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INVERSE PROBLEMS IN ENGINEERING MECHANICS II International Symposium on Inverse Problems in Engineering Mechanics 2000 (ISIP 2000) Nagano,Japan

Editors

M. Tanaka Department of Mechanical Systenns Engineering Shinshu University, 4-17-1 Wakasato, Nagano 380-8553 Japan

G.S. Dulikravich Department of Mechanical and Aerospace Engineering University of Texas at Arlington Arlington,TX, 76019, USA

2000

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Preface Inverse Problems can be found in many areas of engineering mechanics. There are numerous successful applications in the fields of inverse problems. For example, nondestructive testing and characterization of material properties by ultrasonic or X-ray techniques, thermography, etc. Generally speaking, the inverse problems are concerned with the determination of the input and the characteristics of a system given certain aspects of its output. Mathematically, such problems are ill-posed and have to be overcome through development of new computational schemes, regularization techniques, objective functional, and experimental procedures. Following the first lUTAM Symposium on these topics held in May 1992 in Tokyo, another in November 1994 in Paris, and also the last ISIP'98 in March 1998 in Nagano, we concluded that it would be fruitful to gather regularly with researchers and engineers for an exchange of the newest research ideas. The proceedings of these symposia were published and are recognized as standard references in the field of inverse problems. The most recent Symposium of this series "International Symposium on Inverse Problems in Engineering Mechanics (ISIP2000)" was held in March of 2000 in Nagano, Japan, where recent developments in inverse problems in engineering mechanics and related topics were discussed. The following general areas in inverse problems in engineering mechanics were the subjects of the ISIP2000: mathematical and computational aspects of inverse problems, parameter or system identification, shape determination, sensitivity analysis, optimization, material property characterization, ultrasonic non-destructive testing, elastodynamic inverse problems, thermal inverse problems, and other engineering applications. A number of papers from Asia, Europe, and North America were presented at ISIP2000 in Nagano, Japan. The detailed data of the ISIP2000 is available on the Internet (http://homer.shinshu-u.ac.jp/ISIP20Q0/). The final versions of the manuscripts of sixty-two papers from these presentations are contained in this volume of the ISIP2000 proceedings. These papers can provide a state-of-the-art review of the research on inverse problems in engineering mechanics. As the editors of the topical book, we hope that some breakthrough in the research on inverse problems can be made and that technology transfer will be stimulated and accelerated due to its publication. As the chairpersons of the ISIP2000 Symposium, we wish to express our cordial thanks to all the members of the International Scientific Committee and the Organizing Committee. Financial support from the Japanese Ministry of Education, Science, Sports and Culture (Monbusho) is gratefully acknowledged. Co-organizership by the University of Texas at Arlington, USA and Ecole Polytechnique, France is heartily appreciated. Also, co-sponsorship by the Japanese Society for Computational Methods in Engineering (JASCOME) and helpful support by the staff of Shinshu University in managing the financial support from Monbusho are gratefully acknowledged. June 2000 Masataka TANAKA, Shinshu University, Japan George S. DULIKRAVICH, The University of Texas at Arlington, U.S.A.

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Symposium Chairpersons Prof. Masataka TANAKA Department of Mechanical Systems Engineering Faculty of Engineering Shinshu University 4-17-1 Wakasato, Nagano 380-8553, Japan Fax: +81-26-269-5124, Tel: +81-26-269-5120 E-mail: [email protected] Prof. George S. Dulikravich Department of Mechanical and Aerospace Engineering The University of Texas at Arlington Arlington, TX 76019, U. S. A. Fax: +1-817-272-5010, Tel: +1-817-272-2603 E-mail: [email protected]

International Scientific Committee Prof. Masa. Tanaka (Chair), Shinshu University (Japan) Prof. G.S. Dulikravich (Co-Chair), The University of Texas at Arlington (USA) Prof. H. Adeli, The Ohio State University (USA) Prof. C.J.S. Alves, Technical University of Lisbon (Portugal) Prof. S. Aoki, Tokyo Institute of Technology (Japan) Prof. M. Bonnet, Ecole Poly technique (France) Prof. H. D. Bui, Ecole Polytechnique & Electricite de France (France) Prof. T. Burczynski, Silesian Technical University of Gliwice (Poland) Prof. G. Chavent, Universite Paris-Dauphine and INRIA (France) Prof. L. Elden, Linkoping University (Sweden) Prof. H.W. Engl, Johannes-Kepler-Universitaet (Austria) Prof. D.B. Ingham, University of Leeds (UK) Prof. A.J. Kassab, University of Central Florida (USA) Prof. M. Kitahara, Tohoku University (Japan) Prof. S. Kubo, Osaka University (Japan) Prof. P. Ladeveze, ENS de Cachan (France) Prof. A.K. Louis, Universitaet des Saarlandes (Germany) Prof. G. Maier, Politecnico di Milano (Italy) Prof. S. Migorski, Jagiellonian University (Poland) Prof. N. Nishimura, Kyoto University (Japan) Prof. K. Onishi, Ibaraki University (Japan) Dr. R. Potthast, University of Goettingen (Germany) Prof. M. Reynier, ENS de Cachan/CNRS/Universite Paris VI (France)

Vlll

Prof. H. Sobieczky, DLR German Aerospace Research Center (Germany) Dr. B.I. Soemarwoto, National Aerospace Laboratory (The Netherlands) Prof. V.V. Toropov, University of Bradford (UK) Prof. N. Tosaka, Nihon University (Japan) Prof. I. Trendafilova, Katholieke Universiteit Leuven (Belgium) Prof. Z. Yao, Tsinghua University (China) Prof. W. Zhong, Huazhong University of Science & Technology (China)

Organizing Committee Prof. Masa. Tanaka (Chair), Shinshu University (Japan) Prof. G.S. Dulikravich (Co-Chair), The University of Texas at Arlington (USA) Prof. T. Matsumoto (Secretary), Shinshu University (Japan) Prof. K. Amaya, Tokyo Institute of Technology (Japan) Prof. S. Aoki, Tokyo Institute of Technology (Japan) Prof. M. Arai, Shinshu University (Japan) Prof. H. Azegami, Toyohasi University of Technology (Japan) Prof. H.D. Bui, Ecole Polytechnique & Electricite de France (France) Prof. T. Fukui, Fukui University (Japan) Prof. K. Hayami, University of Tokyo (Japan) Prof. S. Hirose, Tokyo Institute of Technology (Japan) Prof. Y. Honjo, Gifu University (Japan) Prof. M. Hori, University of Tokyo (Japan) Dr. H. Igarashi, Kagawa University (Japan) Prof. F. Imado, Shinshu University (Japan) Prof. Y. Iso, Kyoto University (Japan) Prof. K. Kagawa, Okayama University (Japan) Prof. J. Kihara, Himeji Institute of Technology (Japan) Prof. K. Kishimoto, Tokyo Institute of Technology (Japan) Prof. E. Kita, Nagoya University (Japan) Prof. M. Kitahara, Tohoku University (Japan) Prof. F. Kojima, Kobe University (Japan) Prof. S. Kubo, Osaka University (Japan) Prof. A. Murakami, Kyoto University (Japan) Prof. M. Nakamura, Shinshu University (Japan) Prof. N. Nishimura, Kyoto University (Japan) Prof. K. Onishi, Ibaraki University (Japan) Prof. N. Tosaka, Nihon University (Japan) Prof. M. Yamamoto, University of Tokyo (Japan)

Participants Abboudi, S.

France

Kishimoto, K.

Japan

Alves, Carlos J.S.

Portugal

Kitahara, M.

Japan

Amaya, K.

Japan

Knudby, C.

Spain

Aoki, S.

Japan

Kobayashi, A. S.

U.S.A.

Aral, M.

Japan

Kobayashi, S.

Japan

Azegami, H.

Japan

Koishi, M.

Japan

Bemtsson, F.

Sweden

Kojima, F.

Japan

Burczynski, T.

Poland

Koyama, Y.

Japan

Chen, W.

Japan

Kubo, S.

Japan

Cheng, J.

Japan

Lacour, F.

France

Chouaki, A.

France

Langenberg, K.J.

Germany

Liu, D.

China

Constantinescu, A. France Contro, R.

Italy

Matsumoto, T.

Japan

Dennis, B. H.

U.S.A.

Matsushima, K.

Japan

Deraemaeker, A.

France

Migorski, S.

Poland

Duhkravich, G. S.

U.S.A.

Murakami, A.

Japan

El-Badia, A.

France

Nakahata, K.

Japan

Elden, L.

Sweden

Nakajima, M.

Japan

Imado, F.

Japan

Nishimura, N.

Japan

Furukawa, T.

Japan

Nishimura, S.

Japan

Ha-Duong, T.

France

Nowak, A.J.

Poland

Hayabusa, K.

Japan

Oida, S.

Japan

Hayami, K.

Japan

Okajima, N.

Japan

Hon, Y.-C.

China

Okayama, S.

Japan

Hori, M.

Japan

Onishi, K.

Japan

Igarashi, H.

Japan

Ooki, R.

Japan

Inoue, H.

Japan

Orlande, H.R.B.

Brasil

Kabanikhin, S. I.

Russia

Park, K. C.

U.S.A.

Kagawa, Y.

Japan

Pires, G. E.

Portugal

Kanevce, G.

Macedonia

Qian, Y.-J.

China

Kanevce, P. L.

Macedonia

Rikards, R.

Latvia

Kanoh, M.

Japan

Sato, K.

Japan

Kassab, A.

U.S.A.

Shigeta, T.

Japan

Katamine, E.

Japan

Shiho, K.

Japan

Kawai, R.

Japan

Singh, K.M.

Japan

Kawai, T.

Japan

Shirota, K.

Japan

Kimura, A.

Japan

Sobieczky, H.

Germany

Soemarwoto, B.I.

The Netherlands

Verchery, G.

Suzuki, M.

Japan

Wang, Q.-C.

France Japan

Tanaka, M.

Japan

Wu, Z.Q.

Japan

Tanaka, T.

Japan

Cao, X.

Japan

Tomokiyo, K.

Japan

Yagola, A.

Russia

Toropov, V.V.

UK

Yamannoto, M.

Japan

Tosaka, N.

Japan

Yao, Z.-H.

China

Toyoda, T.

Japan

Yoshida, F.

Japan

Trendafilova, I.

Belgium

Yoshikawa, H.

Japan

Tsutsumi, K.

Japan

Yoshino. H.

Japan

Vena, P.

Italy

Zhang, W.-Q.

Japan

Contents Preface Symposium Chaiq)ersons International Scientific Committee Organizing Committee Participants

v vii vii ^iii ix

Inverse Heat Conduction

A combined use of experimental design and Kalman filter - BEM for identification of unknown boundary shape for axisymmetric bodies under steady-state heat conduction M. Tanaka, T. Matsumoto and T. Yano Estimation of the temperature and the concentration fields in a semitransparent medium: Emphasis on the experimental noise disturbance A. Chouaki, I. Darbord and P. Herve Numerical estimation of the transient heat flux boundary conditions for a flat specimen S. Abboudi and E. Artioukhine Estimation of the thermal state of two bars in dry sliding F. Lacour, Y. Bailly and E. Artioukhine Moisture diffusivity estimation by temperature response of a drying body G.H. Kanevce, L.P. Kanevce and G.S. Dulikravich Parameter estimation in moist capillary porous media by using temperature measurements L.B. Dantas, H.R.B. Orlande, R.M. Cotta and RD.C. Lobo Determination of heat transfer coefficient maps using an inverse BEM algorithm E. Divo, A.J. Kassab, J.S. Kapat and J. Tapley Tracking of phase change front for continuous casting - inverse BEM solution I. Nowak, A.J. Nowak and L.C. Wrobel Application of DRBEM and Iterative Regularization to Inverse Heat Conduction M. Tanaka and K.M. Singh Numerical solution of an inverse steady state heat conduction problem F. Berntsson and L. Elden An inverse heat conduction problem and an application to heat treatment of aluminium F. Berntsson and L. Elden

3 13 23 33 43 53 63 V1 81 91 99

Boundary Data Detection in Solid Mechanics

Inverse analysis to determine contact stresses using photoelasticity H. Inoue, K. Hayabusa, K. Kishimoto and T. Shibuya Identification of tractions based on displacement observations at interior points M. Nakajima, K. Hayami, J. Terao, S. Watanabe and S. Ando Identification of dynamic pressure distribution applied to the elastic thin plate M. Arai, T. Nishida and T. Adachi Determination of unsteady container temperatures during freezing of three-dimensional organs with constrained thermal stresses B.H. Dennis and G.S. Dulikravich Analysis of inverse boundary value problem by the alternating boundary element inversion scheme and its improvement using boundary division S. Kubo and A. Furukawa

109 119 129 139 149

Xll

Material Property Determination

Method for identification of elastic properties of laminates R. Rikards A single integral finite strain characterization of soft connective tissues and parameter identification V. Quaglini, R Vena and R. Contro Identification strategies for recovering material parameters from indentation experiments A. Constantinescu and N. Tardieu Identification of material parameters in constitutive model for shape memory alloy based on isothermal stress-cycle tests F. Yoshida, V.V. Toropov, M. Itoh, H. Kyogoku and T. Sakuma

161 171 181 191

Defect Detection

On the identificafion of a crack in 3D acousdcs H.D. Bui, A. Constantinescu and H. Maigre On the identification of conductive cracks C.J.S. Alves, T. Ha-Duong and F. Penzel Inversion of defects by linearized inverse scattering methods with measured waveforms K. Nakahata and M. Kitahara Fast recovering algorithm for crack shape of steam generator tubes using geometric approach F. Kojima and N. Okajima Structural damage idenfificafion using static test data and changes in frequencies X. Wang, N. Hu and Z.H. Yao Inverse analysis for fracture process zone characterizadon D.K. Tran, M.T. Kokaly and A.S. Kobayashi Idendfication of delamination in bonded dissimilar materials with orthotropic electric conductivity by the electric potendal CT method S. Kubo, T. Sakagami and N. Tanaka Damage detecdon of structure using image processing (1st Report, Detection of joint failure) T. Kawai, N. Ikeda and M. Ito

203 213 219 229 239 249 257 267

Shape Determination

Solution to boundary shape idendficadon problems in eUipdc boundary value problems using shape derivatives H. Azegami Shape optimization of transient response problems Z.Q. Wu, Y Sogabe, Y Arimitsu and H. Azegami Soludon to shape determination problem on unsteady heat-conduction fields E. Katamine, H. Azegami and Y Matsuura

Parameter Identification in Solid Mechanics

277 285 295

Muld-objective parameter idendficadon of unified material models T. Furukawa, S. Yoshimura and G. Yagawa 307 The polar method as a tool for solving inverse problems of the classical laminated plate theory G. Verchery, R Vannucci and V. Person 317 Verification of ultrasonic transducer characteristics determined in an inverse problem based on laser measurements T. Kanbayashi, H. Yoshikawa, N. Nishimura and S. Kobayashi 327

Xlll

Identification of elastodynamics load using DRBEM and dynamic programming filter M. Tanaka and W. Chen A geometrical design technique for impact pistons based on stress waveforms D. Liu, A. Chen and P. Zhu Inversion of stress and constitutive relations using strain data for Japan Islands M. Hori Estimation of parameters in tank model analysis using least squares of residuals with constraints M. Kanoh, T. Hosokawa and T. Kuroki Identification of damped joints parameters using the error in the constitutive relation A. Deraemaeker, P. Ladeveze, E. Collard and P. Leconte Condition monitoring and damage quantification in robot joints using nonlinear dynamics characteristics and inverse classification methods I. Trendafilova and H. Van Brussel

333 343 349 359 367 377

Inverse Problems in Aeronautics and Fluid Dynamics

Waverider design with parametric flow quality control by inverse method of Characteristics Y. Qian, H. Sobieczky and Th. Eggers Improper integrals in the formulation of a supersonic inverse problem K. Matsushima A function estimation approach for the identification of the transient inlet profile in parallel plate channels M J. Cola90 and H.R.B. Orlande Inverse aerodynamic shape design for improved wing buffet-onset performance B.L Soemarwoto, Th.E. Labrujere, M. Laban and H. Yanshah The application of modified output error method and its verification on inverse problems in aeronautics (No.2) Y Koyama and F. Imado

389 399 409 419 429

Inverse Problems in Electromagnetics and Acoustics

Optimization of electroplating on silicon wafer K. Amaya, S. Aoki, H. Takazawa and M. Miyasaka Defect shape recovering for electromagnetic problem using HTS-SQUID gradiometer F. Kojima, R. Kawai, N. Kasai and Y Hatsukade Design optimization of electromagnetic devices with neural network A. Kimura and Y Kagawa On the reconstruction of magnetic source in cylindrical permanent magnets H. Igarashi Extraction of transfer characteristics of vocal tract from speech signals K. Tsutsumi and Y Kagawa

439 449 459 467 477

Uniqueness, lU-posedness, Regularization

Regularizing procedures for solving the general inverse problem of structural chemistry and their applications I.V. Kochikov, G.M. Kuramshina, V.R Spiridonov, Yu.L Tarasov and A.G. Yagola Numerical solution of a Cauchy problem for an elliptic equation J. Cheng, YC. Hon, T. Wei and M. Yamamoto

487 493

XIV

Variational and D-N approaches for a magnetostatic Cauchy problem T. Shigeta

501

Numerical and Computational Algorithms

Identification coefficient problems for elliptic hemivariational inequalities and applications S.Migorski Variational approach for the problem of coefficient identifcation of the wave equation K. Shirota On an inverse phase change problem A. El-Badia On some inverse EEC problems M. Chafik, A El-Badia and T. Ha-Duong Numerical solution of under-determined 2D Laplace equation with internal information Q. Wang, Y. Ohura and K. Onishi

513

529 537 545

Applications of Computational Algorithms

Evolutionary methods in inverse problems of engineering mechanics T. Burczyhski, W. Beluch, A. Dtugosz, P. Orantek and M. Nowakowski Screen detection with near field measurements C.J.S. Alves and G.E. Pires Bayesian estimation for nonlinear inverse problems M. Suzuki and A. Murakami Structural optimization using cellular automata T. Toyoda and E. Kita Author Index

553 563 573 583 591

Inverse Heat Conduction

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

A COMBINED USE OF EXPERIMENTAL DESIGN AND KALMAN FILTER-BEM FOR IDENTIFICATION OF UNKNOWN BOUNDARY SHAPE FOR AXISYMMETRIC BODIES UNDER STEADY-STATE HEAT CONDUCTION Masa. TANAKA , T. MATSUMOTO and T. YANG Department of Mechanical Systems Engineering, Faculty of Engineering, Shinshu University 4-17-1 Wakasato, Nagano City, 380-8553 Japan E-mail: [email protected] ABSTRACT This paper presents a combined use of the experimental design and the boundary element method (REM) for the inverse problem of boundary shape determination in rotationally axisymmetric bodies under steady-state heat conduction. The REM andfiltertheory have been so far applied to a number of inverse problems, but their successful applications depend on initial guess of parameters to be identified. This paper aims at constructing a more robust method of inverse analysis in this direction. Inverse analysis is performed by using thefiltertheory-REM after an approximate set of parameters are estimated through the experimental design. Such a two-step method of inverse analysis is proposed in this paper, and its advantages are demonstrated through numerical computation of a few examples. KEYWORDS Inverse Problem, Boundary Shape Determination, Experimental Design, Filter Theory, Boundary Element Method, Rotationally Axisymmetric Rody, Steady-State Heat Conduction INTRODUCTION A two-step method of inverse analysis is proposed in this paper for the inverse problems of boundary shape determination in steady-stae heat conduction fileds. First, a method of experimental design is applied to estimating a set of parameters defining the boundary shape to be identified. An approximate set of parameters thus estimated are then used for the inverse analysis which combinedly uses the boundary element method and the Kalman filter theory [1]. It is assumed that rotationally axisymmetric bodies under consideration are in steady-state heat conduction and that the temperature is measured at a number of points located on the outer boundary. The inverse problem can be stated such that the unknown internal boundary shape should be identified by using the measured temperatures on the boundary. In application of the experimental design [3], wefirstassume that each parameter may have a discrete value belonging to the levels we can assign based on a priori information. According to the number of levels and that of parameters (=factors), we choose an appropriate table of orthogonality [3], and carry out influence analysis of parameters. Analysis of variance is performed for a reduced number of parameter combinations, and then we can know which parameters may

4

M. Tanaka, T. Matsumoto and T. Yano

influence more than the other. The most appropriate parameter combination is thus selected so that a cost function is minimized. Boundary element analysis is carried out for each parameter combination to evaluate the cost function. An application of this method to identification of the blast-furnace refractory was reported in [2]. In this paper, the extended Kalman filter is employed in the second step of inverse analysis, in which the parameter values estimated through the experimental design are used in the second step as their initial guess. In the proposed two-step inverse analysis, the set of parameters estimated by means of the experimental design are used as the initial guess for a second step of inverse analysis, and they are modified through iterative computations as shown in [1]. Some numerical examples are computed by the proposed method of inverse analysis and the results obtained are discussed, whereby the usefulness of the proposed method of two-step inverse analysis are revealed. THEORY Influence Analysis by Experimental Design Experimental design [3] has recently been attracting attention of engineers to select effectively appropriate data sets among a number of given data [4]. Though the experimental design using orthogonal tables we can caluculate effects of factors and find a minimal value of the characteristics. The orthogonal table used in this method is chosen by considering the numbers of factors, levels and interactions. The factors will be assigned in the column, and calculate the characteristic value (in our case the cost function). When there are the interactions between factors, they are assigned based on the linear graph of the table. We shall take the orthogonal table L27(3^^) for example [4], where the superscript 13 denotes the number of rows of the table, the number enclosed with brackets denotes the number of levels of each factor and the maximum number of available factors. Namely, L27(3^^) consists of 13 rows and can treat 13 factors each of which has 3 levels. The number in the table denotes the level of each factor and the characteristic value which is used in the influence analysis is calculated using a set of levels which forms the row of the table. Using the orthogonal table, we can drastically reduce the total number of computations fo the influence analysis. For instance, in a case of three levels and eight factors, only 27 computations of influence analysis is required if the orthogonal table L27(3^^) is employed, whereas 3^ = 6561 computations should be done without use of the orthogonal table. In the variance analysis the influence of a factor on the characteristic value is computed by comparing the ratio of variance (F-ratio). Let us consider the case of a levels. Through r-times computations we can obtain n characteristic values yij. The total sum of square ST has fr degrees of freedom. That is,

1=1 j = i

fT =

(1)

n-l

where y is the totally averaged value of yij. The sum of square .94 and its degrees of freedom are

A combined use of experimental design and Kalman filter

5

given as follows:

tr

\

(2)

J

fA = a-l

where yi. is the averaged value of yij in each level i. The total sum of square ST can be resolved into square sum SA and square sum of errors SE as follows: ST

= SA + SE

(3)

The square sum of errors SE and its degrees of of freedom JE are given as SE

== ST-

IE

= IT-

PSA

PIA

(4)

The variance V is written as a value of each component of square sum divided by corresponding degrees of freedom / , which are written for factors and errors, respectively as follows: VA - SA/JA

(5)

VE = SE/IE

(6)

The F-ratio can be computed using VA and VE as follows: FA = VA/VE

(7)

When F-ratio FA is smaller than 1, we may think that the influence of factor A is small and its effect on the characteristic value is negligible [4]. The F-distribution table is used to examine whether the effect of the factor is significant or not. When the F-ratio is smaller than a critical value given in the F-distribution table, this factor is negligible, and vice versa. For the significant factors the following pure sum of square S'A and effective ratio PA are defined, i.e. S'A

= SA - VE

PA = S'AIST

(8) (9)

The effect of each level Xi can be evaluated as follows: r

x^^Y^mj'

(10)

Boundary Element Analysis The boundary integral equation for the three-dimensional problems of steady-state heat conduction without heat source is given in the Cartesian coordinate system (xi, X2,2:3) by [5] c{yHy) + /J —j^u{x)dT j u'{x,y)q{x)dT =J u*{x, y)qix)dT ^"]^'^^"(^)^r ^

(11)

where u is the temperature at a point on the boundary F, q the outward normal derivative of the temperature, u' the fundamental solution of the Laplace equation, y the source point and c{y) a

6

M Tanaka, T. Matsumoto and T. Yano

constant which depends on the geometry of the boundary at y. The fundamental solution u* is given by u*{x,y) = -—, r = \x-y\ (12) 47rr In this study, we assume rotationally symmetric distributions of variables for rotationally symmetric bodies, and follow the solution by Wrobel [6]. Expressing equation (11) in the cylindrical coordinate (/?, ^, Z), we obtain c{yHy)

+ / / " ^''*^^^^K{x)JdedT

= 1 1 \*{x,y)q{x)JdedT

(13)

where F denotes the meridian of the axisymmetric body and J is the Jacobian which is equal to R in this case. The integrals in equation (13) with respect to 0 can be evaluated analytically. Thus, we obtain c{yHy)

+ JQ* (x, y) u{x)dr = fu* (x, y) q{x)dT

(14)

The details of fundamental solutions U* and Q* can be seen in [1] and [5]. Parameter Identification Using Filter Theory In order to identify the parameters which express the geometry of unknown internal boundary, we combine the boundary element method with the extended Kalman filter [7]. In the following an outline of this procedure [1] will be explained. In the filter theory, we may assume that the measured temperatures compose the observation vector and the unknown parameters defining the geometry of the unknown internal boundary do the state vector. The observation equation which describes the relation between the state vector X and the observation vector y is written as follows: yk = hk{xk) + Vk

(15)

where v is the observation error vector and the subscript A: denotes time. Since there is no temporal change in the temperature, we may regard the subscript k as the iteration number of computation. The state equation describing the transition of state vector about the time can also be assumed as follows: Xfc+i=Ixfc

(16)

where I denotes the unit matrix. Based on a linearized expression of the observation equation (15) and the state equation (16), we can identify the unknown parameters in an iterative manner. The main flow of the iterative inverse analysis can be summarized as follows: 1. Assume the initial state vector Xo/_i and the covariance matrix of estimation errors Po/-i in the extended Kalman filter. 2. Input the measured temperatures y^ and the covariance matrix of observation errors R^.

A combined use of experimental design and Kalman

filter

1

3. Calculate the temperature at observation points using the BEM and the previous estimated state vector i^k/k-i as well as the sensitivity matrix H^, which is defined by

Hfc =

dui dxi duo

^hfc(xfc) dy.k

IXfc=Xfc/fc_i

dui dx2 duo

fi.

_

__

OXi

0X2

duN L dxi

,^^ . . .

dui dxn duo

__

(17)

OXn

duA

duM dx2

dXn

k/k-1

First-order derivatives in equation (17) are calculated by a finite difference scheme. 4. Calculate the filter gain Kk and renew the estimated state vector yik+i/k- The filter gain in the extended Kalman filter is computed by Kfc = Pfc/fc-iH^. [H-kPk/k-i^k

+ ^fc]

(18)

5. Check convergence. Unless convergence is realized, go to the step 3 and repeat the computation. NUMERICAL ANALYSIS Analysis Model In this study the parameters in the inverse analysis are used as factors and the cost function as the characteristic value in the terminology of experimental design. The cost function is defined as the square sum of residuals between the temperatures computed by the BEM analysis and the observed temperatures. In actual analysis, the two-step method of inverse analysis is applied: First, the best level in each parameter is selected by influence analysis of experimental design, and then a precise inverse analysis is performed by a combined use of BEM and Kalman filter with the parameter values estimated by the experimental design. Instead of using equation(lO), the best level in each parameter is found as what minimizes the following expression:

W^ = J2 ViJ

(19)

For the parameters which are judged to be not significant, the level selected by the influence analysis is regarded as the estimated value, if the sensitivity for the temperatures are sufficiently small. Figure I shows the analysis model in which the blast furnce hearth is modeled as a rotationally symmetric body about Z axis. The boundary conditions used in this numerical analysis are also shown in the figure. As a parameter we take the distance between a fixed point (0,10) and a point on the internal boundary which lies on the ray issued from the fixed point with a given angle (Figure 1). It is assumed that the ray is placed with an equal angle with the adjacent ones. In addition, we assume that 10 thermocouples which are shown in Figure 1 by o symbol are located, from which we can measure the temperature. The internal surface is produced by interpolating the eight points using cubic spline curve. Three levels are assumed for each parameter based on a priori information, x symbol denotes the first level of each parameter, • the second and • the third.

8

M. Tanaka, T. Matsumoto and T. Yano

In estimation by the extended Kalman filter, it is assumed that the initial values of diagonal arrays of the covariance matrix for estimation errors is assumed to be 1.0 x 10^, and that the corresponding values of the covariance matrix for observation errors is to be 1.0. Two examples are analyzed to check how the assumed levels infulence the estimation results: The first example in which the target geometry lies within range of the assumed levels and the second example in which the target geometry lies outside the range of the assumed levels.

q = 0 [Wm ' ] X •r

/i = 70[Wm ^K •] Ua = 30 PC]

Fig. 1. Inverse analysis model

An example in which the target geometry lies in the assumed levels. Figure 2 shows the estimated geometry through influence analysis of experimental design. Table 1 summarizes Wi for each level and the values denoted by the circle o indicate the selected levels by influence analysis. The solid line in Fig. 2 denotes the estimated geometry of refractory erosion by influence analysis. Figure 3 shows the estimated geometry after two-step inverse analysis in which the result of influence analysis is used as the initial guess of parameters in the combined Kalman filter-boundary elememt method (KF-BEM) of inverse analysis. Only the eighth parameter x^ is not significant and its sensitivity is very small. Therefore, we assume that this value obtained by the influence analysis for the 8th parameter is the final result and no modification is made in the (KF-BEM) inverse analysis. Figure 3 shows an excellent agreement between the target geometry and the estimated geometry by the proposed two-step inverse analysis. An example in which the target geometry lies outside the assumed levels. In this numerical analysis the levels are assumed outside the target geometry. Table 2 summarizes the results obtained by influence analysis, and Fig. 5 shows the estimated geometry through the two-step inverse analysis. A good agreement can also be recognized. Even if the levels are assumed

A combined use of experimental design and Kalman filter

9

outside the target geometry, the second-step of the BEM-F inverse analysis can provide a very good estimation as a converged solution.

10-

ytk^

Estimated geometry

9-

o

Target geometry

8-

xA

76N

o

jl

%1

1

'.•

;

54-

\

X

iI

3^

\ •

X

^^^^^" -* -

•

^

.

-

'-'

\ •

21in

\ )

1

(

2

3 R

5

4

6

•

Fig. 2. Approximate estimation by experimental design (1st step analysis)

Table 1. Effectivity of each level

0-0 I I I I I I M I I I I I n M I M I I M I I I M I I I I 123 1 2 3 1 2 3 1 2 3 1 2 3 123 1 2 3 123 xl

x2

x3

x4

x5

x6

x7

x8

10

M. Tanaka, T. Matsumoto and T. Yano lU

Target geometry

98-

Initial geometry

7-

Estimated geometry

6N

54-

2-

\

\ \ 1

\ \

J \ —-—V \ \

10- ^

^

Fig. 3. Final estimation by Kalman filter-BEM inverse analysis (2nd step analysis)

lU-

Estimated geometry

9-

X

•

«

•

(>

Target geometry

8-

)

76N

X

5-

' 4-

^ •

'.

• 11

1 ' -''

^

\

\ \

2100

1

Fig. 4. Approximate estimation by experimental design (1st step analysis)

A combined use of experimental design and Kalman filter Table 2. Effactivity of each level

0-6 1 I I I I I M I I t I M 1 I I I 1 I I I I I I M M I M I 123 123 123 123 123 123 123 123 xl

x2

x3

10-

x5

x6

x7

x8

Target geometry

9-

Initial geometry

87-

\

Estimated geometry

6N

x4

5-

/

4-

J\

\ \

2-

\ \

()

1

2

3 R

^t

;i

6

Fig. 5. Final estimation by Kalman filter-BEM (2nd step analysis)

11

12

M. Tanaka, T. Matsumoto and T. Yano

CONCLUSION The two-step method of inverse analysis, first by experimental design and then by Kalman filterBEM, has been proposed for estimating the erosion line of a blast-furnace model. A computer code was developed and applied to a few examples, whereby it was revealed that an accurate solution of the inverse problem could be obtained by the proposed two-step method. It can be mentioned that the proposed method of inverse analysis is very effective and robust, because a priori information can be used as much as possible for the first-step analysis by experimental design. As a future work in the direction of the present paper, it can be recommended to apply the proposed method of inverse analysis to predicting the solidification layer which reportedly is produced on the surface of blast-furnace refractory. On the other hand, we may hope that since the proposed method has wide applicability, it could open a door to successful applications to a wide variety of inverse problems.

References (1) M. Tanaka, T. Matsumoto and S. Oida, Identification of unknown boundary shape of rotationally symmetric body in steady heat conduction via BEM and filter theories. In: Inverse Problems in Engineering Mechanics, ed. by M. Tanaka and G.S. Dulikravich, Elsevier Science, Amsterdam, (1998), pp. 121-130. (2) H. Yoshikawa, M. Ichimiya, S. Taguchi, M. Tanaka, Estimation of erosion line of refractory and solidification layer in blast furnace hearth. In: Proc. 4th Conf. on Simulation Technology, Japan Soc. for Simulation Technology, (1984), pp. 75-78. (3) G. Taguchi (1976). Experimental Design Vols. 1 and 2, Maruzen, Tokyo. (4) T. Kashiwamura, M. Shiratori, Q.Yu (1998). Optimization by Experimental Design for NonLinear Problems, Asakura-shoten, Tokyo. (5) M. Tanaka, T. Matsumoto and M. Nakamura (1991). Boundary Element Methods, Computational Mechanics and CAE Series 2, Baifukan, Tokyo. (6) L. C. Wrobel (1981). Potential and Viscous Flow Problems Using the Boundary Element Method, PhD Thesis, Southampton University, UK. (7) T. Katayama (1989). Applied Kalman Filter. Asakura-shoten, Tokyo.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

13

ESTIMATION OF THE TEMPERATURE AND THE CONCENTRATION FIELDS IN A SEMITRANSPARENT MEDIUM: EMPHASIS ON THE EXPERIMENTAL NOISE DISTURBANCE A. CHOUAKI, I. DARBORD AND R HERVE L3E, Labomtoke d'Energetique et d'Economie d'Energie (Universite Paris X) 1, Chemin Desvallieres 92410 Ville d'Avray FRANCE ABSTRACT This paper deals with the estimation of the steady distributions of the temperature and the concentration fields (T,C) of H2O in a vapor state in a hot gas. Our approach is a non intrusive and a non active method, emission spectroscopy involving an identification algorithm. The (T,C) fields are estimated through the comparison of the analytical intensities with the experimental intensities collected at several locations on the edges of the gas. The analytical spectral intensity transmitted by the gas is given by the equations of radiation transfer for a semitransparent medium. These equations are non-linear Fredholm-type equations. In order to solve them, the (T,C) fields are described using analytical basis functions enabling to recover a wide range of typical profiles encountered in the scope of applications. A special care is given to the sensitivity of the identification procedure to noise influence. First, the sensitivity of eight selected wavelengths with respect to the noise is achieved. Then, the identification procedure described below is applied to a real case study (methane flame). The identification process makes use of the Extended Weighted Least Squares estimator combined to a quasi-newton minimization method. KEYWORDS Semi-transparent medium, emission spectroscopy, identification, temperature, concentration. INTRODUCTION The development of processes of high technology in the scope of hypersonic flows occurs in severe conditions of temperature, pressure (up to 3000 K, and up to Mach 6) and vibration level (more than 180dB). A related issue is the development of hypersonic ramjet launcher. It is performed through the measurement of temperature and concentration of water {H2O) in the exhaust combustion gas. Such measurements could be undertook using intrusive probes. Yet, in the drastic conditions cited above, they are rapidly destroyed. Hence, their success is highly dependent on the development of high technology materials. An alternative to that damagingijetable) technology is optical methods because they operate without any contact with the flow.

14

A. Chouaki, I. Darbord and P. Herve Optical methods may be classified into two categories: passive ones and active ones.

The main active methods are CARS (Coherent Anti Stokes Raman Scattering), DFWM (Degenerate Four Wave Mixing techniques [1], LIIF (Laser Induced Iodine Fluorescence) [2] and Rayleigh Diffusion [3]. Active methods need a light source (more often laser beam) and analyse the incidental light transmission through the medium. The measurements are obtained in one point of the flow so the spatial resolution is very high. Yet, 2D cartography then requires several successive shots. Hence, a though problem encountered is the alignment of the laser beams, especially when the vibration level is high. With CARS or DFWM, local unsteady measurement of temperature and concentration of the main species can be performed. CARS is based on the focus of two different monochromatic laser beams in the medium. The diffused light has properties related to the frequencies of the incidental beams and to the nature of the studied specie. DFWM is founded on the same technique as CARS but mixing four wave laser beams instead of two. LIIF, initially developed to visualize flows is now often used to quantitative metrology. The flow is seeded with a tracer, iodine, and lit by a laser beam. The principle is that the incidental radiation absorption by iodine particles is accompanied by a emission of fluorescence. The concentration of tracer which is activated is function of the flow characteristics (temperature, speed and pressure fields). The absorption lines of iodine are well-known in the visible part of the spectrum. Each line has a more or less large sensitivity to the pressure or the temperature. This method requires an iterative processing of the signal and the knowledge of pressure and temperature in a point of the flow in order to evaluate the constants of the theoretical formula. The use of LIIF for concentration measurement is not possible. In Rayleigh Diffusion, the medium is lit by a monochromatic laser beam. The resulting diffusion is proportional to the density of particules in the gas medium. Assuming the perfect gas equation of state for the medium, the collected signal is then inversely proportional to the temperature. So this method can't perform the measurement of the concentration of a particular specie in the gas. Moreover, to collect the diffused light in only one position, it is necessary to be sure that the laser beam is well focused on a point of the flow. On the contrary, the incident beam is partially absorbed by the studied medium, and the measurements must be carried out on the total length of the beam across the medium. Passive methods don't have need of external sources. Emission Spectrometry (for example Fourier Transform emission spectroscopy) is the method presented in this paper. It consists in the measurement of spectral intensity transmitted by a volume of gas. The measurement is made at the edges of the flame, so it does not change the state of the medium or disrupt the flow. In this paper, our experimental method will be described, before focussing on the inverse problem that is solved as a post processing of the measurement of the intensity, to obtain the (T,C) fields. The paper then recalls the main features of the identification, especially the choice of the experimental wavelengths, the analytical representation of the unknowns and the determination of the cost function. At the end, the results at 8 wavelengths are presented pointing out the sensitivity of the method to experimental noise. This study was carried out on simulated measurements for the

Estimation of the temperature and the concentration fields

15

whole suggested wavelengths. A result starting from a real case of measurement of intensity (5.15mm) on a methane flame is also presented. THE PROPOSED APPROACH The method was first applied in [4] in order to identify the (T,C) fields in a ramjet with hypersonic combustion, then, in [5] in order to determine the pollutant concentration in a turboreactor. Emission spectroscopy consists of the measurement of spectral intensities in different locations at the edge of the gas flow (see Fig. 1). The signal results of the radiative propagative phenomenon emission, absorption and transmission in the semi-transparent medium and which is described by the equation of radiation heat transfer. The equations are written for each measured volume >k

/^

s

1 \ 0

\ Sigr

\\

// 1

\—

\

lAy^

1 1 1 loi 1

(Signal L ( X ) ^

\

>^

Fig. 1. The flame cross section as follows :

j\\X^,T{x,Y^)

i

9r{K.T{xX^C{x,yi))

' L«(A.,T(X.,y)) dr{Xn.nX.y)MX.y)) dy

^^ ^

L.^^^X.^Y,)

^^ ^ L^^^^X^^X.)

(1)

(2)

where L^{Xn^T{x^y)) is the. Planck function, r(An, r ( x , y), (7(a:, y)) is the transmissivity, Lexvi^n-, Yi) is the experimental spectral intensity on the edge x at the Yi location and Le^j,{\n^Xi) is the experimental spectral intensity on the edge y at the Xi location. The whole quantities are given at the wavelength A^. Seeing the form of the equation, a post processing based on the solution of an inverse problem or identification is required to determine simultaneous (T,C) fields starting from measurements. The identification method requires the knowledge of the emission spectra of the studied species, giving the evolution of emission lines with respect to temperature and pressure. The emission spectra is obtained either with the help of a database or with a molecular model [6] and enables to select the most interesting wavelength. Hence, two difficulties appear : As the measurement is carried out on a volume of gas, emission spectroscopy's spatial resolution is not so accurate than with active methods. Moreover, emission spectroscopy can be used to detect gases present in sufficient quantity in the medium or gases at a sufficiently large temperature so as to collect a significant signal. Yet, the performance of actual detectors is increasing and enables to apply the method even for unsteady flows.

16

A. Chouaki, I. Darbord and P. Herve

Emission spectroscopy also answers to several constraints of the problem: the experimental device is reliable, simple and non intrusive. It can also be noted that compared to active methods, methods based on emission are not so sensitive to particles diffusion coming from erosion of the combustion chamber. THE IDENTIFICATION PROCESS In the direct problem, the simulation of the (T,C) fields remains a difficult task, even knowing the combustion and the geometrical parameters. A turbulent flow and the air induction at the interface of the flame are not always well modeled. In order to obtain a better representation of the temperature and the concentration fields, they are derived directly from the spectral intensities measurements. Then, these (T,C) fields could be used in order to improve the analytical parameters involved in the direct problem. The achievement of the identification process is performed around three key-points: • the choice of the experimental measurements, • the analytical representation, • the characterization of the test/analysis distance. The choice of the experimental measurements A first choice consists of defining efficient sensors' locations with respect to the model to identify. In our case, as it will be seen further (), the sensors' locations are not critical to determine. A more difficult task is the choice of the experimental wavelengths. Actually, the experimental tests are achieved at several wavelengths. Generally, one tries to choose the wavelengths which produce high temperatures and concentrations. Nevertheless, the sensitivity of the chosen wavelengths with respect to the temperature and the concentration fields is rarely taken into account. One originality of this work is that the experimental wavelengths are selected considering the sensitivity of the spectral intensities to the (T,C) fields. Below is shown a typical curve of the spectral intensities with respect to the temperature at several wavelengths. Hence, the wavelengths 5.15 fim and 1.82 fxm are chosen because the collected signal is high. Though the signal is low, the wavelength 0.72 fim is also selected because of the high evolution of the spectral intensity with respect to the temperature. The analytical representation The temperature and the concentration fields have to be projected over a function basis. This basis should be able to represent most of the usual profiles encountered in the field of interest. In our case, it means that the basis allows us to represent most of the profiles encountered for example in the field of propane or hydrogen-air gas jet, waste incineration and charcoal power station fires. Hence, the optimization parameters p are the ones involved in the basis' functions and the contributions of these functions to the corresponding field: nt

T{x, y,z) = Y^ Pi^i{x, y, z)

(3)

1=1

nc

C{x,y,z) = Y,Pj'^ji^^y^^) with:

(4)

Estimation of the temperature and the concentration fields • ?li=5.15|a,m A.2=1.82nm X3=0.72 ^im

Spectral intensities (W.m-2.Sri) 40-^

Spectral intensity ^Qo (W.m-2.Sri)

.^'

6.10*

^f 1000

17

4.10* 2.10-"

1500

2000

2500

Temperature (K)

Fig. 2. A typical evolution of the spectral intensities with respect to the temperature at several wavelengths T C nt nc Pi Pj ^i

the temperature field the concentration field the number of the temperature basis functions the number of the concentration basis functions the temperature optimization parameters the concentration optimization parameters the temperature basis' functions the concentration basis' functions

It should be noted that other optimization parameters could be added inside the basis' functions (for example: a parametrized sine or a parametrized normal law). The characterization of the test/analysis distance The identification process seeks to fit the experimental results to the analytical ones through the optimization of the parameters. Nevertheless, the experimental data are incomplete and corrupted by the experimental noise. In the control theory for system identification, the influence of the noise has been investigated by many authors such as [7], [8] and [9]. Hence, the identification process becomes an estimation process where "estimators" are used depending on the a priori known information (for example the density probability function of the the noise, the density probability function of the parameters, the density probability function of the measurements ... ). The estimators can be classified from the one requiring the most amount of a priori information to the one requiring less, as follows: 1. the bayesian estimator, 2. the maximum likehood estimator, 3. the markov estimator and 4. the ordinary least squares estimators.

A. Chouaki, I. Darhord and P. Herve Moreover, the estimation problems can be written in terms of a minimization of a cost function. In the case where the measurements and the parameters are supposed to follow normal laws, the bayesian estimator leads to the Extended weighted Least Squares (E.W.L.S.) estimator characterized by the following cost function ([10]): J{P) = r{pr[Gr]r{p) with:

p. . \Gr]

[GA a

if

+ a{p-

^i—

p,J*[G,](p

^—xni ^

-

p j

(5)

residual vector depending on p optimization parameters vector initial values of the parameters to estimate the inverse of the covariance matrix of r the inverse of the covariance matrix oip a ponderating scalar Hermitian quantity

The first term of J(p) is called the penalty term. It controls the estimator and makes the minimisation problem convex. The second term of J(p) is called the regularization term (in the Thikhonov sense). The scalar a is a ponderating factor which determines the weight of the terms involved in J(p). The values of a enables to control the confidence that has the user in the measurement. For example, in the case where the experimental noise is high, a will have a high value. In the case where the initial parameters are far from the realistic ones, the minimization process becomes very critical. Hence, an alternative to this problem is to decrease the weight of the penalty term at the beginning of the minimization by increasing the value of a. In tough situations, a is determined using a cross validation check [10]. It should be noted that the E.W.L.S. estimator is known to be unbiased and with minimal variance which are very convenient properties. In our case, the residual vector r(p) is defined as follows: r^-^(p)

i^-^(p)

exp

(6)

J-'exp

where L^''^{p) is the analytical spectral intensity at a given wavelength A^ at a given location j , and Lg^'^ is the corresponding experimental spectral density. Hence, the cost function J{p) becomes: n\

j(p) = — nX nl

nl

EEE'"(P)1^^]^'"(£) ^-{P-2.J*\G,]{p-p_J

Lt=i

il)

i=i

where n\ is the number of the involved wavelengths, nl the locations' number and np is the number of the parameters. From the definition of J(p), knowing that the parameters p are initialized to 1, it is obvious that for a = 1 the penalty term has the same weight as the regularization term. Hence, the determination of the value of a is not critical. EXAMPLES As mentioned above, the choice of the wavelengths is of a paramount importance. After choosing the wavelengths with respect to their sensitivity to the temperature and the concentration, it

Estimation of the temperature and the concentration fields

19

remains to select those of them that are less sensitive to the experimental noise. Sensitivity study This study deals with the sensitivity of the identification process with respect to the experimental noise, in the case the 7^20 molecule in methane combustion. After first tests, the following wavelengths: [0.0072 0.0113 0.0117 0,0468 0.0515 0.0546 0.1041 0.1048] lO^^m have been selected. First, the relation between the black body spectral intensity and the temperature has been established using the Hitemp software database database. Then, six analytical basis functions have been chosen (three for each field). Next, a set of "experimental" parameters (p ) is selected in order to simulate spectral intensities. Finally, a noise (with a zero mean, following a uniform law) is added to the simulated spectral intensities. The covariance matrices are taken to be equal to the identity matrix. Two ways have been investigated in order to check the sensitivity of the identification process with respect to the noise: 1. the value of the cost function: the analytical spectral intensities are computed using (p ). Since the experimental spectral intensities are corrupted by the added noise, the value of the cost function is not equal to zero. Hence, Table 1 has been obtained. Each calculation presented in Table 1 has been Noise levels (%) 0.72 1 0.0088 5 0.020 10 0.083 15 0.19 20 0.34 30 0.72

Table 1 The values of the cost function. Wavelengths (^m) 1.13 1.17 4.68 5.15 5.46 10.41 10.48 0.0083 0.0080 0.0093 0.0083 0.0080 0.0084 0.0081 0.021 0.021 0.023 0.022 0.021 0.021 0.021 0.078 0.090 0.088 0.085 0.086 0.083 0.082 0.20 0.20 0.19 0.19 0.21 0.19 0.19 0.35 0.36 0.33 0.31 0.35 0.34 0.35 0.85 0.76 0.81 0.75 0.85 0.76 0.82

obtained as a mean of 40 runs. From this first study, it possible to check the sensitivity of the wavelengths with respect to the noise. The Table 1 shows that the sensitivities are very close, hence, no wavelength will be privileged in the identification process. 2. The values of the identified temperature and concentration fields: Here, the values of the identified temperature and concentration fields are studied in terms of the maximal values and the mean values. For each noise's level, the (T,C) fields have been identified after the minimization of J{p) using the Broyden Flecher Goldfarb Shano (B.F.G.S.) method. The derived (T,C) fields using p have been compared with those obtained using the identified parameters. It leads to Table 2 where: D

D

\J-ana~ -texpl __

I J- exp I

\Cana-Cexp\ \Cexp\

max{) : the maximum value. mean{) : the mean value. From Table 2, one can conclude that the selected wavelengths are less sensitive to the concentrations than to the temperatures.

20

A. Chouaki, I. Darbord and P. Herve Table 2. Comparison of the values of the (T,C) fields. Noise (%) max{RT) (%) mean{RT) (%) max (Re) (%) mean (Re) (%) 0 0.3 0.0 1.3 0.4 1 5.4 1.4 19.3 4.9 5 34.7 9.5 71.6 15.8 10 49.8 12.5 113 26.3 20 85.0 22.0 271 74.0

A real case study The experimental setup The objective is to measure the spectral intensity emitted by a flame simultaneously at 2 wavelengths. The whole optical bench is mobile in one direction in front of the flame, to get the intensity from several locations. Here, the (T,C) distributions are assumed to be axisymmetrical. A diaphragm selects a small volume of gas. A separating blade and two detectors (photomultiplier and INSB) are used to collect simultaneously the signal of the same volume of gas at 0.72fim and 5.15//m. The experimental noise is limited by the use of a modulator and a synchrodectector. 40 measurements are made with an interval of 1mm. Calibrated hole

i

0

^

i

Modulator

c

0

.4!^

i

i

3 0 <

1

Fig. 3. The experimental setup Identification results In order to identify the temperature and the concentration fields, four optimization parameters have been used for each field as follows: T{x, y) = {x - a){x + a){y - b){x + b) C{x, y) = {x - a){x + a){y - b){x + b)

Pi

1 + P2 y\^{x - xoY + (2/ - yo)^ - PsJ

2+P4

(8)

+i

(9)

P5

1 + P6 (\/(^ - ^0)^ + (y - yoY - Pr)

where a, 6, —a, —6 are the frontiers of the studied domain (a = 2.mm and b — 2.m,m). The functions involved in (8) and (9) are enoughflexiblein order to recover the current profiles. In the case of more complex profiles, the (T,C) functions can be complicated by adding appropriated analytical functions.

21

Estimation of the temperature and the concentration fields

When this study was performed, due to technical problems, only the measurements at the wavelength A = 5.15 iim were successful. The minimization process has been initialized using constant profiles (pi and p^ are taken to be equal to zero). Fig. 4 shows the convergence of the cost function with respect to the number of the gradient estimation during the minimization. The initial profiles are shown in Fig. 5 while the final profiles

Number of gradient estimations

Fig. 4. The convergence of the cost function are shown in Fig. 6. The final temperature profile is not very realistic. This could be expected since the wavelength A = 5.15^m is not enough sensitive to the temperature field. Concentration

Temperature

xedge (mm)

y edge (mm)

X edge ^^ (mm)

y edge (mm)

Fig. 5. The initial profiles CONCLUDING REMARKS In this paper, a method enabling to identify the Temperature and concentration (T,C) distributions of steam in a cross section of a hot semi-transparent medium is presented. It comes as a complementary approach to methods such as the finite elements method and the boundary elements method. It makes use of emission spectroscopy associated with an identification algorithm based on the extended weighted least squares estimator. In the proposed approach, a special care is given to the selection of the measurements. They are

22

A. Chouaki, I. Darbord and P. Herve

Temperature

xedge (mm)

Concentration

y edge (mm)

xedge (mm)

y edge (mm)

Fig. 6. The final semi-profiles not only selected with respect to the signal's level, but also with respect to their sensitivity to the (T,C) fields, even if the signal's level is low. Hence, the selected measurements are first checked with respect to the experimental noise before their exploitation. A first example concerning a methane flame has been presented, illustrating the robustness of the proposed method. In order to improve the efficiency of the method, and in order to extend it to a wide range of applications (e.g. high pressures), the basis' functions should be enriched automatically. A work is in progress on this topic.

BIBLIOGRAPHY 1. Grisch, F. and Attal-Tretout, B. (1996). Journal of Nonlinear Physics and Materials, 5(3) pp. 505-526. Pergamon Press. 2. Lefebvre, X. and Leporcq, B. (1996). In: Proceedings ASME Fluids Engineering Division, Summer meeting, San Diego, U.S.A. 3. Mansour,M. (1993). Optics Letters, 18(7) pp. 537-539. 4. Herve, P., B. Serio, M. Loulou, and Dechanterac, L. (1996). In: Proceedings 2nd International Conference on Inverse Problem in Engineering, Le Croisic, France. 5. Darbord, I. and Vally, P. J. June 1999. In: Proceedings 3rd Inverse Problems in Engineering, Port Ludlow, Washington. 6. Fournet,D. (1997) PhD Thesis, Universite Paris X, France. 7. Astrom, K.J. and Eykhoff, P (1971). Automatica, 1 pp. 123-162. Pergamon Press. 8. Eykhoff, P. (1974) System Identification, Parameter and State Estimation. Wiley Interscience, Great Britain. 9. Soderstrom, T. and Stoica, P. (1989) System Identification. Systems and Control Engineering. M.J. Grimble, Germany, Prentice Hall International edition. 10. Tarantola, A. (1987) Inverse Problem Theory, Methods for Data Fitting and Model Parameter Estimation. Elsevier, Netherlands.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

23

NUMERICAL ESTIMATION OF THE TRANSIENT HEAT FLUX BOUNDARY CONDITIONS FOR A FLAT SPECIMEN

S. ABBOUDI*, E. ARTIOUKHINE** * LERMPS, EA 1701, Institut Polytechnique de Sevenans B.P. 449 - 90010 Belfort Cedex, France E-mailf: [email protected] ** Institut de Genie Energetique, 2, Avenue Jean Moulin, 90000 Belfort cedex, France E-mailf: [email protected] ABSTRACT The objective of the proposed study is to analyze numerically transient heating of a flat specimen. The analysis is based on the one-dimensional heat conduction model. The main problem of the analysis is to estimate the heat flux absorbed by a specimen. Temperature evolutions measured inside the specimen are used to solve the inverse heat conduction problem. Iterative algorithms are discussed for estimating the transient heat flux density. The solution is approximated by cubic B-splines. The approximation parameters are estimated by minimizing the residual functional with respect to desired function which is considered as an elements of the function space L2. The residual criterion is used to choose the optimal number of parameters. Results of computational experiments are demonstrated. NOMENCLATURE specific heat c M number of approximation parameters NT number of time step N number of temperature sensors (iV=2) T T To temperature, ambient, initial temperature e thickness of the specimen measured temperature at position X. fi h heat transfer coefficient time, final time t , tf heat flux density Q ^ , ^ adjoint variable, temperature variation density P A,a thermal conductivity, diffusivity 6' estimated error or criterion At, AFo time step, delta Fourier number Es estimated Ex exact KEYWORDS Inverse problem, parametric regularization, heat flux, numerical simulation, estimation.

24

S. Abboudi and E. Artioukhine

INTRODUCTION The ill-posed inverse problem of estimating the surface heat flux from transient temperature histories measured in a heat conducting solid is constantly of a great interest during three last decades. A literature review and a presentation of different methods is given, for example, in Tikhonov and Arsenin [1], Ahfanov et al. [2,3], Beck et al. [4,5,6], Hensel [7] and Murio [8]. Different applications of various methods are presented, in particular, in Zabaras et al. [9] and Delaunay et al. [10]. In this paper, results of numerical analysis are reported, the goal of which is to estimate the heat flux absorbed by a flat specimen cooled at the back side and insulated at its lateral surface. A parameterized representation of unknown function is used with respect the parametric regularization criterion Artyukhin [11] and Artyukhin et al. [12]. The numerical algorithm is based on the minimization of the residual frmctional which is the integrated difference between temperature histories measured and those calculated by solving the direct problem. The conjugate gradient method is used to solve the inverse problem. The residual functional gradient is computed by solving the adjoint problem and the optimal descent parameter is calculated by solving the problem for temperature variations. The heat flux evolution is approximated by cubic B-splines (Alivanov et al 1987. The finite difference method is utilized for solving the direct problem. The influence of the number of parameters is analyzed for small Fourier number {^Fo = a^tle^). INVERSE PROBLEM FORMULATION The specimen is heated by a heat flux of unknown density at the active surface and cooled by a forced convection flow at the opposite surface. The following hypotheses have been taken into account: - thermophysical properties are supposed to be constant, - heat transfer is one-dimensional. - heat transfer coefficient is constant at the cooled surface. Under these conditions, the heat transfer process in the specimen can be described by the following system of equations : — = a^,0<x<e,

0
-x^^^=m

ac - A ^ = h[ne,)-TA ac T(xfi) = T,, 0<x<e

(1)

(2) (3) (4)

In the model (l)-(4), the heat flux density q(t) is unknown. To get additional information about the temperature distribution in the specimen, temperature histories are measured in the specimen at a certain number of points N with coordinates x = X^, n = 1,2, ...,N: T^easi^n^O=fn(t),n

= ia,--;N

This information, together with the model (l)-(4), is used to solve the inverse problem.

(5)

25

Numerical estimation of the transient heat flux boundary conditions ITERATIVE ALGORITHM To build a computational algorithm, we use the variational formulation of the inverse problem of interest. The problem is to fmd such unknown function q{t) for which temperature histories computed from the mathematical model (1) to (4) at the sensor locations are close to measured histories. That leads to the problem of minimizing the residual functional:

J{q) = f, \[T(X„,t-q)~fnit)fdt n=\

(6)

0

where T{X^,t\q), n = 1,2,..., A^, are temperature histories computed at the sensor locations with the heat flux density q{t) given. Unknown function is parametrized in the form of a cubic B-spline : M

(7)

m = TPm^mit)

where p^, w = 1,2,...,M, are unknown parameters, ^^(t), m = 1,2,...,M, are given basis cubic B-splines. The number of approximation parameters M is usually fixed a priori. As a result, the inverse problem is reduced to the estimation of a vector of parameters P = [PI,P2^"'^PM] • Unknown function is considered as an element of the function space L2[0,tj-] of parametrized functions. We use the unconstrained conjugate gradient method of optimization. The residual functional gradient as well as the descent direction in Z2 space have the form : M

m=l M

(9)

D{t) = Y,d^^M

So, the gradient is characterized by the vector g = [gi,g2'---'^M]^ ^^^ ^^^ descent direction by the vector d = [di,d2,.-.,dj^]^. It is easy to show that the residual functional minimization with respect to desired parametrized functions is reduced to those with respect to unknown parameters. The successive improvements of desired parameters are constructed as follows : p'^^=p'^^'d\

5 = 0,1,...

(10)

where s is the number of the iteration under way, /^ is the descent parameter, p is an initial guess for the vector of unknowns parameters given a priori. The vector d^ is computed as follows : d'=-g'+/rd'-\ J = 1,2,

(11)

26

S. Abboudi and E. Artioukhine

where :

^ = 0, /?' = ^ ^ - ^

^

lk*1l

^

(12)

The realization of the iterative procedure (10) is based on computing the vector g at each iteration. This vector is determined by the relationship for the residual functional variation : dJ{qM = {J'p.^)j,M ={JqM^^

(13)

where J'p is the residual functional gradient m R^ space of approximation parameters and J'^ is the gradient in Li space of parametrized functions, (•,) is the scalar product. By using the parametric form (7), it can be shown that (Alivanov et al., [2]): (14)

J'p = Gg where G is the Gram's matrix for basis functions :

This matrix is symmetric and positive definite. The Cholesky decomposition is used to solve the system (14). The most effective method for calculating the gradient J'p m R^ space is based on introducing an adjoint problem. The following expression for the gradient components can be derived : j ; ^ =ji|/(0,/)(p,(/>//, A« = l,2,...,M

(16)

0

where i^{x,t) is the solution of the following adjoint problem (Alivanov et al, [3]): _ ^ ^ ^ ^

0
-i^(Q'^)=0 3c A

(18) A

=2[r(^„,0-/„(0],

« = 1,2,...,7V-1 r „ ( ^ „ , 0 = !^„.,(X„,0. n = \X:;'N -X^

OK.

(17)

= hyf(efy

V/{x,t^) = Q,0<x<e A linear approximation is used to estimate the parameter /* (Alivanov et al., [2]):

(19) (20) (21) (22)

27

Numerical estimation of the transient heat flux boundary conditions

Y,\[T{x,,t)- fn{x„,t)f{x„,tyit y ' - - ^

(23)

—

f^j[HX„,t)pt n=\ 0

where t9{x,t) is the solution of the following boundary-value problem : — = a^,0<x<e,

Q
(24)

- i ^ ? M = ^(,)

(25)

_^^?M = /,^(,,,)

(26)

dx, ^(x,0) = 0, 0 < x < e

(27)

where D(f) is the descent direction in ^2 space of parametrized functions (9). REGULARIZATION To obtain stable solutions of the inverse problem under consideration, the parametric regularization is used (Artyukhin [11]). The principle is based on the residual criterion. The numbers of approximation coefficients M is considered as the regularization parameters. The minimum numbers M^ is shown for, that the residual criterion in the following form is satisfied: J\q{M^)-\<5'

(28)

where 5 is the total (integrated) measurement error defined by :

^ ' = Z \cyl(t)dt

(29)

n=l 0

cr^ (f) is an estimate of the time-dependent standard deviation for the wth temperature history measured. This procedure gives the most stable solution. This number has to be chosen so that the residual criterion would be verified. In this case, the flexibility of the solution is good enough for using the iterative regularization method One iteration of the numerical algorithm is composed by the following steps : - solution of the direct problem and computation of the residual functional, - verification of the residual criterion, - solution of the adjoint problem and computation of the residual functional gradient in ^2 space, - computation of the descent direction, - solution of the problem for temperature variations and computation of the optimal descent parameter, - calculation of the heat flux approximation.

28

S. Abboudi and E. Artioukhine

NUMERICAL SIMULATION To simulate the numerical solution, we have supposed in the problem (1) to (4) that A = a = e = \, h=l00,T,=20 and T^ =10. Sinusoidal heat flux evolution with decrease amplitude is studied :

q(t) = qo(l + sin(ojt))e'''\

0
^0=1000 W/m', ci) = 20/tj. rd/s, ^=0,1 s"', A/= 0,005 s, tj.=l s In these conditions, the different Fourier numbers are : AFo = aAt /e^ = At, AFo, = aAt / X,' =0.5, bJ^o^ =aAtlX^ =0.006 In the above test cases the measured temperatures evolutions was simulated numerically at positions Z, =0.1 and X^ =0.9 with a random noise of 4% of the maximal value of the temperature. In figure 1, we show an example of the noisy simulated temperature histories for the heat flux analyzed.

t

Criterion

c 5 200

-.^f^^^r^X

0

wvi^

0

Fig. 1. Simulated temperature histories ( r j a t ^ , andX,

20

40

60

80

100

M

120

140

160

180 200

Fig. 2. Residual functional in function of the number of approximation parameters

In figure 2, we present the evolution of the residual functional in function of the number of approximation parameters. To obtain this figure, for each value of the number M, we have used a stop test of the iterative algorithm corresponding to a relative error inferior to 8=0.00001 : (30)

where s is the iteration number. It can be seen from the figure 2 that the minimum number of approximation parameters with which the regularizing condition (28) is satisfied is M^=21. The value of integrated measurement errors is J^ = 213 in the test inverse problem under consideration. For an error of 5% on the criterion, the minimum value of M varies between 17 and 40 approximately. Estimated and exact heat flux with M^ = 21 are shown in figure 3 a. There is a good agreement between exact and estimated results in this case. For M<M^, the estimated solution is very stable but it is a bit offsetted as compared to the exact solution; this phenomenon amplifies as M

29

Numerical estimation of the transient heat flux boundary conditions

decreases (see figure 3b for M =\\). For M>M^, a oscillations begins to appear and amplifies as M increases., The figures 3c shows, for M=201, well the existence of these oscillations linked to the influence of local systematic errors. The difference between exact and estimated solutions shows that the average fluctuation is very zero neighbor. 2500

:

:

^2000 E1500

;

(3)

:/ ; \

t

;

;

; ivi=2i

"W]'\

7 ; \ '/ ^ E s \

x1000

:;=

"Jo 500 o I 0 -500 0

^_^\/i_Ex-|s_

",,., l,,,.l . . , , 1 , . , , ! , . , ,

xiooo

7

rr'rE^rj']

JX]^

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

-kL

J'"

AA

i W' Ji

•Art

0

0.1

0.2

0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

Es

Ex-Es\

h h A k%i\!\

0.4

0.5 0.6 time (s)

0.7

\.

Ex

0.8

/I 0.9

Fig. 3a, b and c. Comparison between exact (Ex) and estimated (Es) heat flux density For the three values, M=21, M=ll and M=201, of the number of approximation parameters, we present respectively, in figure 4 a, b and c, at the locations X^ and X^, the temperatures histories calculated from the estimated heat flux density and in figures 5a, b and c, the difference between simulated and estimated temperatures histories at the same locations. These curves explain remarks describe above mentioned and confirm numerically the high efficiency of the methods of parametric regularization developed in this work for to estimate the heat flux density. 1000 800 O

600

f

400 200

;

;

;

:

i^^)

-Tcl.J^

;{b)

IVI=21

=11

;

1 ! 1 \ 1

,Tc2

...;.-.

0 Lr-rT+rrTiT.... 1,,., 1,,., , , , , i , . , , i , . . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

;

itz

(.,^-771',,,, 1,,,, 1,,,, 1,,,, , , , , t , , , , l , , , .

0

0.1

0.2

0.3

0.4

0.5 0.6 time (s)

0.7

0.8

0.9

1

30

S. Abboudi and E. Artioukhine ;(c)

;

;

;

' ' ^ Z " ^ " ' "ITCIT;

Mf201 —

L/i.-iV. Tc2 ;

[/ 0

1 ^j 0.1

0.2

"~'~C .. L... L.. J ^ ^ ^ . . . . 1 , . , . 0.3

0.4

0.5 0.6 time (s)

0.7

0.8

0.9

1

Fig. 4 a, b and c. Evolution of the estimated (or calculated T^) temperatures at locations X^ and X^

{

(b)

'\

Tc1-Tm1

X^ 1

IAA imm V ¥f

o -20 -40

M=11

\

Tc2-Tm2

-80 0

0.1

0.2

0.3

0.4 0.5 0.6 time (s)

0.7

0.8

0.9

-100

1

0

(C)

80

0.1

0.2

0.3

0.4 0.5 0.6 time (s)

0.7

0.8

0.9

1

M=201

60 40 S

20

•7

0

J -20

?^\ii^'^l\tf*A*Atf#/^^^

-40 -60 -80 0

0.1

0.2

0.3

0.4

0.5 0.6 time (s)

0.7

0.8

0.9

1

Fig. 5a, b, and c. Evolution of the difference between estimated and simulated temperatures

CONCLUSIONS To estimate the heat flux evolution from measured internal temperature histories, a method and iterative algorithms are proposed. The method is based on the residual functional minimization in the L2 space of parameterized functions. The number of approximation coefficients is the regularization parameter in the method. The residual criterion in the form (28) is used to choose the regularization parameter. An example of computational experiments was presented for estimating the evolution of heat flux density. The comparison between results, obtained for different values of the number of approximation parameters, show a high efficiency of the iterative algorithm for the optimal value

Numerical estimation of the transient heat flux boundary conditions M^ = 21 for the example studied. For M>M^, instability of the solution appears as M increases and for M <M^, the solution begins to deform as M decreases. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Tikhonov, A., Arsenin V., 1977, Solution of Ill-posed Problems, Wiley, New York. Ahfanov, O. M., Artyukhin, E. A., and Nenarokomov, A. V., 1987, Spline-Approximation of the Solution of the Inverse Heat Conduction Problem, Taking Account of the Smoothness of the Desired Function, High Temperature, Vol. 25, No.5, pp. 520-526. Ahfanov, O. M, Artyukhin, E. A and Rumyantsev, S. V., 1995, Extreme Methods for Solving III Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, Inc., New York. Beck, J. v., Blackwell, B., St. Clair, C. R., 1985, Inverse Heat Condition : Ill-Posed Problems, Wiley Intersc, New York. Beck, J. v., 1993, Comparison of the Iterative Regularization and Function Specification Algorithms for the Inverse Heat Conduction Problem, in Zabaras et al., 1993. Beck, J. v., Blackwell, B., and Hajisheikh, A., 1996, Comparison of Some Inverse Heat Conduction Methods Using Experimental Data, International Journal of Heat and Mass Transfer, Vol. 39, No. 17, pp. 3649-3657. Hensel, E., 1991, Inverse Theory and Applications for Engineers, Prentice Hall, Englewood Cliffs, New Jersey. Murio, D., 1993, The Mollification Method and the Numerical Solution of Ill-Posed Problems, John Wiley and Sons, Inc., New York. Zabaras, N., Woodbury, K., and Raynaud, M., (Eds), 1993, Proceedings of the First International Conference on Inverse Problems in Engineering: Theory and Practice, Palm Coast, Florida, USA, June 13-18 1993, ASME. Delaunay, D., Jamy,Y., and Woodbury, K., (Eds), 1996, Proceedings of the 2nd International Conference on Inverse Problems in Engineering: Theory and Practice, Le Croisic, France, June 9-14 1996, ASME. Artyukhin, E. A, Iterative methods for estimating temperature-dependent thermophysical characteristics. High Temperature- High Pressures, 1997, Vol. 29, pp. 533-539. Artyukhin, E. A. and B. Gamier, Estimating of the thermophysical characteristics of a composite material. High Temperature- High Pressures, 1997, Vol. 29, pp. 541-547.

31

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

33

ESTIMATION OF THE THERMAL STATE OF TWO BARS IN DRY SLIDING

F. LACOUR, Y. BAILLY, E. ARTIOUKHINE Institut de Genie Energetique - Universite de Franche-Comte 2, Avenue Jean MOULIN 90 000 BELFORT - FRANCE Tel: (33) 03 84 57 82 00 - Fax : (33) 03 84 57 00 32 - Email: [email protected]

ABSTRACT In this communication, a thermal study of two bars in dry sliding is presented. The experimental device has been designed by carrying out tri-dimensional computations with Fluent software in order to apply a one-dimensional thermal model. The heat transfer within materials and the thermal conditions at the sliding interface are obtained by solving the inverse heat conduction problem with the use of the iterative regularization method. KEYWORDS Dry sliding , heat generation, inverse problem, measurements, estimation, numerical solution.

INTRODUCTION The problem of analyzmg the thermal state of two solids in dry sliding is of a grate interest during about three part decades. This interest is connected with different applications such as the thermal analysis of brake systems, clutch systems, bearing and others. The physical basis of the thermal conditions during dry sliding is given in [1,2]. It is shown that the main difficulty of the thermal analysis is that two thermal phenomena have to be taken into account: the heat generation at the friction surface and the thermal resistance of sliding contact. This study deals with the problem of estimating the thermal conditions between two solids which are in dry sUding. This work is a part of a thermomechanical characterization of devices used m brake systems. Because in practice the braking operation is generally short, the thermal study is essentially a transient phenomenon. That is why the goal is to estimate the evolution of thermal conditions at the contact surface by measuring temperature histories inside sliding bodies. The main peculiarity of the system under analysis is that no sensor can be placed at the sliding interface during the braking. The developed method consists in using

34

F. Lacour, Y. Bailly andE. Artioukhine

computational and experimental approach based on solving the inverse heat transfer problem. Experimental data are obtained by the use of thermal micro-sensors located in each solid and provided the data which are necessary for solving the inverse problem. EXPERIMENTAL SETUP The materials used for the dry sliding tests are copper and ferrodo. The purpose of using of copper is to have good homogenization of the heat flux near the sliding interface because of its high thermal conductivity. In brake systems, the observed temperature region is quite large, that is why the temperature dependence of the thermophysical characteristics of materials has to be taken into account. The characteristics of copper are well known and largely described in the literature. To obtain the temperature-dependent thermophysical characteristics of ferrodo, preliminary experiments have been carried out with the use of an experimental device similar to that described in [3]. Experimental data processing was realized by solving the non-linear inverse heat conduction problem of estimating simultaneously two temperature functions.

Thermal considerations for the experimental setup To simplify the interpretation of experimental results and numerical calculations, the first step was to define the geometrical parameters of the specimen required to use a one-dimensional model. Many numerical calculations of non-stationary heat transfer were performed by using the Fluent software. The main difficulty consists here in obtaining one-dimensional configuration during long enough time period (a few minutes after the contact) while conserving a possibility to make the specimen mechanically. Fig. 1 gives a typical sample of the geometry simulated with Fluent. A bar made of copper and another one made of ferrodo are in sliding contact. Their lateral surfaces are insulated with the first layer of a special "high temperature" insulator (up to 1700°C) which is based on aluminum oxide. The second insulation layer made of a ceramic which has the thermal conductivity about 0.3 W/m°C near 1000°C (and < 0.6 W/m°C near 250°C) completes the insulation. The fig. 2 shows typical results provided by Fluent with bars of diameter 10 mm. Of course, the temperature distributions are different for two deferent materials under consideration but inside the bars one can see that the cross-sections are isothermal. This actually a characteristic of a onedimensional thermal problem. Fig. 3 shows the temperature profiles calculated along the bars for various radial positions. All curves are superposed. This fact proves that the specimen can be conceded as one-dimensional. Experimental device The experimental apparatus is based on a machine tool used for industrial activities. This system has the property to ensure a nearly perfect alignment of the bars. The copper bar is in rotation while that of ferrodo is fixed. An additional mechanics is used to apply an axial pressure between the bars to generate the friction. The temperature measurements are carried out by means of a set of K type micro-thermocouples (50 jim) [4]. To limit measurement uncertainties related to the use of a turning collector, a specific system of rolling up the wire has been used. This device allows to perform experiments during 90 about seconds at a rotational speed of approximately 2500 rpm. The data acquisition is carried out by using a set

35

Estimation of the thermal state of two bars in dry sliding of amplifiers with cold welding compensation. A PC with the LabVIEW software permits to drive the experiments and the temperature histories are recorded with a samphng rate of 1 kHz for each channel. The sensors are located both at the lateral surface of the bars and inside of them (fig. 5) to verify if the heat transfer process is actually one-dimensional or not. In order to avoid problems of electrical insulation of the thermocouple wires and to obtain good enough precision of determining the sensor locations inside the bars, the wires are inserted through a rigid ceramic sheath. The diameter of the sheath is 800 \xxrL (fig. 4).

INVERSE PROBLEM To simplify the use of the measured temperature histories in computations, the direct problem is formulated by considering the bars as a multi-layer system. The interfaces of this system coincide with the thermocouple locations. The contact between the successive layers is rigorously perfect (fig. 6). At the interface between copper and ferrodo two thermal phenomena are taken into account: the presence of the thermal contact resistance of sliding and the heat source generated by the friction. The mathematical model of the process under analysis is based on the nonlinear heat conduction equation. This model is formed by the following system of equations :

^*^^*^^ = £ ( ^ * ^ ' ^ ^ ] - 4-1 < ^ < 4 . ^ = 1.2,-,^ .t,
= T^{x), i^., < X < i^ , k = 12,...,K

(1) (2) (3)

T,{L,.t) = v,{t)

4(r,)^(L„r) = 4.,(r,.,)^(4,0. ^ = i,2,...,c-i,c + i....,/:-i ox

(4)

ox

T,{L„t) = T,^,{L,.t) ,

k = l,2....,c-l,c

+ l,...,K-l

(5) (6)

T,(L^.t) = v,(t) Thermal conditions at the interface copper-Ferrodo are the following:

^,AT)^{L^,t)OX

Xc^T)^

ox

(L^.t) = S(t)

- A,.,(r)^(4,0 = «s(t) + ^£^4::^(i.,r) dx

R,^{t)

(7)

(8)

36

F. Lacour, Y. Bailly andE. Artioukhine

where S{t) is a point heat source at the copper-ferrodo interface, a is the fraction heat partition at sHding surface which is the ratio between the heat flux entering into two soHds in contact, Rsc{t) is the thermal resistance of the sUding contact. In the system of equations (1)(8) the temperature evolutions Vi(/) and V2(^) are measured by thermocouples. Temperature histories measured at a number of points K -2 inside the specimen are used as the additional information on the temperature distribution in the specimen which is necessary to solve inverse problem : « = 0,1,...,/: - 2

Tn.eas{L,^t)=fM

(9)

The general formulation of the inverse problem of interest is to estimate the heat source S{t) and the thermal resistance of the sliding contact R^X^) from the conditions (l)-(9).

NUMERICAL METHOD The inverse problem (l)-(9) can be solved by estimating the evolution of heat flux densities and temperatures at the sliding interface separately for each bar. Then the sum of heat flux densities gives the density of the point heat source generated by the friction. The thermal resistance of the sliding contact can be determined by the use of the condition (8). Thus, the main problem is to estimate the evolution of the boundary condition at the sliding surface for two bars. To build a computational algorithm of estimating boundary condition, we use the variational formulation of the inverse problem of interest. The problem is to find such unknown function u{t) for which temperature histories computed from the mathematical model (1) to (8) at the sensor locations are close to measured histories. That leads to the problem of minimizing the residual functional : N

rr

J{u) = f^ Z J] [T{L^J'^u)-fM'dt

(10)

where T(L^J;u), n = 1,2,..., N, are temperature histories computed at the sensor locations with the heat flux density u(t) given. Unknown function is parametrized in the form of a cubic B-spline M

u{t) = Y.Pn,
(11)

where ;?„,, m = 1,2,..., M, are unknown parameters, ^^(0, ^ = 1,2,..., M, are given basis cubic B-splines. The number of approximation parameters M is usually fixed a priori. As a result, the inverse problem is reduced to the estimation of a vector of parameters

Estimation of the thermal state of two bars in dry sliding P =

37

(12)

[P\,P2'-^PM}^

The details of the numerical minimization procedure are given in [5]. For each iteration it is necessary to solve three boundary-value problems: • The direct problem (1 )-(8), • The adjoint problem which in our case is: - C „ ( 7\^Vn ; ) ^ = XCrj^9V„ , 4 , dt

vX^Jf)

• ••'

dx'

= 0 , i„., <x
n = 1,2,...,7V + \ .t,
<X
n = 1,2

N+1

(13) (14) (15) (16)

OX

ox

= 2[T„{L„,t;q)-fM.

n = l.2,...,N (17)

(18)

OX

to estimate the heat flux density evolution, (19) to estimate the temperature evolution. The problem in variations which in our case is: d'0„

^ dX ,^^ dr

c ( r Jdt^ = A ( rdxJ ^ + 2dT^ ( r dxJ ^dx^ + ^"^ dx'

dT'^"\dx^

-^^"'' ' ^^ dT dt

!„., < X < L„, n = \,2,...,N + 1 , t^ < t < t^

(20)

e„{x,0) = 0, I„,

(21)

<x
38

F. Lacour, Y. Bailly andE. Artioukhine ^,(l„r) = 0

(22)

0„{L„j)=e„,,{L„.t)

,

n = \,2,...,N

(23)

f^ ( V ) = % i (£„,/),« = 1,2,..., yV

(24)

-A(rJ^(4,/)-§(rJ^^(i„0 = MO

(25)

OX

ox

to estimate the heat flux density evolution, dXL,,t) = Su[t)

(26)

to estimate the temperature evolution. The inverse problem under analysis is ill-posed. To stabilized the numerical solution, we use the iterative regularization [6]. RESULTS A typical example of the temperature histories measured in the bar during sliding is shown in fig. 7. Considering the thermocouples closest to the interface, the temperature increases very intensively. This indicates that the thermal effects of sliding appear in a very short time. Of course, the other thermocouples, located behind, react with a time delay characterizing the heat diffusion. The obtained maximum temperature is about 400°C and remain below the expected values. In fact, ferrodo appears to be a very abrasive material, but it doesn't present sufficient mechanical characteristics to reach very high temperatures with our experimental device. The tangential stresses generated during rotation become too big. The different samples used show that the ablation phenomena can occur during experiments. This may deeply modify the geometry of the bar of ferrodo and prevent to use the temperature histories obtained in Ferrodo with a good efficiency. Nevertheless, the measured data given in fig. 7 can be used to solve the inverse problem for the copper bar. The heat flux density evolution estimated at the sliding interface from the side of the copper bar is shown in fig. 8. At the beginning of the experiment, the heat flux increases very intensively and reaches a maximum value after about two seconds. Then the heat flux decreases.

CONCLUSIONS AND PERSPECTIVES The work presented in this paper gives some results of our first step in the thermal characterization of two solids in sliding contact. The main problem of the analysis is to estimate the evolution of the boundary condition at the sliding surface. A numerical algorithm and a code were developed to solved the non-linear inverse problem of estimating the transient boundary conditions.

Estimation of the thermal state of two bars in dry sliding

39

The realized experiments showed that the friction phenomena induce a very intensive heat transfer process. The dynamic characteristics of the temperature sensors used in this kind of experiment have to be taken into account carefully. The choice of thin enough thermoelectrical sensors and the implementation technique used are particularly efficient to obtain a small time constant and an accurate determination of the sensor locations. We are going to continue these research. It is necessary to modify the geometry of the experimental setup so that the desired temperature range could be reached. The onedimensional analysis permits to verify the general methodology for analyzing the thermal state of solids in dry sliding. In parallel way, we are preparing the experimental setup and numerical tools to analyze two-dimensional systems. REFERENCES 1. F. E. Kennedy, JR., Thermal and thermomecanical effects in dry shding, Wear, Vol. 100, pp. 453476, 1984. 2. J.P. Bardon, Bases physiques des conditions de contact thermique imparfait entre milieux en glissement relatif, Revue General de Thermique, N° 386, fevrier 1994 3. E. Artyukhin, B. Gamier, Estimation of the thermophysical characteristics of a composite material. High Temperature - High Presures, vol. 29, pp 541-547, 1997. 4. Y. Bailly, F. Lanzetta, L. Thiery, L. Girardot, Dynamic calibrations for micronic and submicomic thermoelectric sensors, Microscale Heat Transfer (Eurotherm 98), Poitiers France, 1998. 5. S. Abboudi, E. Artioukhine: Numerical estimation of the transient heat flux boundary conditions for a flat specimen (presented in this conference). 6. O. M. Alifanov, E. A. Artyukhin and S.V. Rumyantsev, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfert Problems, Begell House; New York, 1995.

Ferrodo ®

L.

Fig. 1 Geometry simulated with FL UENT

Fig. 2 FL UENT simulation, isothermal lines

40

F. Lacour, Y. Bailly andE. Artioukhine

Thermocouple

-connexion Fig. 3 FLUENT simulation: temperature at Fig. 4 Thermocouple device different radius (in copper and Ferrodo)

I Thermocouples CO =2500 rpml

I

Copper

Fig. 5 Experimental setup

Thermocouples Copper 1^

ferrodo

B!^

X

0 Fig. 6 Multi-layers configuration

Lc = 80 mm

160 mm

Estimation of the thermal state of two bars in dry sliding

300

-i

---

y'''^

200

%

1/

\

\

f^

100

50

'-""^

_

^

—

at 44 mm

^

to interface

1

i.^--'-^''"^ 1

1

\

!

,i

i

1 ,

\

i

1 —

- ^

^ ^^^^^^^^^^

Y^^^^^^

1 1="

0

f

1 __^^

- — --"

250

E-

H

41

_

—

I 10

12

14

16

18

20

Time [s]

Fig. 7 Temperature measurement at different position in the copper

E ^

3.00E+06

'55 g

2.50E*06

5

2.00E+06

re X

' 50E+06 100E+06

Time [s]

Fig. 8 Estimated heat flux density evolution obtained in the copper at sliding surface

This Page Intentionally Left Blank

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

43

MOISTURE DIFFUSIVITY ESTIMATION BY TEMPERATURE RESPONSE OF A DRYING BODY G. H. KANEVCE and L. P. KANEVCE Faculty of Technical Sciences, St. Kliment Ohridski University, Bitola, Macedonia G. S. DULIKRAVICH Department of Mechanical and Aerospace Engineering, Box 19018 The University of Texas at Arlington, Arlington, Texas 76019, U.S.A.

ABSTRACT The paper presents a method for the moisture diffusivity estimation from experimentally measured temperature response of a drying body by using inverse approach. The mathematical model of drying and the numerical procedure for solution of direct drying problem are given. The Levenberg-Marquardt method is used for estimation of the moisture content and temperature dependent moisture diffusivity. Experiments using numerically simulated temperature data are presented to verify the applicability of the method. Good agreement between the exact and estimated parameter values were obtained. KEYWORDS Parameter estimation, drying, moisture diffusivity, inverse approach INTRODUCTION Drying is a complex process of simultaneous heat and moisture transport within material and from its surface to the surroundings caused by a number of mechanisms. There are several different methods of describing the drying process, but there is no single theory for wet material drying prediction, which encompasses all transfer mechanisms. In the approach proposed by Luikov [5] based on the concepts of irreversible thermodynamics the moisture and temperature fields in the dried body are described by a system of two coupled partial differential equations. The system of equations incorporates coefficients, which are functions of temperature and moisture content. These coefficients must be determined experimentally. For practical calculations, the influence of the temperature and moisture content on all the transport coefficients except for the moisture diffusivity is small and can be neglected. The moisture diffusivity dependence on moisture and temperature exerts a strong influence on the drying process calculation. This effect can not be ignored for the most of practical cases. All the coefficients except for the moisture diffusivity can be relatively easily determined by experiments. The main problem in the moisture diffusivity determination by classical or inverse methods is the difficulty of moisture content measurements.

44

G.H. Kanevce, LP. Kanevce and G.S. Dulikravich

The main idea of the present work is to take advantage of the interrelation between the heat and mass (moisture) transport processes within the drying body and from its surface to the surroundings. The objective is the development of a method of the moisture diffusivity estimation on the basis of a temperature response by using inverse approach. Local moisture content measurements are practically unfeasible especially for small drying objects. Standard drying curves (body mean moisture content during the drying) are complex and of low accuracy, hi this paper, a substitute method is proposed that is based on easy and accurate temperature measurements. The temperature response during convective drying is obtained by numerical experiments. As a representative drying body, a mixture of bentonite and quartz sand with known thermophysical properties has been chosen. MATHEMATICAL MODEL OF DRYING In the case of an infinite flat plate of thickness 2L, if the shrinkage of the material during drying can be neglected (ps = const), the resulting system of equations for the temperature, T(x, t), and moisture content, X(x, t), can be expressed as cps

d\ ~ dx\

dx

at

dx

+ 8psAH

dx

dX at

dx

(1)

(2)

Here, t, x, c, k, AH, s, 5, D, ps are time, distance from the mid-plane of the plate, heat capacity, thermal conductivity, latent heat of vaporisation, ratio of water evaporation rate to the reduction rate of the moisture content, thermo-gradient coefficient, moisture diffusivity, and density of the dry plate material, respectively. As initial conditions, uniform temperature and moisture content profiles are assumed t =0

T(x,0) = To, X(x,0) = Xo

(3)

The boundary conditions on the free plate surface (x = L) are:

"f Dps

+ Jq-AH(1-S)jro=0 x=L

^5X^ V^'^/x=L

+ D8ps

/gj\

(4)

+ Jm=0

V^''yx=L

In the case of convective drying of the sample, the convective heat flux, j<,(t), and mass flux, jni(t), on the surface of evaporation are: Jq=h(Ta-Tx=L) Jm = h D ( C x = L - C a )

(5)

Moisture diffusivity estimation by temperature response of a drying body

45

where h is the heat convection and ho the mass transfer coefficient, Ta is the drying air bulk temperature, and Ca is the concentration of water vapour in the drying air. The water vapour concentration of the air in equiUbrium with the free surface of the body is calculated by C^=L = a(T^=L,X^=L)-Ps(Tx=L)/461.9/(T^=L +273)

(6)

where ps is the saturation pressure and a is the water activity. The water activity or the relative humidity of the air in equilibrium with the sample at the surface temperature and moisture content is calculated from experimental water sorption isotherms. The problem is symmetrical, and boundary conditions on the axis of the plate are: rQj\

0,

dx v^^yx=o

^ax^

ax

=0

x=0

(7)

The system of equations (1) and (2) with the initial (3) and the boundary conditions (4) and (7) has been solved with experimentally determined thermophysical properties. In order to approximate the solution of the equations (1) to (7), an explicit numerical integration procedure was used. The non-linear term has been expanded to

_a_rD ax ax dx

^a^X dx^

dDdX 5x dx

(8)

The derivatives with respect to time have been represented by forward difference form at the grid-point (i, j). All first and second derivatives in space have been approximated at time level G) by central difference forms. The moisture diffusivity, D, in the first term of (8) has been assigned its value at the grid point (i, j). The applied procedure leads to the difference equations Xy+l = (^D - RAD)Xi-lJ + (1 - 2RD)Xij + (RD + RAD)Xi+iJ

(9)

F AH (^ij+i - X y )

(10)

Ty+i =RaTi-ij +(l-2Ro^)Ti j +RaTi+ij + where K-D-- ~ 7 ^ , j ' (Ax)2

^AD

T (Ax)2

4

(11)

and R„=

(Ax)2

ja.

Thermal diffusivity has been calculated as a = k/(cps).

(12)

46

G.H. Kanevce, LP. Kanevce and G.S. Dulikravich

In order to approximate the boundary conditions, central difference form has been appHed to the grid points (1, j) and (M, j) that lie on the boundaries x = 0 and x = L, respectively. A sufficient condition for the numerical stability of this simple explicit method is that the values of the ratios RD and Ra are less or equal to 0.5. The computations were performed with the time increment, At, corresponding to the maximal value of 0.4 for RD and Ra. While Ra is a constant, RD varies with the moisture difftisivity. At most of the grid-points (i, j), Ra was below 0.01. THE DRYING BODY PROPERTIES The presented idea of the moisture diffusivity estimation by temperature response of a drying body has been tested for a model material, a mixture of bentonite and quartz sand [4]. From the experimental and numerical examinations of the transient moisture and temperature profiles [3] it was concluded that for practical calculations, the influence of the thermal diffusion is small and can be ignored. It was also concluded that the Luikov's system of simultaneous partial differential equations could be used by treating of the transport coefficients as constants except for the moisture diffusivity. The appropriate mean values for the model material are: the density of dry solid, ps= 1738 kg/m^, heat capacity, c = 1550 J/(kgK), thermal conductivity, k = 2.06 W/(mK), latent heat of vaporisation, AH = 2.31.10^ J/kg, ratio of water evaporation rate to the reduction rate of the moisture content, 8 = 0.5, thermo-gradient coefficient, 5 = 0.0. The following expression can describe the experimentally obtained relationship for the moisture diffusivity [4]: D =9.0.10-'2.X-2.fli|^f

(13)

The dependence of desorption isotherms of the model material on temperature and moisture content is [2]: a = 1 -exp(-l.5• 10^(T + 273)-"-'l • x-0005.(T+273H3.91)

( 14)

where a is the water activity or the relative humidity of the air in equilibrium with the sample at temperature T and moisture content X. NUMERICAL DRYING EXPERIMENTS The drying experiments have been simulated numerically. The numerical drying experiments have been carried out for the plate of thickness 2L = 3.0 mm, with initial moisture content of X(x, 0) = 0.20 kg/kg and initial temperature T(x, 0) = 20.0 ^C. The drying air bulk temperature was Ta = 80.0 ^C and the relative humidity was s = 0.12. The convection heat transfer coefficient was h = 45.1 W/(m^K) and the mass transfer coefficient was ho = 4.87.10"^ m/s. Satisfactory accuracy was achieved with 41 grid points.

Moisture diffusivity estimation by temperature response of a drying body 47

X[kg/kg]

lOcoincDinoiocMinic

O

O

O

T-

T-

Fig. 1. Transient moisture content profiles Numerical results of the transient moisture profiles inside the body are shown in Fig. 1. The sigmoid moisture content profiles with a large concentration gradient at the evaporation front are obtained.

1300

Fig. 2. Transient temperature profiles This can be explained with reference to the moisture diffiisivity that changes from 10"^^ m^/s at the inner parts of the body to 10"^ m^/s at the outer parts and to the large mass transfer Biot number that is of the order 95 to 5 x 10^ depending of moisture and temperature values. Figure 2. shows predicted temperature variation with time. The temperature gradient is of order 0.3 ^C/mm at the surface and becomes zero at the centre of the body. The heat transfer Biot number is 0.03.

48

G.H. Kanevce, LP. Kanevce and G.S. Dulikravich

Figure 3. shows the volume-averaged moisture X and temperature T changes during drying as well as the drying rate I d X /dt I or mass flux, jm(t) = -ps L d X /dt.

X [kg/kg]; Idx/dtl x10 ^ kg/kg/s] 35 n

500

1000

1500

2000

time [s]

Fig. 3. Predicted moisture, temperature, and drying rate curves

MOISTURE DIFFUSIVITY ESTIMATION The estimation methodology used is based on minimisation of the ordinary least square norm: E(P) = [Y-T(P)r[Y-T(P)]

(15)

Here, Y^= [Yi,Y2, ... ,Yimax] is vector of measured temperatures and T^ = [Ti(P), T2(P), ..., Timax(P)] vector of estimated temperatures at time ti (i = 1, 2, ..., imax), while P = [Pi, P2, ..., PN] is the vector of unknown parameters, imax is total number of measurements, and N is the total number of unknown parameters (imax > N). A version of Levenberg-Marquardt method was applied for the solution of the presented parameter estimation problem [6]. This method is quite stable, powerful, and straightforward and has been applied to a variety of inverse problems [7]. It belongs to a general class of damped least square methods [1]. The solution for P is achieved using the following iterative procedure

P^^' = r + [ ( / V / +^^ir(/V[Y-T(P')]

(16)

where I is identity matrix, ji is damping parameter, and J represents the sensitivity matrix defined as:

Moisture diffusivity estimation by temperature response of a drying body 49

ax,

ar,

9P, 8X2 3Pl

9PN

9P2

9PN

aii,

aX;,

aPo

5Pi

axi

aP2 OT2

3X2

(17)

ar.. aPN

The term |il damps instabilities due to the ill-conditioned character of the problem. Near the initial guess, the problem is generally ill conditioned and damping parameter is chosen large making term \il large as compared to term J^J. Therefore, the matrix J^J is not required to be non-singular at the beginning of iterations when the procedure tends towards a steepest descent method. As the iteration process approaches the converged solution, the damping parameter decreases, and the Levenberg-Marquardt method tends towards Gauss method [1, 7]. In fact, this method compromises between the steepest descent and Gauss method choosing \i so as to follow the Gauss method to as large an extend as possible, while retaining a bias towards the steepest descent direction to prevent instabilities. The presented iterative procedure stops if the norm of gradient of E(P) is sufficiently small, or if the ratio of the norm of gradient of E(P) to the E(P) is small enough or if the changes in the vector of parameters are very small [8]. PIESULTS AND DISCUSSION For the model material considered here the moisture diffusivity could be represented by the following function of temperature and moisture content D:

Di

T + 273\10

D2+X^

303 J

(18)

where Di and D2 are constants. For the inverse problem investigated here, values of Di and D2 are regarded as unknown and all other quantities involved in direct problem formulation were assumed to be known. Thus, P^ = [Di, D2]. Numerical experiments have been conducted to verify the proposed method. Since the sample is very thin (3.0 mm), a single thermocouple has been located in the middle of the infinite flat plate. Two analyses of the simulated data have been considered. In the first analysis, parameters have been estimated using "exact" temperature data at the sensor location obtained by the solution of direct problem for the exact values of the parameters. Adding an error term to exact temperature response then simulated the data for the second analysis. The errors were additive and normally distributed with zero mean and a standard deviation a = 0.5 ^C generated by RANDN routine of the MATLAB program for normally distributed random numbers. The results obtained by applying the present method to the estimation of the unknown parameters for the case with exact data and the data with noise are shown in the Table 1.

50

G.H. Kanevce, L.P. Kanevce and G.S. Dulikravich

Table 1. Performance of the method without and with the simulated measurement noise Iterations

3.7796-10-^

4.3955-10'^

21

1.327710"^

4.6684-10'^

14

D2

9.0. 10-'

0.

