aspects of truss topology optimization. Struct Optimization 1995;10(3):173-179. [2] Cheng G, Jiang Z. Study on topology optimization with stress constraint. Eng Optimization 1992;20:129-148. [3] Rozvany G, Birker T. On singular topologies in exact layout optimization. Struct Optimization 1994;8(4):228-235. [4] Chai S, Shi L, Sun H. An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables. Struct Optimization 1996;12(l):45-56. [5] Wang YF, Sun HC Optimal topology design of trusses with discrete size variables subjected to multiple constraints and loading cases. Acta Mech 1995;27(3). [6] Ringertz UT. On topology optimization of trusses. Eng Optimization 1985;9:209-218. [7] Guo X. A Study on the Singular Optimum Solution of Topology Optimization of Structures. Doctoral thesis, Dalian University of Technology, 1998.

737

Analysis and design of two-dimensional sails Sriram Shankaran *, Antony Jameson Aeronautics and Astronautics Department, Stanford University, Palo Alto, CA 94305, USA

Abstract The development of a computational tool to design and analyze two-dimensional sail geometries is presented in this paper. The behavior of this coupled fluid-structural system is modeled by numerically solving the Navier-Stokes equations for the flow around the sail, and by using a finite element method to estimate the static deflections and stresses on the sail. Hence, the flow solver imposes a loading on the sail and the deflections produced on the sail are incorporated into the flow solver and the process is iteratively repeated until the deflected shape of the sail is obtained. This analysis tool is then embedded in an automatic design environment where changes to the sail geometry are predicted to achieve improvements in a specific cost function. Specifically, the design process simplifies the coupled optimization problem by eliminating structural performance measures from the cost function but takes into account the aeroelastic deflections while performing the aerodynamic shape optimization. Keywords: Sails; Anisotropic unstructured grids; Computational aeroelastics; Automatic shape optimization

1. Introduction

2. Aeroelastic computations

The ability to analyze the flow around complex aerodynamic configurations with fast computational models [1] and the development of novel ideas to automate the design process [2] have placed computational techniques at an enviable advantage in a design environment. Steady state solutions to the Reynolds Averaged Navier-Stokes equations can be obtained, on a parallel computer, in a few hours [4,5] and the use of control theory to formulate the design problem has lead to a drastic reduction in the number of flow computations required to improve an existing design [3]. These algorithmic advances provide a robust stepping stone to tackle more complex physical phenomena and optimization problems, in particular aeroelastic computations and multidisciplinary optimization. The design of sails is a particular example of problems of this nature. In the following sections the development of computational tools to analyze and design sails is outlined.

The physics governing the aeroelastic behavior of the sail is modeled by approximating the sail to a two-dimensional flexible membrane of known length and prescribed end conditions. The Navier-Stokes equations govern the flow around the sail and they are discretized on an unstructured mesh using a finite volume method. An explicit, multistage time stepping scheme is used to integrate the solution in time. A full approximation multigrid method and implicit residual averaging techniques are used to accelerate convergence to steady state. The deflections and stresses that develop on the sail are numerically evaluated by using a finite element technique with linear or bilinear shape functions for the displacements. The structural model is completely linear and does not change during the computations. The flow solver prescribes the loading for the structural simulation and the deflections obtained from the structural computations are incorporated back into the flow solver. This coupled simulation is iterated until the system converges to produce the deflected shape of the sail. The computational tool for the analysis of the fluid flow and the structural simulation package are being developed.

* Corresponding author. Tel.: +1 (650) 723-9564; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

738

S. Shankaran, A. Jameson /First MIT Conference on Computational Fluid and Solid Mechanics

3. Multidisciplinary optimization Although it is impossible to separate the performance requirements of the sail from those of the hull and the keel of the boat, certain aerodynamic characteristics of the sail can be optimized to improve the polars of the boat. For example, to obtain the maximum forward thrust with minimum heeling moment, the moment that tends to overturn the boat, the sail has to be trimmed at an optimum angle to the apparent wind and should produce maximum lift in this position. The Frechet derivative of the cost function is determined by the solution of an adjoint partial differential equation and the shape of the sail is then modified to achieve an improvement in the cost function. The optimization strategy used in this study simplifies the fully coupled optimization problem by eliminating the structural performance measures from the cost function in the design. Therefore, the model of the structure is fixed, but the aerodynamic shape optimization technique takes into account the aeroelastic deflections as the computations proceed.

4. Conclusions The development of a computational tool to design two-dimensional sail geometries was outlined. The various

components that constitute the design package are being developed and validated. The eventual aim of this study is to provide a better understanding of the physics governing this problem and an improved head and main sail configuration.

References [1] Jameson A. MulUgrid algorithms for compressible flow calculations. Second European Conference on Mulfigrid Methods, Cologne, October 1985. [2] Jameson A. Aerodynamic design via control theory. J Sci Comput 1988;3:233-260. [3] Jameson A. Optimum aerodynamic design using CFD and control theory. AIAA Paper 95-1729, AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, June 1995. [4] Jameson A, Alonso JJ. Automatic aerodynamic optimization on distributed memory architectures. AIAA Paper 96-0409, Reno, NV, January 1996. [5] Martinelli L, Jameson A. Validation of a multigrid method for the Reynolds Averaged Navier-Stokes equations. AIAA Paper 88-0414, Reno, NV, January 1988.

739

Existence of a lift plateau for airfoils pitching at rapid pitching rates S.R. Sheikh'''*, Mao Sun'', Hossein Hamdani^ ^ College of Aeronautical Engineering, National University of Sciences and Technology, Rawalpindi, Pakistan ^Beijing University of Aeronautics and Astronautics, Beijing, 100083, People's Republic of China

Abstract In this research effort, the phenomena of dynamic stall in case of airfoils pitching at extremely rapid rates (Q^ > 0.72) have been investigated using numerical flow simulations, with the aim of gaining more insight in to the events that occur when airfoils are made to pitch at such high rates. Flow around a constant-rate pitching NACA 0012 airfoil was studied in detail for this purpose. The force and moment coefficients and detailed flow structures were studied specifically in conjunction with the existing vortical theory to provide insights into these events. It was estabhshed that the sudden changes, in the behavior of the force and moment coefficients of airfoils pitching at extremely high rates, are caused not only due to the shedding of the clockwise (negative) vortex from the airfoil upper-surface, but also due to the cavitation effects, which figure prominently at these rates. Keywords: Navier-Stokes equation; Dynamic stall; Oscillating airfoil; Pitching airfoil; Vortex shedding; Lift plateau

1. Introduction

2. Numerical method.

Dynamic stall is the term used for the deep stall, which occurs in oscillating airfoils during the retraction cycle. When an airfoil is pitching up, the flow separation and hence the stall, is delayed resulting in a higher maximum lift coefficient. However, once the airfoil nears the end of its pitching-up movement and starts the retraction (pitchdown) cycle, a separation region is rapidly formed near the leading edge of the airfoil. This separation region quickly grows till it bursts, causing a massive drop in lift. A similar effect is also seen in case of airfoils moving in the pitch-up mode only. Only difference being that the stall occurs at a much higher angle of attack. However, if the pitching rate of the airfoil is extremely rapid the stall angle is delayed so much that normal stall characteristics are not visible even at as high angle of attacks as a = 80 ~ 90". The CLU-curve also shows a trend, which is quite different from the behavior seen for airfoils pitching at relatively lower rates. The study endeavors to look in detail at this area, and tries to establish plausible explanations for the occurrences.