0

9.0019-lO"^^

0.5

9.0602-10'^^

Exact values Values obtained with "exact" T data Values obtained with added noise

RMS error

Di

It can be seen that a good agreement is achieved in both cases. The root mean squared difference between experimental and obtained values tends to zero in the first case and tends to standard deviation of measured temperature data in the second case. Figure 4. shows the residual. Normally distributed errors can be seen. 1.5-1 Residual fC]

1tOO 1400 time [s]

Fig. 4. The residual The initial guess (Diinit= 5.0-10'^^ and D2init= 0.1) for both cases was the same and it was very far from the exact values of parameters in order to test the stability of the proposed method. Consequently, the number of iterations is high. Figure 5. shows the RMS changes and Fig. 6. illustrates the approaching of parameters to the final values during iterative process for the case with the added noise. 1.0E-09

10 Iterations

Fig. 5. RMS-Error

T 1.0E-01

10

1.0E-05 Iterations

Fig. 6. Moisture diffusivity coefficient

51

Moisture diffusivity estimation by temperature response of a drying body

Since the exact value of D2 for model material equals zero and the performed analysis shows that the second parameter value is zero compared to the X^ values in this case, the analysis with one parameter has been conducted also. The relation for moisture diffusivity (18) was used with D2 = 0.0, and the parameter Di was evaluated by the same method. The similar results were obtained. In order to investigate the influence of drying time, determinant of the sensitivity matrix J^J (17) with normalised elements

"-'p„aiLl(p^ dV^ aRny i=l

m,n= 1, ...,N

(19)

has been calculated. Figure 7. shows | J^J | for one parameter analysis for the entire drying time until the equilibrium temperature and moisture content of the body were reached. Figure 8. illustrates changes of the relative sensitivity coefficient during drying process. Jil=Pi—i-, 1,1 1 ^ . i=l,2,...,imax;

1=1,2,...,N

(20)

. 2.E+03 n c 2.E+03

I 1.E+03^ I 5.E+02 ^

5001

O.E+00^ 500 1000 1500 time [s]

-7-"

Fig. 7. Sensitivity determinant

X

25 n

1000

1500

time [s]

Fig. 8. Sensitivity coefficient Sensitivity coefficient

Determinant xlO

x5 T 1 0 0 — 4-80 Determinant + 60 + 40

Sensitivity coefficient

500

1000

1500

4-20 +0

+ -20 + -40

2000

time [s]

Fig. 9. Comparison of moisture, determinant, sensitivity coefficient, and temperature

52

G.H. Kanevce, LP. Kanevce and G.S. Dulikravich

Two plateaux can be seen on the presented curve. The first plateau corresponds to the moment (Fig. 9.) when the body moisture content is nearly equal to the equilibrium. After this instant in time, rapid body temperature increase occurs associated with a rapid evaporation rate decrease. The second plateau corresponds to the moment when nearly equilibrium temperature has been reached. In all analyses presented here, drying time corresponding to the end of the first plateau (720 s) has been taken. For the drying times shorter than that, the problem is ill posed and local minimum can bee obtained depending on the initial guesses. CONCLUSIONS A method is presented for estimation of moisture diffusivity on the basis of temperature transient response of a drying body by using an inverse approach. The numerical procedure for solution of direct drying problem with appropriate convergence conditions was outlined. Numerical results for transient moisture content and temperature profiles as well as predicted moisture, temperature and drying rate curves are shown. The iterative Levenberg-Marquardt method is applied for evaluation of unknown constants in moisture diffusivity model of its dependence on moisture and temperature. The estimation method is demonstrated using data obtained from a simulation of the experimental design. The results, in the case with errorless measurements as well as with measurements containing random errors, show good agreement between evaluated and exact parameter values and confirm the validity of the proposed method. From the conducted sensitivity analysis the proper duration of the drying experiment can be estimated. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

Beck, J.v., and Arnold, K.J. (1977). Parameter Estimation in Engineering and Science, Wiley, New York. Kanevce, H. G. (1981). PhD Thesis, University of Novi Sad, Yugoslavia. Kanevce, H. G. (1998). IDS '98, Vol. A , pp. 256-263, Halkidiki, Greece. Kanevce, G., Stefanovic, M. and Pavasovic V. (1980). Drying' 80, Vol. 1, pp. 128-131, Hemisphere. Luikov, A.V. (1968). Drying Theory (in Russian), Moscow. Marquardt, D.W. (1963). J. Soc. Ind Appl Math 11, 431. Meijas, M.M., Orlande, H.R.B., and Ozisik, M.N. (1999). 3''' International Conference on Inverse Problems in Engineering, 1999, Port Ludlow, WA, USA. Pfafl, R.C., and Mitchel, J.B. (1969). AIAA Paper, No.69602.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

53

PARAMETER ESTIMATION IN MOIST CAPILLARY POROUS MEDIA BY USING TEMPERATURE MEASUREMENTS L. B. Dantas, H. R. B. Orlande and R. M. Cotta Department of Mechanical Engineering, COPPE Federal University of Rio de Janeiro, UFRJ Cidade Universitaria, Cx. Postal 68503 Rio de Janeiro, RJ, 21945-970, Brazil

P. D. C. Lobo Research and Development Institute, IP&D-UNIVAP Av. Shishima Hifumi, 2911, Sao Jose dos Campos, SP, 12244-000, Brazil

ABSTRACT This work deals with the solution of inverse problems of parameter estimation involving heat and mass transfer in capillary porous media. The physical problem considered here is described by the linear one-dimensional Luikov's equations. The associated direct problem is solved by the Generalized Integral Transform Technique. An analysis of the sensitivity coefficients and the use of the D-optimum criterion permit the design of the experiment with respect to the final experimental time, number and locations of sensors and the number of parameters that can be estimated. The present parameter estimation problem is solved with Levenberg-Marquardt's method of minimization of the ordinary least-squares norm, by using simulated temperature data with random errors. KEYWORDS Luikov's equations. Capillary-porous media, Heat and mass transfer. Parameter estimation, Levenberg-Marquardt's method. INTRODUCTION The phenomena of heat and mass transfer in capillary porous media has practical applications in several different areas including, among others, drying and the study of moisture migration in soils and construction materials [1-3]. For the mathematical modeling of such phenomena, Luikov [1] has proposed his widely known formulation, based on a system of coupled partial differential equations, which takes into account the effects of the temperature gradient on the moisture migration.

54

LB. Dantas et al

The computation of the temperature and the moisture content fields in capillary porous media, from the knowledge of initial and boundary conditions, as well as of the thermophysical problems appearing in the formulation, constitutes a Direct Problem of heat and mass transfer. Such type of direct problem, based on Luikov's theory, was solved analytically through the use of the Classical Integral Transform Technique [2]. Later, Lobo et al [4] found that the associated eigenvalue problem had complex eigenvalues, that were not accounted for in previous works. The effects on the solution of including one pair of conjugate complex eigenvalues were critically addressed by Lobo et al [4], while the inclusion of complex eigenvalues of higher order were discussed in references [5,6]. The use of the Generalized Integral Transform Technique, with simpler eigenvalue problems involving analytical eigenfunctions, can avoid the calculation of complex eigenvalues for the drying problem based on Luikov's formulation. For more details on the use of such hybrid numerical-analytical technique, the reader is referred to references [3,7-9]. We note that numerical techniques [10,11] have also been used for the solution of Luikov's system of equations. The numerical modeling of heat and mass transfer in capillary porous media requires the accurate knowledge of several thermophysical and boundary condition parameters that appear in the formulation. The use of inverse analysis techniques [12-15] permits the estimation of such parameters, from the knowledge of temperature and moisture content measurements taken in the media [16]. In this paper, we present the solution of the inverse problem of estimating three of the several thermophysical and boundary condition parameters, appearing in Luikov's dimensionless formulation for the heat and mass transfer in a one-dimensional capillary porous media. Such three parameters are the Luikov number, the Kossovitch number and the Biot number at one of boundaries. The present inverse problem of parameter estimation is solved by using the Levenberg-Marquardt method of minimization of the least-squares norm [12,13,17,18]. Simulated temperature measurements are used here in order to verify the accuracy of the present parameter estimation approach. By using only temperature measurements in the parameter estimation procedure, very involved techniques for the measurement of moisture content, such as the use of gamma rays, can be avoided. Thus, simpler and faster experiments can be run for the estimation of parameters associated with Luikov's formulation. DIRECT PROBLEM The physical problem under picture involves a one-dimensional capillary porous medium, initially at uniform temperature and uniform moisture content. One of the boundaries, which is impervious to moisture transfer, is put in contact with a heater. The other boundary is put in contact with the dry surrounding air, thus resulting in a convective boundary condition for both the temperature and the moisture content, as illustrated in Fig. 1. The linear system of equations proposed by Luikov [1] with associated initial and boundary conditions, for the modeling of such physical problem involving the heat and mass transfer in a capillary porous media, can be written in dimensionless form as [1]: —^-^-^ = dr —^ dr

^^-^-^ - sKo ^^ ^ dX^ dr

^ = Lu —^^ ^ ^ - LuPn 1——^ dX^ dX^

m 0<X<1, for t>0 in 0<X<1, for T>0

(la) (lb)

Parameter estimation in moist capillary porous media

for T=0, in 0<X<1

e{x,0) = 0, (^{X,0) = 0, djM^^Q, dX

^JM_P„SJM

8X

^^^-Bi^[l-d{l,T)j

.^iM

.tX^O,forr>0

= O,

= 0, atX=l, for r>0

+ (l-s)KoLuBi„[l-^{l,r)j

+ Pn^jM

dx

dX

atX=7,forr>0

+ BiJ(l-{l.r)] = 0,

"

dx

^^ '^

55

(lc,d) (le,f) (ig) (Ih)

where the following dimensionless groups were defined 4x,r)=^^^^, u-u

d^X.T)J^^^^^. s-Tn

BiA

Pn=s'^. u^-u

Bi^M, lu = '^, . 4 ' ,

Ko='-'^^^^. cT,

X='-. Q^-^—.

(2a-d)

(2e-j)

The properties of the porous medium appearing above include the thermal diffusivity {a), the moisture diffusivity {am), the thermal conductivity (A:), the moisture conductivity {kn^ and the specific heat (c). Other physical quantities appearing in the dimensionless groups of eqs. (2) are the heat transfer coefficient Qi), the mass transfer coefficient (hm), the thickness of porous medium (/), the prescribed heat flux (g), the latent heat of evaporation of water (r), the temperature of the surrounding air (Ts), the uniform initial temperature in the medium {To), the moisture content of the surrounding air {u*), the uniform initial moisture content in the medium {Uo), the thermogradient coefficient {5) and the phase conversion factor {s), Lu, Pn and Ko denote the Luikov, Possnov and Kossovitch numbers, respectively. Flow of dry air ^ ^ Heat and moisture transport Moist porous sheet Hot plate

Heat supply rate: q

T

Fig. 1. Geometry for the drying of a moist porous medium Problem (1) is referred to as a Direct Problem when initial and boundary conditions, as well as all parameters appearing in the formulation are known. The objective of the direct problem is to determine the dimensionless temperature and moisture content fields, %{X,f) and ^{X,f), respectively, in the capillary porous media. INVERSE PROBLEM For the inverse problem of interest here, the parameters Lu, Pn, Ko, e. Big and Bim are regarded as unknown quantities. For the estimation of such parameters, we consider available the transient temperature measurements Ytm taken at the locations Xm, m=l,...,M The subscript /

56

L.B. Dantas et al

above refers to the time at which the measurements are taken, that is, r^ for z-1,...,/. We note that the temperature measurements may contain random errors, but all the other quantities appearing in the formulation of the direct problem are supposed to be known exactly. Inverse problems are ill-posed [12-15]. Several methods of solution of inverse problems, such as the one used here, involve their reformulation in terms of well-posed minimization problems. By assuming additive, uncorrected and normally distributed random errors, with constant standard deviation and zero mean, the solution of the present parameter estimation problem can be obtained through the minimization of the ordinary least-squares norm. Such a norm can be written as 5(PHY-e(P)]^[Y-0(P)]

(3)

where P=[Iw, Pn, Ko, e, Big, Bim] denotes the vector of unknown parameters. The superscript T above denotes transpose and [Y - 0(P)]^ is given by [Y-e(P)f-[(f, -0,),{Y, -e,),...

,(f, -0,)]

(4a)

where (}^ - 6^ j is a row vector containing the differences between the measured and estimated temperatures at the measurement positions JC;,, 7W=1,...,M, at time /,, that is,

^-9)AYa-0.vY„-e,„...,Y.^

-9,^)]

(4b)

The estimated temperatures ^tm are obtained from the solution of the direct problem, Eqs. (1), at the measurement location Xm and at time ti. The present inverse problem of parameter estimation is solved with the Levenberg-Marquardt method of minimization of the least-squares norm [12,13,17,18]. The iterative procedure of such method is given by p^+l = p^ + [(j^)^j^+^^Q^]-l(j^)^[Y-e(P^)]

(5)

where J*^ is the sensitivity matrix, p. is a positive scalar named damping parameter, Q is a diagonal matrix and the superscript k denotes the iteration number. k

k

The purpose of the matrix term // Q in equation (5) is to damp oscillations and instabilities due to the ill-conditioned character of the problem, by making its components large as compared to those of J^J, if necessary. The damping parameter is made large in the beginning of the iterations. With such an approach, the matrix J\F is not required to be non-singular in the beginning of iterations and the Levenberg-Marquardt Method tends to the Steepest Descent Method, that is, a very small step is taken in the negative gradient direction. The parameter // is then gradually reduced as the iteration procedure advances to the solution of the parameter estimation problem and then the Levenberg-Marquardt Method tends to the Gauss Method [12]. However, if the errors inherent to the measured data are amplified generating instabilities on the solution, as a result of the ill-conditioned character of the problem, the damping parameter is automatically increased. Such an automatic control of the damping parameter makes the Levenberg-Marquardt method a quite robust and stable estimation procedure, so that it does not require the use of the Discrepancy Principle in the stopping criterion to become stable, like the conjugate gradient method [13,14].

Parameter estimation in moist capillary porous media

57

The sensitivity matrix is defined as

dO[_

dol_

del dOl del_

del_

de^ j(p)-

9e^(P) ap

dP\

36^ dP2

dP,

dP2

dej

deJ deJ

dPi

dP2

dPN

dp^

dP^

(6a)

dOj dPM

The elements of the sensitivity matrix are the sensitivity coefficients. They are defined as the first derivative of the estimated temperatures with respect to each of the unknown parameters Pj, j=\,...,N. The sensitivity coefficients are required to be large in magnitude, so that the estimated parameters are not very sensitive to the measurement errors. Also, the columns of the sensitivity matrix are required to be linearly independent, in order to have the matrix J J invertible, that is, the determinant of J^J cannot be zero or even very small. Such a requirement over the determinant of J^J is better understood by taking into account a statistical analysis, as described below. STATISTICAL ANALYSIS AND EXPERIMENTAL DESIGN After the minimization of the least squares norm given by Eq. (3), a statistical analysis can be performed in order to obtain confidence intervals and a confidence region for the estimated parameters [12]. Confidence intervals at the 99% confidence level are obtained as: Pj-2.576a p


(7a)

where P are the values estimated for the unknown parameters. The Confidence Region can be computed from

(P-p)^v-\p-p)<4

(7b)

where j ^ is the chi-square distribution for N degrees of freedom, for a given confidence level and V is the covariance matrix of the estimated parameters given by [12]: 7 - 1 2

V = (J J)

a

(8)

where a is the standard deviation of the measurement errors. Optimum experiments can be designed by minimizing the hypervolume of the confidence region of the estimated parameters, in order to ensure minimum variance for the estimates. The minimization of the hypervolume of the confidence region given by equation (7b) can be obtained by maximizing the determinant of V ^ in the so-called D-optimum design [12,19]. Since the covariance matrix V is given by equation (8), we can then design optimum experiments by maximizing the determinant of the matrix J^J, also referred to as the Fisher Information Matrix [19]. For cases involving a single sensor, each element F;^ , m,n = 1, ..., A^, of the matrix F = J J is given by:

58

L.B. Dantas et al. T

J J

-I

for m,n = 1, ..., A^

(9)

where / is the number of transient measurements and A^ is the number of unknown parameters. If we take into account restrictions, such as a large but fixed number of transient measurements of M sensors, we can choose to maximize the determinant of an alternative form of F, here denoted as Fi [12], the elements of which are given by [F/:

1 MT^

M V

P

V5=lr=0V

- ^ ^P

dT dr "" dPnj

for m,n = 1, ..., A^

(10)

where z/is the final experimental time. RESULTS AND DISCUSSION In this paper we consider a test-case involving the drying of ceramics. The physical properties for such test-case are taken from reference [11] as: po"^2000 kgW, kg=0.34 W/mK, kn, = 2.4x10-^ kg/ms°M, Cg=607 J/kgK, Crn=\.SxlO'^ kg/kg^M, S=0.56 ^M/K and r=2.5xl0^ J/kg. The length of the sample is assumed to be 0.05 m. The sample is initially at the uniform temperature To=24 °C and at the uniform moisture content wo = 80 °M. The air conditions are supposed to be 7^=30 °C and u* =40 ^M. The coefficients of heat and mass transfer are assumed to be 17 W/rn^K and 1.2x10^ kgWs^M, respectively. Such values for the physical quantities result on the following dimensionless numbers: Big=2.5, Binj=2.5, Lu=0.2, Pn=0.0S4 and Ko=49. The phase conversion factor e gives an indication if the moisture flow occurs in the form of vapor or liquid [1]. If ^=1, the moisture flow takes place in the form of vapor. Since the moisture content of the body is equal to the liquid content (vapor content is neglected), then, for ^=1 changes in the moisture content are due only to the evaporation of the liquid or condensation of vapor, since there is no flow of liquid. On the other hand, for ^= 0 changes of the moisture content are due to the liquid flow and evaporation takes place only at the body surface. For the test-case examined below we assumed 6:= 0.2 [11]. For the solution of the direct problem given by equations (1), we considered the Generalized Integral Transform Technique [3,7-9]. The sensitivity coefficients were computed by finite differences, by using a forward approximation for the first derivatives [12,13]. Figs. 2.a,b present the temperature and moisture content variations at several points inside the body, respectively. Fig. 2.a shows a large temperature drop for small times, due to the evaporation of the liquid in the body. As a result of the evaporation and moisture transfer to the surrounding air, the moisture content is smaller (larger (t)(X,r)) near the open boundary for up to r= 10. Around r= 10 the moisture content inside the whole body approximately approaches the moisture content of the surrounding air ((t)(X,r)=l). For larger times, the moisture content near the heated boundary becomes smaller than that near the open boundary, as a result of the moisture flow from the region at higher temperatures. Note also in Fig. 2.a that, for r> 6, the body temperature becomes larger than its initial temperature as it is heated, because moisture changes are now being controlled by the moisture flow, instead of evaporation.

Parameter estimation in moist capillary porous media

4

6 8 10 12 14 16 18 20 Dimensionless time, X

4

59

6 8 10 12 14 16 18 20 Dimensionless time, T

Fig. 2. Temperature (a) and moisture content (b) variations inside the body. We present in Figs. 3.a-c the relative sensitivity coefficients for temperature, for positions X=l 0.4 and 0 respectively, with respect to the different parameters appearing in Luikov's formulation. The relative sensitivity coefficient is obtained by multiplying the sensitivity coefficient by the value of the parameter that it is referred to. Therefore, the relative sensitivity coefficients can be compared to the magnitude of the measured temperatures and, as a result, it is easier to detect relative small magnitudes and linear dependence. Figs. 3.a-c show that the relative sensitivity coefficients are very little affected by the sensor location. Therefore, the temperatures in the body are equally sensitive to variations on the parameters and the sensors can be located at any point in the region. An analysis of Figs. 3.a-c reveals a strong linear dependence of the sensitivity coefficients with respect to Bim and Lu. Thus, the simultaneous estimation of Bim and Lu is impossible if only temperature measurements are used in the estimation procedure. Also, the relative sensitivity coefficients with respect to Pn and if are quite small. Hence, temperature measurements do not provide useful information for the estimation oiPn and e. From the foregoing analysis of figs. 3.a-c, it appears possible to estimate simultaneously Big, Lu and Ko. The relative sensitivity coefficients of such parameters are not linearly-dependent and they attain values of the same order of magnitude of the temperatures in the region, as can be noticed in fig. 2.a. Therefore, the results presented hereafter are based on the estimation of such three parameters. Figure 4 presents the variation of the determinant of the matrix Fi, the elements of which are given by eq. (10), for the cases of a single sensor located atZ=0 or aXX=l, and of two sensors located atX=0 andZ=l. Fig. 4 shows that basically the same values of det(Fi) are obtained for the cases examined. Such behavior is due to the fact that the relative sensitivity coefficients are basically independent of the sensor location, as observed in Figs. 3.a-c. We note in Fig. 4 that the curves for det(Fi), for each of the cases examined, reach a maximum value around r= 6, because the sensitivity coefficients tend to zero for larger times. Therefore, the final experimental time shall be taken as T/ = 6, in order to obtain accurate estimates for the parameters, without performing too long experiments. For the physical case under study, involving the drying of ceramics as described above, such dimensionless final time corresponds to 14.4 hours. An interesting fact is that, if the six parameters appearing in Luikov's formulation, i.e., F=[Lu, Pn, Ko, £, Big, Bim], are considered as unknown, the

60

L.B. Dantas et al.

maximum value of det(Fi) is of the order of 10"^^. Such quite small value for det(Fi) reflects the linear dependence of the sensitivity coefficients with respect to Lu and Bim, as well as the small magnitudes of the sensitivity coefficients with respect to Pn and s. X = 0.4

X = 1.0

- I — I — I — I — I — I — I — i -

2 4 6 8 10 12 14 16 18 20 Dimensionless time, T

0 2 4 6 8 10 12 14 16 18 20 Dimensionless time, T

X = 0.0

Biq Bin Lu Pn Ko T

1

1

1

1

r

1

1

1

1

0 2 4 6 8 10 12 14 16 18 20 Dimensionless time, T

Fig. 3. Relative sensitivity coefficients for temperature at different positions: (a)X=l, (b) X=0.4 and (c) X=0. lE+O

lE-1

I lE-2

« lE-3

Q

1E-4H

lE-5

0

2

4

6 8 10 12 14 16 18 20 Dimensionless time, T

Fig. 4 - Transient variation of det(Fi).

For the results presented below, involving the estimation ofV=[Lu, Ko, Big] by using simulated measurements, we considered available the transient readings of a sensor located at X=0. The

Parameter estimation in moist capillary porous media final experimental time was taken as ^ 6 , as a result of the analysis det(Fi) discussed above. During the time period 0 < r< 6, we considered available 100 transient measurements. For the estimation of the unknown parameters, we utilized the Levenberg-Marquardt method of minimization of the least-squares norm. The subroutine DBCLSJ of the IMSL [20] based on such method was used in the present work. Table 1 illustrates the results obtained for the estimated parameters by using measurements with different levels of random error, including a=0 (errorless measurements) and a=0.01 YmaxThe 99% confidence level intervals for the estimated parameters are also included in Table 1. The initial guesses used in the iterative procedure of the Levenberg-Marquardt method, were taken as Lu^ = 1.0, Ko^ = 60.0 and Bi^^ = 5.0. An analysis of Table 1 reveals that accurate estimates can be obtained for these three parameters. They were exactly recovered when errorless measurements were used in the analysis and the exact parameters generally fall inside the estimated confidence intervals, for the case involving measurements with random errors. Table 1. Estimated parameters and 99% confidence intervals obtained with the LevenbergMarquardt method Confidence intervals Parameters Exact Estimated Estimated or=0 O-=0.01 Ymca (2.500, 2.502) 2.500 2.5 2.5 Bi, (0.19,0.21) 0.20 0.2 0.2 Lu (48.92, 49.05) Ko 48.99 49.0 49.0 CONCLUSIONS In this paper we presented the solution of an inverse problem of parameter estimation for a one-dimensional capillary porous media, by using Luikov's model for the heat and mass transfer processes. The associated direct problem was solved with the Generalized Integral Transform Technique. The present parameter estimation problem was solved by using the Levenberg-Marquardt method of minimization of the least-squares norm. An analysis of the sensitivity coefficients and of the determinant of the inverse of the covariance matrix shows that it is possible to estimate simultaneously the Luikov number, the Kossovitch number and the Biot number for heat transfer, by using only temperature measurements. For the case under picture, involving the drying of ceramics, the temperatures at any point in the medium are equally sensitive to variations in the unknown parameters and the optimum experimental time is TJ = 6. The use of simulated temperature measurements containing random errors shows that accurate estimates can be obtained for the unknown parameters with the present approach, by using measurements taken atX^O. REFERENCES 1. 2. 3.

Luikov, A. V. (1966) Heat and mass transfer in capillary-porous bodies, Pergamon Press, Oxford. Mikhailov, M. D. and Ozisik, M. N. (1984) Unified Analysis and Solutions of Heat and Mass Diffusion, John Wiley, New York. Ribeiro, J.W. and Lobo, P.D.C. (1998) In: The Integral Transform Method in Thermal and Fluids Science and Engineering, R.M. Cotta (Ed), Begell House Inc. Publishers, NY, pp. 125-143

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62 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

L.B. Dantas et al. Lobo, P.D.C., Mikhailov, M.D. and Ozisik, M.N., (1987), On the Complex Eigenvalues of Luikov System of Equations, Drying Technology 5(2), 273-286. Guigon, S.M.S., Dantas, L.B., Scofano Neto, F. and Cotta, R.M. (1999) Exact solution of Luikov's Equations for Drying in Capillary Porous Media, Hybrid Methods in Engineering (in press) Pandey, R. N., Srivastava, S.K. and Mikhailov, M.D. (1999) Solutions of Luikov Equations of Heat and Mass Transfer in Capillary Porous Bodies through Matrix Calculus: a New Approach, Int. J. Heat and Mass Transfer, 42, pp. 2649-2660. Ribeiro, J.W., Cotta, R. M. and Mikhailov, M. D. (1993) Integral Transform Solution of Luikov's Equations for Heat and Mass Transfer in Capillary Porous Media, Int. J. Heat and Mass Transfer 36, 4467-4475. Ribeiro, J.W. and Cotta, R.M. (1995) On the Solution of Nonlinear Drying Problems in Capillary Porous Media Through Integral Transformation of Luikov Equations, Int. J. Num. Meth. Eng. 38, pp. 1001-1020. Cotta, R. M. (1993) Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Boca Raton, FL. Lobo, P.D.C., Boles, M.A. and Ozisik, M.N. (1995) The effects of geometry for multiregion heat and mass transfer governed by the nonlinear Luikov system. Proceedings of the International Symposium on Two-Phase Flow Modeling and Experimentation, Rome, Italy, pp. 1243-1247. Comini, G. and Lewis, R.W. (1976) A Numerical Solution of Two Dimensional Problems Involving Heat and Mass Transfer, Int. J. Heat Mass and Transfer, 19, pp. 1387-1392. Beck, J.V. and Arnold, K.J. (1977) Parameter Estimation in Engineering and Science, Wiley, New York. Ozisik, M. N. and Orlande, H. R. B. (2000) Inverse Heat Transfer: Fundamentals and Applications, Taylor and Francis, New York. Alifanov, O.M. (1994) Inverse Heat Transfer Problems, Springer-Verlag, New York. Dulikravich, G.S. and Martin, T.J. (1996) In: Inverse Shape and Boundary Condition Problems and Optimization in Heat Conduction in Advances in Numerical Heat Transfer, W. J. Minkowycz and Sparrow, E. M. (Eds.). Taylor and Francis, New York, 1, pp.3 81-426. Dantas L.B., Orlande, H.R.B., Cotta, R. M., Souza, R. and Lobo, P.D.C. (1999) Inverse Analysis in Moist Capillary Porous Media, 15^^ Brazilian Congress of Mechanical Engineering, Sao Paulo. Levenberg, K. (1944) A Method for the Solution of Certain Non-linear Problems in Least Squares, Quart. Appl. Math, 2, pp. 164-168. Marquardt, D.W. (1963) An Algorithm for Least Squares Estimation of Nonlinear Parameters, J. Soc. Ind. Appl. Math, 11, pp.431-441. Emery, A.F. and Nenarokomov, V. (1998) Optimal experiment design, Meas. Sci. Technology, 9, pp. 864-876. IMSL Library, MATH/LIB, Houston, Texas, (1987).

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

63

DETERMINATION OF HEAT TRANSFER COEFFICIENT MAPS USING AN INVERSE BEM ALGORITHM E. DIVO, A. J. KASSAB, J. S. KAPAT, and J. TAPLEY Mechanical, Materials, and Aerospace Engineering Department University of Central Florida, Oriando, Florida 32816-2450 ABSTRACT We present an algorithm designed to extract accurate multi-dimensional heat transfer coefficient maps from surface temperature measurements which may be provided by non-invasive techniques such as infrared tomography or thermochromic liquid crystals. The approach is based on a boundary element method (BEM) which uses either transient or steady temperature inputs and incorporates a smoothing technique borrowed from inverse problem methods to provide robustness to input temperature uncertainties. As it requires only surface mesh, the method is flexible and easy to implement for examination of arbitrarily complex geometries encountered in engineering practice. KEYWORDS Heat transfer coefficient, thermochromic liquid crystals, inverse problems INTRODUCTION Determination of convective heat transfer coefficients (h) is important in the design and analysis of thermal systems exposed to convective heat transfer. Typically transient temperature measurements provided by experiments have been used along with one-dimensional semi-infinite models to extract film coefficient values. This approach is lacking in certain practical applications where heat transfer is actually two- or three-dimensional. In this paper, we present a general purpose algorithm based on a Boundary Element Method (BEM) solver and inverse problem methods to retrieve multi-dimensional heat fiuxes, and consequently heat transfer coefficients distributions using either transient or steady temperature inputs. The inverse method can be used with broad-band or narrow-band thermochromic crystal or laser induced fiuorescence thermal imaging techniques and incorporates a technique which provides robustness to input temperature uncertainties. Furthermore, the proposed method is veryflexibleand can be readily implemented for arbitrarily complex geometries as it requires only a surface mesh. A common approach for measurements of convective heat transfer coefficients relies on thermochromic liquid crystal (TLC) thermography, Hippensteele et al. [1] and Camci et al [2], thermographic phosphor (TOP) thermography, Bizzak and Chyu [3], and temperature-sensitive paints (TSP). Temperature levels are indicated by: (1) color (for TLC), (2) intensity of fluorescent emissions (for TGP and TSP). These methods can provide a broad-band response (detect

64

E. Divo et al

temperatures over a range of nearly 30°C) or narrow-band response (typically single color response to a given temperature). Diagnostic applications of TLC, TGP, or TSP-based methods usually rely on transient measurements where the temperature is measured twice by detection of color or fluorescence intensity, in quick succession at two different times over such a short interval from the beginning of the transient process that heat conduction into the surface is assumed as one-dimensional. Assuming the fluid mean temperature and h are invariant over the time interval, one can write e{t) = 1 - exp{hH/kpc)erfc[h(t/kpc)^

(1)

where 6{t)= [T^^^f^,{t) - Ti]/[Tfiuid - Ti], Ti is the initial temperature of a semi-infinite medium impulsively subjected to a convective boundary condition with the surroundings at temperature Tfiuid, and k and a are thermophysical properties. There are only two unknowns, h and T/Md in the above equation, and hence, by measuring Tsurface(t) at two different instants, one can calculate both unknowns at every point on the surface. A transient experiment is thus used to calculate the steady-state heat transfer coefficient h (considered constant in the above equation). There are more complicated variations of these methods where assumptions of absolute invariance of h and Tfi^id can be relaxed. Nevertheless, all of these methods for measurement of surface distributions of h involve transient experiments and assumption of one-dimensional conduction into the substrate of the heat transfer surface. As a result, there are three serious limitations in application of TLC, TGP, or TSP: (1) in time-dependent flows, heat transfer coefficients are not steady and for these cases, the existing practice of using transient experiment to calculate steadystate heat transfer coefficient cannot be used; (2) sudden start-up may be difficult, or even impossible, to achieve; (3) if test surface has comers or sharp curvature, then spatial resolution is limited near those points by the speed of capturing the video frames of the test surface because of the assumption of one-dimensional conduction into the test plate. It is precisely due to the 1-D approximation that Plexiglass is used as the common medium in experiments. However, Plexiglas is a very poor conductor, and its use in experimental models of components that have a higher conductivity inherently neglects multi-dimensional heat transfer effects. These features are thus avoided by design of experiments, while they are actually an important part of the heat transfer phenomenon under investigation. Although severity of the above limitations depends on actual test configuration, experimental procedure becomes more complicated due to these limitations. On the other hand, all these shortcomings can be avoided if thermographic methods can be used in experimental procedures which avoid reliance on the assumption of 1-D conduction. The algorithm we discuss in this paper provides such an alternative. Two types of problems are identified and associated with specific information provided by thermometry: (1) Type-I: determine h given time histories of temperature at various locations on a convective surface-these data can be provided by broad-band TLC, laser fluorescence, or infrared thermographic methods, and (2) Type-II: determine h given steady-state temperature at various locations on a convective surface provided say by broad-band TLC or infrared thermography. A third problem can be identified as Type-IIL determine h given information with regards to the time it takes to reach a specific temperature at each of various locations on a convective surface, and these data are typically provided by narrow-band TLC thermography. This is by definition an ill-posed inverse problem. However, we do not consider Type III in this paper, although an algorithm is under development for its solution. These three problems cover

Determination of heat transfer coefficient maps

65

the range of inputs provided by experimentalists, and we present a robust tool which is based on inverse methods to provide h values given characteristic inputs of type I and II problems. Inverse methods, Beck et al [4] and Alifanov [5] are adopted in a BEM-based algorithm. The BEM offers several advantages in solving the problem under consideration: (1) the BEM only requires a surface mesh, (2) boundary temperatures and fluxes appear as unknowns in the BEM. Pasquetti and LeNiliot[6] formulate an inverse BEM-based approach to retrieve angular distributions of h from numerically simulated steady-state temperature measurements from a cylinder surface, and they adopt a second order regularization scheme. Hsieh and Farid[7] use an inverse BEM approach to retrieve the angular variation of h over a rough heated horizontal cylinder in an experiment where steady-state surface temperatures are measured non-intrusively by infrared scanning, however, they do not introduce regularization. Martin and Dulikravich [8] use a steady-state BEM-based approach in a numerical study in which h values are retrieved from simulated steady-state temperature measurements, and the authors rely on Singular Value Decomposition (SVD) to stabilize the retrieval process when noisy input data are simulated. We present an algorithm to solve the problem of/i-retrieval using transient input temperatures [9,10] and show how to use this algorithm with steady temperature inputs. BEM-BASED INVERSE ALGORITHM The BEM is used to determine the temperature field in this study. Here, the governing difRision equation for the temperature T(x,t) is first converted to a boundary integral equation (BIE), taking the initial condition to be either zero or a constant, adopting a convolution marching scheme dividing the time domain of interest is divided into F constant time elements, the BIE is then approximated as, Brebbia etal [11] pcC.T[

= f±{T^H^l-
(2)

where the density, p, and the specific heat, c, are taken as constant, Ci is a function of the local boundary geometry when the collocation point ^^ lies on the boundary, and the influence coefficients H^-" and G^f are routinely evaluated using quadratures. The discretized BIE relates the temperature at any collocation point ^i (either on the boundary or in the domain) and any collocation time tp with boundary temperatures and heat fluxes at current and previous time levels. Collocating ^^ at the N boundary nodes, and setting the current time level to tp, leads to

[HY^{TY

- [GY^{qY = J2 {[Gf'UV - [HfiTV)

(3)

where the modified influence coefficient matrix [H]^^ = [H]^^ - ^pc[I], with [/] denoting the identity matrix. The right-hand-side contains convolution information and can be collapsed into a F-l

vector {c}^ = X) {[G]^P{qy -

[H]^P{T}P).

The left-hand-side is re-ordered according to boundary

conditions to obtain a simple algebraic form, [A]{x}^ = {b}^, which is solved at every time level F by direct or iterative methods to obtain unknown boundary values.

66

E. Divo et al.

In the case of broad-band inputs, heat transfer coefficient retrieval is actually a direct problem since the time histories of temperature measurements are used as imposed first kind boundary conditions at convective boundaries. However, uncertainties in temperature measurements, even after filtering of the input data, translate into large deviations in computed heat fluxes and consequently erroneous h values. As a remedy, re-formulation as an inverse problem with regularization is adopted with a least-squarefijnctional,Z^, defined as

z^

M /ff,F

FY

M f

F

7fF\2

where M is the number of temperature measuring points along convective boundaries, T^ are measured temperatures, Tf are computed temperatures given current estimates of h distribution, and AT^ is a normalizing temperature difference equal to the maximum range of temperature on the convective boundaries at the current time level. The term /? is a regularization/tuning parameter, qf are computed fluxes given current estimates of h distribution, 'q f is provided from a mean heat flux distribution obtained by a least-squares fit through currently-computed fluxes [9,10], and A^^ is a normalizing heat flux magnitude equal to the maximum range of flux values at the current time level. The addition of the second term in Eqn. (4), allows minimization to be performed, and /? plays an important role as it controls solution smoothing while suppressing large deviations from measured temperatures. The term^ f, is the value of the least-squares fit of current flux distribution over the convective boundary evaluated at the measuring point i, and it can be expressed as a linear combination of computed nodalfluxesas, g[ = A,q^

(5)

The approximation matrix Aik is derived through an orthogonal polynomial expansion with the polynomials generated by the Gram-Schmidt process. The regularization term drives oscillatory values of currently computed fluxes at measuring points, qf, towards their smoothed values provided by the least-squares fit evaluated at the point "i". Evidently, the choice of the value for /3 is problem-dependent: the higher the input error, the more damping will be required. Furthermore, /3 is also a function of geometry and boundary conditions. A linear relationship between the standard deviation of input errors and /? is often employed in practice. The standard deviation is directly related to input error as: a = lemox 1/2.576 for a level of confidence of 99%, where Emax is the maximum expected error in measurement. Introducing 'q f into the functional Z^, defining the matrix Bik = 6ik - Aik, with 6ik as the Kronecker delta, minimization is accomplished by differentiating Z with respect to the heat fluxes qf at each of the convective nodes and by setting the result equal to zero results in [9,10] {X,,X,k + m^/? Cjk ) q^ = X,, ( r f - d[)

(6)

This is solved at each time F for the boundary heat fluxes at convective boundaries. The sensitivity coefficients Xij = dTf/dqf, are computed directly using the BEM influence, m^ = (AT^/Aq^) , Cjk = BikBij, Bik = kk - ^ik and the previous time history of the boundary conditions, {c}^ is used to define df. Once the above equation is solved at time level F for the boundary fluxes q^ . Newton's cooling law is invoked next to obtain the h values as

Determination

of heat transfer coefficient maps

h = q/{Tyjall - Tref)

61

0)

where q is the surface heat flux obtained from Eqn. (6), T^aii is the surface temperature obtained at time F from the BEM solution, and Tref is the appropriate problem dependent reference temperature. For instance, if the flow-field is external, then T^ef is the temperature of the incoming flow often denoted by Too • Details of this formulation for Type-I problems and aspects of numerical implementation is provided in [9,10], where several numerical examples are presented. In addition, results from a transient experiment validate the procedure and show that when a conductive material is used in experiments, unlike the Plexiglas models commonly encountered, two dimensional effects are indeed pronounced and not captured by traditional methods.

DEVELOPMENT OF TYPE II SOLUTION STRATEGY A steady-state method to obtain surface heat transfer coefficient distributions eliminates drawbacks of the transient approach. In the transient method, 1-D conduction into the test object is assumed over the duration of the experiment. However, this assumption is inappropriate near sharp corners and edges. Second, the transient method requires impulsive start of the flow system, which can be difficult in many cases and hence will lead to experimental error. In the steady state method, surface temperature distributions are measured with TLC on all those sides of the test object where boundary conditions are not known a priori. This thus motivates the solution of Type-II problems. The method of Type-I problems can be applied to the steady-state case of Type-II problems with one of two possible modifications. Here, the governing equation for heat conduction is the Laplace equation for the temperature, in the case of linear heat conduction, or the Laplace equation for the Kirchhoff transform, in the case where the variation of the thermal conductivity with temperature is taken into account. In either case, BEM equations relating surface temperatures to surface heat fluxes are, [H]-{T} = [Gr{q}

(8)

where [HY^ and [GY^ are influence matrices whose individual elements are the steady-state influence coefficients which are routinely computed numerically in the BEM. Comparison of this equation with the transient equation reveals that, at a given time level F, both have the same form with the above missing the right-hand vector {c}^. Consequently, in the steady-state case the above developments hold, with the steady-state matrices [HY^ and [GY^ replacing the transient matrices [HY^ and [G]^^, and the vector {d} set to zero in the final equation for the convective boundary fluxes, Eqn. (6). This requires coding of the steady state influence matrices and verification of the codes. An alternative, is to run the transient solution to steady state under an assumed initial condition. This method requires no modification of the code. Furthermore, as the BEM equations are unconditionally stable, very large time steps can be taken to reach steady state. Thus, the same code can be used to solve the problem of heat transfer coefficient retrieval using either steady or unsteady temperatures as inputs. This alternative approach is taken here.

68

E. Divo et al

RESULTS FROM SIMULATION The steady-state TLC method discussed above will be implemented in a laboratory experiment which simulates tip flow over a cooled vane in a turbomachine. Here a stainless steel block, see Figure 1, with one dimension relatively large compared to the other two, cooled by three internal cooling channels flowing in chilled water was placed in a windtunnel in which the temperature of the air stream can be raised above ambient. The coolant flows in a closed loop along the length of the block. The two end surfaces of the block in the length-wise direction are insulated as well as one of the other four surfaces. The heat transfer boundary conditions within the cooling channels are known as the bulk water temperature as well as water flow rates are to be measured. Furthermore, the flow rate will be adjusted to ensure laminar flow in the cooling channels. The It is 101.6 cm tall, 63.5 cm wide, and steel block has a known conductivity of A: = 14.6W/mK. the cooling holes are square and measure 0.3175 cm on a side. The tip gap H = bcm,. Temperature distributions over the three non-adiabatic external surfaces are obtained with a wide bandwidth TLC. In order to reduce experimental uncertainties, point-by-point in-situ calibration of TLC will be used to correlate hue with temperature. The calculated temperature distributions, obtained from measured hue distributions and the hue-temperature calibration, are provided as inputs to retrieve surface fluxes (and therefore heat transfer coefficients) along those boundaries where the temperature is measured. In order to first ensure the algorithm proposed above provides accurate results, a numerical simulation was undertaken. Here the computational fluid dynamics code FLUENT was used to obtain a conjugate solution for the configuration shown in Figure 1. The inlet air was taken as 53°C, the inlet velocity of Viniet = 7 m/s corresponds to tip gap Reynolds number, RCDJJ, of57,000 based on the hydraulic diameter DH = 2H. The boundary condition at the cooling holes is set to be convective in the numerical simulation. The film coefficient is evaluated as hcooiant = S63W/m^K, based on a laminar flow and assuming a constant temperature at the cooling hole walls. The bulk temperature in the cooling hole is taken as Tfouik = 293.3K. These inputs are representative of the values expected to be found in the experiment. There were 21,000 cells used for the solid and 40,000 for the fluid, and the standard k-e model was used to model turbulence. Once the conjugate solution was obtained, the wall temperatures at the convective surfaces, see Tbem in Figure 1 was used as input to the /i-retrieval algorithm described above. A boundary element model of the block was generated using equal-length constant elements with 80 elements on right and lefl walls, 50 elements on top and bottom walls, 8 elements on each cooling hole. 325

^ 1a '"' 1°

-w L-H

Air

324 : 323

Tdbem TIbem

1

322

Sted block with internal holes

321 320 319 318 : 317 316 315

^^ , 20

_ , , 40

60

80

100

,,,, 120

140

^ 160

180

200

^,,,, 220

240

(a) (b) Fig. 1. (a) test configuration for simulation and (b) input surface temperature distribution.

260

69

Determination of heat transfer coefficient maps

The right wall was modeled as adiabatic, and the cooling holes were modeled with convective boundary conditions. The initial condition was set as T = 0, and two time steps, each of 1500s, were used to reach steady state. The obtained isotherms are plotted in Figure 2. Examination and comparison of the isotherms with those predicted by the FLUENT conjugate solver reveal excellent agreement. This solution was obtained with /3 = 0.02, and the computed surface temperature distribution is dispalyed as Tibem in Figure 1. The computed wall fluxes are plotted in Figure 3 (a) and the corresponding predicted wall temperatures at the adiabatic wall are plotted in Figure 3 (b). Good agreement is found between the two predictions as well.

'15-

[n \^

!

1

"

/ •^ /*

:-g''

(a) (b) Fig. 2 Isotherms obtained by (a) BEM using FLUENT conjugate solution computed surface temperatures as inputs compared to (b) isotherms predicted by the FLUENT conjugate solution.

(a) (b) Fig. 3 Comparison of (a) CFD conjugate solution computed wall fluxes (solid line) with inverse method computed wall fluxes (dashed line) with /5 = 0.02 and (b) CFD conjugate solver computed temperature at the adiabatic wall and BEM prediction of the temperature at that wall.

70

E. Divo et al

CONCLUSIONS An algorithm is presented for accurate determination of multi-dimensional heat transfer coefficient maps from surface temperature measurements which may be provided by non-invasive techniques. The approach is based on a boundary element method (BEM) which uses either transient or steady temperature inputs and incorporates a smoothing technique borrowed from inverse problem methods to provide robustness to input temperature uncertainties. In this study, a steady state application is considered in which thermochromic liquid crystals are used for thermographic measurements. Results are presented from a numerical simulation of a forthcoming experiment validate the proposed approach. Results from the experiment will be reported elsewhere. REFERENCES 1. Hippensteele, S.A., Russel, L.M., and Stepka, F.S., (1983), Evaluation of a Method for Heat Transfer Measurement and Visualization Using a Composite of a Heater and Liquid Crystals, AMSEJ. of Heat Transfer, Vol. 105, pp. 184-189. 2. Camci, C. and Kim, K., (1992), A New Hue Capturing Technique for the Quantitative Interpretation of Liquid Crystal Images Used in Convective Heat Transfer Studies, ASMEJ. of Turbomachinery, Vol. 114, pp. 765-775. 3. Bizzak, D.J. and Chyu, M.K., (1995), Use of laser-inducedfluorescencethermal imaging system for local jet impingement heat transfer measurements. Int. J. of Heat and Mass Transfer, Vol. 38, No. 2, pp. 267-274. 4. Alifanov, O., (1994), Inverse Heat Transfer Problems, Springer-Verlag, New York. 5. Beck, J. v., Blackwell, B., and St. Clair, C.R., (1985), Inverse Heat Conduction: Ill-Posed Problems, John Wiley and Sons, New York. 6. Maillet, D., Degiovanni, A., and Pasquetti, R., (1991), Inverse Heat Conduction Applied to the Measurements of Heat Transfer Coefficient on a Cylinder: Comparison Between an Analytical and a Boundary Element Technique, ASMEJ. of Heat Transfer, Vol. 113, pp.549-557, 1991. 7. Farid, M.S. and Hsieh, C.K., (1992), Measurement of the free convection heat transfer coefficient for a rough horizontal nonisothermal cylinder in ambient air by infrared scanning, ASMEJ. of Heat Transfer, Vol. 114, pp. 1054-1056. 8. Martin, T.J. and Dulikravich, G.S., (1998), Inverse Determination of Steady Heat Convection Coefficient Distributions, ASMEJ of Heat Transfer, Vol. 120, pp. 328-334. 9. Kassab, A.J., Divo, E., and Chyu, M., (1999), A BEM-Based Inverse Algorithm to Retrieve Multi-dimensional Heat Transfer Coefficients from Transient Temperature Measurements, In: BETECH99, Proc. 13th International Boundary Element Technology Conference, Las Vegas, June 8-10, 1999, Chen, C.S., Brebbia, C.A., and Pepper, D, (eds.). Computational Mechanics, Billerica, pp. 65-74. 10. Divo, E., Kassab, A.J., Kapat, J.S., and Chyu, M.K., (1999), Retrieval of Multidimensional Heat Transfer coefficient Distributions Using an Inverse BEM-Based Regularized Algorithm: numerical and experimental results, ASMEPaper HTD-Vol 364-1, pp. 235-244. 11. Brebbia, CA., Telles, J.C.F., and Wrobel, L., (1985), Boundary Element Techniques in Engineering: theory and application in engineering. Springer-Verlag, New York.

INVERSE PROBLEMS IN ENGESTEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

71

TRACKING OF PHASE CHANGE FRONT FOR CONTINUOUS CASTING - INVERSE BEM SOLUTION I. NOWAK^ A.J. NOWAK^ and L.C. WROBEL* t Institute of Mathematics^ ^ Institute of Thermal Technology, Technical University of Silesia, 4-4-101 Gliwice, Konarskiego 22, Poland Brunei University, Dept. of Mechanical Engineering, Uxbridge, Middlesex, UBS SPH, United Kingdom ^ corresponding author: [email protected] ABSTRACT This paper discusses 2-D numerical solution of the inverse geometrical thermal problem consisting of an estimation of the position of the phase change front in a continuous casting process. Proposed algorithms utilize temperature measurements inside the solid phase and the sensitivity coefficients. Solution procedure also involves the application of the Boundary Element Method which simplifies modifications of numerical discretization when position of the front is updated within the iterative process. Intensive computations showed that the phase change front should be modelled by the boundary elements possessing C^ continuity. This is why the quadratic boundary elements with special placement of the midnodes are used in this work. In order to demonstrate main features of the developed algorithms test case from continuous casting of copper is included. KEYWORDS Inverse geometry problem, sensitivity coefficients, BEM solution of continuous casting INTRODUCTION The continuous casting process of metals and alloys is nowadays frequently utilized in the metallurgical industry, and in materials engineering in general. Typically, the liquid material flows into the mould (crystallizer) having the walls cooled by flowing water. The solidifying ingot is pulled out by the withdrawal rolls. The side surface of the ingot below the mould is very intensively cooled by water flowing out of the mould and sprayed over the surface. The transport phenomena encountered in such solid-liquid phase change systems are of the great importance for fundamental research as well as for engineering and technological practice. Mathematical models of encountered phenomena can be formulated as direct thermal problems {i.e. finding the results from the known cause) and/or inverse

72

L Nowak, A J. Nowak and LC. Wrobel

problems {i.e. discovering the cause from a known result). In this work results of ongoing research consisting of a 2-D solution of the inverse geometry problem in continuous casting are presented. Solution procedures involve the application of the Boundary Element Method to estimate of the location of the phase change front making use of temperature measurements inside solid phase. These measurements can be obtained by immersing some thermocouples into the melt and letting them travel with the solidified material (until they are damaged). From natural relationship between time and locations of nodes in continuous casting process, even limiting number of thermocouples recording temperatures can provide substantial amount of information regarded as internal temperature responses. End nodes of quadratic boundary elements used to approximate of the phase change front are searched iteratively considering solid phase only. Iteration loop starts by assuming a trial position of the front being subject to the melting temperature. Now the temperature field can easily be calculated and squares of the temperature differences of values calculated at sensor locations and measurements are used to define an objective function. It is then minimized to correct position of the phase change front. Optimization process is based on the concept of the sensitivity coefficients which are determined in the work again by the Boundary Element Method. It is shown in the paper that fairly standard BEM computer code requires only appropriate set of boundary conditions to performed required sensitivity analysis. In order to demonstrate main features of proposed approaches and developed algorithms some test cases from continuous casting of copper have been worked out. Advantages as well as problems faced during the analysis are reported and some possible improvements of the methodology are suggested. PROBLEM FORMULATION The 2-D inverse geometrical problem discussed in this paper refers to continuous casting process shown schematically in Fig.l. Following equations define the steady-state diffusionconvection heat transfer in a solid phase marked in this figure as ABCDO: • governing equation consisting of a partial differential operator acting on temperature r(r) V2T(r)-ir;.f^=0 a

(1)

OX

where Vx stands for the casting velocity and a is the thermal diffusivity • known boundary conditions defining heat transfer processes along the side of the ingot ^(r) = - A | ^ = qBc{x), q(v) = h{T T{T)

-Ts),

= T,(r),

q{r) = 0,

r G TBC re

Ten

r G Tno

(2)

r G FQA

where qBc{oo) is any but known distribution of heat flux along the side of the ingot, the heat transfer coefficient is marked by h and Ts{r) stands for the temperature of the lowest considered face

73

Tracking ofphase change front for continuous casting

TTTTZ

_j

1

y ^

mould

^

water

XAT"^ spray

'V

0

|D

Figure 1: Scheme of the continuous casting process and the domain under consideration. • constant temperature of the phase change process taking place along the front. Its position is obviously unknown T(r)

^p/l5

r G rAB

(3)

where T(r) represents the melting temperature. To make the mathematical description of the problem complete, some temperatures, say [/j, measured at selected locations are provided. If these sensors are placed inside the body the values IJi are referred to as internal temperature responses. The objective is to estimate unknown x-coordinates of nodes along the phase change front (collected in vector of design variables Y) and calculate the temperature field within the body under consideration. The ill-posed nature of all inverse problems requires that the number of temperature sensors should be appropriate to make these problems overdetermined. Thus, in general, inverse analysis leads to the optimisation procedures in which the objective function A consisting the following least squares is evaluated A

\5f W-i (T,cal

(T,cal

u)

mm

(4)

where vector Tea/ contains temperatures calculated at temperature sensor locations, U stands for the vector of temperature measurements and superscript T denotes transpose matrices. Symbol W represents the covariance matrix of measurements. Thus, contribution of more accurate measured data is stronger than those which have been obtained with lower accuracy. Additional terms intended to improve stability and accuracy of the solution can also be added, e.g. (Teai - U ) ^ W - 1 (Teai - U) + (Y

-

Y)^

Wy'

(Y

-

Y)^

-^ min

(5)

74

L Nowak, A.J. Nowak and L C. Wrobel

where known prior estimates are collected in vector Y, and W y stands for the covariance matrix of prior estimates. It should be remembered that coefficients of matrix W y have to be large enough to catch the minimum (these coefficients tend to infinity if prior estimates are not known). It was found that the additional term in the objective function in many cases plays essential role in inverse analysis. SOLUTION PROCEDURES The estimation process of the unknown position of the phase change front involves iterative loop consisting of two major steps: • position of the front is assumed and the temperature field Tcai inside the body is calculated. It should be noted that this stage of analysis is equivalent to solving the direct problem • the relevant objective function (4) or (5) is minimized [2, 3, 4] and position of the phase change front is updated. The minimization procedure in this work utilises the concepts of sensitivity coefficients Zji, defined as derivatives of the measured quantity, i.e., temperature Tj at a certain location, with respect to an assumed and then identified design variable Yi 3T

(6)

Z„ = §

It is quite easy to show that making use of the sensitivity coefficient matrix Z the following system of equations is obtained to update location of the phase change front (Z^ W " ^ Z + W Y ^ ) Y

=

Z^ W-^

{TF W - ^ Z ) Y *

+

Wy^ Y

(U

- T*) + (7)

As design variables x-coordinates of end nodes along the front, noted as x^, have been selected. The iterative loop is repeated until the convergence criterion (limit for the increment of coordinates) is satisfied. Determining sensitivity coefficients Sensitivity coefficients can easily be calculated solving the boundary problem obtained by differentiation of system (l)-f(3), i.e., • governing equation

1

ri7

V ' Z { T ) - - V , ^ = 0 a OX

(8)

• boundary conditions along all faces but the phase change front qz{r) = 0

r G TBC

qzir) = hZ

T e TCD

Z(r)

=

0

qz{r) - 0,

r ETDO

r G FQA

where qz is the quantity equivalent to the heat flux

(9)

75

Tracking of phase change front for continuous casting

• boundary condition along the phase change front TAB- Differentiation of eqn (3) gives &T_ dT^dx_ dT_dy_ (10) dxi dx dxi dy dxi which immediately produces Z

-

dT dx _ dT^dy_ dx dxi dy dxi

(11)

Introducing now the Fourier's law into account the above equation can be written as follows 1 ( dx dy (12) X \ "^ dxi dxi. where q^ and qy are x- and y-components of the heat flux vector. Taking into account relationship between global and local coordinate systems (see Fig.3) these components can be expressed as

^olid^ Figure 2: Geometrical relations on the phase change front.

(

Qx =

-Qn cos(a) - qr c o s ( | + a)

Qy =

-Qn sin(a) + qr s i n ( | + a)

(13)

where normal and tangential components of the heat flux vector are expressed by qn and qr, respectively, whereas [cos(a), sin(a)] are components of the normalized outward normal vector. Thus, finally the boundary condition along the phase change front takes the following form i f dx Z = - - I [qn cos(a) + qr sm{a)]

\- [qn sin(a) - qr cos(a)]

dy —

(14)

Further particularization of this expression requires definition of the front geometry and will be discussed within next subsections.

76

/. Nowak, A.J. NowakandL.C. Wrobel

BEM solutions Since both boundary problems for the temperature and sensitivity coefficient are very much the same, both fields are determined using the same numerical method. In this work the boundary element method [5] has been applied. This method simplifies considerably numerical discreatization during iterative process of updating of the phase change front location. Additionally, the temperature sensors can take any position, independently on the discretization. The method results in the following two systems of equations HT

=

Gq

(15)

HZ

=

Gqz

(16)

where the vectors q and <\z contain heat fluxes and derivatives of sensitivity coefficients, respectively. Symbols H and G stand for the BEM influence matrices. The coefficients of these matrices result from integration of the fundamental solution u* (and its normal derivative q*) along the boundary elements only. Depending on the dimension of the problem, the fundamental solution is expressed by the following formulae 1

f

VxTa:

Ko ( ^ ^ 2a

2a u

I 2-D problems (17)

= < 47rrA exp

3-D problems

2a

where KQ stands for the Bessel function of the second kind and zero order, r is the distance between source and field point, with its component along the x-axis denoted by r^. It should be noted at this stage that in order to avoid problems with uniqueness of the outward normal along piece-wise line, the phase change front (TAB in Fig.l) should be modelled by a smooth curve. This requires utilization of at least quadratic elements resulting in C^ continuity [1]. Considering a quadratic element whose limiting nodes 1 and 2 (in local numbering) have co-ordinates denoted as Xi, ?/i, X2 and 2/2, respectively, one can determine the location of the midnode 3 from the relationships (although some conditions have to be fulfilled [1]) X3

=

ys = where ^2

= =

1 2 (51 - 52;

(ai -\- a2 - hsi

1 2 ^^^^^ + 251X3 -

- 1 . 5 t / i - 0.5 ?/2 0.5 2/1 + 1.5 7/2

^1

b2

- 62 S2) (18)

ai) — 1.5xi — 0.5^2 0.5xi -h 1.5x2

and 5i and 52 are slopes of the tangents to the boundary element at nodes 1 and 2. Derivatives occurring in equation (I4) Once type of the elements along the phase change front are decided derivatives appearing in equation (14) can be discussed in more details. Let's consider a quadratic element spanned on nodes 1, 2 and 3 (local numbering). Additionally, let's consider a design variable Xi being the x-coordinate of an extreme node i. In Fig.3 both nodes 1 and i are identical and further expressions refer to such configuration.

Tracking of phase change front for continuous casting

11

/+2. /+1 Figure 3: Global and local numbering of nodes along quadratic element. For quadratic boundary element both %- and y-coordinates as well as the temperature vary with a position according to the following relationships Ai) = M ( 0 ^i + ^ 2 ( 0 ^t+2 + Af3(0 a^i+i

(19)

J/(?) = M ( 0 Vi + iV2(0 2/i+2 + A^3(e) J/i+1

(20)

T{i) = NiiO Ti + N2{0 Ti+2 + NsiO Ti+x

(21)

where ^ stands for the local coordinate. Indices z, i + 1 and i + 2 refer to the global numbering whereas quadratic shape functions read NiiO N2{0

= 0.5^(^-1) = 0.5^(e + l)

Nzio =

(22)

i-e

Baring in mind that midnode ( i + l in global numbering or 3 in a local one) is positioned to guarantee C^ continuity, differentiation of eqns (19)-f-(21) results in the following two key expressions

(24)

dxi dxi Introducing into above equations expressions (18) one immediately arrives at the following formulae dxi.+1 dxi

1

dSi^2

fyi+2 - V i - h S i -

2{Si - 5i+2)

dxi

^i ~ ^i-\-2

Si- Si+2

axi \ % 'i+l

(6.+ 2 . . , 0 g

dxi

J

b2Si^2

2

+ . ( 2 ^ - 1 . 5

(25) (26)

Obviously, the quantities Si and 5^+2 require averaging of slopes of appropriate secants, I.e.,

Si

dsi dxi

--=

0.5

= 0.5

yi-2 - Vi

Vi -

Xi—2 ~ Xi

Xi —

yi-2 - Vi {Xi^2 - XiY

yi+2 Xi^2_

yi - I li-\-2 {Xi - X i+2)\

(27) (28)

78

I. Nowak, A J. Nowak and L.C. Wrobel

Si+2 dSi+2

dxi

0.5 •0.5

Vi -

yi+2

yi+2 - Ui+A Xi+2 ~

Vi -

(29)

Xi+4

yi+2

(30)

[Xi — Xi+2)

Under recursive changing of subscripts very similar relationships can be developed for element spanned on nodes z + 2, z + 3, z + 4 and all other. In this way the complete set of the boundary conditions for sensitivity coefficients can be obtained. NUMERICAL EXAMPLE In order to demonstrate the main features of the proposed methodology a test case from the continuous casting of the copper was considered. The results obtained were compared to reference values produced by direct BEM modelling, cf. [1]. In these calculations the following heat fluxes QBC = 4-10^ W/m^ and qcn = 4000 {T — Tg) W/m^ were assumed. All results were obtained for the melting temperature T^h = 1083°C whereas the end temperature Tg was assumed to be 50°C. The phase change front has been discretized into 25 quadratic boundary elements. The temperature measurements inside the ingot were simulated numerically by adding random errors to selected internal temperatures of reference field. The assumed measurement errors did not exceed 2% (namely, they were considered as 0.1%, 0.2%, 0.5%, 1% or 2%). In the calculation, results of which are shown in Fig.4 all extreme boundary nodes but one have been fixed at their exact location. The position of the node no 15 was estimated iteratively utilizing 5 internal error free temperature measurements and objective function (4). Although fifteen iterations have been made only 6 ^ 7 of them were required to find the practically exact position of the node. 1100 1080 exact position I L^ calc. position I |

T r exact temp, calc. temp.

1020 4 8 12 iteration number

16

n ' \ ^ \ 4 8 12 iteration number

^

I 16

Figure 4: Estimation of the location and temperature of one node on the phase change front. The influence of the measurement errors in that test case is shown in Fig.5. To obtain satisfying results objective function (5) was utilized. Certainly, the bigger measurement errors are the worse estimation accuracy of the position of the node is achieved. Neverthe-

79

Tracking of phase change front for continuous casting

less, these inaccuracies are always of order of measurement errors. It should also be noted that iterative process converges fairly well. 0.22

1100

: 0.216 " + • • X ^ m

0.212 ^0.208 0.204 0

1 ' r 10 20 30 iteration number

exact locatior^ err. 2% err. 1% err. 0.5% err. 0.2% err. 0.1% error free

40

10 20 30 iteration number

Figure 5: Estimation of the location and temperature of one node on the phase change front. The next figure shows the same test case but this time position of the whole phase change front has been estimated. Twenty five extreme nodes were considered as design variables. The remaining nodes {i.e. midnodes) were positioned according to formulae (18). A parabola was assumed as an initial guess (i.e., the position of the front) to initiate the iterative loop. Graph in Fig.6 compares the accurate position of the front with the initial guess and numerical results obtained after the first and tenth iterations. 0.02

y co-ordinate 0.04 0.06 0.08

0.1

0

0.02

y co-ordinate 0.04 0.06 0.08

0.1

exact position #" A -—5 - starting position H - 1 iteration - * - 20 iteration

Figure 6: Estimation of the location and temperature of whole phase change front. The careful analysis of the results brought to light that in this case it was impossible to locate all midnodes according to formulae (18) (certain conditions [1] could not be fulfilled). Therefore, this test case suggests extending the developed algorithms for higher order boundary elements, i.e., for Bezier splines composed of the cubic boundary elements.

80

L Nowak, A.J. Nowak and L. C. Wrobel

CONCLUSIONS Analysis performed in this work showed that the Boundary Element Method is a convenient and effective numerical tool for solving inverse geometry thermal problems. Only boundary of the solid phase has to be discretized which is particularly desirable during iterative process of updating of the location of the phase change front. It is also important to note that because of lack of any domain cells the temperature sensors can take any position. The paper demonstrates also applicability of the sensitivity analysis to the phase change heat transfer processes. Intensive calculations showed that the C^ continuity in a geometrical representation of the phase change front, the number of sensors and the measurement errors have the most significant influence on the accuracy of the results. The procedure described in the paper is currently being extended for cubic boundary elements generated by Bezier splines. Such attempt not only guarantees unconditioned C^ continuity but also reduces the number of design variables. ACKNOWLEDGEMENTS The financial assistance of the National Committee for Scientific Research, Poland, within the grants no 8 TlOB 050 15 and SPUB-2/RIE-6/99 coordinated by European Union within the action COST P3 is gratefully acknowledged herewith.

REFERENCES [1] Fie A., Nowak A.J., Bialecki R.A.: Heat transfer analysis of the continuous casting problem by the front tracking method and the BEM, Engineering Analysis with Boundary Elements^ in press. [2] Kurpisz K., Nowak A.J.: Inverse Thermal Problems, Comp. Mech. Publications, Southampton 1995 [3] Nowak A.J.: BEM Approach to Inverse Thermal Problems, Chapter 10, in Ingham D.B., Wrobel L.C.: Boundary Integral Formulations for Inverse Analysis, Comp. Mech. Publications, Southampton 1997 [4] Beck J.v., Blackwell B., Inverse Problem, in Minkowycz W.J., Sparrow E.M., Schneider G.E. and Fletcher R.H., (eds.): Handbook of Numerical Heat Transfer, Wiley Intersc, New York 1988. [5] Brebbia C.A., Telles J.C.F., Wrobel L.C., Boundary Element Techniques - Theory and Applications in Engineering. Springer-Verlag, 1984.

INVERSE PROBLEMS E^ ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

81

APPLICATION OF DRBEM AND ITERATIVE REGULARIZATION TO INVERSE HEAT CONDUCTION MASATAKA TANAKA and KRISHNA M. SINGH Department of Mechanical Systems Engineering, Faculty of Engineering Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan. ABSTRACT This paper presents an application of the dual reciprocity boundary element method to inverse heat conduction problems. To deal with the ill-posed nature of the problem, iterative regularization methods of conjugate gradient type have been used. Preconditioned biconjugate (PBCG) method has been used for the solution if the discretized boundary element system has as many equations as the total number of unknowns. Preconditioned conjugate gradient (PCG) method has been utilized for the general case in which more equations are available than unknowns. Numerical results show PBCG algorithm has very limited applicability for inverse problems. Further, the PCG algorithm performs better than the PBCG method, and is insensitive to measurement errors. However, for a general inverse problem, DRBE-PCG algorithm is seen to be extremely sensitive to the location of measurement points with linear element mesh. At the same time, DRBE-PCG algorithm works particularly well with constant elements, and is insensitive to measurement errors. Further, for the same degree of mesh refinement, constant element mesh yields more accurate and stable results than linear elements. KEY WORDS: Boundary Element Method, Inverse Heat Conduction, Gradient Method , Biconjugate Gradient Method.

Iterative Regularization,

Conjugate

INTRODUCTION Time dependent inverse heat conduction problems frequently require the reconstruction of the unknown boundary conditions using the discrete temperature history at interior locations. Numerical solution of these problems consists of two parts: (i) a numerical method for discretization of the continuum problem, and (ii) a regularization procedure to stabilize the usually ill-conditioned discretized system obtained in the first part. The regularization of the discretized system is re-

82

M. Tanaka and KM. Singh

quired to avoid the extreme sensitivity of the solution of this system to the errors in the measured data. In the analysis of inverse heat conduction problems, all of the three major discretization methods, viz. the finite difference (FDM), the finite element (FEM) and the boundary element (BEM) have been used by various researchers. A very good exposition of these methods as applied to inverse thermal problems, various regularization techniques and related references are available in Kurpisz and Nowak [ 1 ]. For the multidimensional boundary inverse heat conduction problems, advantages of the BEM over FEM or FDM are apparent, and are clearly articulated by Ingham and Yuan [2]. Amongst the various contributions on the application of BEM to inverse heat conduction, particularly noteworthy are those of Pasquetti and Niliot [3], Kurpisz and Nowak [4], and Huang and Chen [5] which specifically deal with multi-dimensional problems. Pasquetti and Niliot [3] propose the use of spatial and temporal regularization methods, whereas Kurpisz and Nowak [4] present a combination of the function specification and regularization method. The regularization method requires use of a regularization parameter whose optimal value must be determined as a part of the solution process. To alleviate this difficulty, one can employ an iterative regularization [4]. Kurpisz and Nowak [4] suggest the use of gradient type methods for this purpose. Huang and Chen [5] make use of the conjugate gradient method in conjunction with time-dependent boundary element formulation, and show that this procedure is less sensitive to measurement errors as compared to the regularization method, and obtains more accurate solutions. For this reason, the conjugate gradient based regularization procedure has been used in this work. All previous applications of the BEM to time-dependent inverse heat conduction problems employ the BE formulation based on the time-dependent fundamental solution, which requires incorporation of the initial conditions through domain integrals (alternatively, it would require the storage of all the coefficient matrices and data from the initial time to the current time step). Dual reciprocity formulation eliminates this drawback by approximating the temporal derivative by a product type approximation, and results in a boundary-only formulation. Application of the standard discretization procedure yields a differential-algebraic system which can be easily solved in a step-by-step manner. This method is well-established for direct heat conduction problems [6], but has not been exploited for inverse heat conduction problems. With this in view, an application of the dual reciprocity BEM to inverse problems is attempted in this work. The one step ^-method has been employed for temporal discretization and, preconditioned conjugate or biconjugate method for iterative regularization of the resulting algebraic system.

DREE FORMULATION The temperature distribution inside a homogeneous isotropic domain ^ with constant material properties, unit diffusivity, and absence of sources or sinks is governed by

(^'0-

(ic,O = 0

in Q,

(1)

with associated initial and boundary conditions. For inverse boundary problems, boundary conditions are not known on the inverse boundary segment F/.

83

Application of DRBEM and Iterative Regidarization

Use of the fundamental solution to Laplace equation, M*, and Green's second identity transforms eq. (1) into following integral equation: (2) where subscript / denotes a point of F or Q . In order to transform the domain integral in (2) to the boundary, the temporal derivative is approximated by a set of A^^ coordinate functions f^ {x) as follows:

(3)

•^^J^fKx)a\t).

where a-^ are unknown functions of time, and f-^{x) satisfy V^i/fJ = fj Use of approximation (3) in eq. (2) yields the following boundary integral equation:

Ui + f (q^'u - u'^q) d r = ^ where rj-f = d^/J/dn.

L-i/r/ + j {q'^ir^ - u*rjj) d r l a^

(4)

Use of standard discretization procedure results in the relation

Hu-Gq

= {H^

(5)

-GE)oL.

One can use eq. (3) to eliminate a. and thereby obtain the differential algebraic system: Cu + Hu-Gq

(6)

= {i,

where C = {GE - H^) F~\ F being the coordinate function matrix composed of column vectors fj and u denotes the nodal vector of u for all the freedom nodes only (in contrast to u which denotes the vector of u for all nodes — freedom nodes plus internal nodes where measurements are made).

TIME INTEGRATION AND ESTIMATION PROCEDURE For the temporal discretization of eq. (6), one can use one step ^-method and obtain the algebraic recurrence relation: C

— Un+l + OHUn-^i

Ar

C

- OGqn+X = — I/„ - (1 - e)HUn

Ar

+ (1 " 0)Gqn.

(7)

In the preceding equation, one can rearrange the unknowns in a vector x and obtain a system of linear equations:

Ax=:h.

(8)

Let Nm denote the number of internal points where measurements are made, and Ni denotes the number of nodes on the inverse boundary segment F/. The system (8) has Np + A^^ equations for Np + Ni unknowns. To find a solution, Np + Nm > Np-\- Ni .If Np-\- Nm = Np-{- Ni, then

84

M. Tanaka and K.M. Singh

matrix A in eq. (8) is regular, but ill-conditioned. For the general case of Np -\- Nm > Np -\- Ni, one can use a least squares estimation employing the functional (9)

A = \\Ax-bl in which || • || denotes the Euclidean norm. Minimization of A leads to the regular system A^Ax

= A^b,

or

5x = d,

(10)

where S = A^ A, and d — A^b. To take care of the damping and lagging effects of the heat conduction operator, use of future time steps is normally recommended. Let r denote the number future time steps used to determine the solution at time level n + 1. At any time level n-\- p, p = 1,... r + 1, eq. (7) can be written as

C

— Un^p + OHUn+p

C

- OGqn-^-p = — Un-^-p-l - (1 - 0)HUn+p-\

+ (1 -

0)Gqn+p-\. (11)

Collecting the unknowns at time level n -j- /? in vector x„+p, the preceding equation can be expressed as A\ Xn+p + AoXn+p-X

= bn-\-p.

(12)

Let us note that matrices Ao and A\ depend on the time step Ar. If a constant step size Ar is used in computations, then Ao and A\ are independent of time level n. In any case, for computations at time level n + 1, same value of Ar would be normally used for all r future time steps. Thus, gathering the unknowns for all future time steps at time level n + 1, following linear system can be obtained: AX

= B.

where X = [x„+i,... , x„+,]^, ^ = [6„+i,... , 6„+.]^,

(13)

and A is a block triangular matrix in which main diagonal is [ A i , . . . Ai], subdiagonal is [Ao,... Ao], and rest of the entries are zero. Once again, the preceding system can be solved as such if A^^ + A^^ = Np + Ni, and employing the least squares estimation procedure for the general case of Np + A^^ > Np + Ni.

ITERATIVE REGULARIZATION In analysis of inverse problems, use of some regularization procedure is generally necessary to minimize the sensitivity of solution to experimental and numerical errors. Two most popular such procedures are — (i) Tikhonov regularization, and (ii) iterative regularization. Of these, the iterative regularization procedures construct the solution by generating a sequence of vectors approximating the solution. In the presence of measurements errors, the number of iterations is set according to the data errors using discrepancy principle [1,5], and this number serves as a regularization parameter. Iterative procedures of the steepest descent and the conjugate gradient type are the most suitable for inverse analysis [1]. Two methods of latter family are outlined below. Starting from an initial guess xi, both of these methods construct construct a sequence of improved estimates as Xk-^i =Xk-\-akPk-

(14)

85

Application ofDRBEM and Iterative Regularization Recurrence relations for scalar ak and vector pk are given below for each method. To improve the convergence of the iterative method, these methods are applied to the preconditioned form of eq. (8), (10) or (13).

Preconditioned conjugate gradient method (PCG) The system matrix S in eq. (10) is symmetric and positive definite. Hence, the conjugate gradient algorithm can be used for solution of eq. (10). Let 5 be a preconditioner matrix for S. The preconditioned form of eq. (10) is 5~^5x = S~^d, Scalar ak and vector pk are constructed using the following recurrence:

Oik =

Vie • Z/c

^—,

Pk' S Pk

n + 1 = n - oikSpk,

f*k-^l

Pk =

' Zk-{-\

Vk • Zk

, Pk-\-\ = zk + PkPk (15)

with the starting values r\ = d—S x\, andpi = zi. Here, vector Zk is the solution of S Zk = n .

Preconditioned biconjugate gradient method (PBCG) The biconjugate gradient method is applicable to general linear systems which may not be positive definite or symmetric. This is the case with the system (8) for Np + A^^ = Np + Ni. This method constructs four sequences of vectors, rk, Vk, Pk and pk to obtain an improved estimate of the solution vector using relation (14) [7]. Let us consider the preconditioned form of eq. (8) or (13) given by A~^Ax = A~^b, where A is a preconditioner matrix. Define vectors Zk and Zk satisfying Azk = Vk, and A^ Zk = Vk- Let ri =b — Axi, where x\ is the initial guess, and r\ = r\. Setpi = z\ and p\ = z\. The recurrence relations for vectors n , n , Pk and pk, and scalar a^^: are [7]: ^k ' ^k

ock = T Pk'A'Pk Pk = —-:

J.

,

n+1 = n - oikApk

—.

Pk+\ =Zk-\- PkPk,

-

-

AT-

n+1 = n - oikA pk

Pk+\ =Zk+ PkPk

/^^\

(16) (17)

Vk ' Zk

The diagonal part of S (or A) has been used as the preconditioner matrix S (or A) in this work.

NUMERICAL EXPERIMENTS To assess the accuracy of the dual reciprocity boundary element procedure in conjunction with iterative regularization described in the preceding section, it is applied to two test problems. Both of these inverse problems have been formulated from a direct problem on unit square with unit initial condition. Boundaries x\ = 0 and X2 = 0 are insulated, whereas q(x, t) = —h u(x, t) along x\ = \ and X2 = 1. The exact solution of the direct problem is given by [8] u{x,t) = w{x\,t) w{x2,t),

(18)

86

M. Tanaka and KM. Singh h=2

O

•

q=0

•

Freedom Poles

O Measurement points

k

o i o i o • o if o i o i o * o i

q=0

Figure 1: Example 1 — problem definition and discretization.

where ^(z,

t)=2h^

COS(PnZ)

Sec(Pn)

expi-Pn

t)-

(19)

n=\

In the preceding equation, fin are the positive roots of the equation pianp — h = 0. In all the subsequent computations, /z = 2 has been used. To simulate the measured data for inverse analysis, the exact solution at the interior points is modified as ,(Xi, t) = «(x/, t)[\ + d(\ - 2/7)],

(20)

where d is the maximum relative measurement error and p e [0, 1] is a random number [1]. Computations have been performed for d = 0, 10~^, 10~^, 10~^, which correspond to measurements with errors, of high accuracy, standard accuracy and low accuracy, respectively. Uniform linear or constant boundary elements have been used for spatial discretization, and one step backward difference method (0 = 1), for time integration.

Example 1 Let us assume that flux is unknown along the boundary jci = 1 in the problem mentioned above. Figure 1 presents the definition of this inverse problem and the boundary element discretization. Figure 2 presents the distribution of estimated flux along jci = 1 obtained using the PBCG and PCG methods. The results with PCBG algorithm are seen to be sensitive to measurement errors, whereas those obtained with PCG algorithm are insensitive to measurement errors. This trend is clearly reflected in the numerical results for flux presented in Table 1. An interesting feature is the accuracy of the temperature estimates which are almost identical with both the schemes.

Application ofDRBEM and Iterative Regularization Exact

: en-0.1

•

*

Exact

O

d=1.0e-5

•

d=1.0e-4

X

d=1.0e-3

X

^ ^ " X^

^ ^ ^ - ^

X

—'"T^ *

87

«

1

O

d=1.0e-5

•

d=1.0e-4

X d=1.0e-3

O^-O.l

* *

X

*

*

^

X -0.2 0

0.2

0.4

0.6

0.8

-0.2

0

0.2

0.4

0.6

0.8

1

y

y

(a) PBCG method

(b) PCG method

Figure 2: Example 1 —fluxdistribution along x = 1 at f = 1.0 obtained with Ar = 1/32. Table 1: Errors in boundary element solution of Example 1 (r = 1, r = 1.0, Ar = 1/32). Measurement Error 0 10-^ 10-4 10-3

L2 Error in u (in %) PBCG PCG 4.08 4.08 4.08 4.08 4.08 4.08 5.18 4.08

L2 Error in q (in %) PBCG PCG 7.54 7.56 7.54 7.54 11.2 7.57 110. 10.1

Example 2 Let us assume that flux is unknown along non-insulated boundary in the direct problem defined earlier. Figure 1 (b) presents the definition of this inverse problem and two boundary element discretizations employed. For this problem, PBCG method produces unstable results. With Mesh 1, flux estimates obtained with the PCG method show an oscillatory behavior irrespective of the degree of the measurement errors, which is reflected in poor results in Table 2. With Mesh 2, in which the measurement points are farther from the boundary, results obtained are simply inaccurate. This indicates the PCG-DRBE algorithm presented in Section 3 is too sensitive to the location of measurement points. Possible remedies to reduce the sensitivity to measurement errors and location of measurement sensors include (a) use of future time steps, and (b) reduction in the number of unknowns to be estimated using constant element discretization. Table 3 presents compares the results obtained using linear element Mesh 1 with and without future time steps. It can be clearly observed that use of future time steps does not help in this case. Next, let us consider the usefulness of constant element discretization in Figure Aa which is of the same degree of refinement as the linear element mesh of Figure 3. Flux estimates in

M. Tanaka and K.M. Singh

iAA

o o o o o o

•

A

A

A

A

•

fl A

OAOT AAf A

o1

f O 1 A f

o1

A 4 o1 A 4

•

q=0

•

Freedom Poles

O Measurement Points (Mesh 1) A Measurement Points (Mesh 2)

oT A • o1 A 4 oT A 4 o1 A 4

Cr

0

-0.5

q=0

(b) Flux distribution

(a) Geometry and discretization

Figure 3: Example 2 — (a) problem definition and linear element discretization, (b)flux distribution along jc = 1 with PCG and Mesh 1 (r = 1, r = 1.0, Ar = 1/32).

Table 2: Example 2 — Errors in boundary element solution with linear elements {t = 1.0, At 1/32). L2 Error in q (%) L2 Error in u (%) d 0 10-^ 10-^ 10-3

Mesh 1 2.91 2.91 2.91 2.98

Mesh 2 91.5 88.7 90.5 97.3

Mesh 1 28.4 29.5 26.8 38.5

Mesh 2 2.01e4-3 1.95e+3 1.98e-f3 2.14e+4

Table 3: Example 2 — Errors in boundary element solution with linear elements and future time steps ( r = 1.0, At = 1/32). d 0 10-^ 10-4 10-3

L2 Error in u (%) r = 1 r =3 2.86 2.91 2.86 2.91 2.91 3.09 3.45 2.98

L2 Error in q (%) r=\ r=3 28.4 21.1 29.5 20.1 26.8 52.3 38.5 77.8

Application ofDRBEM

and Iterative

-0.05

o o o o o o o o o o o

- Exact

od oi ok o* oJ

q=0

•

Freedom Poles

O Measurement Points

89

Regularization

q=0

(a) Constant element discretization

d=1.0e-5

X

d = l.Oe-3

d = l.Oe-4

C3^ -0.1 -

»

^^^-^^.

OI

oi

o •

» -0.15

•

« 0.2

^

-^

* 0.4

'—"-i 1

0.6

'

• • • '

0.8

2/ (b) Flux distribution

Figure 4: Example 2 — (a)finerconstant element discretization, (b)flux distribution along x = 1 with PCG and constant element mesh {t = 1.0, At = 1/32). Table 4: Example 2 — Errors in boundary element solution with linear elements and future time steps (r = 1.0, At= 1/32). d 0 10-^ lO-'* 10"^

rms] irror in u (%) r = 1 r =3 2.62 2.62 2.62 2.62 2.62 2.62 2.62 2.59

rms]Error in q(%) r = 3 r = 1 6.37 6.38 6.37 6.37 6.38 6.56 7.30 8.99

Figure 4b can be seen to be very good and insensitive to measurement errors. Use of future time steps brings a slight improvement in estimates of boundary fluxes. No oscillatory behavior is seen with constant elements. Further, though the results are sensitive to location of measurement points, still the no instability is observed in this case. These observations clearly indicate that the constant element discretization should be preferred over linear elements in DRBE analysis of inverse heat conduction problems. Preceding results clearly indicate that the DRBE-PBCG algorithm presented herein are not robust enough for general inverse boundary problems. It would work well only if the inverse segment forms a small part of the boundary, as is the case in Example 1, but is likely to fail if the inverse segment forms a major part of the boundary. On the other hand, DRBE-PCG methods works fairly well, especially in conjunction with constant element mesh. Further, use of future time steps makes only a marginal difference to the solution, and hence, is not required.

90

M Tanaka and KM. Singh

CONCLUDING REMARKS This paper has presented an application of the dual reciprocity boundary element method in conjunction with iterative regularization for solution of time-dependent inverse heat conduction problems. For iterative regularization, the preconditioned biconjugate (PBCG) and conjugate gradient (PCG) algorithms have been presented. Both the algorithms obtain very good estimate of flux and temperature if this inverse boundary segment is only small fraction of the total boundary. In otherwise case of larger part of the boundary having unspecified boundary conditions, DRBE-PBCG algorithm fails, whereas DRBE-PCG algorithm obtains good results. However, DRBE-PCG algorithm is seen to be extremely sensitive to the location of measurement points with linear element mesh. At the same time, DRBE-PCG algorithm works particularly well with constant elements, and is insensitive to measurement errors. Further, for the same degree of mesh refinement, constant element mesh yields more accurate and stable results than linear elements.

REFERENCES [ 1 ] Kurpisz, K. and Nowak, A. J. (1995). Inverse Thermal Problems. Computational Mechanics Publications, Southampton. [2] Ingham, D. B. and Yuan, Y. (1994). The Boundary Element Method for Solving Improperly Posed Problems. Computational Mechanics Publications, Southampton. [3] Pasquetti, R. and Le Niliot, C. (1991). Numerical Heat Transfer PartB: Fundamentals, 20, 169. [4] Kurpisz, K. and Nowak, A. J. (1992). Engineering Analysis with Boundary Elements, 10, 291. [5] Huang, C. H. and Chen, Chih-Wei (1998). InternationalJournal for Numerical Methods in Engineering, 42, 943. [6] Partridge, P. W., Brebbia, C. A., and Wrobel, L. C. (1991). The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton and Elsevier Applied Science, London. [7] Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. R (1992). Numerical Recipes in FORTRAN. Cambridge University Press, Cambridge, 2nd edition. [8] Carslaw, H. S. and Jaeger, J. C. (1959) Conduction of Heat in Solids, Clarendon Press, Oxford, 2nd edition.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights resei-ved.

91

NUMERICAL SOLUTION OF AN INVERSE STEADY STATE HEAT CONDUCTION PROBLEM

FREDRIK BERNTSSON and LARS ELDEN Department ofMathematics, Linkoping University S-581 83 Linkoping, Sweden ABSTRACT We consider a two dimensional inverse steady state heat conduction problem. The Laplace equation is valid in a domain with a hole, and temperature and heat-flux data are specified on the outer boundary. The problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. By discretizing the differential equation using finite differences, and using Tikhonov regularization on the discrete problem, we get a large sparse least squares problem. Alternatively we can use a conformal mapping, and solve an equivalent problem on an annulus, using a technique based on the Fast Fourier transform. Numerical results using both methods are given.

KEYWORDS ill-posed, Cauchy problem, Tikhonov regularization, conformal mapping, spectral method

INTRODUCTION We consider a two dimensional inverse steady state heat conduction problem. The Laplace equation is valid in a domain 0 c [0,1] x [0,1], that is not simply connected. The domain is displayed in Figure 1. Find a temperature distribution T e (7^(0) Pi (7°(0) that satisfies, AT = 0, inO, onFi, T = g. n-VT = h, onFi,

(1)

where ft denotes outward unit normal, and g,h are the temperature and heat-flux data on the outer boundary Fi. Our aim is to compute an approximation of T on the inner boundary r2. A similar problem was studied in [2]. The problem is known to be ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We discuss the nature of the ill-posedness, by computing approximately the singular values and singular functions of a certain operator related to the problem.

92

F. Berntsson and L. Elden

1

X

Figure 1: The two dimensional domain H. The hole corresponds to the set {0.2 < x,y < 0.8}.

We investigate numerical aspects of Tikhonov regularization applied to a discretization of the problem. Thefinitedifference method leads to a sparse, very ill-conditioned and over-determined system, which we regularize on the nodes on the inner boundary boundary r2. Algorithms for sparse least squares problems are crucial for the efficiency of this method. An alternative, and potentially more efficient, method is to treat (1) explicitly as a Cauchy problem and solve it using a marching method (in analogy to methods for the sideways heat equation [3]). This is done byfirstmapping Q, conformally onto an annulus, and then writing the equation in polar coordinates. The ill-posedness can now be dealt with by filtering away high-fi*equency components in the solution.

ILL-POSEDNESS AND STABILIZATION In this section we make the additional assumption that the heat-flux data, in (1), is /i = 0. This is no restriction since a general problem can always be reduced to this case: Let Ti be a harmonic fiinction satisfying n • VTi = h, on Ti, and Ti = 0, on r2. Then T = Ti-\- T2, where T2 solves (1) with boundary data ft • VT2 = 0 and T2 = ^r - Ti, on Fi. The fiinction Ti can be computed by solving a well-posed problem. Consider the following boundary value problem: Find T e C^ifl) f) C^{(l) satisfying.

Ar = o,

n . VT = 0,

T = f,

inQ, onFi, onr2,

(2)

Let g = T\r^ denote the restriction of T to Fi. Since (2) is a well-posed problem, the linear operator Kf = g, is well defined. Solving the inverse problem (1), with /i = 0, is thus equivalent to solving the operator equation, Kf = g. By discretizing (2) we can compute an approximation of the operator K. Here we use a finite difference method and an equidistant grid on ft. Note that by using an equidistant grid we get more nodes on Fi than on F2. Thus the matrix K approximating K will be rectangular. Using the singular value decomposition (SVD) we can write the solution of (2) in the form. /

Numerical solution of an inverse steady state heat conduction problem

93

Figure 2: The singular values of the matrix K. The sequence {(7^) tend rapidly to zero, and several singular values are of the same order of magnitude as the machine precision. where p and / are vectors containing the values of/ and g on the grid, and {(7^, Wn, ^n} are the singular values and singular vectors of the matrix. In a numerical example we computed the SVD of a matrix K G R504x304 jj^^g ^ ^ ^idA 504 nodes on Fi and 304 on F2. The computed singular values are displayed in Figure 2, and a few singular vectors are presented in Figure 3. The rate of decay of {a^} determine the degree of ill-posedness. For this problem we have exponential decay. Thus the problem is severely illposed, and any attempt to numerically solve the linear system Kf = g would fail. There are several methods available for regularizing the problem [4] so that a numerically stable solution is obtained. Here we describe two such methods. Typically in inverse problems, the singular vectors (functions) corresponding to large singular values are very smooth. Here, however, we see that the vector vi has spikes at the comer locations. This illustrates the fact that derivatives of the solution of a well-posed boundary value problem on 0 can have singularities at the comers of the inner boundary [1, page 34]. Since these are associated with the large singular values they cause no difficulties in the solution of the inverse problem. We also see the typical smooting properties of the operator illustrated in v^o, whose corresponding singular value is of the order 5 • 10"^. Tikhonov regularization In the previous section we defined an operator K by the equation Kf = g, where g =T\r^ and T solves (2). By discretizing (2) using^finite differences we approximate the operator equation b j an overdetermined linear system Kf = g. Since the original problem is ill-posed, the matrix K^ is very ill-conditioned. Thus regularization is needed. Tikhonovs method can be written: Find /

94

F. Berntsson and L. Elden

100

200

300 U(:.1)

400

500

100

200

300 U(:,5)

400

500

100

200 300 U(:,50)

400

500

100

200 V(:,50)

Figure 3: Left and right singular vectors of A\ corresponding to (Ji, CTS, and a^o.

minimizing, mm\\Kf

-gf

^^'\\Lff

= mm

K XL

f-

(3)

where m is the number of nodes on r2, A is a regularization parameter, and L G E^^"^ is either the identity or a discretization of afirstor second derivative. This approach makes expHcit use of the assumption that h = 0. However, the computation of the matrix K requires solving as many well-posed boundary value problems on ft as the number of points on the inner boundary, which may be a prohibitively high cost. An alternative approach is to consider the discretization on the whole domain 0, which leads to the sparse system of linear equations Af = b, where T is a vector containing approximate values of T on the equidistant grid in ft. Here the assumption h = 0 is not neccesary. Let CJ be a set of indices such that T(3) = f G I^"^. Then Tikhonov's method, applied only to the nodes on r2, can be written.

mm\\AT-bf^X'\\LT{3)\\'. f

(4)

95

Numerical solution of an inverse steady state heat conduction problem In our numerical implementation of this method we use a matrix L, which corresponds to a discretization of the second derivative. By this choice we strongly favour a smooth solution. The problem (4) is equivalent to solving a large, sparse, and structured least squares problem.

A spectral method Let 0 ' be the annulus {z = re'^ : ri < r < 1}, and assume that (l){z) is an analytic function mapping 0 conformally onto H'. For simplicity in notation we will identify E^ with the complex plane C. Two properties of conformal mappings are of interest here: If IIJ{Z) is harmonic in 0' then ^(^) = 'ij;{(j){z)) is harmonic in 0. Also n • V^ = 0 on the boundary dVt' impHes that n • VW = 0 on 90. Thus if we find a function ^(2:), harmonic in 0', such that ^[e')^g{cj,-\e')),

0<^<2^,

(5)

then T = ^(^(•)), solves (1), with /i = 0. The Cauchy problem for the Laplace equation in the annulus is easy to solve. For this purpose we use a Fourier series expansion of the solution ^ and the boundary values il^\r=i=go 0~^, 00

V'(r,e)= ^ A;=—oo

CXD

^,(r)e-'=^

and, ^(^'(e-')) = ^

W*'-

(6)

k=—oo

Writing the Laplacian in polar coordinates we get a set of ordinary differential equation for the sequence of Fourier coefficients {^jb},

' i (4)-'*=«• -<-'• ||t'(i)?o. The problem of computing t/' (r i, •) from z/? (1, •) is severly ill-posed, thus regularization is needed. However, for a fixed k the ordinary differential equation can be solved using, for example, a standard Runge-Kutta method. Thus by introducing a "cut off frequency ^c, and using only terms with |A;| < (Jc in (6), we get a stable solution. Now we describe the numerical implementation of this method. The change of variables (5) can be implemented as follows. Let {e'^^}^i be an equidistant grid on the unit circle. Then {(i>~^{d'^^)} is a non-uniform grid on Fi. By approximating the temperature data using a cubic smoothing spline, we find approximate values oig on the non-uniform grid. Thus we have the temperature data on the annulus. In our implementation we use the MATLAB routine s p a p s , for computing the approximating cubic spline. By using the Fast Fourier transform (FFT) we compute approximately the sequence {^k}- The solution at the inner radius is computed by solving (7) and using the inverse FFT. The frequency components corresponding to |A;| > ^^ are explicitly set to zero. We use the MATLAB code ode45 for solving the system of ordinary differential equations. Finally, the nodes at the inner radius of the annulus are mapped onto a set of non-uniform grid points on V2, using the conformal mapping. The solution T|r2 is then obtained by cubic spline interpolation. For this scheme to work we need a method for computing numerically the conformal mapping n' = 0(0). To find such a mapping is in general a difficult problem. For simply connected

96

F. Berntsson and L. Elden

-1

-0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1

Figure 4: An orthogonal grid on an annulus is mapped conformally onto il. polygonal regions this problem is solved by the Schwarz-Christoffel mapping function. A similar mapping function exist for doubly connected domains [5, 6]. A software package for this case exists, and is freely available from Netlib^ The mapping displayed in Figure 4 was computed using this code.

NUMERICAL EXPERIMENTS In this section we give numerical results intended to demonstrate that the proposed methods work well. The test problems were created in the following way: First we selected the function T, which is harmonic in fi, and computed the corresponding temperature and Jieat-flux data, o and /i, on Fi. Normally distributed noise was added giving vectors g^, and h^. By solving numerically the Cauchy problem (1) we obtained approximations of the steady state temperature on r2. We conducted two separate tests, using different functions T as exact solutions. In Figure 5 we display the results obtained using Tikhonov regularization on the finite difference matrix, i.e. by solving (4), where iy is a discretization of a second derivative. In this experiment we used a grid with 504 nodes on Pi and 304 nodes on r2. Thus we solved a least squares problem with a matrix of size 16633 x 16129, and with 57541 non-zero elements. In Figure 6 the results obtained by solving an equivalent problem on the annulus are displayed. This method requires that the heat-flux h is zero. Thus, as a preprocessing step, we solved a well-posed problem boundary value problem. When solving the Cauchy problem on the annulus we needed to compute the F F J of vectors of size 4096. From these tests we conclude that the proposed methods work well. In all cases we managed to solve the inverse problem with accaptable accuracy.

^http://www.netlib.org/toms/785

Numerical solution of an inverse steady state heat conduction problem

0

0

0

0.5

97

1

0

Figure 5: Results obtained using the finite difference method, and Tikhonov regularization. We display the numerical solution in H (left), and at r2 (right,solid). Also we display the exact solution at r2 (right,dashed). The vertical lines mark the locations of the comers. We solved two different problems, in both cases the noise were of variance 10"^ and we used the regularization parameter A = 10"^.

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F. Berntsson andL. Elden

I

'

I

>

I

I

Figure 6: Numerical results using conformal mapping, and solving an equivalent problem on the annulus, using the spectral method. Two different problems were solved, and we display the approximate (solid) and the exact solution (dashed). The vertical lines mark the locations of the comers. Two different problems were solved, in both cases the noise were of variance 10"^ and we used the "cut off" frequency ^c = 8.

ACKNOWLEDGEMENT The work of Fredrik Berntsson is supported by a grant from the National Graduate School for Scientific Computing.

REFERENCES 1. Dietrich Braess. (1997). Finite Elements. Cambridge University Press, Cambridge. 2. B. H. Dennis and G. S. Dulikravich. (1998). In: Inverse Problems in Engineering Mechanics. International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP 98), Nagano, Japan, M. Tanaka and G.S. Dulikravich (Eds.), Elsevier, Oxford, pp. 61-72. 3. L. Elden, F. Berntsson, and T. Reginska. (2000). SIAMJ. Scient. Comput., to appear. Also: Technical Report LiTH-MAT-R-97-22, Department of Mathematics, Linkoping University, Sweden. 4. R C. Hansen. (1997). Rank-Deficient and Discrete Ill-Posed Problems. Numerical Aspects ofLinear Inversion. Society for Industrial and Applied Mathematics, Philadelphia. 5. P. Henrici. (1993). Applied and Computational Complex Analysis, volume 3. Wiley. 6. ChenglieHu. (199S). ACM Transactions on Mathematical Software, 24(3):317-333.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

99

AN INVERSE HEAT CONDUCTION PROBLEM AND AN APPLICATION TO HEAT TREATMENT OF ALUMINIUM

FREDRIK BERNTSSON and LARS ELDEN Department ofMathematics, Linkoping University S'581 83 Linkoping, Sweden ABSTRACT We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors. Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation. KEYWORDS ill-posed, inverse heat conduction problem, heat treatment INTRODUCTION In several industrial applications it is desirable to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements. Another reason is that locating a measurement device on the surface would disturb the measurements and an incorrect temperature is recorded. In such cases one is restricted to internal measurements, and from these one wants to determine the surface temperature. The sideways heat equation is a model of this situation. Find the unknown temperature T that satisfies, ( {kTx)x = pCpTt, —CO < t < oo, { T{L,') =g, -oo
0 < X < L,

(1)

where A: is the thermal conductivity, p is the density, and Cp is the specific heat of the material. The coefficients of the equation are allowed to depend on the temperature, thus the problem is non-linear. The heat equation [3] is assumed to be valid (at least) in an interval 0 < x < L.

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F. Berntsson andL. Elden

Water nozzles

xr^=^

0 0 6 6 <\6 6 0 6 6 6 6 6 0 0 6 6 6 6 0 0 0 6 6 ^ Perforated conveyer

Figure 1: Cooling of an aluminium plate. The thickness of the plate is 100-200 mm. Temperature and heat-flux data are given along the line x = L. The problem (1) is severely ill-posed, thus special numerical methods are required for solving the sideways heat equation in a stable way [2, 5]. We discuss the theoretical properties of the problem (1), as well as numerical methods for solving it. Heat treatment of metallic materials is usually performed in order to obtain materials with suitable properties. Age hardening of so called heat treatable aluminium alloys starts with a solution heat treatment. The material is then held at a high temperature close to where melting starts until an appropriate microstructure is obtained. In order to have good mechanical properties after the following ageing treatment, the material should be cooled as fast as possible. High cooling rates are achieved by spraying water on the metal surface. In order to get the desired effect of the heat treatment, it is important to have good control of the temperature and the cooling rate during the process [1]. In an experiment we found that it is possible to compute the temperature and the heat-flux on the surface, by solving numerically the sideways heat equation. During the experiment an aluminium block is cooled rapidly, by spraying water on one surface. Thermocouples, located inside the material, record the temperature history at a small distancefi*omthe cooling surface. The experimental situation is such that the simple one dimensional model (1) is not valid. However it is possible to modify the model in order to get acceptable accuracy. By solving a Cauchy problem for a more general parabolic equation we are are able to find the temperature distribution in the region between the thermocouple and the surface of the material. The experimental work were carried out at the Department of Mechanical Engineering, Linkoping University. ILL-POSEDNESS In this section we make the additional assumptions that the coefficients in (1) are constant. Also, for simplicity [2], we assume that the differential equation is valid in the half plane, :r > 0. If we require that the solution is bounded, then only one boundary condition at x = L is needed. Thus we consider the following problem: r KTEX = Tf,

-00

< ^ < 00,

X > 0,

\T{L,t) = g{t), -oo
.-.

^^

where K = k/pCp is the thermal diffusivity. The ill-posedness of the problem (2) can be seen in the frequency domain. Let 1 r°° g{0 = - 7 = / e-'^'g{t)dt,

-00 < ^ < 00,

An inverse heat conduction problem and an application to heat treatment of aluminium

101

be the Fourier transform of the data function. Then, formally, we can write the solution of (2) as

£

T(x,t)-

e'«*e(^-^'' Ae/K^i m)di-

(3)

Since the real part of v ^ is non-negative, (3) defines an unbounded operator: If we try to solve (2) numerically, the ill-posedness will manifest itself in the blow-up of high frequency perturbations in the data. The integral (3) exists only for functions 5 with rapid decay in the frequency domain. A natural way to stabilize the problem is to introduce a "cut off" frequency ^c and eliminate all highfrequenciesfromthe solution. Thus we get a regularized solution, 1

/•«' g»?tg(L-x)v'i«/«^(^)^^.

(4)

Note that the degree of ill-posedness depend on /^ and L. If the ratio L/^ is small then a large value for fc can be used without causing the data errors to destroy the accuracy. There is an optimal cut off level, but unfortunately we cannot compute it explicitly since it depends on unavailable information about the exact data g{t), e.g. the smoothness. However it is possible to give a parameter choice rule which, given some a priori information about the solution, leads to a convergent scheme [7, Sec. 3]. One such parameter choice rule was given in [6]. 80

n65

Figure 2: Principal design of the blocks used in the quenching experiments. The block is quenched by a continuous water flow applied on the lower surface. EXPERIMENTAL DETAILS AND MATHEMATICAL MODELLING Blocks of the design shown in Figure 2 were quenched from one end by a continuous water flow. The water cooled surface has the dimensions 65 mm x 65 mm. During the quenching the sides of the block were exposed to air of room temperature. If the sides are cooled by air this will not introduce a large temperature gradient across the thickness of the test specimen. Initially the blocks were heated to 475-480''C, and after a few minutes of (water) cooling the temperature had dropped to below 150 ""C This wide range of temperatures does not permit a model with constant thermal properties to give good accuracy. The experiments are described more in detail in [4]. The quenching equipment can be seen in Figure 3.

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Figure 3: The quenching equipment. The aluminium block has the dimensions 65x65x185 mm. Here we will give a mathematical model that can be used in order to compute the temperature history close to the water cooled surface. Introduce a coordinate system where the x-axis is orthogonal to the cross-section of the aluminium block. Then the heat-flow is essentially directed along this axis. However, there is a small temperature gradient resulting from convection and radiation heat exchange between the sides of the block and the surrounding air (and walls). The cross-section of the aluminium block is square, with side length s = 65 mm, and we consider the energy balance within a thin slice. Assuming that the temperature is constant within the slice, i.e. T = T(x,t),weget pCj^Tt = {m), - 4s-'h,f,{T - Too), (5) where Too is the ambient air temperature (approximately 18 ^C), and /letr is an effective heat transfer coefficient [8]. The first two terms correspond to the usual one dimensional heat equation, and the third term describe the heat loss from the sides of the aluminium block. These losses are due to free convection and thermal radiation. The density p and the specific heat Cp are fairly constant in the temperature range that is considered, but the thermal conductivity is treated as non-linear, k = k{T). The effective heat transfer coefficient is also considered as temperature dependent. In particular the contribution from thermal radiation is strongly non-linear. The differential equation (5) contain several problem specific parameters. The material properties for aluminium alloys can be found in e.g. [8]. The values that was used are, p=27S7 kg/m^, Cp = 8S3J/kgK. The thermal conductivity is temperature dependent and varies between 154 and 194 W/mK. The effective heat transfer coefficient /leff consists of two parts. For the free convection we use as an approximation a constant value 7 W/m'^K, and for the radiation part we use an emissivity coefficient e = 0.8, which gives a value between 4.5 W/m'^K at 20 ^C, and 33 VF/m^ii: at 500 ^C. We conclude this section with a brief discussion of the implementation of numerical methods for solving the Cauchy problem for the parabolic equation (5), with boundary values given at

An inverse heat conduction problem and an application to heat treatment of aluminium 103 x — L. The thermal conductivity k and the convective heat transfer coefficient h^n are temperature dependent. We rewrite (5) as a system of first order differential equations, T \ ^/

0

n-^\(

0

T \_(

0 < X < L, -oo < t < CO,

(6)

where K, = k/pCp, and ip — Aheff/spCp. The initial values are given by, T{L,t) = g{t),

and T,(L,t) == /i(t),

-oo < t < oo,

The implementation of space-marching methods for solving the sideways heat equation were discussed in [5,6]. The suggested numerical method can easily be modified to solve more general problems. In a practical situation we have measurements available only for a finite p ^ o d of time ti
D = F^AcF,

where Ac is a diagonal matrix corresponding to differentiation of the trigonometric interpolant, but there thefi-equencycomponents with ^ greater than a cut off level ^c are explicitly set to zero. The matrix F is the Fourier matrix, and the multiplication of F and a vector can be computed using the Fast Fourier transform (FFT). Note that by using the FFT we implicitly assume that the data vectors represent periodicfiinctions.This is notfiillyrealistic in our application, and we risk to introduce wrap-around effects in the numerical solution. This problem is discussed more in detail in [6]. 450 400 350

2300 00 250

300

Time [s]

Figure 4: The measured temperature data. From the top we display the measured temperature at 165 mm, 125 mm, 85 mm, 45 mm, and 5 mmfi*omthe water cooled surface. The thermocouples are placed at the center of the block. These measurements are quite noisy.

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NUMERICAL RESULTS During the experiment temperature histories were recorded by thermocouples, at certain locations within the aluminium block. The measurement period were approximately 10 minutes, and the results are displayed in Figure 4. The measurements are quite noisy, and therefore we initially approximated the data using a cubic smoothing spline. These measured temperatures can be used to verify that the numerical model gives sufficient accuracy. The tests were performed in the following way: Let Xi, X2, and x^ be the location of three of the thermocouples. By solving a well-posed boundary value problem for the equation (5) in the interval [a:i, ^2] we computed the heat-flux at X2. The temperature history at X3 was then computed by solving the inverse problem, with temperature and heat-flux data given at X2. The approximate solution was then compared with the measured temperature history at X3. In Figure 5 we display computed temperature histories at x = 85 mm and a; = 45 mm. We used a spectral approximation of the time derivative, with cut off level ^c = 300, when solving the inverse problem. The numerical solution is not very sensitive with respect to the choice of the cut off level. The results are surprisingly good. The difference between the computed and measured temperatures are of the order 10 °C. Considering that the temperature range is 20-500 °C, this is a good result. Also we tried to compute the temperature history 5 mm from the water cooled surface. The results are displayed in Figure 6. The cut off level was fc = 300. This time the results are not as good. The reason for this is probably that the sides of the aluminium block are also cooled by the water close to the end of the block. Thus the mathematical model is not valid close to the water cooled end of the block. Note that we have a large numerical error at ^ = 0. This is due to wrap-around effects caused by the use of the Fast Fourier transform.

Figure 5: Approximate temperature histories computed by solving the inverse problem. We display the numerical solution (solid) and the corresponding measured temperatures (dashed). Measurements at 165 mm and 125 mm were used to compute the solution at 85 mm from the surface (left). Also we used measurements at 125 mm and 85 mm to compute the temperature history at 45 m.m.fromthe water cooled surface (right).

An inverse heat conduction problem and an application to heat treatment of aluminium

105

Figure 6: Approximate temperature histories computed by solving the inverse problem. We display the numerical solution (solid) and the corresponding measured temperatures (dashed). Here we used measured data at x = 45 mm and x = 85 mm to compute an approximate temperature history at 5 mm from the water cooled surface (solid). Also we display the measured data at X = 5 m.m. (dashed). CONCLUDING REMARKS In this paper we have demonstrated that the experimental situation, where an aluminium block is rapidly cooled by a continuous water flow, is described by a parabolic differential equation (5), with temperature dependent coefiicients. The model is similar to the sideways heat equation, previously studied in [2, 6], and the proposed numerical methods are easy to modify to cover the more general equation used here. The assumption that the heat transfer within the aluminium block is essentially one dimensional is reasonable, see the discussion in [4]. This is not true, however, close to the water cooled surface. Thus the computed temperature history displayed in Figure 6 has much larger errors than those displayed in Figure 5. Also, the material used during the quenching experiments is the high strength aluminium alloy AATOIO. The material properties used in the numerical model is for an alloy with a very similar composition. However, this is not believed to introduce any essential errors in the results. ACKNOWLEDGMENT The work of Fredrik Bemtsson is supported by a grant from the National Graduate School for Scientific Computing. This work is a result from a fiuitful cooperation between the authors, and the Department of Mechanical Engineering, Linkoping University, and we thank Dan Loyd for helpfiil discussions, and Nils-Eric Andersson for making the experimental data available.

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REFERENCES 1. Nils-Eric Andersson. (1998). Licentiate thesis, Department of Mechanical Engineering, Linkoping University, Sweden. 2. F. Berntsson. (\999), Inverse Problems, 15:891-906. 3. H. S. Carslaw and J. C. Jaeger. (\9A6). Conduction of Heat in Solids. Oxford University Press (Clarendon), London. 4. loannis Efstratiou. (1999). MSc. Thesis LiTH-IKP-Ex-1608, Department of Mechanical Engineering, Linkoping University, Sweden. 5. L. Elden and F. Berntsson. (1998). In: Inverse Problems in Engineering Mechanics. International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP 98), Nagano, Japan, M. Tanaka and G.S. Dulikravich (Eds.), Elsevier Science, Oxford, pp. 3-10. 6. L. Elden, F. Berntsson, and T. Reginska. (2000) SIAMJ. Scient. Comput., to appear. Also: Technical Report LiTH-MAT-R-97-22, Department of Mathematics, Linkoping University, Sweden. 7. H. Engl, M. Hanke, and A. Neubauer. (1996). Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, the Netherlands. 8. J. R Holman. (1992). Heat Transfer, 7th ed McGraw Hill.

Boundary Data Detection in Solid Mechanics

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

109

INVERSE ANALYSIS TO DETERMINE CONTACT STRESSES USING PHOTOELASTICITY H. INDUE, K. HAYABUSA, K. KISHIMOTO and T. SHIBUYA Dept. of Mechanical and Intelligent Systems Engineering, Tokyo Institute of Technology 2-12-1 0-okayama, Meguro, Tokyo 152-8552, Japan

ABSTRACT In this paper, a technique is developed for determining contact stresses from photoelastic image data of one of the contacting bodies. The contact stress is determined by solving two problems: (1) a problem to determine the distribution of the difference of the principal stresses inside the specimen from isochromatic fringe image data and (2) an inverse problem to estimate the unknown boundary values from the difference of the principal stresses inside the specimen. The inverse problem is formulated and solved by using BEM and a nonlinear optimization method. The effectiveness of the present technique is verified by determining the contact stress between a rectangular steel stamp and an epoxy plate. KEYWORDS Contact stress, Photoelasticity, Boundary element method. Inverse problem. Optimization.

INTRODUCTION Evaluation of contact stresses is an important issue in engineering mechanics. Methods for predicting the contact stress have been developed well with the advance of numerical techniques such as FEM or BEM. However, it is still difficult to predict the contact stress accurately when unknown factors such as friction at the contact interface are involved. It is desired to evaluate the contact stress experimentally in order to verify the result of prediction and to investigate unknown factors involved in contact phenomena. The contact stress is sometimes measured directly by some sensor inserted between contacting bodies. However, contact state is more or less altered by the sensor itself. Therefore, indirect methods that do not affect the contact state are needed for accurate measurement of contact stresses. One of the most favorable techniques is the technique based on the concept of inverse analysis in which the contact stress is estimated from experimental data obtained at some points away from the interface of the contacting bodies.

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H.Inoueetal.

Oda et al. [1, 2] proposed a technique to estimate distributions of the contact stress between elastic bodies from experimental data obtained by strain gages attached to one of the contacting bodies. The inverse problem to estimate the contact stress from strain gage data was formulated and solved by the FEM. However, there are some demerits in using strain gage data. The distribution of the contact stress is estimated rather coarsely because the number of available data (i.e. the number of strain gages) is limited to some extent (at most 10 to 20). In addition, the accuracy of estimation is limited by errors involved in strain gage data since the strain gage provides an average strain over the gage area that is not very smaller than the contact area. Oda and Hattori [2] proposed another technique which utilizes photoelastic fringe orders obtained at several points of one of the contacting bodies. Since photoelastic image data are usually much more dense than strain gage data, it is expected that above demerits would be overcome. However, they did not utilize the merit of denseness of experimental data very well. This may be due to that the inverse problem was formulated and solved by the FEM. On the other hand, Hayabusa et al. [3] proposed a technique for stress separation of twodimensional photoelastic data. In this technique, unknown boundary values of the specimen are first estimated from photoelastic data obtained inside the specimen. Then all stress components inside the specimen are determined by solving the ordinary boundary value problem. These two problems were formulated and solved by the BEM. The estimation of unknown boundary values in this technique may be readily applicable to the measurement of contact stresses. However, the technique adopted in Hayabusa et al. [3] did not work very well because of the difficulty in the optimization procedure that minimizes a cost function in terms of light intensities at many points of the image data. In their subsequent paper, Hayabusa et al. [4] introduced a preprocessing to determine the distribution of the difference of the principal stresses from image data. The unknown boundary values were then estimated by an optimization procedure that minimizes a cost function in terms of the differences of the principal stresses at many points of the image data. The accuracy of estimation of unknown boundary values has improved remarkably by introducing the preprocessing. In this paper, the technique developed in Hayabusa et al. [4] is apphed to the measurement of contact stresses using two-dimensional photoelasticity. The contact stress is determined by solving two problems: (1) a problem to determine the differences of the principal stresses at many points of one of the contacting bodies from isochromatic fringe image data and (2) an inverse problem to estimate unknown boundary values (i.e. contact stresses) from the differences of the principal stresses given at many points of the specimen. In order to fully utilize the merit of denseness of image data, the inverse boundary problem is formulated and solved by the BEM. The contact stress between a rectangular stamp and a large epoxy plate is measured to examine the effectiveness of the present technique.

TWO-DIMENSIONAL PHOTOELASTICITY In the circularly polarized light field, the relationship between the Hght intensity at a point of isochromatic fringe image and the retardation phase at that point is given by Id = Asm'O-^B,

(1)

111

Inverse analysis to determine contact stresses using photoelasticity

HDOCXX>-

-OOCXXH

ooooo Applied load (a) Variation of light intensity.

Applied load (b) Variation of phase.

coooo

Applied load (c) Increment of relative phase.

Fig. 1. Determination of the difference of the principal stresses. //

=

Acos^O^B,

0 = Tratad^

(2) (3)

where A is a constant of intensity modulation, B is the background light intensity, a is the photoelastic sensitivity, t is the thickness of the specimen, ad is the difference of the principal stresses, 0 is the retardation phase, and Id and // are hght intensities of dark and hght fields, respectively. The normalized light intensity of dark field /^ is represented by T*

h — ho Iio — ho

sin' 0,

(4)

where ho and //o are light intensities of dark and hght fields at no loading, respectively.

DETERMINATION OF THE DIFFERENCE OF THE PRINCIPAL STRESSES The technique developed in Hayabusa et al. [4] is employed for determining the difference of the principal stresses from a series of isochromatic fringe images obtained by changing the load incrementally. As shown in Fig. 1(a), the normahzed hght intensity at a point of the specimen varies sinusoidally with the load according to Eq. (4). The inverse relation to Eq. (4) is formally expressed as Orel = Sin-^ y/lj. (5) Substituting the hght intensity at each loading level into Eq. (5), the "relative phase" is obtained as shown in Fig. 1(b). However, in order to determine the difference of the principal stresses, the absolute retardation phase shown in Fig. 1(b) is required. If the increment of the relative phase at each load increment is calculated, it is understood that the variation of the phase increment is stepwise and that the absolute value of the phase increment is constant for all load increments as shown in Fig. 1(c). Therefore, the "absolute phase" may be obtained by taking a cumulative sum of the absolute value of the phase increment as shown in Fig. 1(b). Once the absolute retardation phase is obtained, the difference of the principal stresses is readily determined by Eq. (3).

ESTIMATION OF THE CONTACT STRESS Suppose that the difference of the principal stresses is given over a region scanned with a camera (referred to as measured region). An analysis region 0 is defined inside the

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H. Inoue et al

Load Contact region To Analysis region

Measured region Fig. 2. Description of contact problem. measured region and its boundary is referred to as F = To U Fi as shown in Fig. 2. It is assumed that the boundary condition is known on boundary Fi (ex. free surface) while it is unknown on boundary FQ (ex. contact region or virtual boundary). From Somigliana's equation, the stress component crij at any point p inside 0 is given by a.,(p) = j ^ D.,k{p, Q)h(Q)dT

- ^ 5.,,(p, Q)uk{Q)dT

(6)

where ^A:(Q) and Uk{Q) are traction and displacement components at a point Q on F, respectively, while AjA;(p,Q) and Sijk{p, Q) are derived from the fundamental solutions of two-dimensional elasticity. Discretizing the boundary into elements in a similar manner as in the BEM, the following equation is obtained: (Tij = Gc^ — HcU

(7)

where t and u are vectors composed of tractions and displacements at all nodal points on the boundary F, respectively, and Gc and He are known matrices (row vectors) derived from the fundamental solutions. The boundary integral equation is used as an additional requirement. It is given as

c.,(P)«,(P) = J^ t/.,(p, QMQ)dr - J^ r,,(p, Q)u,{Q)dr,

(8)

where both P and Q are points on the boundary F and Cij is a constant determined by the geometry of the boundary. Discretizing Eq. (8) in the same manner as in Eq. (7), another system of equations with respect to t and u is obtained:

Gt-Hu

=0

(9)

where G and H are matrices derived from the fundamental solutions. Substituting Eq. (9) into Eq. (7) and applying boundary conditions on Fj, the following equation is obtained: Cto

(10)

where ^o is a vector composed of unknown traction components on the boundary FQ and C is a known matrix (row vector) derived from known boundary conditions and the fundamental solutions.

Inverse analysis to determine contact stresses usingphotoelasticity

113

Light Source ^Quarter wave plate ] Quarter wave plate

Polarizer CCD camera Specimen

Fig. 3. Experimental setup for two-dimensional photoelasticity. The difference of the principal stresses cid is expressed in terms of stress components as CTi = ^(^11 - a22f + 4
(11)

From Eqs. (10) and (11), it is understood that the difference of the principal stresses at any point inside H can be expressed as a nonlinear function of IQ. Since the differences of the principal stresses are given at many points inside 0 , the inverse problem to estimate the unknown tractions (i.e. the contact stresses) can be formulated as (Tdito) = GTd

(12)

where
U=\\&d-(rd{to)\\

(13)

is employed in order to estimate the unknown traction IQ.

EXPERIMENTAL VERIFICATION Experimental Setup and Photoelastic

Images

Experimental setup for two-dimensional photoelasticity is shown in Fig. 3. Isochromatic fringe image is obtained under the circularly polarized light field. Figure 4 shows the specimen used in this experiment. The specimen was supported by a steel stage at Boundary DE and subjected to compression in y-direction by a steel stamp on Boundary AB. It was made of epoxy resin and 6-mm thick. The photoelastic sensitivity of this specimen was a = 0.0959 mm/N. The specimen was loaded incrementally from 0 N to 926 N with increments of 25.7 N and, as a result, 37 images were obtained. Some examples of fringe images are shown in Fig. 5.

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Fig. 4. Indentation of a rectangular epoxy plate with a steel stamp (dimensions in mm)

(c) 695 N

(d)926N

Fig. 5. Examples of isochromatic fringe images at various loading level. Numerical Model Figure 6 shows the numerical model used for estimating the contact stress between the steel stamp and the epoxy plate. The boundary conditions of this model were assumed as summarized in Table 1. It was assumed that Boundary DE was friction free and constrained in y-direction in order to simplify the numerical analysis. This assumption is not correct but permissible when estimating the contact stress because Boundary DE is sufficiently far from Boundary AB. The analysis region and boundary element meshes were configured as shown in Fig.6. Since Boundary AB was discretized into 10 quadratic elements, the number of unknowns in the inverse analysis was 2N = 42. On the other hand, the isochromatic light intensities at M == 3,094 points indicated by • in Fig.6 were used for estimating the contact stress.

115

Inverse analysis to determine contact stresses using photoelasticity

F

Fi 11 elements

10 elements A TQ O B

11 elements Tj

c

Fig. 6. Numerical model used for estimating the contact stress. Table 1. Boundary conditions of the numerical model.

Boundary AB BC CD DE EF FA

*: unknown Determination

Uy

Ux

* * * * * *

* * * 0 * *

of the Difference of the Principal

^a;

ty

* 0 0 0 0 0

* 0 0 * 0 0

Stresses

Based on a series of 37 isochromatic fringe images, the difference of the principal stresses at 926 N was determined by the method described above. Figure 7 shows the result. In order to check this result, an ordinary stress analysis was conducted by BEM to predict the distribution of the difference of the principal stresses. In this prediction, the contact stresses were approximated by the exact solution for the elastic half plane indented by perfectly bonded rigid flat stamp given by y{^) and

r{x) =

Po /c + 1 27r V ^

1 \/A? Va^ - x'^

Ell^ + 1 27r

1 2 _ X2 A/A

log AC ^ a-\- X log 27r a —X log AC ,

——log 27r

a-\- X

a — X

(14)

(15)

where K = (3 —z/)/(H-i/) [P: Poisson's ratio), a is the half width of the stamp and PQ is the applied load (see e.g. [5]). Figure 8 shows the result of ordinary BEM analysis. Comparing Fig. 7 with Fig. 8, the result of Fig.7 seems inaccurate near the edges of the stamp where the stress is highly concentrated. This is due to the fact that the load increment was too large to evaluate the phase increment accurately. However, Fig. 7 mostly agrees with Fig. 8, which means that the present method is valid.

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y[mm]

0 -46

x[mm]

(a) 3 dimensional plot.

(b) Contour plot.

Fig. 7. Result of determination of difference of the principal stresses (in MPa).

y[mm]

^

-46

x[mm]

(a) 3 dimensional plot.

(b) Contour plot.

Fig. 8. The difference of the principal stresses predicted by ordinary analysis, (in MPa)

-5

0 jc[mm]

5

(a) X direction

10

-10

0 5 jc[mm] (b)>^ direction

-5

10

Fig. 9. Contact stresses on FQ estimated from Fig. 7. (A: x-direction, Q - y-direction)

Inverse analysis to determine contact stresses using photoelasticity Estimation

111

of the Contact Stress

Inverse analysis based on the difference of the principal stresses shown in Fig. 7 was conducted. The tractions on Boundary AB, that is the contact stresses were estimated as shown in Fig. 9. The curves with plots represent estimated tractions while the curves without plots represent the exact solutions given by Eqs. (14) and (15). Although the estimated results are somewhat oscillatory, they agree fairly well with the exact solutions for the elastic half plane indented by perfectly bonded rigid flat stamp. It can be considered that the friction between the steel stamp and the epoxy plate was relatively high.

CONCLUSIONS In this paper, a new technique has been developed for measuring contact stresses using photoelastic images of one of the contacting bodies. The contact stress is obtained by determining the distribution of difference of the principal stresses from a series of isochromatic fringe images and then by estimating the unknown tractions. The inverse problem for estimating the tractions are efficiently formulated and successfully solved by BEM. Although the estimated contact stresses were somewhat oscillatory, it may be improved to some extent by applying some regularization technique.

REFERENCES 1. Oda, J. and Moto, S. (1990). Trans. JSME 56A, pp.1479-1484 (in Japanese). 2. Oda, J. and Hattori, M. (1992). Trans. JSME ^8A, pp.106-111 (in Japanese). 3. Hayabusa, K., Inoue, H., Kishimoto, K. and Shibuya, T. (1998). In: Inverse Problems in Engineering Mechanics, M. Tanaka and G.S. Duhkravich (Eds). Elsevier Science, Amsterdam, pp.603-612. 4. Hayabusa, K., Inoue, H., Tanaka, S., Kishimoto, K. and Shibuya, T. (2000). Trans. JSME^ accepted for pubhcation (in Japanese). 5. Gladwell, G.M.L. (1980). Contact Problems in the Classical Theory of Elasticity. Sijthoff & Noordhoff, Alphen aan den Rijn.

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

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I D E N T I F I C A T I O N OF T R A C T I O N S B A S E D O N D I S P L A C E M E N T O B S E R V A T I O N S AT I N T E R I O R P O I N T S Masayuki NAKAJIMA, Ken HAYAMI, Jiro TERAO, Seigo WATANABE* and Shigeru ANDO Department of Mathematical Engineering and Information Physics Graduate School of Engineering, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan e-mail: [email protected] * Nissan Corporation, Yokosuka, Japan ABSTRACT This paper discusses the inverse problem of estimating the tractions on the boundary of an elastic body from displacement information observed at interior points. The problem arises, for instance, when estimating the tractions and the position where they are applied on the surface of a tactile sensor made of elastic body, from displacement observations obtained by ultrasonic cells inside the sensor body [1, 2]. We will formulate this inverse problem using the three-dimensional Boundary Element Method (BEM) and propose an algorithm to solve it. Finally, we will demonstrate the effectiveness of the algorithm through some numerical experiments. KEYWORDS Inverse Problem, Elastostatics, Boundary Element Method, Tactile Sensor, Generalized Inverse.

INTRODUCTION Previous research on inverse analysis in elastostatics have mainly dealt with estimating unknown boundary quantities (displacement and/or traction) from over-specified boundary conditions for two-dimensional elasticity such as in [3] or [4]. In this paper we will deal with the inverse problem of identifying boundary tractions from displacement observations at internal points. Related work has been done in [5], where displacements on the boundary are estimated using displacement information at interior points using BEM for plane elasticity. The present work [6, 7] identifies concentrated tractions and the positions where they are apphed on the boundary of a three-dimensional elastostatic body with the aim of applying it to tactile sensors [1, 2].

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M Nakajima et al

T H E FORWARD PROBLEM Let r = r ( ^ ) U r(^) be the boundary of the region (elastic body) 0 C R^, where traction is given on F^^^ and displacement is given on V^^\ Assume that F is discretized into boundary elements 7ri{l
t = Tj (j = l , . . . , m ' ) . t = 0, u = 0.

Here, t and u are the traction and displacement on the boundary, respectively. Solving this forward problem by the boundary element method, one first obtains the unknown traction on F^^^ and unknown displacement on F^^^ Then, the displacement at an arbitrary point in H can be determined.

Traction

^y/

Observation Points (Displacement ?)

lyjijJjjL*^ Traction

ilLljiJM^ (D) Figure 1: The forward problem. THE INVERSE PROBLEM Next, we will consider the corresponding inverse problem of estimating the tractions and the positions where they are applied on the boundary, from information of displacement at interior (observation) points. The problem is not an imaginary one. For instance, in the tactile sensor made of elastic material proposed in [1,2], the ultrcisound displacement sensor imbedded in an elastic body detects the displacement at an interior point due to the contact traction on the surface of the elastic body with high precision. From this internal displacement information, the surface traction and its point of application must be estimated in real time. The boundary and internal conditions for this inverse problem (cf. Fig.2) are given as follows. Boundary Condition On boundary elements in F^^^ other than7r„_^ (j = 1,..., m') : t = 0. where the tractions are applied On boundary elements in F^^^ : u = 0. Internal Condition At interior points z^- G H (z = 1, • • •, m) : u = iii.

121

Indification of tractions

Here, ui denotes the given (observed) displacement at the internal point Zi. In this paper, we will formulate this problem using the boundary element method, and then propose an algorithm for the solution of the inverse problem.

Observation Points (Displacement)

Traction ?

Traction ?

Figure 2: The inverse problem.

THE BOUNDARY ELEMENT FORMULATION OF THE FORWARD PROBLEM The Boundary Integral

Equation

The boundary integral equation for isotropic linear elastostatic materials is given by ^u^{x) + c.p. j^tl^ix,y)up{y)dr(y)

= J^ulp{x,y)my)

AT{y)

( a - 1,2,3),

(1)

where the boundary F is assumed to be smooth at aj, and the body force has been neglected. Ua and ta are the displacement and traction components, respectively, and Einstein's summation convention for repeated indices are used. w*^(aj,i/), ^*^(aj,T/) are the fundamental displacement and traction components, respectively.

Discretization

Using Boundary

Elements

The boundary F is then discretized into N planar boundary elements, and the boundary integral equation (1) is approximated as follows by the collocation method, where the displacement u and t are assumed to be constant on each element. oWa(«i) + C.p. Y, M^j)

/ C/?(«n y) d r ( y ) = Y. tp{xj) (a = l , 2 , 3 ,

ul^{xi, t/) dF(y) i = l,2,...,Ar)

(2)

Here, Xi is the node representing the i-th. element. Let the 3 x 3 matrices Hij = {hijap) and Gij = (gijap), where hijap=

I tlp{xi,y)^T(y),

gijcyfi = I

<^(iCi,i/)dF(2/),

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M. Nakajima et al.

and let / be the 3 x 3 identity matrix. Also let ti = {ta{xi)) and Ui = {ua{xi)). equation (2) can be written in matrix form as

y-\-Hn

HiN

Ui

Gu

G IN

UN

GNI

GNN

Then,

(3) H,Nl

"

2 ^ "*" ^NN

•

t]\f

Here, Hij and Gij are known quantities, and from the boundary condition of the forward problem, either it or t is given at each node. Hence, one can determine the unknown displacements and tractions from equation (3). That is, denoting the set of nodes with known displacement and unknown traction by 1, and the set of nodes with unknown displacement and known traction by 2, equation (3) gives the system of linear equations ^11 ~ ^ 1 2

Hn —G\2

G21

H21

—H22

— G22

T2

where, for instance, Ui is the displacement vectors u{x) of the set of nodes 1 aligned in a column. Hence, the unknown Ti and U2 are given by

f/2

^11 ~ ^ 1 2

Hi\

—G12

G2I

H21

—G22

—H22

(4)

T2

SO that all the displacement and traction at all the boundary nodes are now known.

Computation

of Displacement at an Interior

Point

Similar to the boundary integral equation (1), the displacement at a point z interior to the region H is given by M^)

+ liU^,y)My)dr{y)

= J^u:^{z,yMy)dr{y)

{a = 1,2,3).

(5)

Using the same discretization as in equation (2), equation (5) can be approximated by u^iz) + J2 M^i) I

C/JC*, *) d r ( « ) = Yl hi^') I

<0i^^^)

dr(a:)

( « = 1,2,3).

Defining the 3 x 3 matrices Hi{z) = (^.^^(z)) and Gi{z) = {giap{z)), where <,.„^(z) = ^ ^ « : ^ ( z , 2 / ) d r ( y ) ,

hi.0{z) = J^ tl0(z,y)dr{y),

the displacement at the internal point z is approximated by Ui

u{z)

= [-H,{z)

...

+ [Gi(z) . . .

-HM{Z)] UN

(6)

GN{Z)] IN

using the displacement and traction at each boundary node, which was given by equation (4).

123

Indification of tractions

ALGORITHM FOR THE INVERSE ANALYSIS In [8], the three displacement components ^1,^2^ % and -r—, ——, u^ot = 77 I T; T;— axi 0x2 2 \oxi 0x2 (six components altogether) at a single internal point were given as measured data, and an algorithm was proposed for the inverse analysis. In this paper, we will assume that the displacement components (1*1,^27^3) are measured at more than one internal points. Case when Traction is Applied at a Single Point We will follow the argument in the previous section, with the traction r and its point of apphcation on the boundary x^ unknown. In equation (4), all the unknown tractions and displacement vectors in T i , U2 can be expressed as affine transformations of r . Moreover, since in our case, u = 0 on T^^\ and t = 0 on F^^^ except for TT^ where r is applied, in fact T i , U2 can be expressed as linear transformations of r . (Note that u{zi) = 0.) Hence, from equation (6), the displacement u{zi) at each internal observation point Zi, (z = 1, • • • ,m) can be expressed as a linear transformation of r , that is u{Zi) = Ai{Xn)T, where Ai{Xn) is a 3 x 3 matrix that does not depend on r . Thus, the appHed traction r and the position Xn where it is apphed on the boundary can be determined as the r and Xn which reahze (7)

mm \\u - u\\

T,Xn

Here, u is the vector obtained by aligning the displacement vectors iii observed at the internal points z^ (z = 1, • • •, m) in a column, that is. Ui

u is the vector obtained by aligning u[zi) vertically, that is.

U = A{Xr,)T =

r.

(8)

When Xn is fixed, A is a constant matrix whose column vectors are linearly independent. Thus, as in [8], the traction r which minimizes (7) is given, using the generaHzed inverse of A, by r = {A^A)-^A^u. (9) Hence, equation (7) becomes

nnn\\{I-A{A'Ar'A'}u\\\ Xn

(10)

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M Nakajima et al

where we have used the matrix defined in equation (8), unlike in [8]. The Xn which minimizes equation (10) among all the position of the boundary elements aj^(l < n < TV) in V^^\ is the position where the traction is applied, which was to be determined. Substituting this Xn in equation (10), we obtain the traction r which was also to be determined.

Case when Tractions are Applied at More than One Points When tractions are appHed at more than one points, a similar procedure can be performed. Assume that m' tractions T J (j = 1, • • • ,m') are appHed at unknown positions x^^ on the boundary. Then, from equation (4), all the unknown quantities on the boundary can be expressed as a sum of hnear transformations of TJ. Hence, from equation (6), the displacement u{zi) at each observation point 2;, (z = 1, • • •, m) can be expressed as a sum of linear transformations of r^, that is. (11) where Aij{Xn^) is a 3 x 3 matrix which does not depend on TJ. Hence, the applied tractions Tj and their respective positions Xnj {j = 1, •' * 7^') where they are appfied on the boundary are those that reaHze (12) mm \\u-u\\ Here, u is the vector obtained by aligning the observed internal displacements in a column. u is the vector obtained by aligning u{zi) of equation (11) in a column, that is, • Aij

m'

E

'

^i =

u = Y.A{Xn^)Tj=Y^ i=i

j=l

^1

^ 2

A21

A22

Aim' A2m'

T2

-^mj

Next, let A =

All A21

A12 A22

Aim' A2m'

T2

Tm'

and fix all the positions Xn^ where the tractions are applied. Then, since A is a constant matrix whose column vectors are linearly independent as in equation(8), the tractions r ' which minimize equation (12) is given using the generalized inverse of A as

r' =

{A^A)-'A^u.

(13)

Hence, equation (12) can be rewritten as

mm\\{I-A{A^A)-'^A^}u\\

(14)

Xnj

The positions (x^^^Xn^ • • • ,«n /) which minimizes equation (14) among all combinations for choosing m' elements from all the boundary elements in V^^\ are the positions where the tractions are applied.

Indification of tractions

125

Evaluating equation (13) for these (^^15^712 * * * ?^n /) gives the applied tractions at these points.

NUMERICAL EXPERIMENTS The above algorithms for the inverse analysis were implemented for the three-dimensional problem, and numerical experiment results will be presented in order to demonstrate the effectiveness of the algorithm. The experiment model is a hemisphere of radius 22mm made of elastic material (silicon rubber) with Young's modulus 5.0 x 10^ Pa and Poisson's ratio near 0.5, just as in the actual tactile sensor proposed in [1,2]. The base of the hemisphere is fixed (no displacement) and tractions are applied at some points on the hemisphere. For the boundary element discretization of the model, 8 elements along the radius of the base of the hem^isphere, 16 elements along the longitude and 8 elements along the latitude of the hemisphere were taken, resulting in 64 elements on the base and 128 elements on the hemisphere with a total of 192 elements, and the node of each element was taken at the centre of each element, as shown in Fig.3.

Figure 3: Boundary element discretization (left: hemispherical surface, right: base), where the points denote the nodes. Case when Traction is Applied at a Single Point The numerical experiment was done as follows. First, the forward problem was computed for a point traction of magnitude 300 Pa and arbitrary direction appHed at some point on the hemisphere. The displacements caused by this traction at several (two or four) internal points were then computed. The coordinates of the internal points were: (Case of 2 internal points) • (Case of 4 internal points) •

(1,1,17),(-1,-1,17), (1,1,17), ( - 1 , - 1 , 1 7 ) , (1, - 1 , 1 7 ) , ( - 1 , 1 , 1 7 )

(unit: mm).

reflecting feasible positions for placing the ultra sound cells for displacement measurements in the actual tactile sensor. The magnitude of the resulting displacements at the internal points were 0 ( 1 )mm.

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Then, for the inverse problem, 'measurement' errors of either O(10~^)mm, O(10~^)mm or O(10"^)mm were added to the displacement results obtained by the forward analysis, and then the inverse problem of estimating the applied traction was solved using the proposed algorithm. The same experiment was done, changing the position Xn where the traction r was applied. As a result, when the error of the displacement measurement was O(10~^)mm or O(10~^)mm, the position where the traction was applied was correctly estimated, and the relative error of the estimated traction (size and components) were 0(10"^) and O(10~^), respectively, whether the number of internal points for displacement measurement were two or four. This is reasonable, since the number of unknowns for the traction and its position is six altogether, which can be matched by the number of components of the displacement information for two internal points. Fig.4 shows the relative error of the estimated traction as the latitude of the position where the traction was appHed changed, with the longitude fixed. Displacement was given at two interior points with 'measurement' error of O(10~^)mm. The relative error of the estimated traction size and x,y,z components were derived by dividing the respective absolute error by the size of the traction. The relative error of the estimated traction is seen to be 0(10"^).

Figure 4: Case when traction is applied at a single point. ( Relative error of estimated traction vs. Latitude of the position where the traction is applied. Two interior points for displacement measurement with error: O(10~^)mm. ) When the error of the displacement measurement was O(10~^)mm, the estimated position of the traction Xn was dilferent from the true position, and the estimated traction T was far from the true traction, whether the number of internal points for displacement measurement were two or four. In practice, the usual resolution of the sensor for displacement measurement embedded in the tactile sensor in [1, 2, 8] is O(10-2)mm for each displacement component (ux^Uy^Uz), and it takes a long time to improve that resolution to O(10~^)mm by taking averages over a large number of measurements. Therefore, we still need to improve the inverse analysis for the case when the error of the displacement measurement is O(10~^)mm. This may be done by using regularization

Indification of tractions

111

techniques, refining the discretization, or optimizing the position of the internal points for displacement measurement. The computations were done on a 200MHz PC. The precomputation for the matrix set up took 19 seconds, while the actual search for the inverse analysis took only one second, which is sufficiently small for real-time application.

Case when Tractions are Applied at Two Points Next, we will test the inverse algorithm for the case when two tractions are applied on the same hemispherical model. This time displacement measurements are given at four interior points, which are now necessary to identify the two tractions and the points where they are applied. Similar to the previous case, 'measurement errors' are added to the forward analysis results, and the measurement accuracy required to identify the position of the two tractions will be observed. The size of the two tractions are 300 Pa, and the directions of the tractions are taken at random. The position where one of the tractions is applied is fixed at longitude= 7r/16 and latitude= 137r/32 radians. The position of the other traction is varied.

I

I

0.001

Figure 5: Case when tractions are appUed at two points. ( Lines: Relative error of the estimated tractions. Bar: Necessary precision (mm) for the displacement measurement at the internal points vs. Latitude of the position where the second traction is applied. Four interior points for displacement measurement.)

In Fig. 5, the relative error of the estimated traction (size and x^y^z components) are shown in lines, and the necessary precision (mm) for the displacement measurement at the internal points is shown in bars, as the latitude of the position where the second traction is applied is varied. For most positions, the position of the two tractions are correctly estimated and the relative error of the estimated traction is O(10~^) if the measurement precision for the internal displacement is O(10~^)mm. However, more precision is required when the two points where the tractions are applied are close together. In this case the

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M. Nakajima et al

inverse analysis is ill-posed and regularization is required. This time, the computation time for the exact search in the inverse analysis was 160 seconds, since the total number of combinations for the positions where two tractions are appHed is significantly larger than for one. Heuristic iterative algorithms were attempted to reduce the number of searches without success. This W2is because the optimization problem for equation (14) has many local minimums. Therefore, a more efficient implementation of the exact search algorithm is required to speed-up the process so that it may be used in real-time application.

CONCLUSIONS In this paper, we proposed an inverse algorithm using the boundary element method for estimating tractions and the position where they are applied on the boundary of a three-dimnsional elastic body, from information of displacement at internal points of the body caused by the tractions. Numerical experiments for a model tactile sensor showed that for a single traction, when the measurement error of the internal displacement is less than O(10~^)mm, the appHed traction and its position can be estimated with sufficient accuracy, with two internal points for displacement measurement. However, when the measurement error of the internal displacement was O(10~^)mm, the estimation was not satisfactory, and one needs to improve the inverse analysis using regularization techniques, refinement of the mesh, or optimization of the position of the internal points for measuring the displacements, etc. For the case when tractions are appfied at two points, similar results were obtained, although more measurement precision is required when the points are close together. The search for the inverse analysis also needs to be made more efficient for this case.

REFERENCES [1] Naruse, M., Tamayama, K., Shinoda, H., and Ando, S., (1994) Proc. 33rd Conf. of SICE^ pp. 123-124, (in Japanese). [2] Morohashi, T., Imai, R., Muramatu, G. and Ando, S., (1996) Technical Digest of the 14th Sensor Symposium, pp. 109-112. [3] Dennis, B.H. and Dulikravich, G.S., (1998) Inverse Problems in Engineering Mechanics, Proc. Int. Symp. on Inverse Problems in Eng. Mech. (ISIP'98), pp. 61-70. [4] Kubo, S. and Kawaguchi, M., (1998) Inverse Problems in Eng, Mech., pp. 353-361. [5] Ohura, Y., Kobayashi, K. and Onishi, K., (1998) Engineering Analysis with Elements, Vol. 20, No. 4, pp327-335.

Boundary

[6] Nakajima, M., Watanabe, S., Hayami, K., (1999) Proc. 1999 Anual Meet of JSIAM, pp. 104-105, (in Japanese). [7] Nakajima, M., Terao, J., Watanabe, S., Ando, S. and Hayami, K., (1999) Proc. 16th Japan Nat. Symp. on Boundary Element Methods, pp. 103-108, (in Japanese). [8] Terao, J., Abe, M., and Ando, S., (1999) Proc. 38th Conf of SICE, pp. 343-344, (in Japanese).

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

129

IDENTIFICATION OF D Y N A M I C PRESSURE DISTRIBUTION A P P L I E D TO T H E ELASTIC THIN PLATE M. ARAI, T. NISHIDA and T. ADACHI Department of Mechanical Engineering and Science, Tokyo Institute of Technology, 2-12-1,0-okayama, Meguro-ku, Tokyo, 152-8552, Japan

ABSTRACT This paper deal with a strategy for identification of dynamic pressure distribution applied to the elastic thin plate. Boundary element method is adopted to obtain discretized matrix relation between input loading and output signal, where strain information is used as a supplementary information of the inverse analysis in the present study. Laplace transform and numerical inverse Laplace inversion are introduced in order to treat the dynamic behavior. The coefficient matrix to be solved is given on the Laplace-transformed domain as the form of transfer function relating input and output signals. Since the inverse analysis requires regular iz at ion to stabiHze the ill-posed solutions, Tikhonov regularization has been employed with singular value decomposition, where the optimal parameter of Tikhonov method was determined by Hansen's L-curve method. Through some numerical simulation on the circular plate subjected to dynamically distributed loading, the usefulness of the present method based on the Laplace-transformed-BEM is demonstrated in detail. KEYWORDS Inverse Problem, Boundary Element Method, Plate Bending, Pressure Distribution, Dynamic Loading, Laplace Transform INTRODUCTION It is widely recognized that a fracture on the aeronautical structure caused by collision of a foreign object such £LS birds or heilstones is one of the most important problems in this fields[l]. Not only in the aeronautical field but also in general engineering fields, the collision between structures and foreign objects involve some difficulties in the numerical treatment, because it is very difficult to predict exactly the intensity, contact area and the time history of the impact load. Already some experimental techniques have been suggested to measure the distributed load history by a piezoelectric sensor, for example[2]. However, such a direct measurement still remain a problem, because the geometry or rigidity of the contact surface can would be varied owing to putting the sensors on the surface. In recent years, some useful technique to obtain impact load without direct measuring has been developed[3,4,5,6]. In this approach, the applied loading can be measured by indirect manner, that is, inverse calculation is applied to determine the load history using supplementary informations of output signals such as strain or displacement on the elastic body. However a formulation to determine the pressure distribution, especially dynamically

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M. Arai, T. Nishida and T. Adachi

changed pressure, has not been investigated yet that we know of. In this study, the inverse analysis to estimate the unknown dynamic pressure distribution applying on the elastic thin plate is studied. The transmission matrix, which describe a linear relation between distributed load and internal strain, is introduced. The boundary element analysis [7,8] are adopted to compute the transmission matrix on the Laplace-transformed domain, and the numerical inversion of Laplace transform is employed according to the formula of Krings and Waller's method[9]. Tikhonov regularization[lO] is applied to stabilize the ill-posed and unstable inverse solution for the transmission matrix and the operator of numerical Laplace inversion. A typical demonstration about the circular plate subjected to transverse, impact and distributed loading is conducted to illustrate the effectiveness of the present approach. BOUNDARY ELEMENT ANALYSIS Consider a linear elastic, homogeneous and isotropic thin plate subjected to unsteady transverse pressure / . According to classical plate bending theory, a deflection w is governed by the following biharmonic differential equation, DV'w

+ ph^^

= f,

(1)

where f, p and h are time, density and thickness of the plate, respectively. D = Eh^/12{1 — i/^) denotes the flexural rigidity of the plate with the elastic modulus E and Poisson's ratio u. V4(.) is V2(V2(.)), where V^{') is the Laplacian d^dxl -h d^dxl Let Q be a bounded region with total boundary F, where normal rotation T^, bending moment Mn, twisting moment Mns and equivalent shear force Vn are defined by the diflFerential forms T„ = ^ on

^" = -^^^

(2)

- -dT

^')

at a smooth regular point on the boundary F, where d/dn and d/ds denote normal and tangential derivative operators, respectively, and d/dF is derivative operator for the coordinate along the boundary F. In the Laplace-transformed domain, Eq.(l) can be rewritten as DV^W

-h phf^'^w = / ,

(6)

with a Laplace transformation parameter K,. Then, let us employ a fundamental solution '^*(P5Q) satisfying £>V%*(p, Q) + phK'iv*{p, Q) = S{r), r = |pQ|,

(7)

where S{r) is a Dirac's delta function. Namely, w*{p^ Q) is a deflection on the source point p when a unit concentrate and impulsive load 6{i) is applied on the point Q. The concrete form of w* is expressed by Kelvin's function:

Identification of dynamic pressure distribution

**(p,Q) = - ^ ^ K e i o ( Z ) ,

131

(8)

where 77 = KJph/D and Z = yjr\r. Multiplying the fundamental solution 'ii)*(p, Q) with Eq.(6) and expanding it according to Gauss's divergence theorem, we have the following boundary relation between physical quantities on F and the internal deflection it'(p) in Q.

^(p) - /^{^*(P, Q)K(Q) - f:(p, Q)M.(Q) +

M:(p,Q)fn(Q) - K * ( P , Q ) ^ ( Q ) } ^ r ( Q ) + |^7i;*(p,R)/(R)dO(R),

(9)

where Q is a integral point defined on the boundary F. T^, M* and V* are fundamental solutions derived by substituting Eq.(8) into Eq.(2),(3) and (5), respectively. In the present study, the total boundary F is treated as smooth without corner point, therefore we may neglect a discontinuous term about twisting moment in Eq.(9). Moving the source point p to the point P on the boundary, the following boundary integral equation can be derived: i ^ ( P ) = / ^ { ^ * ( P , Q)V;(Q) - f : ( P , Q ) M . ( Q ) 2

+

M : ( P , Q ) f . ( Q ) - F ; ( P , Q)^(Q)}tiF(Q) + \^ ^ * ( P , R)/(R)ci^(R),

(10)

Then, let us approximate the density functions vf^^.. .y* and transverse pressure /(-R) in Eq,(10) by a suitable boundary element or regional element with a polynomial function, respectively. According to the conventional BEM formulation for the plate bending problems, we have the matrix relation as the form: A{K)X^K)

= y(Ac) + B{K)f{K),

(11)

where ^ is a unknown vector of the density functions, y is a known vector derived by the boundary conditions, A and B are coefficient matrix, respectively. Note that the all vector and matrix in the above equation defined on the Laplacd-transformed domain and pressure / is still unknown in the present inverse analysis. Here we let employ the bending strain as the supplementary infromation for the inverse analysis. Differentiating the Eq.(9) according to the definition of bending strain, we have the following boundary integral representation:

e-(p) = /^{^'(P,Q)K(Q) - fI(p,Q)M„(Q) +

Mlip, Q)f„(Q) - flip,

Q)w(Q)}dr(Q) + / r ( p , R ) / ( R ) d n ( R ) ,

(12)

where notation "^' indicates differentiation according to the definition of strain. The above equation can be rewritten as following matrix form: 5(1^) =C{I
+ Z{K) + D(A€)/(AC).

(13)

Eliminating unknown boundary quantities x from Eq.(11) and Eq.(13), we have s{K)=E{K)f{K).

(14)

By the above equation, the discretized transverse pressure /(/^) and strain S(K,) are

132

M Arai, T. Nishida and T. Adachi

related by the coefficient matrix E {K) on the Laplace-transformed domain. Note that a constant term included in the above equation is neglected here assuming that the deformation are uniformly 0 if the transverse pressure is 0. As a result, solving the simultaneous equation (14), the unknown transverse pressure / can be determined from supplementary information s. And then applying the numerical Laplace inversion, the time history of the pressure f{t) can be obtained consequently. In the present paper, the Krings and Waller's methods[5] based on the fast Fourier transformation (FFT) is used for the numerical inversion of Laplace transform. TIKHONOV REGULARIZATION Since inverse analyses essentially have ill-posedness, the numerical solutions generally become unstable and deteriorate owing to the noise and estimating error included in the supplementary information. In order to overcome the difficulties and stabilize the solutions, Tikhonov regularization[lO] is adopted for the determination of the inverse matrix of ^ . At first, let us apply singular value decomposition on the coefficient matrix in Eq.(14). Supposing the number of discretized pressure and supplementary information of strain to be A^ and M, respectively, the matrix E can be decomposed as: (15)

UAV^f=8,

where U, A and V are two dimensional matrix of M X M, M X N and A^ X TV, respectively. A is written as: Ai

0 (16)

0 where the number of m denotes the rank of the matrix E, and the A^ are singular values satisfying: Ai > A2 > • • • > A^ > 0, m< min(M, N).

(17)

In the above equation, the number of non-zero coefficients are at most m in the diagonal coefficients of A. Therefore the eq.(15) can be simplified to (18)

UmAmVlf =8,

where Um and Vm are the matrix of M X m and m X N, respectively. Am is a m by m diagonal matrix and the inverse one is: "l/Ai

0 I/A2

A-} = 0

(19) 1/A,

Finally, in the sense of the solution derived by least square methods, the unknown pressure

133

Identification of dynamic pressure distribution distribution / can be determined as following equation. / = Vr„A-^Uls.

(20)

By the way, the least squares solutions of Eq.(20) minimize the following functional 11 of the residual norm: (21)

U=\\Ef~-s\\\

Nevertheless in the inverse problems, the matrix E sometimes become ill-posed, so the obtained solutions generally tends to unstable. In order t o increase the accuracy and stability of the solutions, Tikhonov regularization is employed here. In the present paper, the following functional ft is selected adding a stabilizer term on the functional of Eq.(20):

(22)

n=\\Ef-s\? + a\W where a denotes a optimal parameter. As a result, the optimal solution /

given by Tikhonov regularization are given as

/ * = {ifE + aiy^lfe.

(23)

Applying singular value decomposition, we have 0

r=

AfTa

Ule.

(24)

Comparing Eq.(19) with Eq.(24), we can easily find that singular values l/Aj are replace by Aj/(Af -ha) in Eq.(24). Namely, giving the appropriate parameter a, it is expected that the the inverse solution obtained by Eq.(24) would be smoothed because the difference of the maximum and minimum value of the diagonal elements become smaller. In the Tikhonov regularization, we must select the value of parameter a appropriately. Generally, it is well known that the optimal parameter ao would exist in the range: A^ < ao < A?.

(25)

In the present paper, Hansen's L-curve method[11,12] is adopted to determine the aoThis method is based on the behavior of the L-curve which is a plot of the norm of the solution 11/11 and the residual \\Ef — s\\. The optimal value ceo is supposed to locate at a corner point of the L-curve. The corner point of the L-curve can be determined by spline interpolation[13,14] and finding out a maximum curvature point of the profile. On the other hand, in terms of Laplace transform, the numerical inversion can b e written as the following matrix operation: / = L-'f

(26)

where L~^ is a transfer function of the numerical Laplace inversion. It is well known that, the numerical Laplace inversion becomes ill-posed and sometimes gives unstable results, especially when the operation is adopted to the inverse analysis as the present study. Namely, the operator L~^ of inverse Laplace inversion should be determined carefully as the same t o the computation of matrix E (K) on the Laplacetransformed domain. Therefore, the Tikhonov regularization is employed again on the

134

M. Arai, T. Nishida and T. Adachi

Fig.l A circular plate with clamped edge subjected to transverse pressure

Fig.2 Division of boundary elements

t {Dlphf^la-

Fig.3 Division of regional elements for the pressure

Fig.4 Time histories of the transverse pressure (Exact histories)

numerical Laplace inversion. The optimal solution can be determined as those which minimize the following functional IT* [3]:

W = \\Lf-ff+a\\ff

(27)

The optimal parameter of a can be determined also by L-curve method according to the procedure described above. NUMERICAL EXAMPLES A basically numerical simulation on a circular plate with clamped edge subjected to unsteady transverse pressure was conducted to confirm the validity of present method. The analyzed model is shown in Fig.l, of which the boundary is divided into 32 isoparametricquadratic elements for the BEM analysis(Fig.2). Unknown transverse pressure is discretized by 54 linear triangle element as shown in Fig.3. Supplementary information of the strain Si and £2 sire given on the 37 nodal points of the triangle elements, so the total number of the supplementary information of strain becomes 74. The histories of the strain are calculated by BEM previously to the inverse

Identification of dynamic pressure distribution

• 0

0.5

135

1

t ipiphy'la'

Fig. 5 Time histories of identified pressure using Tikhonov regularization on the estimating the inverse matrix E {j = noise level included in supplementary infrormation [%])

Fig.6 Time histories of identified pressure using Tikhonov regularization on the estimating the inverse matrix E and inversion of Laplace transform (7 = noise level incluided in supplementary information [%])

analysis and added the artificial random noise. The amplitude of the noise is varied as 1%, 2% and 3% of the maximum strain. In the present demonstration, the dynamic pressure distribution is defined as f{xi,X2]t)

(28)

= g{xi,X2)h{t).

g{xi,X2) and h{t) are given as g{xi,X2) =g{r) = {

0

3a in r > -—,

4

(29)

136

M. Arai, T. Nishida and T. Adachi

Fig.7 Variation of the transverse pressure (Exact history)

Fig.8 Variation of the transverse pressure (Identified history ; noise level 7 = 1%)

where r = ^Jx\ -h x^/a, and

h{t) =

5t — - 1 in 0.2T < t < 0.4T T 3 - — in 0.4T < t < 0.6T T 0 m t<0.2T,t>o.er

(30)

where T is defined as T = TmaxI JD/pha'^^ Tmax is non-dimensional time specified as 2.0 in the study. Fig.4 shows exact time histories of the pressure on the nodal points, where the subscript (1, 19, 31, 37) of / denotes the location of the nodal point indicated in Fig.3. Histories of the pressure computed by inverse analysis at the node 1, 19, 31 and 37 are shown in Fig.5 and Fig.6, respectively, where Fig.5 shows the case using Tikhonov regularization on estimating the matrix E, while Fig.6 shows the case using the regularization additionally on the process of numerical Laplace inversion.

Identification of dynamic pressure distribution

137

As shown in Fig.5, if the Tikhonov regularization is applied only to the determination for the matrix E^ the accuracy of the solutions is decrease at the closing time rage. On the other hand, as shown in Fig.6, adopting the regularization on the Laplace inversion too, we can confirm that the stability of the solutions at the closing part can be increased well. Finally variation of the pressure distribution is shown in Fig.8 compared with exact distribution in Fig.7. Although the center part of the pressure seems to be identified relatively small owing to the discretization error, we can conclude that the present method is generally valid to identify the dynamic-unsteady pressure distribution applied to the elastic thin plate. CONCLUSIONS In the present paper, a strategy for identification of dynamic pressure distribution applied to the elastic thin plate was investigated in detail. Laplace transform and numerical inverse Laplace inversion are introduced in the present formulation to deal with the dynamic identification. It was shown that the transfer function derived on the Laplace-transformed domain as the matrix form, which relate the input load and output strain histories, can be regularized by Tikhonov method, and the optimal parameter of the Tikhonov regularization can be determined by Hansen's L-curve method appropriately. Furthermore in the process of the numerical Laplace inversion, the Tikhonov regularization is valid to stabilize the unstable behavior at the closing time range. Through some typical demonstration about a circular plate subjected to dynamically distributed pressure, the usefulness of the present method based on the Laplace-transformed-BEM was confirmed. ACKNOWLEDGEMENT The authors wish to acknowledge the helpful advice of Prof. H. Inoue in Department of Mechanical and IntelUgent Systems Engineering, Tokyo Institute of Technology.

REFERENCES 1. Morita, H. Hamamoto, A., Adachi, T., Nishimori, K. and Matsumoto, H. (1995), Influence of Impact Velocity and Lay-up Parameter on Impact Damage Resistance of CF/PEEK Laminates, AIAA/ASME/ASCE/AHS Structures^ (Structural Dynamics & Materials Conference), 2, pp.1093-1100. 2. Oda, K., Moto, S., Inoue, J. and Tabota, J. (1989). Development of Sensing System of Piezoelectric Ceramics, Trans. JSME, Ser.A, 55-513, pp.1230-1235, in Japanese. 3. Inoue, H., Kishimoto, K., Shibuya, T. and Harada, K. (1997), Regularization of Numerical Inversion of Laplace Transform for the Inverse Analysis of Impact Force, Trans. JSME, Ser.A, 63-616, pp.2503-2510, in Japanese. 4. Doyle, J. F. (1984), Determining the Dynamic Contact Law, Experimental Mechanics^ 24-1, pp.10-16. 5. Doyle, J. F. (1987), Determining the Contact Force During the Transverse Impact of Plates, Experimental Mechanics, 27-1, pp.68-72. 6. Chang, C. and Sun, C. T. (1989), Determining Transverse Impact Force on a Composite Laminate by Signal Decomposition, Experimental Mechanics, 29-4, pp.414-419.

138 7.

8. 9.

M. Arai, T. Nishida and T. Adachi Arai. M., Adachi, T. and Ujihashi, S. and Matsumoto, H. (1994), Highly Accurate Analysis for Thin Elastic Plate Bending Problem by Boundary Element Method, Trans. JSME, Ser.A, 60-570, pp.436-443, in Japanese. Arai, M., Adachi, T. and Matsumoto, H. (1997), Elastodynamic Plate Bending Analysis by Boundary Element Method with Laplace Transform, JSME Int. J., Ser.A, 4 0 - 1 , pp.65-73. Krings, W. and Waller, H. (1979), Contribution of the numerical treatment of partial differential equations with the Laplace transformation ( An application of the algorithm of the fast Fourier transformation). Int. J. Num. Meth. Eng., 14, pp. 1183-1196.

10. Hayabusa, K., Kishimoto, K, Inoue, H. and Shibuya, T. (1997), Boundary Element Inverse Analysis for Stress Separation in Thermoelastic Stress Analysis, Proc. ICM & M 97, (JSME), Tokyo Japan, pp.809-816. 11. Hansen, P. C. (1992), Analysis of Discrete Ill-posed Problems by Means of t h e L-Curve, SIAM Rev., 3 4 , pp.561-580. 12. Hansen,P.C.and 0'leary,D.P. (1993), T h e Use of the L-Curve in the Regularization of Discrete Ill-posed Problems, SIAM J. Sci. Comput., 14, pp.1487-1503. 13. Kishimoto, K., Inoue, H. and Shibuya, T . (1996), Determination of Stress Components from Thermoelastic Data by Boundary Element Inverse Analysis, Boundary Element Methods, M. Tanaka and Z. Yao (ads), (Proc. 7th Japan-China Symp. on Boundary Element Methods), Fukuoka, Japan, Elsevier, pp.561-580. 14. Kishimoto, K., Inoue, H., Shinbo, H. and Shibuya, T. (1997), Inverse Analysis Related to Stress Separation in Thermoelastic Stress Analysis, JSME Int. J., Ser.I, 4 0 - 2 , pp. 108-116.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

139

DETERMINATION OF UNSTEADY CONTAINER TEMPERATURES DURING FREEZING OF THREE-DIMENSIONAL ORGANS WITH CONSTRAINED THERMAL STRESSES B.H. DENNIS and G.S. DULIKRAVICH Multidisciplinary Analysis, Inverse Design, and Optimization (MAIDO) Department of Mechanical and Aerospace Engineering, Box 19018 The University of Texas at Arlington, Arlington, TX 76019-0018, U.S.A. ABSTRACT A fully automatic procedure for maximizing the minimal local freezing rate in an organ while maintaining the local thermal stresses below a specified level throughout the arbitrarily shaped and sized three-dimensional organ made of different materials has been developed. This was achieved by determining correct variation of unsteady temperature distribution on the walls of the three-dimensional freezing container. A time-accurate finite element computer program was used to predict unsteady heat conduction with phase change and thermal stresses within the realistically shaped organs. A micro-genetic algorithm was then used to achieve nonlinear constrained optimization of time-varying container wall temperature distribution so that the prescribed maximum allowable thermal stress levels are never exceeded throughout the organ. KEYWORDS Inverse unsteady boundary conditions, optimization, thermoelasticity, cryobiology. INTRODUCTION One of the serious difficulties encountered by surgeons involved with organ transplantation is caused by the shortage of available organs. Using current organ preservation protocols which have the organ packed in ice at 4 degrees Celsius with a solution called UW (for University of Wisconsin) that is a mix of electrolytes, organs have the following average shelf lives from harvest to implantation: heart 4-6 hrs, lungs 4-6 hrs, kidney 24-48 hrs, and liver 36-48 hrs. A possible solution would be to establish an organ bank that could store organs with different immunological properties in a frozen state for lengthy periods of time. When preserving living human organs (kidney, heart, lungs, spleen, liver, bone, etc.) for the purpose of performing transplant surgery, the organ could be cooled in a special cryo-protective agent (CPA) gelatin while perfused by a cooling CPA liquid to a prescribed subfreezing temperature and kept at this temperature until used. If the cooling rate is too high, strong residual thermal stresses will cause fractures in the frozen tissues. If the cooling rate is too slow, chemical decomposition in the tissue will make the organ useless [1]. Experiments have shown that although a whole organ does not survive freezing, cells and parts of the organ survive [2]. Thus, there has been a common belief that there is an optimal cooling rate for each particular type of tissue of an organ in order to maximize the survivability of the living cells and reduce the problem of future

140

B.H. Dennis and G.S. Dulikravich

rejection by the organ recipient's body. Most of the controlled rate cooling devices currently in use [3] employ either a liquid cooling bath with ethanol or liquid nitrogen as the heat-exchange medium or a cooling chamber with vaporized liquid nitrogen as the coolant. Freezing protocols that use a single optimal cooling rate at every point on the outside surface of an organ [2,4] result in considerably different values of local cooling rates inside the organ [5]. This situation, coupled with the possibility of rapidly changing thermal properties such as specific heat and thermal conductivity [5] near the phase front, makes the numerical modeling more difficult than in the case of simple heat diffusion. One method that offers a possible practical solution to freezing and thawing of organs is to immerse them in a gelatin thus assuring that the heat transfer from the outer surface of the organ will occur by pure conduction. Numerical analysis of organ freezing [6] predicts that thermal boundary conditions are not propagated uniformly into the interior, resulting in a nonuniform distribution of temperature histories and cooling rates throughout the spatial domain. A plausible objective could be to find the proper time variation of surface thermal conditions of the freezing container so that the optimal local cooling rates are achieved at each instant of time at every point in the organ. This concept was demonstrated as numerically feasible [7-12]. However, it has been impossible to preserve large organs even when the local cooling rates are apparently identical to those proven successful for small samples from the organ [2] because of the fractures caused by the thermal stresses. Other significant damaging mechanisms are ice crystal formation, blood vessel deterioration, and the toxicity effect of the CPA [13-16]. Thus, the objective during the freezing or thawing should not be to enforce experimentally obtained local optimal cooling rates since they apply to small tissue samples rather than whole organs. One of the main objectives should be to limit the thermal stresses that cause the fractures in the organs [13,15]. This paper deals with this particular objective. MATHEMATICAL MODEL The Navier equations for linear unsteady deformations u,v,w in three-dimensional Cartesian x,y,z coordinates are (inertia terms are expected to be negligible during the freezing or thawing) /.

( 1 ^x 3 u I 3x

/. ^ J (;i+G)

/

2 3 u

2 3 V 3x 3y

2 ^ 3 w -hGV^u + X = 0 3x 3z

9 2 A 3 V 3 w ' -HGV^v-hY = 0 "^-^"^ I 3x 3y 3y 3y 3z .

(1)

(2)

Determination of unsteady container temperatures

+

(X + G) 3x 3z

3y 3z

+ —— + GV^w + Z = 0 3z^

141

(3)

Here, A.=

E 2(l + v)

Ev (l + v)(l-2v)'

(4)

where E is the Young's modulus and v is the Poisson's ratio. The body forces per unit volume due to stresses caused by thermal expansion/contraction over the temperature range AT are 3(PAT) 3x

(5)

Y = -{3?I + 2 G )

3(PAT) ay

(6)

Z = -(3>. + 2 G )

a(PAT) az

(7)

This linear thermoelasticity system also includes unsteady energy conservation equation with latent heat of liquid/solid phase change lumped together with specific heat, that is.

at

:V-(kVT)

(8)

The effective specific heat is a combination of the actual specific heat and the temperature variation of the latent heat, L, which is incorporated in the volumetric enthalpy, H, so that

P^effective ~

an VH VH -srj^ ~ aT VTVT

Nl/2

(9)

All physical properties in this model are allowed to vary as function of space and temperature [17]. The latent heat was applied only in the mushy region, that is, at the points where the local instantaneous predicted temperature was between liquidus and solidus values. In systems of impure chemical composition, such as tissue, the phase change process involving release of latent heat is distributed over a range of temperatures rather than occurring at a single specific temperature. The nonlinear latent heat release can be modeled [6] by the application of the lever rule to the phase diagram for a binary, aqueous solution with a given initial composition. During CPA solidification the pattern of latent heat release implemented in the model may exhibit a strong influence on the calculated thermal histories.

142

B.H. Dennis and G.S. Dulikravich

DISCRETIZATION AND OPTIMIZATION Due to its simple shape and the relative availability of thermophysical data, we chose to demonstrate this optimized freezing protocol concept on an example of a dog kidney. The kidney inner region (cortex), kidney outer region (medulla), and the spherical container shapes were created by first generating three concentric cubes. Each of the six faces of each of the three cubes was discretized with a structured grid of quadrilateral cells. The three concentric cubes were then transformed into three concentric spheres by dividing x,y,z coordinates of every grid point on every original cube with the radial distance of the corresponding point on the most outer cube. The six faces of the most outer cube then became the six deformed quadrilateral patches making up the surface of the spherical freezing container. The kidney medulla and the imbedded cortex spherical shapes were then analytically transformed into concentric ellipsoids that were consequently analytically bent. The surface grids consisting of deformed quadrilaterals (Fig. 1) were then connected with quasi-radial lines thus creating a fully boundary conforming three-dimensional structured grid. Each deformed hexahedron type grid cell was then partitioned into 24 tetrahedron type grid cells. The surface variation of temperature on the spherical container wall was approximated with biquadratic Lagrange polynomials using 9 control points for each of the large six deformed quadrilateral patches forming the container wall thus resulting in a total of 26 design variables. Then, the transient temperature distribution was computed at every point of the organ. From this, the actual local cooling rates and thermal stresses were determined at each point of the organ. A nonlinear constrained function maximization method based on a genetic algorithm [20] was used after certain time interval, At, to optimize the temperature at each of the control points on the spherical container surface. That is, the new temperature distribution on the container walls was determined so that it maximizes the average cooling rate in the organ for the given time interval while keeping the local thermal stresses in the organ below a user specified maximum allowable value. The algorithm is outlined in Figure 2. The objective function, also referred to as fitness, F, that was maximized every time interval. At, was

F=

(10)

— + p

Ar

y^yield

J

where AT = Tfj^ai -Ti^itiai and At = tfj^ai - tjnitiai ^^^ P is a user specified penalty term. Notice that the cooling rate is a negative number. PHYSICAL PROPERTIES OF THE ORGAN AND THE CRYOPROTECTIVE GELATIN Although the actual kidney is composed of four distinctive tissues: cortex (the most outer layer), medulla (the congruent inner layer), pelvis (the central domain), and fat (the domain that connects the pelvis with a part of the concave portion of the kidney surface), in this proof of concept example we have chosen to model the kidney as composed of only medulla and cortex.

143

Determination of unsteady container temperatures

Table 1. Specific heat and latent heat for unfrozen dog kidney tissues and the gelatin [17]

Cortex

Specific heat per unit volume: pC = k/a [J m-3 C-1 ]

Latent heat per unit volume: pAH [J m-3 ]

3.68 X 10^

250 X 10^

3.88 X 10^

240 X 10^

2 x 10^

100 X 10^

Medulla Gelatin

Table 2. Thermal conductivity and diffusivity for unfrozen dog kidney tissues and gelatin [17] Coefficient of Coefficient of thermal conductivity: k thermal diffusivity: a [J s-1 m-1 C-1 ] [m2 s-1 ] Cortex

0.4905

Medulla

0.5065

Gelatin

0.1333 X 10"^ 0.1305 X 10"^

0.3

0.13 X 10'^

Table 3. Physical properties for frozen dog kidney bulk tissue and the gelatin Reference Properties & symbols Kidney tissue Gelatin 16; assumed heat capacity per unit volume (frozen): 1.8 X 10^ 1.5 X 10^ pCs [J m - ^ K - 1 ]

Thermal conductivity (frozen): ks [J s-1 m-1 K-1 ]

2.0

4.0

16; assumed

Temperature (liquidus): T [C] Temperature (solidus): T^ [C]

0.0 -2.0

0.0 -2.0

assumed assumed

0.1 xlO""^

16; assumed

Thermal expansion coefficient(frozen): 0.6 X 10""^

p[c-M

Young's modulus of elasticity (unfrozen): E [N m'^ ] Young's modulus of elasticity (frozen): E [N m"^ ] Poisson's ratio (unfrozen): v Poisson's ratio (frozen): v Max. stress (unfrozen): a [N va'^ ]

10 X 10^

^ x 10^

assumed

10 X 10^

1 X 10^

16; assumed

0.333 0.333

0.333 0.333

assumed assumed assumed

I x 10^

10.5 X 10^

Max. stress (frozen): a [N m'^ ]

132.2 X 10^

Yield stress (unfrozen): a [N m"^ ]

0.5 X 10^

1 0.4 X 10^

Yield stress (frozen): a [N m"^ ]

65 X 10^

1 50 X 10^

50 X 10^

14; assumed assumed 14; assumed

144

B.H. Dennis and G.S. Dulikravich

NUMERICAL RESULTS The system of equations (1-3) and (9) was integrated numerically using a finite element method on a tetrahedral non-structured grid, ILU and preconditioned Krylov subspace methods, objectoriented programming in C++ [18]. The accuracy of the finite element code for heat conduction involving solidification was verified though comparison with a known analytic solution [19]. The solidification of a liquid rod, which has an analytic solution, was simulated using a 3-D mesh of a rod composed of 480 parabolic tetrahedral elements. Figure 3 shows the variation of temperature with time for a specific point on the rod for both analytic and numerical solutions. The temperature was taken over a given time interval at a specific internal point inside the mesh. The numerical solution, though very accurate is not exactly the same as the analytic solution. In our FEM implementation of phase change, the temperature interval during which phase changes must be larger than zero. For calculating the rod case, a temperature interval of 1°C was used. However, the analytic solution corresponds to phase change of a pure substance for which this temperature interval is zero. This non-zero temperature interval in the FEM model is the most likely source of the slight discrepancy between the numerical and the analytic results. A very small temperature interval for the phase change can be used. However, in this case it was found that an unreasonably small time step was required to obtain solutions with excellent agreement with the analytic solution. The three-dimensional freezing protocol simulation and optimization algorithm was then applied to a geometry composed of three domains. The outer boundary was a spherical freezing container. Within the container was a kidney consisting of a medulla and the imbedded cortex (Fig. 1). The optimizer was applied after every At = 5 minutes. The penalty term in the objective function, P, was fixed at P = 100 when the maximum von Mises stress in the kidney domain, Omax, was greater than the local yield stress, ayieid- The penalty term was P = 0 for all other situations. The genetic algorithm (GA) used 4 bit strings to represent each of the 26 design variables (container surface node temperatures). Each of these variables was allowed to vary from 20°C to -30°C. A uniform crossover operator was used with a 50% chance of crossover. A fixed population size of 31 was used with a 2% chance of mutation. Each optimization cycle was run for 20 generations. Each optimization cycle was executed on our distributed memory parallel computer made of commodity computer components. Each analysis run, which was composed of simulation of three-dimensional heat conduction with a moving freezing front in this multi-domain region including thermal stress analysis, required 3.5 minutes on a single CPU. Parabolic elements were used for heat conduction and linear elements were used for the stress analysis. The heat conduction grid was composed of 8245 nodes and 5184 tetrahedral elements. The stress analysis grid was composed of 1315 nodes and 5184 tetrahedral elements. A time step of 15 seconds was used for each transient heat conduction analysis. Figure 4 shows the variation of the maximum stress found in the kidney domain for each of the optimized temperature distributions over time. Figure 5 shows the variation of the average cooling rate in the domain for each of the optimized temperature distributions over time. Notice that the cooling rate needs to be significantly reduced in the advanced stages of freezing. Figure 6 shows the temperature distribution for each time interval along a line formed by the

Determination of unsteady container temperatures

145

intersection of two cutting planes. Notice that by 50 minutes, the kidney has been completely frozen. It can also be seen that at some locations on the container surface the optimizer found that temperatures should increase after some time intervals (Figs. 7-9). This is caused by the imposed constraint on the maximum allowable local thermal stresses. A typical convergence history for the GA (Fig. 10) applied to the optimization of container surface temperature distributions shows that the GA finds improvements in only twenty generations. Similar proof of concept with the same software is possible for the optimization of thawing protocols where thermal stress is affected by the residual stresses developed during freezing [16]. CONCLUSIONS In this first of a kind proof of concept simulation it has been demonstrated that it is possible to control the damage caused by the thermal stresses during freezing of organs by periodically optimizing temperature distribution on the surface of the freezing container. Using more accurate thermophysical data, finer spatial and temporal discretization, more diverse tissue subdomains, and more geometrically complicated geometries of organs and containers is a relatively straightforward future extension of this work. A more challenging extension of this work would be to add internal perfusion during the freezing. ACKNOWLEDGEMENTS We would like to express our gratitude to Profs. Y. Rabin and P. S. Steif for the copies of their excellent papers, Dr. G. M. Fahy for the discussions of modern trends in cryobiology, Dr. Craig Bates for useful clinical information, and Mr. R. Krishnamurti for structured grid generation. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8.

9.

Mazur, P. (1970). Science, Vol. 168, pp. 939-949. Jacobsen, LA. and Pegg, D.E. (1984). Cryobiology, Vol. 21, pp. 377-384. Kelley, F.D., Phelan, R. M. and Levin, R.L. (1982). Cryobiology, Vol. 19, pp. 372-391. Fahy, G.M. (1981). Cryobiology, Vol. 18, pp. 550-570. Hayes, L.J., Diller, K.R., Lee, H.S. and Baxter, C.R. (1984). Cryo-Letters, Vol. 5, pp. 97-110. Hayes, L.J. and Diller, K.R. (1983). ASME J. of Energy Resources Technology, Vol. 105, pp. 431-435. Hayes, L.J., Dulikravich, G.S. and Chiang, T.L. (1987). In: Proceedings oflndASMEJSME Thermal Engineering Joint Conference, Honolulu, Hawaii, March 22-21, 1987. Madison, J.V., Dulikravich, G.S. and Hayes, L.J. (1987). In: Proceedings of International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES-II), G.S. Dulikravich (Ed.), The Pennsylvania State University, University Park, PA, October 26-28, 1987. Dulikravich, G.S. (June 1988). Applied Mechanics Rev., Vol. 41, No. 6, pp. 270-277.

146 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

B.H. Dennis and G.S. Dulikravich Dulikravich, G.S. and Hayes, L.J. (1988). In: Proc. of the Symposium on Computational Methods in Bioengineering, ASME BED-Vol.9, pp. 255-265, R.L. Spilker and B.R. Simon (Eds.), ASME Winter Annual Meeting, Chicago, IL, Nov. 27 - Dec. 2, 1988. Dulikravich, G.S., Madison, J.V. and Hayes, L.J. (1989). In: Proc. of 1st Pan-American Congress of Applied Mechanics (PACAM-II), pp. 420-423, C.R. Steele and L. Bevilacqua (Eds.), Rio de Janeiro, Brazil, January 3-6, 1989. Ambrose, C , Hayes, L.J. and Dulikravich, G.S. (1989). In: Proc. of National Heat Transfer Conf, AIChE Symposium Series 269, Vol. 85, pp. 440-405, S.B. Yilmaz (Ed.), Philadelphia, PA, Aug. 6-9, 1989. Rabin, Y., Steif, P.S., Taylor, M.J., Julian, T.B., and Wolmark, N. (1996). Cryobiology, 33, pp. 472-482. Rabin, Y., and Steif, P.S., (1996). Cryobiology, 33, pp. 276-290. Rabin, Y., and Steif, P.S., (1998). ASME J. ofAppl Mechanics, Vol. 65, pp.328-333. Rabin, Y., and Steif, P.S., (2000). Int. J. of Solids and Structures, 37, pp. 2363-2375. Valvano, J.W., Cochran, J.R. and Diller, K.R., (May 1985). International Journal of Thermophysics, Vol. 6, No. 3, pp. 301-310. Dennis, B.H. and Dulikravich, G.S. (1999). ASME Journal of Heat Transfer, Vol. 121, pp. 537-545. Carslaw, H.S., and Jaeger, J.C, (1959). Conduction of Heat in Solids, Clarendon Press, Oxford. Dennis, B.H., Dulikravich, G.S. and Han, Z.-X. (1999). ASME paper 99-GT-441. START

C Ti

=

J

T ,ni,,al

t = 0

5

tf= t , + A t run G A with a n a l y s i s u s i n g t = t | t o t , a n d T = Tj

Figure 1. Surface grids for outer domain (medulla) and inner domain (cortex) of a kidney configuration.

Figure 2. Algorithm for inverse determination of unsteady thermal boundary conditions.

Determination of unsteady container temperatures

20

30

Ul

40

time (min)

Figure 3. Comparison of analytic and numerical solutions for the freezing of a 1D slab.

Figure 5. Time variation of average cooling rate in the kidney for optimized container surface temperature distributions.

20 15 10

Q.

S

5

15

0 "-

-5 -10 -15

-20[

, 1 , ,

10

, , 1

20

,,,!,, 30

time (min)

,

, 1 . , , , 1

40

Figure 4. Variation of maximum stress with time for optimized temperature distributions.

-25

50

Figure 6. Temperature distribution in time along a line formed by the intersection of x-z plane and x-y plane (see Figure 1).

148

B.H. Dennis and G.S. Dulikravich

Figure 7.Surface temperature distribution at 5 minutes.

Figure 9.Surface temperature distribution at 50 minutes.

Figure 8. Surface temperature distribution at 30 minutes.

Figure 10. Typical convergence history for GA applied to the optimization of surface temperature distributions.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

149

ANALYSIS OF INVERSE BOUNDARY VALUE PROBLEM BY THE ALTERNATING BOUNDARY ELEMENT INVERSION SCHEME AND ITS IMPROVEMENT USING BOUNDARY DIVISION

Shiro KUBO, and Akira FURUKAWA Department of Mechanical Engineering and Systems Graduate School of Engineering, Osaka University 2-1, Yamadaoka, Suita, Osaka 565-0871 Japan Phone: +81-6-6879-7304, Facsimile: +81-6-6879-7305 e-mail: [email protected]

ABSTRACT In the present paper an inverse boundary value problem related to crack identification was solved using the alternating boundary element inversion scheme proposed by Kolzov et al. It was found that the conventional alternating boundary element inversion scheme gave a reasonable solution, which converged slowly to the real one. The Fourier expansion was applied to examine the convergence rates of high and low frequency modes of electric potential distribution. It was found that higher order modes converged more slowly. The methods of boundary division and division-site shift were then proposed for achieving better convergence. It was found these methods worked well for obtaining better convergence and better accuracy in estimated boundary conditions. KEYWORDS Inverse Boundary Value Problem, Alternating Boundary Element Inversion Scheme, Laplace Field, Boundary Division, Division-Site Shift, Crack Identification INTRODUCTION Inverse boundary value problems deal with the estimation of boundary conditions on incompletely prescribed boundaries. To make the problem solvable, over-prescribed boundary may be introduced, where boundary values are excessively prescribed. The boundary element method or the finite element method can be used for estimating the boundary

150

S. Kubo and A. Furukawa

values on the incompletely-prescribed boundary from the over-prescribed boundary values. The boundary element method has an advantage over the finite element method, since the over-prescribed values and the values to be estimated are defined on the boundary and these can be directly interrelated by the boundary element method. The inverse boundary integral equation method [1, 2] is based on this idea. The boundary condition prescribed on the over-prescribed boundary can be introduced in the boundary element matrix equation, which can be solved for the unknown boundary values on the incompletely-prescribed boundary. Since the matrix is severely ill-conditioned due to the ill-posedness of the problem, the solution of the matrix equation gives an unstable solution. The regularization is then necessary for obtaining a reasonable solution. The alternating boundary element inversion scheme, which used the well-posed direct analyses only, was proposed by Kolzov et al [3] as an iterative method for solving the inverse boundary value problems. They discussed the convergence of solution obtained by using the scheme. Lesnic et al [4-6] examined the applicability of the scheme to the identification of boundary values for the Laplace field. In the present paper an inverse boundary value problem related to crack identification is solved using the alternating boundary element inversion scheme. For achieving better convergence the methods of boundary division and division-site shift are proposed. Numerical simulations are made to examine the applicability of the alternating boundary element inversio scheme and the usefulness of these methods. ALTERNATING BOUNADARY ELEMENT INVERSION SCHEME Inverse boundary integral equation method uses the over-prescribed Neumann condition and Dirichlet condition on the over-prescribed boundary, together with the Neumann condition on the Neumann boundary and the Dirichlet condition on the Dirichlet boundary. On the other hand in the alternating boundary element inversion scheme, the boundary conditions on the incompletely-prescribed boundary are estimated using one of the Neumann condition and the Dirichlet condition on the over-prescribed boundary alternately and iteratively. As an example consider a two-dimensional region ABCD, where the Lapalce field develops: the governing equation for potential 0 in the Cartesian coordinates ^1X2 is given as,

dmdx\ -h d^ldxl = 0.

(1)

AD is the incompletely-prescribed boundary, where neither potential 0 nor flux q is prescribed, where flux q is defined as the normal derivative of (j) as, q = d(l)/dn = ^—rii, dxi with Hi being the unit normal vector on the boundary.

(2)

Analysis of inverse boundary value problem

151

^ =0

Fig. 1

Two-dimensional region with incompletely-prescribed boundary.

To estimate the boundary condition on the incompletely-prescribed boundary, the overprescribed boundary BC is introduced, where (j) and q are simultaneously specified. On the rest of the boundary the Dirichlet condition or the Neumann condition is prescribed. In the present example the Neumann condition ^ = 0 is given. Alternating boundary element inversion scheme consists of the following steps. Step 1 On the over-prescribed boundary the Neumann condition is employed. Give an initial guess of the Dirichlet condition on the incompletely-prescribed boundary. On the rest of the boundary the prescribed boundary condition is used. Then solve the well-posed direct boundary value problem with these boundary conditions using the boundary element method. Step 2 On the over-prescribed boundary the Dirichlet boundary condition is used. On the incompletely-prescribed boundary the Neumann boundary condition obtained in the preceding step is used. On the rest of the boundary the prescribed boundary condition is used. Then solve the well-posed direct boundary value problem with these boundary conditions using the boundary element method. Step 3 On the over-prescribed boundary the Neumann boundary condition is used. On the incompletely-prescribed boundary the Dirichlet boundary condition obtained in the preceding step is used. On the rest of the boundary the prescribed boundary condition is used. Then solve the well-posed direct boundary value problem with these boundary conditions using the boundary element method. Step 4 Iterate the direct boundary element analyses as in Steps 2 and 3. NUMERICAL SIMULATION OF INVERSE BOUNDARY VALUE ANALYSES USING ALTERNATING BOUNDARY ELEMENT INVERSION SCHEME The problem of crack identification can be reduced to an inverse boundary value problem, when the plane containing a crack is known in advance. The crack can be identified as flux-

152

S. Kubo and A. Furukawa

free part on the crack-existing plane. A rectangular region shown in Fig. 1 is considered. The aspect ratio of the region is 0.2 and height is 20 and the width is 100. The crack-existing plane coincides with AD, which is regarded as the incompletely-prescribed boundary. Two types of cracks were employed in the numerical simulations: a central crack whose half length is 20, and an eccentric crack whose tips are located at x = 30 and x = 85. In the boundary element discretization 20 boundary elements were used for each of the top boundary AD and the bottom boundary BC, while each of the side boundaries AB and CD were discretized using 10 elements. To generate the over-prescribed boundary values on BC, direct pre-analyses were made using the boundary element method, employing the following boundary conditions. For the cracked part of AD ^ = 0 was used, and for the uncracked part of AD (/) = 0 was employed. On AB and CD flux q was set to be 0, and ^ = 1 was employed on BC. Values of (j) and q calculated by the direct pre-analyses were used as the over-prescribed boundary values on BC. The alternating boundary element inversion scheme was applied to the analysis of the inverse boundary value problem. In the first step of the analysis (j) = 1 was used as the initial guess on AD. Potential cj) and flux q estimated by the alternating boundary element inversion scheme after 300,000 steps are shown in Figs. 2(a) and (b), respectively. It is seen that the estimated potential (j) agrees well with the real one. The estimated flux agrees well with the real one except near the crack tips, where flux shows singularity. The convergence is, however, very slow. Similar results were obtained for the eccentric crack. EXAMINATION OF CONVERGENCE MODES USING FOURIER EXPANSION To examine what kind of modes of electric potential distributions converged faster than other modes, the estimated distributions were expressed in terms of the finite Fourier expansion of the form: Ao/2 + ^ {AnCos{n7rx/L) -h BnSm{n7rx/L)), n

(3)

where L is the half width of the region and L = 50 in this study. It was found that the low frequency terms tended to converge fast. On the other hand the high frequency terms converged slowly or showed tendency to diverge. PROPOSAL OF METHODS OF BOUNDARY DIVISION AND DIVISION-SITE SHIFT AND THEIR EXAMINATION From the results shown in the foregoing, it is suggested that the method of boundary division described in the followings may be effective to improve the convergence of high frequency mode, and therefore to improve the total convergence and accuracy of the so-

153

Analysis of inverse boundary value problem

X Real n Estimated

30.00 g ^ ^ - 5 1 — —

20.00

I

^ lg-

10.00

-M

gg as ^ gs X X ^

% -?(, m ^ as-g^

0.00

-10.00

50

100

Location , x

(a) Potential 1.00 0.00 C5H - 1 . 0 0

X

IT^^

WWW

LH -2.00

-3.00 -4.00

50

100

Location , x

(b) Flux Fig. 2 Potential (j) and flux q estimated by conventional alternating boundary element inversion scheme.

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S. Kubo and A. Furukawa

lution. The over-prescribed boundary is divided into two parts, Fai and r32, and the incompletely-prescribed boundary is divided into FQI and ro2. In the 2A:-th step of iterative calculations with k being an integer, the Neumann boundary condition and the Dirichlet boundary condition of the over-prescribed boundary values are used on Fsi and on r32 respectively, while the Dirichlet boundary value and the Neumann boundary value obtained in the preceding step are used on FQI and on F02 respectively. In the (2A: + l)-th step the Dirichlet boundary condition and the Neumann boundary condition of the overprescribed boundary values are used on F31 and on F32 respectively, and the Neumann boundary value and the Dirichlet boundary value obtained in the preceding step are used on Foi and on F02 respectively. Numerical simulations were made to examine the usefulness of the method of boundary division. The left half and the right half of BC are taken as F31 and F32, respectively, and the left half and the right half of AD are taken as FQI and F02, respectively. An immediate application of the method of boundary division from the 2nd step did not give good results, and then the method was applied from the 1000-th step. The potential and flux estimated after 300,000 steps agreed better with the exact ones than those in Fig. 2. For further improvement of the convergence and the accuracy, the method of successive boundary division was considered, in which the over-prescribed boundary and the incompletely-prescribed boundary were subdivided into 4, 8 and 16 parts successively. At the 1000-th step boundary AD and BC were divided into two, and the boundaries were subdivided and the number of divisions were doubled in the following three 10,000 steps. The location of division-site may have an effect of the convergence and the accuracy. The shift of division-site together with boundary subdivisions may be effective. The divisionsites were shifted at the same steps of the boundary subdivisions. The numbers of divisions of the boundaries AD and BC were doubled after 1000 steps, and following 2000 steps, 4000 steps and 8000 steps of calculations. The potential and flux estimated after 300,000 steps are shown in Figs. 3(a) and (b), respectively. It is seen that the estimated distributions agree much better with the exact ones than those in Fig. 2. To compare the effectiveness of the methods of boundary division and division-site shift, the squared residuals were evaluated between the estimated boundary values of 0 and flux q after 300,000 steps and real ones. Table 1 shows the comparison. The variations of the residuals for potential ^ and flux q with the number of steps are shown in Figs. 4(a) and (b), respectively. It is seen in Table 1 that the residuals after 300,000 steps for the values of (f) and q estimated using the methods of boundary division and division-site shift are one-tenth of those obtained by using the conventional alternating boundary element inversion scheme without

155

Analysis of inverse boundary value problem

X Real n Estimated

30.00

^

gg M asas

20.00 10.00 0.00

as as SB §K ^ >^-

-IB-W-I&-*

-10.00 50

100

Location , x

(a) Potential 1.00 0.00 c^ -1.00 X

iS y&

H E ^

^ g

i^ -2.00 -3.00 -4.00

50

100

Location , x

(b) Flux Fig. 3 Potential 0 and flux q estimated by alternating boundary element inversion scheme with boundary division and division-site shift.

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S. Kubo and A. Furukawa

Conventional ^ Single boundary division Successive boundary division •Successive boundary division and division-site shift

50000

100000 150000 200000 Number of steps

250000

300000

250000

300000

(a) Potential

0

50000

100000

150000

200000

Number of steps

(b) Flux Fig. 4

Variations of residuals of estimated i)()tential and flux with steps.

157

Analysis of inverse boundary value problem Table 1

Residuals of estimated potential and flux after 300,000 steps.

Conventional Single boundary division Successive boundary division Successive boundary division and Division-site shift

Residual of potential

25.86 20.72 6.789 2.740

Residual of flux

5.553 4.662 1.618 0.6753

the boundary division and the division-site shift. From Fig. 4 the rate of convergence is higher when the boundary division and the division-site shift are used, even though the residuals increase just after the boundary divisions. When simulated random noise was introduced in the over-prescribed boundary values, the boundary values estimated by the alternating boundary element inversion scheme approached the real ones for small numbers of calculation steps, and then showed the tendency to diverge [7]. There was an optimum number of steps, which gave the best results in the calculation using the alternating boundary element inversion scheme. The number of steps can be regarded as an regularization parameter of the alternating boundary element inversion scheme. The optimum number of steps was successfully evaluated using the discrepancy principle [8]. CONCLUSIONS The alternating boundary element inversion scheme was proposed by Kolzov et al as an iterative method for solving the inverse boundary value problems. The inverse boundary value problem related to crack identification was solved using the alternating boundary element inversion scheme. It was found that this conventional scheme gave a reasonable solution, which converged slowly to the real one. For achieving better convergence the methods of boundary division and division-site shift were proposed. It was found that these methods worked well for obtaining better convergence and accuracy of the solution. ACKNOWLEDGMENTS This study was partly supported by the Grant-in Aid for Scientific Research, Ministry of Education, Science, Sports and Culture.

REFERENCES 1. Ohji, K., Kubo, S. and Sakagami, T. (1985), Trans. Jpn. Soc. Mech. Eng., Ser.A, 5 1 , pp.1818-1825. 2. Kubo, S., Sakagami, T. and Ohji, K. (1986), Computational Mechanics '86, (ed. by Yagawa, G. and Atluri, S.N.), Springer-Verlag, Berlin, pp.V-339-344. 3. Kolov, V.A., Maz'ya, V.G. and Formin, A.V. (1991), Comput. Maths. Math. Phys., 3 1 , pp.45-52. 4. Lesnic, D., Elliot, L. and Ingham, D.B. (1997), Eng. Analysis with Boundary Ele-

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ments, 20, pp.123-133. 5. Lesnic, D., Elliot, L. and Ingham, D.B. (1997), Inverse Problems in Eng., 5, pp.l45168. 6. Lesnic, D., Elliot, L. and Ingham, D.B. (1998), Inverse Problems in Eng., 6, pp.255279. 7. Kubo, S. and Furukawa, A. (2000), Preprint of Jpn. Soc. Mech. Eng., (to be published). 8. Tikhonov, A.N. and Arsenin, V.Y. (1977), Solutions of Ill-Posed Problems, John Willy and Sons, New York.

Material Property Determination

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

161

METHOD FOR IDENTIFICATION OF ELASTIC PROPERTIES OF LAMINATES

R. RIKARDS Institute of Computer Analysis of Structures Riga Technical University, Latvia, Email: [email protected]

ABSTRACT A numerical-experimental method for identification of elastic properties of laminated composites from the experimental results is developed. It is proposed to use the method of experiment design and the response surface approach to solve the identification (inverse) problems. The response surface approximations are obtained by using the information on the behavior of a structure in the reference points of the experiment design. The finite element modeling of the structure is performed only in the reference points. Therefore, a significant reduction (about 50-100 times) in calculations of the identification functional can be achieved in comparison with the conventional methods of minimization. The functional to be minimized describes the diff^erence between the measured and numerically calculated eigenfrequencies of structure. By minimizing the functional the identification parameters are obtained. It was shown that identification functional is convex if the stiffness matrix linearly depends on unknown parameters. The method is employed to identify the elastic properties of the laminates from the measured eigenfrequencies of the cross-ply composite plates. KEYWORDS Identification, laminated composites, elastic properties, vibrations. 1

INTRODUCTION

During the last years investigations for developing a new technique for material identification, the so-called mixed numerical-experimental technique, have started [1,2]. Numericalexperimental identification methods are mainly used in structural applications. For example, elastic properties of laminated composites have been identified by using experimental eigenfrequencies [2]. The stiffness parameters were identified from the measured natural frequencies of the laminated composite plate by direct minimization of the identification functional. In the present study a numerical-experimental method for identification of mechanical

162

R.Rikards

properties of laminated composites from the experimental results of the structure response has been further developed. The difference between the conventional [2] and the present approach is that, instead of direct minimization of identification functional, experiment design is used, by which response surfaces of the functional to be minimized are obtained. The response surface approximations are obtained by using information on the behavior of a structure in the reference points of the experiment design. The finite element modeling of the structure is performed only in the reference points. The functional to be minimized describes the diflference between the measured and numerically calculated eigenfrequencies of structure. It is proved that the identification functional is convex if the constitutive matrix of material linearly depends on unknown parameters. By minimizing the functional the identification parameters are obtained. The method is employed to identify the elastic properties of the cross-ply laminates from the measured eigenfrequencies of plates. The elastic constants of a single transversely isotropic layer have been determined. The main advantage of the present method is a significant reduction of the computational effort. Previously this method was used for solution of the optimum design problems of laminated composite and sandwich plates [3]. 2 PARAMETERS OF IDENTIFICATION AND CRITERION The numerical-experimental method proposed in the present study consists of the following stages. In the first stage the physical experiments are performed. Also the parameters to be identified, the domain of search and criterion containing experimental data are selected. In the second stage the finite element method is used in order to model the response of the structure and calculations are performed in a reference points of the variables to be identified. The reference points are determined by using a method of experiment design. In the third stage the numerical data obtained by the finite element solution in the reference points are used in order to determine simple functions (response surfaces) for a calculation of the structure response. In the fourth stage, on the basis of the simple models and experimentally measured values of the structure response, the identification of the material properties is performed. For this a corresponding functional is minimized by using a method of non-linear programing.