A finite-difference code [1] based on Beam-Warming [2] block approximate factorization solution of Reynold's averaged Navier-Stokes equations using Baldwin-Lomax turbulence model has been developed and validated and is used to carry out all the numerical computations.

* Corresponding author. Tel.: +92 (923) 631-499; Fax: +92 (923) 631-351; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

3. Lift mechanism in light of vortical theory. To study this phenomenon and the underlying events, flow around NACA 0012 airfoil is studied in detail using a highly dense grid (214 x 300) for numerical simulation of these events. Using this grid, different phenomena of interest such as, trailing edge vortices from the lower surface boundary layer can be captured. The force coefficients computed consequently have been discussed in relation with the flow structure in light of the theory of vorticity dynamics (for details see [1,3]), resulting a better understanding of the flow mechanism. Simply put the vortical theory states that, the aerodynamic force exerted on the airfoil by the fluid is proportional to the sum of time-rate of change of the first moment of vorticity and the inertia force of the displaced fluid mass. For constant rate pitching

740

S.R. Sheikh et al /First MIT Conference on Computational Fluid and Solid Mechanics

airfoil, it can be shown that: F=

- ^

(1)

where Yf is the first moment of the vorticity in the fluid. The relationship used for calculation of the pitching rate is given by 0^' = ^+(1 + txp(-4.6t/to), where the constant to specifies the time for Q+ to reach 99% of its final value Q'^. Due to the uncertainty posed by turbulence models this study has been carried out at a lower Reynold's number of Re = 8, 600. A free-stream Mach number M^c = 0.2 has been used. For this study a conventional pitch-axis located at a distance of 0.25c from the airfoil leading edge is chosen. The flow structures in this paper are represented by the equal vorticity lines calculated in the inertial frame and the streamlines that are seen in a frame rotating with the airfoil.

4. Flow mechanism at high pitching rates Fig. 1 shows that the d vs a curve for NACA 0012 airfoil undergoing constant-rate pitching with Q'^ > 0.72 has a distinctively different behavior than what is seen for the low and medium pitching rates. This phenomenon has not been highlighted by any of the previous studies and the current research effort endeavors to put forward a plausible explanation. It has been observed that for rapidly pitching airfoils the lift-curve shoots up at a tremendous rate during the initial stages of pitching. Even before the airfoil reaches an angle of attack of 10 degrees, a very high value of lift is attained. This value can be as high as about 90% of the Cimax- After the initial, sudden rise in the CL VS a curve, the lift-slope tapers-off to a nearly constant value. It is observed that the airfoil maintains lift values of above 90% of Cimax for a considerable time, typically from a ~ 10° to a % 60°. For the purpose of this study this flat region of CL VS a curve will be referred to as the lift plateau. Generally after reaching an angle of attack of a % 60° the airfoil starts to lose lift gradually. No stall like behavior is

observed and even at such high values of angle of attack as, Of = 80° the airfoil is able to maintain considerably high values of lift. These occurrences show that in rapidly pitching airfoils the mechanism leading to dynamic stall is checked and thus no conventional stall like behavior is visible. For identifying the causes of this a detailed study of the airfoil flow structure and the pressure distribution is undertaken. 4J. Case 1 First the case with ^ ^ = 0.72 is considered. From Fig. 1 it can be seen that the CL VS a curve rises sharply as the airfoil starts to move at a high pitching rate from a = 0°. This sharp rise continues till about a = 8°, at which stage the airfoil has attained a lift value of CL = 2.8 or nearly two-thirds of the CL max- The corresponding pressure distribution is given in Fig. 3 (a = 8°). As the airfoil starts to pitch-up instantaneously at high pitching rates with pitch-axis at 0.25c, while the airfoil leading edge moves up, the airfoil trailing edge, being much further away from the pitch-axis, moves down at a much higher velocity. This sudden down-wards movement causes cavitation on the airfoil upper surface near the trailing edge. This in turn develops a large suction region on the trailing edge. This suction region is clearly visible in the pressure curves. In this case upper suction region, especially in the trailing edge area, is clearly much greater than the upper surface suction seen for lower pitching rate cases. Due to the cavitation on the upper surface trailing edge very large counter-clockwise or positive vorticity will be shed into the flow field. The vorticity shed from the trailing edge is much stronger than that for the lower pitching rates and also moves downstream rapidly. This can be seen by comparing the vorticity plot for Q^ = 0.094 and ^+ = 0.72 given by Fig. 4. The numbers in the plots represent the vorticity value. It is seen that at a = 8°, the positive vorticity shed from the trailing edge for the case of Q^ = 0.72 is much larger than that for Q'^ = 0.094. Moreover, for the high pitching rate, the vorticity is shed in a much shorter time (about one eighth of that for the case of lower pitching rate), resulting in

Fig. 1. Lift and drag coefficients for constant-rate pitching NACA 0012 airfoil for rapid pitching rates (Q^ = 0.72, 1.68 and 2.40).

S.R. Sheikh et al /First MIT Conference on Computational Fluid and Solid Mechanics

741

00 II

in

1.00

X

1.50

o

o

1.00

1.50

o

Fig. 2. Flow structures for constant-rate pitching NACA 0012 airfoil (^^ = 0.72). a very large time-rate of change of the first moment of vorticity. This might explain the high lift produced in the stage between a = 0 - 8°. From a = 8° to a = 25° lift increases gradually to Ci = 3.8 or about 85% of CLmaxThe maximum lift of CLmax = 4.38 is achieved at o? = 40°. From Of = 40° to a = 60° the lift decreases gradually to CL = 3.8. After a = 60° the lift drops a little more rapidly, though not sharply like post-stall behavior. Even at a = 80° the airfoil is able to maintain a Hft coefficient of C^ = 1.0, a value greater than the normal post-stall lift for the NACA

0012 airfoil. The CL VS a curve displays the lift plateau between about a = 8° to a = 60°. During this period the change in lift force is gradual and a lift of about 80% of the ^Lmax is maintained. Looking at the flow structure (Fig. 3), it is observed that after the initial shedding of the vortex sheet (i.e., after a is larger than about 8°), the vortex sheet turns to roll back and form a vortex, termed as the starting vortex. Therefore, as a whole the positive vorticity will not move downstream as fast as if without the rolling back, resulting in the Ci vs a curve slope tapering off. When

742

S.R. Sheikh et al /First MIT Conference on Computational Fluid and Solid Mechanics a=8 0.0

-0.2

--""=0.4^

^ ^

a = 40

-0.8 -1.0

Fig. 3. Pressure distribution for constant-rate pitching NACA 0012 airfoil (Q+ = 0.72). angle of attack is larger than about 40°, it seems that the leading edge vortex starts to grow a little faster. During this time period, from a = 8° to a = 60°, the starting vortex (with positive vorticity), gaining strength gradually, moves slowly downstream due to the blockade of the airfoil (the starting vortex is on the back of the airfoil). While the leading edge vortex bubble (with negative vorticity) on the upper surface also gains strength slowly. The result of these occurrences might be that the time-rate of total first moment of vorticity remains nearly constant and hence produces the lift plateau in the CL VS a curve. It is also seen that after a = 57°, as the upper surface clockwise vortex starts to gather strength more rapidly, the lift force also starts to drop at a slightly higher rate. However, since this vortex (dynamic stall vortex), remains attached to the airfoil and does not show the tendency to breakaway even as the airfoil reaches an angle of attack of of = 90°, the drop in lift remains slow and no stall like conditions are observed. Thus in the conventional sense a full stall does not

a = 90

V^fe

-1.2

SIJ:^

-1.4

^^^i^p^

^^Q*o

^^^y ^^^^^^^^^^^

Fig. 4. Comparison between vortical structures for NACA 0012 airfoil at various angles of attack.

occur even at a = 90°. The decrease in lift is mainly due to the slowed downstream motion of the starting vortex and the dynamic stall vortex gaining strength and its center moving a little away from the airfoil surface. This in general is an occurrence, which precludes the onset of dynamic stall. It can therefore, be concluded that while undergoing constant-rate pitching at very high, pitching rates, the airfoil does not experience complete stall.