Fig. 1. Geometry of laminated plate

Methodfor identification of elastic properties of laminates

163

The present numerical-experimental approach is employed for identification of the elastic properties of laminated composite plates. For this the experimentally measured eigenfrequencies are used. It is assumed that the plate dimensions (see Fig. 1), plate mass, the layer angles pi and the layer stacking sequence are known. The parameters to be identified are five elastic constants of a single transversely isotropic layer in the laminated composite plate: Ei,E2,Gi2,023,1^12• The plate is composed of unidirectionally reinforced layers. Directions of material axes of the layer are denoted 1-2-3, where 1 is the fiber direction and 2, 3 are the transverse directions. The unidirectional layer is assumed as homogeneous and transversely isotropic with respect to the fiber direction. In general, the ith layer of the laminated plate can be oriented at an arbitrary angle /9j. The angles of the layers are assumed to be fixed. For example, the cross-ply laminate consists of the layers with angles Pi = 0° and Pi = 90°. The five material parameters of the single layer can be expressed in terms of dimensionless constants a^ (see Ref. [2]). The vector of parameters x to be identified is defined through these dimensionless quantities ai X = [xi, X2, X3, XA] = [a2, as, 0^4,0^5]

(1)

These parameters can be evaluated through the identification procedure using the experimental eigenfrequencies of the laminated composite rectangular plate of constant thickness /i, length a and width b. Let the experimental angular eigenfrequencies be designated by 0)1,0)25 ••• 5^/3 where / is the number of measured eigenfrequencies fi{CJi = 2nfi). The corresponding numerical eigenfrequencies fi {ui = 27rfi) for the set of material parameters a^ are represented by 0)1,0)2,... ,a)J. Let us consider the scaling parameter C which is chosen according to the relation [2]

where 0)1 is the first numerical eigenfrequency calculated with the prior selected longitudinal Young's modulus E^ of the layer. The functional to be minimized describes deviation between the measured o)j and numerically calculated uji{x) frequencies [2]

»w = f(t£*W!)! i=2

(3)

^i

The identification of the elastic constants x is performed on the basis of information obtained from the measurements of the / lowest frequencies. The identification problem is formulated as follows min <^(a;) (4) X

subject to constraints, which define the domain of interest and conditions of a positive definite elasticity matrix. 3

VIBRATION TEST OF PLATES

The glass/epoxy cross-ply laminated plates consisting of 8 unidirectional layers with a layer stacking sequence [90/0/90/0]s were tested. A geometric dimensions and density of plates are presented in Table 1. The eigenfrequencies of the test plates were measured by a

164

R. Rikards

real-time television (TV) holography. The samples were hung upon two threads in order to simulate free-free boundary conditions. The sample was located in front of the holographic testing device. A piezoelectric resonator (in the following called "sensor") was glued in one corner to excite the sample plate with increasing frequency. The sensor is of circular shape with a diameter 25 mm located at the coordinates x = a — 12.5 mm and y = 6 — 12.5 mm. Mass of the sensor is mg= 3.5 g.

m

) " ..,,.

9 \

/

L _[ 13

1

6

10

14

12

11 J

\ \

^ r u^^

}\j

V

;

a4 15

•^

r—^^

19

Fig. 2. Vibration modes of cross-ply plate PUIO Table 1 Parameters of the cross-•ply plates Sample PUIO PUll

a, m 0.1401 0.1401

6, m 0.1401 0.1401

/i, mm 2.011 1.981

p, kg/m^ 1884 1908

The plate is illuminated by laser light and imaged by CCD (Charged Couple Device) array, resulting in speckled image on the PC monitor. When the plate is deformed (exited), this interference pattern is slightly modified. Digital subtraction of two consecutive interference patterns yields a fringe pattern depicting the surface displacements of the plate. The nodal lines of the vibration modes can be easily identified on the monitor. The measurement technique more detailed was described in [4]. Experiments were performed for both plates considered (see Table 1) and about 20 flexural eigenfrequencies were measured. The mode shapes also were recognized in the experiment. In Table 2 the experimental plate flexural frequencies are presented. The frequencies marked with the sign (*) were taken into account in the identification. In Table 2 the quantity m denotes the wave number in the x direction and n denotes the wave number in the y direction (see Fig. 1). The mode shapes (nodal lines) for the plate PUIO are presented in Fig. 2. These mode shapes were recognized in the experiment and most of them were used in identification. However prior to experiment the mode shapes were calculated by the finite element method using the initial guess values

165

Methodfor identification of elastic properties of laminates of the elastic constants. Such calculations is necessary in order to recognize all frequencies in the range and corresponding mode shapes. The mode shapes presented in Fig. 2 were calculated using the elastic constants obtained through identification (see Table 3 below). In calculations of the present mode shapes the mass of sensor was not taken into account. Calculating the plate with sensor the frequencies are slightly different and the mode shapes are similar but no more symmetric. Note that in the identification procedure each experimental frequency should be related to the numerical frequency having the same mode shape. Table 2. Experimental plate flexural frequencies /« [Hz] Mode

Mode shape

i

771,71

"1

M

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

2,0 0,2 2,1 1,2 2,2 3,0 3,1 0,3 1,3 3,2 2,3 4,0 4,1 3,3 0,4 4,2 1,4 2,4

Specimens PUIO PUll 159* 166* 332* 341*

-

-

484* 542* 902* 971* 1090* 1155* 1273* 1523* 1643* 1898* 2003* 2290

464* 529* 869*

-

938 1050* 1143* 1240* 1470*

1838* 1940* 2180

-

2418

2338

-

-

2733* 2665*

4 FINITE ELEMENT SOLUTION The eigenvalue problem for harmonic vibrations of the plate can be represented by Ku = (J^Mu

(5)

Here K is the stiffness matrix of the plate, M is the mass matrix and u is the displacement vector. The eigenvalue relation (5) for the mode Ui which corresponds to the first experimental eigenfrequency Ui can be written in an equivalent form placing Ei in evidence ErK*m = ulMui

(6)

Here EiK* = K is the stiffness matrix. Taking into account relation (2) this equation can be written as CE^^K^'ui = CUJIMUI

(7)

166

hence

R. Rikards

El = CEl

(8)

where E^ is the initial guess value given to the Young's modulus in the fiber direction of the layer and E\ is the corresponding identified mechanical property. After evaluation of the optimum value of x the remaining mechanical properties are calculated by inverse relations [2]. The eigenvalue problem (5) was solved by using a triangular finite element of laminated thick plate with a shear correction [5]. In order to avoid 'shear locking' a selective integration technique was applied. A 22 x 22 regular mesh (968 finite elements) was considered in order to achieve the necessary accuracy for at least 20 first eigenvalues of the laminated plate with FFFF (all edges free) boundary conditions. In calculations the mass of sensor should be taken into account. 5 METHOD OF EXPERIMENT DESIGN Let us consider a criterion for evaluation of the experiment designs which is independent on the mathematical model of the object. The number of variables n and the number of experiments k are the only initial information. The main principles in the present approach are as follows 1. The number of levels in the design space for each variable is equal to the number of the experiments, and for each level only one experiment is performed. 2. The reference points (experiment points) in the design space are distributed as regular as possible. To realize the second principle it is suggested to use a criterion

where Lij is a distance between the reference points having numbers i and j {i ^ j). The problem of minimizing the criterion (9) together with the first principle leads to the non-linear integer programming problem. The experiment designs were determined for different number of the design variables n and diflE'erent number of the experiments k. The experiment design is characterized by the matrix Bij. The domain of interest is determined as Xj ^ [xY^^xf^], where a;™'° and xf^ are respectively the lower and the upper bound on the design variables. Thus, in this domain the reference points, where the numerical experiments must be performed, are calculated by the expression xf = xf + -j^ixj'^ - xfniBij - 1)

(10)

Here i = 1,2,...,/;; and j = 1,2,..., n. Since the matrices B^j of the experiment design are universal, they may be used for the various identification or optimum design problems.

167

Methodfor identification of elastic properties of laminates 6 APPROXIMATION OF RESPONSE SURFACES Techniques from experiment design and response surface methodology are used to build the approximate models from the data in the reference points. The selection of the 'best' regression equation (response surface) in the subregion defined by the lower and upper bound on the design variables is performed by the following procedure. The response surface is built by using data obtained by computer simulation in all points of experiment design. First, consider the approximating function (model) of the following form m

y{x) = Y.Mii^) 1=1

(11)

where Aj are unknown coefficients and (j)i{x) are the basis functions that constitute the model. These functions are built from the set of simple functions <^i5<^25 • • • ^^R- The functions (^^ (r = 1,2,..., i?) are assumed to be in the form

^r(x) = n4"' 3=1

(12)

where n is the number of variables and a^j are positive or negative including zero integers. The form of the approximating function (11) is determined in two steps. First, the perspective functions i(x) are selected by using the least-squares estimation. Then a stepby-step reduction procedure of the number of terms in the model is applied and further reduction of the selected functions is performed. Similar method based on the response surface approach was used in [4,6] to solve the optimum design problems. 7 DETERMINATION OF ELASTIC CONSTANTS Let us consider the procedure of identification for the cross-ply composite plates. The experiment design with four variables (n = 4) and 35 reference points {k = 35) was selected. The upper and lower limits (domain of interest) of the identification parameters are taken as follows 5 GPa < £"2 < 20 GPa, 3 GPa < Gi2 < 9 GPa 3 GPa < G23 < 14 GPa, 0.2

<

1/12

<

(13)

0.4

By using the matrix Bij of the experiment design forn = 4 and k = 35, and the expression (10) the values of all the four identification parameters are calculated for 35 reference points in the domain of interest. Employing these values of parameter x and the initial guess value E^ = 35 GPa in each reference point the first 20 natural frequencies were calculated. The data of numerical experiment were used to determine the response surfaces (11). For this the RESINT program [7] was employed and approximating functions for all frequencies were obtained. These functions were used in the functional (3). In the identification at all 12 (plate PUll) to 14 (plate PUIO) experimental eigenfrequencies were used. Minimization of the functional (3) subject to constraints was performed by the random search method [7]. Results of identification of the layer stiff"ness properties for the two plates considered

168

R.Rikards

(PUIO and PUll) are presented in Table 3. It is seen that the transverse shear modulus G23 is overestimated. The transverse shear modulus can not be reliable determined from the measured frequencies since the plates were too thin {h/a = 1/70) for identification of this property. In this case more thick plates should be used. Table 3. Elastic properties of single layer Property ^i,GPa £;2,GPa Gi2,GPa G23,GPa ^\2

PUIO 38.89 12.78 5.06 11.70 0.304

PUll 38.20 11.95 4.77 9.55 0.392

Verification of the results was performed by the finite element method and through the independent experiments. For the finite element analysis the elastic constants obtained by identification were used (see Table 3) and diflferences between the experimental and numerical frequencies were calculated. For all frequencies the calculated residuals were less than 2%. It is of interest to compare the elastic constants of the single layer of the cross-ply laminate with the properties obtained for the unidirectionally reinforced transversely isotropic plate made from the same material [8]. Results are presented in Table 4. Good agreement of the results for the constants Ei, E2 and Gu is observed. However, there is some difference for the Poisson's ratio. It can be explained since this property is less sensitive to frequencies as modulus of elasticity, especially for the cross-ply laminate. Table 4. Comparison of results for cross-ply and unidirectional laminates Property Ei, GPa ^2,GPa Gi2,GPa ^12

Cross-ply plate 38.55 12.36 4.92 0.347

UD plate [8] 38.81 12.12 5.09 0.255

8 CONVEXITY OF IDENTIFICATION PROBLEM It can be proved the uniqueness of solution of the problem of identification of elastic constants. Actually, the objective function $(x) in Eq. (3) is convex if the eigenvalues A(a;) is a linear functions of x. The proof of this statement is as follows. The eigenvalue problem (5) can be written in the form K{x)u = X{x)Mu, X(x)=u''(x)

(14)

Here the I x I stiffness matrix JFT is a linear function of the parameters of identification x if the constitutive matrix of material linearly depends on unknown variables. Matrices K and Ai" is a symmetric positive definite matrices. The matrix K depends on parameter x, while the matrix M is independent of it.

Method for identification of elastic properties of laminates

169

Vector u e E^ characterizes all possible vibration modes. In [10] it was shown that the extreme eigenvalue of the problem (14) can be determined through the relation '^min ~ niin(lCn,u), u 6 E\

(15)

= 1

{Mu,u)

Here {Ku, u) is notation for the quadratic form. Now the following property of concave functions [11] is employed. Assume that di{x) {i = 1,2, ...,m) are concave functions of parameter x. Then the function (16)

d{x) = min[di(x), d2{x),..., dm{x)]

is also concave, i.e., the minimum of set of concave functions of parameter x forms a concave function of x. 50

45

^

r^^^-^

0.3 0.2

f

0.1 0.05

40

\ 35

r-0 05

_ 0.1

30

\,.,,,, _ L , I 5

7

9

1 fS.

1 1 1 J —_ l

11

13

L_J—I—

15

• 1 1 .1 . X . .

17

19

^2, GPa Fig. 3. Contour plot of identification functional Eq. (3) for plate PUIO For all possible vectors u the quadratic form {Ku, u) in (15) forms a set of functions of parameter x. Here vectors u in (15) are arbitrary in E^ under condition that they are orthonormalized in M-metric, i.e., {Mu,u) = 1. Moreover, they are independent of x. The conditions described above can be summarized in the following statement. The lowest eigenvalue of the problem Ku = XMu, where K is a symmetrical positive definite matrix that depends on parameter x, and M is a symmetric positive definite matrix that is independent of a;, is a concave function of x if quadratic form {Ku, u) is a concave function of X. This statement firstly was proved in connection with solution of the optimum design problems of composite shells [9]. Prom the above statement it follows that if the elements of K are linear functions of x, then the quadratic form {Ku,u) is also a linear function of x. Since a linear function is simultaneously concave and convex, the lowest eigenvalue Ajj^jj^ of the problem Ku = XMu is a linear function of the parameter x. Similarly can be proved a statement that the second, third and higher eigenvalues are also linear functions of parameter x. For this a condition of orthogonality of eigenvectors should be used [10].

170

R.Rikards

The properties of eigenvalues can also be analyzed numerically. Prom analysis of contour plots of the eigenvalues it is seen that these are linear functions of parameter x. Identification functional (3) is a quadratic function of eigenvalues. Thus the functional is convex. It is seen in Fig. 3, where the contour plot of the identification functional for the plate PUIO is presented. 9

CONCLUSIONS

Elastic constants of a single layer of the cross-ply laminate have been determined by using the identification procedure based on the method of experiment design and the response surface approach. It was shown that identification functional is convex if the stiflFness matrix linearly depends on unknown parameters. ACKNOWLEDGEMENTS The investigations concerning development of method of identification of properties of laminated composites are sponsored through Contract No. G4RD-CT-1999-00103 by the Commission of European Community. Thanks are also due to Japanese Ministry of Education, Science, Sports and Culture for a generous support of this paper through the travel grant to the conference ISIP 2000. REFERENCES 1.

2. 3. 4. 5. 6. 7.

8. 9. 10. 11.

Pedersen, P. (1989). In: Discretization Methods and Structural Optimization Procedures and applications, H. A. Eschenauer and G. Thierauf (Eds). Springer Verlag, Berlin, pp. 277-283. Mota Soares, C. M., Moreira de Preitas, M., and Araiijo A. L. (1993). Composite Structures 25, 277. Rikards, R. and Chate, A. (1995) Structural Optimization 10, 46. Rikards, R., Chate, A., Steinchen, W., Kessler, A. and Bledzki, A. K. (1999). Composites. Part B 30 279. Rikards, R., Chate A. and Korjakin, A, (1995). Engineering Computations 12 61. Roux, W. J., Stander, N. and Haftka, R. T. (1998) Int. J. Numer. Meth. Engng. 42 517. Rikards, R. (1993). In: Optimal Design with Advanced Materials. Proceedings of the lUTAM Symposium, Lynghy, Denmark, 18-20 August, 1992, P. Pedersen (Ed.). Elsevier, Amsterdam, pp. 148-162. Bledzki, A. K., Kessler, A., Rikards, R. and Chate, A. (1999). Compos. Sci. and Technol. 57 2015. Rikards, R. (1980). Mechanics of Solids. Translation from Mekhanika Tverdogo Tela 15 130, (translated from Russian by Allerton Press Inc.). Gantmakher, F. R. (1967). Theory of Matrices. Nauka Publishing House, Moscow (in Russian). Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press, Princeton, New Jersey.

INVERSE PROBLEMS IN ENGESTEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

171

A SINGLE INTEGRAL FINITE STRAIN CHARACTERIZATION OF SOFT CONNECTIVE TISSUES AND PARAMETER IDENTIFICATION

Quaglini V., Vena P., Contro R. Department of Structural Engineering, Laboratory of Biological Structures Mechanics Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano - Italy ABSTRACT A Single Integral finite strain constitutive model is proposed for the soft connective tissue capsule that forms around breast prostheses. The mechanical behavior of the tissue is analyzed in vitro tests. A characteristic feature is the independence of the amount of stress relaxation (actual value at time t versus initial value of the stress) by the applied strain. On this basis the integrand function can be splitted into the product of an "elastic" function (variable: strain) and a "reduced relaxation" function (variable: time). Two different sets of parameters, related to the two functions, are involved in the formulation of the constitutive laws. The decoupling of the stress from the time dependence allows to identify separately each set of parameters by using experimental data from distinct tests: tensile tests and relaxation tests. The identification procedure is based on the comparison of experimental data and predictions by the constitutive model. An error function based on the measure of the difference between observed and calculated data is constructed and minimized through the identification process. KEYWORDS Viscoelasticity; finite strains; variable separation. INTRODUCTION The replacement of pathological organs and tissues with artificial devices (like heart valves, blood vessels, skin, tendons, ligaments, etc.) is today a common clinical practice. Living materials are the constituents of the replaced organs, as well as the interfaces of the prosthetic substitutes: the knowledge of their mechanical behavior turns to be necessary for the reliability of the replacement; moreover living materials can be used for bioprostheses manufacturing. Therefore, formulations of constitutive laws for living tissues are nowadays gaining increasing interest. Soft connective tissues consist of fibers of collagen and elastin, embedded in a ground substance of proteoglycans and water. They are characterized by great extensibility, timedependent behavior and nonhnear stress-strain relationship [1]. Accurate constitutive laws of the tissues must be capable of containing such characteristic features, but this had led in the

172

V. Quaglini, P. Vena and R. Contro

past to formulations [2-6] which were difficult to handle especially in the parameter identification process. In order to simplify either the experimental or the numerical demands of the identification procedure, phenomenological equations had been often employed for soft tissues but the counterpart lay in low consistency and in general in a limited range of validity of the models. In the present paper, a constitutive model for a soft connective tissue is proposed, which, while it is consistent with continuum-based theories, it simplifies the identification process by an appropriate choice of the constitutive equations, modvated by experimental evidences, that decouples the strain- fi-om the time-dependent behavior. Mechanical tests on tissue samples yield the main features of its mechanical behavior, which in turn lead to the formulation of constitutive equations and suggest the identification strategy. EXPERIMENTAL RESULTS The tissue considered for the development of the model is the connective fibrous capsule which forms around implanted mammary prostheses. Tests were carried out on periprosthetic fibrous capsules that had been removed after surgical prosthesis correction. Seven capsules fi*omfivepatients were included in the experimentation. Strip specimens (6 mm wide, 50 mm long) were cut out from the capsules along different directions; the number of strips obtained fi-om each capsule depended on the dimension of the disposable tissue. As the excised capsular tissue can be approximately considered as a flat membrane, in-plane characterization was accomplished. Uniaxial tests on the strips were carried out on a MTS 858 MiniBionix (MTS, Minneapolis, USA) equipped with a 25 N load cell. On the specimen surface markers were drawn with black ink and their positions were recorded on a videotape during the test, for strain measurement processing. Each specimen underwent the following experimental program: (a) preconditioning, by cyclic loading and unloading fi-om 0 to 6 N, until repeatability of the response was attained [1]; (b) tensile tests consisting of loading to 6 N and unloading to 0 N, at different strain rates: 3 30 - 300 %/min; (c) stress relaxation tests at three different stretch levels; a 15 minutes recovery in saline solution elapsed before each successive test. The stretch ratios to which these tissues were subjected could be more than 10 percent. The tissue proved to be transversally isotropic, since no difference were detected among different directions in the plane of the samples. From load-stretch data, stress-strain curves were obtained (Fig. l[Left]); the uniaxial stress-strain curves are not linear, with increasing stifftiess at higher stresses; furthermore the curves are low sensitive to the strain rate (Fig. 1 [Right]). The results from stress relaxation tests are reported in terms of normalized relaxation curves, obtained by dividing the actual load at time t by the maximum load at the instantaneous application of the stretch (Fig. 2). In our tests normalized curves at different stretch levels overlap; this feature has been proved to be common to most soft connective tissues [1]. THE CONSTITUTIVE MODEL As in its physiological range the capsular tissue is subjected to large deformations, constitutive laws should be formulated in the framework of finite strain theory, especially in view of two and three dimensional problems. Objective (material frame indifferent) Cauchy-Green finite strains E\^ and material Piola-Kirchoff stresses S^ are employed [7].

173

A single integralfinite strain characterization 1.00-r

1.00

0.80 .

0.60

(L>

1

0.40

'B

o ^ 0.20 0.00 1.00

1.04

1.08

1.12

1.12

Fig. 1. [Left]: A typical stress-strain curve from a tensile test on periprosthetic capsular tissue. The nominal stress in the loading direction is plotted against the stretch ratio A; [Right]: Uniaxial stress-stretch curves are insensitive to the strain rate. 1.00-t +

100

Stretch=1.04

O

stretch=1.07

^

stretch=l.l

200 time (sec)

300

400

Fig. 2 Stress relaxation curves at different stretch levels can be reduced to a unique normalized stress relaxation function, independent from the applied stretch A. The behavior of a nonlinear viscoelastic material undergoing finite strains is fully described by the sum of the series of integrals

^(o=t^[,--iy,,^,^..^^^^^^^^

(1)

where S and E are the stress and strain tensor respectively, R^^"^ is the stress relaxation response from a single-step relaxation test and R^""* is the difference between the actual relaxation function from a n-steps test and the prediction based on (n-l)-steps data [8]. For many soft collagenous tissues the time dependence of the response is adequately well defined by a single-step relaxation test, and, as a consequence, the stress can be expressed as a function of one time variable only r^ = T . Then the first term of the integral series (1) yields an adequate approximation of the material behavior in other than single step histories [1,6].

174

V. Quaglini, P. Vena and R. Contro

A Single Integral model is therefore proposed for the behavior of periprosthetic capsular tissue. The model is expressed by the relation: S0)= r

rf,R[E(r);r-r]

J-QO

collagenous

(2)

where R is the response to a step strain history in which the strain tensor is changed from 0 to E at time t = 0\ it is called the stress relaxation tensor and it is in general a nonlinear function of E, with R ( 0 , r ) = 0 . At times r for which E(r) is differentiable, d^K can be written as (dot denotes time derivative) ..R[E(r),.-r]=^Bfe^E(r>/..

(3)

At jump discontinuities of E(r) the expression J E R is defined through the integral property J'_JER[E(r),r-r]-R[E(5"),/-5"]-R[E(5-)^-5-] .

(4)

For a smooth strain history the Single Integral model takes the form

S(0= f d^m^-rh

r

'^^^tiryir

(5)

or, by integration by parts

S(0=R[E(4o].j:^5^g^^r.

(6)

It should be noted that, as the model employs objective stress and strain measures, equations (5) and (6) are objective too. Constitutive equations are defined on the basis of the experimental results from the mechanical tests on the tissue. As in stress relaxation tests at different strain levels the ratio between the stress at time t and the stress at time zero is independent from the magnitude of the applied strain, R can be split into the product of two functions, one of strain only and the second one of time only: R(E,/) = S^(E).G(0 .

(7)

S^(E) can be defined as the response to an instantaneous strain history: it is uniquely related to the strain tensor and it is called the "elastic response function". The temporal response is carried by the "reduced relaxation function" G{t): its components Gij(0 are dimensionless and subjected to the condition Gjj(0 = 0. Elastic response function. As it appears from constant strain rate tensile tests along different directions, the capsular tissue is isotropic in the plane of the samples. The existence of a potential W^ for the elastic response S^ is assumed of the form

175

A single integral finite strain characterization

(8) / i , h, h are the invariants of the metric tensor C. The typical shape of the stress-strain curves (Fig. 1) suggests the following choice of the potential W^ =a{exp[^(/, - 3 ) ] - l } - a M / 2 - 3 ) + / f e ) + g f e + 4

+^1,).

(9)

which corresponds to an exponential form for the elastic function S^. In the above expression, the assumption of incompressibility I3 = 1, which is very common for soft living tissues [1] is made. The principal stresses can be obtained by differentiation of eq. (9):

SI, =«^(exp[/?(£„ + £ , , + £ 3 3 ) ] - £ „

81.=-^.-^ a^33

5^33

-EJ-

2^22 + 1

(10,a-c)

2^33 + 1

In uniaxial tensile tests, S^^'is drawn from the applied force and the stretch ratio /li [7], while 1^22 = 0 . From eq. (10,b) the hydrostatic pressure;? is obtained and after substitution in eq. (10,A) it yields SI, = a ^ : ^ ^ _ ^ { - l - 2 £ 3 3 +2exp[y9(£„ 2^jj +1

+E,,+EJI

(11)

Due to the low thickness of the tissue, the hypothesis of plane state of stress 1^33 = 0 is made. Functions/and g must be chosen in order that eqation (10,c) is identically satisfied. Reduced relaxation function. Stress relaxation is found to be isotropic in the plane of the sample; it is assumed the isotropy of G(t) along the direction of the thickness too, and thus the tensorial function G(t) reduces to the scalar-valued function G(t). As a consequence of the scarce sensitivity of the tissue to the strain rate, the following form of reduced relaxation function, which corresponds to a uniform relaxation spectrum, is chosen [2] \-\-K{E^

^1^ -E,

^1 v^iy

G{t): 1 + ^ln with E^{y) =

/•oo

Jy

^

7

dz .

' t ^

(12)

176

V. Quaglini, P. Vena and R. Contro

PARAMETER IDENTIFICATION The proposed Single Integral viscoelastic model is conceived as a multiplicative decomposition of the elastic response and the time-dependent response. This feature allows to split the whole set of parameters into two distinct sets. The first set contains the parameters a, P of the elastic function S^ only, the second set contains the parameters AT, 6\, Bi of the reduced relaxation function G{t). The identification procedure relies in the minimization of an error function which is a measure of the difference between the measured response yk, on laboratory tests, and the model prediction hk(x) calculated by using equations (11) or (12). The elastic function parameters can be identified through eq. (11) fi*om the loading branch of a tensile test at infinite strain rate. Actually, tensile tests at the highest strain rate (300%/min) were used to approximate the infinite strain rate test. The reduced relaxation function parameters of equation (12) were identified on the basis of normalized stress relaxation data. The two sets of parameters were estimated for each patient. In order to take into account data from different specimens from the same capsule, a two steps procedure had been adopted. An estimation of the parameters is firstly obtained by using data from the test on one specimen. The estimation was performed by minimizing the following function: Min % = (y, - h ( x ) ) " w ( y , - h ( x ) ) S.T.

^^^^

g(x)<0

g(x) = -j3 when the parameters of the elastic function are considered, while g(x) = ^^-^2 ^s the relaxation parameters are to be identified. In equation (13), the entries of vectors y i and h 1 are the measured and the calculated stresses, respectively, at each sampled time instant in the test of the first specimen. The function h i(x) is the nonlinear function of the unknown parameters vector x given by the constitutive equations (11) or (12). The constrained minimization problem (13) provides a first estimation Xi of x. A refined identification of the parameters is then carried out by including in the error function the whole set of experimental responses {k = 2, ..., N) which are available. The second step consists in minimizing the following nonlinear function: Min ^,={x,^, S.T.

-xrP(x,_, _x)+(y^ - h J x ) ) ^ W ( y , -h^(x))

g(x)<0

(14)

The subscript K refers to the k-th experimental set of results. The entries of vectors XK-I and x are, respectively, the estimation of the parameters at the previous experiment and the unknown parameters to be estimated in the current step. The problem (14) can still be interpreted as the minimization of the differences between experimental and numerical results, but a priori information on the parameters value is now taken into consideration by means of the matrix P. If the element Pu of P tends to zero, no a priori information is available on the i-th parameter of x, while a large P w means that accurate a priori information on such parameter is already available. The functions ^K are nonlinearly dependent on the parameters which have to be identified. To solve the optimization problems (13-14), a first order method is adopted consisting in a recursive procedure, based on the linearization of the functions and constraint equations at the current values of the parameters. This linearization is adopted for searching the minimum value of ^K in the neighborhood of the current values Xj^ of x.

A single integral finite strain characterization

111

In order to obtain a fast convergence in the minimization process, analytical derivatives of the objective functions were provided. RESULTS The optimization problem (14) applied to the identification of the relaxation function parameters leads to the evaluation of the parameters 61, 62 and K. Parameter 61 is related to the stress relaxation which takes place during the first time instants after stretching of the tissue, whereas 02 is related to the long term relaxation, and therefore the duration of the observation could affect the identification of this latter parameter. The value of parameter K influences the total amount of relaxation during the whole test. The values of the parameters as provided from the identification process allowed a good fit of the experimental curves. Figure 3 [Left] shows the experimental results and the model prediction for one patient. Figure 3 [Right] shows the history of the cost function during the optimization process. The iteration histories of the parameters 0i, O2 and K are shown in Figure 5c, 5d and 5e. The optimization procedure converged with monotonic decrease of the objective functions. However the parameters iteration histories show that for parameter K some oscillation can occur. Good fitting was also attained for the stress-strain curves from the tensile tests at 300%/min (Figure 4 [Left]). Figure 4 [Right] reports the iteration history of the cost function, while the iteration histories of the parameters a and p during the minimization process are reported in Figures 5 a and 5b. 1.00-

10.00

0.90-

8.00

I

0.80-

I

0.70-

g

6.00 4.00

0.60

2.00

0.50

0.00 0

100

200 time (sec.)

300

400

•

A

I

10 15 Iterations

20

25

Fig. 3 [Left]: Fit of calculated to experimental curves for the relaxation response in terms of normahzed stress versus time; optimal values of the parameters 0i, 62 and K are employed; [Right]: Iteration history of the error function for identification of parameters 9i, 62 and K.

178

V. Quaglini, P. Vena and R. Contro

0.80

0.30 k

\

0.60 c ^

0.40

E Z

0.20

0.10

0.00 0.00

0.20

0.04

0.08 E.,

AAA

0.00

0.12

0

10

20

Itarations

30

40

Fig. 4 [Left]: Fit of calculated to experimental curve for the elastic response in terms of nominal Stress vs Green strain; optimal values of the parameters a and p are employed; [Right]: Iteration history of the error function for identification of parameters a and p. The identification of the elastic parameters provided a unique solution regardless of the initial values given as the input of a recursive procedure. Different results were obtained in the identification of the relaxation parameters: more than one local minimum could be found for different initial values of the parameters. A possible explanation of the difficulties in identifying the parameters of the relaxation function could be the considerable noise that affected the experimental relaxation data. Table 1 summarizes the results for the 5 patients. The same constitutive model qualitatively describes the mechanical behavior of the same tissue from different patients, but different values of the material parameters are needed to account for the hetero-individual differences. Table 1: Results of the identification process for five different patients. Patient 9i (sec) 02(sec) g (MPa)

1 2 3 4 5

0.032 0.133 0.042 0.67 0.24

A 28.125

66.648 84.539 20.263 10.987

4.71 2.26 1.08 0.22 0.39

293.57 127.93 1366.6 10000. 4000.

K

0.12 0.13 0.12 0.06 0.13

CONCLUSIONS A Single Integral finite strain viscoelastic model is proposed for the soft connective tissue of mammary periprosthetic fibrous capsules. Experimental tests on tissue samples pointed out that the tissue is isotropic in the plane of the sample and that the elastic response is nonlinear with increasing stiffness at increasing strain; moreover the amount of stress relaxation is not influenced by the applied strain. A constitutive law was chosen in order to take into account all of these main features of the observed response. The integrand function was split into the product of an "elastic function" (depending on strain only) and a "reduced relaxation function" (depending on time only); a continuous spectrum was considered for the stress relaxation function and a nonlinear elastic relationship was drawn by the definition of an elastic potential for a transversely isotropic solid.

179

A single integral finite strain characterization 0.50

T'--]

30.00

y--^-\

0.40-

0.30-

^

0.20

25.00

c^ 20.00 4

15.00

0.10 0.00 10

20 Iterations

30

10.00

40

10

5.00-

0.40-

4.00 4

0.30-

3.00-

0.20-

0

30

(b)

(a)

O

20 Iterations

O

0.10

2.00

0

10

4 6 Iterations

2

4 6 Iterations

10

(d)

(c)

0.16-

0.15

^

0.14

0.13-

0.120

2

4 6 Iterations

8

10

(e) Fig. 5 Iteration history of parameters a (a), P (b), Gi (c), 62 (d) and K (e) during the identification process. The separation of stress and strain variables in the constitutive equations allowed to share the material parameters into two distinct sets, relative to the elastic and to the reduced relaxation function respectively. The identification process takes advantage from such partition, as the

180

V. Quaglini, P. Vena and R. Contro

material parameters can be identified separately from two distinct test data: high strain rate tensile tests and stress relaxation tests. The tests were performed on several specimens from each capsule in order to provide a first estimation of the parameters and subsequently a refinement of their values by recursive application of the optimization procedure. Since no previous studies were found on breast capsular tissues, it was not possible to make comparison with the results obtained in this study. Uncertainties in the experimental measures, which were affected by a considerable noise especially in the relaxation tests, brought the optimization procedure to have more than one local minimum. However the choice of the absolute minimum provided good fit of the experimental data. The results obtained from the identification procedure show that the proposed model is well suited for the description of the mechanical behavior of connective tissue as observed in tensile tests and relaxation tests. It still has to be proved that the model is capable to reproduce the mechanical behavior in a rather general strain history. To accomplish such analysis, the Authors believe that a more reliable identification of the relaxation parameters, from less noisy experimental data, should be required. In order to get a tractable identification process, both the material and the geometrical properties of the tissue were assumed to be homogeneous in each whole capsule. Further work with different numerical tools are needed in order to take into consideration the non homogeneities in the properties of samples, as well as in the strain field. In this case finite element analyses could be adopted for a more refined identification of material properties and their local variation.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

Fung Y.C. (1993) Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag, New York. Fung YC (1972) (Fung YC, Perrone N and Anliker M eds.) Prentice-Hall, Englewood CHffs, New Jersey, Ch.7, pp.181-208 Young JT, Vaishnav RN & Patel DJ (1977) J. Biomech. 10: 549-559 DeHoffP.H., (1978) J. Biomech., 11: 35-40. Bingham D.N., and DeHoffP.H., (1979) J. Biomech. Engng. 101: 15-22. Johnson GA, Liversay GA, Woo Sl-Y and Rajagopal KR (1996) J Biomech. Engng. ASME 118:221-226 Fung Y.C. (1965) Prentice Hall, Englewood Cliffs, New Jersey. Pipkin AC & Rogers TG (1968) J. Mech. Phys. Solids 16: 59-74

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

181

IDENTIFICATION STRATEGIES FOR RECOVERING MATERIAL PARAMETERS FROM INDENTATION EXPERIMENTS ^ Andrei CONSTANTINESCU, Nicolas TARDIEU

Laboratoire de Mecanique des Solides (CNRS UMR 7649) Ecole Polytechnique - 91128 Palaiseau cedex e-mail: c o n s t a n t ; tardieu@lms . p o l y t e c h n i q u e . f r ABSTRACT This papers reports the progress obtained by the authors on the development of a numerical method for the identification of material parameters from indentation experiments. The proposed technique minimizes a least squares cost functional using gradient descent methods. It has previously been shown that the adjoint state method can be extended to variational inequalities and this permits to compute the gradient of the cost functional with low numerical cost. Using this technique a series of identifications from indentation experiments have been done and their results are discussed in the sequel. KEYWORDS indentation / elastoviscoplasticity / identification / contact problem / adjoint state method INTRODUCTION An indentation experiment is performed by pressing a punch on a material sample and measuring the applied force and the indentation depth. The underlying inverse problem treated in this paper is: The identification of the parameters of the constitutive law of the sample from the simultaneous knowledge of the displacement and the resultant force on the indenter This inverse problem is of practical interest as the indentation test does not demand the creation of complicated specimens and is simple to perform. However the applications are restricted by the difficult mechanical interpretation of the test. One can differentiate two types of indentation experiments: ^presented at IS IP 2000, International Symposium of Inverse Problems in Engineering, 6-10 march 2000, Nagano, Japan

182

A. Constantinescu and N. Tardieu • In the Hardness Test the punch is pressed with a given force on the sample. The punch is then lifted and the vertical displacement is deduced from geometrical measurements on the residual inprint. This test is also known as the Brinell, Rockwell or Vickers hardness test, the name denoting just the shape of the indenter: three face pyramid, sphere or respectively a four face pyramid. • In the Continuous Indentation Test the punch is pressed into the sample and then lifted under continuous force or displacement control and the dual quantity is measured. An example of measured indentation curve is given in Figure 1.

The drawback of the hardness test is the final displacement measurement, which does integrate in one global answer loading, unloading as well as the rate effects during the indentation. However, due to its experimental simplicity compared to the continuous indentation test it is wide used in the industry. The results presented in this paper are focused on the interpretation of the continous indentation test but they can easily be extended to hardness tests.

Figure 1: Experimental displacement-force indentation curve The state of the art in mechanical interpretation of the indentation test is based on a series of closed form solutions and flow rules for the elasto-plastic constitutive behaviour. The monographs of Tabor [16] and Johnson [6] give an overview of the techniques used and the analytical solutions implied. A series of more recent papers like [4, 7, 13, 15] have introduced new semi-empirical formulas based a series of constitutive assumptions and an interpretation of the deformation process under the indenter. One of the less covered areas is the one of viscoelastic and elastoviscoplastic constitutive laws, where the rate dependence of the behaviour is important. Unfortunately even for rate dependent laws, the classical methods do not provide a systematic tool to identify constitutive parameters to be finally used in industrial structural computations. A state of the art of the mathematical formulation of the direct problem is presented in the monograph by Kikuchi and Oden [8]. One can remark a series of difficulties related to the contact conditions inherent to the indentation problem. From the point of view of the inverse problem we dispose of a small number of results. The existing studies formulate the inverse problem as the minimization of a cost functional. The cost functional is generally a least squares distance between computed and measured indentation forces or displacements. As theoretical results we can mention the works of Hasanov, who showed in a series of papers [9, 10, 11, 12] that the inverse problem admitsits a solution. He assumed that the material has a nonlinear elastic behaviour. The numerical identification method proposed was based on a step wise trial an error search of the tangent elastic moduli with increasing vertical displacement

Identification strategies for recovering material parameters

183

of the punch. The disadvantage of his reconstruction approach is that it can not be appUed to rate dependent behaviours or to inhomogeneous sample (for example functionally graded materials). A numerical approach, also based on the minimization of a least squares functional, is that of Koguchi [14]. He identified the elastic moduli of a layered material. His minimization was based on a simplex method and a closed form solution of the direct problem. In the same global framework of the minimization of a least squares cost functional the authors have obtained a series of theoretical and numerical results. They have shown that the adjoint state method can be extended to variational inequalities in order to compute the gradient of the cost functional [18] and used this technique to identify elastic [18] and elastoviscoplastic [2] coefficients from simulated indentation experiments. This paper presents on overview of these identification techniques and discusses a series of identifications from experimental indentation curves. THE INVERSE INDENTATION PROBLEM The inverse problem formulated as: Identify the parameters c of the constitutive law of the sample from the knowledge of the indentation curve, given by the displacementforce measurements pairs: [U^^^^ F/^^), i = 0 . . . /. The framework proposed here to solve this inverse problem was the minimization of the least squares functional:

where p denotes the contact pressure on the contact surface TcThe direct dependence of the computed forces Fi{c) of a direct solution of the indentation problem, with material parameters c and imposed punch displacements given by Ul'^'^, i — 0 .. .1, leads to a constrained minimization problem. dj In order to compute the gradient of the cost functional -7— a Lagrangian C functional has oc been constructed by adjoining to J the variational formulation of the direct problem. Supposing the existence of a saddle point of the Lagrangian C it has been shown that the gradient of J can be computed from the solution of the direct contact problem and the solution of an adjoint problem. For the sake of simplicity we shall not present here the complete deduction and the equations of these problems. Two classes of constitutive behavior have been treated under the assumption of small displacements and rotations: linear elasticity in [17] and in elastoviscoplasticity [19, 2]. The passage to large displacements and small rotations for these behaviours in the formalism of standard generalized materials is presented in [20]. A schematic view of the direct and the adjoint problem are presented in Figure 2. Some important properties of the adjoint problem: • It is not a contact problem. For the cost functional (1), its loading is given by a Dirichlet condition (imposed displacement) on the effective contact surface of the direct problem. • It is a time dependent system of partial differential equations and a final condition given by the solution of a well-posed elasticity problem.

184

A. Constantinescu and N. Tardieu

Adjoint Problem

Direct Problem

IJ^^i -. f <'Om^ pexp

I S

\ U ^*P

Ijadi ^

p <^<3^

p,

pexp

.E o

2

I fj^dj ^

^c

pexp

•D

I

5 Figure 2: Indentation test: The Direct problem • Interpreted in the reversed time, the adjoint problem is a linear elastic problem if the direct problem is elastic [17], and a viscoelastic problem if the direct problem is elastoviscoplastic [2]. • The parameters of the adjoint constitutive law are determined by the coefficients of the direct constitutive law and the solution of the direct problem. Under the assumptions of small strains and an elastic constitutive law: o-(ii) = A{c) : £{u)

(2)

where u,E,a,A denote respectively the displacement vector the strain and the stress and the elasticity tensor; the gradient of the cost functional takes the following form: VcJ

=

dc

fe{u):

Jn

dA : e(u*) dQ dc

(3)

where u* is the solution of the adjoint equation. In the computations, the direct and the adjoint problem have been integrated using the finite element method and coupled with a gradient descent algorithm as presented in the next section. THE NUMERICAL COMPUTATIONS The direct and the adjoint problem have been programmed in the finite element oriented language Castem 2000 [1]. The adjoint viscoelastic problem has been integrated using an implicit (Forward Euler) integration scheme. The commonly used gradient descent method was the BFGS algorithm with an Armijo line search rule [3].

Identification strategies for recovering material parameters

185

The finite element meshes were refined in the contact area in order to keep results precise in a reasonable computational time. From a series of meshes, the one with the largest density and not affecting the resultant forces on the indentation curves has finally been chosen (see Figure 3).

/ y

/

/

/

y 0.005

0.01 Pentration depth (mm)

Figure 3: Comparison of computed indentation curves for a coarse and a fine mesh A series of identifications using simulated measurements and the previous technique have already been presented for elastic materials in [18] and for Maxwell and Norton-Hoff viscoelastic materials in [2]. In the sequel we shall present a series of identifications using the same technique and real experiments. DISCUSSION OF EXPERIMENTAL RESULTS A series of indentation experiments were performed on three different materials: nylon, duraluminium and polyethylene. The choice of the materials was justified by their behaviour varying from elastoplastic to viscoelastoplastic at normal room temperature. The material samples were cylinders with ^ 2.5 x 2.5 cm, diameter x height. As the maximum indentation depth was about 25 /im the faces of the samples have been polished and assured to be parallel. The indentation was done with a Vickers pyramid by a displacement controlled machine. The identification were done supposing that the material behaves under the elastoviscoplastic Norton-Hoff law and under the assumption of large displacements ans small rotations. The Norton-Hoff law is a standard generalized constitutive law [5] without work hardening. The flow rule can be written as: £:(AiXi) = 5 : A(Ti + Aef

OCT

(4)

where the pseudo potential $ is given by K m +1

K

)T

where a^ is the elasticity limit, ( • )4. is the positive part operator and ( • )eg is the equivalent Mises stress. Finally the plastic strain increment is determined by : ' 2 ^

K

>"(S;^'

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(fi is the deviatoric part of o-j. The Young modulus, the yield limit, the viscosity and the power coefficient have been identified. The Poisson coefficient has been fixed at the a priori value i/ = 0.3. The row data from the indentation machine has been used for the identification. The last part of the unloading has been neglected, due to a convergence problems in the direct algorithm in unloading at large step sizes. The indentations on the Nylon sample have been performed at Zfim/min. The experimental and identified indentation curves are presented in Figure 4. The identified parameters (after 53 iterations) are given in Table 1 It is difficult to judge the validity of the identified parameters as we did not dispose of other types of experimental curves in order to determine classically the material parameters. The Young modulus and the yield limit are in the intervals commonly reported in the literature for this material. Table 1: Identified parameters for the Nylon Initial Values

Final Values

E (MPa)

1000.

Iteration 56 1930.

K (MPa.si/")

100.

83.95

4.

7.34

n ay (MPa)

30.

15.59

J

2.08

0.020

4

Experimental curve Identified curve

/ i

/

/

y

/

-

Experimental curve Identified curve

/ ' / / / / / /

f

/

_ ^.^.fr^^""' __.r»-^-^/

.

0.02 Penetration depth (mm)

/

, 0.002

/

y

/

/

, , , 0.004 Penetration depth (mm)

/-

/

// //

A i

/' / •

^ /i \(

f

, 0.006

Figure 4: Experimental and identified indentation curve for the Nylon and the Duralumin The second sample was a duraluminium (Au4G) and the penetration rate was Ijim/min. The experimental and the identified curve are in good agreement (see Figure 4). However a direct inspection shows for example that the identified Young modulus of ^ 35 GPa (see Table 2) is at half the common sense value of J^ 70 GPa for this material. In making this comparison one should also keep in mind that the duraluminium is almost elastoplastic at room temperature and that is does not correspond exactly to the chosen material law. Another difference comes from

Identification strategies for recovering material parameters

187

the role played by the Young modulus in different constitutive laws which can lead to different interpretations for this parameter. Table 2: Identified parameters for the duraluminium

E (MPa) K (MPa.si/^) n

Initial Values

Final Values

50000.

Iteration 43 37471.8

1500.

2750.76

5.

4.60

cjy (MPa)

100.

52.09

J

0.22

0.0034

The third material, the Polyethylene was indented with a rate was G/jtm/min. The identified indentation curve is good agreement with the experimental one even if the loading and the unloading are under and respectively overevaluated (Figure 5). In this case, we did dispose of an experimental traction curve and the identified values (see Table 3) were used to simulate the same traction experiment. The traction curves are pretty different even if the shape and the order of magnitude are similar. A possible explanation justifying the difference could be the order of magnitude of difference in the global strain rates of the experiments: ^ = 3 • 10~^s~^ for the traction and 6 = 2- 10~^s~^ for the indentation. It is also possible that the friction between the diamond indenter and the polyethylene sample has not been negligible. Table 3: Identified parameters for the polyethylene Initial Values Final Values E (MPa)

200.

Iteration 53 520.

K (MPa.s^/")

10.

7.90

n

4.

7.93

a^ (MPa)

J

6.

9.28

3.85

0.039

The performed identification can be qualified as encouraging taking into account that no a priori information has been used and that a rate dependent constitutive law has been identified from a single experiment at practically constant loading rate. It is important to notice that the indentation curves have always been well represented in spite of the differences between identified and expected material parameters. A global drawback comes also directly from the Norton-Hoff constitutive law itself. As shown in a previous work [2], the cost functional presents a long valley if expressed in well

A. Constantinescu and N. Tardieu

0.02

004 Enfoncement (mm)

Figure 5: Polyethylene. Left: Experimental and identified indentation curves for the Polyethylene. Right: Experimental traction and computed traction with the parameters identified from the identation definite parameter plane. This would suggest that the material parameters are slightly correlated. \ s one can remark, a decrease in the yield limit could be compensated by an increase of the viscosity and its power coefficient. OTHER IDENTMCATION STRATEGIES Besides the identification problem reported in the last section, there are two essential drawbacks of the identification strategy used before. Both are related with the incomplete usage of the available informations. Their resolution could conduct to computationally faster identifications. The first drawback comes from the evaluation of the cost functional itself. The cost functional and its gradient are integrated on the whole time interval and therefore the descent algorithm can not take into account different sensitivities of the material parameters in the different sections of the loading-unloading curve. A typical method taking into account the "local" sensitivity of a nonlinear function in a least squares distance is the Levenberg-Marquardt method. A direct application of this method is not possible in our case due to the current gradient computation using the adjoint state method translated in the difficulty of computing directly —^^^ instead of — ; r ^ . This can be overcome oc oc by changing the expression of cost functional in order to obtain the desired result. The first tests done using a Levenberg-Marquardt descent algorithm are promising. The second drawback is related to the number of evaluation of the cost functional in the line search of the gradient descent algorithm. For a descent step, one needs a first evaluation of the functional and its gradient ; several evaluations of the functional are necessary in order to determine the optimal step size. In our case, we have remarked that the evaluation of the cost functional is ?^ 3 more expensive as the one of the gradient. It is therefore interesting to compute also the gradient at each evaluation of the cost functional and then to use the obtained information more effectivelly. In order to solve this problem, we proposed to estimate a model function for our cost functional from a small number of cost functional and gradient computation and then minimize this model function. An example of the computed cost functional and its model are represented in Figure 6. The

Identification strategies for recovering material parameters

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Figure 6: The cost functional and its polynomial model in the case of a Maxwell constitutive law constitutive model was a Maxwell viscoelasticity law and the model function was obtained by fitting the cost functional over a quadratic polynomia with material parameters as variables. With 9 random evaluations of the cost functional and its gradient, the identified parameters where 5% different of the real values. Both strategies, the Levenberg-Marquardt descent algorithm and the model functions are currently under development for the inverse indentation problem and more results will be provided in the near future. CONCLUSION This paper described an original method for identifying material characteristics from indentation tests. Based on efficient mathematical methods (adjoint state methods and optimization) the method applied to different behaviours ranging from elasticity to elastoviscoplasticity (sufficiently smooth standard generalized constitutive laws). The efficiency of this technique has been proven through numerically and experimentally based studies. The perspectives are the combination of the existing results with more efficient identification stategies like Levenberg-Marquardt algorithm or polynomial models and the generalization of the adjoint state method to elastoplastic constitutive laws. ACKNOWLEDGEMENT The authors would like to thank Said Taheri for continuous support and Jose Spino for providing the indentation experiments. They gratefully acknowledge the financial support of Electricite de France for this work. [1] Castem2000 Finite Element Code h t t p : / /www. e a s t e r n . o r g : 8001 [2] Constantinescu A Tardieu N 1999 On the identification of nonlinear constitutive laws from indentation tests Submitted to Inverse Problems in Engineering [3] CULIOLI J.C. - Introduction a roptimisation Ellipses, 1994

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[4] Doener M F, Nix WD 1986 A method of interpreting the data from depth-sensing indentation measurements J. Mater. Res., 1 (4), 601-609 [5] B. Halphen, Q.S. Nguyen, Sur les materiaux standards generalises, Journal de Mecanique, 1975. [6] Johnson K L \9S5 Contact Mechanics Cambridge University Press [7] S. Jayaraman, G.T. Hahn, W.C. Oliver, C.A. Rubin, P.C. Bastias, Determination of monotonic stress-strain curve of hard metals from ultra-low-load indentation tests, Int. J. Solids Structures, Vol. 35, Nos. 5-6, 1998. [8] Kikuchi N , Oden J T 1988 Contact Problems in Elasticity:A Study of Variational Inequalities and Finite Element Methods SIAM, Philadelphia [9] Hasanov A, Seyidmamedov Z 1995 The solution of an axisymmetric inverse elasto-plastic problem using penetration diagrams Int. J. Non-Linear Mechanics, 30 (4), 465-477 [10] Hasanov A 1995 An inverse problem for an elastoplastic medium SIAM J. Appl. Math., 55(5), 1736-1752 [11] Hasanov A 1997 Inverse coefficient problems for monotone potentials operators Inverse Problems, 13 (5), 1265-1278 [12] Hasanov A 1998 Inverse coefficient problems for elliptic variational inequalities with a nonlinear monotone operator Inverse Problems, 14 (5), 1151-1170 [13] Loubet J.L, Georges J.M, Marchesini O., Meille G., Vickers indentation curve for a magnesium oxyde (MgO) J. ofTribology, 106, 1984 [14] Koguchi H 1999 Determination of mechanical properties of thin films and functional gradient materials using an inverse technique Conference on Inverse Problems in Engineering: Theory and Practice, ASME [ 15] Oliver W.C, Pharr G.M., An improved technique for determining hardness and elastic moduli used load and displacement sensing indentation experiments, J.Mater.Res., 1{6), 1992 [16] Tabor D 1951 Hardness of metals Oxford University Press [17] Tardieu N Constantinescu A 1999 On the determination of the elastic coefficients from indentation tests accepted for publication in Inverse Problems, [18] Tardieu N Constantinescu A 1999 On the identification of nonlinear constitutive laws from indentation tests Conference on Inverse Problems in Engineering: Theory and Practice, ASME [19] Tardieu N Constantinescu A Taheri S 1999 Identification inverse de lois de comportement standard generalisees sans ecrouissage par indentation Colloque National en Calcul des Structures, Giens, France [20] Tardieu N 2000 Identification des lois de comportement elastoviscoplastique par inentation. These de Doctorat, Ecole Polytechnique, France

INVERSE PROBLEMS IN ENGINEERESTG MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

191

IDENTIFICATION OF MATERIAL PARAMETERS IN CONSTITUTIVE MODEL FOR SHAPE MEMORY ALLOY BASED ON ISOTHERMAL STRESS-CYCLE TESTS F. YOSHIDA^>, V. V. TORGPOV^), M. ITOff>, H. KYOGGKU^^ and T. SAKUMA^^ 1) Department of Mechanical Engineering, Hiroshima University, Japan 2) Department of Civil and Environmental Engineering, University of Bradford, UK 3) Department of Mechanical Engineering, Chiba University, Japan 4) Department of Mechanical System Engineering, Kinki University, Japan 5) Central Research Institute for Electric Power Industries, Japan ABSTRACT A set of material parameters in a thermo-mechanical constitutive model for a shape memory alloy (SMA), that can describe the pseudoelasticity and shape memory effect, were identified using experimental data obtained from isothermal stress-cycle tests conducted at various temperatures for Ti-Ni-Cu SMA. The identification was performed by minimizing the difference between the test data and the results of the corresponding numerical simulation using an advanced optimization technique based on iterative multipoint approximation concept. This approach allows to determine the phase transformation temperatures directly from the identification, while they are usually obtained only by some special experimental techniques such as DSC (differential scanning calorimetry). KEYWORDS Shape memory alloy, Ti-Ni-Cu alloy, constitutive model, material parameters identification, isothermal stress-cycle, phase-transformation temperature INTRODUCTION When a shape memory alloy (abbreviated to SMA) is isothermally loaded, it exhibits pseudoelasticity at high temperature and reversible shape memory effect at low temperature [14]. Such thermo-mechanical behaviour, which is due to the martensitic transformation (austenite [=parenf (p)] —*'martensite[m]) and its reverse transformation {m-^p), is described by a thermomechanical constitutive model. In general, all constitutive models incorporate some material parameters to be determined from experiments. However, especially for SMAs, the identification of material parameters in a constitutive model by conventional trial-and-error (curve fitting) approach is almost impossible because the constitutive equations are higly nonlinear, and also the number of material parameters in the model is quite large. Therefore, the identification problem should be treated as an optimization problem.

192

RYoshidaetaL

The authors [5, 6] recently proposed a method of the identification of a set of material parameters in a constitutive model of cyclic plasticity for monolithic, as well as bimetallic, sheet metals from cyclic bending tests. In these research works, material parameters were successfully identified by minimizing the difference between the test results and the results of the corresponding numerical simulation using an advanced optimization technique based on iterative multipoint approximation concept presented by Toropov et al, [7-9]. Furthermore, the authors [10] applied this optimization technique to the identification of material parameters in a thermo-mechanical constitutive model for SMA [11-12], and six material parameters were determined using stressstrain data obtained from an individual isothermal stress-cycle test. As an extension of the above-mentioned study, in the present work, a set of material parameters in a modified constitutive model for Ti-Ni-Cu SMA were identified. For the identification, the whole experimental data obtained from isothermal stress-cycle tests conducted at various temperatures were employed simultaneously. It is worth notinig that this approach allows to determine the phase transformation temperatures directly from the identification as two additional material parameters, while they are usually obtained only by some special experimental techniques such as DSC (differential scanning calorimetry). Furthermore, thermo-mechanical behaviour of the SMA in another type of experiment, so-called stress-recovery test, is discussed by comparing the experimental observation with the prediction by the constitutive model incorporating thus identified material parameters. THERMO-MECHANICAL BEHAVIOR OF SMA AND ITS CONSTITUTIVE MODEL Shape Memory Effect and Pseudoelasticity Uniaxial tension-unloading tests for a wire of 3-mm diameter of Ti-Ni-Cu SMA (Ti-49.8Ni41.7Cu-8.5at%) were performed in a water pool for several temperatures ranging from 20 to 80 °C. Figure 1 shows the experimental results of stress-strain responses in uniaxial tension and the subsequent unloading under various isothermal conditions. In this figure, two typical stressstrain responses are observed. At low temperatures (20 and 60 °C in Fig. 1), a certain amount of residual strain remains after unloading. After that, when the specimen is heated up above a certain temperature (e.g., 80°C), the strain completely recovers, as illustrated with the dotted line in Fig. 1. The phenomenon of the recovery of residual strain is called shape memory effect. At high temperature (70 and 80 °C in Fig. 1), the stress-strain response during loading process looks like elastic-plastic behaviour, however, the tension-induced large strain recovers in the subsequent unloading process. This behaviour is cdlXtd pseudoelasticity. The above-mentioned thermo-mechanical behaviour of SMAs results from the martensitic and reverse transformations. The temperature- and stress-dependent phase transformation of a SMA is schematically illustrated in Fig. 2. At high temperature, the SMA has fully austenitic phase (sometimes we call this austenite phase 'parent'). When it is cooled down, the martensitic transformation starts at temperature M^ and becomes fully martensitic phase at M^ (the transformasion can be indicated as /? -*^ m). Starting with the full martensite, the austenitic transformation (reverse transformation of m-^p) starts at A^ when it is heated up and finishes at A^ In our SMA, the orthorhombic martensite transformation and its reverse transformation takes place. The transformation temperatures under stress-free condition (hereafter they are denoted by M^^, M^^, A^^ and A^^) are 46, 51, 59 and 62 °C, respectively. It should be noted that the inelastic deformation of martensite is due to the reversible movement of the twin or intervariant

Identification of material parameters in constitutive model

193

350

CO Q.

CO

en 0

L_ +-•

0.01

Strain

0.015

0.025

Fig. 1 Stress-strain responses of Ti-Ni-Cu SMA under isothermal uniaxial tension and the subsequent unloading at various temperatures.

CO CO

CO

(p^m)

Temperature

Fig. 2 Schematic illustration of temperature- and stress-dependent phase transformations of SMA.

boundaries under applied stress (m^^m), The shape memory effect results from the (m -^ m) deformation (from A-^B-^A at T^ in Fig. 2) and the subsequent {m-^p{-^m)) transformations during heating up followed by cooling down (A - • C at T^ (-^ A)). The phase transformations are strongly stress-dependent, as schematically shown in Fig. 2. For example, let us consider the phase transformations in isothermal loading cycle starting from a stress-free point at a high temperature above A (point D at T^in Fig. 2). In this case, the stressinduced (p—*'m (—*^m)) transformation occurs during the loading process (E—»'F in Fig. 2), and

194

F.Yoshidaetal

(m -*p) reverse transformation during the unloading process (G -* H in Fig. 2). The above stress-induced transformations accompanied by (m —^ m) deformation are the cause of pseudoelasticity. Constitutive Model In order to describe the thermo-mechanical behaviour of the SMA, a constitutive model proposed by Tanaka, etal.[\\-\2] was used. In this model the uniaxial stress-strain relation in rate-type is given by the equation: (1)

G = De-\-pDt + Q,^

where
(2)

Here, D^ and D^ denote the Young's modulus values for the /7-phase and the m-phase. The transformation kinetics is given by the equations:

yi^ = V«r-fc>>o

(3)

for the martensitic (p-^m) transformation, and

_i = b,cj-b,a>0

(4)

for the reverse (m^^-p) transformation. Here, ^^, c^, b^ and c^ are the additional material parameters. If all the material parameters are constant, then the volume fraction of martensite ^ is explicitly given by the following equations using the transformation temperatures M^ and A^: ^ = l - e x p [ V M ( M , , - r ) + Z7^CT]

(5)

for the martensitic (p-*m) transformation, and ^ = oxp[b,cMso-T)

+ b,a]

(6)

for the reverse (m-^p) transformation. The whole set often material parameters consists of the two partly different subsets [D^,Z)^,M,^,j3,Q,^^,c^J and [Z)^,Z)^,A^^,j8,Q,Z7^,c^J for the stressstrain calculations for the loading and unloading processes, respectively. FORMULATION OF THE MATERIAL PARAMETER IDENTIHCATION PROBLEM General Formulation Let us consider the phenomenological coefficients of the constitutive model to be identified

195

Identification of material parameters in constitutive model

as components of the vectors G R ^ . Then the optimization problem can be formulated as follows [7-9]: Find the vector x that minimizes the objective function

F{x) = J^e^F%x) A,<x.
^^^

{i = l,...,N)

where L is the total number of individual specific response quantities (denoted by a) which can be measured in the course of experiments and then obtained as a result of the numerical simulation. F"(jc) is the dimensionless function

nx)={i[/?f - ^i^.<)r}/{iK" r}

(8)

which measures the deviation between the computed a -th individual response and the observed one from the experiment. Here, the following notation is used: T" : a parameter which defines the history of the process in the course of the experiment (e.g. the time or the loading parameter), T" (a=l,..., L, s =1,..., 5 ) : the discrete set of S^ data points, R" : the value of the a-th measured response quantity corresponding to the value of the experiment history parameter rf, R"(x,t"] : the value of the same response quantity obtained from the numerical simulation, Q" : the weight coefficient which determines the relative contribution of information yielded by the a-th set of experimental data, i4, , B. : side constraints, stipulated by some additional physical considerations, which define the search region in the space R^ of optimization parameters. The optimization problem (7) has the following characteristic features: - the objective function is an implicit function of parameters x, - to calculate values of this function for the specific set of parameters x means to use a nonlinear numerical (e.g. finite element) simulation of the process under consideration, which usually involves a large amount of computer time, - function values present some level of numerical noise, i.e. can only be estimated with a finite accuracy. The direct implementation of any of conventional nonlinear mathematical programming techniques would involve too large amount of computer time and, moreover, the convergence of a method cannot be guaranteed due to the presence of numerically induced noise in the objective function values and (or) its derivatives. To solve the problem, the iterative multipoint approximation concept presented by Toropov et al [7] is used here. The technique is based on the iterative approximation of computationally expensive and noisy functions, F"(jc), (a=l,..-, L) by the simplified noiseless functions. To construct such functions, the methods of regression analysis are implemented. Further details of the identification technique can be found in references by Toropov ^r a/. [7,8]. The specific form of the multipoint approximation technique, used in this work, is based on

196

F.Yoshidaetal

the algorithm of sequential quadratic programming (SQP) which builds up an approximation of the inverse of the Hessian matrix using the objective function values and its derivatives. The multipoint approximation of the objective functions are used iteratively instead of original function values. It should be noted that in problems of parameter identification for nonlinear constitutive models the traditional use offinitedifferences for evaluation of derivatives would almost certainly spoil the convergence of the optimizer because the accuracy of derivatives is severly affected by the noise. As the simplified functions are used to evaluate the derivatives only, the requirment for simplified functions to allow for internal minimum can be waived. As the approximations are constructed in each iteration to evaluate the derivatives only, a fairly simple approximation types can be used, e.g. a quadratic polynomial without cross-terms. F^{x,a'') = < + £(a«,,,,,^,. + a3",,„,,xf)

(9)

/=l

which contains 2N+1 tuning parameters to be found in each iteration by the least-squares surface fitting. The Present Identification Problem The full set of material parameters in the constitutive equation and the transformation kinetics equations [Eqs.(l)-(3)] includes nine parameters [PM^^A^^SO^\O^^^^M^^M^^A^^A\^ because the thermal expansion coefficient j3 cannot be identified from isothermal experiments. These nine parameters were identified using the stress-strain (A (in this work, the experimental results for 70 and 80 °C were chosen). The individual loading-unloading diagrams (a =1,..., 4) are regarded as individual response quantities. Here in the idenfification problem for the constitutive equations and the transformation kinetics equation: - the optimization variables JC = 1 x ,jC2,...,JC9jare the material parameters - the set of values of R^ (a=l,..., L) corresponds to values of the stress a" found from the experiment, - the function /?"(JC,T") corresponds to the calculated stress cr", - the experiment history parameter T" is the strain f", - the index a in Eq. (7) is 1 for the uniaxial tension and 2 for the subsequent unloading at 70 °C, and a = 3 and 4 correspond to the tension and the unloading at 80 °C. RESULTS AND DISCUSSIONS Figure 3 shows an example of the comparison of the experimental stress-strain curves for the Ti-Ni-Cu SMA tested at 70 and 80 °C and the corresponding results simulated with the constitutive models, incorporating the identified set of material parameters. In this figure, the calculated results (the solid line for 80 °C and the broken line for 70 °C) have been obtained using the identified material parameters. Since this type of constitutive model cannot describe the experimentally observed gradual stress decrease during unloading [10], in this material parameter

197

Identification of material parameters in constitutive model

0.005

0.01

0.015 Strain

0.02

0.025

Fig. 3 Experimental stress-strain response for Ti-Ni-Cu SMA tested at 70 and 80 °C and the corresponding results of simulation with the constitutive models incorporating the identified set of material parameters. Identifications were performed under the weighting condition of 0^ (70 °C) = e^ (80°C) =1 for loading and 6^ (70 °C) = 0^ (80 °C) =0.25 for unloading.

identification, we put a larger weight for the loading process (a= 1 [for 70 °C] and 3 [for 80°C]) than for the unloading (a= 2 [for 70°C] and 4 [for 80°C]), i.e., 0^ = 0^ = 1 and ^^ ^ ^ = 0.25. In this problem, some parts of a set of material parameters were not used for the calculation of a certain response, i.e. only seven parameterts [D^,D^,M^^,Q,Z?^,C^J out of nine parameters D^,D^,M^^,A^^,Q,Z?^,c^,Z?^,c^J were used for the loading process, and seven parameters D^,D^,A^^,Q,^^,c^J influence the unloading process. Even so, the material parameters were well identified without any special treatment within an acceptable computation time (e.g., a few minutes by EWS SUN Ultra 1-140 MHz). It is worth noting that this approach allows to determine the phase transformation temperatures M^^ and A^^ directly from the identification, while they are usually obtained only by some special experimental techniques such as DSC (differential scanning calorimetry). Thus determined M^^ and A^^, 50. PC and 58.9 °C, respectively, are close to the values obtained by DSC, 5 PC and 59 °C. Furthermore, the determined Young's moduU for thepphase and the m-phase, 71.6 and 23.9 GPa, are in good agreement with the experimental observations [4]. As an example of the application of the constitutive model incorporating the identified material parameters, let us predict another type of thermo-mechanical behaviour of the SMA, i.e., the thermo-mechanical cycling as shown in Fig. 4(a). The SMA is uniaxially loaded up to a certain strain level at low temperature {M^< T^^f) keeping the strain constant. After that, the specimen is isothermally unloaded 2iiT=T^. Figure 4(b)

198

F. Yoshida et al 250

c

4

- - - ^i

- Experimental data -Numerical simulation

'—•——•——*—•—

/?.r. M., Ti Aso Af, T2 Temperature (a) Test condition

0.005

0.01

0.015

0.02

Strain

(b) Stress-strain response

Fig. 4 Experimental stress-strain response for Ti-Ni-Cu SMA in a thermo-mechanical cycling and its numerical simulation.

shows the comparison of the predicted and experimantal stress-strain curves for the Ti-Ni SMA tested at T^ = 55°C and T^- 80°C. The stress increases during heating up (this phenomenon is sometimes called 'stress recovery'), and the strain recovers in unloading process. It is found in Fig. 4(b) that the model describes fairly well the behaviour in the thermo-mechanical cycling, while the calculated stress level is slightly smaller than the experimental result. CONCLUDING REMARKS The present approach allows to determine the material parameters in a constitutive model describing the complicated thermo-mechanical behaviour of SMAs including the shape memory effect and pseudoelasticity. In the present research work, a set of parameters, which includes the phase transformation temperatures, for a Ti-Ni-Cu SMA has been successfully identified, using the experimental results of isothermal uniaxial tension and the subsequent unloading conducted at various temperatures. It should be noted that an advanced optimization technique based on the iterative multipoint approximation concept made it possible to solve the present identification problem within an acceptable computation time.

Identification of material parameters in constitutive model

199

ACKNOWLEDGEMENT The authors are grateful to Dr T. Okada, Mr T. Hatakeyama and Mr T. Watanabe for their help with experiments and computation. The contribution of The Royal Society is acknowledged for the Joint Project Grant offered to the authors. REFERENCES 1. 2. 3. 4.

Delaey, L., Krishnan, R. V., Tas, H. and Warlimont, H. (1974) /. Mat. Set., 9(9), 1521. Fischer, F. D., Sun, Q-P. and Tanaka, K. (1996), Applied Mechanics Reviews, 49(6), 317. Funakubo, H. (ed.) (1987), Shape Memory Alloys. Gordon and Breach Science PubUshers. Tanaka, K., Tobushi, H. and Miyazaki, S. (1993), Mechanical Properties of Shape Memory Alloys. Yokendo Ltd. 5. Yoshida, F., Urabe, M. and Toropov, V. V. (1998) Int. J. Mech. Sci. 40(2-3), 237. 6. Yoshida, F., Urabe, M. and Toropov, V. V., (1998) In: Inverse Problems in Engineering Mechanics, Tanaka, M. and Dulikravich, G. S, (Eds), Elsevier Science, pp. 243-252. 7. Toropov, V. V., Filatov, A. A. and Polynkin, A. A. (1993) Structral Optimization 6, 7. 8. Toropov, V. V. and van der Giessen, E. (1993) In: Optimal Design with Advanced Materials, Pedersen, P. (Eds.), Elsevier Science, pp. 113-130. 9. Toropov, V. V., Yoshida, F. and van der Giessen, E. (1997) In: Proceedings of EUROMECH Colloqium on Material Identification Using Mixed Numerical Experimental Methods, Sol, H. and Oomens, C. W. J. (Eds.), Kluwer Academic PubUsher, pp. 81-92. 10. Yoshida, F., Toropov, V. V., Itoh, M., Kyogoku, H. and Sakuma, T. (1999) In: Identification in Engineering Systems, Friswell, M. I., Mottershead, J. E. and Lees, A. W. (Eds), The Cromwell Press Ltd, pp. 150-159. 11. Tanaka, K. (1986) Res. Mech., 18, 251. 12. Tanaka, K., Kobayashi, S. and Sato, Y. (1986) Int. J. Plasticity, 2, 59.

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Defect Detection

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INVERSE PROBLEMS IN ENGINEERE^JG MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

203

ON THE IDENTIFICATION OF A CRACK IN 3D ACOUSTICS Huy Duong Bui^'^, Andrei CONSTANTlNESCu\ Hubert MAIGRE^ ^ Laboratoire de Mecanique des Solides (CNRS UMR 7649) Ecole Polytechnique - 91128 Palaiseau cedex e-mail: bui; constant ;maigre@lms .polytechnique. fr

^ Divion R&D - Electricite de France 92141 Clamart cedex

ABSTRACT This paper addresses the problem of the identification of a planar crack in 3D acoustics. The measurements are arbitrary transient inputs applied to the boundary of the solid and the corresponding response. We show that the overdetermined boundary data allows the explicit reconstruction of the crack plane and the crack geometry. The approach is based on the construction of a reciprocity gap the definition of a series of well chosen adjoint states. This approach conducts conducts to direct identifications of the host plane of the crack and to a closed-form formula for the geometry of the crack. Compared to classical solution of similar inverse problems in acoustics this method applies to bounded bodies and does not use the Bom approximation in the frequency domain. KEYWORDS scattering / bounded body / planar crack / inverse problem INTRODUCTION Identifying the shape of a crack from the scattered pattern of an acoustic wave is an industrial problem with a complex and interesting mathematical background. In the last decades a large research effort has been deployed in order to solve the numerical and mathematical questions reveled by the underlying acoustic inverse problem. An overview of the subject is presented in the book by Colton and Kress [10] and in some recent papers [3, 4, 5, 11]. On the one hand most of the studies are concentrated on the Helmholtz equation in the frequency domain and not on the initial transient acoustic problem. On the other hand, both in the numerical or mathematical analysis, the far field pattern of the acoustical wave is analyzed supposing that the body is of infinite extent. Moreover using a frequency analysis most works accept the Born approximation. This can be summarized by a series of limitations: unbounded

204

H.D. Bui, A. Constantinescu and H. Maigre

domain; known Green function; source in the interior of the body; source far away from the crack in order to consider plane waves, etc., which should be avoided for different applications. When the body is supposed to be bounded, the employed methods consists of minimizing a cost functional defined by the distance between measured and computed data by different techniques. One can remark that in other crack identification problems a series of direct identification formulas have been obtained. We cite for example Alexandrini [2], Andrieux [6], or Friedman [12] for the electrostatic crack identification problem or Andrieux [7] for the elastostatic case. Some of these results have been obtained using the reciprocity principle and the techniques initially developed by Calderon [9]. The purpose of this paper is to show that the reciprocity principle and the underlying techniques can also be applied for identification of crack from acoustic measurements. We shall show that this technique permits to remove the limiting assumptions: Helmholtz equation, infinite body and the Bom approximation. We shall consider a bounded body, subject to Dirichlet or Neumann boundary conditions and assuming that the dual boundary condition is measured. The main result is the identification of a crack from a single transient measurement pair. The result is constructive for a planar crack. The position and the normal of the host plane are determine as the solutions of a series of simple equations involving explicit adjoint fields and boundary measurements. Once the host plane is determined, the geometry of the crack, considered as the support of the discontinuity of the acoustic field is expressed as an inverse Fourier Transform. The paper presents first the acoustic problem and the classical equations involving the Helmholtz equation. Then the reciprocity principle is introduced and finally the host plan and the crack are identified.

THE DIRECT AND THE INVERSE ACOUSTIC PROBLEM Let us consider a body occupying a domain Q in its reference configuration and having a crack inside. The transient acoustic problem on r2 F is gouvemed by the following equation: C{w) = dtdtu - A?z = 0

(1)

with the following initial condition. u(x,t<0)=0

dtu{x,t<{))^^

(2)

and boundary conditions: dnu\en = ^'^

or

u\dn^i^^

(3)

The boundary conditions on the crack faces are: anU+'-|F = 0

(4)

The direct problem seeks to determine the complete acoustic field from a given acoustic field ip^ or a pressure distribution ip'^ on the boundary dVt. The inverse problem treated here seeks to determine the position and the geometry of the crack from the simultaneous knowledge of ip^ and ^"^ on the whole boundary dO..

205

On the identification of a crack in 3D acoustics Measurement O

Incident wave

Diffracted wave

a^R-"

Cracli

Figure 1: The classical problem configuration in an infinite domain CLASSICAL RESULTS IN THE FREQUENCY DOMAIN The classical results for a special class of acoustic solutions: u{x,t) = (j){x)tx^{—ikt)

(5)

Supposing this form of the transient solution, u{x) becomes a solution of the Helmholtz equation:

) + A;V = 0

(6)

The acoustic u{x) field is now supposed to be written as the sum between the incident and the diffracted wave:

' - 0' + 0^

(7)

In this case, the inverse probem seeks to identify the crack F from the knowledge of 0 \ generally a planar wave and from the measurements of ucf)^ in a certain number of points. In the case of an infinite body Q. = B? the diffracted wave admits an integral representation:

\x) =

l^^4,{y)-^G''{x,y)-G'ix,y]

dri'y-Hy)

dS'^ y

(8)

where F is the crack surface and G'' is the Green function of the Helmholtz equation: G'{x,y)

=

(9)

-\Qxp{ik\x — y\' 4i7r\x — y\

Taking into account the boundary condition on the crack F (-Q^{y) — 0) one obtains: ct>\x)

L ^f-w'*''"

(10)

dS'u

After a limiting process x -> F the last equation becomes:

cj>\z) = 0W + ( H

/,

^(y)-^G*(z,y) ony

dS'u

(11)

(pv) means that the integral should be take in the sense of Cauchy principal value. This equation is usually solved using the Born approximation. The Bom approximation is founded on two simplifying assumptions:

206

H.D. Bui, A. Constantinescu and H. Maigre

• the diffracted wave is created just by the lightened part of the crack F+ • the integral term in the right hand side of (11) is negligible compared with the first one. This conducts to the final equation: ^"(z)

=

2(j>v) /

| r ( y ) - ^ G * ( z , y ) dSy dny

(12)

which will be used afterwards in the identification problems. PERTURBATION OF THE TRANSIENT EQUATION In order to obtain general identifiability results and to eliminate the limitations created on the one hand by the treatment in the frequency domain due to the Helmholtz equation and on the other hand by the Bom approximation let us consider the complete transient problem. This will also enable a the construction of an exact solution in the case of a bounded domain. The reasoning presented in the sequel is constructed using a perturbed acoustic problem. The perturbation, a small viscosity term, does vanish for the asymptotic solution of the physical problem and is used only as a mathematical artifact. The perturbed equations of the direct problem have now the following form: Ce{w) - dtdtu -Au

+ edtu = 0

(13)

with initial conditions: u{x, t<0)

=0

dtu{x, t<0)

=0

(14)

and boundary conditions: dnu\dn - (p^

dnu\F = 0

(15)

We shall also assume that the behaviour of the solution and its velocity for large times is the following: lim t^ \u,{x,t)\ = 0 lim t'^ \dtUe{x,t)\ = 0

(16)

This can be achieved using an adequate boundary condition as proven by optimal control techniques by Lions [13]. For a given Neumann boundary condition we shall suppose that the Dirichlet boundary condition is measured for all times. The problem addressed here is the reconstruction of a planar crack F from the knowledge of Dirichlet - Neumann boundary data pair. The crack will be considered fixed over time and it will be simultaneously characterized by a free boundary and a jump in the acoustic solution, denoted by [u] at its passage.

On the identification of a crack in 3D acoustics

207

THE BETTI RECIPROCITY GAP AND THE ADJOINT PROBLEM The presence of the viscosity term in (13) is the key point of the presented solution of the inverse problem. This regular perturbation of the scalar wave equation permits to avoid the harsh severe conditions imposed generally to the adjoint fields. Let us introduce an adjoint field w defined on Q x [0, cx)[, bounded and with C^ continuity. Multiplying (13) with w gives after a series of integrations by parts: /•oo

I Jo

n

/»oo

I {{dnu\w - lu}dnw) ds dt = JF

JO

+ -\-

Jn\F

n

/ u {dtdtw -AwJn\F

sdtw) dxdt +

(17)

[w dtu - u dtw + euwj^ dx +

Jo JdQ

{w dnU - u dnw) dsdt

If w is chosen to be the solution of: C*{w) = dtdtw — Aw — edfW = 0

(18)

taking into account the initial and boundary conditions (14), (15 and the behaviour at infinity (16), one obtains: roo

n

poo

r

(19) "/ / [^I ^nWdsdt = / {udnW — wdnu) dsdt JO Jdn Jo JF This equation relates boundary values of the solution and of the adjoint field with the jump of the solution on the crack. We shall denote the integral on the right hand side of the last equation Betti Reciprocity: HE. TIB is a Hnear form of w and depending of the measured boundary fields

nB{w; i/;^, 0^) =^ r [ {ip'^dnW - w(r) ds dt Jo Jdn

(20)

The preceding reciprocity equation (19) is a straightforward generalization to the transient case of the reciprocity gap already used for a series of static problems by a series of authors Calderon [9], Andrieux et al. [6, 7], Alessandrini et Diaz [2]. Measured response Measured excitation

Figure 2: The problem configuration in a bounded domain

208

H.D. Bui, A. Constantinescu andH. Maigre

Figure 3: The crack in the host plane IDENTIFICATION OF THE PLANE OF THE CRACK IDENTIFICATION OF THE NORMAL OF THE PLANE In order to identify the plane of the crack, denoted by P , we shall use usual adjoint fields, i.e. time retrograde planar waves and generalised acceleration solution. The fields will be considered to be solutions in the sense of distribution of the wave equations (13) and (18) with e = 0'^. Let us define the projection in Q, of such a wave by:

w^{x,t)

f{x + kt) 0

iftG[0,T[ lite [T,oo]

(21)

where k denotes the direction of propagation of the wave. The integral operator on the jump [u], on the right hand side of 19, is identically zero on P X [0, oo[, if 5n^i^ is identically zero on P x [0, oo[. As the direction of Vw^ is k, dnW =^ 0 implies k - n — 0. Thus two planes waves, characterized by fci, ^2, chosen such that: 2=

1,2

(22)

determine the normal of the plane P of the crack: (23) This means that two distinct solutions of KB{w^] ip'^^cj)'^) = 0, function of k determine the normal vector to the plane of the crack through the direction of propagation of the two plane waves. IDENTMCATION OF THE POSITION OF THE PLANE Once n has been determined, let us consider for the sake of simplicity that the direction n corresponds to the third coordinate axes Ox^. In this configuration, the plane is described by: x^ = c,c e R and the complete determination of the position of the plane is given by the knowledge of c.

On the identification of a crack in 3D acoustics

209

In the sequel we shall present two different families of adjoint fields leadinging to the position of the plane. First let us consider the adjoint field: w,{x,t)^\{x,-hf^\[t~Tf

(24)

This field is considered time retrograde and parametrized hyh^R, satisfyes (18) with £ = 0"^. The normal derivative multiplying the jump on the crack in the reciprocity relation (19) is: dnWb{x, t) = dx^Wb{x, t) = {x3 - b)

(25)

The reciprocity equality becomes: nB{wb; ^"^, (/^^) ^ -

r

I H ( ^ 3 - h) ds dt

(26)

This shows that TZ8{wb; ip^,(j)^) = 0 just if 0:3 — b for the points of the crack F. Therefore the position of the host plane is determined by searching the solution of the equation nB{wb; V^"", 0"") = 0, as a function of b. It is important to remark that the chosen adjoint field Wb, which is not wave, could be used without imposing conditions at t = 0 or t = 00. The behaviour at the times t == 0 or t = 00 is controlled by the behaviour of the solution u assured through the conditions (14) and (16) The second possibility is to chose two families of adjoint fields corresponding to plane waves of the following type: w+4x,t)

= n{{t-t+)-xs)

w;_{x,t) = n{{t-r) + {x^-L))

(27)

(28)

parametrized by t+, t~ G R, where % is the unit step Heaviside function and L denotes the dimater of the domain in the direction n of the unit normal of the plane of the crack. On can remark that the two families of waves are travelling in opposite direction. Each wave will highlight the crack only once in its trajectory. The normal derivative multiplying the jump on the crack in the reciprocity relation (19) are: dnW^^{x,t) dnw;_{x,t)

= S{{t-t+)-X3) = 6{{t-t-)-{xs-L))

(29) (30)

with 6 the Dirac function. The reciprocities corresponding to the two families are: 7ei5(t(;+;^^,0^)

-

nB{w;_-ilj'^,(r)

= - l[u{r

- flu{t+^c,Xi,X2,c)]dxidx2 ^{L-c)),xuX2,c)\dxidx2

(31) (32)

JF

We now have to find t+ and t~ such that the waves highlight the crack at the same moment t" + (L - c) - t+ + c Then the jump {u\ is therefore the same in the two reciprocities and the reciprocities are equal: •RBiwt,; V", r ) = TlB{w;.; V", r)

210

H.D. Bui, A. Constantinescu and H. Maigre

Once f^ and t~ are found position of the plane is then determined by: 1

c = -{L + t-

These results are similar to the ones obtained by Alves and Ha-Doung in their scattering analysis in case of acoustic [5] and elastic [4] waves. They have shown, using a different technique, that in an infinite domain the plane of the crack can be identified from a single experience if the wave does not see the crack transversally, i.e. if the wave really opens the crack. The method presented here is different, being based not only on the experimental wave and its diffraction, but also on the adjoints fields w^ or respectivley Wt+ and Wt-. The fields w identifying the crack, is the one for which the crack is invisible and who gets a zero reciprocity gap: TZB — 0, or the ones getting the same reciprocity on the crack, i.e. destroying each other on the crack. IDENTMCATION OF THE CRACK GEOMETRY As the plane P of the crack has already been determined, one can perform a change of variables such that P = OxiX2. Let us consider the following adjoint field solution of (18) depending of three parameters i'=^{^u^2)eR'mdqeR: Wq^>{x, t) =^ t\p{iqt) exp(-z($' • x')) exp((|$'|^ - 9^ - ieq^^x^)^

(33)

where the prime ' denotes points of the plane P, i.e. x' — (xi, X2). The reciprocity gap associated with this adjoint field, w £/, will be denoted by:

7^B(r,g;^'",r) The jump of the solution on the crack | u | will be extended with the value zero to the whole plane, defining a function D : P x [0,00] -> /?: Vix',t)

= [uix',t)]

(34)

We shall now analyze the temporal Fourier transform [14] associated with D: /•OO

K[x',q)=

I

V(x',t)cxp{iqt)dt

(35)

The inverse Fourier transform writes : 1 f^ V(x', t) = — exp{-iqt)}C{x', q) dq (36) 27ry_oo On can remark that the spatial support of V is independent of the time t and bounded in P by the extension of the crack surface ( suppX>(x', t) C F ). This implies that the spatial supports of /C(x', q) and P(x', t) are exactly the same. The identification problem of the geometry of the crack is now equivalent with the identification problem of the supports of K, and V. In order to determine the support of /C, which characterizes also the geometry of the crack, its sufficient to study the behaviour of )C{x', q) with ^ > 0 arbitrary fixed. As: dnWq^'\x^=Q = 5x3^9 e'1x3=0 = ( I ^ T " 9^ " ^^?) ^

(37)

211

On the identification of a crack in 3D acoustics

The reciprocity gap can be rewritten an allows the determination of /C(x', q) from the following equation:

/.

/C(aj', q) exp(-z i' • x') dx' - nB{i', q)(\if

- q^ - leq)"^ : - Q{i')

(38)

The right hand side of the preceding equation denoted by 6(^') is a C°° function. This is assured by the term eq> ^. Let us continue analytically all the functions depending on (J in a neighborhood of the real axes in a complex space, i.e. s' — ^' + ir)', r}' e B?. Equation (38) becomes: /

/C(x',^)exp(-zs'. x')dx' - 7^i3(s^^)(|sf - q^ - leq)''^

(39)

As the body has been supposed to be bounded in space, Q C [-a, a] , one can obtain for sufficiently large \s'\ the following inequality: \G[s')\ < Cexp(a(|5i| + |s2|))exp (a (l^il^ + \s2\^)'^) < Cexp(2a(|5i| + \s2\))

(40)

with C > 0 is a real constant. We recall the inequality (40) assures that a distribution is of exponential type < 2a. Therefore G{s^) is a distribution of an exponential type and applying the Paley-Wiener theorem ([14], p. 271) implies that Q{s') is the Fourier transform of a tempered distribution with a compact support. Passing to the limit for e ^ 0+, with a fixed q > 0 gives the following expression of the function /C : ^x',

9) = ^

/ ^ 7lB(r, 9)exp(i x' • ^'Mf

- q'-

*0+)-5 d^'

(41)

The explicit computation of this integral gives the support of /C, consequently the support of V = [w] and therefore the geometry of the crack. CONCLUSION This work proposes a reconstruction method for the position and shape of a planar crack from transient acoustic measurements in a bounded body. The work has been based on the reciprocity principle and can be extended to elastodynamics. ACKNOWLEDGEMENT This paper was partially supported by EC Contract no. ERBIC15 CT97 0706. REFERENCES [1] ACHENBACH J.D. 1980 Wave Propagation in elastic solids North Holland Publications, Amsterdam

212

H.D. Bui, A. Constantinescu and H. Maigre

[2] ALESSANDRINI G . et DIAZ VALENZUELA A., 1994. Unique determination of multiple

cracks by two measurements. Quad. Mat. Univ. Trieste.

[3] ALVES C . J . S . et HA-DUONG T., 1997. On the far field amplitude for elastic waves. Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, Ed. E. Meister (Frankfurt: P. Lang) [4] ALVES C.J.S. et HA-DUONG T., 1997. On inverse scattering by screens. Inverse Problems, 13, p. 1161-1176. [5] ALVES C.J.S. et HA-DUONG T., 1999. Inverse Scattering for elastic plane waves. Inverse Problems, 15, p. 91-91. [6] ANDRIEUX S., BEN ABDA A. et Bui H.D., 1997. Sur identification de fissures planes via le concept d'cart la rciprocit en lasticit, C. R. Acad. Sci. Paris, 324, Srie I, p. 1431-1438. [7] ANDRIEUX S., BEN ABDA A. et Bui H.D., 1999. Reciprocity principle and crack identification. Inverse Problems, 15, p. 59-65. [8] Bui H.D., CONSTANTINESCU A., MAIGRE H., 1999. Diffraction acoustique inverse de fissure plane: solution explicite pour un solide borne C. R. Acad. Sci. Paris, 327, Srie lib, p.971-976. [9] CALDERON, A . , 1980. On an inverse boundary problem, Seminar on Numerical Analysis and Application to Continuum Physics, Rio de Janeiro, p.65-73. [10] COLTON D. AND KRESS A., 1992. Inverse Acoustic and Electromagnetic Scattering Theory, (Applied Mathematical Sciences 93), Springer, Berlin [11] HA-DUONG T., 1997. On boundary integral equations associated to scattering problems of transient waves. ZAMM, 6, p. 261-264. [12] FRIEDMAN A. et VOGELIUS M . , 1989. Determining cracks by boundary measurements, Indiana Univ. Math. J., 38, 3, p. 527-556. [13] LIONS J. L., 1986 Controlabilite exacte des systemes distribues, C. R. Acad.Sci. Paris, 302, seriell, 1986, p.471-475 [14] SCHWARTZ L., 1978. Thorie des distributions, Hermann, Paris.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

213

ON THE IDENTIFICATION OF CONDUCTIVE CRACKS

C. J. S. ALVES ^^\ T. HA DUONG ^2) and F. PENZEL ^^^ ^^Wentro de Matemdtica Aplicada, Instituto Superior Tecnico Av.Rovisco Pais 1, 1049-001 Lisboa, Portugal Email: calves@math. ist. utl.pt ^^^ Universite de Technologie de Compiegne, BP 20529, 60205 Compiegne, France Email: e-mail: [email protected] ^^^ Technische Universitdt Darmstadt, Schlossgartenstrasse 7, Fachbereich Mathematik, Arbeitsgruppe 12, 64289 Darmstadt, Germany, Email: e-mail: [email protected]

ABSTRACT We consider the problem of identification of a connected crack in a bounded domain. Conditions on the boundary data are presented such that the crack can be identified by the corresponding measurement. An admissible crack (or screen) is considered to be a part of a boundary of an open set with Lipschitz regularity. We show that in the case of admissible connected shapes, a single measurement is enough to determine the position and the shape of a conductive crack, or an acoustic screen. KEYWORDS Inverse problems, uniqueness, cracks, screens. INTRODUCTION Recently, results have been obtained on the identification of cracks in unbounded and bounded domains (an account of the state of the art can be found in [1]). The case of cracks in an infinite domain was first considered in [2] and in [3]. In [3] (Remark 5, Proposition 4) it was proved that the knowledge of three far field patterns generated by incident plane waves are necessary and sufficient to determine the location of a plane crack in R^. In this paper we shall study the problem of identification of a single crack in a bounded domain (see the references [4], [5] and [6] for recent developments). If one prescribes an infinite set of boundary data, the crack can be uniquely determined, see the citations no. 6-11 in [7]. In the paper by Friedman and Vogelius ([7]) it was proved that two measurements are necessary and sufl&cient to determine an insulating crack in a two-dimensional domain. More recently, new results were obtained by Alessandrini and

214

CJ.S. Alves, T. Ha-Duong and F. Penzel

Diaz Valenzuela [8], and Alessandrini and DiBenedetto [9], proving that in the case of conductive cracks two measurements are also sufficient to establish uniqueness in the three dimensional case. However like in [7], the authors obtain these results using input data which are differences of two Dirac masses on the boundary, therefore excluding more regular data such as continuous or integrable functions (see also [10] where other measurements are proposed). In this text we prove that for the case of a connected conductive crack (or acoustic screen) one measurement is enough to determine the crack if we prescribe any data that is continuous and non vanishing on the boundary-. CONDUCTIVE CRACK PROBLEMS Throughout the paper we denote by Q an open, bounded, simply connected set in R**, (usually d = 2, or d = 3) and by F its boundary, which we assume to be Lipschitz. For the definition of an open set with Lipschitz boundary we refer to [11]. Let cj"^ C n be a connected, open set with a Lipschitz boundary and CJ~ = ft\uj'^. Let n be the outer normal vector-field defined a.e. on F or on du;'^ with direction into R^\Q on F and with direction into LJ~ on duj^. We define an open submanifold of dcu^ to be an admissible crack 7. In the two dimensional case this means that any piecewise C^ curve inside Q is an admissible crack, and in the three dimensional case, the main restriction is on the surface orientation, excluding, for instance, the case of a Mobius strip. For smooth manifolds 7 the traces of smooth functions u for XQ € 7 are defined as usual by ^"^(0:0)== lim u{x), d^u{xo) =^ lim n(xo)'Vu{x). If d:^u{xo) = d~u{xo), we define ^^^(^o) = d^uixo). Then each function u G ^^"2(^X7) which satisfies Au e L^(Q\7) has traces u"^ 6 H^^^{^/) and d^u G i/~^/^(7) obtained by continuous extension of the trace operator (cf. [11]). Note that the spaces H^^l'^[^i) can be identified with the spaces defined by taking the restrictions of distributions from H^^l'\d'^^) and that if w G W:^{Q\-i) and iif Aw G L'^i^Vi) then the jumps [u] = u+ - uand [dnu] = d:^u — d~u can be extended by zero to distributions in H'^^l'^{duj'^). We consider the following Dirichlet problem

I

-Au -: 0 1/^ = 0

in fi\7, on 7,

u=f

onF,

The problem (D) is well posed in ^^^2(^X7) for any given / G /f^/^(F), and one can define the jump [dnu] G H~^^'^{^/) across the admissible crack 7, as well as the normal derivative g = dnU G i/~^/^(F). The function / will be called the input data and g the output data. The inverse problem that we consider is to determine a crack 7 from the couple (/, g) of input and output data. We emphasize that here we will consider connected cracks, and we will prove that, using positive (or negative) input data, one measurement is enough to determine a conductive crack. Difficulties with identification. First, we notice that it is rather simple to consider an example where a single measurement is not sufficient to locate a crack. In fact, just consider the input data f{xi,X2) = :ri3:2, and any crack located on the xi or on the X2

On the identification of conductive cracks

215

axis. It is clear that u{xi,X2) = 2:1X2 is a solution of problem (D), no matter what set Q is considered, as long as it contains the cracks. Since the solution is the same, this input data does not allow to distinguish between two different conductive cracks if they are located in the axes. Moreover, supose that the two diflFerent cracks are both located in the xi axis, if the experiments are made using input data /(xi, X2) = {xi — c)x2, where c is any constant, the solution is u{xi, X2) = {xi-c)x2, no matter where the cracks are, since we have u{xi, 0) = 0, for all Xi. This shows that even an infinite number of measurements may be not sufficient to identify the crack, if the the input data is not well chosen. In the previous examples the input data changes sign, and we will now prove that if we provide input data that does not change sign, only one measurement will be enough to identify any conductive crack. To do this we begin by proving a crucial lemma that connects the support of the jump [dnu] with the crack itself. Notice that in the previous examples this support was void, because the solution was in fact analytical inside the all domain Q.

Figure 1: 3D profiles of the solution in two different cracks. One can see the discontinuities of the normal derivative on the crack. Under the hypothesis of positiveness on F it is clear, by the maximum principle, that no analytical solution would be possible. It is sufficient to change the sign of /, considering f{xi,X2) = |3:iX2| to see that the support of the [dnu] coincides with the crack. Two simple experiments with a finite difference method are presented in Fig.l. In both cases we considered Q —] - 1 , I p . The plot on the left shows the results for a crack 7i = [ - | , | ] x { 0 } , and the one on the right was made for a smaller crack 72 = {0} x [—i, 0], orthogonal to 71. In both cases it is clear the discontinuity of the normal derivative along the crack, which shows the relation between the support of [dnu] and the crack itself. In Fig.2, on the left, we plot out the difference between the output data dnU on the edges of the square for 71. It is clear that a 90 degrees rotation would produce a similar result, concerning a crack 73 = {0} X [—|, | ] , therefore it is this difference in F, the edges of the square ] — 1, I p , that allows the distinction between an horizontal or a vertical crack. It is worth noting that this difference is relatively small, in this case it is about 8% of the absolute values measured, and this is probably due to the fact that we impose / = 0 on the crack lines. More clear is the distinction between 72 and 73, as one can see in Fig.2, on the right. The dashed line shows the information on the edges which are parallel to the crack (notice that only one line is needed since the results turn to be the same either on {-1} x [-1,1] or on {1} X [-1,1]). One can see that the difference is more significant on the right, which

216

C.J.S. Alves, T. Ha-Duong and F. Penzel

corresponds also to the difference between 72 and 73 in what concerns the X2 axis. The thick line shows the result on the [—1,1] x {1} edge, where the difference is larger, and the the thiner one on the [-1,1] x {-1} edge, where the difference is smaller. This may still be explained to the fact that the difference between 72 and 73 can be resumed to the segment {0} x [0, \] which is closer to the [-1,1] x {1} edge. ,_,^^____^^

£0

15

^ ^ . - - ^

0.05 -0.1 0.15 -0.2

\

~ \

-

\y

-

-

/ /

Figure 2: The difference between the boundary profiles of dnU of the cracks 71 and 73 (on the left), and of 72 and 73 (on the right). Lemma 1. Let u he the solution of problem (D). If^ is an admissible crack and we consider f ^ 0,f ^ 0 onT, then supp{[dnu]) = 7. Proof. Suppose supp([^n^]) = 7i / 7, then 70 = 7\7i has a non void interior with respect to the topology of 7 and 70 Pi (^\7i) ^ 0. Since in 70 we have [dnu] = 0 and [u] = 0 (in fact u = 0), one can deduce that Aw = 0 in Q\7i. Thus, the maximum principle proves that there can not be a point XQ in ^ \ 7 i such that u{xo) — 0. That contradicts the fact that u is null on 70. • Remark 1. This support lemma also holds for non connected cracks and can be stated even for more general cracks with bifurcations, however we can not avoid the orientation of the crack, since otherwise we would not be able to define [dnu]Theorem 1. Let 71,72 be two admissible connected cracks such that, for an input data / > 0 , / 7^ 0, one has dnUi = dnU2 = ^, where u\,U2 are solutions of (D) for 71 and 72 (respectively). Then 71 — 72. Proof. We consider the open set r^\(7iU72). We define Qc to be the connected component of Q\(7iU72) with r C dO-c (there is only one component in this situation because 71,72 C Q), and we define Q* — Q\Qc (see Fig. 3). We have v = ui - U2 —^ inVtc because of Holmgren's theorem, because ui — U2 = f and dnUi = dnU2 = g on V implies v = 0 in a neighborhood of F inside Qc and by analytical extension i' = 0 on Qci) Suppose Q* = 0. This is the case when n\(7i U 72) = Qc is connected. In this case 1; = 0 on Q\(7i U 72), therefore [^„wi] — [dnU2\ on 71 and on 72. Outside 72 we know that U2 is analytic, meaning that 0 — [^„if2] = [dnU\] on 7i\72 and therefore supp([^n«i]) = 7i n 72- Using the Lemma we know that supp([^nWi]) = 71 and conclude that 7i = 7i n 72. The same argument gives 72 = 71 H 72, thus 71 = 72-

On the identification of conductive cracks

217

Figure 3: The filled area denotes the Q* set. This area can be crossed by 71 defining connected open components. One of this components will be the open set u. a) Suppose Q* y/^ 0. Let 7^ — Q* n 71, i.e. a part of 71 that divides Q* in open connected components and we take uu to be one of that components (if 7^ — 0 this means that uj = n*). Since dcu C Yi U dn\ this means that du = 7* U {dQ* n 71) U {dQ* n 72) and the part 72 = dQ* n 72 can not be void (otherwise du: C 71). Now, since ui = U2 on Q.c we have Ui — U2 — ^ on 72 C d^c and the condition Ui = 0 on 71 imply ui ::= 0 on duj. By uniqueness of the interior Dirichlet problem Ui — 0 in a; C Q\7i and therefore by analyticity Ui = 0 in Q\^fi which implies / = 0. This contradicts the hypothesis and therefore Vt* = 0, which brings us to case i). D Remark 2. An extension of the proof of this theorem to non connected cracks is under current research. SCREEN IDENTIFICATION We now extend this results for the Helmholtz equation, concerning the identification of acoustic screens. The proof of the Theorem 1 can follow the same steps, however, the proof of the support lemma must be different, since the maximum principle is no longer available! We now have

r -{A + k^)u=^0 {H)< u^ = 0

[ u=f

in Q\7, on 7,

onF,

where A; > 0 is the wavenumber. Our proof of the support lemma will have two restrictions: (i) 7 must be connected, and (ii) 0 < k < kc. Here ko is the smallest eigenvalue of the interior Dirichlet problem in Q. Lemma 2. Let u he the solution of problem {H). Suppose 0 < k < ko and that 7 is an admissible connected crack. If we consider / > 0 on F, then supp{[dnu]) — 7. Proof. Suppose again that supp([5„w]) = 71 ^ 7, then 70 — 7\7i still has a non void interior with respect to the topology of 7 and 70 Pi (0\7i) ^ 0. Since in 70 we have [^„w] •=• 0 and [w] = 0, one can deduce that (A + k^)u = 0 in Q\7i. Therefore 70 must be a part of a level line, and therefore it has an analytical extension, 7o where w — 0. It is clear that 70 can not cross F because we assume / > 0. Thus 70

218

C.J.S. Alves, T. Ha-Duong and F. Penzel

must intersect 7, and since 7 is connected this intersection defines an interior open set u, with border du) C 7 U 70, with Dirichlet boundary conditions w = 0. Since we suppose that k < ko and since a; C Q, by the strong monotonicity property of the eigenvalues we conclude that k is not an eigenvalue of the Dirichlet problem in cj, and therefore u — ^ in uj. The result now follows immediately, since by analytic continuation, if = 0 in Q\7, because u C n \ 7 , and this contradicts / > 0. • Remark 3. This proof can not be extended to non connected cracks. Consider the situation where 7 is defined by three components 7^ - [-3, -1] x {1}, 75 = [1,3] x {1}, 7^ = ^B(0,2)n R X R~. Suppose now that the support of [dnu\ is 71 = 7a U 76. Therefore 70 = 7c, and the analytic extension of the circle intersects 71 in the two disjoint parts 7a and 7^ without defining an interior open set uo, since the analytic extension can not cross 71 (see Fig.4). Remark 4- The restriction on k is merely for the sake of uniqueness in u;. An extension of the proof to any k is under current research.

Figure 4: In the case of non connected screens, it is not possible to ensure the existence of a;, as one can see in this example, since the analytical extension of 70 crosses 71 in two non connected components. Theorem 2. Suppose 0 < k < ko and let 71,72 be two admissible connected cracks. If for an input data / > 0, one has dnUi = dnU2 — g, where Ui,U2 are solutions of (H) for 71 and 72 (respectively), then 71 = 72. Proof. Immediate consequence of the proof of Theorem 1 and of Lemma 2. • REFERENCES [I] Isakov, V. (1998). Inverse Problems for Partial Differential Equations. Springer, New York. [2] Kress, R. (1995) Math. Meth. Appl. Sci. 18, 267. [3] Alves, C. J. S. and Ha-Duong, T. (1997) Inverse Problems 13, 1161. [4] Bryan, K. and Vogelius, M. (1992) SIAM J. Math. Anal. 23, 950. [5] Andrieux, S. and Ben Abda, A., (1996) Inverse Problems 12, 553. [6] EUer, M. (1996) Inverse Problems 12, 395. [7] Friedman, A. and Vogelius, M. (1989) Ind. Univ. Math. J. 38, 497. [8] Alessandrini, G. and Valenzuela, A. D. (1996) SIAM J. Contr. Opt. 34, 913. [9] Alessandrini, G. and DiBenedetto, E. (1997) Indiana Univ. Math. J. 46, 1. [10] Kubo, S. (1991). InJnverse Problems in Engineering Sciences, pp. 52-58; M. Yamaguti (Ed.), Springer, New York. [II] Grisvard, P. (1985). Elliptic Problems in Nonsmooth Domains. Pitman, London.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

219

I N V E R S I O N OF D E F E C T S B Y L I N E A R I Z E D I N V E R S E S C A T T E R I N G METHODS WITH MEASURED WAVEFORMS K. Nakahata and M. Kitahara Department of Civil Engineering, Tohoku University Aoba-yama06, Aoba-Ku, Sendai 980-8579, Japan

ABSTRACT The linearized inverse scattering methods are investigated to reconstruct the shape of defects in the two-dimensional elastic body. The methods are based on the Born and Kirchhoff approximations for unknov^n displacements in the integral representation of the scattered field. To show the performance of the methods, the specimen that has the circular cavity with cracks is prepared and the backscattering data are measured. The measured waveforms are processed and fed into the inverse methods. The results show the capability of the methods to reconstruct the shape and size of defects. KEYWORDS Ultrasonics, inverse scattering method, elastodynamics, measured backscattering data.

INTRODUCTION The size, shape and location of defects are a fundamental information to estimate the residual life of the structural component. The linearized inverse scattering methods to reconstruct the image of defects from the ultrasonic backscattering waveforms are investigated in this paper. The comprehensive review of the linearized inverse scattering methods has been given by Langenberg[l]. The Born inversion method has been studied by Rose et a/.[2,3] and KirchhofP inversion method by Cohen and Bleistein[4,5]. The performance of the methods to reconstruct the shape of defects has been examined by using the numerically calculated backscattering data in Ref.[6] for the 2D problem and in Ref.[7] for the 3D problem. The experimental performance of the methods has been shown in Ref.[8] for an elliptical cavity model and a notch model. In this paper, the aluminum specimen which has the circular cavity with notches is prepared as the combined defect model and the backscattering waveforms from the defect are

220

K. Nakahata andM. Kitahara

acquired by the immersion ultrasonic testing in the pulse-echo transducer configuration. The performance of the linearized inverse scattering methods is examined by using the measured backscattering waveforms.

SCATTERING PROBLEM It is important to distinguish the crack-like defects from the volumetric defects like a circular cavity. Here we consider the combined defect model as shown in Fig.l. The defect D^ exists in the two-dimensional elastic body D. The elastic modulus and mass density are denoted by Cijki and p for the host matrix D\D^ and by Cijki + ^Cijki and p + Sp for the defect D^. The measurement point y is far from the defects in the usual transducer arrangement. In this measurement situation, the L-L pulse echo methods for the ultrasonic testing is adopted. Here, the longitudinal(L) wave is transmitted as the incident wave and the same transducer receives the backscattered longitudinal(L) wave from defects.

Fig.l, Defect D"^ and wave fields in the host matrix

D\D^

LINEARIZED INVERSE METHOD In this study, two inversion methods are applied to reconstruct the shape of defects from the ultrasonic backscattering data. These methods are based on the elastodynamic inverse Born and Kirchhoff approximations. The details of two inversions have been given by Kitahara et a/. [6,7]. Here the brief outline of inversions are summarized. In the usual transducer configuration for the ultrasonic testing, the measurement point y is far from the surface of defects. The far-field expression of scattered wave is obtained as < ^ - ( y ) =. D{kL I y \)Am{if) + D{kT \ y \)B^{y)

(1)

where D{z) — J2/{Tiz)e'^^~''/'^\ In Eq.(l), Am and Bm are scattering amplitudes for longitudinal and transverse waves, respectively. In this study, we use the longitudinal

Inversion of defects

221

scattering amplitude in the actual experimental measurement A^[y) = ^^%ym 4/i

j qi{x)e-"''-y-^dV

(2)

JD

where K = ki/kr and y is the unit vector pointing to the measurement point y. In Eq.(2), qi{x) is the equivalent source which represents the characteristics of the defect. There are two ways to get the explicit form of the equivalent source. From the volume-type integral formulation, we get the expression qi{x) = r{x){Spuj'^Ui{x) - SCijkiUk,i{x)d/dxj}

(3)

where r{x) is the characteristic function of the defect D^ and has a unit value in the defect. From the surface-type integral formulation, the following form of qi can be obtained qi{x) = -j{x)Cijki{nj{x)uk,i{x)

- ni{x)uk{x)d/dxj]

(4)

where 7(x) is the singular function which has a value on the surface of the defect. When the scattered longitudinal wave component is measured, the left-hand side of Eq.(2) is known. If we can solve this integral equation for P or 7 in the equivalent source g^, the shape of the defect can be reconstructed. The problem is that both the geometrical function F (or 7) and the displacement field Um are unknowns in Eq.(2). For this reason two approximations are introduced for displacement fields to linearize Eq.(2). Born Inversion The Born approximation is introduced to the scattering amplitude in Eq.(2). Here the volume type of the equivalent source in Eq.(3) is adopted. The Born approximation is to replace the displacement field u by the incident wave u^ in the defect D^. The incident wave n° is assumed to be the longitudinal plane wave u°(x) = ^Od%xp(z^V.ic)

(5)

where vP is the amplitude, d is the unit polarization vector, k^ is the wave number of the incident wave, and p^ is the unit propagation vector. In the case of L-L pulse echo method, k^ = ki, d = p^ = —y. For voids, the elastic modulus SCijki and mass density Sp are set to be —Cijki and —p. The longitudinal scattering amplitude in Eq.(2) is reduced to the following form

AUh, y) = ^^%^ / r{x)e-'"'^y-^dv. Z

JD

(6)

In Eq.(6), the integral is the Fourier transform r{K) \j^^2k ii ^^^^^ characteristic function with K = 2kLy- If we can obtain the scattering amplitude Am{kL,y), the characteristic function T{x) is reconstructed by the inverse Fourier transform

r(x) = - 4 r

TT 70

r

^0

%iAm{kL,y)e''''y-''kLdk,dy. U KIT

(7)

222

K. Nakahata and M. Kitahara

Kirchhoff Inversion In the Kirchhoff approximation, the surface type of the equivalent source in Eq.(4) is adopted. The Kirchhoff approximation is to replace the displacement field u on the defect surface by the incident wave u^ and the reflected waves at the tangent plane of the surface. In this case, the scattering amplitude in Eq.(2) is reduced to Am{kL,y)

u^ymki = --

(8)

JD

where 7//(x) = ^{x)H{y • n{x)) is the function which has values in the illuminated side of the defect surface and H{') is the Heaviside step function. From Eq.(8), JH{X) is obtained by the inverse Fourier transform 2

r27r

TT^ Jo

Jo

(9)

U^KL

P R O P E R T Y OF B O R N A N D K I R C H H O F F I N V E R S I O N The basic property of two inversion methods is shown first by using the numerically calculated backscattering waveforms. The defect model is shown in Fig.2. In this model, the cracks of length a are attached to a cylindrical cavity of radius a.

o a

(y/a)

(y/a) oJ

- 2 - 1 0 1 2

-

3

-

2

B o m inversion

-

1

0

1

2

3

(x/a)

(x/a)

Kirchhoff inversion

Fig.2. Born and Kirchhoff inversions from numerical waveforms.

Inversion of defects

223

The backscattered waveforms are calculated by the boundary element method in all directions. The Born inversion reconstructs the characteristic function F which takes the value in the defects. The volumetric part is reconstructed well in Fig.2. The Kirchhoff inversion reconstructs the singular function 7 which takes the value only on the surface. The surface of the defects is reconstructed clearly in Fig.2.

EXPERIMENTAL SETUP The experimental setup is shown in Fig.3. The cylindrical aluminum specimen with a defect model is immersed in water. The pulser generates the electric pulse signal. This signal makes transducer vibrate and the ultrasonic waveform is transmitted into water. The scattered waveform from the defect is measured by the same transducer. The waveform transformed to the electric signal is branched to the receiver, and recorded on the digitaloscilloscope as the time-averaged data. The data in time domain are transformed to the frequency domain. The frequency domain data are used as the input for the linearized inverse methods.

Pulser-receiver

Fig.3. Experimental setup.

Digital-oscilloscope

pecimen

In this study we use the immersion type transducer whose center frequency is IMHz. The normalized frequency spectra of reflected waveform from the aluminum plate is shown in Fig.4. The frequency range from O.lMHz to 1.5MHz is used for the inversion of defects.

224

K. Nakahata andM. Kitahara 1.0-

— Absolute value - - Real part Imaginary part

y\ 1 ' 1 111/ 1

u 0 "

X vX

I

1

1

1

1

/

\

I

1

1

1

i

I

1

1 1

II II 1J

M

1 1

^

ly*—7-*,-^ \ /

/

V/

~

\ J \\

—

'

—

'

—

1

—

'

—

•

—

'

—

1

—

1

Frequency(MHz) Fig.4. Frequency range of transducer.

DATA P R O C E S S I N G The longitudinal scattering amplitude ym^m in the elastic solid is required for the shape reconstruction from Eqs.(7) and (9). To extract the waveform in the solid, the following data processing[9] is adopted here. The received waveform 0*^(/) in the experimental system is expressed in the frequency domain

o-(/) = /(/)T(/)py(/)H„,(/)£-(/)//,„(/)Ty(/)ii(/)

(10)

where / ( / ) is the input-signal, T{f),W{f),H^s{f),J^sw{f)yR{f) are the effects of the transmitting transducer, water path, transmission from water to the solid, transmission from the solid to water, receiving transducer, respectively. The term E^^{f) represents the scattering effect in the solid. The reference signal 0^^^{f) is measured as shown in Fig.5 and it can be written

O^'Hf) =

nf)T{f)Wif)HUf)E''^H,Uf)Wif)R(f)

(11)

where E^^^ is the reflection coefficient at a planar free surface. It is to be remarked that the measured reference signal in E q . ( l l ) is same as the measured signal in Eq.(lO) except for terms E^^^ and E^^. Therefore the scattered waveform in the elastic soHd is obtained from pref O'V) £-(/) (12) Oref{fy The processed waveform E^'^{f) can be used as the longitudinal scattering amplitude yraAm[kL,y)

in Eqs.(7) and (9).

Inversion of defects

lis

Fig.5. Measured and Reference waveforms.

MEASUREMENT AREA AND ANGLE In the actual NDT, it is not easy to measure the scattering ampHtudes from defects in all directions and frequency range. Here we examine the resolution of defects that the linearized inverse methods have from the following two view points. Measurement area The measurement area is defined in Fig.6. Here the measurement area is chosen in all direction or in the one side of the specimen. Measurement angle The measurement angle A^ is defined in Fig.6. Here the shape reconstruction of defects are performed by the waveforms with the measurement angle of 10° or 20° intervals.

Measurement area

Fig.6. Measurement area and angle.

226

K. Nakahata and M. Kitahara

SHAPE RECONSTRUCTION OF DEFECT We used a specimen made of aluminum as shown in Fig.7. In this specimen, two notches are attached to the circular cavity.

80r

Fig.7. Aluminum specimens for experiment. Figs.8 ~ 10 show the results of the shape reconstruction in the measurement angle of A^ — 10°. Fig.8 is the result from measurements in all directions and the shape of defect is clearly reconstructed. Fig.9 is the case of upper side measurement and Fig. 10 is the left side measurement. In Figs.9 and 10, the side of illuminations is almost reconstructed. Fig. 11 shows the result for the measurement angle of A^ = 20°. The rough shape of the defect is reconstructed in this case.

^G=w

All direction

i5i 10

(mm) 0 -5-

-20 -15

Bom inversion

-10

-5

0

Kirchhoff inversion

Fig.8. Reconstruction by Born and Kirchhoff inversions (measurement area : all direction)

Inversion of defects

227

A0=1O° Focus Upper side

15-|

(mm)

iHi

0-

-5-

•

-10-

10

^i

14 1i t|1111

5-

5

<^\

(mm)

oi -5

'•i

-10

-15-

-15-| 0

-15

1

1

1

1

1

1

-10

-5

0

5

10

15

-20

1

—

20 -15

20

Bom inversion

I

—

-10

I

—

-5

I

—

0

I

—

5

I

—

10

I

—

15

I

20

Kirchhoff inversion

Fig.9. Reconstruction by Born and KirchhofF inversions (measurement area : upper side).

A0=1O(

Left side

15105-

(mm) 0-5-10-15-20-20 -15 -10

Bom inversion

-5

Kirchhoff inversion

Fig. 10. Reconstruction by Born and KirchhofT inversions (measurement area : left side).

228

K. Nakahata andM. Kitahara

AO =20°

All direction 2015105-f

(mm) 0

J (nini) o-^P'

fl^H^

-5-10-

-15H

-15 +

-20 -15 -10

-5

5

10

Bom inversion

15

20

-20 -15 -10 -5

Kirchhoff inversion

Fig.ll. Reconstruction by Born and KirchhofT inversions (measurement angle : A^ = 20°). ACKNOW^LEDGEMENT A part of this work was supported by the Grant-in Aid for COB Research(No.llCE2003), The Ministry of Education, Science, Sports and Culture. REFERENCES 1. 2. 3. 4. 5. 6.

Langenberg, K.J.(1989). Wave Motion 11, 99. Rose, J.H. and Krumhansl, J.A.(1979). J. AppL Phys. 50, 2951. Rose, J.H.(1989). J. Pure AppL Geophys. 131, 715. Cohen, J.K. and Bleistein N.(1979). Wave Motion 1, 153. Bleistein, N. and Cohen, J.K.(1980). Wave Motion 2, 75. Kitahara, M., Hirose, S. and Achenbach, J.D.(1996). In: The First US-Japan Symposium on Advances in NDT, The American Society for Nondestructive Testing, Columbus, pp.187-192,

7. Kitahara, M. and Hirose, S.(1998). In: Inverse Problems in Engineering Mechanics, M. Tanaka and G.S. Dulikravich(Eds). Elsevier Science B.V., Amsterdam, pp.163-170. 8. Nakahata, K. and Kitahara, M.(2000). ln:Review of Progress in Quantitative Nondestructive Evaluation 19, D.O.Thompson and D.E.Chimenti(Eds). American Institute of Physics, New York, in press. 9. Schmerr, L.W.(1998). Fundamentals of Ultrasonic Nondestructive Evaluation, Plenum Press, New York.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

229

FAST R E C O V E R I N G ALGORITHM FOR CRACK S H A P E OF S T E A M GENERATOR TUBES USING GEOMETRIC APPROACH ^ Fumio KoJiMA^ and Nobuyuki O K A J I M A ^ ^ Graduate School of Science and Technology, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, JAPAN e-mail: kojimages, kohe-u. ac.jp

ABSTRACT This paper is concerned with a computational method for recovering a crack shape of steam generator tubes of nuclear power plants. Eddy current testing is used for in-service inspection of tubes in steam generators, The crack shape can be detected as voltage trajectories by scanning a transmitter-receiver pancake type probe coil. First, a mathematical model of the inspection process is derived from the Maxwell's equations. Second, the input and output relation is given by the approximate model by means of the finite element method. Numerical experiments for characterizing outer crack of tubes are demonstrated. Finally, a parameter estimation technique is proposed for recovering the crack shape using data from the probe coil.

KEYWORDS Parameter estimation, nondestructive evaluation, eddy current testing, spline function, finite element method.

INTRODUCTION Recently, the demand has grown for quantitative nondestructive evaluation (QNDE) of nuclear power plants. The detection of a crack on steam generator tubes is a critical issue for the structural integrity of the plants. In QNDE, the objective is not only to find out flaws but also to determine the size, shape, and orientation of the flaw. Eddy current testing (ECT) is used for in-service inspection of tubes in steam generators because of high detectability and rapid scanning process. Although there exist the wide range of tube degradation mechanisms, it is important to detect the surface crack inside and/or outside tubes. A transmitter-receiver pancake type probe coil is scanned along the inner presented at ISIP 2000, International Symposium of Inverse Problems in Engineering, 6-10 march 2000, Nagano, Japan

230

F. Kojima and N. Okajima

surface of steam generator tubes. Defects can be detected as a probe impedance or voltage trajectory from the receiver by exciting the transmitter. The inversion techniques by parameter estimation problems have been studied computationally and experimentally by many authors (e.g. [1][2]). Parameter estimation techniques can be effectively used if an accurate forward problem is carefully selected. However, those are computationally intensive iterative procedures in which the accurate forward problems must be solved numerous times. Recently, most efforts in computational saving are directed to the use of database for an ensemble of the forward solutions [3] [4]. In this paper, a new approach for parameter estimation is proposed for the sake of computational savings. The model output is directly reconstructed by parameter space with data sets of the forward solutions.

MATHMATICAL MODELING O F INSPECTION The eddy current problems considered here involve a transmitter-receiver pancake type probe coil as shown in Fig. 1. The mechanism of eddy current testing is described in the following way. First, the force current density vector is applied to the transmitter and then the corresponding eddy current is generated inside steam generator (SG) tubes. The voltage of the receiver is affected due to the existence of the eddy current. The eddy current testing is performed to detect and characterize the crack outside the conductor by analyzing the coil voltage change due to the eddy current. The mathematical model of eddy Steam Generator Tube Crack

Receiver

Transmitter Fig. 1. Sample material and measurement technique current testing can be described by the magnetic vector potential A = {A^.A^.A-^ and the electric scalar potential (\) in three spatial dimensions. Let (t.x) be the time t G [0, oo) and the location x — [xi.x^.x-^ in B?. Let V be the bounded domain in F?, V -^ \ x = (xi,a:2,X3)

"•^

2

/

2 <x\ 2 +. xl<{'^]

1^0

\ 2

,Q<x,
(1)

where VQ and vi are sufficiently large and where Vi is sufficiently small satisfying 0 < v^
Fast recovering algorithm for crack shape of steam generator tubes

231

Vo c V

Fig. 2. Inspection area

V such that < Xi + X2 <

Vo={xeV

—

,dr <X3


(2)

where di ( » Vi) and do ( ^ Vo) denote the inner and outer diameter of the tube. Figure 2 depicts the domain of the inspection area. As is well-known, the mathematical model of eddy current testing can be derived by Maxwell's equations. For convenience of discussions, the time integration of the electric scalar potential is defined by

#(i)

(r) dr.

(3)

J —(X

Then, the eddy current model considered here can be represented by 1 /^o

V^A + juja^{A

+ V $ ) = XTJT

V • jojcr^{A + V $ ) = 0

in V

(4)

in V

(5)

with the boundary condition A = 0

on a F

(6)

where uj denotes the angular frequency of AC current and where ac 0

inVi otherwise

(7)

In the above equation, ac implies the electrical conductivity of INCONEL600. The voltage change in the receiver coil can be computed as dlds

J JCR

(8)

where [V x A]|c7;^ imphes the magnetic flux measurement on the receiver coil surface and where dl and ds denote the fine element of circumferential direction of the receiver coil and the surface element of the horizontal sectional plan of the receiver coil, respectively. Thus the relation between the input and the output is described by the following map: A^;.

(9)

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F. Kojima and N. Okajima

EDDY CURRENT ANALYSIS Finite element model The inspection area V is divided into parallelepiped rectangular elements as shown in Fig. 3. Let A^e be the number of finite elements in V, and Nh be the number of nodes in V.

Vo V Fig. 3. Division of inspection area Let ip be an element of //o = {^ I (^, Vif e L^V),ip\dv

= 0}.

(10)

The magnetic potential A = {v4i,yl2,^3} and time differentiation of the electric scalar potential $ are approximated by Nh

(A:-1,2,3)

Al = J2^z^i

(11)

1=1 Nh

(12) where t/^^ denotes a basis function in HQ. Let

H^^ = K,u>^.--,
{k = 1,2,3,4)

be the Fourier coefficient vector of Ai, A^, A-^ and $ , respectively. The finite Galerkin model of Eqs.(4) and (5) can be given by

- [K^\ 0 0

AK,f

0 [Ki] 0

0 \K^\ 0 [K^] [Ki] [K,]

[K,]^ lK,f IK,]_

' W

w

< W' H/4

(13)

Fast recovering algorithm for crack shape of steam generator tubes

233

where the (j, z) G {!,••• , A^} component of each block matrix is denoted by (14) (15)

[^4 = J^ [ff ^x^i^rdV /v

^ 9xi

(/ = 1,2,3)

(16)

and the component of each vector is given by {F,},=

///

XrJ'o^l'jdV

(; = 1,2,3),

(17)

respectively. Simulation

studies

In the numerical experiments, the region V and the tube VQ defined by Eqs.(l) and (2) were preassigned as Vi = 0.01 [mm], Vo = 44.46 [mm], vi = 227.67 [mm], di = 19.69 [mm], do = 22.23 [mm], dr = 84.47 [mm], di = 143.20 [mm]. Since INCONEL600 is used for SG tube, the conductivity of the tube dc and the permeability of the tube //Q are taken as cTc = 1 X 10^ [S/m] and /IQ = 4TT x 10"'^ [H/m]. The number of finite elements and nodes were set as N^ = 13464 and Nh = 15552, respectively. The current density / and frequency / = Lj/2n of transmitter coil were / = 9.5238 x 10^ [A/m^] and / =150 [kHz] or 300 [kHz], respectively. The coils were moved along the crack as shown in Fig. 4. The scanning positions of transmitter and receiver coil were set as j,P ^ {^rcos6i,rsinOi,oc^)

(i = 1,2, • • • ,mp

(18)

where r = 9.145 [mm], Xg = 113.836 [mm] and rrip denotes the number of observation points. More precisely, the scanning procedures in Eq.(18) were set as rrip = 8 and {^^}Ll = {60.88, 66.18, 71.47, 74.12, 76.76, 79.41, 84.71,90} [deg]. The difference of angle between transmitter and receiver is a = 29 [deg].

Crack

Receiver Transmitter

Fig. 4. Scanning position of transmitter-receiver probe coil

(19)

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F. Kojima and N. Okajima

The defect considered here was assumed to be a circumferential crack with 0.2 [mm] width and to be exhibited outside the SG tube. The crack profile (of varying size and depth) is represented as shown in Fig. 5, where Nc denote the number of division of crack length and (7J (?' = 1, 2, • • • , Nc) denote the depth profile parameters (the percentage of thickness of SG tube). Integration in Eqs.(14)-(17) can be computed numerically by Gauss-Legendre formula for each finite element. The incomplete conjugate gradient method is adopted to the hnear system (13). The numerical results of the voltage trajectories corresponding to the tube with 25% crack and with 75% crack are demonstrated in Figs. 6 and 7, respectively.

PARAMETER ESTIMATION BY GEOMETRICAL APPROACH Our aim is to recover defect geometries using probe voltage trajectories. Parameter estimation problems are applied to our inversion technique. The usual procedure can be formulated as a minimization of the fit-to-data functional: £:,{q) = ^ | A £ ( q ) - A E , p

(20)

with respect to some geometrical parameters q. Here /S.Ed denotes measurement data, i.e., the set of voltage trajectories, while A E ( q ) implies the model output corresponding to the defect parameter q. The model output can be computed from the finite elements model (13). More specifically, it is described by AiJ(q) = / / ( q ) { A t ( q ) . ^ ( q ) , /lj(q), ^ ' ' ( q ) } ^ € « " "

(21)

where / / ( q ) denotes the appropriate interpolation matrix and where {•} denotes the finite element solution with respect to the defect parameter q. As is well-known, the minimizer of the fit-to-data functional E{c\) is to implement qC+i) = qC) _ A('>d<'>

for/ = 0 , l , 2 . - - -

(22)

where d^'^ denotes the feasible direction vector and where E,(q(') + A(')d(')) =

min E,(q(')-f r/d^'^). (23) r?e[o,oo) The practical implementations of the algorithm (22) involve the crucial part of computational volumes. Namely, those are computationally intensive iterative procedures in which the accurate forward problems (13) and (21) must be solved numerous times. Recently, most efforts on computational saving are directed to the use of database for an ensemble of the forward solutions. The model output is directly reconstructed by parameter space with data sets of the forward solutions of Eqs.(13) and (23). Let Qad be the class of admissible parameters given by Qad = {<\eR'''^\^
z = 1, 2, • • • , iVe}

(24)

where tc denotes the thickness of the SG tube and qy denotes the maximum depth of the crack. Let {BQ(X)}^^^ be the bilinear sphne functions [5] of the form

B,(x)=^^^Y[(l±x,)

in(-l,l]^^

(25)

Fast recovering algorithm for crack shape of steam generator tubes

Fig. 5. Parametrization of crack profile

> ^ O

>

2.4e-06

Cl-.

O ^

2.3e-06

dD 2.2e-06

2e-06 I 5.6e-07

L_ 5.8e-07

6e-07

6.2e-07

6.4e-07

6.6e-07

Real part of voltage (V)

6.8e-07

7e-07

Fig. 6. Numerical results of voltage trajectory (Depth: 25% {?>^^KHz))

5.6e-07

5.8e-07

6e-07

6.2e-07

6.4e-07

6.6e-07

Real part of voltage (V)

6.8e-07

7e-07

Fig. 7. Numerical results of voltage trajectory (Depth: 75% {Z^OKHz))

235

236

F. Kojima and N. Okajima

Furthermore the interval [0, qu] is assumed to be divided into Nd subintervals, Ai : 0 = 9 ^ < < ? i " < - - - < < ? r ^ ' = 9 ( / < i o . Then we can reconstruct the series of basis function {Bf^{q)}^^i^

(26) with knot sequence,

{A4 = nA^. Let us collect data set {^Ei}^^^^

(27)

given by G /?-^

AE, = AE{q^)

for^Xe{A4.

(28)

Then the model output can be approximated by Wd + l

AE(q) ^ Y. ^E^^f'i^)-

(29)

1=1

Thus, instead of solving the output least square problem for (23), we solve the following approximate parameter estimation problem: A p p r o x i m a t e Identification P r o b l e m : Find the optimal q = q which minimizes the approximate fit-to-data functional 1,^<^ + 1

^approx(q) = -

5 ] A E , 5 f ( q ) - AEA

|2

(30)

1=1

with respect to q G QadComputational

experiments

with simulated

data

In our numerical experiments, we chose a 'true' parameter vector q°, generated the corresponding solution numerically, added random noise, and then used this as 'data' for our inverse algorithm. The value of the true parameters and the initial guesses were chosen as q"" = I 25.0, 50.0, 50.0,25.0 ],

^'"'^ = [ 37.5, 37.5, 37.5,37.5 ].

For the optimization routine of (30), we used the gradient projection method. Table 1 reports the estimated parameter results for the data without noise and with 2,3,4, and 5% relative noise. In Table 1, the accuracy of the estimated value was evaluated as Accuracv=

'''"I

|q - q1

Figures 8 and 9 represent the value of cost function and the estimated crack shape, respectively.

Fast recovering algorithm for crack shape of steam generator tubes Table 1. True and estimated values of q at iteration 100 91 92 93 94 papprox

Accuracy

True 25.0 50.0 50.0 25.0

Noise free 24.76 49.75 49.69 24.61 8.882

2% noise 3% noise 4% noise 5% noise 24.94 25.28 26.19 29.39 50.62 50.17 49.29 50.49 49.87 50.18 53.35 56.55 30.55 32.79 36.44 37.61 6.165 1.387 2.463 3.849

X10-20

xlO-i^

X10-14

X10-14

X10-14

130.8

14.16

10.14

6.589

5.313

Fig. 8. Estimated crack depth (noise free)

True Shape Inital Guesses Estimate

Fig. 9. Estimated crack depth (2% noise)

Fig. 10. Estimated crack depth (5% noise)

IZl

238

F. Kojima andN. Okajima

CONCLUDING REMARKS A computational method was proposed for recovering a crack shape of steam generator tubes of nuclear power plants. Eddy current model was developed for the crack shape profile identification. A mathematical model of the inspection process was derived from the Maxwell's equations. The finite element model was effectively tested for the forward analysis of SO tube inspections. The validity of the proposed parameter estimation technique with experimental data is under study. ACKNOWLEDGEMENT This study was supported in part by the Research Committee on the Nondestructive Evaluation Technology by ECT of the Japan Society of Applied Electromagnetics and Mechanics through a grant 5 PWR utilities and Nuclear Engineering Ltd.

REFERENCES [1] BOWLER, J. R . , 1995. Studies in Applied Electromagnetics and Mechanics, 8, 31, lOP Press, Amsterdam. [2] KOJIMA, F . , 1997. Studies in Applied Electromagnetics and Mechanics, 12, 279, lOP Press, Amsterdam. [3] CHEN, Z . and MIYA, K . , 1998. J. Nondestructive Evaluation, 17, 167. [4] BANKS, H.T., JOYNER, M.L., WINCHESKI, B . , and WINFREE, W.P.,

nical Report CRSC, TR99-30, North Carolina State University.

[5] CHUI, C.K., 1986. Multivariate Splines, Vol. CBMS54, SIAM.

1999. Tech-

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

239

STRUCTURAL DAMAGE IDENTIFICATION USING STATIC TEST DATA AND CHANGES IN FREQUENCIES

X. Wang, N. Hu and Z.H. Yao Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China ABSTRACT In this paper, a two-stage identification algorithm for structural damages is presented using measured static displacements and changes in natural frequencies. In the first stage, a damage signature matching (DSM) technique is improved to detect the damage locations. In the damage signatures, the data of static deformation and natural frequencies are employed jointly and the effect of damage severity is eliminated effectively. The damage locations, then, can be detected successfully by matching measured damage signature (MDS) with predicted damage signature (PDS). After obtaining the possible damage locations, an iterative estimation scheme for nonlinear programming, based on the quadratic programming technique, is proposed to predict damage extents. Two examples are given to illustrate the effectiveness of the present algorithm. The first one is a planar truss and the test data are simulated numerically. The second one is a beam with two clamped ends, in which the data are obtained experimentally. The results show that the proposed algorithm is efficient for the damage identification of certain structures. KEYWORDS Damage assessment, static test, natural frequency, damage signature matching, nonlinear programming, planar truss, beam INTRODUCTION Within recent ten years, the damage assessment of structures has drawn wide attention in various engineering fields. The approaches in this area can be classified into two major categories, i.e. dynamic identification and static identification. Therein the dynamic ones have been developed more rapidly. For the detection of damage locations, Chen and Garba [1] suggested to adopt the well-known residual force vectors. The flexibility matrix is also used for damage detection [2, 3]. For the estimation of damage extent, Ricles, Kosmatka [4] and Lim [5] presented methods for correlating measured modal data with analytical finite element models. Kaouk and Zimmerman [6] identified both the damage location and extent using a generalized minimum rank perturbation theory. Although the dynamic identification algorithms are rapidly developed, many difficulties still should be overcome: (1) Most of the dynamic identification algorithms adopted numerical models, which cannot simulate the

240

X. Wang, N. Hu andZ.H. Yao

dynamic characteristics of the practical structure accurately. (2) For modal testing, it is difficult to obtain the obvious and accurate mode shapes on some stiff civil structures. (3) Some methods based on the changes of stiffness properties, require the modal data of high-orders, which are difficult to obtain accurately. Static damage identification methods are usually simpler than the dynamic ones. These methods have attracted much attention in civil engineering. Sanayei et al. presented a static parameter identification technique using the FEM, they further adopted the idea of model modification in an iterative identification procedure for damage detection [7]. More other static parameter assessment methods are based on the optimization algorithms [8-10]. Hjelmstad and Shin [11] presented an adaptive grouping algorithm of parameters for detecting and assessing the damages from static response. The main problems in the static identification methods come from the following two aspects: (1) There is less information used in the static damage identification methods, compared with those used in the dynamic ones, which makes the damage identification more difficult. (2) The effect of the damages may be concealed due to the limited load paths. It is very difficult to identify the damaged components, which have fairly little contribution to structural deformation under the applied load cases. Some researchers studied the algorithms using the dynamic and static test data together [12]. In this paper, a two-stage identification algorithm for identifying the structural damages is presented by employing the changes in natural frequencies and measured static displacements. A planar truss is used to numerically verify the effectiveness of the proposed algorithm. Furthermore, the experimental data obtained from a beam with two clamped ends are directly employed to assess the damage. THEORY Identification of damage location For healthy structures, the analytical model for static response can be described as follows: (1)

Ku = p

where K is the stiffness matrix and u is the displacement vector. Generally, the structural damages can cause changes in the stiffness matrix by an amount AiT, then the equilibrium equation can be expressed as (2)

{K^^K)u=p The displacement vector «* can be evaluated in first-order approximation as u={K + AK)-' p «{K-' - K'AKK')p

(3)

Then the change of the displacements due to the existing damages is described as Au = u-u

=K'AKK'p

(4)

By using the finite element method, the change of system stiffness matrix can be expressed as the summation of the changes in stiffness matrices of the elements. ND

AK = J^BjAk;B,

(5)

Here ND is the number of damaged elements, 5, is the Boolen matrix corresponding to the ith element and M / is the stiffness change of the /th element, which can be expressed as A^; = a^E^k,

(6)

where a. is a scalar denoting the damage fraction or damage extent of stiffness (-1.0
241

Structural damage identification

Poisson's ratio probably. For structures with single damage or multiple damages with the same severity, the scalar a. can be extracted from the equation (represented as a) for separating the influence of damage extent and damage location. The free vibration of an undamped structure is described by the following eigenvalue problem: [K-CO]M)I>^=0

(7)

where K is the stiffness matrix of the structure; M is the mass matrix; cOj is the yth natural circular frequency and ^^ is theyth mode shape. The change in square of natural frequency, ^co], can be evaluated in first-order approximation: AcD]^f.AK(l>j{f^M(l>^

(8)

In order to identify the damage locations, the idea of damage signature matching (DSM) technique is adopted [13]: If the changes of the static or dynamic response for all possible damage cases are predicted with an analytical model, the measured response changes can be compared with them one by one. The set of predicted changes, which are well matched with the measured values, can be identified and the corresponding damage case can be considered as the real damage of the structure. However, the magnitude of static or dynamic response changes due to damage depends on both the damage location and extent. The effect of damage severity should be eliminated before using the approach of DSM. Synthetically using the information of natural frequencies and static responses, the following vector is defined as the Damage Signature (DS): {DS}=A«/(A^;)

(9)

where is^co. is the change in anyone of all natural circular frequencies. According to equations (4) and (8), for the case of single damage or multiple damages with the same severity, the scalar a denoting damage extent could be eliminated in equation (9). The DS depends only on the damage location. Then, matching Measured Damage Signatures (MDS) with Predicted Damage Signatures (PDS) can assess the damage location. The MDS of load case i is defined as {MDS,}=[A«,/(A^^)L

(10)

where the subscript m denotes that the measured data are employed. The change in the first order frequency is generally used as the reference value. Similarly, the PDS under load case /, assume element k is damaged, can be calculated from equations (4) and (8) as: {PDS,}=[(A«,y(A«^)J^ (11) where the subscript p denotes that the value is predicted analytically. Finally, the summation of the norms for all load cases is defined as the total discrepancy for the possible damage case to evaluate the possibility of existing damage in element k £.=II|{PDSj-{MDS,,||

(12)

7=1

where nl is the total number of the static load cases. It should be noted that AM in the above equations includes only those degrees of freedom, where measurements have been made. The unmeasured components can be set to be zero or removed from the global vector using an appropriate Boolean matrix. The possible damage location in the structure is indicated by the smallest discrepancy calculated from equation (12). The following damage index Dek is defined for element k\

D.,=\IE,

(13)

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X. Wang, K Hu and Z.H. Yao

Estimation of damage extent When the possible damaged members are detected by employing the algorithm described above, the estimation of damage extent is the next important step. The present algorithm is based on the minimization of a squared error function, which has the following general form for every load case, where the variables x are linearly constrained between bounds: minimize J(x) = \e(x)f jl

s.t x<x<x

(14)

where e{x) is a measure of discrepancy between an observation and its counterpart computed by the parameterized model. For a certain load case of load vector /?, the measure of error is the difference between the change of the displacements computed from the FE model and the measured response: e,{a) = [K-'p,-{K^^Kyp,]-Mi,

/ = 1,2,...,«/

^K-'/^KK-'p,-^u,

(15)

where AM is the change of the displacements experimentally acquired under the load vector p. The design variable vector a includes the damage extents of all possible damaged elements. If there are NP possible damaged members at the first detection step, from equations (5) and (6), it can be obtained: A7^

K^KK'p

NP

= Y,a,E,K-\B]^k,B,)K-'p

= Y.a,Z,

(16)

where Z, = E.K'iB'^Mi^E^K'p

(17)

By employing the least-square method, the following optimization model can be established. minimize ^ ^ (a,Z/ - Aw ^ j s.t. -1.0
s.t.

+ C^a +D (19)

-d
a = {a, a, ... a^,,} A = [A„\ A,=±ZiZ;,

(20a) kJ = \,2,-.,NP

(20b)

7=1

C'=\-±Z,'Au' -iz^Au'

-

-i;z;,A«4

(20c)

7=1

and d, 0 and / are a vector with all components of unit value, a vector with all components of zero, and unit diagonal matrix, respectively. The algorithm presented by Goldfarb and Idnani [14] is adopted to deal with this quadratic programming problem. The constrained optimization problem in equation (14) is intrinsically nonlinear, and the first-order approximation is introduced for linearization. However, if the absolute value of at is not small enough, the precision of linear approximation cannot be

Structural damage identification

243

guaranteed. Then, there will be extraordinarily large discrepancy between the true value of the damage extent and the calculated one. The following iterative process is employed to correct the conventional algorithm. Step I: Initialization. Initialize the variables, vectors and matrixes that will be utilized when solving the quadratic programming problem. Initialize the iteration index / = 0. Start with a working set that there is no damage in the structure: AJSTO = 0 ,

a, =Aa, =0

(21)

Step 2: Solve quadratic programming problem. Calculate the change of the stiffness matrix A^, according to the current increment of damage extent A a/, which is used to replace a in equations (5) and (6). Add up the iteration index (/=/+l), then update the stiffness matrix: K. = K,_j + AK._, (22) Introduce into ^Aw)/, which is expressed in equation (23), to replace A5 in equation (18). S(Aul=Au-Au, (23) Solve the quadratic programming problem (19) using Goldfarb and Idnani' s algorithm to get the increment of damage extent A at instead of the damage extent. In equation (23), AS is the vector of the change of the displacements experimentally acquired, and Au. = K~^p - K~^p is the one calculated using equation (1), where there is no approximation in it. Step 3: Evaluate the discrepancy of the linear approximation. Update the value of damage extent as follows a. = a,_, + Aa. (24) If A ai is less than a pre-set threshold, it can be considered that the obtained damage extent at is sufficiently close to the real value, then terminate the analyzing procedure. Otherwise, go back to step 2. Because the increment of the damage extent Aat in equation (19) is used instead of the damage extent at, the definition region of components in Aat should be [-1.0,1.0], instead of [-1.0,0.0] as expressing in equation (19). NUMERICAL EXAMPLES Planar truss structure

Fig. 1. A five-bay planar truss structure for damage identification A five-bay planar truss shown in Fig. 1 is used to verify the damage identification methods described above. The FEM analysis is carried out to simulate the experimental data by using two-noded linear bar elements. The lumped-mass representation is adopted to generate the mass matrix during calculating the natural frequencies. Total numbers of elements and nodes are 26 and 12, respectively. The diagonal and side dimensions of each square bay are 0.7071 m and 0.5 m, respectively. The truss is constructed from mild steel truss members, whose material constants are: E = 2.1 x 10^ GPa, p = 7.9 x 10^ kg/m\ the section area is ^ = 1.0 x 10""^ m l Two load cases are shown in Table 1, the value of the loads are: Pi=1.0x lO^N, ^2=3.Ox lO^N, P3=1.0xlO^N.

X. Wang, N. Hu and Z.H. Yao

244

Table 1. Load cases investigated for the planar truss Load case

Loads P2,P3

Five damage cases shown in Table 2 are investigated, including the longeron and batten damage cases. Cases A and B denote the single damage location problems, cases C and D denotes the multiple damage cases. Case E denotes the seriously damaged components to verify the correctness of the estimation of damage extent. Table 2. Damage cases investigated for the planar truss Damage case

Damaged element

A B C D E

Damage condition 20% reduction 20% reduction 20%) reduction 20%) reduction 50%) reduction

10 11 10, 11 10,20 10

in E in E in E in E in E

Damage location. The damage index Dek calculated from equation (13), which is normalized by the largest one in all candidates and denoted by NDE, is displayed in the bar charts for four kinds of damage cases. In equations (10) and (11), the change in the first order frequency is used as Aco. From Fig. 2(a) and 2(b), it can be found that the proposed approach is very effective for the cases of single damage. For damage case C, the result of damage location assessment is shown in Fig. 2(c). The damaged batten element 11 cannot be detected, although the damaged longeron element 10 can be identified obviously. In fact, the load carried by element 11 is much less than that of element 10 for two assumed load cases. For damage case D, element 10 can be detected clearly, although the damage location figure becomes obscure in Fig. 2(d) due to the similar roles played by elements 10 and 20 in the structural deformation. 100.00 iij 90 00 (a) z 80.00 g 70.00 1 60.00 §> 50.00 1 40.00 •o 30.00 •= 20.00 'r i 10.00 2 0.00 L A ^ ^ A J . 1 3 5 7 9 11 13 15 17 19 21 23 25 Element number

g ^ 5 ~ 1 5 ^ 1 S ^

100.00 90.00 80.00 L 70.00 60.00 t50.00 40.00 i30.00 2000 10.00 0.00 1 3

(c)

5

7

9 11 13 15 17 19 21 23 25 El ement number

g 1 o Z 1 5 ^ 1 1

100.00 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00

(b)

9 11 13 15 17 19 21 23 25 Element number 100.00 iU 90.00 z 80.00 2 70.00 1 60.00 1 50.00 i 40.00 1 30.00 % 20.00 1 10.00 z 0.00 9 11 13 15 17 19 21 23 25 Element number

Fig 2. Identified damage location using complete and accurate test data Damage extent. For assessing the damage extent, in equation (15), we employ two load cases independently. The corresponding results of estimated damage extent (see Table 3) show that the exact value can be obtained using the proposed optimization technique with iterative

Structural damage

245

identification

process, even in the case of 50% reduction of the Young's modulus. The undamaged elements can be removed effectively, though they are chosen as the possible damaged members. Table 3. Results of the damage extent for the planar truss Damage extent

Damage case

Possible damaged element

Load case 1

Load case 2

A

10

0.1999

0.2000

B

11

0.2000

0.2002

C

10

0.2000

0.1962

10/11

0.1998/0.2086

0.1999/0.2000

10

0.2221

0.2414

7/10/20

-0.7 X10~V0.1999/0.2001

0.6X10"V0.2000/0.2000

10

0.5003

0.5006

D

Beam with two clamped ends -25.5 cm -

, Saw cut

I PI

I 0.6

x^j 1 \ I \ i \ A \ s \ k \ 1 \ % \9\ \n \n \n \\3 \u ^iS: \\6\\i \w \\9\ 1^

I

Fig. 3 . A b e a m with t w o clamped ends A n aluminum b e a m with t w o clamped ends is used to validate the present method (Fig. 3). T h e material constants are: E= 70.0 GPa, /7=2.70x 10^ kg/ml A saw-cut d a m a g e w a s introduced b y cutting t w o cracks on the top and bottom surface of the intact specimen. T h e depth of one crack is a quarter of t h e total thickness of the beam. Experimental modal analyses and static tests were conducted on intact and damaged beam specimens. T h e F E M computation w a s performed employing 2 0 two-node b e a m elements. T h e saw-cuts are located at the center of the 9th element. O n each node of this element, there are t w o degrees of freedom, i.e. deflection and rotation. T h e saw-cut damage in the F E M computation is approximately simulated using reduction of the bending rigidities in the 9th element, i.e. 8 7 . 5 % reduction of flexural rigidity. The experimental frequencies and the static responses in t w o load cases of the intact and damaged structures were obtained (Table 4). The above information, combined with the F E model, w a s used to identify the damage location and assess the damage extent. Table 4. Load cases for the b e a m with two clamped ends Load case

Node

Force

11

50.0 N

6

50.0 N

Test data and filtering using wavelets. For the intact and damaged structures, the lowest two frequencies obtained experimentally and numerically are shown in Table 5. It can be found that the numerical results agree with t h e experimental ones very well for t h e intact structure. H o w e v e r , t h e differences b e t w e e n t h e n u m e r i c a l a n d e x p e r i m e n t a l results b e c o m e m o r e significant for t h e d a m a g e d structure. T h e measured static responses are plotted in Fig. 4. The deformations obtained from the F E M analysis are also shown for comparison. A n important thing to b e note is that t h e test data include only the deflection of all the nodes. There is n o

246

X. Wang, N. Hu and Z.H. Yao Table 5. Results of the frequency for the beam with two-fixed ends Intact FEM Experiment

2nd Frequency Intact Damaged

1 St Frequency Damaged

85.969 85.726

236.657 238.572

74.710 83.984

223.760 235.792

information of rotations could be measured. Furthermore, the deflection at nodes 5, 6 and 7 in load case 1 and nodes 10, 11 and 12 in load case 2, respectively cannot be measured for the geometrical interference of the experimental apparatus. The values at these nodes in Fig. 4 are obtained by polynomial fitting. From this figure it can be found that there are little differences between the numerical and experimental results in the load case 2. But the differences are more obvious in load case 1, although the fundamental shapes are similar. These differences may be caused by the boundary condition, material properties, approximation in numerical model and by the noises in the experiments. (intact) Measured (damaged) I FEM (intact) FEM (damaged) -^ 0.4 I E I "^ 0.3 i 5

0.2 I

6

i

Number of node

Number of node

Fig. 4. Measured static responses (left: load case 1; right: load case 2) 0.040

^

0.07

/ \ // //

| . 0.06

I

0.04

•5

0.03

§> 0.02 5

0.01

/ \

0.035 0030

'o \

—•—TMtdata — o — Numerical result

0.025 0.020 0.015 0.010

/y

0.005 0.000

0.00

o^«^'

-0.01 Number of node

Number of node

Fig. 5. Changes of the displacements (left: load case 1; right: load case 2) The changes of the displacements obtained experimentally are shown and compared with the numerical ones in Fig. 5. In this figure, the changes of the displacements of the beam can be considered as low frequency component of the wave in space field. The measuring error can be regarded as high frequency noise. Then, a wavelet filter is employed to process the test data. Because the test data is decomposed into approximation coefficients and detail coefficients by performed one-dimensional wavelet decomposition, the influence of different coefficients to the identification results can be separated and verified respectively. As a well-known mathematical tool for the frequency spectrum analysis, wavelet transform inherits and develops the basic idea of conventional Fourier analysis. It is more suitable for reflecting the local characters of the signal and has the higher distinguishing ability for higher frequency. In the toolbox of the software MATLAB, four kinds of wavelets have been selected to process the test data. The parameters adopted in these wavelets are shown in Table 6. The universal threshold V21og(«) is selected by the reason of more efficiently removing the noise. The filter values using "sym6" and "db4" wavelets make the wave shape smoother and more

Structural damage identification

247

close to the numerical one, while the results using "haar" and "sym2" wavelets keep only the main and rough trend of the test data. The filter result using wavelets are shown and compared with the test data in Fig. 6. Table 6. Wavelet and the parameters adopted using in MATLAB Wavelet

Multiplicative threshold rescaling

Sym6 Db4 Haar Ssym2

mln sin sin mln

Decomposition level

Number of node

Number of node

Fig. 6. Filter results of the displacement changes (left: load case 1; right: load case 2) Damage location. In equations (10) and (11), the change of frequency, i.e. Aco is used the changes in the first and second order frequencies respectively for comparison. The identification results of damage location, representing by the normalized damage index NDE, are shown in Figs. 7 and 8. From Fig. 7(left), the damaged element 9 can be detected very clearly when using the original test data and the first order frequency. Few undamaged 1.00E+02 uj- 9.00E+01 i . 8.00E+01 -| g 7.00E+01 -j ~

6.00E+01

I

5.00E+01

E 4.00E+01 Z

3.00E+01

•^ 2.00E+01

11 13 Element number

15

17

19

i

1.00E+01

Z

O.OOE+00 13 Element number

15

17

19

Fig. 7. Damage location results using original test data (left: 1^ frequency; right: 2" frequency)

Fig. 8. Damage location using "sym6" filtered data (left: 1 frequency; right: 2" frequency) elements may be estimated as damaged ones. In Fig. 7(right), the identification effect is better when using the 2nd order frequency. The reason may be that the relative error in the experimentally obtained first order frequency is larger than that in the 2nd one. Fig. 8 show the

248

X. Wang, N. Hu and Z.H. Yao

computational results of damage location after the processing using "sym6" of wavelets. The Figure illustrate that the results using "sym6" wavelets are almost the same with that in Fig. 7, although some specific figures may be better in the results of "sym6" wavelet. Damage extent. The damage extent is estimated by choosing element 9 as the possible damaged member. The results of the iterative optimization process are shown in Table 7. It can be seen that there are obvious errors in the damage extent, about 100 % error by comparing with the accurate extent, i.e. 87.5 % reduction in Young' s modulus. This deviation may mainly be caused by the difference between the numerical model and the practical damaged beam. Actually, the beam with saw-cut should be modeled as a two dimensional problem. Table 7. Results of the damage extent for the beam with two-fixed ends Damage extent

Load case 1

Load case 2

0.406

0.420

CONCLUSIONS An efficient and simple damage identification technique of two stages is proposed, which employs the structural static deformation and the first several natural frequencies. The basic idea of the DSM technique is adopted and improved for effectively detecting the damage locations. An iterative optimization algorithm is presented to assess the damage extents. The efficiency of the proposed technique has been demonstrated using the planar truss structure and the aluminum beam with two clamped ends. From the identification results, it can be found that the present method is efficient for locating the single damage in the structure. Furthermore, it is successful when using the experimentally obtained data. For multiple damage case, usually only one most important damaged location to the structural deformation can be identified. Furthermore, although the proposed technique can estimate the damage extent very accurately when using the numerically simulated data, only approximate even poor damage extent can be obtained when using experimentally obtained data, if the difference between the numerical model and the practical structural member is considerable. ACKNOWLEDGEMENTS Financial support from the fiind of National Key Lab. on Mechanical Strength and Vibration in Xi'an Jiaotong University is gratefiilly acknowledged. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Chen, J.C. and Garba, J.A. (1988). AIAA Journal 26(12), 1119-1126. Lin, C.S. {\995). AIAA Journal 1^{9\ 1650-1654. Pandey, A.K. and Biswas, M. (1994). Journal of Sound and Vibration 169(1), 3-16. Ricles, J.M. and Kosmatka, J.B. (1992). AIAA Journal 30{9), 2310-2316. Lim, T.W. (1995). AIAA Journal of Guidance, Control and Dynamics 18(3), 411-418. Kaouk, M. and Zimmerman, D.C. (1994). AIAA Journal 32(4), 836-842. Sanayei, M. and Scampoli, S.F. (1991). Journal of Engineering Meek 117(5), 1021-1036. Sanayei, M. and Onipede, O. (1991). American Institute of Aeronautics and Astronautics Journal 29(7), 1156-1179. Banan, M.R., Banna M.R. and Hjelmstad, K.D. (1994). J Struct. Engrg, 120(11), 3243-3258. Banan, M.R., Banna M.R. and Hjelmstad, K.D. (1994). J Struct. Engrg, 120(11), 3259-3283. Hjelmstad, K.D. and Shin, S. (1997). Journal of Engineering Mechanics 123(6), 568-576. Hajela, P. and Soeiro, F.J. (1989). AIAA Journal 28(6), 1110-1115. Lam, H.F., Ko, J.M. and Wong, C.W. (1998). Journal of Sound and Vibration 210(1), 91-115. Goldfarb, D. and Idnani, A. (1983). Mathematical Programming 27, 1-33.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

249

INVERSE ANALYSIS FOR FRACTURE PROCESS ZONE CHARACTERIZATION D. K. TRAN\ M. T.KOKALY' and A.S. KOBAYASHI' 'The Boeing Commercial Airplane Group, Seattle, Washington 98124-2207, USA ^University of Washington. Dept. of Mech. Eng., Seattle, Washington, 98195-2600, USA.

ABSTRACT A hybrid experimental-numerical procedure based on inverse analysis is used to characterize the fracture process zone in semi-brittle material. The crack bridging forces and the dissipated fracture energies in concrete and high density polycrystalline alumina were determined and the trailing fi-acture process zone was shown to be the dominant source of fracture energy dissipation. KEYWORDS Inverse analysis, hybrid experimental-numerical analysis, concrete fracture, ceramic fracture, fracture process zone.

INTRODUCTION For concrete and single phase ceramics, the two toughening mechanisms, which augment Griffith's surface energy dissipation rate, involve possible energy dissipation in the frontal and trailing process zones. In addition, phase transformation in some ceramics can reduce the crack tip state of stress and hence increase its fracture toughness. Observed micro-cracking in the frontal process zone, such as those generated by aggregate-mortar debonding in concrete and by differential thermal contraction during the processing of ceramics have led to theoretical estimations of the increased fracture toughness due to the "crack shielding" caused by the increased compliance in the frontal crack tip region. These microcracks, however, must be within two micro-crack distance of each other for any measurable increases in comphance. On the other hand observed bridging of ceramic grains across the opposing crack faces in single phase ceramics has led to numerous theoretical modeling in the trailing wake zone of an extending crack. Similar theoretical modelings of crack bridging in concrete have been conducted for a longer period. Historically, Hillerborg [1] was the first to advance the concept of a cohesive zone for a trailing fracture process zone of a stably growing crack in a concrete

250

D.K. Tran, M.T. Kokaly andA.S.

Kobayashi

fracture specimen to explain the "strain softening behavior" observed in such specimens. This cohesive zone concept, which derives its origin from the ideahzation of ductile fracture of metals, was based on a postulated fracture process zone (FPZ) where microcracks form and coalesce ahead of a growing traction-free macrocrack. Later studies have, however, related that cohesive zone to the fracture surface immediately behind the crack tip and where aggregate bridging for concrete fracture, and grain bridging for fracture of monolithic ceramics, restricted the crack opening. For mode I fracture, this resistance can be characterized by a crack closure stress (CCS) versus crack opening displacement (COD) relationship which was modeled by a series of nonlinear spring straddling the crack face as shown in Fig. 1. This model smeared the discrete nature of the bridging mechanism.

Force

Nonlinear Springs Bridging Crack Faces

Displacement

Fig. 1. Force-displacement relation for nonlinear spring. Most of the above studies involved an inverse analysis where a postulated CCS versus COD relation was varied until the computed and measured remote loading parameters, such as the applied load and load-line displacement, coincided. These remote parameters are, however, relatively insensitive to the small changes in the CCS versus COD relation as will be shown later. Experimental studies involving direct measurements of the crack tip parameters are relatively few since the COD in the vicinity of the crack tip is difficult to measure and the CCS cannot be determined directly. In the following, the utility of the inverse technique is demonstrated in a hybrid experimentalnumerical analysis of the fracture process zone in semi-brittle material. HYBRID EXPERIMENTAL-NUMERICAL ANALYSIS Hybrid experimental-numerical stress analysis provide more and correct information, which otherwise cannot be obtained separately from the two procedures. The hybrid technique is particularly usefril in interjecting physical reality in the theoretical/numerical analyses associated with linear elastic fracture mechanics as well as with nonlinear fracture mechanics involving plastic yielding in front of the crack or crack bridging stress trailing the crack tip. Nonlinearity in the latter often requires the use of inverse analysis in seeking an optimized material properties or constitutive relation associated with the fracture process.

Inverse analysis for fracture process zone characterization

251

Experimental procedure The experimental procedure consists of determining sequentially the vertical and horizontal displacements, u and v, surrounding the extending crack with a four beam moire interferometer [1]. An orthogonal moire grating of 600 or 1200 lines/mm with a fringe multiphcation of two provided the two displacement fields from which the crack opening and crack sliding displacements, (COD and CSD), were determined. The grating for room temperature testing was affixed by the well known replica technique [2]. Briefly, the procedure consisted of transferring a master grating of 600 or 1200 lines/mm together with a highly reflective aluminum coating to a thin half-cured epoxy film coated on the specimen surface. Phase shifting was used to further increase the sensitivity of the room temperature moire analysis [2]. For elevated temperature testing, a layer of photoresist coating on the specimen surface was exposed to the +/- diffraction order of a collimated light from a He-Cd laser. After developing, the coating was etched with an oxygen plasma, nichrome sputtered and the photoresist mask was removed. The remaining islands of nichrome dots formed a 1200 lines/mm orthogonal grating. Phase shifting could not be used for elevated temperature testing since the CCD camera was not fast enough to freeze the moire fringe patterns oscillating with the air current surrounding the furnace during the elevated temperature testing. Thus a 35 mm camera with a shutter speed of 1/1000 second was used to capture the oscillating moire which was then analyzed by a nonlinear regression analysis. Numerical procedure The numerical procedure consisted of a propagation analysis [3] of the fracture specimen by a two-dimensional, linearly elastic, finite element (FE) model. The measured load and the associated moire fringe pattern provided the prescribed vertical load and horizontal displacement at the contact point between the loading pin and the specimen. The instantaneous crack tip location was determined from the moire data with an accuracy of 0.1 mm. Since the boundary conditions were completely prescribed by the experimental data, this finite element boundary value problem could have been solved directly. In practice, however, the relative displacement field obtained from the moire grating did not provide the COD/CSD with the necessary accuracy for a prescribed displacement boundary value problem. Thus the time consuming inverse analysis was used to determine the crack closing stress (CCS) versus COD and the crack shearing stress (CSS) versus CSD relations which provided an agreeable match between the computed and measured COD and CSD. Figure 2 shows the flow chart used in the trial and error process of matching the measured (by moire) and the computed COD's (by FE analysis) in this boundary value problem. The iteration process was stopped when good agreement between the computed and measured displacement of at least three crack extension was reached. The CCS versus COD and the CSS versus CSD relations and the modulus of elasticity were the three disposable parameters in this inverse analysis. The sum of the work done by the CCS acting on the COD and the CSS acting on the CSD for unit crack extension represented the energy dissipation rate in the fracture process zone. The energy release rate was computed directly by incrementing the crack tip, one finite element node and computing the difference in elastic energies stored in the specimen prior to, and after, incremental crack extension.

252

D.K. Iran, M.T. Kokaly andA.S. Kobayashi

MIXED-MODE CONCRETE FRACTURE While it may be argued that mixed mode, i.e. a combined Kj and Kjj, fracture does not occur in brittle material, the mere existence of interlocking aggregates/grains generate interface shearing force along the straight crack. The most common mixed mode fracture in concrete structures is the diagonal tension failure of a reinforced concrete beam. Such cracking usually develops from flexural cracks generated along the tension edge of the beam. The initial crack extend perpendicularly from the edge of the beam and kinks only after extending beyond the reinforcement.. In this study, the influence of interface shearing force due to aggregate interlocking on crack kinking was quantified by offset loading plain concrete beam specimens. Three-point bend concrete specimens, 406 x 95 x 50 mm, were fabricated with a high early strength Portland cement, an aggregate of 6.4 mm maximum size and a local Seattle sand. The concrete strength at the time of the test, as determined from cylinder tests, was 48 MPa. A shallow notch of 0.3 mm width and 19 mm depth was machined in the bottom of the specimen and an initial crack was extended an additional 19 mm depth from that notch by Mode I loading. Mixed mode loading was applied by rearranging the loading points so that the notch 'vas offset from the central loading point. The test procedure consisted of applying an mcrementally increased load-point displacement, photographing the moire fiinges and recording the applied load and displacements. Displacements were increased until the microcrack tip had almost penetrated through the depth of the specimen. A total of five specimens were tested for stable crack growth under mixed mode fracture. The kinked crack paths of the first batch of two specimens differed from those of the second batch of three specimens as shown in Fig. 3. This difference was attributed to the greater Mode I crack extensions applied to the second batch.

C START J FEM INPUT - Load and Displacement History - Crack Length - Poisson Ratio

OUTPUT - Stress - Strain - Displacement • Energy

c

END

1st BATCH KINKING ANC5LE: 30*

2nd BATCH KINKING ANGLE: 19« I

J

Fig. 2. Flow chart for calculating COD and CCS.

Fig. 3 Kinked crack paths in five concrete fracture specimens.

Inverse analysis for fracture process zone characterization

253

Figure 4 shows the average measured load versus computed load-Hne displacement relations for the second batch of specimens. It was necessary to use the computed rather than the measured load-line displacements since the latter, which was recorded only on one side of the specimen, did not represent the overall deflection due to the uneven local crushing at the load point. Figure 5 is a composite figure showing the COD and CSD variations along the initial and kinked crack for increasing crack extensions expressed in terms of the total crack length in three specimens from the second batch. The curves in this figure are the displacements obtained from moire interferometry and the circles are computed results. The CSDs along the kinked portion of the crack are negligible, but are appreciable along the initial crack region where appreciable rigid body motion is generated.

CRACK EXTENSION

• •

LOAD(w/FPZ) LOAD(w^oFPZ)

Computed Load-Line OispL (^m) Fig. 4. Average load vs. load-line displacement. 2nd batch. "T" 120

CRACK KiNKiNQ

100

1 . ®° Q CO

— •— • O

-r-

MEASURED COD MEASURED CSO COMPUTED COD COMPUTED CSD

60

8 = -20 -40

INITIAL

SAWED CRACK

oeooo KINKED CRACK REGION

Total Crack Length (mm) Fig. 5. Measured and computed COD/CSD. 2nd batch.

254

D.K. Tran, M.T. Kokaly andA.S. Kobayashi

Figure 6 shows the CCS versus COD and the CSS versus CSD relations which provided the best fit between measured and computed CODs and CSDs. The CCS increased with increasing CSS thus indicating that additional frictional forces were generated by the interlocking forces in the more severely kinked crack for the two specimens from the first batch.

<0

a

CO

o

CO

o o

60

80

COO ancj CSD (^m) Fig. 6. CCS vs. COD and CSS vs. CSD relations. The differences in the crack kinking angles in the two batches of concrete specimens resulted in the differences in the input work and hence in the partitioned energies. However, the fracture energies and the fracture energy rates dissipated in the FPZ for the two batches of specimens coincided thus suggesting that these quantities are intrinsic material property of the concrete studied. For a frilly developed FPZ, the dissipated energy rate at the FPZ was about 70 percent of the released energy rate. CRACK GROWTH IN ALUMINA AT HIGH TEMPERATURE Wedge-loaded, double cantilever beam (WL-DCB) specimens, 26.6 x 7 x 3.8 mm, were machined from a high-density (AL23) commercial alumina with 99.7% purity. The grain size ranged from 3 to 28 jam with an average size of 10.4 ^im. The specimen was side grooved in order to channel the stably growing crack along its intended straight crack path. The fracture specimen was loaded in a rigid displacement controlled loading fixture and the sequentially changing moire fringes associated with a stably growing crack were recorded. Recordings of the load were generally made at the peak load of each increment of increasing displacement loading. Load line displacement was not recorded A total of 12 specimens were tested at room temperature, 600, 800 and 1000 C. Moire fringe patterns obtained from each test provided the crack opening profile and the wedging displacement at the pin/specimen contact for several crack lengths and loading. Figure 7 shows the COD profiles obtained from the moire fiinge patterns for five crack extensions at 25 C. Crack tip blunting is noticeable, possibly due to the viscous flow of the glassy interfaces between alumina grains.The CCS versus COD relations for room, 600, 800 and 1000 C obtained by the inverse analysis are shown in Figure 8. The lack of a trailing tail in the CCS versus COD relation of the high density alumina is probably due to the lack of a broad grain size distribution containing large grains. Unlike Roedel [4] who used a small crack extension to derive the bridging stress with an assumed power law, the bridging stress in this study was

Inverse analysis for fracture process zone characterization

255

calibrated over a longer crack extension. Typical example of the resultant CODs obtained by finite element analysis with the CCS versus COD relations of Fig. 8 are shown in Fig. 7.

FE COD/2 without CCS FE COD/2 with CCS —

Exp. COD/2

o u

O

Fig. 7. COD profiles. Testat25C.

T = 25C T = 600C T = 800C T =1000 C

1.5

2.5

COD (microns)

Fig. 8. CCS vs. COD relations. Figure 9 shows the resistance curves in terms of the stress intensity factor K^. The stress intensity factor was computed by the standard strain energy release rate procedure. The leveling off of the KR curves is consistent with the full development of the FPZ with crack extension. The rate of energy dissipated in the FPZ was about 95 percent of the energy release rate.

256

D.K. Iran, M.T. Kokaly andA.S. 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

+

Kobayashi

at«2s*^5*«ft«^ *^^!^

••-•

-AL23-1998atT=25C -AL23-1998atT=600C -AL23-1998atT=800C -AL23-1998atT=1000C

H

3

1

1

4 5 A a (mm)

1

6

H

7

Fig. 9. Resistance, Kj^, curves. CONCLUSIONS A hybrid experimental-numerical procedure based on an inverse analysis was used to study the fracture process zone in two semi-brittle material, i.e. concrete and high density alumina. The crack closing stress versus crack opening displacement relations governing the fracture process zones trailing stably growing cracks in concrete and high-density alumina, specimens were quantified. The jfracture process zone was shown to be the major energy dissipation mechanism in semibrittle fracture specimens. ACKNOWLEDGEMENT This research was supported by the Air Force Office of Scientific Research grant nos. AFOSR91-0218 andF49620-93-l-0210. REFERENCES 1. Hillerborg, A., Modeer, M. and Petersson, P.E. (1976). Cement Concr, Res. 6, pp. 773-782. Post, D. (1993). In: Handbook on Experimental Mechanics, 2nd Revised Edition, A.S. Kobayashi (Ed.), VCH Publishers, pp. 297-364. 2. Tran, D.K., Kobayashi, A.S. and White, K.W. (1999). Experimental Mech. 39, pp. 20-24. 3. Kobayashi, A.S. (1979). Nonlinear and Dynamic Fracture Mechanics, N. Perrone and S.N. Atluri (Eds.), ASME AMD-36, pp. 19-36. 4. Roedel, J., Kelly, J.F. and Lawn, B.R. (1990). J. Am. Ceram. S. 73, pp. 3313-3318.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

257

IDENTIFICATION OF DELAMINATION IN BONDED DISSIMILAR MATERIALS WITH ORTHOTROPIC ELECTRIC CONDUCTIVITY BY THE ELECTRIC POTENTIAL CT METHOD Shiro KUBO, Takahide SAKAGAMI, and Noriaki TANAKA Department of Mechanical Engineering and Systems Graduate School of Engineering, Osaka University 2-1, Yamadaoka, Suit a, Osaka 565-0871 Japan Phone: +81-6-6879-7304, Facsimile: +81-6-6879-7305 e-mail: [email protected] ABSTRACT The identification of delamination defects on an interface in two-dimensional and threedimensional orthotropic bonded dissimilar materials was made, by applying the electric potential CT (Computed Tomography) method proposed by the present authors. The orthotropy of electric conductivity was taken into account in the boundary element calculations of electric potential distribution. For the three-dimensional materials, a hierarchical inversion scheme was applied, which consisted of two-dimensional scanning analysis and fully three-dimensional analysis. In the two-dimensional scanning analysis the least residual method was appHed to several scanning planes for obtaining a rough estimate of the delamination. For further detailed identification of delamination, the least residual method based on the three-dimensional boundary element electric potential analysis was applied. Numerical simulations of identification of delamination defects in a two-dimensional dissimilar body and a three-dimensional dissimilar body were made. The location and size of the delaminations were determined with good accuracy, demonstrating the applicability of the proposed method. KEYWORDS Non-destructive Testing, Electric Potential CT Method, Inverse Analysis, Defect Identification, Boundary Element Method, Orthotropy, Bonded Dissimilar Materials, Delamination Defect

258

S. Kubo, T. Sakagami and N. Tanaka

INTRODUCTION Bonded dissimilar materials and composite materials can be advantageously used to satisfy recent very severe and complex requirements, which are difficult to be satisfied simultaneously by using monolithic materials. On the other hand bonded dissimilar materials and composite materials have stress concentrations or stress singularities at points of geometrical irregularity, such as the intersection of the interface and free-edges and corner of constituent materials. The mechanical properties of the interface are not so good as those of the constituent materials. In many cases there are defects on the interface of bonded dissimilar materials and composite materials, which may develop in service condition and weaken the materials significantly. The quantitative nondestructive inspection of the interface defect or delamination is indispensable for the quantitative evaluation of the strength of bonded dissimilar materials and composite materials. Many nondestructive inspection techniques can be found. The electric potential method has been applied successfully for identifying cracks embedded in electric-conductive monolithic materials [1-7]. The problem of crack identification can be regarded as an inverse problem [8]. The present authors [9, 10] proposed the electric potential CT (computed tomography) method for the crack identification. In this method cracks are identified using inverse analyses from electric potential distributions observed on the surfaces of the body. They have shown the applicability of the method for the identification of cracks in homogeneous isotropic materials using numerical simulations and experiments [11-13]. They also applied the method to the identification of interface defects in bonded dissimilar materials [14-16]. The constituent materials were assumed to be homogeneous and isotropic. It was found that the quantitative identification of interface defects in bonded dissimilar materials can be made by the electric potential CT method. Bonded dissimilar materials and composite materials often show anisotropy in material properties. The anisotropy have to be taken into account for the quantitative identification of defects. There is a possibiUty that the anisotropy is advantageously used in the identification. In the present paper the electric potential CT method is applied to the identification of the delamination defects on the interface of two-dimensional and three-dimensional orthotropic bonded dissimilar materials. Numerical simulations are made to examine the applicability of the electric potential CT method. INVERSE METHOD FOR DEFECT IDENTIFICATION Least Residual Method To identify a delamination defect, measured electric potential distribution c/)^^^ on observation surface Fob is compared with calculated electric potential 0^^^. Residual Rg between (j)^^^ and 0^^) is defined by the following integral over Fob-

259

Identification of delamination R,=

f

(1)

[ct>^^)-cj>^^)fdT

Based on the least residual method, residual Rg is used as a criterion for the defect identification. The defect giving the smallest value of Rg is taken as the most plausible one. Boundary Element Analysis of Electric Potential Distribution The boundary element method [17] has been extensively applied to the analyses of electric potential distribution in homogeneous, monolithic and isotropic materials. Now consider a bonded dissimilar body, which is consisted of materials 1 and 2 and has a defect on the interface of two materials. The electric conductivity is assumed to have orthotropy, whose axes coincide with the direction of the Cartesian coordinates XiX2Xz [xyz). The conductivity of material m in xi direction is denoted by cJi- Then the governing equation of the distribution of electric potential (j)^'^^ in material m is written as, d {(jf^^d^'^'^'^/dx,) jdxi = 0

in material m.

(2)

The summation convention is applied in the expression. On the outer boundary of the materials, potential (j)^^"^ or flux q^^"^ is prescribed, where q^^^ is defined as,

with Ui denoting the outward unit normal vector on the boundary. In this study it was assumed that the bonding is complete and the continuity of potential (j) and flux q across the interface holds on the interface of the dissimilar materials:

^''-^'' ^int ~ V^int ? 4^ ^int ~= -4t yint '

(4)

where (')i^ denotes the value on the interface defined in material m. The boundary element equation states that the value of electric potential (jy^ at point A on the boundary of material m can be expressed by an integral over its boundary T^'^'> as [10], CAA™'

+ / < „ , (9*4™^ -
(m = 1,2),

(5)

where 0* and q* denote the fundamental solutions of the Laplace equation, and CA being a constant dependent on the shape of boundary around point A and a^ . The fundamental solutions 0* for the three-dimensional and two-dimensional bodies are expressed as, 0* - 1/ (47rroVcr^^V^^Vf ^)

for

three - dimension

(6)

^* = \n{l/rQ)l h-K^a^^^^a^^A

for

two - dimension

(7)

where TQ is defined in terms of the distance ri in the Xi direction between points A and B

260

S. Kubo, T. Sakagami and N. Tanaka

on boundary F^"^^ by The fundamental solution q* is related to (f)* as, (9)

q*=cj^;^\dct>*/dxi)ni.

The entire boundary of material m is discretized into boundary elements with A^^^ nodes. When point A is set to be at the z-th nodal point on boundary Tm of material m, Eq. (5) is discretized in the form,

c^^^ + E ^h^^r^ - E ^h^t^ = 0

(m = 1, 2),

(10)

where 0*^ and ql^ are determined by integrals expressed in terms of the fundamental solutions (j)* and q* over elements containing point j . The quadratic element was used in this study. Equation (10) gives a matrix equation for (/> and q at nodal points. The boundary values at nodes, and therefore the distributions of potential (j) are obtained by solving the matrix equation for materials 1 and 2, together with prescribed boundary conditions on the outer boundary concerning 0 or q, and the compatibility of 0 and q written by Eq. (4). NUMERICAL SIMULATIONS Identification

of Two-Dimensional

Delamination

Defect

Numerical simulations of identification were made for two-dimensional delamination defects in bonded dissimilar materials shown in Fig. 1. The axes of orthotropy were taken to coincide with X2X3 (yz). The values of a^""^ and cr^"^^ shown in Table 1 were used. The interface was assumed to be parallel to the y direction. The potential distribution on GH, which was obtained when potential 0 was prescribed to be 1 and 0 at points G and H, was used for the identification of the defect. Location of defect center yc and half defect length a were estimated from the potential distribution.

The boundary element analyses were made to generate measured data of electric potential 0^^^ on the observation surface Fob- The symbol " • " in Fig. 1 shows the location of observation points. To investigate the eflfect of measurement noise in 0^^\ artificial random noise of the level of 0.1%, 0.5% and 1.0% was introduced in the calculated potential values.

An example of the results of identification for the noise level of 0.1%, 0.5% and 1.0% is shown in Table 2. It is seen that a reasonable estimation can be made even for the noise level of 0.5% although the estimation is deteriorated with increase in the noise level.

261

Identification of delamination

H 22. 5

f

22. 5

Ma t e r i a 1

1

O

F ^ II

yo

Ma t e r 1 a 1

3:

2

V ^

y

W= 1 0 0

^

C

Fig. 1 A delamination defect in two-dimensional bonded dissimilar materials used in numerical simulations.

Table 1 Orthotropic electric conductivity used in numerical simulations.

Material 1 776 Material 2| 0.53

0.53 776

3.74 3.74

Table 2 Parameters of a two-dimensional delamination defect estimated from noisy data. Noise[%]

Estimated defect parameters Center yc Ayc/W[%] Half length a Aa/W[%]

0.00

50.000

0.000

0.10

49.391

0.50

49.183

1.00

[Actual value

1

Rs

1

O.OOE+00

7.500

0.000

0.609

7.736

0.236

1.20E-05

0.817

8.553

1.053

1.14E-04

48.988

1.012

9.193

1.693

50.000

0.000

7.500

0.000

4.55E-04 1 0.000 1

262

S. Kubo, T. Sakagami and N. Tanaka

Xc

\

= 35

''illllll

.

^

X c=15

(a) Defect shape I

•11^1-4 J (b) Defect shape II

Fig. 2 Delamination defects in three-dimensional bonded dissimilar materials used in numerical simulations. Identification

of Three-Dimensional

Delamination

Defect

Numerical simulations of identification were made for elliptical delamination defects in three-dimensional bonded dissimilar materials shown in Fig. 2. The interface was assumed to be perpendicular to one of the axes of orthotropy. The value of af^' shown in Table 1, which corresponded to that of layered CFRP [18], was used. The electric potential distributions on side surfaces for 75 < 2: < 105 were used, which were obtained when 0 == 1 and (/) = 0 were prescribed on lines y = 16.5 and y = 3.5 on the top surface ABCD. The defects were assumed to be eUiptical, and their locations and sizes were estimated.

The boundary element analyses were made to generate measured data of electric potential (/)^^^ on the observation surface. The symbol " •" in Fig. 2 shows the location of observation points. To investigate the effect of measurement noise in (t)^^\ artificial random noise of the level of 0.1%, 0.5% and 1.0% was introduced in the calculated potential values.

To speed up the identification the hierarchical inversion scheme [19] was employed, in which a two-dimensional scanning analysis was followed by a three-dimensional analysis. In the two-dimensional scanning analysis the least residual method was applied to a cross section shown by dotted area in Fig. 3. The residual Rs between (/)(^^ and ^^^^ was evaluated along lines MN and PO on the free surfaces. The potential distribution (jP^'> on side surfaces, which was obtained when potential (j) is prescribed to be 1 and 0 at points Ti and T2 on the top surface, was used for identification of the defect. The calculated electric potential (t)^^\ was obtained by applying the two-dimensional boundary element analysis. The defect

263

Identification of delamination

A

P

^/W A \n

= 1

D

C

^ = 3-5

<|) = 0

T2

Material 1

l[

P i

A i 1

El. y -'

K

\

y0

v

Mat er i a l 2

Fig. 3 Two-dimensional scanning plane for a rough identification of delamination defect in a three-dimensional dissimilar body. location and size giving the smallest value of Rs were taken as the most plausible ones for each scanning plane. By combining the results of identification obtained for several scanning planes, a rough estimate of the defect shape and location can be obtained. Then the three-dimensional inverse analysis was made based on the least residual method for a detailed identification. It was assumed that the defect was elliptical. The defect parameters giving the smallest value of R^ were sought by applying the modified Powell optimization method [20]. The defect identified by the two-dimensional scanning analysis was used as the initial guess in the three-dimensional analysis.

In Fig. 4 solid fines show the defects estimated by the two-dimensional scanning analysis for seven scanning planes when the noise level is 0.1%, thin dashed lines being the actual defect shapes. It is seen from the figure that a rough estimate can be obtained for the large defect, while the estimated shape for a defect long in y direction is elongated in the X direction.

Figure 5 shows the defect shapes estimated by the three-dimensional analysis for the noise level of 0.1% and 1.0%. The large defect can be identified with good accuracy even in the presence of 1% noise, while the defect long in y direction is estimated reasonably only when the noise level in (\P^^ was 0.1%. These results show that good estimation of delamination defect can be made when the noise level is of the order of 0.1 %, which can be achieved without difficulties by using commercially available equipments.

264

S. Kubo, T. Sakagami and N. Tanaka

•

Estimated - Actual

With error 0.1% 10

20

30

40

50

60

(b) Defect II, noise level 0.1%

(a) Defect I, noise level 0.1% Fig. 4

Three-dimensional delamination defects estimated by two-dimensional scanning

analysis from noisy data.

20

- Estimated -Actual

- Estimated 16

'Actual

12 1-

With error 0.1 X 10

20

Krth error 0.1% 30

40

50

60

10

20

30

40

50

60

(c) Defect II, noise level 0.1%

(a) Defect I, noise level 0.1%

- Estimated -Actual

Krth error 1.0%

With error 1.0% 0

10

20

30

40

50

60

10

20

30

40

50

60

X

(b) Defect I, noise level 1.0%

(d) Defect II, noise level 1.0%

Fig. 5 Three-dimensional delamination defects estimated by three-dimensional analysis from noisy data.

Identification of delamination

265

CONCLUSIONS The electric potential CT method was applied to the identification of delamination defects in two-dimensional and three-dimensional orthotropic bonded dissimilar materials. As an inverse analysis method for identifying the defect, the least residual method was used. For the identification of three-dimensional defect the hierarchical inversion scheme incorporating the two-dimensional scanning analysis and the three-dimensional analysis was employed to achieve an efficient identification. The orthotropy of the electric conductivity was taken into account in the boundary element analysis. Numerical simulations have shown that the delamination defects can be identified reasonably by the electric potential CT method. ACKNOWLEDGMENTS This study was partly supported by Grant-in Aid for Scientific Research, Ministry of Education, Science, Sports and Culture.

REFERENCES 1. Aronson, G.H. and Ritchie, R.O. (1979), J. Testing & Evaluation, 7, pp.208-215. 2. Gangloff, R.R (1981), Fatigue of Eng. Materials k Structures, 4, pp.15-33. 3. Hicks, M.A. and Pickard, A.C. (1982), Int. J. Fracture, 20, pp.91-101. 4. Wilkowski, G.M. and Maxey,W.A. (1983), Fracture Mechanics 14th Symposium, ASTM STP 791, Amer. Soc. Testing Mater., pp.II-266-294. 5. Murai, T. and Kagawa, Y. (1986), Int. J. Numerical Methods in Eng., 23, pp.35-47. 6. Coffin, L.F. (1988), Fracture Mechanics 19th Symposium, ASTM STP 969, Amer. Soc. Testing Mater., pp.235-259. 7. Abe, H., Saka, M., Wachi, T. and Kanoh, Y. (1988), Computational Mechanics '88, Vol. 1 (ed. by Atluri, S.N. and Yagawa, G.), Springer, Berlin, pp.12.ii.1-4. 8. Kubo, S. (1988), JSME Int. J., Ser.I, 31, pp.157-166. 9. Ohji, K., Kubo, S. and Sakagami, T. (1985), Trans. Jpn. Soc. Mech. Eng., Ser.A, 51, pp.1818-1825. 10. Kubo, S., Sakagami, T. and Ohji, K. (1986), Computational Mechanics '86, (ed. by Yagawa, G. and Atluri, S.N.), Springer-Verlag, Berlin, pp.V-339-344. 11. Kubo, S., Sakagami, T. and Ohji, K. (1988), Computational Mechanics '88, Vol. 1 (ed. by Atluri, S.N. and Yagawa, G.), Springer-Verlag, Berlin, pp.12.i.1-5. 12. Sakagami, T., Kubo, S., Hashimoto, T., Yamawaki, H. and Ohji, K. (1988), JSME Int. J., Ser.I, 31, pp.76-86. 13. Kubo, S., Sakagami, T. and Ohji, K. (1991), Fracture Mechanics (Current Japanese Materials Research, Vol. 8) (ed. by Okamura, H. and Ogura, K.), Elsevier, London, pp.235-254. 14. Kubo, S., Sarasinpitak, J. and Ohji, K. (1996), Mater. Sci. Research Int., 2, pp.26-32. 15. Kubo, S., Sakagami, T., Kitaoka, H. and Ohji, K. (1998), Proc. Int. Symp. on Inverse Problems in Engineering Mechanics ISIP 98 (ed. Tanaka, M. and Dulikravich, G.S.), Elsevier, pp.91-100.

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16. Kubo, S., Sakagami, T., Sarasinpitak, J., Kitaoka, H. and Ohji, K. (1998), Experimental Mechanics, Advances in Design, Testing and Analysis (ed. Allison, LM.), Balkema, pp.933-938. 17. Brebbia, C.A. and Walker, S. (1980), Boundary Element Techniques in Engineering, Butterworth, London. 18. Todoroki, A., Suzuki, H., Kobayashi, H., Nakamura, H. and Shimamura, Y. (1998), Trans. Jpn. Soc. Mech. Eng., Ser.A, 64, pp.1654-1659. 19. Sakagami, T., Kubo, S. and Ohji, K. (1990), Int. J. Pressure Vessels k Piping, 44, pp.35-47. 20. Powell, M. J. D. (1964), Computer J., 7, 155.

INVERSE PROBLEMS ES[ ENGE^EERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

267

DAMAGE DETECTION OF STRUCTURE USING IMAGE PROCESSING 1ST REPORT, DETECTION OF JOINT FAILURE T.KAWAI Department of Mechanical Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan N. IKEDA Department of Mechanical Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan M. ITO Production Engineering Laboratory, Matsushita Electric Co. Ltd., 2-7 Matsuba-cho, Kadoma, Osaka 571-8502, Japan ABSTRACT The parameter identification was very important in the research areas including the parameterisation of joints and boundary conditions, model structure determination and modelbased damage detection. We previously proposed new techniques to identify parameters of a finite element model, i.e. the density and the modulus of elasticity of a cantilever beam, by using an image processing technique. Using our technique, a small crack in the uniform beam could be detected. Also our technique indicated that the deflection of the beam was well measured by processing the captured image data and that the parameters of the beam were identified by the measured deflection with high accuracy. In this paper we tried to detect the joint failure in the structure by processing the captured joint image data. The joint failure was precisely identified as the reduction of the rotational stiffness of the joint. KEYWORDS Parameter identification, Structure, Rotational stiffness. Joint failure. Crack, Inverse problem L INTRODUCTION In identifying the parameters of mechanical or structural systems, it is highly important that we are able to identify modal or finite element parameters, and to detect cracks or failures. Because

268

T. Kawai, N. Ikeda and M. Ito

noise in the experimental data degrades the estimation of these factors, we must precisely measure the deflection or position of each element of the system and regulate the data processing. Many researchers have devoted themselves to the detection of cracks or other damage in structural systems. Since any crack in a system reduces the total stiffness and results in a corresponding change in natural frequencies and modal shape, the main topic of this study is to propose a method for measuring natural frequencies and their modal shapes, and for identifying the size and location of cracks[l][2][3]. J. E. Mottershead and M. I. Friswell [4] surveyed a wide range of reports and emphasized the importance of model updating in structural dynamics. The areas of recent advancement in model updating research include: (a) parameterisation of joints and boundary conditions using generic elements[5], (b) adaptive excitation and selective sensitivity[6], (c) model structure determination[7], and (d) model-based damage detection[8]. Image processing has a number of attractive features; e.g., it allows easy measurement of the deflection of each point of a whole system. On the other hand, this method cannot offer high accuracy in both time and space. This is the cheap reason that this method is not applied for the parameter identification of mechanical systems. In our previous reports [9,10], we captured the image of the cantilever beam and attempted to identify the density and the modulus of elasticity of a uniform beam and of a combined beam, and found that (l)Using Spline smoothing with symmetrization, the precise shape of the beam was obtainable; (2)Using Spline smoothing with symmetrization and the weighted Tikhonov regularization technique, the precise parameters of the beam could be estimated. Further, a new method utilizing SA (Simulated Annealing) could detect a small crack in a uniform beam quantitatively. In this paper, we intended to develop a new method for detecting a join failure in the structure by using image processing technique. The failure reduces the stiffness at the joint and affects the vibration mode of the structure. First, the deflection and the rotation of the joint were measured by processing the captured joint image. Then, the stiffness of the joint was estimated by calculating the stiffness of the finite element model of the structure. 2. PROBLEM DESCRIPTION The model of the structure is illustrated in Fig. l(a)(b)(c). The failure is supposed to exist at the joint and to reduce the stiffness. As shown in Fig.2, the joint failure reducing the bending stiffness was assumed to be the combination of the health beam and the rotational spring at the joint. We defined this reduction as the semi-rotational stiffness coefficient k. Fig.3(a) illustrates the damaged joint and Fig.3(b) shows the deflection dy, dz and the rotation 6 caused by the forces/? *,p* and the moment m* at the joint. In general. Equation 1 relates the force vector p to the deflection vector d by theflexibilitymatrix F. d = Fp d^=«<^*),p^ = ( ; ; » * )

(^)

The damaged segment A-B has negligible length and has only rotational stiffness 4EIk/L. The semi-rotational stiffness coefficient k represents the degree of damage, i.e., the infinity value for

269

Damage detection of structure using image processing

the perfect joint and a low value for a damaged joint. The non-damaged segment B-C has the constant parameters, i.e., the flexural rigidity EI, the sectional area A and the length L. The flexibility matrix F^^ for the segment A-B is derived as Equation 2 and the flexibility matrix F^^ for the segment B-C is derived as Equation 3. 0 0 0 0 0 0

0 0 L AEIk

L EA

0

0

L' 3EI L' 2EI

L' 2EI L EI

0 0

(2)

(3)

^

Failure

r ^

^

+

Rotational Spring Ni+l ... -

Ni+2

Fig. 2 Joint failure Al B

C

A. (a)

Fig. 1 Model of structure (a) arrangement (b) node definition (c) deflection

Fig. 3 Definition of failure (a) model of failure (b) deflection

270

T. Kawai, N. Ikeda and M. Ito

The combined flexibility matrix F

for the segment A-C is calculated by Equation 4.

f^AC =H^F^^BH + FB^

(4)

where "l 0 0" H= 0 1 0 0 L 1 As the result, the stiffness matrix K^^ for the segment A-C is derived as Equation 5.

K,

E4 L :F~' = 0 0

0

0 \2EI Ak-\-\ L' 4{k + l) 6EI 2k+1 1} 2{k + \)

6EI 2/: + l L- 2{k + l) 4EI 4k+ 3 L 4(^ + 1)

(5)

Finally we can get the constitutive equation 9 for the structure illustrated in Fig.l by using Equations 6,7,8. P. = - H p ,

(6) (7)

p, = Ke

(8)

HKH"^

-HK d,

(9)

Here, the matrix K in the equation 8 is the matrix K^^ expressed by Equation 5. The suffix 1 or 2 in Equation 9 must be assigned to a pair of nodes, ^. ^, A^^^ and so on in Fig 1(b). When the suffix 1 or 2 is assigned to the joint, A^^ in Fig. 1(b), k has the finite value. Otherwise, k is the infinity. To estimate the coefficient k, the structure is excited at the point P by the harmonic oscillator with a frequency of (o. According to the FEM method, the whole structure is divided into 2N-\-M segments. The equation of motion 10 of this structure is derived using the characteristic matrix A, of which the components are the mass matrix M and the stiffness matrix K, the deflection vector u( which involves the deflection y^, z^ and the angle of deflection 9^^ of the nih element), and the dynamic external force vector F, for which Au = F where u = {J, z^ ^,

(10) y^z^d^

... >'2yv.M.i ^zyv.M.i 6>2^.M.I}. A = K-ft>'M

Damage detection of structure using image processing

111

To detect a joint failure, five points marked on the Fig. 1(b) were picked up and relating equations in matrix Equation 10 were extracted. For the equations having the three node points^.^,A^. and N.^^ in itself reduced the stability of the derived equation, these equations were not used in estimating the coefficient k. Finally the derived matrix Equation 11 relates the rotational stiffness coefficient k of the joint to the deflection and the angle of deflection of five selected points. a^=b (11) where a = (K^, - 4ft;'M J x - 4F,, b = [^co^M^ - K,)x + 4F^, In the next step, we intend to measure the deflection and the angle of deflection of selected points and solve the above Equation 11. Generally speaking, the deflection is easily measured by ••{y>48E/ l^ 2AEI

/-

0 0 0

[-

8EI I

96EI l^

0

0 48£/ l^

0

0

0

0

0

2AEI l0

0 0 0 AEI I 0 0

0

0

48E/

24EI

60EI

/' 2AEI

/' 8EI /

/' UEI

I'

0 0

I' UEI

0

0

0

0

0

0

0

0 0

AEI I 0

0

0

0

0

0

M.

2AEI

0

0

K.-

-2 ^ , - 2 y>-X 2,-, 6,-, y, Z, &! y , , , Z;„ BuX y^l

1 420

0

/'

0

48E/ l^ 24 El

0 32EI I 24EI

4S EI

1'

/?

—

0 24 El

0

0

f-

1-

4EI 1

r-

4EI

/'

0 0

0 0

0

0

/

0

0

0

0 4EI I

0

0

0

0

0 48£/

0 24 EI

0

0

0

0

0

/•^

24EI

r-

96EI I'

0

0

r-

8EI I

0

32EI I

48EI I' 24 El l-

0

0 24EI l8EI I

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 4EI

—-— + —-

0.I UEI

UEI

0

0

0

I

UEI

e

0

0

1-

24EI

UEI

24EI 32EI I 24 El I' SEI

f-

UEI — /' UEI

0

u

48EI

UEI

0

1

24EI

24EI

0

/'

1-

1^

0

I' 28EI I UEI

48£/

)

SEI

0

4 EI

^M

24 El

0

1-

0 8EI I

^Ul

~r

4EI 0 I ^ 0 u 0 0 0 0

0

0

0

0

UEI

UEI

0/' 60EI

/'0 UEI

0

0

0 48EI

0 24EI

/^

0 0

/2

/2

UEI I'

28EI I

24EI l\

8EI I

'54/

-13/'

312/

0

0

54/

0

13/'

0

0

0

0

0

13/'

-3/^

0

0

8/^

-13/'

0

-31'

0

0

0

0

0

0

0

54/

0

-13/'

296/

0

22/'

70/

0

0

0

0

0

0

0

70/

0

0

296/

22/'

0

54/

-13/'

0

0

0

0

13/'

0

~3l'

22/'

22/'

8/'

0

13/'

-31'

0

0

0

0

0

0

0

0

54/

13/'

0

312/

0

54/

-13/'

0

0

0

0

0

0

-13/'

-31'

0

0

8/^

13/'

-31'

using the laser displacement sensors or the eddy current type displacement sensors and so on. On the contrary, it is very difficult to estimate the angle of deflection because of the requirement of calculating the derivative of the deflection. If the discrete type of displacement sensors, i.e., the

272

T. Kawai, N. Ikeda andM. Ito

laser displacement sensors and like that, is used, the curve fitted to the discretely measured points must be calculated and then estimated the slopes at these points. Above approach is applicable very well in the case of noise free measurement, however, both of the noise in an actual measurement of the deflection and the restriction that the fitting curve is controlled only at the measured discrete points cause the systematic error in the angle of deflection. To avoid this problem, the image processing technique was applied in this paper. 3. IMAGE PROCESSING The image captured by the CCD camera is expressed by the pixels, for example, image of 512 x 512 pixels and the size of pixel 0.2 x 0.2 mm. The image was first processed by basic image enhancements, i.e., the noise reduction, the edge enhancement and so on, and then was estimated to get the central line of each beam in the structure by using the Spline smoothing. After this image processing, the deflection and the rotation of the joint was calculated. A lot of pixels and the Spline smoothing reduced the noise and gave the precise deflection data. 4. DAMAGE DETECTION As already stated, using Eq. 11, the stiffness of the joint could be estimated from the exciting force, the deflection and the angle of deflection at selected points. The deflection data is estimated by the above image processing technique and the exciting force is given in the experiments. In the simulation, the structure specified by Table 1 was excited under the condition of Table2. Fig. 4 illustrates the relation between k and the failure of the joint under the above simulation condition. In this case, the rectangular crack with fixed length of 4 mm is assumed to exist in the structure of height 500 mm and width 400 mm (see Fig.3). Following computer simulation was carried out to evaluate our proposed technique. l.Give the exciting force and the geometrical parameters, i.e., height, width and like that, of the structure

4.0[mm]

4.0[mm]

3.5[mm]

^

Thickness of joint [mm]

\^-^

3.5[mm]

log,.k

Fig. 4 Joint failure

Fig. 5 Relation between k and actual damage

Damage detection of structure using image processing

273

Table 1 Specification of structure Length

Beam 1,3 Beam2

Width of beam Thickness of beam

L (mm)

500 400

b (mm)

30

p (kgluv')

Modulus of elasticity

E (Pa)

Number of division Size of pixel

3.5

h (mm)

Density of beam

Beaml,3 Beam2 Px^Py

7700 2.06X10" 10 10 •

M N

0.2X0.2

(^"^ ^ mm)

Table 2 Exciting force No.