References [1] Sheikh SR, Mao Sun. Dynamic stall suppression on an oscillating airfoil by steady and unsteady tangential blowing. Aerospace Sci Technol 1999;6:355-366. [2] Beam R, Warming R. An implicit factored scheme for the compressible Navier-Stokes equations. AT AA J 1978;16(4):393-402. [3] Wu JC. Theory of aerodynamic force and moment in viscous flows. AIAAJ 1981;19(4).

743

Shape optimization for crashworthiness featuring adaptive mesh topology Nielen Stander*, Mike Burger Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA 94550, USA

Abstract A successive linear response surface method (SRSM) is applied to the shape optimization of a vehicle crashworthiness problem in which a preprocessor is used to parameterize the geometric model and mesh topology of the vehicle instrument panel. An upper limit on the element size is used as a criterion for the mesh adaptivity. Simulation is conducted using the explicit dynamic analysis code, LS-DYNA. The study demonstrates the effectiveness of adaptive meshing and simulation-based shape optimization in problems of complex behavior such as crash simulation. Keywords: Crashworthiness optimization; Response surface methodology; Adaptive mesh refinement; Experimental design; Successive approximations; Mesh topology

1. Introduction The explicit dynamic analysis method has become a standard approach for solving nonlinear dynamic problems involving crash and impact simulation. At the same time, simulation-based optimization is increasingly being adopted as an aid to explore the design space effectively and with minimal user intervention. In this endeavor, the Response Surface Method [1] as adapted by e.g. Roux et al. [2] for structural optimization provides an option for addressing the 'step-size dilemma' in sensitivity analysis and optimization. The method, which avoids the necessity for analytical or numerical gradient quantities, has been incorporated by Kok [3] in a successive approximation scheme for finding converged optima, while attempting to avoid local optima and spurious gradient information. The successive response surface method (SRSM) as applied in crashworthiness design by Akkerman et al. [4] uses a similar adaptive windowing scheme which involves panning and zooming to position and size the region of interest in the design space. In this case, the windowing parameters adapt according to (i) the move characteristics of the current iteration (difference between the optimal and starting design values in relation to the size of the region of interest), and (ii) detection of oscillation, a phenomenon pecu* Corresponding author. Tel.: 449-2500; Fax: 449-2507; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

liar to successive linear approximation methods. The linear surfaces are fitted to the responses of the design points determined by the experimental design. To determine suitable points within the region of interest, the Z)-optimality experimental design criterion is used. The above methodology has been incorporated in the LS-OPT® ^ code to analyze an example in the crashworthiness design of a vehicle instrument panel. The design problem, which relates to occupant safety, has eleven design variables. Some of the variables are shape parameters which have been defined in a geometric preprocessor, TrueGrid®^ code to analyze an example in the crashworthiness design of a vehicle instrument panel. The design problem, which relates to occupant safety, has eleven design variables. Some of the variables are shape parameters which have been defined in a geometric preprocessor, TmeGrid®^ which is a standalone program and which can be incorporated in the design cycle. It was shown in a previous study by Akkerman et al. [4] that shape optimization is important but that a large number of variables introduces a redundancy in variability, and that certain variables are more significant than others for reducing the knee forces. A particular feature of that study ^ LS-OPT is a registered trademark of Livermore Software Technology Corporation. ^ TrueGrid is a registered trademark of XYZ Scientific Applications, Inc.

A^. Stander, M. Burger/First MIT Conference on Computational Fluid and Solid Mechanics

744

was also that a constant mesh topology resulted in a poor appearance of the mesh in extreme comers of the design space. The present study focuses on this aspect only and it is shown that an excellent result can be achieved over a wide range of shape variation when maximum element size is used as a criterion for adaptation of the topology.

2. Finite element model and parametrization of example Fig. 1 shows the finite element model of a typical automotive instrument panel (IP). The spherical objects represent simplified knee forms which move in a direction as determined from prior physical tests. The system is composed of a knee bolster that also serves as a steering column cover with a styled surface, and two energy absorption steel brackets attached to the cross vehicle IP structure. The brackets absorb a significant portion of the lower torso energy of the occupant by deforming appropriately. A steering column isolator (also known as a yoke) is used as part of the knee bolster system to delay the wrap-around of the knees around the steering column. The simulation is carried out for a 40 ms duration by which time the knees have been brought to rest. The brackets and yoke are non-visible and hence their shape can be optimized. For this purpose, eleven design variables, depicted in Fig. 2 have been chosen to represent the design. Some of these design variables involve simple sizing such as the gauges of the brackets and bolster while all the others are geometric in nature. The mesh topology of selected parts is adapted by specifying an upper limit on element size of 9 mm in these areas. The preprocessor then adapts the mesh size and topology according to the geometric changes required by the various designs. Fig. 3 illustrates the ability

of this feature to adapt the mesh over a wide range of shape variables. In this case the range is defined by the upper and lower bounds of the design space. Based on the over-sampling rule of 1.5 used by Roux et al. [1], 19 simulations are conducted for each iteration. The computations were performed on an HP V Class server (16 processors) running 10 processes in parallel. A single simulation requires about 3.7 hours, resulting in 33 hours total including the final verification run.

3. Design formulation 3.1. Constraints For optimal occupant kinematics, it is essential that knee intrusion into the IP be limited to desired values. Upper bounds of the left and right knee displacements, DL and DR, are used to limit the maximum knee intrusions to 115 mm. The yoke displacement is limited to 85 mm by specifying the displacement of a node at its axis of symmetry. 3.2. Objective The selection of a low force constraint value forces the optimization formulation to minimize the maximum knee force subject to the constraints above, i.e. min(max(FL, F/?)) where the subscripts L and R refer to left and right respectively. The response is also maximized over time so that only the peak knee forces are used. The knee forces have been filtered, SAE 60 Hz, to improve the approximation accuracy.

Non-visible, optimizable structural part

Simplified knee forms

Styled surface, non-optimizabie

Fig. 1. Typical instrument panel prepared for a 'Bendix' component test.

A^. Stander, M. Burger/First MIT Conference on Computational Fluid and Solid Mechanics

745

Width Gauge

Depth idlu«

Width Depth

Fig. 2. Design variables of the knee bolster system.

\

-

•

••-

Left Right

1.2 • D

1^ 0.8

—•

[ J

iJ j j

0.6 0.4 0.2 0 1

0

•

1

2 Iteration Number

3

4

Fig. 4. Optimization history of knee forces.

Fig. 3. Mesh adaptivity: smallest and largest shapes in the design space.