Exciting frequency a>(Hz)

Exciting force F(N)

Exciting point

Expl

8

8

10

Exp2

70

26

20

Exp3

90

27

30

Table 3 Simulation result Given value k, (Damage ratio) Identified value

2.Assume the index k

0.1 (68%)

1.0 (36%)

3.0 (20%)

5.0 (14%)

10.0 (7%)

100.0 (1%)

0.091

1.03

2.85

4.43

8.41

NA

8

3.Calculate the deflection of the structure by the FEM method 4.Capture and process the structure image to estimate the deflection and the angle of deflection 5.Solve the Eq.ll by the least square method to estimate the index k^ 6.Compare the given value k and the estimated value k^ After several simulations, our proposed technique showed the good detection ability of failure. Table 3 demonstrates the identified value for several damage level. For example, the value of it^ =3 ± 0.5 was estimated for k =3. 5. CONCLUSIONS In this paper, we tried to detect the joint failure in the structure by using image processing technique and get the following results. (l)The deflection and the rotation of the joint were well estimated by using image processing

274

r. Kawai, N. Ikeda and M. Ito

technique. (2)The joint failure modeled as the reduction of the rotational stiffness was well detected by the proposed technique. REFERENCES 1. T. Y. Kam and T. Y. Lee (1994) J. Sound & Vibration, 31(7), pp.549-558. 2. F. Vestroni, M. N. Cerri and E. Antonacci (1996) Pro. 3rd EURODYN, pp.41-50. 3. Y. Narkis and E. Elmalah (1996) Int. J. Mech. Sci., 38(5), pp.499-507. 4.J. E. Mottershead and M. I. Friswell (1993) J. Sound & Vibration, 167(2), pp.347-375. 5.G. M. L. Grandwell and H. Ahmadian(1995) Mechanical Systems & Signal Processing, 9, pp.601614. 6.Y. Ben-Haim(1992) Mechanical Systems & Signal Processing, 6 pp.85-96. 7.H. G. Natke(1995) Inverse Problems in Engineering, 1 pp.267-272. 8. H. G. Natke and C. Cempel(1997) Model-Aided Diagnosis of Mechanical Systems, Berlin, Springer-Verlag. 9. Kawai, Ota, Ito and Ito(1997) Proc. 17* Biennial Conf. on Mechanical Vib. and Noise, VIB8371, pp.1-7. 10. Kawai, Ota, Ito and Ohyama,(1998) Inverse Problems in Engineering Mechanics, Elsevier, pp.235-242.

Shape Determination

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

277

SOLUTION TO B O U N D A R Y S H A P E IDENTIFICATION PROBLEMS IN ELLIPTIC B O U N D A R Y VALUE P R O B L E M S U S I N G S H A P E DERIVATIVES Hideyuki AzEGAMi Department of Mechanical Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku-cho, Toyohashi, 441-8580, Japan e-mail: azegamiOmech.tut.ac.jp ABSTRACT Identification problems of geometrical boundary shapes of domains in which elliptic boundary value problems are defined are treated. These problems can be formulated as minimization problems of squared error integrals between the actual solutions of the eUiptic boundary value problems and its reference data with respect to perturbation of the uncertain boundary. The fundamental theory concerning with the shape derivatives of functionals with respect to domain perturbation and the gradient method in Hilbert space have been presented by mathematicians. Based on the theories, this paper presents a concrete solution to the geometrical domain identification problems. It briefly describes the derivation of the shape gradient functions for the shape identification problems of two types referring to boundary value on subboundary and referring to gradient in subdomain and introduces the definition of the gradient method in Hilbert space. Using these shape gradient functions and the concept of the gradient method in Hilbert space, a concrete solution to the geometrical boundary identification problems is presented. This solution coincides with the traction method that had been presented by the research group of the author.

KEYWORDS Shape identification, elliptic boundary value problem, domain perturbation, material derivative, shape gradient function, gradient method in Hilbert space, traction method. INTRODUCTION Inverse problems with respect to geometrical boundary shapes of linear elastic bodies, heat transfer field, viscous flow fields, ideal fluid flow fields, sound pressure fields, etc. in steady state can be generalized as geometrical shape identification problems of domain in which elliptic boundary value problems are defined. These problems can be formulated as minimization problems of squared error integrals between the actual solutions of state functions and its reference data with respect to perturbation of the uncertain boundary. This domain

278

H. Azegami

perturbation with moving boundary can be formulated by selecting a one parameter family of mapping from current domain to perturbed domain as design variables [1]. The fundamental relations for the shape derivatives of functionals of state functions with respect to the domain perturbation was derived heretofore [2, 3, 4]. Moreover, the basic concept of the gradient method in Hilbert space has been presented [5, 3]. However, concrete solution to the geometrical domain identification problems using the shape derivatives has not been investigated intensively. This paper is devoted to presenting a concrete solution to the geometrical domain identification problems that had been presented by the research group of the author [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] but keeping the mathematical background. In this paper, n dimentional domain in /^", n = 2,3, is considered, where R denotes the real number. For vectors in i?", the arrow notation (•) and the suflftx notation (• )j are used for convenience in writing. In the arrow notation, V is used for spatial gradient, the dot notation is used for scalar product, (•) indicates n x n matrix, and (•) denotes the transpose. In the suflftx notation, (•),i = d{')/dxi and the summation convention are used. (•) \ (•) denotes the subtraction between sets. (•) denotes the closed set of (•). The Sobolev space that consists of those n dimensional vector functions of p-th power integrable in the sense of Lebesque measure until m-th derivatives defined on (•) is denoted by (py"*'P( • )Y for all integers m > 0 and any number p satisfying 1 < p < oo. / / ( . ) = l^O'P(.) and i/"^( •) = 1^"*'2(.). (C"^(.))" denotes continuous function space of n dimensional vector functions until m-th derivatives defined on (•). The word of vector is used in the sense of elements in BJ^. The word of function is used for mappings from BJ^ to R or R^, especially vector function is used for mappings from R^ to R^, that are elements or {W^^^(R^))^. For mappings from function spaces in function spaces such as W^'P{R^) to R the word of functional is used.

DOMAIN PERTURBATIONS Let i?Ci^'*, n = 2,3, b e a given bounded domain and df2 = F he its boundary. One way to generate the small perturbations of i? is to use a one parameter family of one-to-one mappings, i.e. vector functions, fs.R^^ i?", 0 < 5 < e, where 6 is a small positive number and (1)

fs : Q 3 X \-^ X e Os

f-^ :ns^x^x

(2)

ef2

To keep the one-to-one property, it is assumed that the following conditions are satisfied [4]. (i) fs and f-^

belong to (W^i'°°(i?'*))" or {C^R''))''

(ii) The mappings s ^ f , ( f ) and s H-> f-^{x)

for all s € [0,e).

belong to {C^{[0,e)))'' for all x € i^".

The derivative of Tg with respect to s defined by V{x)^^{T-\x)) is called velocity.

Sen,

(3)

279

Solution to boundary shape identification problems

Considering a domain functional JQ{Q) of a distributed function (t)Q € W^'^{Q) in a domain Q with the C^ class boundary F and a boundary functional Jr{^) of a distributed function (j)r G VF^'^(i7) on the C^ class boundary T, their Gateaux derivatives, that are called shape derivatives, Jn{Q) and Jr{^) with respect to the velocity V{f2s) is given by the formulae of the material derivative: for Ja{f2) = / (f)ndx, Jn = lim -{Jn{Qs) ~ Jo{0)) s^o s for Jr{f2)= I (j)rdr, Jn{n)

(4)

= [ (l)'^dx-h f (t>nn -VdV Jn Jr

(5) = lim -{M^s) - Jr{f2)) = f U'r + {Vnr + c/^r^) n-V] dP s-vo s Jr ^ ^ where n denotes the outer normal vector well defined at almost all points of T, Vn( •) = V( •) • n, K denotes the mean curvature, and (j)'^ and (l)'p indicate the derivatives under a spatially fixed condition during domain perturbations of the distributed functions (j)n and 0 r respectively [4]. Mf2)

SHAPE IDENTIFICATION PROBLEMS Using the definitions and formulae with respect to domain perturbations, shape derivatives for boundary shape identification problems can be derived. For the simplicity, let us consider a strong elliptic boundary value problem of the second order related to a realvalued scalar state function. This problem is described in the strong form: - V • A{x)Vu{x) + ao{x)u{x) = / ( f ) u{x) = uo{x) X e To A{x)Vu{x)

• n(f) = g{x)

(6) (7)

x e ^

f G T \ fo

(8)

uoeUo

(9)

and in variational form: a{u, v) = l{v)

u-uoeU

yv eU

where the bilinear form a( •, •) and the linear form /(•) are defined by a{u, v) = l(v)=

{Vu • AVv + auv) dx Jn f fvdx-\- [ gvdr Jn Jr\ro

(10) (11)

and UQ{X), A{X) = A{x)'^ and ao(f), x G R^, denote given functions. ellipticity, the following conditions are required. 3a > 0 :

ao{x) >a

and

z- A{x)z > a\z\'^

For the strong

Vf G R""

\/z e R"

(12)

To assure existence of a unique solution for state function u, it must be at least that A G (L°°(i?^))'*^'', ao G L°°(i?^), / and ^ G i/^(i?^) and the admissible sets UQ and U are given by Uo = {ue

H\Q)\

U = {ueH\f2)\u(x)

u{x) = 0, f G r \ A } = 0, xeTo,

f udx = Owhen

Jn

(13) f JTO

dP = 0}

(14)

280

H. Azegami

Referring to boundary value on subboundary A shape determination problem in which the state function u is specified with a given function w on s. subboundary / D C T \ fo can be formulated as a minimization problem of a squared error integral: Err,{u-w,u-w)=

(15)

f {u-wfdr

and described by min

Epn (u — w,u — w)

such that

a(u,v) = l{v)

u — UQ e U

(16)

UQ e UQ ^V e U

For the sake of simplicity, let the coefficient functions of up, A, ap, f and g be fixed in R^ during domain perturbations and the velocity 1/ = 0 at To D T \ To and the singular points on r . The shape derivative of the objective functional can be obtained as follows. Applying the Lagrange multiplier method, or the adjoint variable method, the Lagrange functional L(i7, u, v) of this problem is defined by L = Epj^ {u — w,u — w) — a{u, v) + l{v)

(17)

where v e U was introduced as the Lagrange multiplier function, or the adjoint function, with respect to the weak form in Eq. (16). Using the formulae of the material derivative shown in Eqs. (4) and (5), the shape derivative of the Lagrange functional is obtained by L = 2Er^ {u - w, u') - a{u', v) - a{u, v') + l{v') + {Grr,n, V)

(18)

where the linear form {Gr^'^, V) with respect to the velocity V is defined by {Gr^fi,V)=

f

G^ft-Vdr^

j Gun'Vdr+

Gyj = 2{u - w)S/n(u - w) -\- {u- w)'^K,

I _ GgU-VdE

+ f GfU-VdP

{19)

Gu = - V ( w - Up) • AVv - ap{u - up)v,

Gg = Vngv + ^Vn^ + gvi^, Gf = fv

(20)

Considering the invariability of L with respect to any perturbation of v' E U and u' e U at the optimal condition, the governing equations with respect to u and v are given by a{u,v') = l{v') WeU a{u\v) = 2Eroiu-w,u') WeU

(21) (22)

Eq. (21) gives the same solution as the week form of Eq. (9). Eq. (22) is called the adjoint equation of the present problem. Using the solutions u and v of Eqs. (21) and (22), the derivative of the Lagrange functional agrees with that of the objective functional and the linear form {GTC^, V) with respect to the velocity V: L\u,v^ErM,^={Gr^n,V)

(23)

From the fact that the vector function Gr^n is a coefficient function with respect to velocity V that is the derivative of the design function Tg, GrD^i is called the shape gradient function of the present problem. The scalar function Gpo is called the shape gradient density function.

281

Solution to boundary shape identification problems Referring to gradient in suhdomain

Another shape identification problem is considered in which the gradient of the state function V ^ is specified with a given vector function w in subdomain QD C O. Using the objective functional defined by EQ^ {VU -W,VU-W)=

[

{Vu - w) ' {Vu - w) dx

(24)

JQD

this problem is formulated by min

EQJ^ {VU

such that

— w, Vu — w)

a{u, v) = l{v)

u - UQ e U

UQ e UQ \/V e U

(25)

Let us assume the same conditions for UQ, A, ao, f, g and V. The Lagrange functional of this problem is defined by L = EQJ, {VU -W,VU-W)-

a{u, v) + l{v)

(26)

Using the formulae of the material derivative shown in Eqs. (4) and (5), the shape derivative of the Lagrange functional is obtained by L = 2EQ^ {VU - w, Vu') - a{u', v) - a(u, v') + l{v') + {Gn^n, V)

(27)

where the linear form {GQJ^U^ V) with respect to the velocity V is defined by {Gn^n,V)=

[

G^n'Vdr+

G-uj = {Vu - w)' (Vu ~ w), Gg = Vngv + gVnV H- gvi^,

[ Gun'Vdr+

f _ Ggn'Vdr+

f Gffi'Vdr

(28)

Gu = -V{u - UQ) ' AVv - aQ{u - uo)v, (29)

Gf = fv

where dQo denotes the boundary of HoConsidering the invariability of L with respect to any perturbation of v' e U and u' e U at the optimal condition, the governing equations with respect to u and v are given by (30)

a{u,v')=l{v') WeU a{u', v) = 2En^ {Vu - w, Vu')

W €U

(31)

Eq. (31) is the adjoint equation of the present problem. Using the solutions u and v of Eqs. (30) and (31), the derivative of the Lagrange functional agrees with that of the objective functional and the linear form (G^/^^n, V) with respect to the velocity V: L\u,v = Ea,\u,v = {Gann.V) GQ^U is called the shape gradient function of the present problem.

(32)

282

H. Azegami

GRADIENT METHOD IN HILBERT SPACE With shape derivatives the gradient method in Hilbert space has possibility to be applied to reshaping algorithm. Let ^ be a real Hilbert space with scalar product (•, • )$ and norm II • ||<^. Let J : ^ 1-^ i^ be a real valued functional. We wish to find a unique solution (j)* e ^ such that (33)

J{(j)*) = mm J {(j))

If J is Frechet differentiable in ^ , the gradient of J , denoted by G j , is defined by the unique element of ^ which satisfies [3] (Gj,^)^ = l i m i ( J ( 0 - f C ( p ) - J ( 0 ) )

V^G<^

(34)

The gradient method in ^ advocates to replace the increment of (^, denoted by Ac/), by the unique element of ^ which satisfies 6(Z\0,^) = - ( G j , ^ ) ^

(35)

y^pe^

where 6( •, •) is a uniformly bounded and coercive bilinear form in ^ that satisfies [5] 3a, /? > 0 : b{(t>, (j>) > aUWl

\/(l>e^

and

6(0, ip) < /?||0|U||^|U

V0, ^ G <^^ (36)

Then, it is guaranteed that the ^Acj) decreases the functional: J((/> + C^0) = JW + {Gj, CM)^ + 0(0 = J{cj>) - 6(Z\0, C^0) + o(C) < J{(t>) - aC||zA(^|p + o(C)

(37)

where C is a small positive number and o( •) is the Landau functional, i.e. lim^^o 7 ^ ( 0 = 0Indeed, the second term in the right side of the inequality is strictly negative and the third term can be made very small. TRACTION METHOD Applying the gradient method in Hilbert space to the shape identification problems, a concrete solution can^be proposed by selecting an appropriate Hilbert space. However, the design variables fg and its derivative V were belong to {W^''^{Br')Y which is not a Hilbert space although being a Banach space. Then it is not possible to apply the gradient method in Hilbert space to the shape identification problems directly. A well-advised idea is to select a Hilbert space which includes {W^''^(BP')Y and to find a domain variation V belongs to {W^^'^{BJ')Y. Such a Hilbert space can be found in [H^{BJ')Y that can be defined by D=^[v

e [H\R'')Y

I V[x) = 0, f G {fo n T\TQ

and singular points on T } , and

constraints with respect to domain perturbations > One of the most familiar coercive bilinear forms in {H^{R^)Y continuum problems restricting rigid motions: a{u, ^) = / CijkiUk,iVij dx

(38) is that used in linear elastic (39)

283

Solution to boundary shape identification problems where Cijki € L°°{R^), ij.kj = 1,2, •••,?!, denotes the Hook stiffness tensor. Using a( •, •) for 6( •, •) in Eq. (36), a concrete solution can be presented for determining the velocity V e D by a{y,y) = -{Gr^n,y)

or

- (C^^n,^

Vy€D

(40)

and reshaping with AsV for a given small positive number As. This solution coincides with the traction method that author proposed previously [6, 7, 9]. Whether or not the solution V in Eq. (40) belongs to J9 fl {W^^'^{R'')Y depends on the smoothness of the shape gradient function. The necessary smoothness for the boundary and coefficient functions was discussed in the previous paper [9] using the regularity theorem for elHptic boundary value problems [3, 19]. Reshaped domains by this solution have smoother boundary in differentiability for one time than those by the direct solution moving boundary in proportion with the shape gradient functions [9]. REFERENCES 1.

CEA, J. (1981). Problems of shape optimization. In: Optimization of Distributed Parameter Structures, Vol. 2, pp. 1005-1048, E. J. Haug and J. Cea (Eds.). Sijthoff & Noordhoff, Alphen aan den Rijn.

2.

ZOLESIO, J. P. (1981). The material derivative (or speed) method for shape optimization. In: Optimization of Distributed Parameter Structures, Vol. 2, pp. 1089-1151, E. J. Haug and J. Cea (Eds.). Sijthoff & Noordhoff, Alphen aan den Rijn.

3.

PiRONNEAU, O. (1984). Optimal Shape Design for Elliptic Systems. SpringerVerlag, New York.

4.

SOKOLOWSKI, J. and ZOLESIO, J. P. (1991). Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag, New York.

5.

CEA, J (1981). Numerical methods of shape optimal design. In: Optimization of Distributed Parameter Structures, Vol. 2, pp. 1049-1088, E. J. Haug and J. Cea (Eds.). Sijthoff & Noordhoff, Alphen aan den Rijn.

6.

AZEGAMI, H. (1994). A solution to domain optimization problems. Transactions of the JSME, Series A, 60, pp. 1479-1486 (in Japanese).

7.

AzEGAMi, H., SHIMODA, M . , KATAMINE, E . and Wu, Z. C. (1995). A domain optimization technique for elhptic boundary value problems. In Computer Aided Optimization Design of Structures IV, Structural Optimization, pp. 51-58, S. Hernandez, M. El-Sayed and C. A. Brebbia (Eds.). Computational Mechanics Publications, Southampton.

8.

AzEGAMi, H. and Wu, Z. C. (1996). Domain optimization analysis in linear elastic problems (approach using traction method). JSME International Journal, Ser. A, 39, pp. 272-278.

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H. Azegami

9.

AZEGAMI, H . , KAIZU, S., SHIMODA, M . and KATAMINE, E . (1997). Irregularity of shape optimization problems and an improvement technique. In Computer Aided Optimization Design of Structures V, pp. 309-326, S. Hernandez and C. A. Brebbia (Eds.). Computational Mechanics Publications, Southampton.

10.

AZEGAMI, H . and SUGAI, Y . (1999). Shape optimization with respect to buckling. In Computer Aided Optimization Design of Structures VI, pp. 57-66, S. Hernandez, J. Kassab, and C. A. Brebbia (Eds.). WIT Press, Southampton.

11.

AZEGAMI, H . and KODAMA, K . (1999). Solution of shape optimization problems to maximize deformation under constraints on stiffness and strength. In Proceedings of the First China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, pp. 17-23, Y. X. Gu, B. Y. Duan, H. Azegami, and E. M. Kwak, (Eds.). Xidian University Press, Xi'an, China.

12.

Wu, Z. C. and AZEGAMI, H . (1995). Domain optimization analyses offlowfields. In Proceedings of Second China-Japan Symposium on Optimization of Structural and Mechanical Systems, pp. 236-241.

13.

Wu, Z. C. and SOGABE, Y . and AZEGAMI, H . (1997). Shape optimization analysis for frequency response problems of solids with proportional viscous damping. Key Engineering Materials, 145-149, pp. 272- 278.

14.

SHIMODA, M . , AZEGAMI, H . and SAKURAI, T . (1996). Multiobjective shape optimization of linear elastic structures considering multiple loading conditions: Dealing with mean compliance minimization problems. JSME International Journal, Ser. A, 39, pp. 407-414.

15.

SHIMODA, M . , AZEGAMI, H . and SAKURAI, T . (1998). Numerical solution for min-max problems in shape optimization: Minimum design of maximum stress and displacement. JSME International Journal, Ser. A, 41, pp. 1-9.

16.

SHIMODA, M . , AZEGAMI, H . and SAKURAI, T . (1998). Traction method approach to optimal shape design problems. SAE 1997 Transactions, Journal of Passenger Cars, 106, pp. 2355-2365.

17.

KATAMINE, E . and AZEGAMI, H . (1995). Domain optimization analyses of flow fields. In Computational Mechanics '95, Theory and Applications (Proceedings of the International Conference on Computational Engineering Science), pp. 229234.

18.

IHARA, H .

19.

LADYZHENSKAYA, O . A. and URAL' TSEVA, N . N . (1968). Linear and Quasi-

and AZEGAMI, H . and SHIMODA, M . (1999). Solution to shape optimization problems considering material nonlinearity. In Computer Aided Optimization Design of Structures VI, pp. 87-95, S. Hernandez, J. Kassab, and C. A. Brebbia (Eds.). WIT Press, Southampton. linear Elliptic Equations. Academic Press, New York.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

285

SHAPE OPTIMIZATION OF TRANSIENT RESPONSE PROBLEMS Z.Q. WU\ Y. S 0 G A B E \ Y. ARIMITSU^ and H. AZEGAMI^ ^Department ofMechanical Engineering, Ehime University, 3, Bunkyou, Matsuyama, Japan ^Department ofMechanical Engineering, Toyohashi University of Technology, 1-1, Hibarigaoka, Toyohashi, Japan ABSTRACT A numerical method of shape optimization for transient vibration problems is presented. The transient responses are expressed as summation of frequency responses of each frequency by using Fourier transform. The shape gradient function is derived as a sum of shape gradient function corresponding to each single frequency and expressed in terms of modal parameters that can be obtained from modal analysis for transient responses. The traction method is applied to shape optimization analysis. It is low computational cost because there is no extra sensitivity analysis necessary. Numerical results are attached at the end of this paper to show the vaUdity of this approach. KEY WORDS Optimum Design, Vibration of Continuous System, Modal Analysis, Shape Optimization, Traction Method. 1. INTRODUCTION In general, the computation cost of the shape optimization problem in transient response is fairly higher than that of the stationary problem. The purpose of this study is to develop a simple and low cost numerical method of shape optimization for transient dynamic problems of soHds with proportional viscous damping. A well-known approach for shape optimization problems, which is called the discrete approach, describes a domain shape with parameters of finite numbers and finds out an optimum solution of the parameters by utilizing mathematical programming methods. This approach, however, becomes quite costly in calculating the sensitivity and searching the optimum when the number of design variables increases. An alternative approach, which is called the continuous approach, describes the domain variation with a mapping fiinction and applies the concept of material derivatives to derive the sensitivity, which we call shape gradient fiinction. The continuous approach is proved to be cheaper in calculating the sensitivity than utilizing the discrete approach. The traction method proposed by one of the authors above is a practical application derived from the theoretical implications of continuous approach[4,5]. The advantage of using this

286

Z.Q. Wuetal.

method is that we can obtain a smooth boundary with less distortion of mesh and lower computational cost. In this paper, the traction method is applied to shape optimization problems of transient dynamic responses. The transient responses are expressed as summation of frequency responses combined with power spectrums of each frequency by using Fourier transform. The shape gradient function is derived as a sum of shape gradient functions corresponding to each single frequency. The shape gradient function of a frequency response is calculated using modal parameters that can be obtained from modal analysis for transient responses. Numerical results are given to show the validity of this approach at the end of this paper. MAPPING AND MATERIAL DERIVATIVE METHOD To obtain a shape gradient function that represents the relation of a variation in a domain and the resulting variation in the objective functional, the material derivative method[l] is used.

Fig.l. 1 to 1 mapping Consider an initial domain i? with a boundary F varying to a domain n^ with a boundary r^ . The domain variation can be expressed in a 1 to 1 mapping T^(X):Xef2\-^xen^ , where the index s denotes the history of the variation. Regarding s as time, the velocity field V can be defined as[2] y -

Considering a domain integral functional J^

ds and a boundary integral functional J ^

(1)

of a

distributed function ^^ (x), their derivatives with respect to s are given by (2) (3)

287

Shape optimization of transient response problems ^; = lim —- (^^^^^ - ^,)

where

(4)

where v^ = n-V- and n is an outward unit normal vector[3]. In this paper, the Einstein summation convention and the gradient notation (•) • •=^d{:)l^x- are used in the tensor notation. The notation K denotes the mean curvature. MINIMIZATION OF FREQUENCY RESPONSES Since an arbitrary time-dependent function can be transformed into a summation of frequency response using Fourier transform, we will treat the frequency response problems at first and then apply the results to an arbitrary response in next section. Problems of Frequency Responses Consider a frequency response problem with proportional viscous damping. can be expressed in continuum form as a(u,v) + c{u,v) + h{Uyv) = d{p,v) where u(t,x)

and v(t,x)

Its state equation (5)

u GU, \fv eU

are the displacement vector and its variation respectively, U is

the admissible set defined as U = {VG(H (R,n^))''\v(R,r^^)

= O} .

a(u,v)

,b(u,v)

and d(p, v) are the bilinear forms defined as ^(u,v)

= j ^ ^ C^j^iUj^iV^jdx

(6)

Ku>v) = ^^ pu-v^dx

(7)

d{p.v)=\

(8)

p.v^dr

Here the damping term c(u, v) is considered as a proportional viscous damping, that is c(u, v) = ab(u, v) + /h(u, v)

(9)

where a and J3 are coefficients of damping. We introduce the following relations to rewrite the state equation in order to be convenient to use latter. p(t, r,2) = P(r^,) cos cut ^ P(r,, ) (e^^ + e-^^' )/2 u = —^-^

2

^

where P is the amplitude of a harmonic force, ( • )* is a conjugate complex number of ( • ). Assuming the arbitrary variable i; as i; = ve~^^^, then the state equation is obtained as

288

Z.Q.Wuetal {a{u,v) + jcoc{u,v) - co'b(u,v) - d{P,v)}e-'^ + {a(u\v)

- j(Dc(u\v) ~ co^b(u\v) - d(P,v)}e-''^

=0

Using modal analysis method, the solution is given as 00

^ = Z^(r)^(r) ^(.)-

(11)

Y^-^ ' i(^) -CD -^ JCO{P\r) + « )

TE {1,2,3,-,^}

(12)

where u^^^ and /l^^^ is the rth eigenvector and eigenvalue respectively, and there are the following relations. Problems of Mwimizirig Frequency Responses In a general way, a frequency response function can be expressed into a function of u integrating in a domain or over a boundary. That is F{u)^\^au)dx or

(13)

F(u) = \j{u)dr

For example, the absolute value of mean-compliance, the strain energy and the kinetic energy mean-comj ll W = —a(u,u*) and K = ~co^h(u,u*) d{P,u-^u*}. can be expressed as C |2 2 2 respectively. These functions are used frequently to reduce the vibration of whole structure. Even the local objective functions, such as stress and displacement, can also be written in integral form by introducing a 5 -function. Optimization problems of frequency response can be formulated as Given Q,M,P,co find Q^ that minimize F{u) subject to

{{\ + j(DP)a{u,i) + -^{{\-j(op)a{u\v*)-{(o^

and

772 = j ^ dx<M

{co^^-jcoP)b{u,i)-d(P,i)) + jcoP)co^b(u\v*)

- d(P,v*)} = 0

(14)

where Eq.(14) is the constraint on volume. By combining objective function and constraints with Lagrange multiplier, the Lagrange functional L is assembled as

289

Shape optimization of transient response problems L = F{u) - {(1 + j(DP)a{u,i) - ((D' - jcoa)b(u,d) - d{P,i)} -{(o)^ -jcofi)a(a\i*)-(cD^

+j(oa)h{u\i*)-d{P,i*)}

^^^^ +

A{m-M)

where v is the Lagrange multiplier, and is also known as adjoint variable here, and A is the Lagrange multiplier of the constraint function. Shape Gradient Function In case of objective function F{u) = J f(u)dx,

using Eq.(2) and Eq(3), the derivative of L

with respect to s is derived as L - -{(1 + j(Dp)a{u,v') - (o)^ - ja)a)h{u,v') )-(6>' +j(oa)h{u\v*

-{{\-jcop)a{u\v*

d(P,v)} )-d{P,v*

- {(1 + jcDP)a{u',i) - (co' - jCDa)h{u\i) - \

%:u'dx)

-{(\-j(Dp)a{u*

-:^^^' •'^^ du

)} (16)

X)-(co^

+jcDa)b(u* .v*)-\^

+ A(m - M)

+

^^)

lG{u,i)nV^dr

where the function G is G{u,i) = f{u)-2Re[{\

+ jcDP)Cj,^iUk^iU^^j -{co^ ^ jcoP)pu^v,] +A

(17)

where Re[ • ] means the real part of a complex number. Here we call the scalar function G the shape gradient density function and call the vector function G = Gn the shape gradient function. Let the 5th term of Eq.l6 be equal to 0, we can obtain the constraint equation on volume. Let the 7st and the 2nd term of Eq.(16) be equal to 0, we get the state equation from which u can be determined. Let the 5rd and the ^th terms of Eq.( 16) be equal to 0, we get the adjoint equation as (l + ja)P)a(a\i})-(a)^-jcDa)b(A\i)-\

^u'dx=0 ^^s du

(18)

The solution v of this equation can also be described in terms of u^^.^ and /l^,.) which obtained by modal analysis. For example, when the objective function is given with strain energy, v can be described as

r=l

r=\ 2 (/t(^) -CO ) +CO (^P\r) + ^)

Eliminating terms that are equal to 0, one can get the derivative of L as

290

Z.Q.Wuetal L=\

G(a,i)nV^dr

(20)

•^^ s

In case of objective function F(u) = J f(u)dr,

the shape gradient density function can be

derived as G(u,i) = -2Re[(\ + jcoP)Cj^kiUkiU,j - (co' + jcofi)pu,v,]-^ A

(21)

while the adjoint variable v can be solved from the adjoint equation below (\^jo)/^)a(u\v)-((o^

-j(oa)h{u\v)-

f ^udF^O ^^s du

(22)

For example, when the objective function is given with the absolute value of mean-compliance, the shape gradient density function is obtained as G{u,6) = 2Re[(l + jcoP)Cj,j,iUf,,u,j -(a)'+ JcoP)pa,v^] + A V - sign(d(P,u

+ u*)) — u*

(23) (24)

MINIMIZATION OF TRANSIENT RESPONSE In last section, a method to derive the shape gradient function of problems of frequency response is presented. The result will be extended to transient response problems in this section. By using Fourier transform, any excitation can be broken up into a sum of sinusoids or complex exponentials. Hence, by the superposition property of liner systems, the response to a sum of inputs will be the sum of the responses to the inputs applied separately. Consider an arbitrary time-dependent exciting load. It can be separated into an amplitude part P{r^2) ^"d a time function p{t) as pit, r^2) = -^(^^2 )P(0 The autocorrelation function of p(t) is defined as Cpp(r) =lim^

f_ll P(t)p(t + T)dt

(25)

By taking the Fourier transform of the autocorrelation function, one obtains the power spectral density function Spp(6;): Sppico) = r CppiT)e-''*dt J-oo

(26)

By using Fourier transform, a response function of p(t,r^2)^ ^^^ instance, the time average of stain energy

291

Shape optimization of transient response problems —

1 rTli

W = lim—\

aiu(t,n,)Mt,^J)dt

(27)

is converted to W = £a(A,(^,/2J,A\(^,/2J)SppM/^l

(28)

which is a sum of frequency response of each frequency co. Therefore, the shape gradient function G for this objective function can be expressed into G{uico,rj,vio^,rj

= £G(^„(a),i2J,fe„(«,/2J)Spp(«)d[^^j

(29)

where h^{(o,f2g)y h^^(co,Ils) are displacement vector and its adjoint variable respectively when Fourier transform of excitation p(t), p(cD) = 1. It is obvious that the shape gradient function is a sum of shape gradient function for each frequency derived in last section. TRACTION METHOD As shown in last two sections, for an objective functional L(M,yl) with a constraint function, applying the material derivative method and the Lagrange multiplier method, the derivative of L to 5- can be derived into a boundary integral function of a shape gradient function G and a velocity field V as L = l(G,V)=j G^Vdr^^^Gn^Vdr

(30)

The traction method was proposed as the method that solves the velocity field V by[4,5] a(y, w) = -l(G, w)yweU

(31)

where a ( - / ) is the bilinear form for the variational elastic strain energy, U is the admissible set of domain variations. Eq.(31) means that the velocity field is obtained as a displacement of a pseudo-elastic body defined in i7^ by loading of the pseudo-external force in proportion to -G under constraints on displacement of the invariable boundaries. That the solution V can decrease the objective functional in convex problems is assured by the positive definite property of the energy bilinear form a ( - / ) as shown in the following equation[4]. L = l(G, V) - -a(V, V)<0

(32)

292

Z.Q. Wuetal

NUMERICAL EXAMPLES

p{t) ^

Boundary condition for response analysis

P

Boundary condition for domain variation

Fig. 2.

150

mm

Boundary conditions of a beam-like plate

: A -

50 \

1 40 :\ Ui

.100 o

50

O

:

1

1, , ,.

_[_

0.01

0.02 0.03 Time [sec] Fig. 3.

0.04

§.30 : \ C/5 fe 20 - \ ^ \ £ 10

' V,

1000 Frequency [Hz]

2000

An impulsive load and its power spectrum

Using the shape gradient functions derived above, some numerical analyses of a beam-like plate are performed with traction method. The initial shape and boundary conditions are shown in Fig.2. The plate is excited by a load at center and clamped at both ends. It is discretized with 4-nodes elements and assumed to be a plane stress problem. The first eigenfrequency and the second eigenfrequency are 901 Hz and 2123Hz respectively. Example 1. The excitation load is given with an impulsive load shown in Fig.3. It can be found that power spectrum distributes mainly in low frequency domain. From the 1st to the 20th modes are taken into account in modal analysis. Fig.4 shows the numerical result. It is found that the objective functions decreased to 61%. Example 2. The excitation load is given with a transient force shown in Fig. 5. The power spectrum covered from OHz to 4000Hz and has a peak at about 1200Hz to 1500Hz. The range of peak is between the first and second eigenfrequency. Fig. 6 shows the numerical result. It is found that the objective functions decreased to 36%. The optimized shape is

293

Shape optimization of transient response problems ^.-.^^

^->^

p-

~prT^5r>->--,.^'-c's::2p\--i''r

p i ^ ^ ^ ^ ^ j f^'---^Lj fi

\j^'^

(a) Optimized shape

b

^

"\

0.9

•V

o •^ 0.8

—^^ c/c- •

0

lilil^

$a - 1

Initial shape Optimized shape

0.7 0.6

1

10 20 Iteration Number

(b) Iteration histories of objective function C and mass M Fig. 4.

,

1

.

0.01 0.02 Time [sec]

30

1

,

0.03

1

0.04

(c) Time history of displacement of loading point

Results of example 1

10000F

o

-10000 0

0.001 0.002 0.003 0.004 0.005 Time [sec] Fig. 5.

1000 2000 3000 Frequency[Hz]

4000

A transient load and its power spectrum

quite different from that of example 1 because the peak of power spectrum distributes in different frequency domain.

294

Z.Q. Wuetal

(a) Optimized shape

Initia shape Optimize4 shape A

^0.0001 " co

i

o

0

^

• \J N

1 \A 1 ^i»

6-0.0001 1

20 40 60 Iteration number (b) Iteration histories of objective function C and mass M Fig. 6.

0

.

1

,

1

,

1

0.001 0.002 0.003 0.004 0.005 Time [sec] (c) Time history of displacement of loading point

Results of example 2

CONCLUSION A numerical analysis technique based on the traction method is developed for shape optimization analysis for transient responses. The shape gradient function is derived using the material derivative method and the Lagrange multiplier method, and solved the domain variation using the traction method. It is low cost because the shape gradient function is derived in terms of modal parameters that are obtained from response analysis. Numerical results for some beam-like plate problems showed the validity of this approach. REFERENCE 1. Zolesio, J. P. (1981). The material derivative (or speed) method for shape optimization, Optimization of Distributed Parameter Structure, Alphen aan den Rijn 2. Haug, E. J., Choi, K. K. and Komkov, V. (1986). Design sensitivity analysis of structural systems. Academic press, Orlando. 3. Sokolowski, J. and Zolesio, J. P. (1991). Introduction to shape optimization: Shape sensitivity Analysis, Springer-Verlag, New York. 4. Azegami, H. (1994). Tran. ofjpn, Soc. ofMech. Engs, 60(A),1479. 5. Azegami, H. and Wu, Z. C. (1994). Tran. ofjpn. Soc. ofMech. Engs, 60(A), 2312.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

295

SOLUTION TO SHAPE DETERMINATION P R O B L E M ON UNSTEADY HEAT-GONDUCTION FIELDS Eiji KATAMINE^, Hideyuki AZEGAMI^ and Yasuhiro MATSUURA^ ^Department of Mechanical Engineering, Gifu National College of Technology Shinsei-cho, Motosu-gun, Gifu 501-0495, Japan ^Department of Mechanical Engineering, Toyohashi University of Technology Tempaku-cho, Toyohashi 44^-^^^0, Japan ABSTRACT This paper presents a numerical analysis method for shape determination problems of unsteady heat-conduction fields in which time-histories of temperature distributions on prescribed subboundaries or time-histories of gradient distributions of temperature in prescribed subdomains have prescribed distributions. The square error integrals between the actual distributions and the prescribed distributions on the prescribed subboundaries or in the prescribed subdomains during the specified period of time are used as objective functional. Reshaping is accomplished by the traction method that weis proposed as a solution to shape optimization problems of domains in which boundary value problems are defined. The shape gradient functions of these shape determination problems are derived theoretically using the Lagrange multiplier method and the formulation of material derivative. The time-histories of temperature distributions are evaluated using the finite element method for space integral and the Crank-Nicolson method for time integral. Numerical analyses of nozzle and coolant flow passage in wing are demonstrated to confirm the validity of this presented method. KEYWORDS Inverse Problem, Optimum Design, Numerical Analysis, Unsteady Heat-Conduction, FiniteElement Method, Domain Optimization, Shape Identification, Traction Method INTRODUCTION On the numerical analyses of inverse problem for the unsteady heat-conduction fields, there was much research that it aimed at identification for the time-histories of temperature and heat flux [1][2], the intensity and positions of heat source [3] [4], the characteristic values such as the heat conductivity coefficient and the heat transfer coefficient [5], etc. However, the research for the boundary shape identification has been far from satisfactory. Even if it is the steady-state problem, there are not many research report [6]-[10]. In the previous study of numerical analysis for the some shape identification, the numerical optimization procedure first involves providing a substitute model with a finite number of degrees of

296

E. Katamine, H. Azegami and Y. Matsuura

freedom as a continuum model before formulation of the optimization problems. Based on this model, a shape optimization (or shape identification) problem, in which the design variables are defined in a finite dimensional vector space, can be analyzed numerically using mathematical programming techniques. This method, however, was not effective in locating an optimal solution to problems with a large number of design variables because of the large number of dimensions in the design space. On the other hand, the present authors have focused on the solution of the shape determination problem based on the distributed sensitivity function which uses adjoint variables. In previous paper [11], we presented a numerical analysis method for these shape inverse problems of steady heat-conduction fields in which the temperature distributions on subboundaries were controlled to the prescribed distributions. Reshaping was accomplished by the traction method [12] [13] which was proposed by one of the authors as a solution to shape optimization problems in which boundary value problems were defined. In this study, we applied the traction method to the prescribed problem of time-histories of temperature distributions or time-histories of temperature gradient distributions on unsteady heat-conduction fields. The traction method is a method of applying the gradient method of the distribution system which directly uses the sensitivity functions (shape gradient functions) of the domain variation which are theoretically derived from the optimization problem. In the traction method, the domain variations that minimize the objective functional are obtained as solutions of pseudo hnear elastic problems of continua defined on the design domains and loaded with pseudo distributed traction in proportion to the shape gradient functions on the design domains. The numerical solutions of both evaluation of the shape gradient function and the pseudo linear elastic problems for evaluation of the domain variation can be obtained using the finite-element method or the boundary-element method. Therefore, the traction method can be apphed to complex shape determination problems with a large number of design variables. Additionally, this method is advantageous in that the oscillating phenomenon does not occur during the iterative process of shape optimization because the design boundaries are not moved by the mere movements of finite-element nodal points but by the traction in proportion to the shape gradient functions [13]. First, we formulate a temperature square error integrals minimization problem and theoretically derive a sensitivity function for this problem. The shape gradient functions are derived theoretically using the Lagrange multiplier method and the formulation of material derivative. In a similar manner, we can derive the shape gradient function for a prescribed problem where the time-histories of temperature gradient distributions are specified in the prescribed subdomains. Then, numerical method based on the traction method for the these prescribed problems are presented. Lastly, the successful results for the two-dimensional problems demonstrate the validity of the presented method. T E M P E R A T U R E SQUARE ERROR INTEGRALS MINIMIZATION PROBLEM Formulation

of

Problem

We consider a domain i? G i?^, (n = 2, 3), with boundaries F = FQD Fi , where we write temperature (j) = ^{x, t) on space x e f2 and time t G [0, T]. R is the set of real numbers.

297

Solution to shape determination problem

(a) Temperature prescribed problem

(b) Temperature gradient prescribed problem

Fig. 1 Determination problem of domain with (a) prescribed temperature on prescribed subboundary FD C A and (b) prescribed temperature gradient in prescribed subdomain QDC Q The temperature 0(ro, [0,T]) distributed on To, the heat flux g ( r i , [0,T]) distributed on A 5 the heat conductivity coefficient A:, density p, capacity c, heat source / ( x , t) distributed in Q and the initial temperature distribution 0(f, 0) = o(^) are given as known functions or values, respectively. A domain variation problem where the temperature distribution (j){rD, [0,T]) is specified with (j)D{rD-> [0)^]) on subboundaries FD C / \ can be regarded as a shape determination problem, as shown in Fig. 1(a). The domain variation of the heat-conduction field from domain Q to Qg = Ts{f2) can be described using a one-to-one mapping Tg. The index 5 denotes the history of domain variation. Ts(i7) is assumed to be the elements of set D of admissible functions which satisfy the restrictions on domain variation. For simplicity, we assume that the subboundaries Fi is invariable, i.e., Ts{Fi) = Fi. This problem is formulated as Given i? and ^5 Pi c, / , q, 0, 00 : fixed in space

(1)

find Hs or fs{f2) e D that

inimize / Er^dt= Jo

(2)

f Err,{(t> - (I)DI (f) - (J)D) dt = ( f Jo Jo JTD

subject to / |a((/), w) + h[(j)^t, w) - c{w) - d{w)^ dt = 0

{(t> - (J^D? dF dt {3)

(1)^%

\/w ^Wt

(4)

Eq.(4) is the variational form, or the weak form using the adjoint temperature w{x,t), of the state equation. The bilinear forms a{^,w), b{(t)^ti'w) and the linear forms c{w), d{w) are defined by a{(j)^w) = /

b{(j)^tiU))= / pc(t)^t'^dx^

k4>jWjdx, c{w) =

fw dx, JQS

d(w) =

JTi

qwdF

The sets ^t and Wt of the admissible temperature (j) and adjoint temperature w are given by ^t ={{x, t)jnj Wt ={wix,t)

t e [0,T],

xefsiFo),

= qix, t), t G [0, T], x G f , ( r i ) ,
G H\ns

w(f, T) =

X [0,T]) I 4,{x,t) = ^{x,t),

WT{X)

(5)

X [0,T]) I w{x,t) = 0, i G [0,T], f G f , ( r o ) , = 0, f

G

i7s}

(6)

298

E. Katamine, H. Azegami and Y. Matsuura

In the tensor notation of this paper, the Einstein summation convention and partial differential notations [^).^ = ^{' )/dxi and {'),t = d{- )/dt are used. H^{ •) and n denote the Sobolev space and the an outward unit normal vector, respectively. Shape Gradient Function Applying the concept of the Lagrange multipher method, this problem can be rendered into a stationary problem of the Lagrange functional L=J^

{Er^{(t>-(t>D, (f>-(t>D)]dt-

J^{a{(l),w)-^b{(t>,uw)-c{w)-d{w)}dt

The derivative L with respect to s is derived using the velocity function V{f2s) = = dfs{f-\f2s))/ds[12][13]as L=-

j

(7) dTs{f2)/ds

{a(0, w') + 6(0,t, w') - c{w') - d[w')] dt

- [{a{ct>\

w) + 6(0;,, w) - 2Err,{ct>'. - M}

dt (8)

+IG{V)

where (•) is the material derivative, and ( • ) ' is the shape derivative for domain variation of the distributed function under a spatially fixed condition. The hnear form IciV) of the velocity function V is given by IG{V)=

G= Prom Eq.(8), (t>{x,t) and w{x,t) f{a{(l), [{a{(l>\

f

(9)

Gfi'Vdr

(10)

f {-k(t)jWj}dt Jo

are obtained by

w') + b{(l)^u ^') - c{w') - d{w')}dt w) + 6((/>;„ w) - 2ErM\

(j> - M]

= 0 dt = 0

\/w' e Wt

(11)

V0' G %

(12)

that indicate the variational form of the original state equation for temperature (j){x^ t) and the variational form for w{x,t), which we call an adjoint equation, respectively. Under the condition satisfying Eqs.(ll) and (12), the derivative of the Lagrange functional agrees with that of the objective functional and the linear form Iciy) with respect to V: L|^,„ = [j'^ Er^ dt) U,„ = IG{V)

(13)

The coefficient vector function Gn in Eq. (9) have the meaning of a sensitivity function relative to domain variation and are the so-called shape gradient function. The scalar function G is called the shape gradient density function. The shape gradient function can be derived theoretically, thus, domain variation can be analyzed by the traction method [12] [13].

299

Solution to shape determination problem

T E M P E R A T U R E GRADIENT SQUARE ERROR INTEGRALS MINIMIZATION PROBLEM In previous section, we formulated a prescribed problem for time-histories of temperature distributions (/)(r/;, [0,T]) on the prescribed subboundaries FD Q A in unsteady heat-conduction fields and derived the shape gradient function for the problem. In a similar manner, we can derive the shape gradient function for a prescribed problem where the time-histories of temperature gradient distributions V(J){QD^ P J ^ ] ) ^^^ specified with 9D{^D, [0?^]) in the prescribed subdomains QD C /2, as shown in Fig.l(b). This problem is formulated as Given i? and A;, p, c, / , q, (^, (j)o : fixed in space

(14)

find Hs or f,(i7) G D that minimize / EQ^dt= Jo = r Jo

subject to j

(15)

/ EQ^^iVcj) — go, V^ — gD)dt Jo

( V ^ - go) • {Vct> - go) dx dt

I

(16)

JQD

{a((/), w) + 6((/),t, w) - c{w) - d{w)] dt = 0

(t)^^t

V^ G W, (17)

For simplicity, we assumed that the subboundaries / Q and the prescribed subdomains QD are invariable, i.e., fs{rQ) = FQ and fs{QD) = ^DThe linear form Iciy) of the velocity function V and the shape gradient function for this problem are obtained in the manner of the previous section as lG{y)= G=

(18)

f Gfi'VdF

Jri

(19)

l—k(j)jWj — pc(j)^t'^ + fw-{-\/niq'w)-{-{qw)K\dt

where Vn{') = ^{-)-n and K, denotes the mean curvature. (/)(x, t) is the solution of the original state E q . ( l l ) and w{x,t) is determined by the adjoint equation: £{a{cl>\ w) + b{% w) - 2E^^{V(P\ V(/> - go)} dt = 0

V(^' G H^t

(20)

MUMERICAL SOLUTION TECHNIQUE Traction

Method

The traction method is a procedure for determining the amount of domain variation (velocity function V) that reduces the objective functional, based on the governing Eq. (21). This method uses the gradient method in a Hilbert space, a technique that is also employed in distributed parameter optimal control problems. a'^iV^y)

=-lG{y),

^yeD

(21)

300

E. Katamine, H. Azegami and Y. Matsuura ri=ro Initial design

Un-steady state heat conduction field analysis

I

(a) Coolant flow passage in wing D

Un-steady state adjoint heat conduction field analysis •^

End )

Calculation of shape gradient function Velocity analysis Updating shape with velocity

(b) Nozzle

Fig. 2 Numerical procedure where the bilinear form a^{V,y) defined in the domain i7o as

Fig. 3 Shape determination problem gives the strain energy of the pseudo linear elastic body

a^{u,v)

= J

AijkiUk,iVij dx,

(22)

where u^ v and Aijki are displacement, variational displacement and an elastic tensor, respectively. The governing Eq.(21) indicates that the velocity field t? is a displacement field when negative shape gradient functions —Gn act on the boundaries as an external force. In other words, with the traction method, domain variation is a displacement field when the shape gradient functions act as an external force in a pseudo elastic problem. Accordingly, Eq.(21) can be solved using a solution to ordinary linear-elastic problems, thus confirming the general applicabihty of the traction method. Numerical

Procedure

A flow chart of the shape optimization system is shown in Fig. 2. The main elements of the system include two unsteady heat conduction analyses, calculation of the shape gradient function, a velocity analysis based on the traction method and shape updating. The shape gradient function is evaluated using the two heat conduction analyses in which the distributions of temperature (j){x, t) and adjoint temperature w{x^ t) are analyzed. The time-histories of temperature distributions are evaluated using the finite element method for space integral and the Crank-Nicolson method for time integral. In the solution of the original state E q . ( l l ) , the temperature (j){x,t) is analyzed using the initial condition (/)(x, 0) = (t>o{x) in the time direction from t = Q to t = T. On the other hand, in the solution of the adjoint Eq.(12) or (20), the adjoint temperature w{x,t) is analyzed using the initial condition w{x, T) = WT{X) = 0 in the time direction from t = Ttot=^0. Then, the shape gradient function is calculated using the results. The domain variation V on the Eq. (21) is analyzed using the finite element method. Shape optimization analysis is performed by repeatedly executing these elements sequentially To perform an analysis, a doniain variation coefficient As is set which adjusts the magnitude of the domain variation AsV per iteration.

301

Solution to shape determination problem

t = 2.5 X 10-2 i

t = 5.0 X 10"^ sec

t = 5.0 X 10-2 sec

Fig. 4 Objective temperature distributions of coolant flow passage problem deg

20 30 40 50 60 70 80 90 100 110

EZII

t = 5.0 X 10-^ sec

^ -= 2.5 X 10-2 sec Initial Domain

t = 5.0 X 10-2 sec deg

I 20 ] 30 40 50 CZZ! 60 nz2 70 80 90 100 110

t - 5.0 X 10-^ sec

^ = 2.5 X 10-2 sec Converged Domain

t = 5.0 X 10-2 sec

Fig. 5 Results of temperature distributions in the coolant flow passage problem N U M E R I C A L RESULTS We present the results of two numerical analyses for 2D shape determination problems using the traction method and shape gradient function derived in previous sections. Coolant

flow

passage

in

wing

We analyzed a shape determination problem of coolant flow passage in wing for the temperature prescribed problem as shown in Fig.3(a). The outer surface boundary of wing was assumed as the prescribed subboundary / D = / \ . The shape shown in Fig.4 was made to be the objective shape, and the temperature distribution in the outer surface boundary was assumed to be the prescribed temperature distribution 4>D{^I t)- The design boundary is the left-side boundary of coolant flow passage. For simplicity, the case we considered had the following conditions : length of wing / = 0.25 m, specified temperature (^ = 20 deg, heat-conductivity coeflficient k = 0.204 k W / m deg, heat-flux q = 150 kW/m2, density p = 2710 kg/m^, capacity c = 0.896 kJ / kg deg, initial temperature distribution (/)o(x) = (j)Q — 20 deg, specified period of time T = 0.05 sec. Numerical results of this problem are shown in Figs. 5 and 6. Figure 5 shows a comparison of the shapes and temperature distributions between the initial domain and the converged domain. Figure 6 shows the meshs, the time histories of temperature at point A in Fig.3 (a), and the iterative history ratio of the objective functional normalized with the initial

302

E. Katamine, H. Azegami and Y. Matsuura

value. Based on these results, it was confirmed that the time histories of temperature in the converged domain exhibited agreement with the time histories in the objective domain, and the value of the objective functional approached zero. According to the numerical results of this basic problem, we confirmed the vahdity of the present method.

Objective domain

A Objective domain 0 Initial domain D Converged domain 0.02

. .0.031 . , . 1 , .

Time [sec]

Time histories of temperature at point A j

Initial domain

1

1 ° Objective Functional |

]

'°° 1

°°< 10

Converged domain Mesh

20

30

40

Number of Iteration

Iteration history of objective functional

Fig. 6 Results of the coolant flow passage problem Nozzle In the prescribed problem for temperature gradient distribution, we analyzed a thickness shape problem in nozzle as shown in Fig.3(b). Considering that the two halves are symmetrical, the upper half of domain A-B-C-D was analyzed. The subdomain in the neighborhood of inner wall in nozzle was assumed as the prescribed subdomain Qr,. The shape and temperature gradient distributions shown in Fig.7 were given as the objective shape, and the temperature gradient distributions, respectively. The design boundary is the outer side boundary A-D. The case we considered had the following conditions : length of wing / = 0.06 m, specified temperature ^ = 100 deg, heat-conductivity coefficient k = 0.204 k W / m deg, heat-flux q = 0 kW/m^, density p = 2710 kg/m^, capacity c = 0.896 kJ / kg deg, initial temperature distribution 0o(^) = 0o = 20 deg, specified period of time T = 0.001 sec. Numerical results of this problem are shown in Figs. 8 and 9 in the same way as the above results. We confirmed that the values of the objective functional approached zero, and the converged domains analyzed by the proposed method exhibited good agreement with the objective domain. CONCLUSIONS This paper derived the shape gradient functions with respect to the shape identification problems of unsteady heat-conduction fields to control temperature distributions and temperature gradient distributions to prescribed distributions. The validity of the traction method using the derived shape gradient functions was confirmed by the numerical results.

Solution to shape determination problem

w^fimwjir**-

[deg/m] , -6300 ! -5600 , -4900 -4200 -3500 EZZ3 -2800 , , -2100 ^ -1400 = r -700

y*->"1^'

t = 1.0x 10-^ sec

303

t = 5.0 X 10"^ sec

t = 1.0 x 10"^ sec

Fig. 7 Objective temperature gradient distributions of nozzle problem [deg/m] -6300 -5600 -4900 -4200 -3500 E=Z] -2800 cm -2100 -1400 -700

t - 1.0 X 10-^ sec

t = 1.0 X 10"^ sec

t - 5.0 X 10-4 sec Initial Domain

t = 1.0x 10-^ sec

t - 5.0 X 10"^ sec

t - 1.0 x 10"^ sec

[deg/m] -6300 -5600 -4900 o:^ -4200 -3500 -2800 -2100 -1400 H ^ -700

Converged Domain Fig. 8 Results of temperature gradient distributions in the nozzle problem ACKNOWLEDGEMENTS This study was financially supported by the Sasakawa Scientific Research Grant from The Japan Science Society. REFERENCES 1. Nakamura M., Tanaka M. and Ishikawa H. (1992). Inverse Analysis Using BEM to Estimate Unknown Boundary Values in Transient Heat Conduction Problems, Trans, of Jpn. Sac. of Mech. Engs., (in Japanese), 58-555, A, 2206-2221. 2. Bai Q. and Pujita Y. (1997). A Finite Element Analysis for Inverse Heat Conduction Problems, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 63-608, B, 1320-1326. 3. Kubo S., Ohnaka K. and Ohji K. (1988). Identification of Heat-Source and Force Using Boundary Integrals, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 54-503, A, 13291334. 4. Momose K. and Kimoto H. (1995). Green's Function Approach to Optimal Arrangement of Heat Sources, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 61-585, B, 1762-1767. 5. Tanaka M., Nakamura M. and Shiozaki A. (1993). A Boundary-Element Inverse Analysis Procedure for Estimation of Thermal Properties in Transient Heat Conduction, J. Soc. Mat. Jpn., (in Japanese), 42-477, 708-713.

304

E. Katajnine, H. Azegami and Y. Matsuura

6 a

Objective domain

i

2

^ sJ^

A Objective domain | • o Initial domain H D Converged domain |

~T .. 1.." Time [sec]

"""^

Time histories of temperature gradient in xi direction at point P Initial domain

" ' T " 'j 1 o Objective Functional ]

2 °-* 2 0.6

\

0

0.4 0

0.2

Converged domain Mesh

..

f'^joL^.^-K-ff

Number of Iteration

Iteration history of objective functional Fig. 9 Results of the nozzle problem

6. Barone M. R. and Caulk D. A. (1982). Optimal Arrangement of Hole in a Two-Dimensional Heat Conductor by a Special Boundary Integral Method, Int. J. Num. Meth. Eng.^ 18, 675685. 7. Kennon S. R. and Duhkravich G.S. (1986). Inverse Design of Multiholed Internally Cooled Turbine Blades, Int. J. Num. Meth. Eng., 22, 363-375. 8. Yoshikawa F., Nigo S., Kiyohara S., Taguchi S., Takahashi H. and Ichimiya M. (1987). Estimation Refractory Wear and Solidified Layer Distribution in the Blast Furnace Hearth and Its AppUcation to the Operation, Tetsu-to-Hagane, (in Japanese), 73-15, 2068-2075. 9. Shau R., Batista J. and Carey G. F. (1990). An Improved Algorithm for Inverse Design of Thermal Problems With Multiple Materials, Tran. ASME J. Heat Transfer, 112, 274-279. 10. Meric, R. A. (1995). Differential and Integral Sensitivity Formulations and Shape Optimization by BEM, Engineering Analysis with Boundary Elements, 15, 181-188. 11. Katamine E., Azegami H. and Kojima M. (1999). Boundary Shape Determination on SteadyState Heat Conduction Fields, Proc. of the First China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, edited by Y. X. Gu, B.Y. Duan, H.Azegami and B.M. Kwak, Xidian University Press, 33-40. 12. Azegami H. (1994). Solution to Domain Optimization Problems, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 60-574, A, 1479-1486. 13. Azegami H., Kaizu S., Shimoda M. and Katamine E. (1997). Irregularity of Shape Optimization Problems and an Improvement Technique, Computer Aided Optimum Design of Structures V, edited by S. Hernandez and C. A. Brebbia, Computational Mechanics Publications, Southampton, 309-326.

Parameter Identification in Solid Mechanics

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

307

MULTI-OBJECTIVE PARAMETER IDENTIFICATION OF UNIFIED MATERIAL MODELS Tomonari Furukawa, Shinobu Yoshimura and Genki Yagawa Department of Quantum Engineering and Systems Science University of Tokyo 7-3-1 Kongo, Bunkyo-ku, Tokyo 113-8656 Japan Ph: +81-3-5802-5334, E-mail: [email protected] ABSTRACT Although the regular! sation increased the popularity of parameter identification due to its capability of deriving a stable solution, the significant problem is that the solution depends upon the regularisation parameters chosen. This paper presents a technique for deriving solutions without the use of the parameters, and further an optimisation method, which can work efficiently for problems of concern. Numerical examples show that the technique can efficiently search for appropriate solutions. INTRODUCTION With the increase of safety design requirements, an inelastic material model, which can describe accurate material behaviours, is indispensable. Originally, such a model was developed to describe a particular material behaviour although various material behaviours such as cyclic, stress-relaxation, creep behaviours are observed in real life. A number of material models have been proposed in the last decades accordingly [1 and references therein]. In order to cover a broader range, recently developed models include terms describing plasticity and viscosity, which are the two major material behaviours, and, unified models, which unify both the plastic and viscous terms as viscoplastisity, have received considerable attention. In accordance, other terms are added to these models for accuracy, and the models resultantly have complex formulations with a number of material parameters. To use the models, this gives rise to developing a technique for identifying material parameters for these models. Considering the characteristics of the models, necessary for such a technique is that the technique can handle: (1) various material models (2) a number of material parameters, (3) various experiments such as cyclic, stress-relaxation and creep tests. The promising formulation for general nonlinear identification problems is the method of least squares where a solution is found by minimising an objective function, the residual between experimental and computed outputs, using an optimisation method. Mahnken and Stein [2] developed a technique for identifying a number of parameters for various models using a gradient-based optimisation method. This technique can however find a solution when the objective function is not complex. The first robust technique was then developed by Furukawa and Yagawa [3] where a Continuous Evolutionary Algorithm (CEA) [4] was used as an optimisation method. Nevertheless, the technique handles plural experimental data by introducing weighting

308

T. Furukawa, S. Yoshimura and G. Yagawa

factors, and the solution is largely dependent upon the weighting factors to be chosen. In this paper, a technique for identifying parameters from various experimental data is first proposed. In this technique, weighting factors are not introduced, and the identification problem is formulated with a multi-objective function. A robust multi-objective optimisation method termed Multi-objective Continuous Evolutionary Algorithms (MCEAs) [5], is then proposed for use to find the solutions for this class of problems efficiently. PARAMETER IDENTinCATION Suppose that we have a set of experimental data [u-*, v,*], where U.*G U and V.*G V , and the corresponding model v having parameters x e X , the experimental data can be related to the model by v(u-*,x) + e. = v- *, (1) where e, represents the sum of the model errors and measurement errors: e. =e,'"^+epp. (2) The parameter identification is typically defined to idendfy the continuous vector in engineering problems X c /?", given a set of continuous experimental data, U,V ^R". In order to solve it, a parameter identification problem is often converted to the minimisation of a continuous functional [6]: / ( x ) -^ min , (3) X

where the f : R" ^ R is most commonly the residuals between the computed and model outputs: (4) / ( x ) = £||(v(u,*,x)-v,*)|| . The parameter set minimising such an objective function is to be found within a search space: X„,in^X<X,3,,

(5)

where K i „ , x „ , J = X . UNIHED MATERIAL MODELS Input and output for identification Constitutive models describe the stress-strain relationship, and are constructed with material experiments, given the time-variant behaviour of one of the strain and stress as input and getting the time-variant behaviour of the other as output. Let the input and the output be expressed as g and h respectively, i.e., [g,h]e {[£,o],[o,£]}. If the time-variant input is given by g = git)^ (6) Constitutive models can then derive the output with respect to the input: h = h(g;x), C^) where x represents the material parameters. The output is also related to time via the input, and can be rewritten implicitly as:

Multi-objective parameter identification

309

h = h\t\x). (^) Cyclic behaviours, for example, are often derived by controlling the strain, and the constitutive models describing the behaviours are therefore constructed by defining Eq. (4) with [w,v] = [£,o]. Meanwhile, the stress relaxation behaviours and the creep behaviours represent the time-variant output with constant strain and stress respectively, i.e., [M,v] = [r,o] for stress relaxation behaviours and [w, v] = [r,£] for creep behaviours. Often, the constitutive models are constructed by separating the effect of elastic behaviour from the overall behaviours. The total strain is thus the sum of the elastic strain e' and the inelastic strain a'", and is given by: e^e'+e'". (9) The stress is hence derived from the strain: (7 = E(e-e'"), (10) and vice versa, where E is Young's modulus representing a linear coefficient, and the key of the constitutive modelling is how to define the inelastic strain £'". Unified models The most distinguished inelastic material behaviours are the viscous and plastic behaviours. Unified models describe the behaviours by a unified viscoplastic strain: £'" =£' -\-£P =£'P,

(11)

where £^\ £" and £ "^ represent the plastic, viscous and viscoplastic strains respectively. Plastic hardening materials perform plastic deformations only upon increasing the stress level and the yield condition for plastic deformations changes during the loading process. The performance of such materials thus depends on the previous states of stress and strain. In such path-dependent cases, the inelastic range of materials is in general expressed by means of the thermodynamic forces associated with the two internal variables, back stress representing kinematic hardening x ^"^ ^^^S stress representing isotropic hardening R : (12) f = J(o-z)-R-k<0, where o and k are respectively the stress and material constant, and / represents a distance in the stress space. The viscoplastic flow follows the normality rule, which states: do where the plastic multiplier X is derived from the hardening rules through the consistency condition f = df = 0. The viscoplastic potential is generally expressed as a power function of / is Eq. (11). Chaboche's model [7], a popular viscoplastic model, uses this flow rule and, under stationary temperature condition, has the form together with the kinematic and isotropic hardening rules:

/\
^'^^ (15)

310

where K,n,H,

T. Furukawa, S. Yoshimura and G. Yagawa

D,h,d are material parameters and (.) represents Macauley bracket. The

dynamics of the equations can be uniquely specified by giving the initial conditions of the variables: (17)

MULTI-OBJECTIVE PARAMETER IDENTIHCATION Problem formulation The objective function of material parameter identification in a conventional sense is represented as ^ A .„ . „2 (20) where m is the number of different experiments, w. and m- are a weighting factor and the number of data for the / -th experiment, and W. represents a weighting factor for each data. The problem of this formulation is however that the solution largely depends upon the choice of the weighting factors. One may therefore want to find a solution independently of the choice. Particularly when handling different experiments, one want to see the solution space without weighting factors rather than finding a solution with weighting factors. This gives rise to a multi-objective formulation without weighting factors for experiments: f(x)^=[/-,(x),...,/^(x)]->min, (21) X

where f (x): R" -^ R"' is given by A

.„

. „2

/,(x) = 2lv^;|p(";*;x)-v;.*)|| .

(22)

7=1

Weighting factor w'j is trivial compared to w., so that it can be set to w'j = 1 unless particular importance needs to be put on some data. The solutions for the multi-objective problem are the well-known Pareto-optimal set [8], and a decision vector x^^ € R" is said to be Pareto-optimal if and only if there is no vector x^ e R" for which V = f ( x j = (v,,...,v„) dominates u = f ( x j = («!,...,M„), i.e., there is no vector x^ such that V,<W.,V/G{1,...,/I} A v.<M.,3/G {!,...,«}.

(23)

Figure 1 illustrates an example where Xg and x,2 satisfy Eq. (23). The parameter identification for unified material models formulated as a multi-objective optimisation problem is characterised as that: • the problem is multi-objective, • the objective function / ( x ) is continuous but can be complex such as non-convex, • the search space is continuous.

Multi-objective parameter identification

311

•9/,

Fig. 1 Parato-optimal set

MULTI-OBJECTIVE OPTIMISATION METHOD Fundamentals MCEA proposed here is basically represented by the following four characteristics; the method • adopts a multi-point search such that it can find the Pareto-optimal set through the evaluation of multiple objective functions, • gives equal evaluation to search points with the same Pareto-optimality, • has probabilistic direct search based on evolutionary computation for robustness, • implements the continuous representation of the points and continuous evaluation for proportional selection in order to search efficiently for this class of parameter identification problems, as pointed out previously. Figure 2 shows the fundamental structure of MCEA proposed by the authors, which is efficient for problems with continuous search space. First, a population of individuals, each represented by a continuous vector, is initially (generation r = 0) generated at random, i.e., P' ={x\,...,x\}e{R'')\ (24) where /I represent the population size of parental individuals [9]. Each vector thus represents a search point, which corresponds to the phenomenological representation of individual. t = o;

Initialise P(t); do{ Recombinate P(t); Mutate P(t); Evaluate P(t); Rank P(t); Select P(t); t++; }while(condition);

Fig. 2

Fundamental MCEAs

k=i; P'(k) == P(t); do{ Find G(k) fromP'(k); Eliminate G(k) from F(k); }while(F(k) != 0);

Fig. 3

Ranking process of individuals

312

T. Furukawa, S. Yoshimura and G. Yagawa

Reproduction The definition of the recombination and mutation becomes the probabilistic distribution of the phenomenological measures accordingly. In the recombination, parental individuals breed offspring individuals by combining part of the information from the parental individuals, thereby creating new points inheriting some information from the old points. The recombination operation is then defined as j x ; = (l-//)x^+//x^ (25) where parameter ju may be defined by the normal distribution with mean 0 and standard deviation o : // = A^(0,C7^) (26) or simply a uniform distribution: // = rand(/i,i„,;W„3,).

The mutation can also be achieved simply by implementing x" = rand(x •„, x „ „ ) . ^

nun'

max'

(27)

(28)

with a small possibility [4]. Note that the mutation may not be not necessary for parameter ju with normal distribution since it can allow individuals to alter largely with small possibility, when the coefficient ju is large. Evaluation, Ranking and selection As the Pareto-optimal set satisfying Eq. (23) is to be found as solutions, the ranking process of individuals is composed of an elimination rule. In the rule, all the points are first concerned and the Pareto-optimal set is ranked No. 1. The points in rank No. 1 are then eliminated, and the points in No. 2 are ranked as the second Pareto-optimal set, and all the other ranked are generated stepwise in the same fashion [10]. The points in rank No. k, G(k), are defined as (29) Oik) = {x, I rank(x,) = kyi e {l,...,n}} for further convenience, and the ranking process of the individuals is summarised in Fig. 3, and the ranked points are illustrated in Fig. 4. Figure 5 illustrates the evaluation of the fitness of each individual. The evaluation process starts with finding the best and worst objective function value of each point: /bes,=minVy(x,)IVi6{l,...,n)}, (30) and

/wo,sv=max{/-/x,)IV,€{l,...,n)].

(31)

If we temporarily define the fitness as J worst/

•' best/

we can get the normalised conditions: 0<
(33)

and this allows us to treat the fitness of each function with the same scale. The fitness of points with the same rank has to be the same, and the true fitness of each objective function is thus defined

Multi-objective parameter identification as:

O .(X.) = O f ' \ x . ) = max^'^. (x.) I x. G G{k)].

313

(34)

The fitness of each individual can be conclusively calculated as:

(35)

m

0(x,) = X^,(x,.), which has the range 0
(36)

The selection operator favourably selects individuals of higher fitness to produce more often than those of lower fitness. As (x.) > 0 is satisfied by this equation, the proportional selection [10], which is reported to be faster in convergence than the other popular selection of the ranking selection [10], can be directly used in the proposed algorithm. In this selection, the reproduction probabilities of individuals are given by their relative fitness:

These reproductive operations form one generation of the evolutionary process, which corresponds to one iteration in the algorithm, and the iteration is repeated until a given terminal criterion is satisfied.

/,

^\i^r)

Fig. 4 Ranks of individuals. Shadowed areas Fig. 5 Evaluation of individuals. Shadowed represent search areas of the points in rank 3. areas each represent search areas of the points in rank 3.

NUMERICAL EXAMPLE Appropriateness of the technique In order to confirm its appropriateness for finding Pareto-optimal solutions, MCEA was first used to identify two parameters by minimising two objective functions where the exact set of solutions is known and can be seen on two-dimensional graphs. In this example, the objective functions are given by (38) /,W = ||xf,

/,(x) = ||x-zf,

(39)

where z'^ = [0.2,0.4]e/?^ and xe R^ is subject to search space (5) with x ^ „ = [ - 5 , - 5 ] and

314

T. Furukawa, S. Yoshimura and G. Yagawa

^Lax - [^'5] • The exact Pareto-optimal set in this problem can be determined analytically and is given by (40) X = { x l x = ^,rG[0,l]}, and we can thus investigate the performance of the proposed technique with the exact solution. Values of major parameters for MCEA used to solve the problem were listed in Table 1. Table 1 Parameters of MCEA for a simple multi-objective optimisation problem Value Parameter 2500 No. of generations 10 Population size 0.02 Mutation rate Figure 6(a) show the Pareto-optimal set computed in /, - fj space at 2500th generation. In accordance with Eq. (40), the exact Pareto-optimal solutions are known to be on the curve shown in the figure, and the comparison indicates that the good approximate solutions are obtained. Figure 6(b) shows the corresponding Pareto-optimal solutions in JC, - X2 space. It looks that solutions can have much improvement, but they have already been close enough to the exact solution when the original search space is recalled.

• Computed Exact

».-$^

r"

: : ^ ^ )

0.1

0.2

x2

0.3

0.4

0

(a) Function space (b) Parameter space Fig. 6 Pareto-optimal solutions of a simple multi-objective optimisation problem Parameter identification of a unified model Applicability of the technique was investigated with the parameter identification of an actual unified model. The model used here was Chaboche's model explained in Section 3, and cyclic and creep pseudo-experimental data, shown in Fig. 7, were used to identify seven material parameters, /?o, K, « , / / , D , h and d . These pseudo-experimental data were also created from Chaboche's model but each with a different parameter set as listed in Table 2. The mutation rate was again 0.2, but the population size and the number of generations were 20 and 5000 respectively.

315

Multi-objective parameter identification Table 2 Material parameters used to create pseudo-experimental data n Other parameters h d K D H ^0 50 Cyclic 50 5 cycles 3 300 5000 0.6 100 60 Creep 40 2.5 4500 0.45 500 sec 150 250

ooooooooooo

>ooooooooo

50 •

2 1-0^04

o t -50 t -100 20

-f50 ' Strain %

Time s

(a) Cyclic (b) Creep Fig. 7 Pseudo-experimental data created by Chaboche's model Figure 8 shows the resultant Pareto-optimal solutions in /j - /2 space. It is seen that the solutions draw a very smooth curve. Figures 9 and 10 show the corresponding solutions in RQ-H and H-RQ space. Both the results clearly indicate that the exact solutions in Table 2 are included in the Pareto-optimal solutions and that the other solutions interpolate linearly between the exact solutions. Pareto-optimal solutions in other spaces also had these expected results, and we may conclude that the technique can find Pareto-optimal solutions appropriately for unified material models.

Fig. 8

0.04 0.08 f2 (creep)

0.12

Pareto-optimal solutions of the identification problem in function space

316

T. Furukawa, S. Yoshimura and G. Yagawa

60

3.5

50

40

2.5

30

5500 4000 4500 5000 H RO (a) R^-n space (b) H -R^ space Fig. 9 Pareto-optimal solutions of the identification problem in parameter space

30

40

50

60

CONCLUSIONS This paper has presented a technique for identifying parameters of unified material models using a multi-objective optimisation method. Unlike conventional single-objective approaches where a solution obtained depends upon the choice of weighting factors, the technique allows multiple solutions to be derived, and the solution set derived by the technique can find the whole solution space. The applicability of the technique was demonstrated with a simple numerical example, and its effectiveness was further confirmed by parameter identification of a unified model with pseudo-experimental data. REFERENCES 1. 2.

Ohno, N. (1990). In: Applied Mechanics Review, 43, pp. 283-295. Mahnken, R. and Stein, E. (1994). In: Bui, T. and Tanaka, M. (eds.), Inverse Problems in Engineering Mechanics, pp. 137-144. 3. Furukawa, T. and Yagawa, G. (1997). In: Int. J. Numerical Methods in Engineering, 40, pp. 1071-1090. 4. Furukawa, T. and Dissanayake, G. (1993). In: Proceedings of the 71st JSME Annual Meeting, 930-71, pp. 509-510. 5. Furukawa, T. (2000). In: Int. J. Numerical Methods in Engineering, under review. 6. Bard, Y. (1974). Nonlinear Parameter Estimation, Academic Press, NY. 7. Chaboche, J.L. and Rousselier, G. (1983), In: Trans. ASME, J. Pressure Vessel Technology, 105, pp. 153-164. 8. Fonseca, CM. and Fleming, P.J. (1995). Int. J. Evolutionary Computation, 3, 1,, pp. 1-16. 9. Furukawa, T. and Yagawa, G. (1995). Trans. Japan Society of Mechanical Engineers, 61, 586, pp. 1409-1415. 10. Goldberg, D. (1989). Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, MA. 11. Furukawa, T. and Yagawa, G. (1998). Int. J. Numerical Methods in Engineering, 43, pp. 195-219.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

317

THE POLAR METHOD AS A TOOL FOR SOLVING INVERSE PROBLEMS OF THE CLASSICAL LAMINATED PLATE THEORY G. VERCHERY, P. VANNUCCI, V. PERSON L.R.M.A. - ISAT, Universite de Bourgogne - BP31, 58027 Nevers Cedex, FRANCE [email protected]

ABSTRACT This paper deals with the resolution of inverse problems of the Classical Laminated Plate Theory. It presents an original method for design of laminates based upon the polar description of tensors. The method is illustrated by three particular inverse problems for which stacking sequences are determined in order to obtain given stretching, bending and coupling behaviors and has permitted to find a great number of new solutions. KEYWORDS Polar representation, inverse problems, composites design, laminates, uncoupling, quasihomogeneity, membrane isotropy. INTRODUCTION Fiber reinforced laminates are widely used in modem applications. For these kinds of structures, the Classical Laminated Plate Theory [I] and its various extensions provide efficient methods for theoretical analysis, i.e. when the stacking sequence, the orientations and the properties of the individual laminas are known. Conversely, for design of laminates, a very limited number of rules are available. For stiffriess design, two are currently known and used in practical applications : the Werren and Norris rule to get membrane isotropy [2], the symmetrical sequence rule to suppress stretching/bending coupling [1]. In fact, inverse problems of the Classical Laminated Plate Theory still appear as mostly unresolved questions. The difficulty lies with anisotropy and heterogeneity, which yield to conceptual and practical obstacles. First, anisotropy for fourth rank tensors such as elasticity tensors introduces many coupling effects, not easy to understand and manage with clear insight. Anisotropy handling is even more difficult when rotations of the material axes are involved, which is a basic feature of the laminate construction. The stacking process also introduces further coupling. All these points result in a very large number of parameters to control in the inverse problems. We have developed a method which is able to face these difficulties in an efficient way. It is based on the polar representation of tensors. This representation, which can receive both a graphical [3] and an analytical application, has been developed and applied to various problems by Verchery during the last twenty years [4]-[12]. It is based on two complementary

318

G. Verchery, P. Vannucci and V. Person

steps: first, it introduces complex components for tensors, so generalizing the complex variable methods (such as the Kolosov-Muskhelishvili representation of symmetrical second order tensors) and associated graphical methods (such as Mohr's circle for symmetrical second order tensors and Tsai's circles for elasticity fourth order tensors [1]), as well as other quantities introduced to make easier the use of the Classical Laminated Plate Theory (such as the Tsai and Pagano parameters Ui, [13]); secondly, it writes these complex quantities with moduli and polar angles, closely related with invariants and symmetries of the tensors. THEORY Polar Description for Laminates The Classical Laminated Plate Theory shows that the general constitutive equation for a laminate made of various superposed plies can be expressed in matrix form: A B

B D

(1)

where N is the tensor of membrane forces and M that of bending moments, while, s and x are respectively the tensors of middle plain strains and curvatures. A, B and D are fourth order tensors : A is the membrane, D the bending and B the coupling stiffiiess tensor; this last describes the coupling between the membrane and bending behaviors for non-homogeneous plates. According to the general sketch of Fig. 1, the components of these three tensors A, B and D, for a N-plies laminate (N = 2p+l if odd and N = 2p if even), are given by:

A=IL-p(-k.i -zjQk, B 4 z L - p k . i -4)Q^'^=\n-M.i

-4)Qk (2)

yi h/2

Zk+i

h/2

Zld

-1

h/2

Zk+l

—•

'k -P

h/2 -P a)"

h

b)

Fig. 1. General sketch ; a) odd number of plies ; b) even number of phes In relations (2), Q k is the stiffiiess of the ply k expressed in the global reference fi*ame of the laminate, that is, obtained from the intrinsic (or on-axis) stiffiiess of the ply by a rotation at an angle 5^ to take account of its orientation. The transformation relations, when the axes are so rotated, are rather cumbersome using Cartesian co-ordinates since they involve fourth-power laws of the director cosines of the angles formed by the new frame to the old one.

The polar method as a tool for solving inverse problems

319

Nevertheless, a great simplification can be achieved using the polar representation technique. Namely, for a plane elastic tensor L, the six Cartesian components Lijki, are expressed by six other quantities. To, Ti, Ro, Ri, O Q and O^ called polar components, according to the following complex equations : 8To = L i l l l +L2222 - 2 L i i 2 2 +4Li2i2 8T1 = L n 11 + L2222 + 2Li 122 8R0 e^^'^o = L i n i +L2222 -2L1122 - 4 L i 2 i 2 + 4 i ( L n i 2 -L2212) 8R1 e ^ 1 =l^\\\\

-L2222 + 2 i ( L n i 2 +L2212)

The scalars To, Ti, the moduU Ro and Ri, as well as the difference of the polar angles Oo - Oi are invariant under frame or material rotation. The scalars To, Ti are the isotropic part, and isotropy is obtained if, and only, if the moduli Ro and Ri are null, while square symmetry is obtained for Ri null; for orthotropy it must be Oo = Oi + k7r/4. From this polar representation, for a frame rotation by an angle 0, the Cartesian components of tensor L are easily expressed with the polar components, as a function of the rotation angle, in the form : Lxxxx = To + 2Ti + R Q COS4(OO - 6) + 4Ri cos2(Oi - 9)

Lxxyy = -To + 2Ti - Ro COS 4(Oo - e) Lyyyy = To + 2Ti + R0 COS4(00 " 6) " 4RJ cos2(Oi - G) Lxyxy=To-RoCOs4(Oo-e)

(4)

Lxxxy = RQ sin4(Oo - 6) + 2Ri sin2(Oi - 0) Lyyxy = - R o sin4(Oo - 6) + 2Ri sin2(Oi - 6) Then, equations (2) may be rewritten in order to obtain the four complex components of each one of the tensors A, B and D. These are expressed from the polar components of the stiffnesses of the layers by the following summation formulas: Polar components of tensor A for a laminate :

Tl=EL-pTlk(^k+l-Zk)

Rie^^^^-EL-pRlk^'^^'^^^'''Wl-^ Polar components of tensor B for a laminate :

T0=|Zk=-pT0k(zLl-4) 1 •5-in

„

4i
2\

320

G. Verchery, P. Vannucci and V. Person

Polar components of tensor D for a laminate :

To=^ZL-pTOkk+l-4) (7)

where Tok, Tik, Rok, Rik, Ook and Oik are the polar components of the stffiiesses of the ply k. In particular, Ook and Oik are the angles defined for the ply k by its own referencefi-ame.6k is the angle that suchfi-ameforms with the global reference frame of the laminate. A further simplification in the analysis, and a necessary hypothesis to have general solutions, is obtained when limiting to laminates made of identical plies, that is, plies with the same thickness and the same material properties. In this case, equations (5), (6) and (7) give simply:

e-k

Roe^^^o = i R o e ^ ' * o ^ P ^ _ ^ e ^ ' « ^ ( z L , - z ^ )

(8)

Roe^'*oaR„e^''^o^P^_^e^'«^(4^,-z3)

R.e^^«. =lR,e2*«^.2:L-p^'''^k.l-4) Herein, h is the total thickness of the laminate and N is its number of plies. We have applied this general approach to three specific inverse problems: i) cancellation of membrane-flexure coupling, ii) search for membrane-flexure homogeneity, and iii) search for membrane isotropy. The first and third problems aim to meet usual requirements in composite applications, while the second, based on a concept introduced by Verchery, provide composites with a simpler behavior than the laminates currently used. Inverse Problems for Cancellation of Membrane-Flexure Coupling and for MembraneFlexure Homogeneity: It may be noticed that the inverse problem of the search for uncoupled laminates (problem i) is in fact a particular case of the more general inverse problem of the search for quasihomogeneous laminates (problem ii). Indeed, a quasi-homogeneous laminate is defined as a laminate with no membrane-flexure coupling and identical elastic properties for tension and

321

The polar method as a tool for solving inverse problems bending and is characterized by the following two equations: C=iA-^D=0 h h^

and B = 0

(9)

Consequently, we will simply present hereafter the general polar equations of the search of quasi-homogeneity. The general uncoupling solutions will be obtained by solving only those corresponding to the condition B = 0. In view of equations (8), the quasi-homogeneity conditions (9) become simply h

"

h^

N'^''=~P

}j3 •^k=-p

V k+1

k/

These equations can be rewritten in the form: ^o+ZL,^k(e''"^+e^-«-k)=0

(Ua)

'=o+ZL,^k(e''"'+e^-«-k)=0

(lib)

lLibk(e'''''-e^'^-'^)=0

(lie)

Zk=i''k(e^'^''-e^'^-0=O

(11^)

where the coefficients Cj^ and h^ are given by: Ck=p(p + l ) - 3 k ^ C o = p ( p + l),bk =k for N odd; C]^ =(p-lXp + l)-3k(k-l),Co =0,bk = 2 k - l for N even.

^^2)

The solution of the inverse problem of cancellation of membrane-flexure coupling leads to consider only the last two of these equations. Inverse Problem for Membrane Isotropy : A laminate is isotropic for membrane effects if and only if: Ro=Ri=0

(13)

In the case of laminates composed by identical plies, this relation gives (see Eq. (8)): Roe4**o=Ro-e4''l'oyP N ^^Rie2i'I>l=RiiLe2i1'iyP

e^'^k ^ Q P e^'^k =o

(14)

322

G. Verchery, P. Vannucci and V. Person

From these equations, it can be seen that, if the single layer is not isotropic, that is, if RQ and R J are not null, the condition for membrane isotropy requires: yP e'"^!' =0 ^k=-p yP e2i8k=o

(15)

Consider that the layers of the laminate are disposed in Ng different orientations, and that there are ni layers in the i^^ group of orientation defined by the angle 6i. Then, equations (15) can be rewritten as: X|!?n,e^^^i-0

(16a)

Xjlf n^e^'^i =0

(16b)

So, the solutions will be obtained as numbers ni of phes disposed at angles 5i; to remark that they will not depend on the positions of the layers in the stacking sequence. Of course, solutions can be looked for analytically, but also by graphical considerations thanks to the polar method [3]. Indeed, these summations of complex numbers can be represented by two summations of vectors, one for equation (16a) and another for equation (16b). The lengths of the vectors are the number of plies n\ in each group of orientation, while the orientations of the vectors are given by the arguments of the complex numbers i.e. 46i and 26i. In order to have null sums the broken lines obtained by summing the vectors have to form polygons, that is, they have to be closed. Particular case: laminate with layers disposed in four different orientations In the case of laminates with four different orientations, equations (16) give : ni e^^^i + n2 e^^^^ + n3 e^^^3 + ^^ ^^^^A = Q ni e^^^i + n2 e^^^^ + ^3 e^'^^ + ^4 e^^^^ ^ Q

(17a) (17b)

In the general case, these two complex equations lead to four algebraic equations. However, assuming that the polygon obtained from the vectors corresponding to the second of these equations is an isosceles trapezium such as ni = n 4 , 61 = 0 , 83 = 7c/2 and 84 = 71 - 8 2 , then equations (17) reduce to only two algebraic equations as follows: n, +n3 +2n2(2x^ - l ) = 0 n| - n 3 +2n2X = 0

(18)

where x = cos 25 2This system of two equations and four unknowns, U], n2, n3 and x, can be solved in terms of n2 and x for instance as follows: ni =n2(l-2xXl + x) n3=n2(l + 2xXl-x)

(19)

323

The polar method as a tool for solving inverse problems

and the total number of plies N is given by: N = 4n2(l + xXl-x)

(20)

From equations (19), it can be seen that, in order to have integer values for n^ and n3, the unknown x has to be a rational number of the type ±p/q, where p and q are integers, and to be such as |x| < 1/2. Then for a given value of p/q, it is possible to calculate the integer value of n2 which gives the minimum total number of plies N for the laminate. Finally, values of numbers of plies Uj and n3 are determined using equations (19). RESULTS AND DISCUSSION Inverse Problem for Cancellation of Membrane-Flexure Coupling : Quasi-Trivial Uncoupled Solutions In order to obtain the uncoupled solutions, the two complex equations (lie) and (lid) involving complex exponentials of the ply angles with integer coefficients have to be solved simultaneously. Except for a very small number of plies, it has not yet been possible to find their general solutions, or even to discuss the existence and the number of these solutions. Nevertheless, we could discover special properties of the integer coefficients in the equations, and from these properties, we were able to find a particular class of solutions. In fact, it can easily be noticed that coefficients bk are anti-symmetric about the middle plane of the laminate. This particularity permits to verify immediately that any symmetrical stacking sequence will automatically satisfy equations (lie) and (lid), as expected. However, this well-known rule of symmetrical sequence to suppress in plane/bending coupling is not a necessary condition but only a sufficient one. A more general condition can be formulated as follows: an uncoupled laminate is obtained each time that the sum of the coefficients bk of plies having the same orientation is zero. Such a solution is called quasi-trivial solution of the uncoupling problem and each of the groups formed by all the plies of the laminate having the same orientation is called a saturated group. Consequently quasi-trivial solutions are obtained without solving completely the system formed by equations (lie) and (lid) and are completely defined by their stacking sequence. To be noticed that the orientations of the phes are not determined. An algorithm based on the previous properties has been built to find all these particular solutions for a given number of plies. Running this algorithm, we got solutions up to 18 phes. The results are presented in Table 1. Compared with the unique solution given by the classical rule of symmetrical stacking sequences, we observed a large number of non symmetrical stacking sequences with no coupling. In fact, the number of solutions, roughly increasing with the number of plies, exceeds hundreds above 15 plies, and thousands above 17 plies: such a fact was absolutely unsuspected in previous works on coupling. Furthermore, as these solutions depend on a certain number of arbitrary values for the angles, they open the possibility of an unlimited number of uncoupled laminates. Table 1. Number of independent uncoupled solutions N. of plies N. of solutions

7 2

8 2

9 4

10 5

11 11

12 15

13 41

14 57

15 174

16 275

17 1033

18 1639

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G. Verchery, P. Vannucci and V. Person

Inverse Problem for Membrane-Flexure Problem of quasi-homogeneity

Homogeneity : Quasi-Trivial

Solutions

of the

The analysis and the principle of the resolution for this case are similar to the above inverse problem for cancellation of membrane-flexure coupling. First, with the polar representation, we got four complex equations (Eq. (1 la) to (1 Id)) involving complex exponentials of the ply angles with integer coefficients bk and Ck. While coefficients bk are anti-symmetric about the middle plane, as already said, and vary linearly, coefficients Ck are symmetric about this plane and vary quadratically. Furthermore, it can be noticed that the sum of coefficients Ck as well as the sum of coefficients bk are zero. Then we will have particular solutions, called quasitrivial solutions of the quasi-homogenous problem, each time that the sums of coefficients bk and Ck of a saturated group will be zero. We also obtained these solutions with an algorithm taking advantage of the structure of the coefficients in the equations. We have not succeeded in a general treatment and discussion of the four complex equations, which seem to be of paramount difficulty, as illustrated by the fact that we found a solution for seven plies (i.e. for only seven variables) but no particular solutions for nine or ten plies or variables. We have run the algorithm up to 30 plies, and we observed many solutions, depending on some arbitrary values for the angles, in a number more or less increasing with the number of plies, with a few tens of solutions over 17 plies, hundreds over 22 plies and thousands over 25 plies as shown in Table 2. It is worth noting from these results that symmetrical sequences represent only a few part of the total solutions. Moreover, comparing results of Table 2 with those of table 1, it can be seen that for a 17-pUes laminate, for example, there are 1033 uncoupled solutions among which 23 have also a similar behavior in membrane and in bending. Experimental evidence was obtained on specimens that we designed and manufactured according to these results. Table 2. Number of quasi-trivial independent solutions N. of plies Total solutions Symetric sol.

7 1 1

8 1 0

9 0 0

10 0 0

11 3 2

12 1 0

N. of plies Total solutions Symetric sol.

19 52 0

20 40 0

21 44 2

22 130 3

23 594 1

24 167 0

13 3 2

14 2 1

15 4 0

16 8 1

17 23 0

18 5 0

30 28 29 27 26 25 2352 1495 1282 5902 45441 6146 3 1 3 0 7 0

Inverse Problem for Membrane Isotropy The inverse problem for membrane isotropy is governed by the general system of equations (16). As it can be seen from these equations, the sum of coefficients multiplying the complex exponentials is no more zero this time but is equal to N. Consequently, it is impossible to have quasi-trivial solutions for membrane isotropy and solutions will exist in this case only for fixed values of the angles. However, we have used two semi-empirical ways, which gave us particular solutions. Starting from the quasi-trivial quasi-homogeneous (QTQH) solutions described above, we have searched to select the ones which meet the Werren and Norris solutions. Remembering that a laminate made of N identical plies satisfying the rule of Werren and Norris when: first, it has an equal number of plies in each group of orientation; secondly the orientations differ of

The polar method as a tool for solving inverse problems

325

an angle equal to 180°/Ng (where Ng is the number of different groups of orientation). Remembering also that the values of the angles are not determined for the quasi-trivial solutions but only the stacking sequences. A QTQH solution is thus able to be a Werren and Norris solution each time that its sequence presents the same number of plies in each saturated group. It is then sufficient to orient the plies so that to have differences of angles equal to 180°/Ng. The QTQH solutions have been examined up to 30-layers laminates and we found: 5 solutions for a 18-plies laminate with 3 different orientations at angles -60°/0°/60°, 1 solution for a 24-plies with 4 different orientations at angles -45°/0°/45°/90°, 219 solutions for a 27-plies laminate with 3 different orientations at -60°/0°/60° and 29 solutions for a 30phes laminate with 5 different orientations at angles 0772°/1447216°/288°. For instance, one of the solutions for a 18-phes laminate is [07-60760707-607607607607-607-6070707 -6070707607607-60°], which is a non symmetric solution as well. It is worth noting also that all these laminates are in fact isotropic for both the in-plane and the out-of-plane behavior, which is a new result: indeed, these laminates are completely isotropic, and this is to our knowledge, the first time they are found. Furthermore, we have verified that combinations of these quasi-homogeneous isotropic solutions are also isotropic, thus generating solutions which are not of the Werren and Norris type. Starting from the graphical application of the polar method, we obtained exact solutions for the case of four orientations of plies with special conditions in order that the polygon of vectors corresponding to equation (17b) is an isosceles trapezium as described previously. The principle of resolution of the system of two integer equations (18) thus obtained is also presented previously by mean of equations (19) and (20). Using a numerical process based on this analysis, we obtained, together with the Werren and Norris solutions, which were once again to our knowledge the only ones known up today, some extra solutions. The first ten of these new solutions, which do not follow the rule of Werren and Norris, are presented in Table 3 in increasing order of the number of plies. Table 3. Isotropic membrane laminate for the isosceles trapezium condition 1 2 3 4 5 6 7 8 9 10

laminate [05,909,icpg] [04,90io,±(P9] [07,90io,±(P9] [0^,9027,±925] [0i3,9028,±(P25] [0i8,9028,±q)25] [022,9027, ± 9 2 5 ] [027,9035, ±932] [040,9054, ± 9 4 9 ] [045,905,, ± 9 4 Q ]

Number of pHes 30 32 35 84 91 96 99 126 192 195

9 37,761 35,264 40,203 33,211 36,271 39,231 42,130 41,410 40,893 42,952

CONCLUSION In these three examples of inverse problems, we have found particular solutions or classes of solutions. Although, from a theoretical point of view, general methods and solutions still remain to be found, it must be pointed out that our work has provided completely new results. Specially, in the first two problems, the numbers of solutions are so high that they certainly can fulfill all practical requirements. Further, the polar method has proved to be very efficient

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and general, so the present results can be easily extended, at the designer's request, to other inverse problems, such as other required properties, approximate solutions with definite precision, and laminates made of different laminas. REFERENCES 1. 2. 3.

4.

5.

6.

7.

8. 9. 10. 11. 12. 13.

Tsai S. W., Hahn H. T., Introduction to composite materials, Technomic, Lancaster, Pennsylvania, 1980. Werren F., Norris C. B., Mechanical properties of a laminate designed to be isotropic, US Forest Products Laboratory, Report 1841, 1953. Vong T. S., Verchery G., line methode d'aide graphique a la conception des sequences d'empilement dans les stratifies (Graphical aided design of stacking sequences for laminates), in French with English summary, Proceedings of the 5^ French conference on composite materials (JNC-5), C. Bathias and D. Menkes eds., Plurahs, Paris, 1986, pp. 267-280. Verchery G., Les invariants des tenseurs d'ordre quatre du type de Velasticite (Invariants of fourth rank tensors with the symmetry of the elasticity tensors). Proceedings of the Euromech colloquium 115, Villard-de-Lans, 1979, J. P. Boehler ed.. Editions du CNRS, Paris, 1982, pp. 141-149. Kandil N., Verchery G., New methods of design for stacking sequences of laminates. Proceedings of the first international conference on computer aided design in composite material technology (CADCOMP 88), C. A. Brebbia, W. P. De Wilde and W. R. Blain eds.. Computational Mechanics Pubhcations and Springer Verlag, 1988, pp. 243-257. Kandil N., Verchery G., Some new developments in the design of stacking sequences of laminates. Proceedings of the 7^^ international conference on composite materials (ICCM-7), Wu Yunshu, Gu Zhenlong and Wu Renjie eds., International Academic Pubhshers & Pergamon Press, 1989, Vol. 3, pp. 358-363. Kandil N., Verchery G., Design of stacking sequences of laminated plates for thermoelastic effects. Proceedings of the second international conference on computer aided design in composite material technology (CADCOMP 90), W. P. De Wilde and W. R. Blain eds.. Computational Mechanics Publications and Springer Verlag, 1990, pp. 227-238. Verchery G., Designing with anisotropy. Textile composites in building construction, Part 3 : Mechanical behaviour, design and applications, P. Hamelin and G. Verchery eds., Plurahs, Paris, 1990, pp. 29-42. Grediac M., Vautrin A., Verchery G., A general method for data averaging of anisotropic elastic constants. Journal of Applied Mechanics, Transactions of the ASME, 1993, 60 (September 1993), pp. 614-618. Verchery G., Designing with anisotropy - Part 1 : Methods and results for laminates. Keynote lecture, 12^^ International conference on composite materials (ICCM-12), Paris, 1999. Vannucci P., Verchery G., Designing with anisotropy - Part 2 : Laminates without membrane-flexure coupling, \i^ International conference on composite materials (ICCM-12), Paris, 1999. Vannucci P., Gong X.J., Verchery G., Designing with anisotropy - Part 3 : Quasihomogeneous anisotropic laminates, 12* International conference on composite materials (ICCM-12), Paris, 1999. Tsai S. W., Pagano N.J., in Composite Materials Workshop, S.W. Tsai et al. eds., Technomic, 1968.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

327

VERIFICATION OF ULTRASONIC T R A N S D U C E R CHARACTERISTICS DETERMINED IN A N INVERSE PROBLEM BASED ON LASER MEASUREMENTS T. Kanbayashi^, H. Yoshikawa^, N. Nishimura^ and S. Kobayashi^ 1. Department of Global Environment Engineering, Kyoto University, Kyoto 606-8501, JAPAN 2. Department of Construction Engineering, Fukui University of Technology, Fukui 910-8505, JAPAN ABSTRACT An inverse problem of determining the elastic v^ave produced by an ultrasonic transducer from laser velocimetry data is considered. The quality of the inversion is tested via a comparison between simulation results obtained with 3 dimensional time domain elastodynamic BIEM and experimental data. The agreements of these results are seen to be satisfactory. Also, the particle velocity of the scattered wave is concluded to have sufficient geometrical sensitivity for determining the unknown shape of defects. KEYWORDS BIE, ultrasonic transducer, laser interferometer INTRODUCTION The present authors have been investigating BIE (boundary integral equation) approaches for solving inverse problems related to ultrasonic non-destructive determination of defects in structures[l] for some time. The setting of these elastodynamic inverse problems assumes that one knows both incident waves to illuminate the defects, and the scattered waves from the defects. In reality, however, neither of these waves are easily measured in experiments using ultrasonic transducers, mainly because the output of ultrasonic transducers has not been related to physical quantities in a quantitative manner. In view of this, we have considered an inverse problem which uses real data obtained with a laser interferometer to determine the waves from the transducer[2]. The present paper continues the efforts made so far by examining the quality of the inversion carried out in the previous paper. Namely, we compare the measured scattered wave from an artificial defect with the corresponding numerical result obtained using the reconstructed elastic wave from the transducer and a 3 dimensional time domain BIEM. This paper begins with the description of the experiment carried out for the present analysis. The specimen used in this experiment has a circular cylindrical hole as an artificial defect. This specimen is designed so that one may obtain data to determine the elastic wave from the transducer, and measure the scattered wave from the defect at the same time. After a brief recapitulation of the inverse problem of determining the elastic wave from the transducer, we proceed to the detail of the BIE analysis for estimating the scattered wave

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T. Kanbayashi et al.

from the defect. Then the comparison of the experimental and numerical results of the scattered wave is made. These results are shown to agree well at least for small t, although some difference develops as time grows. We finally examine numerically the sensitivity of the scattered wave to the geometry of the artificial defect, concluding the feasibility of the final objective of the present series of investigations, i.e., the determination of the geometry of defects using ultrasound and laser velocimetry. EXPERIMENT Figure 1 shows the specimen used in the experiment. This specimen is made of aluminium alloy, and has a circular cylindrical shape with the diameter of 200mm and the hight of 50mm. This specimen has an artificial defect, i.e. a cylindrical hole having the diameter of 20.0mm which opens at the bottom face of the specimen. The depth of the flat end of the artificial defect is 4.8mm deep from the top face of the specimen. We set an ultrasonic transducer (SONIX, central frequency = 500 KHz, diameter = 15mm) at the indicated position (20mm from the centre of the specimen) using silicone grease as the couplant. The ultrasonic transducer is then excited with an electric pulse generated by a pulser (RITEC). We then use laser interferometer (ONO measuring instruments) to measure normal velocities at points denoted by Z/i_4 and Ri in the same figure. The points I/i_4 are taken 5mm apart from each other, and the points Li and Ri are placed 22mm away from the centre of the transducer. The frequency range of the laser interferometer is from IHz to 5MHz. The interferometer produces a voltage proportional to the measured normal velocity, and the output from the interferometer is recorded with a digital oscilloscope (LeCroy). The S/N ratio of the output is enhanced with a stacking of 5000 sweeps. The idea of this experiment is to determine the characteristic of the transducer by using the data obtained at Z/i_4 in an interval of t where the responses at these points are not affected by the reflection from the artificial defect. Our interest is to see if one can use the wave field thus reconstructed to predict the response at the point Ri where the effect of the defect is expected to be significant.

laser interferometer

22

^ ^ ^ ^ 2 ^ 1 V

iUl

^~~""^"\

5 5 5 —

2oh--.! - ^

30

—-——

^ ^ ^ ^ ^ ^ ^

Figure 1: Specimen.

Verification of ultrasonic transducer characteristics

329

INVERSE PROBLEM We now consider the following inverse problem: Find the elastic wave produced by the transducer given the normal velocities V^{t) at several boundary points x^ ioi j — I,..., iV, where N is the number of the boundary points where the measurement is made. In reality the data V^{t) are obtained with laser velocimetry. To solve this problem, one may model the action of the transducer to the specimen by a time varying uniform normal load p{t). With this assumption one obtains an integral equation given by (1) V'{t) = / ' k^{t - s)p{s)ds (i = 1 , . . . , iV) Jo where k^ (t) is the normal displacement at the point x^ produced by a uniform load having a Dirac delta time variation. Since this integral equation is unstable to solve numerically, we convert it into the following stabilised integral equation with the help of Tikhonov's regularisation[2]

ep{s) + j\{s,T)p{T)dT K{S,T)=Y,[

• Jma,x(s,T)

=

E/^'='(*

- s)V^it)dt,

(2)

k'(t-s)k^{t-T)dt

where e is the Tikhonov parameter. The solution p of (2) then determines the elastic wave from the transducer as one solves a direct problem with this p as a part of the boundary condition. Namely, one solves the following initial boundary value problem for the unknown velocity v: /iAv-f (A + /i)VV-V = pij mDx{t>0) Tv - 0 on {dD \ 5) x (t > 0) Tv = -p{t)n on 5 X (t > 0) v\t=o = v\t=o = 0

(3)

in D

where D is the domain under consideration, dD is the boundary of D, S is the part of the boundary where one puts the transducer, n is the unit normal vector to dD and T is the traction operator, respectively. RESULTS We have carried out an inversion to determine p{t) using the velocities V'^{t) (i = 1 , . . . , 4) obtained at Li_4 shown in Figure 1. The data in the interval of to < t < ti are used in the inversion, where to is the rise time at Li, ti — to-\- At and A^ = 6 {fis). It is seen that one may then assume the domain for the analysis to be a half space since the reflection from the side and the bottom of the specimen is negligible. Hence one may use the well-known solution by Lamb to obtain k^{t) in (1) and (2). A numerical solution of (2) is obtained as one discretises (2) in a standard manner for a given s, while the optimum Tikhonov parameter e is obtained from the L curve[3]. With p thus obtained one may solve (3) to compute the velocity at Ri using numerical techniques such as BIEM. In the present analysis we have used a 3 dimensional time domain BIEM with a truncated plane boundary to this end. The number of boundary elements used are 1216, and the velocities of P and S waves are set equal to CL — 6800m/s and CT — 3050m/s, respectively. Figure 2 shows the mesh used for the analysis.

330

T. Kanbayashi et al.

Figure 2: Mesh.

Figure 3 shows the computed velocities at points Li_4 obtained with (1) and the p{t) reconstructed from the measured normal velocities at these points. In this figure the computed velocities are marked "eLi". The integral of Lamb's solution is used to compute k\ We have also plotted the corresponding real data in the same figure, and marked them ''mLi\ This result shows that the quality of the fitting to data is satisfactory.

Figure 3: Computed and observed velocities at Li_4. Figure 4 shows the normal velocity at Ri computed using the inversion result p{t) and BIEM (solid line), and the corresponding experimental results (broken line). These curves seem to be in agreement except for larger t. This discrepancy is considered to be partly due to the coarseness of the mesh used with BIEM. From this result we may conclude that the ultrasonic transducer characteristics obtained in the inverse problem stated above is fairly reliable. The final objective of the present series of investigations is to determine the geometry of

Verification of ultrasonic transducer characteristics

331

Figure 4: Computed and observed velocities at Ri

unknown defects from an ultrasonic measurement using laser interferometer. At this stage it is still premature to try an inversion of this type since the agreement of the theory and experiment in Figure 4 is not completely satisfactory. It is, however, worthwhile to see if the present experimental framework has sufficient geometrical sensitivity for the use in the determination of defects. To this end we have calculated the normal velocities at Ri for various values of the hole depths d and radii r. Figure 5 shows the computed velocities at Ri for d = 2.4, 4.8 and 7.2mm. The experimental result for the true value oi d — 4.8(mm), marked 'mRi\ is also shown in this figure. Figure 6 presents similar results for r == 5, 10 and 15mm, together with the experimental result with the true value of r = lO(mm), also marked 'mRi . These results tell that the scattered waves from the defect are sufficiently sensitive to the geometry of the defect.

Figure 5: Depth dependence of the computed velocity at Ri

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T. Kanbayashi et al.

CONCLUSION In this paper we could confirm the accuracy of the solution of the inverse problem of determining the elastic wave produced by an ultrasonic transducer from laser velocimetry data. It is also shown that the velocity of the scattered wave is quite sensitive to the change of the geometry of the scatterer. From these results we may conclude that we are almost ready for the final objective of the present series of investigations, i.e. the determination of the geometry of the defects from the wave form measurements using laser velocimetry.

Figure 6: Radius dependence of the computed velocity at Ri

REFERENCES 1 Nishimura, N.(1997). In: Theoretical and Applied Mechanics, 46, pp.39-57, G.Yagawa and C.Miki (Eds). Hokusensha Publ., Tokyo. 2 Nishimura, N. and Kobayashi S. (1998). In: Inverse Problems in Engineering Mechanics, pp. 197-204, M. Tanaka and G.S. Dulikravich (Eds). Elsevier, Amsterdam. 3 Hansen, P.C.(1992). SIAM review Z4., 561.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

333

IDENTIFICATION OF ELASTODYNAMICS LOAD USING DRBEM AND DYNAMIC PROGRAMMING FILTER Masataka TANAKA and Wen CHEN Department of Mechanical Systems Engineering, Shinshu University, Wakasato 4-17-1, Nagano 3808553, Japan (E-mail: dtanaka®gipwc.shinshu-u.ac.jp; **[email protected]). ABSTRACT This paper presents the first coupling application of the dual reciprocity BEM (DRBEM) and dynamic progranmiing filter to inverse elastodynamic problem. The DRBEM is the only BEM method, which does not require domain discretization for general linear and nonlinear dynamic problems. Since the size of numerical discretization system has a great effect on the computing effort of recursive or iterative calculations of inverse analysis, the intrinsic boundary-only merit of the DRBEM causes a considerable computational saving. On the other hand, the strengths of the dynamic progranaming filter lie in its mathematical simplicity, easy to program and great flexibility in the type, number and locations of measurements and unknown inputs. The combination of these two techniques is therefore very attractive for the solution of practical inverse problems. In this study, dynamic load is estimated based on noisy measurements of velocity and displacement at very few locations. Numerical experiments involved with the periodic and Heaviside impact load are conducted to demonstrate the applicability, efficiency and simplicity of this strategy. The affect of noise level, regularization parameter, and measurement types on the estimation is also investigated. KEYWORDS Dual reciprocity BEM, dynamic programming filter, regularization, inverse elastodynamic analysis. INTRODUCTION In recent years inverse elastodynamic problems have received increasing attention due to a broad range of engineering necessity. In general, the solution of the inverse dynamic problem is a much more difficult task than the direct problem due to some degree of noise in the measurement data. In other words, the inverse solution is extremely sensitive to measurement errors, namely, the ill-posed nature of the inverse problem [1]. In compared with direct problems, research of inverse dynamic problems is much less reported in literature, especially for inverse elastodynamic problem [2]. There are several methods available now to stabilize and estimate the inverse solutions of dynamic problem [1]. Among them, the dynamic programming filter with regularization, introduced recently by Trujillo and Busby [2,3], is a very competitive technique. The strengths of this approach lies in the mathematical simpHcity, easy to program and its great flexibility in the type, number and location of measurements and unknown excitation sources. On the other hand, an appropriate numerical method is also required in inverse analysis to transfer the continuous models of various practical problems into discretization system. In recent years, the BEM has become increasingly popular in the numerical discretization of

334

M. Tanaka and W. Chen

dynamic partial differential systems occurring in many branches of science and engineering. Transformation of the domain integrals has been a central task in the BEM solution of such problems to preserve its boundary-only merit. There are several different approaches available now for this purpose. However, as was pointed out in [4,5], the dual reciprocity BEM (DRBEM) stands out the method of choice in engineering computations due to its ease of implementation, intrinsic boundary-only merit for general problems and strong flexibility of applying fundamental solutions. Much research has been reported in literature to apply the DRBEM to a variety of direct dynamic problems. In contrast, there has been only very limited amount of research carried out in the DRBEM analysis of inverse dynamic problems. It is also worth stressing that since the inverse analysis usually require recursive or iterative computation many times, the dimension of numerical analogous equations has an especially huge affect on the computing time and storage requirements. The boundary element method enjoys a far more saving in computer resources for inverse analysis in comparison with the domain-type method such as the standard FEM and FDM because the method produces a relatively much less size of numerical modeling for continuous system [6]. However, it is noted that the dynamic progranmiing filter so far is only applied to analyze some structural dynamic problems in the combination with the FEM [2,7]. It was also claimed in [4,5] that the DRBEM is the only BEM method, which does not require domain discretization for general linear and nonlinear problems, although the interior collocation points may be used to improve solution in some cases. The relatively small dimension of the DRBEM discretization equations is especially advantageous for the dynamic programming filter, since its computational effort increases quickly as the size of system equation increases. A combined use of the DRBEM and dynamic programming filter will be very attractive in terms of computational efficiency compared with the FEM and other BEM techniques. The purpose of this study is concerned with the estimation of the input force magnitude of elastodynamic problems by a combined use of the DRBEM and dynamic programming filter. As for the approximate method of time derivative, Zhong and Williams [8] recently presented a so-called precise integration method (PIM). The method is in fact equivalent to the exponential matrix approach used in [2]. The merit of the PIM over the latter is mathematically explicit and easy to use. In this study, we employ this technique to discretize temporal derivative. Nimierical experiments are plates subjected to in-plane periodic load. Dynamic input load is estimated based on noisy measurements of velocity or displacement at very few locations. A computer program generating random number is employed to yield random measurement errors. The exact displacement and velocity responses versus time can be contaminated with various amount of noise to better simulate real measurements. The detailed solution procedure is next explained and some conclusions are finally drawn based on the present work. The main points of the interest are to investigate the affect of noise level, regularization parameter, and measurement types on the estimation. To the authors' best knowledge, this is the fu*st attempt to use the DRBEM combined with dynamic programming filter to handle the inverse dynamic problem. NUMERICIAL MODELING OF PLATE ELASTIC WAVE The cantilever plate subjected to in-plain dynamic load is often used as a benchmark problem in the BEM analysis [4] due to the availability of its analytical solution. In this study, we choose it as the numerical examples. The equation describing a wave propagating through an elastic medium is given by

Identification of elastodynamics load

335

Measurement points

Fig. l.DRBEM discretization and measurement points of a square plate

(1) subjected to the initial conditions (2a) (2b)

M(JC,0) = 0 , M(JC,0) = 0 ,

and the displacement and traction boundary conditions u(Xyt) = ^y

T{x,t) = P,

(3a) (3b) (3c)

^1=1,

jci=0,

where c denotes the wave velocity. P is the external plain traction as shown in Fig. 1. Note that all variables are dimensionless in this study. In terms of BEM, Eq. (1) is weighted by the fundamental solution u* of Laplace operator. By employing Green's second identity, we have diUi+ l^{T*u-uT)dr

=

1 5'M * -j^—-^udQ.,

(4)

where subscript / denotes the source point, T* = du*/dn, n is the unit outward normal; and d- = J 5(^,x)JQ. r means the boundary of plate. The essence of the dual reciprocity BEM is to transform the domain integral on the right-hand side of Eq. (4) by a set of coordinate function/(x) (5)

336

M. Tanaka and W. Chen

where the superimposed dot represents the time derivative of the second order, a^ are unknown functions of time, and A'^ and L are the numbers of the boundary and selected internal nodes, respectively. After some inferences, the DRBEM formulation is given by 4«, + l{ru

- uT)dr = J^d^yfj + l{T'yf' -"V^]-^.

(6)

where 7]^ = dxf/^ /dn, functions ip^(x) are linked with the specified coordinate functions/. The coordinate functions presented in [9] are applied in this study. In this study, the linear element (A r=OA) is employed, and one internal point is placed in the interior domain as shown in Fig. 1. Since displacement boundary conditions are involved in the cantilever plate, Eq. (6) is a differential algebraic system. By using an approach of matrix partition [4], the DRBEM formulation can be reduced to (7)

mu-^ku = f,

where m and k are respectively mass and stif&iess matrices. In what follows, we use the precise integration method to approximate time derivative. The above equation (7) can further be restated as the first-order system, namely, v = //v + r,

(8)

in which r= , / . "{:}• "=[::} -{:}

(9)

/ is the unit matrix. The key step in the PIM is to accurately evaluate the exponential matrix (10)

T = exp{H^T).

By using a recurrence procedure, the exponential matrix T calculated by the P M can possess the highest accuracy of a digital computer. The general solution v to Eq. (8) is given by V.,, = r[v. + H-% + H-'r)\ - H-%, where r^ = (r^^j -

/})/T.

+ H-\],

(11)

Assuming that the outer forcing terms vary linearly within time step

[t^,tj+i], Eq. (11) can be restated as v.,,=rv. + Pr.+ D(r.,,-r.),

(12)

where P = {T-I)H~\ D = {P-r)H~\ The above equation (12) is the numerical discretization equation of the plate elastodynamic wave problem. INVERSE PROBLEMS SOLUTION Trujillo and Busby [2] pointed out that the first-order regularization generally performs better

337

Identification of elastodynamics load than the zero-order one for the dynamic problems. The first-order regularization formulation for elastodynamic equation (12) is given by Z,,, = Rz,^Gq., \T where ^ , = r . , , - r . , R = \

P]

(13)

\D\ fv. I, ^ = L L z,=\

q is actually the first order derivative of the forcing term. The state vector z now includes the forcing term. The essence of the dynamic programming filter [2,3] is to formulate an optimal control problem, namely, Es{Z'
(14)

i=l

where {x,y) denotes the inner product of two vectors, N is the number of measurements, flf/denote the measurement data, and A is the weighting matrix and chosen as the identity matrix in this study. J, represent the state variable corresponding to the measurements. B is the Tikhonov regularization parameter. The L-curve method and generalized cross validation are two approaches in use for selecting optimal regularization parameter. In this study, we apply the former, d can be related to the state variable z by d^^Qzj.

(15)

By applying the least-squares criteria and dynamic programming principle to Eq. (14), we can get two sets of forward and backward recurrence formulas, respectively. The first step in computation is backward recurrence, namely, D^=[2B + 2G'Efi)'\ F„=2GX, 4_, = Q'AQ^R'[E^ - FjD^Fj2)R, s„_, = -IQ'Adl,

+ R'{1 - FjD„G)s„.

(16a) (16b) (16c) (16d)

The above equations are solved with initial conditions starting at the end point n=N E^ = QAQ\ s^=-2Q'Ad;.

(17a) (17b)

Note that all vectors D„G\ and matrices DJ^^ should be stored during the backward sweep. The forward solution is then calculated by the recurrence formulas ^„_,=-D„G\-D„F„/?z

(18)

and Eq. (13) from w=l. One can find from the above recursive formulas that the dynamic programming filter is mathematically simple and easy-to-program. Also the method is not restricted to the numbers and locations of measurements and unknown input terms.

338

M. Tanaka and W. Chen

NUMERICAL RESULTS AND DISCUSSIONS

Fig. 2. Periodic load estimation based on velocity measurement of 10 percent noise level at one point C {B~6A) 3 2.5 ^—s o

0^

pL,

1

o

0.5 0

T3 (U

OH

Estimated

15


P

- Exact

2

-0.5

-1

-1.5

I

4

I

I

I

I

I

6 Time (ct)

I

I

8

I

I

10

12

Fig. 3. Periodic load estimation based on velocity measurements of 10 percent noise level at two points A and C (5=17.6) In this study, the plates subjected to the harmonic and Heaviside impact load are tested. The measured data are artificially produced by corrupting the exact displacement or velocity history at some sample points with different degree of noise, namely, < = c^+e,,

(19)

where Cj is the exact response of velocity or displacement, d* represents the corresponding contaminated data and is considered as noisy measurements, f j is the added noise generated by

339

Identification of elastodynamics load

£.=M»A*(y.-0.5),

(20)

where A is the peak velocity or displacement value of the forced vibration at measurement points, Tj is the normally-distributed random number over the interval [0,1] with zero mean generated by a computer routine [10]. ju is the noise percentage degree of the amplitude. juA actually denotes the given standard variance of random measurement errors. It is noted that dimensionless time step size cAt =0.1 is employed for all results discussed below. The first numerical experiments are concerned with the periodic load. The recognized timeforce curves using one-point and two-point velocity measurements of 10% noise level are respectively illustrated in Figs. 2 and 3. It is seen from Fig. 2 that except for an apparent disagreement in the closing time range, the estimation using a single point measurement agrees very well with the true input force history. In contrast, Fig. 3 shows that the prediction based on two-point velocity measurements is accurate in the whole time range. To provide more insights into the affect of measurements on estimation. Fig. 4 shows the recognized periodic force-time curve by using 20% noisy measurements of displacement at points A and C. It is found that the estimation encounters heavier loss of accuracy again in the closing range as in the previous cases using one-point measurement, although overall estimation is very smooth. In contrast. Fig. 5 reveals that the prediction based on two-point velocity measurements of 20% noisy level remains accurate overall with some amplitude attenuation at the last phase. Therefore, in the case of periodic load, the estimation based on the two-point velocity measurements always works better and is preferred. To demonstrate the ability of the present inverse method to the heavy noise measurements, the noise level is raised to 40%. The estimation using two-point velocity measurements is illustrated in Fig. 6. It is observed that the prediction encounters visible amplitude attenuation. This is because the larger noise level degrades the estimation. However, the overall estimate still well reflects the true nature of periodic load.

^ o e? Cn u o ^3 O

fe

0^

3 2.5 ? 1.5 1 0.5 0 -0.5 -1 -1.5 4

6 8 Time (ct)

10

12

Fig. 4. Periodic load estimation based on displacement measurements of 20 percent noise level at two distinct points A and C (5=19.4)

M Tanaka and W. Chen

340

OH

OH

10

12

Fig. 5. Periodic load estimation based on velocity measurements of 20 percent noise level at two distinct points A and C (B=23.2)

PH

o OH

Fig. 6. Periodic load estimation based on velocity measurements of 40 percent noise level at two distinct points A and C (5=105.5) We also tested the present approach to the estimation of the Heaviside impact. Figs. 7-9 illustrate the predictions based on the one-point or two-point velocity and displacement measurements of 20% noisy level. The measurement errors cause some oscillations in the estimated force curves. However, the overall agreement is pretty good. It is noted that the accuracy performances of three estimations are nearly the same. These results verified the accuracy, efficiency and robustness of the present strategy for impact load where high frequency components play an important role. It is observed that unlike the previous harmonic load case, the measurement type and number have no evident affect on the estimation accuracy of the Heaviside load.

341

Identification of elastodynamics load

^ &H (u p

t

- Exact 1.5

" Estimated

1 h'-A^v'^-Vx^A^^i.. 0.5

•

2

'

4

•

'

•

'

•

•

•

'

•

6 8 Time (ct)

10

12

Fig. 7. Heaviside impact force estimation based on velocity measurement of 20 percent noise level at one point C (5=38.4)

• Exact 1.5

i

^

" Estimated

1 0.5

4

6 Time (ct)

10

12

Fig. 8. Heaviside impact force estimation based on displacement measurement of 20 percent noise level at one point C (5=40.8)

^ ^

U ^

- Exact 1.5

'Estimated

1 l/^A^/^ V^'^ V ^ W - " ^ - S / " s ^ 0.5

•

0

•

•

'

•

2

•

.

4

.

I

6 8 Time (ct)

•y

l

^••"^w^-j"'^

l

10

12

Fig. 9. Heaviside impact force estimation based on displacement and velocity measurements of 20 percent noise level at two distinct points A and C (5=43.4)

342

M Tanaka and W. Chen

CONCLUDING REMARKS The foregoing numerical experiments demonstrated that the DRBEM in conjunction with the dynamic programming filter is an accurate, robust and computationally efficient methodology to identify input load based on velocity and displacement measurements of different noise level. It was found that the present approach is insensitive to measurement errors and can give good estimation even using heavily noisy data. For the periodic load, the velocity measurements in two distinct points produce better prediction in general. Also, it is noted that the performances are not very sensitive to regularization parameter. The present combined approach is mathematically simple and easy to computer programming. The dynamic programming filter is found to satisfy the criteria for a competitive inverse method proposed by Beck et al. [11], while the DRBEM is a powerful technique to transfer a variety of continuous dynamic problems to discrete systems. Therefore, the coupling application of these two methods should be extremely promising for practical inverse elastodynamic analysis. ACKNOWLEDGEMENTS: This work was carried out as a part of the research program supported by the Japan Society for the Promotion of Science. Additional financial support was provided as Grant-in-Aid for JSPS fellows by the Ministry of Education, Science, Sports and Culture, Japan. REFERENCES 1. Kurpisz, K. and Nowak, A.J. (1995) Inverse Thermal Problems. Comp. Mech. Publ., Southampton, UK. 2. Trujillo, D.M. and Busby, H.R. (1997) Practical Inverse Analysis in Engineering. CRC Press, New York. 3. Trujillo, D.M. and Busby, H.R. (1983) ASMEJ. Dynamic Systems, Meas. Contr. 105, 200. 4. Partridge, P.W., Brebbia, C.A. and Wrobel, L.W. (1992) The Dual Reciprocity Boundary Element Method. Comput. Mech. Publ., Southampton, UK. 5. Katsikadelis, J.T. and Nerantzaki, M.S. (1999) Engineering Analysis with Boundary Elements, 23, 365. 6. Ingham, D.B. (1997) In: Boundary Integral Formulations for Inverse Analysis, pp. 1-33, D.B. Ingham and L.C. Wrobel (Eds.), Comput. Mech. Publ., Southampton, UK. 7. Busby, H.R. and Trujillo, D.M. (1987) Computers & Structures, 25-1, 109. 8. Zhong, W.X. and Williams, F.W. (1994) J. Mech. Engng. ScL, Part C, 208-6, 427. 9. Wrobel, L.C. and Brebbia, C. A. (1987) Computer Methods in Applied Mechanics and Engineering, 65, 147. 10. Press, W.H., Teukolsky, S.A., Vetteriing, W.T. and Flannery, B.P. (1992) Numerical Recipes in Fortran. Cambridge University Press. 11. Beck, J.v., Blackwell, B. and St.Clair, C.R. (1985) Inverse Heat Conduction: Ill-Posed Problems. Wiley Intersc, New York.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

343

A GEOMETRICAL DESIGN TECHNIQUE FOR IMPACT PISTONS BASED ON STRESS WAVEFORMS

LIU DESHUN CHEN ANHUA ZHU PINGYU Department of Mechanical Engineering, Xiangtan Polytechnic University 411201 Hunan, China

ABSTRACT Based on impact mechanics model of piston, a geometrical design technique of impact pistons is proposed by using wave mechanics and characteristic line theory, which the geometry of an impact piston can be determined from the stress wave given. Experimental resuhs of impact show that the inverse design method is valid. And the inverse design method of pistons is used to measure the dynamic stress—strain curves. KEYWORDS Impact, piston, wave mechanics, inverse design, stress waveform INTRODUCTION The essential problem in the design of a penetration device, such as a percussive drill or a pile driver to be applied to rock or soil penetration, is that of transmitting the energy produced by rapidly repeated impacts of a piston on one end of a long, slender drill rod or pile, and effectively vitalizing this energy to shatter rock or soil at the other end of the rod or pile. Since the 1950s, the importance of designing highly efficient penetration devices has increased. Firstly, some investigators have built the better understanding of the bit-rock interaction and rock destruction[ 1,2,3]. Then, great efforts have been spent on theoretical studies and numerical simulation of the efficiency of the penetration devices[4,5]. It has been generally believed that the efficiency is determined by the stress waveform generated in the rod by impact of the piston and the control and design of this waveform is of great signification from the viewpoints of both the penetration performance and the life of the devices. Based on the above knowledge, the common procedure of the design for the piston is: (i) the initial geometrical shape of a piston is given; (ii) the efficiency and stress distribution of the penetration system which includes the piston are analysed and evaluated; (iii) the geometrical shape of the piston is changed, i.e.,a new design is formed. In fact, this design method is an analytical approach, interpreted as a direct problem. Its inverse problem, called as inverse design, in which the piston geometry is directly determined from the desired stress waveform, has left to solve. In this paper, based on impact mechanics model of piston, inverse design method of impact piston is proposed by using wave mechanics and characteristic line theory, which the geometry impact piston can be determined from the stress wave given. Then, inverse design method is used to design impact pistons in the measurement of the dynamic properties of rocks.

344

D. Liu, A. Chen and P. Zhu

MATHEMATICAL MODEL In common, the operation of the penetration system is considered as the case of a short piston, of length L, striking the end of a semi-infmitely long rod having the same cross-sectional area, which the piston is circular in cross section and any variation in shape consists of changes in diameter at various positions along its length. And it is assumed that a spring exists between the piston and the rod to simulate the impact contact of them. Based on one-dimensional stress wave theory, a mathematical model describing the impact of a piston on a semi-infinite rod follows.

ot

ax

pc A(x) — + — = 0 dx dt Boundary conditions : F{x = 0,/) = 0 F\t)/pcAJ Initial conditions: A(x) = 7rr\x)

—^^ F\t)

v{0< x < L,t = 0) = v^

— = k[v(x = L,t)-

(1)

= F{x = L,t) F(x>0,t

= 0) = 0

where, F is the normal force, positive in pressure, v is the particle velocity, positive in the direction of increasing x , c is density and elastic wave speed of piston and rod. A(x), r(x) are cross-sectional area and diameter which are function of the coordinate x along the piston, AQ , r^ are cross-sectional area and diameter of rod. L is length of piston, v QIS impact velocity of piston. F'(t)is force wave produced by impact. For the simplicity, the above equations can be re-written in non-dimension . Let X = x/L, t = ct/L, v = vlv^, F = F/(pcA^vJ , k = kL/(pcAJ, F'= F*/(pcA.vJ

,

obtain dt

r^^

dx + ^ ^ 0 dx dt

F(jc = 0,0 = 0 ^ ^ X £ z M = ) t ( 7 ; - r ) F'=F(x dt v(0<xOj = 0) = 0 r = r{x)

= U)

^^^

0<x
For given piston geometry r ( x ) , we shall seek the stress waveform produced by the piston, this problem is called as direct problem. Therefore, we pose the inverse design problem: Given F*(0,find r(x)such that the piston with geometry r(x) shall produces the force wave F* (t) when it impacts on the given rod. INVERSE DESIGN METHOD As a numerical analysis method, characteristic line method is usually used to solve the wave

A geometrical design technique for impact pistons

345

equation According to characteristic line theory the following equation can be obtain d^ + dx = 0

/^x

Let a piston occupies x e [0, L], and consists of TV uniform segments with constant cross-sectional area and equal transit time l/n shown in Fig. 1.So, a characteristic line grid is established. We introduce three points in the grid, a,b,c.F.j,v.j are the normal force and partial velocity at the point a{ij) ,and similarly for the other points. For the three points, the following equations from eq.(3) can be derived.

Fig.l. Impact piston characteristic line grid

(4)

Therefore, =

[F,,_,+F,^+r;iv,^-v,^_,)]/2

(5)

Correspondingly, the boundary conditions and initial conditions in eq.(2) are changed. ^«,=0 F\jln) F\jln)

(6) (7) (8)

= F-=F„^ = F-=F,.

^,0=1,^0=0

(9)

r,=r{i/n)

(10)

Based on stress wave theory and initial conditions, we can gain F „ .

(11)

From boundary conditions, the following equations can be obtained F„^=F;=F'U/n) K,=(F„^-F,.,.,)n/k

+ F„^

(12)

Inverse design of the impact piston, in which the piston geometry r(x) can be determined directly from the given stress waveform F'(t), may be summarized: (i) Putting the given stress

346

D. Liu, A. Chen and P. Zhu

waveform F* in equations(12) , (F.,^'^,.) can be obtained; (ii)Calculating radius of the piston (x = l) r, from equation (7); (iii) Again putting the value of r,, {F^-,v^.) in equations (5) ,(F_,^,7;,,_,^) can be obtained. By using this procedure repeatedly, the radius of the piston geometry r ( / = 0,...,«) can be determined successively from i = n to / = 0 . Finally, we obtain the radius of piston as functions of the coordinate x along the piston. DESIGN EXAMPLES AND EXPERIMENTAL VERIFICATION The experimental set—up is shown in Fig.2, which is called as the Split Hopinson Pressure Bar(SHPB). Cylindrical steel pistons accelerated by means of an airgun and impacted on steel rod with length 130 mm and diameter 22 mm. A specimen was located between a input rod and a output rod with the same geometry. The rods were guided by low—friction slide bearings so that it is free to move axially. A damping device at the end of the rod served to absorb the energy and stop the motion after a text. One pair of diametrically opposite and axially oriented strain gauges is attached the input rod at a distance of 500 mm from the impact end in order to measure the strains ^, (t), which was used to calculate the stress waveforms. Another similar pair of strain gauges was attached to the output rod at a distance of 500 mm from the specimen in order to measure the strains, (0. which was used to determine the parameters of the specimens. The gauges for each pair were connected to a bridge amplifier in opposite branches, so that contributions from small accidental bending strain were suppressed. The strain signals were recorded by means of a two— channel digital oscilloscope with a sampling interval of\/js. The recorded signals were transferred to a computer for evaluation.

Fig.2. Experimental set-up. Dimensions are in mm.

Using a computer program, called as PID, several pistons are designed and made to prove inverse design valid. Two steel pistons shown in Fig.3 were tested. The length of a piston was two of the another. The result show that the practical waveforms which the pistons designed produce are in good agreement with the given stress waveforms. It must be mentioned that the stress waveform used to determine the piston geometry is only a part of the waveform produced by the impact. This part for r =0 to 1 ( 2 1 / c ) can determine the whole piston geometry, the other of the stress waveform is its consequence. Therefore, it may be said that the part for /=0 to 1 {2Lie) includes all information about the piston geometry and determines the other.

A geometrical design technique for impact pistons

ZM

0.8 p^

\L 44^10

TV oqjerimental curve

0.8

§

0.8 0.8

-

r/

""^^^>';iJheoretical curve

"//

(a)

©qDerimental curve theoretical curve

(b) Fig.3. Impact pistons and their stress waveforms

The Split Hopkinson Pressure Bar (SHPB) has been widely used to measure the properties of solid materials, such as metals and rock. The loading stress waveform in the SHPB is initiated by impact between a rod and a cylinder hammer of the same diameter as the rod, which is a rectangle loading stress wave. Numerous investigators have found that oscillations exist in the reconstituted stress-strain curves obtained by the SHPB subjected to the rectangle loading wave [6,7]. Because the existence of oscillation it becomes very difficult to get the exact dynamic parameters of the materials. Therefore, we tried to use an approaching sine loading stress wave shown in Fig.3 in order to eliminate oscillations in the measurement of the dynamic properties of rocks. A series of experiments were performed to measure the dynamic stress-strain curves of many kinds of rocks. Several experimental results for marble specimens subjected to rectangle loading stress wave and approaching sine loading stress wave were illustrated in Fig.4 (R represents rectangle wave and S represents approaching sine wave in figures).

10

15

STRAIN(10"')

Fig.4. the dynamic stress—strain curves of marble specimens

348

D. Liu, A. Chen and P. Zhu

CONCLUSIONS Inverse design of the impact piston makes it possible for designer to produce the desired waveform. PID provides a basis for the determination of efficient penetration devices operating by mechanical impact, such as rock drill or pile driver. Similar studies have been extended to investigate the behaviors of the materials loaded by many kinds of stress waveforms, including studies on the reasonable loading waveform to measure the stress—strain curves to effectively eliminate oscillations. Acknowledgements This study is supported financially by the National Natural Science Foundation of China and Natural science fund of Hunan REFERENCES 1. 2. 3. 4. 6. 7.

Dutta,P.K.(1968), Int. J. Rock Mech. Min. Sci.,5,501-508 Fairhurst,C.( 1961), Mine & Quarry Engineering,27,169-178 Lundberg,B.(1973), Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 10,401-409 Lundberg,B.(1985), Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 22(4),237-249 Seisuke Okubo, Yuichi Nishimatso (1991), Shigen-to-Sozai, 107 (4), 209-214 Liu Deshun, Li Xibing(1996), Trans. Nonferrous Met. Soc. China,6(2),144-147 Liu Deshun, Li Xibing(1999), Mechanical Impact Dynamics,Science Press,China

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

349

INVERSION OF STRESS AND CONSTITUTIVE RELATIONS USING STRAIN DATA FOR JAPAN ISLANDS

M. HORI Earthquake Research Institute, University of Tokyo Yayoi, Bunkyo, Tokyo 113-0032, Japan

ABSTRACT A new inversion method of identifying stress is applied to Japan Islands to estimate regional stress distribution. This method finds a stress distribution from a strain distribution, by solving boundary value problems. Numerical simulation and model experiment verify the validity of the method. The deformation increment of Japan Islands is measured by a global positioning system (GPS) network. We carry out the numerical computation of estimating regional stress increment distribution for Japan Islands, and obtain a self-equilibrating stress increment field. The regional constitutive relations are then estimated from the relation between the computed stress increment and the measured strain increment. Some discussions are made on the usefulness and limitation of the present inversion method. KEYWORDS Stress inversion, constitutive relation estimate, GPS, crust deformation. INTRODUCTION A dense array of global positioning system (GPS) with around one thousand stations has been installed in Japan, in order to continuously monitor crust deformation in highest spatial resolution. While this system has its own significance, measured data of displacement or strain increment can be used for the earthquake prediction. If regional stress is estimated from the GPS data, it will provide valuable information for the occurrence of earthquakes since they are caused by the failure of faults which is mainly driven by stress. The identification of regional stress and constitutive relations is thus needed, even though it is not easy to measure them due to the large length scale as well as regional heterogeneity, anisotropy, and in-elastic behavior. The identification of local material properties is a common concern for various fields of engineering [1,2]. This is because the advanced computational method is able to reproduce and to predict, say, failure processes, if accurate constitutive relations are given. Recently, the author and his colleagues [3,4] proposed a new inversion method that estimates a stress field from a strain field for a body consisting of materials with unknown properties. The key feature of the method is that it does not need a parameterization of constitutive relations like ordinary methods. The method directly computes stress accumulated in linear or nonlinear materials, although a target body must be in two-dimensional plane stress or strain state.

350

M. Hori

boundary dS

thin sample V

surface S

Figure 1: Body in Plane Stress State

We seek to identify a distribution of regional stress increment in Japan, applying the new inversion method to the GPS data from which a distribution of strain increment is obtained [5]. Regional constitutive relations are then estimated by comparing the stress increment with the measured strain increment. In this paper, we first present the formulation of the inversion method. Results of a numerical simulation and a model experiment are shown in Section 3 to demonstrate the validity of the method. In Section 4, results of the stress increment inversion of Japan Islands are summarized. The estimate of regional constitutive relations is also presented, with some discussion on the limitation and modification of the inversion method. In this paper, index notation is used for vector and tensor quantities, with a Cartesian coordinate system, (xi and X2 axes are on the plane of interest and the Xs-axis is normal to it. Summation convention is employed, and an index following comma stands for the partial diflPerentiation with respect to the corresponding coordinate. FORMULATION OF INVERSION METHOD We briefly summarize the new inversion method proposed by the author and the colleagues [3,4]. Let y be a thin body consisting of an unknown elasto-plastic material. The body is in plane stress state, and the surface and its boundary are denoted by S and dS, respectively. Displacement and traction, Ui and U {i = 1,2), are measured on S and along dS, respectively; see Fig. 1. First, we consider three in-plane strain increment components which are determined from Ui. They are decomposed into elastic and plastic parts, e^j = e^j + ef^, where superscript e or p stands for the elastic or plastic part. It is assumed that the elastic tensor Cijki is known, and the stress increment &ij is given as aij — Cijkie\i. Furthermore, it is assumed that the plastic strain increment is in-compressive,

er, = 0.

(1)

Here, the index i takes 1 or 2; the extension to a case when i ranges from 1 to 3 is possible though the expression becomes less transparent. For any plastic constitutive relations, the three in-plane stress components satisfy the two

Inversion of stress and constituve relations

351

equations of equilibrium in the Xi and X2 direction. Their increments, therefore, satisfy (T,,, = 0

0' = 1,2).

(2)

Taking advantage of cjij = Cijki[^ki — ^\i)^ we rewrite Eq. (2) in terms of the total and plastic strain increments, e^j and e?. For instance, the left side of Eq. (2) becomes [cijkihi),i - {cijki^li),i' To simplify expression, we define &fj = Cij^ihi and a*- = -Cijkicli, calling afj apparent stress and a*j eigen-stress,^ respectively. Making use of Eq. (1) and assuming the elastic isotropy, we can derive the following equations for crl^{= —^22) and 0-1%: ^11,2 +

^ 12,2

(3)

^12,2 5

*

<^12,1

11,2

(4)

^2

It follows from aij = afj -h a*^ that the traction along the boundary gives the following boundary conditions: ni&l^ + ^20-12 = ^1 - rii&l^ - n2&i2,

(5)

nicri2 - n2cr*^ = ^2 - ni&l2 ~ ^20^22^

(6)

where n^ is the unit normal. As is seen, Eqs. (3) and (4) together with Eqs. (5) and (6) form a set of boundary value problems for &l^ and al2The above boundary value problems are easily decoupled even though the resulting governing equations become second-order partial differential equations. The decoupled problems are ^ii,u = - ^ n , i i + ^^2,22

on 5,

^11 ~ ^1^1 ~ ^2^2 — ( ^ 1 ^ 1 1 ~ ^ 2 ^ 2 2 )

along

^I2,ii

— ^12,11

^12,22 ~ ^11,11

^22,22

(7) dS,

^ ^ O,

(8) &I2 = ^ i 4 + ^2^1 — (^1^2(^11 + ^22) + ^12) along dS. It should be emphasized that the eigen-stress can be obtained by solving these linear boundary value problems, even though the plastic constitutive relations are non-linear. Indeed, once al^ and &I2 are given, the three stress increments are ^11 C^22 . ^12 _

=

\^ti] ^2%

L'^12 J

+

r <^ii 1 -^22

(9)

^12

NUMERICAL SIMULATION AND MODEL EXPERMENTS In order to verify the validity of the new inversion method, we carried out a numerical simulation and a model experiment. The results of the inversion are briefly summarized in this section; see [4] for a detailed explanation. ^See [6,7] for eigen-stress or polarized stress which is used in micromechanics .

352

M. Hori

Numerical simulation The target of the numerical simulation is a rectangular sample of an elasto-plastic material. The isotropic elasticity with Young's modulus E and Poisson's ratio u is known, but the plasticity that is actually governed by a von-Mises yield function is not known except that Eq. (1) holds for the plastic strain increment. The sample is subjected to a vertical load applied to a part of the upper edge, with the bottom edge being fixed and the right and left edges being traction free; see Fig. 2. Displacement and traction are first computed, and they are used as input data of the inversion. A finite element method is applied for both computations. For each increment of the applied load, the two boundary value problems, Eqs. (7) and (8), are solved and the stress increment is computed by using Eq. (9). A typical example of the stress increment inversion is shown in Fig. 3 where a) and b) are the exact and distributed force

Figure 2: Model of Material Sample

0.001 0 -0.001

a) exact stress increment (da, 2)

b) predicted stress increment (da,j)

Figure 3: Comparison of Stress Distribution

0.001

0.0014

> \

ellipse of yield surface

a)exact distribution of plastic strain increment

magnitude of increment 0.005

ellipse of yield surface b) predicted distribution of plastic strain increment

Figure 4: Comparison of Yield Locus

353

Inversion of stress and constituve relations

computed distribution of cri2. The agreement is satisfactory; in particular, it is seen that the stress concentration near the loading points is well reproduced in the inversion. Using the computed stress and the measured strain, we seek to identify the plastic constitutive relations. A yield locus is shown in Fig. 4); a) is the exact plastic strain increment plotted in the principal stress plane, showing an ellipse that corresponds to the von-Mises yield surface; and b) is the computed plastic strain rate. Although the ellipse in Fig. 4b) is less clear, we can evaluate the yield stress using the ellipse. The estimated value is 0.001 ± 0.0002[MPa], which is close to the exact value of 0.001[MPa]. Model Experiment A model experiment uses a gelatin sample, and the major objective is to estimate local constitutive relations near a crack; see Fig. 5. The sample is disk-shaped, with radius 30[cm] and thickness l[cm], and the torsional loading is applied at the bottom. Displacement on the surface is measured by image analysis that chases markers put on a rectangular region of 7[cm] by 8[cm] to form a gird pattern of 0.5[cm] distance using a CCD camera. A crack appears in this rectangular region. Traction along the boundary of the rectangular domain is set zero, since overall applied load does not increase significantly during the formation of cracks. The elastic properties are measured from a simple shear test. We compute the stress increment using the measured strain increment, and estimate the local constitutive relations. As a typical example, Fig. 6 plots relations between shear strain and stress components (o"i2-ei2) for several points near a crack. Also, Fig. 7 plots relations between the maximum shear strain and stress (7-T) . While the relations between

Riedel shears Torsional Shearing

Figure 5: Schematic View of Model Experiment

0

8,,

Figure 6: Relations between eu and du

0.1

0.2

0.3

Figure 7: Relations between 7 and r

354

M Hori

strain and stress components differ point to point, a common relation is observed for the invariants; for points where the crack passes, the maximum shear strain increases even though the maximum stress attains an almost constant value; and for other elements, the principle stress is compression, and the maximum shear stress increases almost linearly. A fact that similar relations are obtained supports the validity of the proposed inversion method. APPLICATION OF INVERSION METHOD TO JAPAN ISLANDS We apply the new inversion method to Japan Islands, using displacement increment measured by the GPS array. It should be emphasized that, while the stress state of Japan Islands varies vertically, the stress increment during a period of the GPS array measurement is approximated to be in plane stress state. The least square prediction is applied to get rid of measurement noises from the GPS data and to obtain smooth strain increment; see [8]. In applying the new inversion method, attentions must be paid on the assumption of Eq. (1) and the boundary conditions, Eqs. (5) and (6). The assumption is valid if inelastic deformation of Japan Islands is mainly due to the sliding of faults that causes only shear deformation. The boundary traction needs to be estimated to accurately estimate regional stress. The effects of the boundary tractions, however, can be separated due to the linearity of the boundary value problem, Eqs. (7) and (8), and hence U — ^ \s used in the present analysis. Three preliminary checks are made in numerically solving the boundary value problems which use GPS data. The first check is the convergence of numerical solutions. In Fig. 8, respectively, the distribution of the maximum shear stress f is plotted where a) and b) use the discretization of A = 0.25 and 0.125 degree. As is seen, the convergence is satisfactory. Second, two domains of different configurations are computed to examine the effects of the boundary. It is shown in Fig. 9 that the effects of boundary on regional shear stress increment (0-12) are not that significant. Finally, the effects of the reference elasticity is studied; unlike the previous examples shown in Section 3, the elasticity of Japan cannot be measured. Two cases of Poisson's ratio, 1/ = 0.2 and 0.3, are shown in Fig. 10a) and b), respectively. It is seen that overall patterns of the maximum shear stress increment are similar to each other. Now, we compute the stress increment fully, using the element size of A = 0.125 degree, the configuration of Type 1 shown in Fig. 9a) and Poisson's ratio of v = 0.25. In Figs. 11 and 12 plotted are the distribution of the first invariant of strain and stress increment, 6 and a, and the maximum shear strain and stress increment, 7 and f, respectively. The difference between the strain and stress increment distributions clearly shows that, since the apparent stress increment afj = cij^iCki does not satisfy equilibrium, non-zero a*- is generated and produces self-equilibrating stress increment. Regional constitutive relations are obtained from the predicted strain increment and the measured stress increments. As an example, we plot in Fig. 13 the ratio of f/7, which is a measure of the shear stiffness, and the difference in the orientation of the maximum shear and stress increments, which corresponds to the anisotropy. Regional heterogeneity and anisotropy are seen from the

355

Inversion of stress and constituve relations figures.

a)A=0.5

c)A=0.125

Figure 8: Convergence of Numerical Solutions

130

135

a) type I

140

145

130

135

140

145

b) type 2

Figure 9: Effects of Boundary Conditions

Figure 10: Effects of Reference Elasticity In Figs. 11 and 13, high stress concentration near the boundary and some regions of quite large f / 7 are observed. These values are unrealistically high, and are probably caused by the zero boundary traction increment and the assumption of Eq. (1). Traction can be estimated by considering in-situ measurement of stress at several sites or by computing stress increment that is associated with the global crust movement. The assumption of Eq. (1) should be replaced with a more general one. Also, we have to consider the least square prediction of displacement increment which may underestimate the localized regional deformation. In the present boundary value problem, however, higher order derivatives of displacement increment are computed and hence a smoother distribution is required. To resolve these two problems, we propose several modification of the inversion method [9].

356

M. Hori

d)a

c)e

Figure 11: Distribution of e and a

c)Y

d)T

Figure 12: Distribution of 7 and r

d)e-<|)

C)T/Y

Figure 13: Regional Stiffness and Anisotropy

The assumption of Eq. (1) is valid for a material with homogeneous elasticity. Indeed, this leads to the bulk stiffness becomes uniform when Cijki is isotropic, i.e., a ^ e

0-11 -f 0-22 ^ e

^ f 1 + <^22 6

= C i l l l +C1122-

As shown in Fig. 13, Japan Inlands have regional heterogeneity, and hence using this assumption is not appropriate. When the bulk modulus is regarded as an unknown parameter, we need to modify the inversion method in order to efficiently find the parameter. A simple modification is that, instead of using eigenstress, we consider Airy's stress function, a, which produces in-plane self-equilibrating stress components, ^11 = a,22,

cr22 = a,ii,

and

ai2 = -a^u-

For simplicity, consider a case when the material is linearly elastic, isotropic, but heterogeneous. When strain is measured, we can derive the governing equation for Airy's function

Inversion of stress and constituve relations

357

a using an unknown bulk modulus /€, as a^a = i^ea. Boundary conditions of Airy's stress function are determined from boundary traction as well. Considering resultant force along the boundary, ri = J ti d^, we can derive riia^i = —nir2 + n2ri. Therefore, we can pose the following boundary value problem for Airy's stress function: a,ii + «,22 = ^(cu + 622)

on 5,

nitt^i + n2a,2 = —^1^2 + ^2^1 along dS.

(10)

The form of the boundary value problem for Airy's stress function is similar to those for the eigenstress, Eqs. (7) and (8). While we present only the formulation, this modified inversion method appears promising to study a case when a body is elastically heterogeneous, since the boundary value problem does not need higher order derivatives of measured displacement or strain; compare Eq. (7) or (8) with Eq. (10). We are now developing the modified inversion method which seeks the distribution of K, simultaneously. CONCLUDING REMARKS The results of preliminary checks and trial computation shown in the preceding section support the applicability of the new inversion method to Japan Islands. Since the new inversion method takes advantage of a fact that stress is in equilibrium, the distribution of the inverted stress increment is slightly different from the distribution of measured strain increment. While the further investigation is needed, these results indicate potential usefulness of applying the new inversion method to Japan Inlands. The results also show some drawbacks and limitation of the inversion method, such as the usage of smoothened strain data, the treatment of the boundary condition, and the strong assumption given by Eq. (1). To overcome these problems, we are now modifying the inverse method, formulating a boundary value problem for Airy's stress function, which is to be of a more suitable form for the stress inversion of the Japan inlands [9]. Also, we are planning to compare the estimated regional constitutive relations with known crust structures.

REFERENCES 1. Bui, H.D.: Inverse Problem in the mechanics of materials: An introduction, CRC Press, New York, 1994. 2. Tanaka M. and G.S. Dulikravich (eds.): Inverse Problems in Engineering Mechanics (Proceeding of International SyElsevier, New York, 1998. 3. Hori, M. and T. Kameda: Formulation of inverse problem of identifying material properties based on equivalent inclusion method, in Proceeding of International Symposium on Inverse Problems, (ed. by M. Tanaka and G.S. Dulikravich), Elsevier, New York, pp. 225-234, 1998. 4. Hori, M., T. Kameda, and N. Hosokawa: Formulation of identifying material property distribution based on equivalent inclusion method. Structural Eng./Earthquake Eng., JSCE, Vol. 16, No. 1, pp. 21-30, 1999. 5. Kato, T., G.S. El-Firky, E.N. Oware and S. Miyazaki: Crustal strains in the Japanese islands as deduced from dense GPS array, Geophys. Res. Lett., Vol. 25, pp. 3445-

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3448, 1998. 6. Nemat-Nasser, S. and M. Hori: Micromechanics: Overall Properties of Heterogeneous Materials, Second Edition, North-Holland, London, 1998. 7. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc, Vol. A241, pp. 376-396, 1957. 8. El-Fiky, G.S., and T. Kato: Interplate coupling in the Tohoku district, Japan, deduced from geodetic data inversion, J. Geophys. Res., Vol. 104, no. B9, pp. 2036120377, 1999. 9. Hori, M. and T. Kameda: Inversion of stress from strain without fully knowing constitutive relations, (in review process, 2000).

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

359

ESTIMATION OF PARAMETERS IN TANK MODEL ANALYSIS USING LEAST SQUARES OF RESIDUALS WITH CONSTRAINTS

M. KANOH and T. HOSOKAWA Department of Civil Engineering, Kyushu Sangyo University, Fukuoka 813-8503, Japan T. KUROKI Department of Civil Engineering, Fukuoka University, Fukuoka 814-0180, Japan ABSTRACT A tank-model simulation was proposed and developed by Sugawara, M. [1,2] and recently used to demonstrate how rainfall brings about flood, surface, medial, and base outflows to a river. We applied the tank model to Dam A of Fukuoka and its catchment area and investigated the question of how, even with the base outflow to the dam, enough water might be retained for 130,000 citizens for one month, even during droughts such as the one in 1994. In order to resolve such a problem, it was necessary to identify many parameters in the tank-model simulation, such as the coefficients of discharge at the outlet of each tank and the outlet elevations. To determine unknowns in the analysis of pipeline flow, Kanoh [3] proposed a method in which the sum of the constraints and the squares of the residuals are minimized using Lagrange's method of the indeterminate multiplier. In this paper, we apply the same methodology for identification of the parameters in the tank-model simulation. Here, the constraints are adopted to stabilize the solutions against the oscillation in the back-analysis procedure. On the other hand, the residuals are shown as the noises in the observation equation. The observation equations, which state the relationships between observed values and unknown values, are derived by the finite-difference method that is based on the water-mass balance in the tank model. The convergence, stability, and accuracy of the methods are numerically examined. KEYWORDS Tank model simulation, base outflow to a river, new water resource for water supply, coefficients of discharge, the heights of the outlets, constraints, Lagrange's method of the indeterminate multiplier INTRODUCTION We considered the water-mass balance in the whole basin of a river, where rainfall, evaporation, and stream flows are measured in the catchment area, as illustrated in Figure 1. Figure 2 shows the concept of a four-stage tank model that explains the water-mass balance in the field. We applied the tank model to a dam and its catchment area and investigated the quantity of flood.

M. Kanoh, T. Hosokawa and T. Kuroki

360

surface, medial, and base outflows to the dam. The base outflow could be regarded as a kind of groundwater with the potential to serve as a reserve water supply during periods of drought. GOVERNING EQUATIONS Using the tank model, the governing equations for four tanks are d(hj,j)/dt = R - Ev - a j . h^j - p^j. a 3j. (h^,- h^^) - Pso* ^so' (^TI" ^ J

' * ^^'^^ tank

(1),

d(hj,2)/^t = a^. h^, - a 2- h^2 - Pm2- «m2- (^12- ^Li)

• • s^^^^^ ^ ^

(2)'

d(h^)/dt-a,.h^-a3.h^ . p ^ . a ^ . ( h „ - h j

• -thirdtank

(3),

d(h,,)/dt = a 3 . h ^ - a , . h , , - p ^ . a ^ . h ^ ,

• -fourthtank

(4),

Qout = Psi-«si-(hTrhu) + Pso-«so-(hTrhJ + P ^ - a „ ^ - ( h T , - h J + P ^ 3 . a ^ 3 . ( h ^ - h J

where h^.^ ~ hj,4(hj. ^: k = 1, 2, 3, 4) are water depths of the first ~ fourth tanks, a^ ~ a^ (a ^ and a ^ ,ttgj,ttgQ,a are coefficients at the bottom and side outlets of each tank, and R and Ev are rainfall and evaporation amounts. Furthermore, Pj ~ p^ (PJ denote the step functions, Q^^^ is the total discharge that flows out fi-om all the tanks. aii^ %v ^so' ^n^'^ms' ^g ^fc thc dischargcs that evaporation rainfall flowout from the side outlets of each tank.

t

rainfall

First tank (surface outflow)

ttsO qso [ijl

CThLi ai

qi Second tank (medial outflow)

^

I ^m2 'qm2

±j2

a2 q2

Third tank (medial outflow)

^

|0^m3 qm3

^T3

hi3

JL q3

base outflow

Fig. 1 Outflow mechanism of whole basin of a river and the corresponding tank model

Fourth tank (base outflow)

%

11T4

q4

Fig. 2 Four-stage Tank Model simulation

361

Estimation of parameters in tank model analysis BOUNDARY CONDITION The step functions p^ ~ p^ (P^^) have the values as P,=^l in case h^^^^k ' Pk^^ i^^^ase \^<\^

: k=l,2,3,4

(6),

where h^^^ (h^^ ~ h^^) are the heights of the outlets of each tank. OBSERVATION EQUATION Regarding the identification of parameters (p) of the tank model, the observation equations that state the relations between observed and calculated values are described as [3,4] v = Hp-y

(7),

where v is the noise vector, H is the coefficient matrix of p, and y is the vectors consisting of the known terms. In equation (7) let p be p + Ap , we have V = HAp + (Hp-y)

or simply

v = HAp +L

(8),

where p and Ap describe the vectors consisting of the assumed values of p and the correction terms of p, respectively. H should satisfy the following relationship as H = [aQ/8pi 8Q/3p2 • • • 3Q/ap„]

(9),

where n is the number of the parameters. Here the residuals are shown as the noises in the observation equation. The unknown values of the observation equations are derived by the finite difference method that is based on the water-mass balance in the tank model. FINITE DIFFERENCE METHOD Adopting the center time-difference scheme to Equations (1) ~ (4), we obtain the following finite difference expressions: [h^,]"*' = [h,J°+(At/2).{[R -Ev - a, • h „ - P,,• a ,, • (h^,- h,,) - p,„. a^,- (h„- h^„)]"V [R -Ev - a,, h^, - Psi- a SI • (^TI- h j " Pso* "so" iK' Kdt) [ h g " ' = [h j"+(At/2>{[ a,. h,, -a ,• h „ - p ^ . a ^ . (h„- h J ] " ' + ^ [a,.h„-a,.h^2-p^.a^.(h„-hi^)]°}

(10),

(H)'

[ h „ ] " ' = [h„]''+(At/2>{[ a,, h ^ -a 3. h „ - p^,- a^,- (h„- h,3)]"'+ ^ [ ttj. h ^ - a 3. h „ - p^3. a ^ . (h„-h,3)]°} [h„]"' = [hg"+(At/2)-{[ a,. h „ - p^. a^. (h„- hjf*\

[ a,- h „ - p,. a^- (h„- h J]°} (13).

362

M. Kanoh, T. Hosokawa and T. Kuroki

Where At is the time differential and [h^J

stands for the value of h^^ on (n+l)«At time.

CONSTRAINTS In order to stabilize the solutions against the oscillation in the back analysis of the tank model, we consider the following constraints as (14).

g(p) = 0 Differentiating the Equation (14) with respect to p and linearizing it, we have E^Ap + g(p)=0 where E

(15),

is the transposed coefficient matrix of E, and E should satisfy Equation (16). E=[dg/dpi ag/dP2 • • • ^g/^Pn]

(1^)-

DETERMINATION OF Ap Solving the observation Equation (7) with the constraints shown as Equation (14), we determined Ap. Using Lagrange's method of the indeterminate muhiplier, we tried to minimize ^ described as ^ = (HAp + D '

W(H/^

, + L) + 2 KT' /^T ( E ' Ap + g(p))

(17),

where W is the weight matrix and K is the indeterminate coefficient vector. Differentiating Equation (17) with respect to Ap and letting it be equal to zero, we have

d<|) / dAp = 2(HAp + L)'^ PH + 2K'^E'^ = 0

(18).

Equations (15) and (16) yield the following simultaneous equations: N E"^

El Apl 0H K J *

g(F) J

=0

(19),

where N = ffAVH. As the coefficient matrix of Equation (19) is regular, we can put the inverse matrix of it as "N E'^

E 0

-1

M [RT

Rl SJ

(20).

363

Estimation of parameters in tank model analysis

We then obtained the inverse matrix, and we can determined Ap as

Ap = - | ( N + EE'^)"^ - E ( E ' ^ E E ' ^ E ) " ^ E ' ^ | H ' ^ W L - E(E^E)"V-gCp)

(21),

where M=(N+ETE)-I - ECE^EE^E)-^E^ is the Moore-Penrose inverse matrix of N [5]. WEIGHT MATRIX W In this paper, the weight matrix W denotes the weights of observation values of streamflowsand precipitation in the catchment area. The quantity and quality of the observation values in technically advanced countries can be regarded almost in the same degree by writing the unit matrix (I) for W. In cases where the observation values may be unreliable or sometimes missing, the weight matrix W should be used to amend them. NUMERICAL EXAMPLES We applied the tank model to Dam A of Fukuoka and its catchment area and investigated the quantity offlood,surface, medial, and base outflows to the dam. We expected the base outflow to be enough to supply more than 100,000 citizens for a month, even during periods of drought. Figure 3 illustrates the observed values of rainfalls (R) and the total discharge (Q^^^) in the Dam A catchment. The numerical examples concern the identification of the 14 tank-model parameters. In tank model simulation, at first the water depths of the first ~ fourth tanks (h^^ ~ h^^) are analyzed by the finite difference method described as Equations (10)~(13), and then the total discharge Q^^^ is calculated using Equation (5). Hereafter we express the calculate value of Q^^^ as Q^ . In order to obtain Q^^ of the Dam A, it is necessary to identify fourteen parameters (p) in the governing Equations (1)~(4). Solving the observation equations with the constraints by use of Lagrange's method of the indeterminate multiplier, the correction term (Ap) was determined, and then we investigated the unknown values of parameters (p) using the inverse analysis. (mm/day) 0

(m /sec)

8 7 6 5

Qobs I

Qobs 3 i t t

300

2 1

J^ 0

100

200

300 (day) 400

500 549

Fig. 3 Observed values of rainfalls and the total discharge (Q . ) in the catchment of Dam A

364

M. Kanoh, T. Hosokawa and T. Kuroki

RESULTS AND DISCUSSIONS Order ofIdentifying Parameters and Examples of Constraints For stabilizing the solutions against the oscillation in the back analysis procedure, we adopted the constraints and the appropriate order of identifying the parameters. Regarding the coefficient matrix H of Equation (9) that is obtained from the derivatives (^Qcal ^ ^P), there are three kinds of characteristic regarding stability at the back analysis procedure. The first kind is forthe identification of the five parameters a^, a^^, a^, a^^ a , that may be most stable because H has the term of bj.j ~ h^^. The second kind is for the identification of the five parameters h^^^, \^, h^^, \^, h^, that may be stable because H has not the term of hj,, ~ h^^but, instead, the term of a^, a^j, a^, a^3, a . The third kind is for the identification of the four parameters a^, a^, a^, a^, that may be unstable if the parameters a _ ~ a and a^ ~ a are not yet identified because H can not be obtained directly from Equation (5).However, obtaining the values of hj.j ~ lij,^ from Equations (10) ~ (13) and substituting the values for Equation (5), H can be expressed. We propose this methodology to identify theparameters in order of the stability. The order of identifying the parameters may be considered to be a kind of constraint in a broad sense. In tank model analysis, the constraints fill an important role, since the solution here is more likely to be diverse or oscillating than in the analysis of flow in a distribution pipe network if an order of identifying the parameters is not adopted. In this paper we adopt the order of identifying the parameters and examine some constraints numerically so that the back analysis procedure of the methods is stable and convergent. The constraints contain two kinds of conditions. The first is that the heights of side outlets of each tank (h^^^ ~ h^^) should not be negative. The second is that the pivot values of the coefficient matrix of the simultaneous equations should be larger than 0.0001 in determining Ap. According to our experiences with back analysis, we can say that the stability and accuracy of the proposed method are almost same as those of the Kalman filter method, since the fourteen parameters do not change with time. The proposed method also has two advantages over the Kalman filter method: (1), it is easy to adopt arbitrary constraints and the appropriate order of identifying the parameters, and (2), it can be computed quickly. Identification ofParameters and Calculated Discharges We applied the tank model to Dam A of Fukuoka and its catchment and investigated the identified parameters, the calculated discharge, and the surface, medial, and base outflows to the dam for 549 days from 1992 to 1993. We selected specific days because of a serious shortage of water during more than 200 days in 1993, and we wanted to compare the shortage of outflows to the dam with those of 1992, which was an average year. Table 1 shows the initial and identified parameters together with the iterations and the residuals. Table 1 Initial and identified parameters in the tank model analysis of Dam A first kind coefficient «sO

«m2 «g «sl «m3

third kind

second kind ^LO

^L2

^L4 «1

^3

«2 «4 ^Ll ^L3 0.600 0.040 0.700 0.010 0.0001 30.0 10.0 0.0000 Initial value 0.050 O.OlOd 0.100 0.0010 20.0 0.0000 0.054 0.00004 29.5 12.09 0.0000 0.624 Identified 0.038 0.002 value 0.043 0.007^ 0.008 0.006 8.01 0.0000

365

Estimation of parameters in tank model analysis (m /sec) -I

1—I

1—r

Qcal

1.5 h

0.5

0

100

200

300

400

(day)

500

549

Fig. 4 Calculated discharge (Q^^) of Dam A using the assumed initial parameters 3 (m /sec)

300

^'"^-^^-^•^-^J^^*!—^^^'^ 500 (day) 400

549

Fig. 5 Calculated discharge (Q ) of Dam A using the identified final parameters Table 2 Investigated surface, medial and base outflows to Dam A (unit: m^ / sec) Jan Feb Mer Apr May Jun

Jul Aug Sep Oct Nor Dec

rainfall

141

126

68

225

130

336

62

24

190

58

39

62

surface outflow

14.8

9.55

0.28

27.2

12.6

67.0

3.72

000

26.4

0.96

0.00

0.76

medial outflow

50.7

59.6

42.8

68.2

66.0

95.3

59.6

28.1

45.6

48.8

22.5

24.7

base outflow

28.5

18.8

22.2

22.7

24.9

25.5

28.3

29.9

29.6

31.3

31.0

323

Qobs

94.1

87.9

65.4

118

104

188

91.7

57.9

102

81.0

53.5

57.8

base outflow( 1980)

28.0

27.4

30.1

30.4

333

34.2

37.8

42.9

48.1

56.3

58.4

62.4

Qobs (1980)

78.7

62.5

124

159

183

151

564

676

424

225

144

139

Dam A (1994) i

L y_

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M. Kanoh, T. Hosokawa and T. Kuroki

Figures 4 and 5 illustrate the calculated discharges (Q^^j) using the initial and identified parameters, respectively. Comparing them with the observed discharges (Q^^s^ illustrated in Figure 3, Q^^ using the identified parameters agrees well with Q^^^. Table 2 expresses the calculated values of the surface, medial, and base outflows to the dam. Noticing the base outflow to the dam, even during the serious water shortage of 1978, it did not decrease as much as the total discharges and might have retained enough water for 130,000 citizens for a month. Using the back analysis we can save the iteration and computing time by identifying the parameters in tank model simulation of Dam A. The summations of squares of residuals of the converged values are all less than 85.0 throughout the calculations on the identification of the parameters in the numerical examples.

CONCLUSIONS In order to identify the parameters in the tank-model simulation, we proposed a method in which the sum of the constraints and the squares of the residuals were to be minimized using Lagrange's method of the indeterminate multiplier, and in which the simultaneous equations for the correction terms of the parameters could be set up. For stabilizing the solutions against the oscillation in the back-analysis procedure, we adopted the constraints and the appropriate order of identifying the parameters. Considering that the coefficient matrix H consisted of the derivatives (9Qcai l^v) ^ ^ could be divided into three kinds of characteristics for stability in the back-analysis procedure, we identified the parameters in order of their stability. According to our experiences with back analysis, we can say that the stability and accuracy of the proposed method are almost the same as those of the Kalman-filter method, since the 14 parameters do not change with time. The proposed method also has two advantages over the Kalman-filter method: (1), it is easy to adopt the constraints and the appropriate order of identifying the parameters, and (2), it can be computed quickly. We applied the tank model to Dam A of Fukuoka and its catchment area and investigated the possibility that the base outflow to the dam might retain enough water to supply 130,000 citizens for a month, even during seriously dry periods such as the one in 1994. Using back analysis, we saved iteration and computing time by identifying the parameters in the Dam A tank-model simulation. REFERENCES 1. 2. 3. 4.

5.

Sugawara, M., Watanabe, I., Ozaki, E. and Katsuyama, Y. (1977): Method of Automatic Calibration of Tank model (First Report). Research Notes of the National Research Center for Disaster Prevention, N. 17, pp. 43-87, (in Japanese). Sugawara, M., Watanabe, I., Ozaki, E. and Katsuyama, Y. (1984): Tank model with snow component. Research Notes of the National Research Center for Disaster Prevention, No.65, pp. 293-312. Kanoh, M. and Kuroki, T.(1998). In: Proceedings of the International Symposium of Inverse Problems in Engineering Mechanics 1998, M. Tanaka and G.S. Dulkravich (Eds). Elsevier, Japan, pp.255-261. Kanoh, M. and Kuroki, T.(1998). In: Proceedings of 20th International Conference on Boundary Element Method, Estimation of physical unknowns in submerged breakwater using nonlinear least squares-boundary element method, A. Kassab, C.A. Brebbia & M. Chopra,(Eds.). Computational Mechanics Publications, Florida, pp. 483-492. Tajima, M. and Komaki, K.(1996). Theory and Applications ofLeast Squares Method, Toyo Shoten, (in Japanese).

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

367

IDENTIFICATION OF D A M P E D JOINTS P A R A M E T E R S USING T H E ERROR IN T H E CONSTITUTIVE RELATION Arnaud D E R A E M A E K E R \ Pierre L A D E V E Z E ^ Eric C O L L A R D ^ , Philippe L E C O N T E ^ ,

^ LMT-Cachan (E.N.S. de Cachan / C.N.R.S / Paris 6 University) 61 Avenue du President Wilson / 94235 CACHAN CEDEX Tel : 33 - 14740.24.02 Fax : 33 - 14740.27.85 ^ Thomson CSF Optronique Rue Guynemer BP 55 / 78283 GUYANCOURT CEDEX / France

ABSTRACT This paper deals with the modelling and identification of mass, stiffness and damping parameters of joints. We propose to validate joint models based on experimental tests. In these tests, the substructures are assumed to be perfectly known and only the joints parameters need to be identified. Because of the need to have models with few parameters, we propose to start with a very simple model for the joint. An identification of the parameters is carried out. The method used is based on the concept of 'error in the constitutive relation' that has been widely used in the domain of model updating. One of the features of this method is that it gives a quantitative value of the quality of the identified model. By setting up an a priori quality requirement for our model, we are able to decide whether the identified parameters are satisfactory or not. If such is not the case, a new model is chosen and the identification is performed once again. The model will evolve until the required quality value is reached. The validated model can then be incorporated in a full structure and if the correlation with the experimental results is not sufficient, model updating can be carried out. The method is then iterative : one iteration consists of a localisation stage in which the most erroneous regions and parameters are selected and a correction stage in which the parameters of the selected regions are adjusted. The iterations stop when the error gets lower than a prescribed value. In this study, the method is appUed to real experiments performed on an assembly of two bolted aluminium plates.

KEYWORDS Structural model updating, damping, joint identification, error in the constitutive relation, Finite element method

368

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INTRODUCTION Finite element computations have become a common practice for the prediction of the dynamic behaviour of structures. With the growth of computer power, the models have become more and more sophisticated. The uncertainty on the model's parameters and the necessary simplified modelling of some parts of the structure are often responsible for a distance between the numerical predictions and the experimental results. Among these parameters, the least understood part of the structures is probably the joints. These are however very important components since most of the damping is located at these joints, and assuming rigid joints usually gives very poor predictions. The introduction of a joint in a structure consists first in the definition of the joint's model. The second step consists in the identification of the joints parameters. Although the first point seems to be less studied, it is often the most crucial point and identifying parameters for a bad model of the joint gives very poor results. It is therefore of great importance to define a quality measure of the identified model. Choring a priori a required quality for the model enables us to validate or reject our model. If the model is rejected, new models have to be built until the required quality level is achieved. For the modelling of a joint, most approaches are based on the subdivision of the structure into substructures. The joint is then modelled as a massless stiffness matrix between the nodes at each interface. Damping is introduced through the use of a complex modulus approach. The stiffness matrix is mainly made of uncoupled translational and rotational springs (see for example [1]). The high value of the stiffness of such springs is often responsible for bad conditioning and difficulties in the identification process. In [2], an alternative approach using geometrical parameters is presented in order to overcome these difficulties. In some cases, the mass of a joint cannot be neglected and 'lumped masses' are added to the joint model. As mentioned earlier, once the joint model is chosen, the second step consists in the identification of its parameters. It seems that the first methods used the modal information (mode shapes and eigenvalues) [3]. Because of the concentration of damping at the joints location, systems including joints are subject to non proportional damping which makes very difficult the extraction of modal properties (the modes become complex numbers and few methods are able to deal with such data). In the early 80's, new methods emerged dealing with the FRF's (see for example [4]) in order to overcome the diflfiiculty of extracting modal properties. In these methods, it is usually assumed that the rest of the structure is modelled accurately enough so that it allows to identify the joints parameters. As explained in [5], although the problems of joint identification and model updating are not exactly the same, similar methods can be used in order to solve both problems. In this paper, we propose to use the concept of 'error in the constitutive relation' in order to perform joint identification. The method was first developed and applied for model updating in the free vibration case in [6, 7]. It was then extended in order to take into account dissipative effects such as damping based on FRF's information in [8] and was successfully applied in [9] and [10]. One of the main features of the method is that it gives a quantitative value for the quality of the assumed model compared to the measurements. The updating can thus be stopped when a required quality is reached. The method used

Identification of damped joints parameters

369

in order to reach that required quality is an iterative one in which each iteration consists of two steps. The first one, called the localisation step seeks to locate the most erroneous regions in the model. The second step, called the correction step allows to improve the quality of the model by acting on a reduced set of parameters contained in the selected regions. After each iteration, the quality of the model is computed. The iterations stop when the quality has reached the required value. Although joint identification can be performed using this method on a whole structure containing joints, it seems easier to isolate the joint in a much simpler configuration in order to study its characteristics. It is indeed a very hard task to decide on the joint's model without any a priori knowledge on its behaviour and incorporating many parameters with the hope that this will enable us to 'fit the curves' makes the updating process very difficult. We will thus be looking for a simple model of a joint. This model can be enriched or simplified based on experimental tests performed on the joint in a very simple configuration. The proposed approach is thus as follows. The joint needs to be tested in a very simple configuration in which only its parameters need to be identified. Starting off with a very simple model, the identification of the parameters allows to reach a certain quality level. If this reaches the required quality, the model is validated. If such is not the case, the model needs to be changed, based on some physical understanding as will be shown in this paper. This process is repeated until the required quality is reached. Once the model for the joint has been validated, it can be incorporated in a structure. The quality of the whole model can be computed using the same method and if it is not satisfactory, model updating will be carried out following the iterative procedure described here above in which the joint is considered as a region in the structure that will be corrected if it is selected in the localisation step. The small amount of parameters for the joint will make this task fairly simple. In this paper, after a short review of the identification and updating method, we propose to build a model for an assembly of two bolted aluminium plates based on specific experimental tests in the frequency domain. We start off with a very simple model of the joint. Because the identified model does not satisfy the required quality chosen a priori, we introduce a new parameter in the model. We then carry the identification once again and show that our model as reached the desired quality level. The model is thus enhanced based on experimental results and a quality level chosen a priori.

ERROR IN THE CONSTITUTIVE RELATION Let us consider a structure given by a domain Q during the time interval [0,T]. On the boundary dO. of the structure, the displacements U^ and forces F_^ are described on diQ. and ^ 2 ^ , respectively. Body forces f. are given in Q. The reference problem during [0,T] can then be written as follows : Find the displacement [/(M,t), the stress o[(M,t) and the density r ( M , t ) , t G [0,T], M G (^ such that they satisfy • the boundary and initial conditions • the equilibrium equation • the constitutive relations

370

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The concept of the 'error in the constitutive relation' is based on the separation of these equations in reliable equations (boundary and initial conditions -h equilibrium equation) and less reliable equations (the constitutive relations) Drucker Error In this section, we will define one of the variants of the error in the constitutive relation called the Drucker error. In order to define the Drucker error [11], we introduce a triplet s=(f/, flr,r) in Q X [0,T] . This triplet involves two kind of quantities: 'static quantities' (a, r ) denoted (tf^sXs) and a 'kinematic' quantity U_ denoted U^. The constitutive relations are given in a functional form as follows : (1)

t=A{e{U)\r;r
(2)

l-\t= p-n^ We define the quantity :

eAs) = sup r f {(1 - 7)(r, - r j o (f/, - u,) T
Jfl

+ 7 tr[{a, - as){£{U,) - £s)]}dt dQ

(3)

where
•Pind^.^r

^^

(,)

minimising ^^{s') with s' G <S^J A triplet 's' is said to be admissible (s G <S^^' ) if it satisfies the reliable equations of the problem. The minimisation is seeking to satisfy as well as possible the less reliable equations. Note that for problems with multi-excitation, the amplitudes of the forces are considered less reliable quantities. In this case, they are considered as unknowns and are added to the solution 's' to be found. Comparison of the Model with Experimental

Data

In the previous section, we defined one variant of the error in the constitutive relation. The solution of the reference problem (the assumed model) is given by the minimisation of this error. It is now necessary to define a measure of the quality of this assumed model compared to the experimental measurements. The experimental model suff'ers two limitations : it is 'incomplete' meaning that only a few degrees of freedom are measured and it is subject to measurement noise inherent to the acquisition process. Just as we did for the analytical model, we need to distinguish between the accurate and less accurate measurements.

Identification of dampedjoints parameters

?il\

Comparison for a Given Frequency. At a given frequency, we consider to be reliable : the measured angular frequency uj, the position and direction of the exciting forces, and imposed displacements. We consider to be less reliable : the amplitude of the measured exciting forces F^ and imposed displacements U_^ and the measured displacements noted n t / ^ , where 11 is a projection operator that details the measured part of the displacement. ( ~) denotes the measured quantities. These considerations lead us to define a global modified error in the constitutive relation which enables us to compare the analytical model with the experimental results. For a given frequency UJ and in the case of a single unitary excitation, we write : e^{s)=e{s)

+ ^ ^ { \ \ m - m f }

(5)

where ||.|p is an energetic norm equivalent to ^^(5) and the coefficient -^ \s d. weighting factor. The value of r is chosen depending upon the reliability of the experimental information. In the case of multi-excitation, additional terms are added in order to take into account the fact that the amplitudes of excitation are not reliable quantities. The new problem to solve is given by :

Find

seS^^

ad

minimising 6^(5') with s' G S^^

^g^

The quantity ^^(s), with 's' solution of (6) is a quantitative measure of the quality of the model with respect to the measurements. A relative error can be built by dividing this value by the total energy of the solution. Comparison over a Frequency Bandwidth. For a given frequency bandwidth [uJmin-i^rr ^min-) ^max\ we use a pondering factor z(cj) in order to define the error over the frequency bandwidth by

f

z[uj)e^{s) duj

(7)

J UJrr

Note that the function Z{IJJ) usually includes the scaling of the absolute value to the relative value of the error (which is obtained by dividing by the total energy of the solution at each frequency). Implementation

of the Updating Method

The method is iterative. In a first stage, the most erroneous regions are localised by solving problem (6), with fixed parameters given by the initial model at a given frequency UJ. The solution to this problem lies 'in between' the assumed model and the experimental measurements. This solution can be closer to the experimental measurements by acting on the ~ coefficient, depending on the confidence in the measured data. The computation is carried out for all the frequencies in the considered frequency bandwidth. Then the error in the constitutive relation is computed for each substructure : r''\(uj)eEjs)duj where Een

(8)

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A. Deraemaeker et al.

The substructures presenting the highest error are selected for correction. In a second step, the selected regions [JC]s are corrected by solving the following problem : Find the set of parameters {k} G [JC]s minimising e{{k'}),

{k'} G [IC]s

(10)

After each step, the quality of the model is assessed by computing I

z{uj)C{s)duj

(11)

and the updating process is stopped if the required quality level is reached.

APPLICATION T O BOLTED PLATES Description of the

Experiment

Two structures have been tested. The first structure is a homogeneous aluminium plate and the second structure is an assembly of two bolted aluminium plates. Damping in the joint is introduced by means of a viscoelastic film placed between the plates in the overlapping area. The structure under test is bolted to the top of the shaker through which it is excited in the vertical direction as represented on Fig 1. Both structures are equipped with twelve accelerometers as shown on Fig 2 and a reference accelerometer is put on the basis of the shaker. The frequency range for the test is [0 2000] Hz.

Fig 1: Test structures : homogeneous plate and bolted plates

Remarks on the Measurements. can be made:

By looking at the data provided by the test, two remarks

• Torsion appears around 200 Hz and 500 Hz, which should not be the case because we assumed that the shaker imposes a uniform acceleration to the plate. The presence of only one measurement point on the basis of the shaker does not allow us to verify if the acceleration is not uniform along the attached edge of the structure.

Identification of damped joints parameters

^o

^o

'o

'b

^o

=o

'o

^b

^o

*o

'o

"b

373

Fig 2: Position of the sensors on the test structures • The peaks for the 4 rows of accelerometers do not match exactly. This is due to the fact that the measurements were made in 4 different runs including each 3 accelerometers (1,2 and 3, 4,5 and 6 , etc ..). The weight of the accelerometers influenced the resonance frequencies of the structure differently for each run due to their different relative distance to the edge. Even though as we just mentioned, some uncertainties exist in the measurements, we will see that the identification process is robust and gives very satisfactory results. Homogeneous Plate The homogeneous plate was used in order to remove the influence of the joint between the shaker and the tested structure for the identification of the bolted plates. Parameters for this joint were identified in a first stage which will not be discussed here and good agreement was found between the predicted and measured FRF's. Bolted Plates Before we start the identification process, we need to choose a quality level for our model. Considering the problems cited previously for the measurements, 10% seems a reasonable value. We start off by using a very simple model for the joint. We use plate elements with a double thickness in order to represent the overlapping of the plates as shown on Fig 3. The results of the prediction using this model are given on Fig 3. For the identification, three different sets of parameters were chosen and only the data on the frequency bandwidth [0 1000 Hz] were used. The first one is the Young's modulus of the plate elements, the second one is both the Young's modulus and the density of these elements and the last one is the thickness of these elements. The FRF's are shown on Fig 4. The values of the error for the three identified models are respectively 15, 10 and 38%, showing that only the second model satisfies the required quality. It should be noted however that by modifying the mass of the joint elements in this model, the total mass does not correspond to the total mass of the real structure.

374

A. Deraemaeker et al. Channel 12

10%

Joint 2 (double thickness" h=l cm

1 1—

1

1'

experimental 1 FE model |

'

10'

/;

EIOO^M

!

iio4

\

A,' \

\ i 1/

' ',

1

ii

''

10-^

Joint 1 : parameters identified on homogeneous plate.

in-3

800 1000 1200 Frequency (Hz)

1400

1600

1800

Fig 3: FE model for the bolted plates and FE predictions on channel 12

100

200

300

400 500 600 Frequency (Hz)

700

800

900

1000

Fig 4: F R F prediction for the three different identified models The figure also shows that if we use a stiffness parameter, we cannot have a good prediction for modes 2 and 3 simultaneously. If we look at the shape of the bolted plates for all the modes in the frequency bandwidth [0 2000 Hz] (Fig 5), we see that modes 3 and 5 are the only ones that make the joint bend in the yz plane. In fact, if we perform the identification on the frequency bandwidth corresponding to the first two modes instead of the frequency range [0 1000 Hz], we see that modes 3 and 5 are poorly represented while the model gives good prediction for the other modes (Fig 5).

375

Identification of dampedjoints parameters

800 1000 1200 Frequency (Hz)

1400

1600 1800

Fig 5: Modes in the frequency bandwidth [0 2000] Hz and F R F for model identified on first two modes This fact motivates the need for a new model. In order to decouple the bending in the xz plane and in the yz plane, we add a beam with bending stiffness in the yz plane and all other parameters set to zero. Damping is added only to the beam by introducing a complex modulus. The results of the identification of the stiffness and the damping for this model are given on Fig 6 Channel 12 10^

1 ,

1— 1—

ik

10'

fio°

Joint (plate elements)

experimental 1 identified FE model |

•W\

\w

1 10"'

/\

s^

1 Y/ \ /

1

10'^ Beam

element

(flexion)

200

400

600

$00 1000 1200 Frequency (Hz)

1400

1600

1800

2000

Fig 6: F E model and predictions with the identified parameters The error level is now 9.9% and satisfies the required quality level. This new model does not have a 'non-physical' mass which is the reason why we will prefer it. It also allows a better fit of the damping characteristics.

376

A. Deraemaeker et al.

CONCLUSION In this paper, we propose an approach for the modelling and identification of damping, mass and stiffness properties of a joint. The approach is based on a specific experimental test performed on the joint and the use of an identification method based on the error in the constitutive relation. The method gives a quality value for the identified model and allows to either validate the model or change it so that a given quality is reached. It has been applied to an example of two bolted aluminium plates. The identification of the first model did not satisfy the quality level so that a new model was introduced. This new model did satisfy the required quality and was thus validated. The so-identified model can be used in a real structure. The quality of this whole structure can then be assessed using the same approach as for the joint alone. If its quality is not satisfactory, model updating can be carried out.

REFERENCES 1. Ren, Y. and Beards, C. (1995) Journal of Sound and Vibration 186(4), pp. 567-587. 2. Mottershead, J. and Shao, W. (1993) Journal of Applied Mechanics 60, pp. 117-122. 3. Inamura, T. and Sata, T. (1979) Annals of the CIRP 28, pp. 235-239. 4. Mottershead, J. and Stanway, R. (1986) Journal of Sound and Vibration 109, pp. 495-506. 5. Nobari, A., Robb, D. and Ewins, D. (1993) Journal of Modal Analysis 8(2), pp. 93. 6. Ladeveze, R (1983) Tech. Rep. 33.11.01.4 Aerospatiale Les Mureaux (in french). 7. Ladeveze, P. and Reynier, M. (1989) Vibrations Analysis Techniques and Chap. "F.E. Modeling and Analysis" pp. 355-361. 8. Ladeveze, P. and Chouaki, A. (1998) In: Inverse Problems in Engineering pp. 187-196 M. Tanaka and G.S. Dulikravich Elsevier.

Applications, Mechanics

9. Chouaki, A., Ladeveze, P. and Proslier, L. (1998) AIAA Journal 36(6), pp. 1094-1099. 10. Chouaki, A., Deraemaeker, A., Ladeveze, P. and Le Loch, S. (2000) Proc. pp. 140-146. 11. Ladeveze, P. (1993) Internal Report 150 L.M.T. Cachan (in french). 12. Drucker, D. (1964) Journal de Mecanique 3(2), pp. 235-249.

IMACXVIII

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

377

CONDITION MONITORING AND DAMAGE QUANTIFICATION IN ROBOT JOINTS USING NONLINEAR DYNAMICS CHARACTERISTICS AND INVERSE CLASSIFICATION METHODS I.TRENDAFILOVA and H. VAN BRUSSEL

Department of Mechanical Engineering, PMA, Katholieke Universiteit Leuven, B-3001 Heverlee-Leuven, Belgium

ABSTRACT A method is proposed for condition monitoring and backlash detection and estimation of robot joints from their time-domain vibration signatures. The motion of four types of robot joints - without backlash, with a small backlash, with medium backlash and with large backlash - is analyzed using their measured acceleration signals. The application of a nonlinearity check has shown the presence of deterministic nonlinear chaotic motion in the irregular behaviour of the considered joints. The considered time series are proven to be a combination of harmonic and chaotic motion. Accordingly, chaotic invariants are used in order to classify such kind of time series, and the corresponding robot joints that produce them. The maximum Lyapunov exponent and the correlation dimension are used to form features. Different classification rules and different features are applied to build several classifiers. Their performance is estimated on a set of signals from the introduced categories. KEYWORDS Condition monitoring, robot joints, nonlinear dynamics characteristics, backl2ish, damage detection and quantification, pattern recognition. INTRODUCTION The dynamic behaviour of robots is strongly influenced by the condition and the characteristics of their joints (link connections). The robot joints are considered as one of the major sources of nonlinearities in the robot structure. The dynamic behaviour of the robot elements and their vibration signals are clearly affected by the presence and the size of backlash and clearance in various robot links. The presence of backlash in robot joints can result in severe degradation of robot performance. Early damage detection and possible quantification of defects in robots from their vibration signals, which is the primary aim of condition monitoring, is an essential task, since it forms the basis for any consequent decision for part repair, replacement or discharge. In this paper we consider the problem for detecting backlash in a robot joint and characterizing the condition of the joint as an inverse damage detection problem. We try to extract features characterizing the condition of the joint directly from its vibration measurements using the tools of nonlinear dynamics. These features are further utilized to develop classification methods to distinguish among healthy and different damaged conditions of the robot axes. Various techniques can be used in order to distinguish among vibration signatures [1,2,4,5]. In this work we explore the possibility to use the tools of nonlinear dynamics and particularly some characteristics of nonlinear motion for vibration signature comparison and feature extraction [3,6]. The idea is that a defect in a link-joint introduces a nonlinearity in the signal transmission part. The characteristics of the robot link motion in the phase space are expected to differ for signals coming from damaged and non-damaged parts. Accordingly the motion invariants for backlash and no-backlash conditions and for different types of backlash are expected to differ. Our study of the dynamics of robot joints using nonlinear time series analysis proves the difference in the motion character and consequently in its characteristics for the different backlash

378

L Trendafilova and H. Van Brussel

cases considered. Several categories of robot joints are defined considering damaged and healthy joints as well as joints with different backlash extent. The classifiers developed try to distinguish among these categories using different features, formed on the basis of the nonlinear dynamics properties of the motion, and different classification procedures. EXPERIMENTS Experiment J. Measurements were done on a Reis RV robot. The aim of the experiments is to identify backlash in the robot joints. Backlash was introduced in two rotational axes of the robot. Vibration signals were recorded at the end points of both axes for the situations "backlash" and "no backlash", using piezoelectric accelerometers. Measurements were done sequentially, moving only one axis at a time. Experiment 2. Another series of similar measurements were performed on a Puma 760 robot. Different amounts of backlash were introduced in two rotational axes by adjusting the backlash compensation screws of the robot links. Several situations were explored: i) No backlash (N) which corresponds to the normal state of the joint; ii) Small backlash (S); iii) Medium backlash (M) and iv) Large backlash (L) which corresponds to the maximum backlash. These joints are rotational, each of them driven by a servomotor through a gear transmission. The vibration responses are measured using an accelerometer mounted on the end-effector. PROBLEM STATEMENT The aim of this study is to develop procedures to detect backlash and quantify it, using data collected from the acceleration signals of the joints in the time domain. We shall formulate this problem as a categorization one. For the first type of experiments with the Reis robot (Experiment 1) two categories can be introduced: the category of intact (no-backlash) joints, which we shall denote with N and the category of damaged (backlash) joints, which will be denoted by B. For the second kind of experiments with the Puma robot four categories are introduced: i) the category of no backlash joints (N), ii) - joints with small backlash (S), iii) joints with a medium backlash (M), iv) - joints with a large backlash (L). The task is to develop procedures to recognize among the introduced categories collecting information from the time presentation of their acceleration signals. This clearly defines two categorization problems for the two types of experiments considered: Problem I: Find the class ( category) to which a robot joint belong, using its acceleration signal, choosing between no backlash (N) and backlash (B) joints. This is a backlash detection problem. Problem II. Identify the category to which a robot joint belongs, using its acceleration signal, choosing among the four introduced categories N, S, M and L, that correspond to the backlash size. This is a backlash quantification problem. For the second kind of experiments one can also define a detection problem (Problem I), i.e to distinguish between intact (N) joints and the three types of backlash joints. In categorization problems one uses features in order to distinguish among the defined categories. A mapping from the original space into the feature space is used. The feature space is obtained from the original one after a proper selection and transformation of its coordinates. In general the features, which define the new feature space should be independent form the way in which the signal is measured, i.e from the time interval used between the coordinates, from the number of coordinates, etc. The selection and the definition of features is one of the major conditions for the success or the failure of a categorization problem. Thus a careful analysis of the signals is a necessary step towards choosing an appropriate feature space. As a result the number of signal coordinates in the feature space is reduced to a possible minimum and these coordinates facilitate the further classification task since they ensure a better differentiation among the different categories. In the next paragraph we shall analyze the signals coming from the defined categories and discuss the selection of features.

379

Condition monitoring and damage quantification in robot joints

SIGNAL ANALYSIS At first glance the vibration signatures seem periodic (Fig. 1), since the robot joint rotates with constant frequency, but when looking at a single cycle it is substantially transient. The transients look very much random like. Looking at the signal spectra, one can observe that all the spectra are broadband [3]. Those corresponding to N and S show some peaks at certain frequencies, which gradually disappear with the increase of the backlash [3]. Obviously there is another component in the dynamics of the robot joints, except the harmonic motion of the joint. The image of the transients as well as the spectra suggests that this component is either a nonlinear one that causes unstable chaotic motion of the joints, or a random one that can be described using a stochastic model. In our previous research we have reconstructed the embedding space of the motion of robot joints, that is needed to completely unfold the dynamics of the process, for the considered categories.

Figure 1. Typical signals We found a time lag of 8-10 (80-100 jUs) and a space dimension of m7=3(or 4) for N and S, and a time lag of 4-6(40-60 fis) and a space dimension of m2=5 for the M and L signals( as well as for the B signals) [3]. The relatively low dimension of the space also suggests the presence of a nonlinearity and the availability of a deterministic model for this component of the robot joints motion. Thus our next task is to determine the character of the additional non-periodic component, choosing between a random and a deterministic model. Surrogate data series can be used for the purpose [12,13]. A number of surrogate data sets is generated, that mimic only the linear properties of the original time series. This is achieved by Fourier transforming the original time series and substituting the phases with random numbers. A consequent estimation of any of the nonlinear characteristics of the system should yield significant differences for the case of a nonlinear system, since by the transformation we have destroyed its structure. The original time series and the surrogate data sets are compared using discriminating test statistics. For this case we used three discriminating statistics: the correlation dimension, the average mutual information, and the maximum Lyapunov exponent [3,11]. Figures 2 and 3 show the results for the correlation dimension (D2) and the Lyapunov exponent (LE). Obviously the values for the original time series and the surrogates, which are shown by the interval M(*)±2a(*), where M denotes the mean value and a - the standard deviation, differ substantially.

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The surrogate tests with all the statistics suggest the rejection of the hypothesis for linearly correlated noise, since considerable differences can be observed between the characteristics for the original and the surrogate

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series (Figs 2,3) [3,11]. The correlation dimension shows convergence with the increase of the embedding dimension (Fig.6). The Lyapunov exponents for nearly all the signals (except for some N signals) are positive, which is a strong signature for chaos (Figures 3,4b,5b). These two facts come to confirm the hypothesis for the presence of a nonlinear deterministic chaotic component in the motion of robot joints. Thus summarizing our results we characterize the motion of robot joints as a combination of a periodic motion and a chaotic process. For the cases of no backlash and small backlash the chaos and the unpredictability of the process is rather weak, and it gets stronger with the increase of the backlash. This is confirmed by the dimension of the recovered embedding space (m/=3(or 4) for N and S, and m2=5 for M and L) as well as by the Lyapunov exponents for the different signals - they are smaller for the N and S series and gradually increase for the M and L series [3]. The problem of differentiation between different signals is a categorization problem. Classification of dynamical systems and especially feature selection is a critical part of the analysis of the measured time series. The task is to be able to classify a signal to one of several categories. Pattern recognition methods can be used for such purposes. Pattern recognition methods work on the basis of mapping of the signals from their original space into a feature space in which the defined categories become better distinguishable. In the time space a signal is presented by a number of values of its magnitude corresponding to discrete time points, i.e. s = {5i,52,...,^J^ ={s(tO,s(t2)^-,s(t^)f . (1) ' i the feature domain it is supposed to be presented by a smaller number of features/ which will allow us to distinguish among the defined categories. s=>f = ( / i , / 2 , . . . , A ) ' , where it < n (2) When the source is linear we are used to the idea of identifying spectral peaks in the Fourier spectrum of the signal. But chaotic motion produces broadband continues Fourier spectrum, and the resonance frequencies are to be replaced by other invariants [3,7,8]. The two major features that have emerged when classifying nonlinear time signals are fractal dimensions and Lyapunov exponents. These quantities remain invariant under changes of the coordinate system into which the signal is represented, they characterize the signal itself and some of its geometric and dynamic properties. Fractal dimensions characterize the geometric figure of the attractor and relate to the way points on the attractor are distributed [6,9]. Lyapunov exponents tell how orbits on the attractor move apart under the evolution of the dynamics [6,8]. In this application we suggest the use of the maximum Lyapunov exponent and the correlation dimension as features. In the next paragraph we consider several pattern recognition (classification) procedures based on these features for distinguishing between the introduced categories.