4. Results The optimization process required about 4 iterations to converge, using 77 simulations in total. The points and lines in Fig. 4 represent the simulated and predicted knee

forces respectively (scaled by 6500 N). The approximate knee force pairs are equal after the first iteration since min-max problems often equalize the relevant responses (see also Table 1). The computed results also converge to the same number. Table 1 summarizes the baseline vs. optimum results and shows that none of the constraints were activated. The final knee displacement results are accurately approximated by the response surfaces, but the final yoke displacement has very poor accuracy. This is probably due to the fact that a single central point was chosen along the entire length of the yoke and as can be seen from Fig. 5, the deformation of the yoke is generally large and uneven. It could also be that the response of the yoke with respect to the radius design variable is extremely nonlinear. In spite of this deficiency, the computed or predicted yoke displacement is inactive at the optimum. When comparing the present results with those of [4] in which a constant mesh topology was used (results not re-

746

N. Stander, M. Burger/First MIT Conference on Computational Fluid and Solid Mechanics

Table 1 Results of optimum design

Left knee force, FL Right knee force, FR Left displacement, Di Right displacement, DR Yoke displacement

Baseline

Optimum

Computed

Predicted

Computed

Upper Bound

6756 8866 95 98 35

5847 5847 96 100 31

6014 6325 97 99 19

115 115 85

is problem dependent and could affect robustness in some cases. • Certain responses are extremely noisy and need to be redefined in a different way, perhaps by filtering over time or introducing constraints at more than one point on the relevant part. The most important quantities namely the knee displacements and forces appear to be smooth and can be accurately approximated. • The present study confirms that the problem is essentially unconstrained as it seems that the deformation is restricted by parts other than those which have been parameterized. • The optimization methodology is successful in improving the design but can possibly be made more efficient by screening the design variables to reduce the design space. A future study will focus on using multiple starts to test global optimality as well as the reduction of design variables through screening processes.

References Fig. 5. Optimum design: deformed configuration at r = 40 ms. peated here) it appears that both the baseline and optimum results are similar. The model did therefore not improve significantly in terms of its structural integrity.

5. Conclusions The paper demonstrates the use of a successive response surface method to an instrument panel design in order to enhance its crashworthiness. The following conclusions were made: • The mesh adaptivity did not yield significantly better results than a mesh of constant topology, but this aspect

[1] Myers RH, Montgomery DC. Response surface methodology: process and product optimization using designed experiments. Wiley, 1995. [2] Roux WJ, Stander N, Haftka R. Response surface approximations for structural optimization. Int J Numer Methods Eng 1998;42:517-534. [3] Kok S, Stander N. Optimization of a sheet metal forming process using successive multi-point approximations. Struct Optim 1999;18:277-295. [4] Akkerman A, Thyagarajan R, Stander N, Burger M, Kuhn R, Rajic H. Shape optimization for crashworthiness design using response surfaces. Proceedings of the International Workshop on Multidisciplinary Design Optimization, Pretoria, South Africa, August, 2000.

747

Multi-criteria evolutionary structural optimization involving inertia Grant P. Steven^'*, Kaarel Proos*', Y.M. Xie'^ ^ School of Engineering, University of Durham, Durham DHl 3LE, UK ^Department of Aeronautical Engineering, University of Sydney, Sydney, NSW 2006, Australia '^Faculty of Engineering, Victoria University, Box 14428 MCMC Victoria 8001, Australia

Abstract In this paper, we present methodologies for topology structural optimization with the combined optimality criteria of stiffness and inertia. The construction of a global criterion to guide the optimization when the individual criteria are contradictory is resolved. An example is presented and discussed. Keywords: Topology; Structural optimization; Evolutionary method; Multi-criteria; Pareto

1. Introduction Research in structural optimization using a heuristic evolutionary method has proved successful in all single criteria situations, of stress, stiffness, frequency and buckling with single and multiple loads and support environments. This research is directed towards the adoption of multi-criteria structural optimization into the algorithmic framework of the Evolutionary Structural Optimization method of Xie and Steven [1]. This current paper reports on the dual criteria of stiffness and Moment of Inertia (Mol); these are relevant the design of robots and aircraft. Four weighting schemes are examined. A single example is presented that goes some way towards validating the approach and indicating the best weighting scheme.

and [K'] is the stiffness matrix of the /th element. To maximize stiffness elements with lowest sensitivity are removed/altered. Each element contributes to any of the six Mol. The contributions can be ranked from high to low and dependent upon the optimality criterion; appropriate elements can be eliminated in the slowly evolving manner of the ESO process.

3. Weighting method formulation A weighting global criterion produces the, well known, Pareto set [2]. Using normalized individual criterion the weighting formulation can be written as M

^Lticrit = ^iK 2. Determination of sensitivity numbers for element removal In an FEA model, the element sensitivity numbers for each criteria are combined to make the element removal decision. For the stiffness criterion the element sensitivity number is [1]

+ ^2R'2 + . . . + ^MR'M = E "^J^J

(2)

7=1

^liiuiticrit i^ ^^^ multiple criteria function that determines removal, the ic's are selectable weights (they sum to unity) and the /?'s are the ratio of element sensitivity to the global maximum for that criterion. The global criterion method has no optional ingredient and is given by

GL^,.^=[(R\~S[f + {Ri-S^f + .. l/M

(1) a; = -xiu '[K^W] where {u'} is the displacement vector of the /th element

H'^'.-S'.)T = Ei^j-'j)'

* Corresponding author. Tel.: +44 (191) 374-3935; Fax: +44 (191) 374-2550 E-mail: [email protected]

^muiticrit is the multiple criteria function that determines element removal; the 5"s are the ratio of the minimum

© 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

(3)

748

G.R Steven et al /First MIT Conference on Computational Fluid and Solid Mechanics

value to the maximum of the same criteria. Alternatively, each element can have a logical AND or a logical OR scheme applied to determine its removal. These options are self-explanatory.

4. Example of a rectangular plate with fixed supports This rectangular plate (Fig. 1), has been divided up into 45 X 30 square elements. The comers are either fixed or loaded. It is desired to maximize the stiffness across this loaded plate, and to increase the specific Mol about the z-axis, i.e. the squared Radius of Gyration (RoG). Figs. 3 and 4 illustrate the resulting topologies for each of the four ESO multi-criteria methods for the iterations where 50% of the material has been taken away. The dominance of the moment of inertia about the z-axis as

seen in Fig. 3, where the nature of a circle is still evident even at very low weightings of Molz, is also portrayed in the Pareto curves of Fig. 2. This is where most of the points making up each Pareto curve are distributed more towards the Molz end of the curve. The global criterion topology of Fig. 4 closely resembles the 50/50 weighting topology of Fig. 3. The numerical evidence of this is substantiated in Fig. 2 where the global criterion point 4a lies nearest to the displayed weighting point of 50% stiffness/50% Molz. A comparison of the logical AND topology (Fig. 4) to those of the weighting method (Fig. 3) deduces that the AND method produces topologies resembling that based on 1(X)% Molz (Fig. 3). Verification of this can be made with Fig. 2, where point 4b is positioned nearest to point BeA correlation can be made between the topology of the logical OR operator Fig. 4c) and the fully stiff topology (Fig. 3) of the weighting method. This correlation can be

150 mm lOON

jsr

^

Fixed support

Mesh size 45 x 30

100 N

«4sr Fixed support

Fig. 1. Initial design domain of a rectangular plate under loading withfixedsupports. 4.00E-08 3.50E-08 f

3.00E^

I I

"•"•—10 % Material Reiiio>wl Weighted Multicriteria ESO •* "• 2 0 % Material Removed WeigNedMudticriteria ESO • * " ' 2 5 % Material Removed Weighted Multicriteria ESO - •>^ - 30 %Material Removed Weighted N4ilticriteria ESO •^ " " 4 0 % Material Removed Weighted Multicriteria ESO ~^ 50 % Material Removed Weighted Niilticriteria ESO D 5 0 % Material Removed AND Multicriteria ESO ^ 50 % Material Removed OR Multicriteria ESO 50 % Material Removed aobal Criterion Ntilticriteria ESO B3

IB..

•2 2.50&08

ttB,

4b)

B,

2.0OErO8 L50E-08 0.05

0.052

0.054

0.056

0.058

0.06

0.062

0.064

Radius of G^^ticMi about z -axis (m) Fig. 2. Pareto curves for RoG about z-axis versus mean compliance x volume.

0.066

•tl

G.P. Steven et al /First MIT Conference on Computational Fluid and Solid Mechanics

(a)

(b)

(c)

749

(d)

Fig. 3. Optimal designs for different weighting criteria. Material removed: 50%. (a) lOstiff : JCMOIZ = 1:0, (b) 0.95 : 0.05, (c) 0.5 : 0.5, (d) 0.0: 1.0.

—

—

Fig. 4. Fifty percent removal, (a) Global criterion, (b) Logical AND. (c) Logical OR. validated with the graph of Fig. 2, vv^here point 4c lies in the vicinity of A6.

References [1] Xie YM, Steven GP, Evolutionary Structural Optimisation. Springer, London, 1997.

[2] Koski J, Multicriterion Structural Optimisation. In: Adeli H (Ed.), Advances in Design Optimisation. Chapman and Hall, London, 1994, pp. 194-224.

750

Enhancing engineering design and analysis interoperability. Part 1: Constrained objects Miyako W. Wilson^'\ Russell S. Peak^'*, Robert E. Fulton ^'^ " Georgia Institute of Technology, Engineering Information Systems Lab \ 813 Ferst Drive, Atlanta, GA 30332-0560 USA ' Georgia Institute of Technology, Woodruff School of Mechanical Engineering^, 813 Ferst Drive, Atlanta, GA 30332-0405 USA

Abstract The wide variety of design and analysis contexts in engineering practice makes the generalized integration of computer-aided design and engineering (CAD/CAE) a challenging proposition. Transforming a detailed product design into an idealized analysis model can be a time-consuming and complicated process, which typically does not explicitly capture idealization and simplification knowledge. Recent research has introduced the multi-representation architecture (MRA) and analyzable product models (APMs) to bridge the CAD-CAE gap with stepping stone representations that support designanalysis diversity. This paper introduces constrained objects (COBs) as a generalization of the underlying representations. The COB representation is based on object and constraint graph concepts to benefit from their modularity and multi-directional capabilities. Object techniques provide a semantically rich way to organize and reuse the complex relations and properties that naturally underlie engineering models. Representing relations as constraints makes COBs flexible because constraints can generally accept any combination of I/O information flows. This multi-directionality enables design sizing and design verification using the same COB-based analysis model. Engineers perform such activities throughout the design process, with the former being characteristic of early design stages and vice versa. This paper presents basic examples to illustrate the main COB concepts. To validate the COB representation, other work describes electronic packaging and aerospace test cases implemented in a toolkit called XaiTools^. In all, the test cases utilize some 260 different types of COBs with some 370 relations, including automated solving using commercial math and finite element analysis tools. Results show that the COB representation gives the MRA a more capable foundation, thus enhancing physical behavior modeling and knowledge capture for a wide variety of design models, analysis models, and engineering computing environments. Keywords: Constrained object (COB); Constraint graph; CAD-CAE integration; Multi-directional

1. Motivation While computing tools continue to advance, Wilson [1] identifies the need for a unified physical behavior modeling representation with the following characteristics: • Has tailoring for design-analysis integration, including support for multi-fidelity idealizations, product-specific analysis templates, and CAD-CAE tool interoperability. • Supports product information-driven analysis (i.e., sup* Corresponding author. Tel.: +1 (404) 894-7572; Fax: +1 (404) 894-9342; E-mail: [email protected] ^ http://eislab.gatech.edu/ ^ http://www.me.gatech.edu/ © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

ports plugging in detail design objects and idealizing them into a diversity of analysis models). • Has computer-processable lexical forms along with human-friendly graphical forms. • Represents relations in a non-causal manner (i.e., enables multi-directional combinations of model inputs/ outputs). • Captures engineering knowledge in a modular reusable form. This paper overviews recent work that describes the foregoing needs and related literature. The following sections overview the constrained object (COB) representation developed to address these needs [1,2]. Basic examples illustrate the main concepts. Two other papers in this confer-

751

M.W. Wilson et al /First MIT Conference on Computational Fluid and Solid Mechanics b. Constraint Schematic-I Notation

a. Constraint Schematic-S Notation variable a subvariable a.d

pa bC\

subvariable sb

y

-o^

relation rl{a,b,s.c) lr1

e = b-c

h

subsystem s of cob type h

option category 1 dA] I option 1.1

0c k>

l[i-i]/=^^

e=f

equality relation L[j:i,n] —[Q|

100 lbs O

[1-2]

f=^

O^

option 1.2

30e6psi O-—

Result b = 30e6 psi (output or intermediate variable)

hooibsWO—

Result c = 200 lbs (result of primary interest)

o

-V t>

O

aggregate c>v

Input a =100 lbs

^

Q

Equality relation is suspended

Xr1 I I

r\

Relation rl is suspended

o-l_r~°

•o^^ elementw^.

Fig. 1. Basic constraint schematic notation. straint schematic notation that graphically emphasizes object structure and relations among object attributes. Two lexical languages, COS and COI, are the master forms which are computer-interpretable (Fig. 2b, Fig. 3b). Other forms depict subsets of COS and COI model content and include graphical views to aid human comprehension. The structure level languages and views define concepts as templates at the schema level (meta-level), whereas the instance level defines specific objects that populate one or more of these templates. The next sections present several

ence describe using COBs for design-analysis integration [3] and as a step towards multi-functional optimization [4].

2. COB basics 2.1. COB modeling languages and views The COB representation includes several modeling languages and views. For example, Fig. 1 summarizes con-

a. Constraint Schematic»S (also Constraint Graph-S)

Traditional Form Shape Schematic L

o^

AL F

^

Xj

AAA^

-Hi

deformed state

r3

spring constant,/:

F =kAL r2

•F

undeformed length,L^

force, F t

total elongation, AL . length, L

AL=L-

.start, Xi ^end, X2

Relations

r^:^L = L-LQ r,:F = kAL

b. Lexical COB Structure (COS) COB spring SUETYPE_OF abb; undeformed_length, L<sub>0 : REAL; spring_constant, k : REAL; start, x<sub>l : REAL; end, x<sub>2 : REAL; length, L : REAL; total_elongation, &:Delta;L : REAL; force, F : REAL; RELATIONS rl : " == <end> - <start>"; r2 : " == - r3 : " == <spring_constant> * "; END COB;

c. Subsystem-S (for reuse by other COBs)

Fig. 2. COB structure: spring primitive.

Elementary Spring

b^o p^i PX2

ALH

id

752

M.W. Wilson et al /First MIT Conference on Computational Fluid and Solid Mechanics a. Constraint Schematic-I Design Verification

r3

^^s{an,x,

total elongation, A L ^

r2

length, L

O

22 n

L=.v,-.v,

KJ

1

^•send, -y?

example 1, state 5.1

rzZTT. ^i ^ s p r i n g constant,^ 20 N/mm \\ {j ^ f^ '

I

force, F /

F =kAL

^undefomned length,Ln

20 mm

s t a t e 1.0

example 1, state 1.1

nQSETing. constant, k

Design Synthesis

b. Lexicai COB instance (COI)

vundeformed length,L^

r3

r2 AL=L-

total elongation. A L ^ L, | — f

2 fr^rn

length, L

10mm\Q^ 32 mm

Q^

(solved):

INSTANCEOF spring; INSTANCEOF spring; undeformed_length : 20.0 ; undeformed_length : 20.0; spring_constant : 5.0; spring_constant : 5.0; start : ?; total_elongation : ?; end : ?; force : 10.0; length : 22.0; total_elongation :i[_2 . force : 10.0; "'•'•••'^ ENDINSTANCE;

state 5.1 (solved):

state 5.0 (unsolved): force, F /

F =kAL

state 1.1

(unsolved):

IKSTANCE^OF spring; undeformed_length : spring_constant : ?; start : 10.0; length : 2 2.0; force : 4 0.0;

r

INSTANCE__OF spring; undeformed_length._ spring_constant start : 1 0 . 0 ;

20 . 0 "^"•'~

end : 3 2.0; length : 22.0; total_elongation : 2.0; force : 4 0.0; ND INSTANCE;

Fig. 3. Multi-directional (non-causal) capabilities of a COB instance: spring primitive.

of these forms for COBs that represent basic engineering concepts. 2.2. Example: spring primitive The upper-left portion of Fig. 2 shows the traditional form of an idealized spring object. A shape schematic defines the variables and their idealized geometric context, and algebraic equations define relations among these variables. Representation of this object as a COB s p r i n g template is shown in Fig. 2, where the constraint schematic graphically depicts its relations and variables. Fig. 2b is the COS textual form, which is the master template from which the other forms can be derived. Fig. 2c is an encapsulated form known as a subsystem, which is useful for representing this object when it is used as a building block in other COBs (e.g., Fig. 5). In all these forms the relations can support any valid input/output combination. For example, in relation r l , attributes l e n g t h and s t a r t can be inputs to produce end as the output, or end and s t a r t can be inputs to produce l e n g t h as the output. Fig. 3 shows views of an instance of this s p r i n g entity in two main states. In state 1, s p r i n g c o n s t a n t , u n d e f ormed l e n g t h , and f o r c e are the inputs, and t o t a l e l o n g a t i o n is the desired output. The COI lexical form (Fig. 3b) shows state 1.0 as this COB instance exists before being solved. State 1.1 shows it after solution (including constraint schematic form in Fig. 3a), where one can see that l e n g t h was also computed as an intermediate

value, and that end and s t a r t have no values because there was not sufficient data to determine them. State 5 shows this same s p r i n g instance where the desired deformed l e n g t h has been changed to be an input and s p r i n g c o n s t a n t has become the desired output. Peak and Wilson [3] describe how these basic cases characterize design synthesis vs. design verification. 2.3. Example: two-spring system Given the system of two springs in Fig. 4a, with traditional approaches one could draw their freebody diagrams (Fig. 4b), specify their relations and boundary conditions (Fig. 4c), and solve the resulting system of equations for the desired output. One could use computational math tools like Mathematica to aid this process and change input/output combinations. Yet essentially one would have a list of equations whose engineering meaning would not be inherent in their existence (e.g., one could not query relation rl and know that it is part of a spring). Furthermore, adding and deleting equations to change input/output directions for a large system of equations could become unwieldy. When one considers the constraint graph for this two-spring system (Fig. 5b), one recognizes that the shaded portions are essentially duplications of the same kind of relations (e.g., r l l vs. r21). Traditionally, one would have to manually replicate and adjust these similar relations via a potentially tedious and error-prone process. COBs address these issues by grouping relations and variables according to their engineering meaning and placing them into explicit reusable contexts.

M.W. Wilson et al /First MIT Conference on Computational Fluid and Solid Mechanics

753

a. Shape Schematic

•AA/V

U

•AAA^ h ^ - U2

Ui

b. Freebody Diagrams ^

L2

^1

^ AL^

^10

AA *< 1• Avvv

F,

• Xj2

,

• ^ 1 1

>

T

Fj

F2

<

^'

vvv

AL2

• ^ ^^

. ^r

AAA

• X21

^

•^22

c. Variables and Relations ^ ^

Kinematic Relations

t?Cj: Xjj

rj2 :/\Lj

bC2 •' Xj2 ~^21

=Lj -

LJQ

""""""^

' 2 / • -^2 —

r22:AL2 ^23

= F2

22 ~

*^^4 * -* 2 =

2/

=L2- L20 = k2AL2

^^^

Boundary

Conditions

bc^: Fj = = ^2

rj,:Fj =kjALj Constitutive Relations

=0

rj]: Lj = Xj2 - Xjj

P

bc^.-Uj =--ALJ bc^

U2 •-=AL

,+Uj

Fig. 4. Traditional mathematical representation: two-spring system. a. Constraint Schematic-S

b. Constraint Graph»S

spring 1

1^ Elementary Spring Ok Dto

1

FC Ate

bc5

i^ V J "V

11^^' —

>^

i

—

bc2

spring 2

1

bc3

^ Eiementarv

0L„ DAI

bc4 f-\ p \J ^

\

A t e L—Jt<2=AL2 + Mi| 0 "2 / be 6 tc ......#.......,....,/......... c. Lexical COB Structure (COS)

Analysis Primitives \ witii HncapMilated Kioematic & Constitutive Relations

System-l^evel Relations (Boimdary Conditions)

COB spring_system SUBTYPE OF analYsis_system; 'jM*- springl : spring; ••^J^ spring2 : spring; deformationl, u<sub>l : REAL; deformation2, u<sub>2 : REAL; load, P : REAL; RELATIONS bcl <springl.start> = = 0.0"; *bc2 <springl.end> == :spring2.start>"; bc3 <springl.force> = = <spring2.force>" ; bc4 <spring2.force> = = "; bc5 <deformationl> == <springl.total_elongation> <deformation2> == <spring2.total_elongation> bc6 EM3__^C0B;

<deformationl>

Fig. 5. COB structure: two-spring system. For example, by applying object-oriented thinking, the shaded regions in Fig. 5b are represented by two s p r i n g subsystems in Fig. 5a. There is no need to specify these

relations in the corresponding COS lexical form (Fig. 5c), as they are included in the s p r i n g entity per its COS definition (Fig. 2). System level boundary conditions are

754

M.W. Wilson et al. /First MIT Conference on Computational Fluid and Solid Mechanics

the only other relations that need to be specified here. With this definition completed, the constraint graph can now be seen as another view derivable from the lexical form; it essentially is a fully decomposed constraint schematic where no subsystem encapsulations are present. XaiTools™ is an analysis integration toolkit [1,2] that implements these concepts directly from the COS and COI forms. It enables links with design tools and effectively provides an object-oriented constraint-based front end to traditional CAE tools, including math tools like Mathematica and finite element analysis tools like Ansys.

3. Industrial examples and test cases Industrial applications of COBs and other test cases are given in [1,2] along with structure and re-usage statistics (for more than 260 types of COBs). Applications include thermomechanical analysis of printed wiring boards and assemblies (PWA/Bs), structural analysis of airframes, and thermal analysis of electrical chip packages.

4. Summary This paper introduces constrained objects (COBs) as a new representation of engineering concepts that has these characteristics: • Declarative knowledge representation (non-causal) • Combination of objects and constraint graph techniques • COBs = (STEP EXPRESS ^ subset) + (constraint graph concepts and views).

3 STEP EXPRESS (ISO 10303-11) is an object-flavored information modeling standard geared towards the life cycle design and engineering aspects of a product [http://www.nist.gov/sc4/].

Test cases show that COBs provide these advantages over traditional analysis representations: • Greater solution control • Richer semantics • Capture of reusable knowledge Envisioned extensions include capturing assumptions and limitations so that some verification of analysis results might be automated.

Acknowledgements This work builds on analyzable product models (APMs) by D. Tamburini (see [2]).

References ^ [1] Wilson MW The Constrained Object Representation for Engineering Analysis Integration. Masters Thesis, Georgia Tech, 2000. [2] Peak RS. X-Analysis Integration (XAI)^ Technology. Georgia Tech Report EL002-2000A, March 2000. [3] Peak RS, Wilson MW. Enhancing engineering design and analysis interoperability. Part 2: A high diversity example. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001. [4] Dreisbach RL, Peak RS. Enhancing engineering design and analysis interoperability. Part 3: Steps toward multi-functional optimization. In: First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 12-15, 2001.

^ Some of these references are available at http://eislab.gatech.edu/ ^ X = design, manufacture, sustainment, etc.

755

Retrofit design and strategy of the San Francisco-Oakland Bay Bridge continuous truss spans support towers based on ADINA R.W. Wolfe''*, R. Heninger'' ^ California Department of Transportation, 12501 Imperial Highway, Norwalk, CA 90650 USA ^ California Department of Transportation, 1801 30th Street, Sacramento, CA 95819 USA

Abstract Following the Loma Prieta Earthquake, the California Department of Transportation was charged with developing, and implementing retrofit strategies to mitigate seismic effects on bridge structures linking the State highway system. The findings of a 1989 Governor's Board of Inquiry initiated this concerted seismic retrofit effort. The San Francisco-Oakland Bay Bridge (SFOBB) was targeted early in this effort, as it had exhibited vulnerability during the Loma Prieta earthquake. This paper discusses the simulation and design work of the authors on a portion of the SFOBB utilizing ADINA®. Keywords: ADINA; SFOBB; Seismic; Retrofit; Vulnerability; Resonance; Tower

1. Introduction

2. Analysis

The now infamous Loma Prieta earthquake struck the San Francisco Bay area on October 17, 1989 at 17.04 Pacific Standard Time (PST). Three major metropolitan communities were impacted: San Francisco, Oakland, and San Jose. The epicenter of the 7.1 M earthquake was located in a local, sparsely populated, mountainous region. Approximately 20,000 buildings sustained damage, with 1300 destroyed. The human toll counted 62 fatalities, 3757 reported injuries, and 8000 left homeless. Amazingly, of the 4000 bridges in the vicinity of the event, only 18 bridge structures were closed to traffic following the earthquake, Housner et al. [1]. Economic losses, although a much less tangible measure of the earthquake's impact, were substantial. The primary tributary between the cities of San Francisco and Oakland, the SFOBB (reference Fig. 1) was closed for over a month due to damage sustained.

2.1. Towers A and B The continuous portion of the SFOBB west spans consists of a three-span superstructure beginning at the concrete anchorage, extending to Tower A, then Tower B, and terminating at the concrete pier W-1 (reference Fig. 2). These spans have a length of 378 feet, 96 feet, and 378 feet, respectively. Towers A and B (reference Fig. 3) extend vertically 102 feet from the top of the concrete pedestal to the bottom of the superstructure. The tower cross bracing has a single ' x ' configuration with horizontal struts at the top and bottom. The individual members consist of built-up sections with various cross-sectional geometries. The concrete pedestal is roughly 29 feet high, with the top 8 feet exposed, and connects to a 4x29x104 feet spread footing. A very deep, dense sand layer underlies this spread footing. 2.2. Vulnerability analysis

* Corresponding author. Tel: +1 (562) 863-3308; Fax: +1 (562) 863-1542; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

The 'as-built' model of the continuous truss was modeled in ADINA®, Bathe [2], with beam and plate elements representing the superstructure. Towers A and B,

R.W. Wolfe, R. Heninger/First MIT Conference on Computational Fluid and Solid Mechanics

756

Upper Dock Eiev-atcK^ -^ ^ ^^ ^ [Lower Deck Eleva^O'^ = 151 7 f:"]

377'

95'389' W1

Yerba Buetia Island Anchorage

- * Oakland

Fig. 1. Elevation view — SFOBB.

37F»

VVl

m

2310'

m ))

i>a


ws

23tff liAliiii AiiclMinHPi

Fig. 2. Elevation view — SFOBB continuous truss. and the foundations. Rigid links and constraint equations were added in the applicable areas. Nonlinear P-Y and T-Z springs were incorporated along the height of the tower foundations at significant changes in the soil layer, representing the passive pressure exerted by the soil to the foundation. The input excitation time-histories were added to the anchorage end of the truss and to ends of the various P-Y and T-Z springs at the foundations of the towers. Input motions in the form of 80-s time-histories were derived from the global model of the west spans, with critical motions up to 30 s. 2.3. The results of the analysis Computer simulations using ADINA® revealed that the towers seemed to experience a resonant behavior in the longitudinal direction, creating very large displacements at the mid-height of the columns. Due to this violent

resonance, the associated forces in the columns were very large, and the 'demand/capacity' (d/c) ratios, Reno et al. [3], exceeded the elastic regime, ranging from a magnitude of two to three. This behavior resulted from the fact that the excited motion of the heavy superstructure, coupled with the rocking of the relatively large tower foundations overwhelmed the towers' natural inertia.

3. Retrofit strategy The focus of the retrofit strategy was to moderate the resonance in the towers, thereby reducing the d/c ratios in the members. The first strategy included adding dampers to the opposite ends of the superstructure in order to reduce the demand at the tower connections. Thirty cases of dampers were investigated, yielding a 17% demand reduction. A second strategy dramatically increased the column

R.W. Wolfe, R. Heninger/First MIT Conference on Computational Fluid and Solid Mechanics

151

Fig. 3. 'As-built condition' — Towers A and B.

Fig. 4. Artist's rendering — Towers A and B retrofit.

cross-sections, thus increasing the tower capacity, effectively changing their natural period. This approach did not decrease the d/c ratio sufficiently. The third strategy incorporated longitudinal tower bracing from the top of one tower to the mid-height of the second. Displacements were reduced, but large additional undesirable loads were directed to the mid-height brace connections of the columns. Another strategy integrated friction pendulum isolators at the column/superstructure interface. This strategy worked once the friction in the bearing was overcome by the superstructure seismic demand. In other words, the towers were not stiff enough to help break the frictional forces, and therefore experienced some resonance and large displacements before the isolators were activated.

favor of all of the d/c ratios in the columns, braces and struts of the towers (reference Fig. 4). This strategy was implemented into the model to ascertain the impact to the remainder of the structure. Results were favorable and this strategy was reviewed and accepted by a peer review of academic and industry experts.

4. Conclusion Finally, a combination of friction pendulum bearings and longitudinal tower bracing yielded results that were in

References [1] Housner GW et al. Competing Against Time. Report to Governor George Deukmejian from the Board of Inquiry on the 1989 Loma Prieta Earthquake, 1990. [2] Bathe KJ. Finite Element Procedures. New Jersey: PrenticeHall, 1996. [3] Reno M, Duan L. San Francisco-Oakland Bay Bridge West Spans Seismic Retrofit Design Criteria. California: Department of Transportation, 1997.

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Vibration transmissibility of printed circuit boards by calibrated FEA modeling Jingshu Wu ^, Ray Ruichong Zhang ^'*, Steve Radons ^ ^ Medtronic Physio-Control Corporation, Redmond, WA 98073, USA ^Division of Engineering, Colorado School of Mines, Golden, CO 80401, USA

Abstract This study establishes a finite element analysis (FEA) modeling of printed circuit boards (PCB) calibrated with testing data. This calibrated model is then used to examine the vibration transmissibility of PCB within a plastic medical device case. This model-based investigation will help engineers efficiently improve the PCB mechanical design by considering realistically uncertain aspects and environments. Keywords: FE model updating; ANSYS; Printed circuit boards of medical devices; Vibration and shock responses; Static and dynamic tests

1. Introduction With the advances of finite element (FE) modeling, model-based design of structures in general and medical devices in particular is replacing the traditional trial-anderror approach. This study examines the vibration transmissibility of a printed circuit board (PCB) within a plastic case of a medical device, i.e., Lifepak500 Automatic External Defibrillator (AED) which is used more often now in public locations, and vulnerable to vibration and shock in ambulance/medical helicopter transportation and use. Specifically, based on the pertinent research (e.g., [2,4,1,5]), this study first establishes an ANSYS-based FE modeling of PCB calibrated with testing data. This calibrated model is then used to predict, among others, the vibration transmissibility of PCB within a plastic medical device case.

2. Testing As shown in Fig. 1, laboratory tests of PCB were performed first in the rigid fixture that has the same geometry as a medical device case. Static bending/twist test data * Corresponding author. Tel.: +1 (303) 273-3621; Fax: +1 (303) 273-3602; E-mail: [email protected] © 2001 Elsevier Science Ltd. All rights reserved. Computational Fluid and Solid Mechanics K.J. Bathe (Editor)

Fig. 1. Tesfing of PCB with a fixture.

/ Wu et al. /First MIT Conference on Computational Fluid and Solid Mechanics Table 1 PCB material properties determined from static testing data Mass density

Young's modulus

Poisson ratio

0.0002 Ib/in^

2,000,000 psi

0.25

are used to determine the PCB material properties and to establish proper FEA boundary conditions, while dynamic testing data are used to estimate the PCB damping and to calibrate FEA modeling. These tests are described as follows: (1) Static tests: The deflections at various locations of the PCB are measured by the INSTRON test system. With the aid of the so-called 'smeared properties' ([3]), the testing data are then used to determine PCB material properties (e.g.. Young's Modulus) to be used in the FEA model, which is shown in Table 1. Twist tests are also performed to determine the shear modulus of PCB. (2) Dynamic tests: Both rigid and flexible contacts between the fixture and the PCB were considered in the tests, in which the flexible contact simulates a more realistic plastic medical device product case while the rigid contact provides a simpler FEA model.

3. Calibrated FEA modal modeling With the use of the material properties from the static testing data, an ANSYS FEA modeling of PCB is established. The model is then calibrated with the dynamic testing data with consideration of the two contact conditions. In the modeling with a rigid contact, the PCB is actually 'fixed' or 'simply supported' to the fixture, which is determined by the design of the rigid fixture used in the tests. This test condition can be easily simulated by ANSYS with 'fixed' or 'simply supported' boundary conditions. The modal analysis of FEA does not include damping in the structures. In the modeUng with a flexible contact, beams or bar structure elements are used to simulate the mounting points where the PCB is constrained to a medical device plastic case, and the material properties of

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Table 2 First three natural frequencies of PCB with a rigid contact condition Modes

FEA (Hz)

Test (Hz)

Ratio of FEA/Test

1 2 3

169 210 252

171 207 263

0.988 1.01 0.958

Table 3 First three natural frequencies of PCB with a flexible contact condition Mode

FEA (Hz)

Test (Hz)

Ratio of FEA/Test

1 2 3

133 201 225

135 196 240

0.985 1.02 0.938

those bar/beam elements are determined by the static tests of the medical plastic case using an INSTRON system. The damping in the PCB-case system is determined from modal tests, curve-fitting techniques, and a trial-and-error approach associated with parametric change analysis of the FEA model. The FEA mesh is shown in Fig. 2. The comparisons of calibrated FEA modeling and testing data in the first three natural frequencies of the PCB are shown respectively in Tables 2 and 3, indicating the models are favorably consistent with the testing data.

4. PCB vibration transmissibility The calibrated modeling is used to predict the vibration of PCB within a plastic medical device case. To validate the modeling, a sine-sweep test is performed on a medical device. The shaker input (location/channel #1) acceleration level for this test is set as 1.0 G and in the range of 5 to 2000 Hz. The acceleration responses at three locations of the PCB (#2, #3, and #4) are measured (see Fig. 3). The FEA modeled and measured responses are also the transmissibility ratios since the input is 1.0 G as set by control channel, #1. Tables 4 and 5 give the results at PCB locations #2 and #3 with 5% damping assumed. They are well consistent each other. These results also indicate that

Table 4 Acceleration amplitudes at location #2

Fig. 2. FEA mesh of afixtureand PCB.

Mode

FEA

Test

Ratio of FEA/Test

1 2 3

4.5 8.5 1.8

3.12 7.6 4

1.4 1.12 0.45

/ Wu et al. /First MIT Conference on Computational Fluid and Solid Mechanics

760

-4. ^c^

*1:

f-Z-s#Z

Fig. 3. The measurement locations on PCB. Table 5 Acceleration amplitudes at location #3 Mode

FEA

Test

Ratio of FEA/Test

1 2 3

0.95 12 1.8

peak invisible 13 2

N/A 0.92 0.90

the acceleration transmissibility at location #3 is mainly determined by the second torsion mode, while all the first three vibration modes have significant effects at location #2. Displacements for the 1.0 G harmonic acceleration input and 5% damping are mostly under 0.005 inch, which is not shown here due to limited space. The FEA model can be further used to calculate stresses of PCB system caused by vibration or shock.

5. Conclusions The modeling of a vibrating PCB within a plastic case is a complicated process. This study uses the following step-by-step approach for the modeling: (1) FEA modeling of a PCB's static deformation and comparing the theoretical results with the INSTRON test results to determine the material properties.

(2) FEA modeling of a PCB's vibration with rigid boundary conditions and comparing the theoretical results to the test results using a rigid fixture to identify the dynamic properties such as natural frequencies, and finally (3) FEA modeling of a vibrating PCB located on a simulated flexible base (Lifepak500 Automatic External Defibrillator case) by introducing bar/beam structure elements into the FEA model to simulate the PCB/case connections. The FEA modeling results agree well with vibration test results. The established model could be a useful and efficient approach for determining theoretically the natural frequencies, mode shapes, and various responses (e.g., stresses) in PCBs. Therefore, it helps engineers design new PCB products with better reliability by predicting the possible fatigue and failure mechanisms and mitigating those failures through improved designs.

Acknowledgements This work was supported by Medtronic Physio-Control (JW, SR) and by the National Science Foundation with Grant No. CMS 0085272 (RRZ).

References [1] Engel PA. Structural Analysis of Printed Circuit Board Systems. Berlin: Springer-Verlag, 1993. [2] Steinberg D. Vibration Analysis for Electronic Equipments. Chichester: Wiley and Sons, 1988. [3] Pitarresi JM et al. The 'smeared' property technology for the FE vibration analysis of printed circuit boards. ASME J Electron Packag 1991;113(3):250-257. [4] Wong TL, Stevens K. Experimental modal analysis and dynamic response predicdon of PC boards with surface mounted components. ASME J Electron Packag 1991;113(3):244-249. [5] Stevens K, Wu J. Estimation of natural frequencies of printed circuit cards with uncertain properties. In: Mahajan R, Han B, Barker D (Ed), Experimental/numerical methods in electronic packaging, volume 2. Society of Experimental Mechanics, 1998.