CLASSMCATION PROCEDURES FOR SIGNLAS COMING FROM ROBOT JOINTS. Features The Lyapunov exponent. The number of Lyapunov exponents of a system equals the dimension of the state space into which its motion can be represented. The Lyapunov exponents of a system characterise the rates of divergence of its trajectories in the phase space [6,9]. For a non chaotic system the Lyapunov exponents are either zero or negative. Hence if a system has at least one positive LE, then it can be considered chaotic. The Lyapunov exponents of a system are its main dynamic invariants and aie supposed to change if the dynamics of the system changes. The maximum Lyapunov exponent (LE) determines the character of a nonlinear dynamic system. Thus the LE is the most important dynamic characteristic of any motion which can be used to distinguish between non-chaotic and chaotic systems as well as to characterise the degree of chaoticity of a system [6]. If a system is chaotic, then its maximum LE is positive, while for a stable system it will be either zero or negative. As was explained in the previous paragraph, a defect introduces instability in a dynamic system [3] and its size influences its main dynamics characteristics. As was expected the maximum Lyapunov exponent is directly proportional to the backlash size. Thus it can be used to distinguish between backlash and no-backlash conditions of the joint as well as to characterise the backlash extent. The maximum LE X is determined by S.BSoe^"

(3)

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where 5Q «1 is the distance between two nearby points at time 0 and 5^^ is the distance at time n between the two trajectories emerging from these points. An attractive property of the Lyapunov exponent, which is very important for this application, is that it can be determined directly from data. The Correlation Dimension. Another characteristic that describes the geometric properties of the attractor of a dynamic system is the correlation dimension [3,14]. It gives an idea for the active dimension of the attractor of the motion in the phase space. The correlation dimension is a geometric invariant of a system in its phase space and thus is expected to change if a defect is introduced or if the size of the backlash is changed. Our analysis proved that the presence and the size of a defect affect the correlation dimension. In general the correlation dimension is smaller for joints with no backlash and with a small backlash and gets larger with the increase of the backlash. The correlation dimension is estimated from the correlation sum. The correlation sum counts the number of pairs within a distance smaller than E:

N(N-l)

(4)

where 0 is the Heaviside step function. In the limit of an infinite amount of data and for small 8 one expects C to scale like a power low, i.e. C(e)«e^2, (5) defining the fractal dimension D2 where it exists. D2 is the correlation dimension. It gives the slope of the curve In C versus /« e , i.e.

0,= , i n . i ^

(6)

The correlation dimension estimates how the correlation sum scales with the distance. The correlation function can be estimated directly from the time series. When plotting In C{e) v/s In € one usually obtains a curve, which has linear behaviour only within a small range of length scales E. The slope of this linear part gives the value for the correlation dimension. The correlation dimension as a function of the space dimension. For any time series the correlation dimension is a function of the dimension of the embedding space m, i.e. D2=D2(m). For a deterministic chaotic process it converges to a saturation value when the sufficient dimension for the process reconstruction is reached. For a random process, the correlation dimension does not reach a plateau, since no sufficient dimension can be reached. Thus D2(m) can be used to identify and estimate chaos.

Figure 4. Ranges for a) the correlation dimension and b) the maximum Lyapunov exponent - Puma robot We try to utilise as features the maximum Lyapunov exponent A, the correlation dimension D2, as estimated in the embedding state space, and the correlation dimension as a function of the embedding dimension D2(m). Our previous research found [1,2,3] that as expected the Lyapunov exponent changes for

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the different categories of backlash joints introduced. Fig 4 shows the ranges for the Lyapunov exponent for the categories introduced for the Puma robot experiments, and Fig 5 - for those introduced for the Reis robot experiments.

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Figure 5. Ranges for a) the correlation dimension and b) the Lyapunov exponent - Reis robot As can be observed the regions for the LE and for D2 for the category of backlash joints and no-backlash joints are very slightly overlapping (Fig. 5). The ranges for the LE and D2 for the four categories introduced for the PUMA robot experiments also show some overlapping, especially betweeft N and S and between M and L (Fig.4). But in general the LE and D2 can provide a good feature basis for further PR procedures for detection and quantification of backlash. Figure 6 shows some examples of the correlation dimension v/s the state space dimension for the four categories introduced for the Puma robot. One can observe that for the N and S case there is small nearly linear segment, after which the curve converges to the saturation value. For the cases of M and L backlash the initial segment gradually increases and is not linear any more. Accordingly, instead of a number of discrete values to determine such a curve, we introduce four parameters which define the behaviour of the curve: a,b,s and p^ where s is the saturation value, \i gives the value of m where the convergence is reached, i.e. the sufficient dimension of the embedding space, and a and b approximate the initial segment as a parabola: D2=am

+bm if m< ji

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(7)

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Figure 6. Correlation dimension as a function of m- PUMA robot 1. We introduce the first feature vector using the whole signal s without taking into account that it is a deterministic nonlinear chaotic one. In order to chose features for it, i.e. to reduce the number of coordinates, the Karhunen-Loeve transform is applied to the signal s, which is presented as

s=i/jO^-

(8)

7=1

where and <|)k are the first k eigen-vectors of the covariance matrix R R=Rij=E{xiXj}

(9)

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Thus the signal s is substituted by the vector

^^=[f]Y = {fx\fl-JlY

(10)

We used k=5 since 5 is the minimal value for which one does not obtain coinciding vectors from the different categories. 2. As a second feature vector the Lyapunov exponent X and the correlation dimension D2 are directly used as features, i.e. f^=[/i^/2^r=[A,z)2r (11) 3. For the third feature vector the Lyapunov exponent X and the parameters a,b,s,JLi that characterise the relation Diim) are used directly as features, i. e.

f^=[/^...,/5^r=[A,a,^,5,/xr

(12)

The Classifiers. In this study we applied two different classifiers with all feature vectors defined above to be able to compare their performance and eventually choose the most appropriate one. 1-NN classifier [16]. This is a very simple classifier to develop. It uses a vector from each category as a prototype for the corresponding category. The mean vectors from the training sample were used as prototypes for each category, i.e.:

M

fi^=£'(fi)=-^-=i-, M

(13)

where fj** is the prototype vector for the i-th category, fj are the vectors from the training sample that belong to the i-th category. A new vector f is classified to the category j to which its nearest neighbour belongs, i.e. Dj = D(f P ,f) = min D(ff ,f) (14) The 1-NN classifier is a deterministic classifier and accordingly it does not account for the variability of the vectors from the different categories and for their statistical properties. 1) To build the classifier the training sample is used and f.^ are found according to (13); 2) When a new feature vector f is presented the distances to all the prototype vectors A are found; 3) Dj is determined according to (14); 4) f is classified to (Dj. A statistical learning classifier. This classifier works on a Bayesian statistical principle and uses the probability density functions of each category as discriminant functions. It provides an optimal classification rule from a statistical viewpoint since it minimises the classification losses {15]. The discriminant functions for each category are given by the following expansion over a set of functions (py.

diif) = pimAf) = lwij(pjif) = WiV(f) I- J

(15)

where Wf are found as a result of the next training procedure with the vectors f(k) from the training sample: Wi(fc + l) = Wi(^)+a^f(^){r,[f(^)]-Wi(^)'f(^)} (16) and r, is defined as fl iffeCJ; ^/(f) = i (17) ' [0 iff^CJi 1) The classifier is built determining the coefficients Wi according to (16), using the feature vectors from the training sample f(k) 2) When a new vector f is presented, (pj are calculated

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3) The discriminant functions are obtained according to (15) 4) f is classified to Q)J if dj (f) = min di (f)

SOME RESULTS. Here we present and discuss some results obtained using the above classifiers and feature vectors. Two classifiers are built: one using the experiments with the Reis robot and another one which uses the experiments with the Puma robot. The first classifier serves the purpose for backlash detection and distinguishes between N (no backlash) and B (backlash) categories; while the second one quantifies backlash categorising it to one of the four categories introduced - N,S,M and L. The classifiers were built using learning samples, which contained 1015 signals from each category for the first classifier and 1026 signals from each category for the second one. After that the results were tested using test samples containing 926 signals from each category for the first classifier and 987- for the second classifier. 1) Let us first turn to the results obtained using the first feature vector f^ that is obtained by applying the Karhunen Loeve transform to the recorded signal. The results for the PUMA robot are given in Table 1 and the results for the Reis robot are given table 4, which present the corresponding confusion matrices. The confusion matrix gives the percentage of falsely and correctly classified signals form the test sample. As can be observed the highest percentage of correctly classified signals is about 55%. There is substantial divergence in the classification - some signals from N are classified to M, and there are signals from L classified to S. Obviously this feature ensures a rather poor classification. 2) Next we have a look at the performance of the classifiers that utilise the second feature vector f^, fefined by eq (11). It contains the maximum Lyapunov exponent of a signal and its correlation dimension. As was expected (figs. 4a,b and fig 5a,b) these features assure a quite good classification performance with both classifiers used. The confusion matrices are given in Table 2 a,b and Table 4. The percentage of correctly classified signals for both cases with both classifiers is above 90% The performance of the statistical classifier is somewhat better that of the 1-NN classifier, but both classifiers can be considered as successful examples for signal classification used for condition monitoring of robot joints. 3) The third classifier uses the vector f^ defined by eq (12) and contains the maximum LE and the characteristics of the correlation dimension curve. Tables 3 and 4 contain the corresponding confusion matrices for this classifier. Both classifiers can be considered successful with a very good level of performance- all the percentages of correctly classified signals are above 95%. Again the performance of the statistical classifier is slightly better compared to the 1-NN classifier. This classifier performs even better than the previous one, for the Puma and the Reis robot cases. SOME COMMENTS AND CONCLUSIONS The primary task of this paper is to detect and quantify backlash in robot arms using their acceleration signatures. We first offer an in-depth analysis of the signals that come from the considered robot joints. Our analysis proves the nonlinear, deterministic, chaotic character of the motion performed by the joints. The problems for condition monitoring and backlash detection and quantification in robot joints is formulated and considered as a categorisation problem. Two robot joint categories are defined for the Reis robot backlash detection problem. Four robot joint categories to be recognised, corresponding to different backlash extent are introduced for the Puma robot quantification problem. The choice of features is discussed in the light of the obtained results. Some major nonlinear dynamics characteristics are introduced and considered as possible features. Three feature vectors are introduced that are estimable directly from the vibration signals. Two classifiers are used with each of the feature vectors. Our results show that the classifiers based on the Karhunen-Loeve transform of the time series, which accounts for the stochastic properties of the signal, have rather poor performance. This can be explained by the deterministic nature of the motion and hence the signals, which combine a periodic process with a nonlinear deterministic one. The deterministic properties of the signal are not taken into account by the Karhunen-Loeve transform- it reflects only the stochastic relation between the signal components. The two

Condition monitoring and damage quantification in robot joints

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other feature vectors that are introduced using the Lyapunov exponent and the correlation dimension, applied with both classifiers show very good performance for both problems. The statistical classifier proves the better option for all the categorisation programs. This is probably due to the variability in the signal characteristics, which comes from the variability of the signals from the same category. The statistical classifier takes this variability into account and makes the best use of it, providing optimal classification from a statistical viewpoint. The 1-NN classification rule does not take into account the statistical properties of the categories and thus shows a bit worse performance with all the feature vectors. The third feature vector (12) somewhat overrides the second one (11) - both nonlinear dynamics classifiers with f^ show the highest performance level (Tables 3 and 4). This could be explained by the suggestion that this vector contains more information about the correlation dimension, besides its saturation value D2 - this vector characterises the whole curve D2(m) using four parameters /see eq(7) and (11)A Eventually these characteristics turn out more informative and hence the better classification when using the feature vector (11). This paper uses signal analysis in order to select features for robot condition monitoring and damage detection. It proposes a straightforward method for robot joint condition monitoring and backlash detection, which is based primarily on the nonlinear nature of the motion. It shows that the preliminary signal and motion analysis is a powerful tool in the feature selection for condition monitoring. The classifiers based on the nonlinear characteristics of the motion demonstrate very good performance and can be successfully used for detection and quantification of backlash in robot joints. Similar procedures, based on the nonlinear character of the robot joint motion, can be developed and used to detect and quaiitify other defects, such as belt tension, friction, etc. Table la)Confusion matrix for the performance of the 1-NN classifier with the feature vector f^ N L M s True class N 6 52 30 12 4 True class S 37 10 49 8 True class M 20 21 51 2 True class L 3 40 45 Table lb) Confusion matrix for the performance of the statistical classifier with the feature vector f L N S M 4 True class N 25 55 16 5 True class S 30 16 59 20 True class M 6 20 54 True class L 5 5 55 35 Table 2a) Confusion matrix for the performance of the 1-NN classifier with the feature vector f^ N L S M True class N 0 92 7 1 True class S 0 4 90 6 True class M 6 0 3 91 True class L 0 92 0 8 Table 2b) Confusion matrix for the performance of the statistical classifier with the feature vector f L N M S 0 True class N 6 0 94 0 True class S 2 3 95 True class M 3 0 1 96 True class L 0 0 7 93

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Table 3a) Confusion matrix for the performance of the 1-NN classifier with the feature vector f' L N M S True class N 0 96 4 0 True class S 0 2 3 95 True class M 3 0 0 97 True class L 98 0 2 0 Table 3b) Confusion matrix for the performance of the statistical classifier with the feature vector f' L N M s 0 True class N 0 98 6 0 True class S 2 2 98 0 True class M 0 1 99 True class L 0 3 97 0

True class N True class B

Table 4. Confusion matrices for the detection procedure (Reis robot) Statistical classifier 1 -NN classifier j4 f* f' f^ f^ f* N B N B N B N B B N B N 95 5 98 2 45 4 94 55 45 6 55 96 62 3 97 3 97 43 57 5 38 3 97 95

REFERENCES 1. Trendafilova I, H. Van Brussel and B. Verbeure, On feature extraction for condition monitoring using time series analysis and distance techniques, (1999), In: Identification in Engineering Systems., pp 169178, M.L Friswell, J.E. Mottershead, A.Lees (Eds.), Cromwell Press Ltd. 2. Trendafilova I, H. Van Brussel and B. Verbeure, On feature extraction for condition monitoring using nonlinead time series analysis and pattern recognition procedures,(1999). Inverse Problems in Engineering Mechanics /to appear/. 3. Trendafilova I, H. Van Brussel, Nonlinear dynamics tools for the analysis of robot joint motion, 2000, /submitted/ 4. Pan M.- Ch., Van Brussel H., Sas P. Verbeure B., Detection and quantification of the link-joint backlash of a robot- time-frequency signal representation and feature extraction, (1996) ISMA 19- Tools for Noise and Vibration^ pp 407-421. 5. Pan M.- Ch., Van Brussel H., Sas P. Verbeure B., Fault diagnosis of joint backlash, (1998), Trans. ASME, Journal of Vibration and Acoustics, 120(1), 13. 6. Krantz H., Schreiber T., Nonlinear time series analysis, (1997), Cambridge University Press 1. Moon, P., Chaotic vibrations, An introduction for applied scientists and engineers, (1987), J.Wiley & Sons Inc. 8. Hilbom R., Chaos and Nonlinear dynamics, (1994),^Oxford University Press, 9. Abarbanel H., Analysis of observed chaotic data, (1996), Springer 10. Fraser A.M., H.L. Swinney, Independent coordinates for strange attractors from mututal information, (1986) Phvsica A 33, 1134. ll.Gradisek J., E. Govekar, I. Grabec, Time series analysis in metal cutting: Chatter versus chatter-free cutting, (1998), Mechanical Systems and Signal Processing 12(6),839 12. Theiler J., S. Eubank, A. Lx)ngtin, B. Galdrikian,J.Farmer, Testing for nonlinearity in time series: the method of surrogate data, (1992), Phyisca D 58. 77. 13. Palus M., Testing for nonlinearities using redundancies: quantitative and qualitative aspects, (1995), PhysicaDSO, 186. 14. Mathew J, Some recent advances in signal processing for vibration monitoring., (1997), Proc. Fifth Intemat. Congress Sound and Vibration, December 15-18, Adelaide, South Australia, pp 903-918. 15. Fukunaga ,Keinozuke,, Introduction to statistical pattern recognition, (1990) Academic press New York,. 16. Tou J, Gonzales R, Pattern recognition principles, (1978), Addison Wesley Publ. Comp

Inverse Problems in Aeronautics and Fluid Dynamics

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

389

WAVERIDER DESIGN WITH PARAMETRIC FLOW QUALITY CONTROL BY INVERSE METHOD OF CHARACTERISTICS Y. QIAN Beijing University of Aeronautics and Astronautics Beijing, China 100 083 H. SOBIECZKY DLR German Aerospace Research Center Bunsenstr. 10, D-37073 Gottingen, Germany Th. EGGERS DLR German Aerospace Research Center LiUenthalpl. 7, D-38108 Braunschweig, Germany

ABSTRACT This contribution reports about a generaUzation of the successful "Osculating Cones" method for designing waverider forebodies and inlets, to an "Osculating Axisymmetry" (OA) concept for obtaining shapes with higher volumetric efficiency and flow quality control on the inlet plane. The approach is based on an inverse axisymmetric method of characteristics which through the OA concept allows the construction of 3D flows as a solution to the 3D Euler equation within a few seconds on a workstation. Thus we have a useful tool for aerodynamic and multidisciplinary optimization efforts to design high speed configurations. Computational flow analysis examples confirms the inverse approach as illustrated for a number of OC waveriders designed at DLR. The status of applications to full aerospace vehicle integration is shown. KEYWORDS Aerodynamic design; Waveriders; Supersonic Flow; Characteristics; Inverse design; Shock waves.

390

Y. Qian, H. Sobieczky and Th. Eggers

INTRODUCTION

Nearly a decade ago Sobieczky [1] developed a concept for practical methods to compute the flow field behind a given oblique shock wave: cross marching starting from the shock surface with given post-shock flow properties as initial conditions defines a sector of the flow including its boundary compatible with the shock. Jones [2] has obtained 3D waverider forebodies through solving this initial value problem by an inverse numerical marching technique for the 3D Euler equations. A higher order approximative solution to this approach, with the constraint to constant shock strength, exploits a single conical flow solution for the design of waveriders with arbitrary 3D planform, the idea is called "Osculating Cones" concept. This extremely fast method for finding configurations with their surrounding flow field data led to an attractive extension of Bowcutt's [3] code for viscous flow waverider optimization: Center [4] developed a software for rapid intractive computation and optimization of advanced waverider shapes. Various research groups in Germany and in the USA have used this software to design configurations with application to hypersonic transport vehicles and performed numerical and experimental analysis of such waveriders in design and off-design conditions. This has been reviewed in [5] and recendy an extension from the "Osculating Cones" (OC) to an "Osculating Axisymmetry" (OA) concept was proposed [6]. Our effort reported in this presentation is aimed at combining the advantage of allowing for an arbitrary strength shock surface realized in [2], with the basically 2D approach of the OC concept which proved so practical for 3D configuration design. The OA concept seems useful for this goal in most practically relevant cases. INVERSE METHOD OF CHARACTERISTICS Given a limited portion AD (see Figure 1) of a plane 2D or axisymmetric oblique shock wave in known upstream flow defines a limited domain of the downstream flow which can be computed by the method of characteristics in inverse mode. Non-physical solutions showing limit surfaces may occur but depending on the given Shockwave data, a part of the resulting flow model is ready to be used as ramp flow for diffusers, inlets and vehicle forebodies. ^.U

—

••'

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391

Waverider design with parametric flow quality control

A useful modification of this initial-boundary value problem is the prescription of data along an axial station BC to define desired inlet flow quality direcdy, replacing part BD of the input shock wave data and defining solution in the upstream domain BCE. It avoids computing some unneeded parts of the flow and we have a possibility to effectively control flow quality at the axial inlet station where the pre-compressed flow enters the propulsion system. OSCULATING AXISYMMETRY The following steps of creating three-dimensional geometries are equivalent to the well-documented method of Osculating Cones: Prescribing spanwise functions of an Inlet Capture Curve (ICC) (see Figure 2) to define the 3D geometry of an inlet lip and the Flow Capture Curve (FCC) to create the diffuser leading edge geometry. Data for these curves have been found as a very suitable input for a workstation-based automated and interactive optimization software [4] for high speed configuration geometries. In the present contribution an even higher degree of practically relevant shape and flow control is proposed by additionally defining data for the Shock Generatrix Curve (SGC, along AB in each meridional plane) and Inlet Flow Quality (IFQ) at x = const, consisting of given Mach number and flow direction distribution, but observing input to be consistent with post-shock data in B. The additional effort for non-straight SGC in the computation consists of accepting the shock angle to be a variable of the x-direction, preserving the fundamental OC concept of constant postshock pressures in cross section planes x = const. The resulting flux diagram of the resulting computer code is depicted in the Appendix. z, z

M,Q

B,B' Figure 2. Input curves for 3D configuration definition: Inlet Capture Curve (ICC), Flow Capture Curve (FCC), Shock Generatrix Curve (SGC) and Inlet Flow Quality (IFQ) parameters

FLOW CONTROL AT THE EXIT PLANE Application of the inverse method of characteristics requires a solution of the flowfield depending solely on the input shock shape (represented by the SGC) first. Data at the exit characteristic BC,

Y. Qian, H. Sobieczky and Th. Eggers

392

see Fig. 3a, and data at the chosen exit plane BE at x = x^, (understood to be the inlet plane of the propulsive system), define a new boundary value problem for another run of the method of characteristics. Again the marching direction is in the crossflow direction, to fill the gap CBE with a solution for the flowfield and compatible with the upstream post-shock solution. Fig. 3b shows the added part for an axisymmetric case study. Data for Mach number and flow angle need to be consistent in quantity and slopes in point B connecting the solutions at the given shock wave. These constraints reduce the number of input parameters for the design code which is desirable especially for optimization studies.

a

o

Figure 3. Schematic view (a) of characteristic domains of computation, including added Inlet Flow Control (EFQ), special computed case study illustration (b) for an axisymmetric flow pattern.

PARAMETRIC INPUT VARIATION The above illustration depicts separate sets of input data to be defined in respective planes of 3D space: ICC, FCC, SGC and IFQ. Presently we use software with different degrees of automation of the design process, from an adapted list input to the WIPAR [4] interactive program for waverider design optimization. The computer codes are based on tools for rapid general geometry definition [7], with special emphasis to an ability of creating realistic shapes by a small set of input parameters which are tailored to target specific aerodynamic or structural qualities. Introducing the new OA concept to the proven OC software, it is the SGC and IFQ parameters which need to be added to input data. Variation of only a few parameters leads to a rich variety of resulting shapes which may therefore easily be grouped closely around the optimum of a chosen objective function. So far, such variation has been used for transonic and supersonic airfoil and wing optimization in a 'manual' approach; for supersonic and hypersonic applications and using our rapid Euler-accurate inverse method of characteristics we will have an efficient tool for future automated design and optimization procedures of inlet diffuser ramps and waverider forebody geometries. An illustration of the variability of shapes is shown in Fig. 4 by depicting an input data set in the simple "list input" mode, for a typical forebody as depicted in Fig. 4. The advantage of using key curves which may be composed be several intervals of basic functions defined by an input parameter 'G' is the strong shape control avoiding wavy curves between given support points of, for instance, a spline representation.

Waverider design with parametric flow quality control

393

An idea about the variability of shapes obtained by key input data is given by illustration of two different waveriders in Fig. 5, which can be obtained by changing the few numbers of only one line in the key input list Fig. 4. SHOCK GENERATRIX CURVE DEFINITION KEY U F(U) SI G-0.0000 0.0000 1.72 3. 6. 1.0000 1.2000 1.0 4. 6. 6. 10. 10.200 0. 9.

-S2-1.0

PLANFORM AND LEADING EDGE GEOMETRY DEFINITION U -F(U)—- -SlGS2--- ---El--- ---E2 KEY 0. 10. 0.0000 0. 0 0. 1. 2. 10. 1.00 0.7854 0. 9 11. 0.0000 0.0000 1. 8 11. 1.5708 250.00 0. 9 12. 0.0000 -250.0 0. 8 12. 1.5708 0.0000 0. 9 13. 0.0000 100.00 0. 6. 0. 1. 0 13. 0.7854 0.0000 0. 9

Figure 4. Input data definition by the "key functions technique": Shock Generatrix Curve (SGC = key 6), Support points distribution along leading edge (key 10), Inlet Capture Curve coordinates y, z (ICC = keys 11, 12) and Flow Capture Curve (FCC = key 13). These data, with freestream Mach number (M^o = 2), define a waverider forebody geometry and inviscid flow properties completely.

The WIPAR interactive computer code written by Center [4] employs this kind of parameter input for usage on a graphic workstation. The new OA concept still needs to be introduced into this software.

Figures. Variability of waveriders illustrated for two case studies: Geometry and pressure distribution obtained by modification of FCC (key 13 input data).

394

Y. Qian, H. Sobieczky and Th. Eggers

PRELIMINARY CONCLUSIONS FOR THE NEW OA CONCEPT For the new OA extensions of the concept, only a few case studies have been carried out so far. Inlet ramps in 2D or axisymmetric flow as well as with 3D curved shock waves with axially varying strength are being created to investigate robustness of the code applied with a wide variety of input parameters. Curved shock waves allow gaining some advantages over the simple (constant shock strength) OC concept: First, the length of an inlet or forebody ramp may be reduced, for moderate supersonic Mach numbers M^o up to 5%. This will result in reduced skin friction and of course structural weight. Second, the typical conical flow quality with a pressure rise along the wetted surface may be controlled to become accelerated flow due to a resulting convex surface streamlines design. This may influence boundary layer instabilities and transition. Third, convex shape obtained from a curved shock wave input results in a higher volumetric efficiency of the inlet or forebody configuration. All advantages are influencing overall aerodynamic, structural weight and thus economic efficiency: Design and optimization of new configurations using the concept, therefore, could and should be carried out in a multidisciplinary approach. STATUS OF CFD ANALYSIS OF INVERSE OC DESIGN RESULTS AT DLR We can report about some practical applications to hypersonic waverider design, including onand off design CFD analysis, experimental verification and concepts to integrate waverider forebodies to complete aerospace vehicle configurations, using mature OC software tools at DLR German Aerospace Center. Application of reliable CFD analysis will be needed to estimate the practical value of the design results from the new OA procedure. Eggers [8] has applied the OC concept using the WIPAR software for an extensive study of a variety of waverider shapes in the hypersonic Mach number regime. Here we cite results of a waverider 'WRE8' which was investigated both with CFD methods and experimentally. Comparison of input design and analysis shock shapes are depicted in Fig. 6.

CFDiEULER

,

DESIGN

Figure 6. Comparison [8] of design input shock shape trace (ICC) and shock resulting from an Euler CFD analysis, (a). Surface isobars resulting from both OC design and Euler analysis, (b). Configuration WRE8, in inviscid flow with M^, = 8.

Waverider design with parametric flow quality control

395

As verified in this study and also by other authors [9] using the OC concept, Euler CFDflowsimulation confirms the design concept: Both the shock shapes and pressure distributions agree well enough to use OC - and the extension to OA - based software as efficient design tools for early phase practical vehicle design. A more critical look needs to investigate the role of viscous flow effects inevitably disturbing ideal inviscid flow results and strongly influencing off design aerodynamic performance as well as aerothermodynamic effects in the high (hypersonic) Mach number regime. Fig. 7 shows some results for viscous CFD:

EULER EULER

NAVIER'STOKES

NAVIER-STOKES INTERACTION SHOCK

Figure 7. Comparison [8] of inviscid (Euler) and viscous (Navier-Stokes) CFD analysis: Configuration WRE8, in viscous flow Moo = 8, Re = 1.1 Mill., laminar boundary layer.

As expected from flat plate boundary layer models, which show the strongest displacement thickening near the leading edge, we observe an effective body thickening resulting in a shock strengthening and slight detachment in the leading edge area and disturbance at the upper surface which should be undisturbed free stream in inviscid flow. Shock - boundary layer interaction is strongest therefore near the leading edge. Because of the strong thermal loads these effects require both refined CFD analysis and experimental investigations to fully estimate the practical efficiency of waverider design cases. Results of the OA concept allowing for surface pressures expanding along the wetted surface, therefore will be interesting for CFD case studies of viscous flow effects, especially once the numerical simulation will predict transition reliably: Accelerating flow past curved shock waves will interact substantially different with the boundary layer. With the waverider design results presented in [8], configuration WRE8 was tested also in a hypersonic gun tunnel so that design and numerical analysis can be compared with experimental results for lift and drag coefficients, (Fig. 8). The investigation was carried out to develop several methods analyzing configurations in hypersonic flow. With gained confidence in CFD, the designer has a set of computational tools supporting the development of refined design and optimization strategies.

Y. Qian, H. Sobieczky and Th. Eggers

396

Shock expansion analysis

0.25

Stiock exp + B.L. analysis O

EulerCFD

A

Navier-Stokes CFD

A 0.04 r

0.06

Experiment

0.08

Figure 8. Drag polar results: Comparison of Shock expansion ( + B.L.) methods, Euler and Navier-Stokes CFD, and experiment in a gun tunnel. Results for configuration WRE8 [8], in viscous flow Hx, = 8, Re = 1.1 Mill., laminar boundary layer.

INTEGRATION TO FULL AEROSPACE CONFIGURATIONS During the past 15 years hypersonic transport concepts have been investigated, like the American X-30 or the German SANGER configurations, among others. For theoretical modelling, these vehicles were also stimulating the development of geometry preprocessing for design and analysis, as well as for design modifications applying the OC waverider concept, see Fig. 67 in [7]. A waverider is only a part of the full aerospace vehicle configuration, but its role as diffuser for the propulsive unit makes a careful integration of the components extremely important for the whole design success. Propulsion integration of OC-derived waveriders is therefore a natural next step to having operational design software [9]. Recently a convincing demonstration of the aerodynamic superiority of the waverider concept for configuration forebody design was given [10]: Various forebody studies for a ramjet flight test vehicle 'JAPHAR' resulted in the selection of an OC-derived waverider forebody and its integration to a full configuration including the propulsive unit. CFD analysis for the different shapes was carried out, with the aim to provide suitable flow quality at the inlet plane (Fig. 9). Such studies may be extended to include OA-derived shapes, with CFD studies of viscous interaction along the lower surface down to the propulsion inlet plane. CONCLUSION As illustrated for applied studies with CFD and experiment, several practical studies have proven the OC concept as a useful design tool. The OA extension may further enhance flexibility of the method but a more interdisciplinary approach including structural, thermal and payload optimization seems necessary. For this end, it is extremely helpful for a design computation to obtain the forebody component of a vehicle to take only a few seconds for each configuration on an average workstation. Future improvements could implement interactive graphic tools based on modem computer coding like Java. Our development of geometry generators is therefore aimed at combining OC / OA concepts and direct parametric surface definition to become flexible, flow phenomena-guided and rapid preprocessing software tools for multidisciplinary optimization.

Waverider design with parametric flow quality control

perfect gas

397

equilibrium air

Figure 9. Configuration JAPHAR [10] with a waverider forebody designed using the OC concept. CFD analysis including aerothermodynamic effects, here: investigation of high temperature effects on the shock position. Thefigureshows iso-Mach in a cross section at the waverider forebody exit = propulsive unit inlet plane.

REFEEIENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Sobieczky, H., Dougherty, F.C., Jones, K. D.: Hypersonic Waverider Design from Given Shock Waves. 1st Intl. Waverider Symposium, University of Maryland (1991) Jones, K.D.: A New Inverse Method for Generating High-Speed Aerodynamic Flows with Applications to waverider Design. Ph.D. Thesis, University of Colorado (1993 Bowcutt, K.G., Anderson, J., Capriotti, D.: Viscous Optimized Hypersonic Waveriders. AIAA paper 87-0272 (1987) Center, K. B.: An Interactive Approach to the Design and Optimization of Practical Hypersonic Waveriders. Ph.D. Thesis, University of Colorado (1993) Sobieczky, H.: Configurations with Specified Shock Waves. In: Sobieczky, H. (Ed.), New Design Concepts for High Speed Air Transport. Springer - CISM Series Vol.366, Wien, New York: Springer Verlag (1997), pp. 121-136. Sobieczky, H., Zores, B., Wang, Z., Qian, Y.J.: High Speed Flow Design Using the Theory of Osculating Cones and Axisymmetric Flows. Chinese Journal of Aeronautics, Vol.12, No.l, (1999) Sobieczky, H.: Geometry Generator for CFD and Applied Aerodynamics. In: H. Sobieczky (Ed.), New Design Concepts for High Speed Air Transport. CISM Courses and Lectures Vol. 366. Wien, New York: Springer (1997), pp.137-158 Eggers, Th.: Aerodynamischer Entwurf von Wellenreiterkonfigurationen fUr Hyperschallflugzeuge. Dissertation T.U. Braunschweig, DLR-FB 1999-10 (1999) Takashima, N., Lewis, M.: Engine-Airframe Integration on Osculating Cone Waveriderbased Vehicle Design. AIAA 96-2551, (1996). Eggers, Th., Novelli, Ph.: Design Studies for a Mach 8 Dual Mode Ramjet Flight Test Vehicle. AIAA 99-4877, (1999)

398

Y. Qian, H. Sobieczky and Th. Eggers

APPENDIX: C O M P U T E R C O D E FLUX D I A G R A M FOR INVERSE W A V E R I D E R DESIGN

(^

START

^

'' read design parameters including Mach number, ICC, FCC, SGC, grid size,...

'' determine a fixed reference length and reference radius of the configuration

determine shock shape with constant slope at each X=const section of the wave, for each osculating plane using basic shock curve SGC

inverse MOC

determine start point on the leading edge find stream line from the point downstream to the exit plane shift and rotate the data to shape 3D configuration tlow parameter computation: pressure, density, temperature, velocity components

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

399

I M P R O P E R I N T E G R A L S I N T H E F O R M U L A T I O N OF A SUPERSONIC INVERSE PROBLEM

KISA MATSUSHIMA Makuhari Systems Laboratory, Fujitsu Ltd. 1-9-3, Nakase, Mihama-ku, Chiba, 261-8588, Japan. [email protected]

ABSTRACT An inverse design method based on the residual-correction concept has been developed and applied to aerodynamic design of wings for Japanese SST. The method uses an supersonic inverse problem formulated into integral equations. During the process of the formulation, we have encountered mathematical difficulties to deal with improper integrals suffered by singularities. These difficulties have been solved using the extended idea of Hadamard and Cauchy's concepts of integrals. Thus, the mathematical consistency of the inverse problem formulation has been guaranteed. In addition, a satisfactory design result which has been obtained by means of the method is presented .

KEYWORDS Aerodynamic wing design.

Improper integrals.

Supersonic inverse problems

INTRODUCTION A next generation supersonic transport (SST) is of great interest in Japan as well as in Europe and the U.S. because of the projected trends in the world aviation market in the near future. Research to develop a SST has been performed world-widely from various aspects. One of the challenging technologies is to determine an aerodynamic shape of a supersonic wing using an inverse problem. We have developed a new inverse problem for supersonic flows. It is based on Takanashi's method which was devised for transonic wings in 1985 [1]. The new inverse problem has already been applied to a design problem [2, 3]. From the engineering point of view, the inverse problem works well [4]. However, several mathematical difficulties are left during the formulation process [3]. We have encountered singularities on the integral equations which are characteristic of the supersonic inverse problem. In this article, the difficulties are settled so that the formulation guarantees its mathematical consistency.

400

K. Matsushima

AERODYNAMIC DESIGN METHOD The design procedure for supersonic wings is iterative method. Fig.l illustrates the procedure. The method determines the wing section's geometry which realizes a specified target pressure distribution at all span stations of a wing. First, a baseline shape is to be guessed. Then the flow field around the wing is analyzed by flow simulation to get the current Cp distribution on the wing surface. Next the inverse problem is solved to obtain the geometrical correction value A / corresponding to the difference between target and current pressure distributions ACp. The new wing is designed by modifying the baseline shape using A / . Now, the current shape is updated. The next step is to go back to the flowfield analysis. The flow analysis is conducted to see if the current shape realizes target pressure distribution. If the difference between target and current pressure distributions is negligible, the design is completed. Otherwise, the next step is once again to solve the inverse problem and iterate the design loop until the pressure difference becomes negligible. This iterative procedure of reducing the residual is widely used in numerical aerodynamic design.

Lwerse Problem

[For Wing

t^STOP

]

For Wing

Fig.l Design method of residual-correction concept. There are two primary parts; one is a flow analysis part which conducts grid generation and Navier-Stokes flow simulation. It evaluates the residual. The other is a design part where the inverse problem is solved to update the wing geometry. The design part determines the correction which is expected to compensate for the residual. Both parts are completely independent from each other in terms of their algorithms and equations. The accuracy of a design result depends on the analysis part. The flow analysis of the present design method is done by a Navier-Stokes simulation code. So, designed geometry by the method are valid for Navier-Stokes flows. The efficiency of the design method depends on the design part of the inverse problem. The small perturbation of the potential flow approximation is adopted for the formulation of the inverse problem, which will be discussed in the next section.

Improper integrals in the formulation of a supersonic inverse problem

401

To conclude this section, the author would like to mention one important characteristic of the present design method: the design part handles not the flow quantity itself but the A-form value, which is difference between two states of a flowfield. Accordingly, with the present method, one is approximating the slight change of a Navier-stokes flowfield by the potential flow theories with thin wing approximation.

FORMULATION OF AN INVERSE PROBLEM Basic Equations and Greenes Formula As stated in the previous subsection, the inverse problem of the design part should handle the A-form value, which is difference between two states of a flowfield. The goal of the formulation is to obtain the mathematical function to relate A C p to geometrical correction A / The flowfield considered here is shown in Fig 2. There are two projection plots of a 3-D flowfield. It is a supersonic flowfield with freestream Mach number greater than 1.0 ( Moo > !)• A wing is located at the z = {) plane. The a::-axis is streamwise, the y-axis is spanwise. The basic equations are in A-form. Assuming that the small perturbation theory holds for the difference of flow variables and applying a PrandtlGlauert transformation, the equations are expressed as -A(/)^^ H- /\(j)yy -h /^(i>,Z = 0

(2)

~Af^{x,^)=f3'AM^.y.±0) ACp^ix,

| ) = -2f3'AM^,

(1)

z/, ±0)

(3)

where /? = yJM^ — 1 and, '-h' and '-hO' indicate the physical quantity on the upper surface of a wing, while ' - ' and ' - 0 ' do that of the lower surface. Applying Green's theorem for a hyperbolic equation to Eq.(l), A(/> is formulated into the following analytical form, A{x, y, z) = - i - A j / ^ J [ A ^ < ( e , , , , +0) - A^{(, TJ, -0)]^{X,

y, z; (, 77,0)}.^^'^^;

where 'i!{x,y,z;^,r],C)

= cosh-'

"^ ~ ^

(5)

The domain of integration is denoted by r+ where (x - f )^ - {y — vY ^ 0- I^ is shown as a shadowed area on a wing surface in the right-hand-side sketch of Fig. 2. Then the x

402

K. Matsushima

partial derivatives in Eq.(4) is conducted and integral by parts with respect to ^ is done to avoid singularity. Evevtually, one obtains

z{x-i) 27r 27r J Jr^ Jr. [(y - 7?)2 + z^]r,

(AMi, rj, +0) - A
(7)

vhere

i-c = \/(x - 0^ -{y-

vY - 2^

(8) y.

1

2. C

RegMntfsvrbce wiegratian ( n ^ pluiftrM plane)

Region of integration (Wng sectional plane)

Fig. 2 Coordinate system of a supersonic flowfield for formulation. Improper Integrals and Integral Equations for the Inverse Problem In order to expose the boundary condition ACp and the unknown shape function A / explicitly, one does further calculus with Eq.(7). In fact, ACp is associated with Ac^^: and A / is associated with Ac/)^. So, one should differentiate Eq.(7) with respect to x and z respectively, then take the limit as z -^ 0. There needs to be special treatment of the differentiating process, because the integrand encounters its singularity at the boundary or inside of the region of integration. On the top of this, one should be aware of the order to operate the double integration , in ^ and rj directions, when handling improper integrals. Two extended ideas for integration have to be introduced to evaluate the result of a integral whose integrand is not always definite over the domain. They are found in Refs. 5 and 6. and cited here in the next two subsections. The first idea is the finite part and the other is the principal part of integrals, whose notations used in this particular article are ^ J and ^ / respectively. Before describing the extended concepts of integrals, two fundamental equations for the inverse problem are shown. The first one is obtained by differentiating Eq.(7) with respect to X and taking the limit as 2: -^ 0. To lead the differentiated integral equation to the final form blow, one should use the finite part concept for improper integrals as well as a lot of calculus; Aws{x,y)

=

-Aus{x,y)

{x-OAws{C,v) TT J \ Jr.

dr}

di

(9)

403

Improper integrals in the formulation of a supersonic inverse problem where

1 ^ aA/(:r,^,+0) P^

gA/(:r,f, - 0 ) ^ -) dx

dx

(11)

The other one is obtained by differentiating Eq.(7) with respect to z and taking the limit as 2: —)• 0. Using the principal part concept, the equation yields d^

(12)

where

2-^(ACKa;,|,+0)-ACK^,| = -;^(ACKx,|,+0)-ACKa^,f,-0))

Aw,. =

(13)

A<^,(a;,y,+0) + A
" ~;^l,

^x

+

¥x

)

^^^^

In Eqs.(9) and (12), /S.Ws^x^y) and /S.Wa{x^y) are unknowns. Aw;^ is associated with the thickness change at (a:,y) on a wing. Aty^ is associated with the curvature change of the wing section camber, at (a:,y). As for the order of the double integral, the integral with respect to -q is to be performed first in both fundamental equations. Finaly, the geometrical correction. A / , is calculated using At/;^ and ^Wa'. ^f±{x,^)

= ^l^^[Aw,{^,y)±Aw,{^,y)]d^

(15)

Finite Part of an Integral Improper integrals whose integrand becomes singular at the edge of the integral domain commonly appear in supersonic small perturbation theory. In this case, special attention is needed for taking the partial derivative through the integral, because the function value G{x^x) of Eq.(16) is not definite and Leibnitz's theorem cannot be straightforwardly applied. Here is Leibnitz's theorem; ^j;Gix,x)dy = Gix,x) + f'-^dy

(16)

404

K. Matsushima

Hadamard introduced the finite part concept and showed the operation in general to settle the difficulty associating the indefiniteness of G(x,x). However, general formula does not always give the forms whose values can be evaluated for engineering purpose. In order to attain to Eq.(9), the following formula is more practical. This is referred as the extended finite part of an integral by Lomax, Heaslet and Fuller [5, 6]: d ni^) dx Jt

f{x,y)

\[^)

da ax

J

'I

f{x,y)

d

1

Mx)

^{ x)-y

=dy

i:

da J r<^' f(x,y) dy dx {yja{x) - yf

(17)

With this formula, the notation, which means the finite-part of an integral in this particular article, is introduced. For practical uses, it can be calculated as da dx

f /"^"^

I

f[x,y)

-dy

{yja{x)-y)

where the indefinite integral F[x,y)

= -F(a:,t)+

hm

y-^a{x)

V{x,y)

-F{x,y)

\/«(^) - \

(18)

such that

{yja(x) - yf

dy = F ( x , y) -f- constant

(19)

exists. Principal Part of an Integral The next concept is concerning an improper integral confronted during the process of the z partial derivative of A(/) Sit z = ± 0 to lead to Eq.(12). That integral form is shown later as the left-hand side of Eq.(20). Since the differentiation is to be done with respect to z , z cannot be set by 0 in advance of the differentiation. When z = 0^ the integrand is not always definite. So z is set by 0 using the limiting operation after the diflFerentiation. To treat the improper integral on the practical design process, a useful formula for a generalized principal part of integral has to be introduced. The formula was also devised by Lomax, Heaslet and Fuller [5, 6]:

if Hm

f

d dz

zf(v,z)

a < y
+ T/.(3/,0)

(20)

Ja

The first term of the right-hand side of Eq.(20) is the extended concept of Cauchy's principal part. If the denominator of the integrand of the term was the first order, Cauchy's principal part idea would be directly applied. To express the extended concept of an integral, the special notation that is not generally used is introduced here.

Improper integrals in the formulation of a supersonic inverse problem

405

This improper integral can be evaluated as

7' {y - vY

dr]

H{y,b)-H{y,a)

(21)

if an indefinite integral H{y^ rj) exists such that

m I {y-iY

drj = H{y^ rj) + constant

(22)

APPLICATION TO DESIGN The wing for an SST is aerodynamically designed at Moo = 2.0 by means of the present method. The design is to be done with the wing-fuselage combination of the SST. The wing section geometry at the every span station is to be designed while the shape of the fuselage and the wing planform are fixed. The perspective view of the SST roughly discretized with surface panels is presented in Fig. 3. The planform of the wing-fuselage configuration can be seen, there. To design wings for the wing-fuselage configuration of an SST, Navier-Stokes (N-S) simulation about a wing-fuselage configuration is conducted in the analysis part [2, 3], so that the fuselage effect can be taken into for design. The designed wing geometry is for the wing-fuselage configuration. It might be needless to say that if one conducted flow simulation about a wing alone with this design method, he could design a geometry good for a wing alone system. This is one of the evidences to show the wide flexibility of the method.

Fig.3 Wing-fuselage configuration of SST. A target pressure distribution was prescribed to design a high L/D wing for the SST [4]. It has elliptical load distribution along span-wise direction which minimizes the induced drag. Along the chord-wise direction, it has step function type distribution on the upper surface. The target pressure is plotted with chain lines in Figs 4 and 5. The design method determines the section geometry of the wing which realizes the target pressure distribution at each span-station. For the computation to solve the inverse problem, the half span of the wing is divided into 82 (spanwise) x 50 (chordwise) panels.

406

K. Matsushima

The design results at two important span stations are shown in Figs.4 and 5. Figure 4 presents the wing section geometry and the pressure distribution at 30%-span station. Figure 5 illustrates those at 70%-span station. The initial pressure distribution denoted by ' + ' lines. The final converged state of pressure distribution is indicated by lines of diamond. The target pressure is indicated by chain lines, diamond lines show very good agreement with the chain lines. The wing section geometry is also presented in Figs.4 and 5. The dashed lines indicate the section geometry of the baseline (initial) wing, while the solid lines do that of the designed wing section. The resulting wing realizes almost identical pressure distribution to the target. NAL4BT03 (381) 70% SEMI-SPAN

NAL4eT032(3ai) 30% SEMI-SPAN

0.2

"Desigrt 'Desigrf 'Base'

0.15

• 0.15

~

Targe? - - 0.1

0.1 ^**^\*X'»*'>*^*««••».? 0.05

- f/

-0.1

-0.15

•-«. »-•-« *-«.« *-*-**•.•••«^«^'?«?¥e* .^ + + + + * + + + + + 4.+ *

*

*

*

*

^

*

•

*

^

-

•P

• ..,.<*«--^*x

::;=^^5^^^^''"'*^^^~^-^^,_^^^

- -

0

y

-0.05

-0.1

^** .

'Desigrt •Desigrf » Base' 'Basrf * ' Target* - - \

" * * ' ' ^ ^ 4 . » * * + * '^

/ • " "

•

0.05 •

0

•0.05

**^

+ + + + •»• + ++ 4.+

******

* ..^^^"^

•..«>
-

, 5 j ( K * * » * * * <j » v » 5 e V i ' » * <> *

»''

-0.15

\ Fig.4 Design process at 30% semi-span.

Fig.5 Design process at 70% semi-span.

CONCLUSION The inverse problem for an aerodynamic design of a supersonic wing was studied. This article mainly described the mathematical techniques to treat improper integrals which appeared during the formulation process of the inverse problem. Being consistent with the residual-correction concept of the design method, the inverse problem handles not the flow quantity itself but A-form values, which are difference between two states of a flowfield. It wcis formulated into integral equations. To solve the equations, difficulty occurred to calculate the resulting values of improper integrals. This could not be settled even by commonly used numerical approximation technique, but should be settled mathematically. Introducing two extended idea of an integral, the formulation was completed holding mathematical consistency. An application of the method to the practical design of a SST wing was performed. The design method using the inverse problem worked very well for the application. This is only one example. However, this proves that the present design method can be applied to wide variety of wing design in supersonic flows, together with the mathematical consistency assured by this article,

Improper integrals in the formulation of a supersonic inverse problem

407

REFERENCES 1. Takanashi, S.: Iterative Three-Dimensional Transonic Wing Design Using Integral Equations. (1985). J. Aircraft 22, 8, pp.655-660. 2. Matsushima, K., Iwamiya, T., et. al.: Aerodynamic Wing Design for NAL's SST Using an Iterative Inverse Approach. (1998). International CFD Workshop for SST Design, National Aerospace Laboratory. 3. Matsushima, K., Jeong, S. et. al.: Supersonic Inverse Design Method and Its Application to Japanese SST. (1999). Lecture Note in Physics, Proceedings 16th ICNMFD, pp. 79-84. 4. Iwamiya, T., Yoshida, K. et. al.: Aerodynamic Design of Supersonic Experimental Airplane. (2000). International Workshop on Numerical Simulation Technology for Design of next Generation Supersonic Civil Transport, National Aerospace Laboratory. 5. Lomax, H., Heaslet, M. A. and Fuller, F. B.: Integrals and Integral Equations in Linearized Wing Theory. (1951). NACA Rep. 1054. 6. Heaslet, M. A. and Lomax, H.: Supersonic and Transonic Small Perturbation Theory. (1954). Vol. 6 of High-Speed Aerodynamics and Jet Propulsion, pp.122-344, Princeton University Press.

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

409

A FUNCTION ESTIMATION APPROACH FOR THE IDENTIFICATION OF THE TRANSIENT INLET PROFILE IN PARALLEL PLATE CHANNELS M. J. Cola^o and H. R. B. Orlande Department of Mechanical Engineering, EE/COPPE Federal University of Rio de Janeiro, UFRJ Cid. Universitaria, Cx. Postal: 68503 Rio de Janeiro, RJ, 21945-970, Brazil

ABSTRACT This paper deals with the use of the conjugate gradient method of function estimation for the identification of the inlet temperature profile, also supposed to vary with time, in a parallel plate channel. The fluid flow is assumed to be laminar and hydrodynamically developed. The channel walls are subjected to a constant temperature boundary condition. Temperature measurements taken downstream are used in the inverse analysis. The accuracy of the present solution approach is examined by using simulated measurements containing random errors, for strict cases involving functional forms with discontinuities or sharp-corners for the inlet temperature profile. KEYWORDS Conjugate gradient method, function estimation, adjoint problem, forced convection, laminar flow, transient inlet profile INTRODUCTION Heat conduction was the first heat transfer mode to be addressed in the inverse problem literature. This is probably because of the interest of researchers involved with the space program in the 50's and 60's, aiming at the accurate estimation of thermal properties of heat shields, as well as of the heat flux on the surface of re-entrant vehicles. Some heuristic methods of solution for inverse problems, which were based more on pure intuition than on mathematical formality, were developed in the 50's. Later in the 60's and 70's, most of the

410

MJ. Colago and H.R.B. Orlande

methods, which are in common use nowadays, were formalized in terms of their capabiUties to treat ill-posed unstable problems. The basis of such formal methods resides on the idea of reformulating the inverse problem in terms of an approximate well-posed problem, by utilizing some kind of regularization (stabilization) technique. In this sense, it is recognized here the pioneering works of scientists who found different forms of overcoming the instabilities of inverse problems, including A. N. Tikhonov[l-3], O. M. Alifanov[4-6] and J. V. Beck[7-9]. The interest on the solution of inverse heat convection problems is more recent than on the solution of inverse heat conduction problems. To the best of the authors' knowledge, the first article dealing with an inverse heat convection problem is the one due to Moutsoglou, from 1989 [10]. Afterwards, several other articles involving inverse convection appeared in the literature[ 11-23]. In their majority, those papers dealt with forced convection inside tubes or channels [11-19,21,23], with unknown wall heat flux [11,12,14-16,19,21,23] or unknown inlet condition [13,17,18]. With respect to the unknown inlet condition, either a uniform transient inlet temperature [17], or a steady-state inlet temperature profile (varying across the channel), have been considered as the unknown quantity [13,18]. In this paper, we present the solution of the inverse problem of estimating the temperature inlet profile, also supposed to vary with time, in a parallel plate channel. As the solution technique, we apply the conjugate gradient method of ftmction estimation [4-6,12,14-19,24], by assuming that no information is available regarding the time and spatial variations of the unknown function. Simulated temperature measurements taken inside the channel are used in the inverse analysis, in order to address the accuracy of the present solution technique, as well as to design the experiment with respect to the number and locations of temperature sensors. Basic steps of the conjugate gradient method of function estimation include: (i) The Direct Problem, (ii) The Inverse Problem, (iii) The Sensitivity Problem, (iv) The Adjoint Problem, (v) The Gradient Equation, (vi) The Iterative Procedure, (vii) The Stopping Criterion and (viii) The Computational Algorithm. Details of such steps, as applied to the present inverse problem, are described next. DIRECT PROBLEM The physical problem considered in this work involves the laminar hydrodynamicallydeveloped forced convection of a Newtonian fluid in a parallel plate channel. The inlet temperature, f(y,t), may vary with time as well as across the channel. The channel walls, separated by a distance 2h, are assumed to be maintained at a constant temperature Tj, equal to the initial fluid temperature inside the channel. The formulation of such physical problem is given in dimensionless form as:

u( Y ^ 5 0 0-0 0-0 0 = F(Y,T)

0-0

inO0;forT>0

(l.a)

at Y=0, X > 0; for T > 0 atY=l,X>0;forT>0 atX=0, 0 < Y < l ; f o r T > 0 ForT = 0 ; i n O < Y < l , X > 0

(l.b) (l.c) (l.d) (l.e)

where the following dimensionless groups were defined

411

Identification of the transient inlet profile at 4h^ Y=^ ax , , 2 u„4h

X - „

(2.a) (2.b) (2.C)

(2.d) and the dimensionless velocity profile is given by: U(Y) = 6Y(1-Y)

(2.e)

The direct problem is concerned with the determination of the temperature distribution inside the channel, from the knowledge of the velocity profile (2.e) and from the inlet condition F(Y,T). INVERSE PROBLEM The inverse problem under picture in this paper is concerned with the estimation of the time and spatial variations of the inlet temperature, F(Y,T), by using temperature measurements taken downstream. The inverse problem is reformulated as a minimization problem involving the following objective functional: S[F(Y,r)]=

ry{z„(O-0[r„,r;F(Y,r)]}^d.

(3)

where N is the number of sensors used in the analysis, Zn(T) are the measured temperatures at the position rn, while 0[rn,T;F(Y,T)] are the estimated temperatures at the measurement positions. The minimization of the objective functional (3) is obtained through the conjugate gradient method [4-6,12,14-19,24]. Two auxihary problems, known as the sensitivity and adjoint problems, are required for the implementation of the iterative procedure of such a method, as described next. For further details on the derivation of these problems the reader is referred to references [4-6,12,14-19,24]. SENSITIVITY PROBLEM In order to develop the sensitivity problem, we assume that the temperature 0(X,Y,T) undergoes a variation A0(X,Y,T), when the inlet temperature undergoes a variation AF(Y,T). By substituting into the direct problem (1) 0(X,Y,T) by [0(X,Y,T)+A0(X,Y,T)] and F(Y,T) by [F(Y,T)+AF(Y,T)], and then subtracting from the resulting equations the original direct

412

M.J. Colago and H.R.B. Orlande

problem, we obtain the following sensitivity problem for the sensitivity function A 0 ( X , Y , T ) [46,12,14-19,24]: — - T - - ^ — + U(Y)—— OX" ox oX A0 = 0 A0 - 0 A 0 - A F ( Y, T ) A0 = 0

inO0;forT>0

(4.a)

at Y=0, X > 0; for x > 0 atY=l,X>0;forT>0 at X=0, 0 < Y < 1; for x > 0 forx = 0 ; i n O < Y < l , X > 0

(4.b) (4.c) (4.d) (4.e)

ADJOINT PROBLEM The adjoint problem is obtained by multiplying equation (l.a) by the Lagrange Multiplier X(X,Y,x), integrating the resulting expression over the time and space domains and adding the result to the functional given by equation (3). We obtain:

S[F(Y,0]=

f'y{z„(r)-0[r„,r;F{Y,r)]pdr

JY=0 JX=0

Jr=0

I

^ ^

^e(X,Y,r)_^(^)a0(X,Y,r)l

dr

^ ^

+

dX

^^^^y

I

We now perturb F(Y,x) by AF(Y,x) and 0(X,Y,x) by A0(X,Y,x) in equation (5) and subtract equation (5) from the resulting expression to get the variation AS[F(Y,x)] of the functional S[F(Y,x)]. By employing integration by parts, utilizing the initial and boundary conditions of the sensitivity problem and also requiring that the coefficients of A0(X,Y,x) in the resulting equation should vanish, the following adjoint problem is obtained [4-6,12,14-19,24]: d' X dX - + ^ ^ U ( Y ) ^ +2 y { 0 [ r , r ; F ( Y , r ) ] - Z „ ( r ) } ^ ( r - r J =O

^^^

n=l

inO0 where S{-) is the Dirac delta function, and the boundary conditions become A=0 A=:0 A= 0 A= 0 where L is the channel dimensionless length.

at Y=0, 0 < X < L; for X > 0 atY=l,0<X0 atX=L,00 forx=Xf;inO
(6.b) (6.c) (6.d) (6.e)

413

Identification of the transient inlet profile GRADIENT EQUATION In the process of obtaining the adjoint problem, the variation of the functional reduces to AS[F(Y,r)]= j

j U(Y);L(0,Y,r)AF(Y,r)dYdr

(7.a)

By assuming that the function F(Y,T) belongs to the space of square integrable functions in 0 < X < If, 0 < Y < 1, we can write AS[F(Y,r)]= I

I VS[F(Y,r)]AF(Y,r)dYdr

(7.b)

Hence, by comparing equations (7.a) and (7.b), we can obtain the gradient equation for the functional as: V S [ F ( Y, r ) ] = U ( Y ) /I ( 0, Y, r )

(8)

ITERATIVE PROCEDURE The iterative procedure of the conjugate gradient method, as applied to the estimation of the function F(Y,T) is given as [4-6,12,14-19,24]: F^^'(Y,r) = F ^ ( Y , r ) - ^ ' d ' ( Y , r )

(9.a)

where k is the number of iterations. The direction of descent d'^(Y,T) is obtained from d ' ( Y, T ) = V S [ F ' ( Y, T ) ] + Y ^ d '-' ( Y, T )

(9.b)

The conjugation coefficient y^ can be obtained from the Fletcher-Reeves expression as f

y^^

f''{vs[F^(Y,T)])'dTdY

JY-OJJ-O

fork=l,2,...

(9.c)

f f''{vs[F'^-i(Y,T)]}'dTdY J Y=0 J 1=0 with y ^ = 0 for k = 0 The search step size p*^ is obtained by minimizing S[F^^\Y,T)] with respect to p . The following expression results

414

M.J. Colago andH.R.B. Orlande f"y{0[r„,r;F^(Y,r)]-Z„(r)}A0[r„,r;d^(Y,r)]d. (9.d) I

y { A 0 [ r „ , r ; d ^ ( Y , T ) ] } ' dr

where A0[rn,T;d''(Y,T)] is the solution of the sensitivity problem (4), obtained by setting AF(Y,T)=d'(Y,T). STOPPING CRITERION The iterative procedure of the conjugate gradient method does not possess by itself the regularization property required to obtain stable solutions for inverse problems, such as the one under picture in this paper. However, the use of the stopping criterion based on the Discrepancy Principle gives the conjugate gradient method an iterative regularization character [4-6]. In this case, the stopping criterion is given by S[F(Y,r)]
(10)

where 5'[F(Y,T)] is computed with equation (3). The tolerance c is chosen so that smooth solutions are obtained with measurements containing random errors. It is assumed that the solution is sufficiently accurate when |z„(r)-0[r„,r;F(Y,r)]|«c7

(11)

where a is the constant standard deviation of the measurement errors. Thus, s is obtained from equation (3) as .^-No-'r^

(12)

For cases involving errorless measurements, s can be specified a priori as a sufficiently small number, if the sensors are appropriately located. For those cases involving measurements with unknown standard deviation, an alternative approach based on an additional measurement can be used, as described in [6,24], COMPUTATIONAL ALGORITHM Suppose an initial guess F^(Y,T) is available for the function

F(Y,T).

Set k = 0 and then:

Step L Solve the direct problem (1) and compute 0(X,Y,T), based on F'^(Y,T). Step 2. Check the stopping criterion (10). Continue if not satisfied. Step 3. Knowing 0 ( X , Y , T ) and measured temperatures Zn(T), solve the adjoint problem (6) and compute >^(0,Y,T).

Step 4. Knowing A<(0,Y,T), compute VS[F''(Y,T)] from equation (8).

415

Identification of the transient inlet profile Step 5. Knowing the gradient V S [ F \ Y , T ) ] , compute y*" from equation (9.c). Step 6. Set AF(Y,T)=d*'(Y,T) and solve the sensitivity problem (4) to obtain A0[rn,T;d''(Y,T)]. Step 7. Knowing A0[rn,T;d'^(Y,T)], compute the search step size p^ from equation (9.d). Step 8. Knowing the search step size (3^ and the direction of descent d'^(Y,T), compute the new estimate F*^^^(Y,T) from equation (9.a), and return to step 1. RESULTS AND DISCUSSIONS In order to illustrate the accuracy of the present function estimation approach, we used simulated measurements containing random errors, normally distributed, with zero mean and constant standard-deviation (a). Such simulated measurements were obtained by adding a random noise to the solution of the direct problem for an a priori established functional form for the inlet profile F(Y,T). For the results presented below, the inlet profile was taken as F(Y,T)=FY(Y)FX(T). Functional forms containing discontinuities and sharp-comers were examined for FY(Y) and FX(T), because they represent the most difficult functions to be recovered by inverse analysis. Let us consider a test-case with FY(Y) and Fx(x) taken, respectively, in the form: ^

,

,

fl

for

0
0.3
and

0.7
'^^^Hl

for

1 2500r-0.5 F.(Y) = -2500r +4.5

for 0
^''-"^ (13.b)

The values selected for the dimensionless variables correspond to a physical case involving the flow of water, initially at 20°C, flowing at a mean-velocity of O.Olm/s in a channel 0.05m high and 0.36m long. The final physical time corresponds to 35s. For the results presented below, we assumed available 181 temperature readings per sensor, which corresponds to one measurement taken every 0.19s. The direct, sensitivity and adjoint problems were solved by finite-differences, by using an upwind discretization for the convection terms. By examining equations (6.b,c,e) and (8), we note that the gradient equation is null at the final time, as well as at the channel walls. As a result, the initial guess used for the unknown function at such time and locations is not changed by the iterative procedure of the conjugate gradient method. Therefore, instabilities on the estimated function are expected in the neighborhood of such points. One approach to overcome such instabilities at if is to consider a final time larger than of interest for the problem [24]. In the present case we used Tf=0.0024 for the inverse analysis and increased the number of measurements in time accordingly; but only the results in the domain of interest, i.e., 0 < x < 0.002 are presented below. Also, sensors were not located at the channel walls, because no useful information would be recovered from the measurements taken at such positions. Figures 1 .a,b present the results for the spatial and time variations of F(Y,T), respectively, obtained with measurements containing random errors of standard deviation a=0.01Zmax, where Zmax is the maximum measured temperature. An initial guess of F(Y,T)=0 was used for the iterative procedure of the conjugate gradient method. Eighteen sensors uniformly distributed across the channel section, at the longitudinal position X=3.9xl0"'^, were used in the

416

M.J. Colago and H.R.B. Orlande

analysis. Such longitudinal position corresponds to 19% of the channel length (L=2.05x10'); hence the sensors were located sufficiently far from the channel inlet. Figures 1 .a,b show that very accurate estimates can be obtained for the spatial and time variations of F(Y,T), with 18 sensors located at X=3.9xl0''^, for different times and transversal positions. As expected, the estimation of the time variation of the function was much more accurate than of the spatial variation, because the time frequency of measurements was ten times larger than the spatial distribution of the sensors across the channel. Figure 1 .a clearly shows the effect of the null gradient at the channel walls, which caused deviations from the exact solutions at the neighborhood of Y=0 and Y=l. However, note that the effect of the null gradient at the final time is very little noticeable in Fig. 1 .b. Such is the case because a final time 20% larger than that of interest was used in the analysis, as described above. 5-1

Exact

-+

Estimated (T = 3.33x10-4)

A

Estimated (T = 1.00x10'3)

u

Estimated (T = 1 . 7 0 X 1 0 "3)

E

Q

1

'

1

'

\

'

0.2

Fig. l.a. Step

0.4 0.6 0.8 Dimensionless position, Y variation of F(Y,T) across the

I

1.0

channel given by eq. (13.a). Exact

-+

Estimated (Y = 0.21)

A

Estimated (Y = 0.47)

CI

Estimated (Y = 0.74)

Q)

E

Q

n—'—I—'—r O.OE+O

Fig. l.b.

4.0E-4

8.0E-4 1.2E-3 1.6E-3 2.0E-3 Dimensionless time, T Triangular variation of F(Y,T) in time given

by eq. (13.b).

Identification of the transient inlet profile

All

The effects on the inverse problem solution of the longitudinal position and of the number of sensors are now examined. Table 1 illustrates the RMS errors for different test-cases, obtained with the same experimental conditions of the results shown in Figs. l.a,b, except for the number and longitudinal position of the sensors. Note that test-case 2 corresponds to the one illustrated in Figs. 1 .a,b. As expected, the RMS error decreased when the sensors were located closer to the channel inlet and when more sensors were used in the analysis. In fact, completely unrealistic spatial distributions were estimated with only 8 sensors, even when they were located very close to the channel inlet at X= 1.3x10'"^. On the other hand, the time variations were quite reasonably estimated, even for such low number of sensors. The RMS error is defined as y,y[Fex(Y„,rJ-F,3,(Y„,rJ]^ K

(14)

m=l n=l

where M is the number of transient measurements taken per sensor, while the subscripts "ex ' and "est" indicate exact and estimated quantities, respectively. Table 1. RMS errors for different number and longitudinal position for the sensors Test-case Number of Sensors Longitudinal location SRMS 1 0.20 1.3x10-' 18 2 0.27 3.9x10-' 18 0.34 3 5.2x10-' 18 4 0.56 1.3x10-' 8 CONCLUSIONS In this paper we applied the conjugate gradient method for the identification of the transient inlet profile in a parallel plate channel, by using a function estimation approach, where no information is assumed available regarding the functional form of the unknown. Results obtained with simulated temperature measurements reveal that quite accurate estimates can be obtained for the time and spatial variations of the unknown function, if the proper number of sensors is used and if they are suitably located inside the channel. The use of a final time larger than that of interest can reduce the effects of the null-gradient and improve the accuracy of the estimated function. REFERENCES

4.

Tikhonov, A. N., "Solufion of Incorrectly Formulated Problems and the Regularizafion Method", Soviet Math. Dokl, 4(4), 1035-1038, 1963. Tikhonov, A. N., "Regularization of Incorrectly Posed Problems", Soviet Math. DokL, 4(6), 1624-1627, 1963. Tikhonov, A. N. and Arsenin, V. Y., Solution of Ill-Posed Problems, Winston & Sons, Washington, DC, 1977. Alifanov, O. M., "Determinafion of Heat Loads from a Solution of the Nonlinear Inverse Problem", High Temperature, 15(3), 498-504, 1977.

418 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

MJ. Colago and H.R.B. Orlande Alifanov, O. M., "Solution of an Inverse Problem of Heat-Conduction by Iterative Methods", J. Eng. Phys., 26(4), 471-476, 1974. Alifanov, O. M., Inverse Heat Transfer Problems, Springer-Verlag, New York, 1994. Beck, J. v . , "Calculation of Surface Heat Flux from an Internal Temperature History", ASME Paper 62-HT-46, 1962. Beck, J. V. and Arnold, K. J., Parameter Estimation in Engineering and Science, Wiley Interscience, New York, 1977. Beck, J. v., Blackwell, B. and St. Clair, C. R., Inverse Heat Conduction: Ill-Posed Problems, Wiley Interscience, New York, 1985. Moutsouglou, A., "An Inverse Convection Problem", J. Heat Transfer, 111, 37-43, 1989. Moutsouglou, A., "Solution of an Elliptic Inverse Convection Problem Using a Whole Domain Regularization Technique", AIAA J. Thermophysics, 4, 341-349, 1990. Huang, C. H. and Ozisik, M. N., "Inverse Problem of Determining Unknown Wall Heat Flux in Laminar Flow Through a Parallel Plate", Numerical Heat Transfer, Part. A, 21, 55-70, 1992. Raghunath, R., "Determining Entrance Conditions From Downstream Measurements", Int. Comm. Heat Mass Transfer, 20, 173-183, 1993. Machado, H. A. and Orlande, H. R. B., "Estimation of the Timewise and Spacewise Variation of the Wall Heat Flux to a Non-Newtonian Fluid in a Parallel Plate Channel", Proceedings of the Int. Symp. On Transient Convective Heat Transfer, 587-596, Cesme, Turkey, August, 1996. Machado, H. A. and Orlande, H. R. B., "Inverse Analysis of Estimating the Timewise and Spacewise Variation of the Wall Heat Flux in a Parallel Plate Channel", Int. J. Num. Meth. Heat & Fluid Flow, 1, 696-710, 1997. Machado, H. A. and Orlande, H. R. B., "Inverse Problem for Estimating the Heat Flux to a Non-Newtonian Fluid in a Parallel Plate Channel", Journal of the Brazilian Society of Mechanical Sciences, 20, 51 -61, 1998. Bokar, J. C. and Ozisik, M. N., "Inverse Analysis for Estimating the Time Varying Inlet Temperature in Laminar Flow Inside a Parallel Plate Duct", Int. J. Heat Mass Transfer, 38,39-45,1995. Liu, F. B. and Ozisik, M. N., "Estimation of Inlet Temperature Profile in Laminar Duct Flow", Inverse Problems in Engineering, 3, 131-141, 1996. Liu, F. B. and Ozisik, M. N., "Inverse Analysis of Transient Turbulent Forced Convection Inside Parallel Plates", Int. J. Heat Mass Transfer, 39 2615-2618, 1996. Li, M., Prud'homme, M. and Nguyen, T. "A Numerical Solution for the Inverse Natural - Convection Problem", Numerical Heat Transfer, Part B, 307-321, 1995. Szczygiel, I., "Estimation of the Boundary Conditions in Conventional Heat Transfer Problems", in Inverse Problems in Heat Transfer and Fluid Flow., vol. 2, HTD - Vol. 340, 17-24, G.S. Dulikravich and K.A. Woodburry (eds.), ASME, 1997. Moaveni, S. "An Inverse Problem Involving Thermal Energy Equation", in Inverse Problems in Heat Transfer and Fluid Flow., vol. 2, HTD - Vol. 340, 49-54, G. S. Dulikravich and K. A. Woodburry (eds.), ASME, 1997. Aparecido, J. B. and Ozisik, M. N., " Nonlinear Parameter Estimation in Laminar Forced Convection inside a Circular Tube", 3^"^ International Conference on Inverse Problems in Engineering, Port Ludlow, June 13-19, 1999. Ozisik, M. N. and Orlande, H. R. B., Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, 2000.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

419

INVERSE AERODYNAMIC SHAPE DESIGN FOR IMPROVED WING BUFFET-ONSET PERFORMANCE B.I. SOEMARWOTO*, Th.E. LABRUJERE*, M. LABAN* and H. YANSYAH** * Department of Theoretical Aerodynamics and Aeroelasticity^ National Aerospace Laboratory (NLR)^ Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands. "^^Department of Aerodynamics Configuration and Performance, Nusantara Aircraft Industries (IPTN)^ Jl. Pajajaran 154, Bandung 40174, Indonesia. ABSTRACT The paper describes the re-design of a wing/fuselage aircraft configuration in transonic flow with the objective to improve the buffet onset boundary in terms of lift coefficient CL at a cruise Mach number of 0.8. This re-design is done by prescribing a pressure distribution on the wing surface that implies a higher CL at the buffet-onset boundary. The inverse problem refers to the computation of a wing shape that produces the prescribed pressure distribution. A defect correction approach is applied for solving the re-design problem, where an inverse design methodology for isolated wings in inviscid flow is combined with viscous flow analysis code for wing/body configurations. The defect correction makes use of the design code SYN87 for isolated wings in inviscid flow and the analysis code MATRICS-V for wing/fuselage configuration in viscous flow. Computational results are shown for improvement of the buffet-onset performance of a wing/fuselage configuration at the design Mach number of 0.8. KEYWORDS Inverse method, aerodynamic design, shape optimization, buffet-onset, viscous-inviscid interaction, optimal control INTRODUCTION Transonic buffet is an unsteady fiow phenomenon that occurs when transonic shock waves induce a certain type of fiow separation. Figure 1 gives an illustration of buflPet flow around a circular-arc wing section as computed by the NLR's ENFLOW CFD system [4]. Each picture represents instantaneous Mach contours. The upper left picture shows an incipient flow separation induced by the shock wave near the trailing edge on the lower surface. As this lower surface flow separation gets larger as indicated in the upper right picture, the lift on the airfoil changes with the effect of moving the shock wave upstream and reducing the shock strength. This reduction in turn allows the flow to re-attach as shown in the lower left picture. Finally, as the lower surface flow attaches, the shock wave starts to form

420

B.I. Soemarwoto et al.

Figure 1: Buffet flow around a circular-arc wing section computed by ENFLOW (Ref. [4]). again at the trailing edge with increasing strength until it induces another flow separation. The process repeats itself forming a self-sustained oscillation. The same flow mechanism occurs at the upper surface with a phase shift of 180 degrees. The oscillatory aerodynamic loads and structural deformations associated with buffeting flow lead to a so-called buflfet-onset boundary, defined by a curve on the CL-M plane, where CL and M are the lift coefficient and Mach number, respectively. Below this curve, the aircraft operational conditions are considered to be free from buffet. The certification requirement states that CL,buffet > '^-^CL,cruise• The present re-design concerns the improvement of the buffet-onset performance in terms of CL at the design Mach number. This re-design is done by prescribing a pressure distribution on the wing surface that implies a higher CL at the buflPet-onset boundary. The inverse problem to be described refers to the computation of a wing shape that produces the prescribed pressure distribution. The flow associated with the buffet-onset boundary is dominated by turbulence and strongly unsteady. To deal with such a flow directly in a design procedure would entail the use of a flow analysis tool based on the time-accurate Reynolds-averaged Navier-Stokes equations, that is capable of resolving the unsteadiness. Such a direct approach for designing three-dimensional bodies such as wings is unnecessary. In the present study, the inverse problem will be solved using defect correction which is a well-known approach in numerical analysis. This approach is based on the code SYN87 (Refs. [3],[5]) for isolated wing design in inviscid flow, in combination with the code MATRICS-V (Ref. [6]) for the analysis of a wing/fuselage configuration in viscous flow. This paper is organized as follows. First, the design problem will be described, compris-

Inverse aerodynamic shape design

All

ing the definition of the existing geometry and the flow conditions, and the flow analysis for the existing geometry. This analysis is followed by a description of the design algorithm with explanation on the defect correction approach. Subsequently, computational results are demonstrated and conclusions are drawn in the last section. BUFFET B O U N D A R Y ANALYSIS OF EXISTING CONFIGURATION Figure 2 shows the existing wing/fuselage configuration and the wing sections. This configuration represents a typical civil transport aircraft with transonic cruise speed. The design point Mach number has been specified as M = 0.8. The Reynolds number of 3.0 x 10^ typical for wind tunnel measurement is used for the viscous flow computations, with the laminar to turbulent boundary layer transition specified at 30% and 7% of local chord on the lower and upper surface of the wing, respectively.

(a) Half wing/fuselage configuration.

(b) Wing sections along the span.

Figure 2: Existing wing/fuselage configuration. The existing wing/fuselage configuration is analyzed with respect to its aerodynamic characteristics using MATRICS-V, which is a computational aerodynamics code for drag prediction to support the design of jet transport aircraft. MATRICS-V is based on fullpotential flow in quasi-simultaneous interaction with boundary-layers on the wing. Flows with considerable separation can be modeled by MATRICS-V. This feature allows one to compute the off-design aerodynamic boundaries, such as the buflfet-onset boundary. The buffet-onset boundary is determined based on the boundary layer parameter indicating shock-induced flow separation and the extent of the flow separation along the wing span that normally leads to buffet. Figure 3 shows the distribution of the boundary layer shape factor H, (denoted as Hbar in the flgure), for different values of d at the design Mach number M = 0.80, where r] and x/c are the non-dimensionalized length on the wing in the spanwise and chordwise directions, respectively. The criteria for buffet-onset is that the maximum value of H indicating shock-induced flow separation is larger than or equal to 3.3, and that this flow separation occurs over more than 25% of the wing span. In Figure 3 on the left, H reaches a maximum value of Hmax at around x/c = 0.5 corresponding to a location near the shock wave. On the right, the figure shows the spanwise distribution of Hmax- In order to determine the buffet-onset boundary, the curves are interpolated for

422

B.I. Soemarwoto et al.

the CL value that corresponds to Hmax > 3.3 covering 0.25 of ry. This interpolation gives CL ^ 0.56 at the buffet onset boundary. The objective in the present study is to re-design the wing shape by means of an inverse design methodology such that the CL value at the buffet-onset boundary is increased, while keeping the planform fixed. Data obtained from flow solutions at various C, 4 §3.5 2

3

O

i2.5 w c Z 2 E

X (0

1

E0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Data interpolated in C|_ to 25% shock-induced separation in span

0^=0.470

0^=0.435

Figure 3: Buffet-onset boundary of the existing wing/fuselage configuration.

DEFECT CORRECTION APPROACH In order to explain the defect correction approach (Refs. [2], [1]), the following system can be considered: P = Nx, (1) where A^ is a non-linear operator applied to a control variable x resulting in a state variable p . In the case of a flow analysis such as that described in the preceeding section, N, x and p can be considered as representing the code MATRICS-V for the analysis of wing/fuselage configuration, the wing geometry and the pressure distribution, respectively, with x specified. Re-designing the wing geometry by prescribing a target pressure distribution can thus be interpreted as solving the system (1) for x = x* with a given p = p^ : X* = N-'pt.

(2)

For a current approximation x'' of x*, the defect can be defined as: ^(x^) = TVx^ - p ,

(3)

Inverse aerodynamic shape design

423

The error of x*^ with respect to x* is written as: e'^ = x ^ - x * ,

(4)

which can be expressed as: e^ - N-\^t

+ ^(x^)) - N-^^t

= N-'dix'^).

(5)

In the defect correction approach, the error e^ is estimated by e^ by means of an approximate inverse operator N~^: e^ = iV-^d(x'^). (6) In view of equation (3), the above equation can be worked out to yield: e^ = N-^Nx^

(7)

- N-^Pf

A new iterate is determined as follows: x^+i = x^ - e ^

(8)

The new geometry can be substituted into equation (7), forming an iterative procedure until convergence is achieved. In the present study, however, instead of following the iterative procedure, a one-step approach has been chosen, involving only one N and two N~'^ operations, and one geometry update. In this one-step approach, pt in the right hand side of equation (7) is replaced by its approximation pj, and equation (7) is written as: e'' = N-^Nx''-N-%,

.

(9)

where x^ represents the existing wing geometry. In the present study, the approximate inverse operator N"^ is represented by the code SYN87 for geometry design of wing-alone configuration in inviscid flow modeled by the Euler equations. In dealing with the geometry design, the minimization of the following cost functional is considered:

I=U

{p-PifdS

(10)

where pt is the target pressure, p is the pressure of the current geometry, and Bw refers to the wing surface. The iterative solution method for the minimization of the functional is based on optimal control theory using the wing geometry as the control. In each iteration a variation of the wing shape is considered. These variations will cause variations of the actual pressure and consequently variations in the cost functional (10). Each iteration requires the solution of an adjoint problem for the co-state variable which is defined using the flow solution for the current geometry. The pressure distributions coefficient Cp on the wing sections of the wing-alone configuration in inviscid flow (representing a result of A^x^) is depicted in Figure 4 at the design Mach number and at a lift coefl&cient of 0.56 slightly above the buffet-onset lift coefficient. These inviscid flow wing-alone pressure distributions can be compared with the results of MATRICS-V for the wing/fuselage configuration in viscous fiow (A^x"") shown in Figure 5. A comparison suggests that the effect of the wing/fuselage interference as well as the effect of the boundary layer is large. In spite of this large effect, it will be shown that use of the approximate inverse operator N~^ in the defect correction approach can lead to improvement of the existing wing.

424

B.L Soemarwoto et al. Tl=0.1406

,5

Ti=0.3594 ^^

Ti=0.2344 .,

r|=0.4531

Figure 4: Cp distribution on the isolated wing in the inviscid flow at M = 0.8 and CL = 0.56 {dashed line indicates the M = 1 level in Cp).

Ti=0.15.,5

Ti=0.55

-1.5.

Ti=0.35 ,5

T]=0.25 15r

Ti=0.65

Ti=0.85

l]=V.f

-1.5

Ti=0.45

f -0.5

&0

0.5

^5.

0.25

0.5

0.75

1

Figure 5: Cp distribution on the wing of the wing/fuselage configuration in the viscous flow at M = 0.8 and CL = 0.5655.

Inverse aerodynamic shape design

425

The first term in the right hand side of equation (9) represents a geometry design using SYN87, with the target pressure distribution obtained from the viscous analysis of the current wing/fuselage configuration using MATRICS-V. The resulting wing-alone geometry, can be considered as a baseline geometry simulating the viscous eflPect and fuselage effect. A flow analysis for the geometry x^ in inviscid flow gives an inviscid pressure distribution p. In order to determine p^, p is modified such that improvement with regard to buffetonset performance is implied. After p^ is specified, another geometry design using SYN87 (the second term in the RHS of equation (9)) is required in order to obtain e^. The redesigned geometry is finally obtained from x^ = x" - e ^

(11)

which can be analyzed using MATRICS-V with respect to improvement of the buffet-onset boundary in terms of CLCOMPUTATIONAL RESULTS In. order to be able to simulate viscous effects with reasonable success, i.e. to perform N~'^Nx'' in equation (9), the flow about the wing should be fully attached over the whole surface. Therefore, at the design Mach number M = 0.8, the highest lift coefficient for which the flow meets this requirement has been determined and is found to be CL = 0.52. The corresponding pressure distribution is referred to as the current pressure distribution. The resulting baseline geometry (x^) produces an inviscid pressure distribution (p), shown in Figure 6 along with the current (viscous) pressure distribution. The deviation around the shock wave can be attributed to the fact that the inviscid flow model produces sharp (discontinous) shock waves and therefore cannot match the pressure of the shock wave boundary layer interaction with the lower gradient. This local mismatch is compensated by deviations at other locations in order to achieve the same CL- Regardless of the discrepancies, the pressure distributions shown represent a minimum deviation, where the obtained baseline wing geometry can be considered as simulating both the viscous displacement effect and the fuselage effect. The inviscid pressure distribution is modified as follows: (i) the chordwise distribution is altered such that the local lift coefficients are increased, (ii) the pressure jump at the shock wave is decreased so as to get a weaker shock wave, and (iii) a less adverse pressure gradient is applied aft of the shock wave. The modification is performed bearing in mind that it should eliminate the shock-induced flow separation without implying a flow separation in the trailing edge region. This modification leads to a pressure distribution representing pt in equation (9). The geometry design for pt leads to the redesigned wing (x^), which is analysed for M — 0.8 and CL = 0.52 with the fuselage present. It has been observed that the value of Hmax is significantly reduced except in the region near the trailing edge. It is noted that in the geometry design algorithm the traiUng edge is not allowed to move and, in order to avoid "fish-tail" geometries, a rather severe constraint is applied to the geometry close to the trailing edge. As a consequence, the wing geometry in the trailing edge region can hardly change during the inverse design process. The situation is dealt with by a careful manual modification of the trailing edge region of the redesigned wing, while preserving the improvement that has been achieved in other areas of the wing, such as in

426

B.I. Soemarwoto et al Ti=0.0156

71=0.5156

Ti=0.1406

71=0.2656

71=0.3906

71=0.8906

Figure 6: Cp distribution of the inviscid flow at M = 0.8 and CL = 0.52 simulating the Cp distribution of the viscous flow around the wing/fuselage configuration {solid line= inviscid; dashed line= viscous/target). the region near the shock wave. After this modification, the chordwise distributions of H of the redesigned and existing wings are shown in Figure 7, where improvement in terms of Hmax is demonstrated. Figure 8 gives a comparison between the pressure distributions, which indicates reduction in the shock wave strength. Figure 9 shows the geometries of the wing sections. As an improvement in terms of Hmax has been achieved by the redesigned wing for CL = 0.52, it has to be verified whether this implies a higher CL at the buffet onset boundary. The buffet onset boundary is searched for by performing MATRICS-V computations sequentially in a way similar to that leading to Figure 3. The buffet-onset boundary was found to be corresponding to CL ~ 0.62. Compared with CL ^ 0.56 of the existing wing, the redesigned wing represents about 10% improvement of the buffet onset performance for the wing/fuselage configuration. CONCLUSION A redesign procedure has been formulated for improvement of the buffet-onset boundary of a wing/fuselage aircraft configuration. The procedure is based on a standard defect correction approach, formulated in terms of the wing pressure distributions. It has been found that design aimed at increasing CL at the buffet-onset boundary can be done effectively using the flow condition with a lower value of CL at which the flow separation has not yet occurred. It is concluded that the present defect correction approach opens the possibility of using a low fidelity tool for geometry corrections, in combination with a high fidelity tool for flow analysis, for solving inverse design problems of complex configurations. In this respect, research in the future can be carried out to investigate the applicability of the approach for more complex configurations such as those incorporating engine nacelles.

All

Inverse aerodynamic shape design Ti=0.15

.

Ti=0.35

Ti=0.25

^<

a.5

12.5

0.5

0.25

0.75

0.25

0.5

0.25

1

Ti=0.55

0

Ti=0.45

4|

Ti=0.65

Ti=0.75

0.5

0.75

1

Ti=0.85

0.75

Figure 7: H distributions on the existing and redesigned wings of the wing/fuselage configuration for C L = 0.52 {solid line= redesigned; dashed line= existing).

^=0.15

,.

Ti=0.25

15,

r|=0.35 .,5

Ti=0.45

Ti=0.85

Figure 8: Cp distributions on the existing and redesigned wings of the wing/fuselage configuration for C L = 0.52 {solid line= redesigned; dashed line= existing).

428

B.I. Soemarwoto et al. Ti=0.15

Ti=0.35

Ti=0.25

Ti=0.45

02

0.1

^—

g» ^

*~*"~>»^ ^^^^

^ 0

-0.1

-02 . . . . 1 . . . • 1 1 I . 1 1 • 1 i-i-i

0.25

05

0.75

1

Ti=0.65

Ti=0.55

0.25

0.5

0.75

0.25

0.5

Ti=0.75

Ti=0.85

0.75

Figure 9: Geometries of the existing and redesigned wings {solid Une= redesigned; dashed line= existing). Acknowledgment The investigation has been carried out in the collaboration framework of APERT95 JR03, with funding from the Royal Netherlands Academy of Sciences (KNAW) for the NLR part, and from the Indonesian Nusantara Aircraft Industries (IPTN). Thanks are due to B. Oskam and J. van Egmond for helpful discussions and to B.B. Prananta for providing the illustration of buffet flow. REFERENCES [1] E. Arian, A. Batterman, and E.W. Sachs. Approximation of the Newton Step by a Defect Correction Process. ICASE Report No. 99-12, NASA/CR-1999-209099, 1999. [2] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley and Sons, 1966. [3] A. Jameson. Optimum aerodynamic design using CFD and control theory. AIAA 95-1729, 1995.

Paper

[4] J.C. Kok and B.B. Prananta. Demonstration of Feasibility of Unsteady Navier-Stokes Computation for Pitching Delta-Wing at Low Speed. NLR-CR-99551, 1993. [5] Th.E. Labrujere and J.W. Boerstoel. CFD algorithm for the solution of optimum aerodynamic design problems using control theory; The SYN87 code. NLR-CR-98429, 1998. [6] A.J. Van der Wees, J. van Muijden, and J. van der Vooren. A fast and robust viscousinviscid interaction solver for transonic flow about wing/body configurations on the bases of full potential theory. NLR-TP-93214 U, 1993.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

429

The Application of Modified Output Error Method and Its Verification on Inverse Problems in Aeronautics (No.2) Yasumasa Koyama and Fumiaki Imado Dep.of Mec. Systems Eng., Shinshu University, 4-17-1 Wakasato Nagano 380-8553 JAPAN Tel: +81-26-269-5144; Fax: +81-26-269-5145; Email: [email protected] Abstract This is the second report of our study which tries to estimate the aerodynamic characteristics from the aircraft flight data. The algorithm of a modified output error method developed by the authors are presented, and some examples of its application on YF16 aircraft flight data are shown. The method is originally developed for the purpose of analyzing an air accident. As the method has been proven to be quite effective, we hoped to apply it to the other aircraft's flight data for estimating the aerodynamic characteristics and the input control histories. As there were no data available at hand, we first produced the flight data by simulations, then applied the method whether we can reproduce the aerodynamic characteristics and input control histories. The employed aircraft is YF16 about which fairly detailed data are published. Some defects of our former algorithm has been improved and the result shows that the method is quite effective to this purpose. LINTRODUCnON In order to obtain aerodynamic coefficients and their derivatives from flight data, maximum fikelihood method and output error method have been successfully employed until recentiy. In cases v^here we can obtain enough amount of flight data under indicated conditions, tiiese methods can estimate those coefficients with fine accuracy. However in cases of air accidents, we can obtain only one flight data from DFDR (Digital flight data recorder) under very abnormal flight conditions. Therefore, in order to estimate the aircraft aerodynamic coefficients and flight states depending on the DFDR data, we will require some different algorithm from above methods. Motivating by the fact, we have developed an algorithm which perfectiy can reproduce the original flight data with input variables which simultaneously satisfying the aircraft s k degree-of-freedom dynamics. The appUcation of the algorithm to an actual aircraft accident we have been studying showed fine performance, and we could estimate the aircraft condition under the accident without tune delays and large noises. The algoritiim we have developed is in principle depending on the ou^ut error method, therefore we nominated it as the modified output error method. Then we appHed the method to see whether we can reproduce the aerodynamic characteristics and input control histories, on YF16 aircraft flight data produced by simulations. The result shows that the time histories of altitude, total velocity, Euler angles are very precisely estimated vdthout any significant time delay and errors. However, in time histories of three aerodynamic force coefficients Cx,t,Cyj,Cz,t and three aerodynamic moment coefficients C/,/, Cm,t Observation noise P ^ Aircraft Input u — Jpdate (Z-y)

it

L>| l^fy\f.] 1

•

pQ

Flifrht Hfitn 7

-,

fl 1

Fig.l The modffied output error metiiod algorithm

430

Y. Koyama and F. Imado

Initia prcx^ss

-.^i(t\Xl)(t) = n'd(t) Pit)- = h^dit)/^t

K0 = Fu

=

aid g + mg sindd

F„ = -Ondg - nig cosdd cos^d

M'

.^2[ed{i + tL) -ddit)- -q(t)tLVy

nL\

Integrate Eqs (1),(2),(3),(5),(8X(12) r*(/+/z.),//*(/ + /L),0*(/ + /L)

I

t^V^VAt-^ti)-V*(t-¥tL) ^d^ed(t-\-tL)-e*(t-\-tL)

Determine t^Fu.l^Fw.l^^Mv to satisfy AK,A/7,A0 = Q Fu = Fu + A F„, Fw = Fw + A FHMy = Mv + AA/v 1 Mu =

2[(t>d{t-\-tL)-(t>d(t)-p(t)tL]lx/tL^ Integrate Eqs (4), (7) ^()>^(i>d(t-^tL)-(i>*(t+tL)

X

Determine AA/^ to satisfy A 0 = Q Mu = Mu + A M„

1

M / = 2[il)d (t + tL)-rpd (t) - r(t)tL Vz 1 ti^ Integrate Eqs (2), (6), (9) ^'^p=Jpdit + tL)-^*{t 1

Determine

+ tL)

^M ^ to satisfy ^^|J == 0 1

M . = Mw

+AA/M'

Repeat the process A through B by employing all the state and control variables updated. Fig.2

Flow chart of the algorithm

The application of modified output error method

431

and Cn,t, some large discrepancies between the true values and the estimated values were found. After ISIP '98, we investigated the reason, and have improved the algorithm of the method. Then, we have succeeded to estimate these aerodynamic coefficients fairly well. In this paper new results obtained by applying to three maneuvers: Steep Ascent and Roll, High-speed yo yo and Low-speed yo yo, are shown. 2.THE MODIFffiD OUTPUT ERROR METHOD ALGORITHM Figure 1 shows the algorithm of a modified output error method. The method is employed to estimate the input u itself, while a conventional output error method is employed to estimate the parameters of the model. The method is different fi-om tiie output error method, but as the conception is analogous to it, we named it as the modified output error method. In the conventional aircraft, the available datafiromDFDR are, acceleration components of the aircraft three axes utd (X axis), and (Z axis), and am (Y axis) , the Euler angles 0^(roll angle), ^^(Pitch angle) and i/;^(yaw angle), and the angle-of-attack a,the altitude h which is obtainedfi-omboth an aneroid altimeter and a radio altimeter, TAS(True air speed) Vd etc., where the sufBx " J " shows the values are obtained from DFDR. The aircraft two axes acceleration components at, an and three axes torque components Mu,Mv and Mw are selected as control variables, which are determined so tiiat, the integrated output of the aircraft six degree-of-freedom dynamics are to be consistent with the flight data hd, dd,(l)d, and i/;^. As for the lateral acceleration ai, tiiere is no data available directly connecting with it, therefore the data are treated as true values, and other five control variables are determined through five observed data. These five control variables are not determined simultaneously, but at first at, an and Mv are determined to satisfy the aircraft longitudinal equations, next Mu is determined to satisfy the roll equation, and finally Mw is determined to satisfy the yaw equation. The flow chart is shown in Fig.2. Equations appeared in the figure are as follows. II = Fu/ m - qw + rv v = Fv / in - ni + pw w ^ Fw I m - pv + qu

p = {izMu - IxzMw)/{lxIz -Ixz^) q^Mvlly

(1) ( 2) ( 3)

(4) (5)

r - {IxM'v - IxzM'u) /{Ixlz - Ixz^) (6 ) (j) = p + {qs(p + rc4) tan 6 ( 7) e - qc(t> ~ rs(i> (8) ( 9) \p = {qs^ + rc<j>) sec6 X = cOcipii + {-cclxsip + s(j)sOc\p)v + {sc^ksip + c^sdc\p)w (10) V = cdsipii + (c(l)cip + s(lxs6silj)v + (-s(l)C'ip + c<j)sds'ip)w (11) i - -sdii + .v(^ 0v + c(l)cdw (12) M'u = Mu-qHw + rHv (13) (14) Mv = Mv-rHu + pHw (15) Mw = Mw-pHv + qHu Hu = Ixp + Ixzr (16) Hv = lyq (17) Hw = Ixzp + Izr (18) where s(p, c0 etc. show the abreviation of sin 0 , cos0 etc. at, ai and a„ are corresponding to the output of the aircraft three axes accelerometer, which are obtained by subtracting gravitational terms from Fu I m , Fv / m and F^ , respectively. The transformation from the inertial coordinates to the body coodinates are implimented by successive rotation about Z , Y and X axes with ip (yaw), 6 (pitch) and

432

Y. Koyama andF. Imado

0 (roll), respectively. The first step of the algorithm shown in Fig.2 is explained in detail. (a) Calculate following to the initial process, the values of 0,!/;,/;,/% F„ , F^ and My , where 6 (t) and q(t) are current states. (b) Integrate eqs (1), (2), (3), (5), (8) and (12) ti seconds, and let the obtained values of V Ji and ^ at t + tL bt V*(t + tL), h*{t + tL) and 6*(t + tL) respectively. (c) Change the Fu value very small amount A F«, and implement the same process as (b). This time we note the obtained values as Viit + ti), Init + ti) and 0i(r + a ) . (d) Changethe Fy^ and Mv values in the same way as (c), and note obtained state values at t + ti as V2{t + tL),h{t + tL), diit + fi) and Vsit + ti), hU + ti), O-iit^-ti) respectively. (e) Now we have obtained all the effect coefficients of A F „ ( 0 ,AF>v(0,AA/v(r) to K(/ + / i ) , li{t + tL) and 6(t + tL), therefore we can calculate the exact values of A F „ ( 0 , ^FwU) and A A/v(/) so that the values of V Ji and 6 at t + ti are to be coincide with those of DFDR data VdU + ti), hdit + ti) and OdO + ti) by solving the simultaneous equations. (f) Update the F„,Fwand My values by Fu = Fu + A F«, Fyv = Fyv + A Fw, Mv = Mv + A Mv In the same way, we determine the control variables A/„ and A/w as are shown in Fig.2. The whole process from A through B is repeated again by employing the updated state and control variables. Next, by employing the updated variables, eqs (1)^(12) are integrated at the time interval A t, which is far smaller than ti, and the whole process is repeated until the end of time histories of the DFDR data. 3 .YF16 FUGHT DAIA GENERAHON PROGRAM The program is composed of the flight simulation program of YF16 and the aerodynamic coefficients estimation program, which is the same as the modified output error method program stated in the preceding section. The flight simulation program is produced by replacing the flight vehicle model, its control system and aerodynamic model of GPMS ^ by those of YF16. A remarkable difference between these exists in the aerodynamic models, where in GPMS, aerodynamic forces and moments are expressed by aerodynamic derivatives, while in the YF16 program, they are expressed by aerodynamic coefficients which are usually three-dimensional table functions of a , j3 and control surfaces. We will briefly explain about this point. The forces and moments appears in Eqs (1)^(6) and (13)^(15) are expressed as follows. (19) Fw = Fug + Fut + Fua (20) Tv ~ t'vg + ivt + iva (21) r'w — I'yvg + twt + tyva (22) Mu = Mug + Mut + Mua (23) Mv = Mvg + Mvt + Mva (24) Mw = Mwg + Mwt + Mwa where sufficies " g ",'V", and " a " express "gravity", "thrusf and "aerodynamic", respectively. Aerodynamic forces and moments are given 1by (25) Fua = QsCx,t (26) Fva = QsCyj (27) Fwa = Q'^CzJ Mua = QsbCl,t (28) (29) Mva = QscCm,t (30) Myva = QshCn,t where Q is the dynamic pressure, b and c are the wing span length and the mean aerodynamic chord, respectively.

The application of modified output error method

433

4 .RESULTS AND DISCUSSION Several maneuvers are implemented by employing YF16 flight data generation program, and the generated flight trajectories are supplied to be analyzed by the improved algorithm of the modified output error method. Among them, three maneuvers are selected and the results are shown in the following. The first maneuver, fig.3 shows a trajectory of a steep climbing and succeeding positive rolling at first until 445 degrees, then, negative rolling until -45 degrees. Figures 4 through 14 show time histories of tme variables which are obtained by YF16 flight data generation program, and the estimated values by the modified output error method. In these figures, solid lines show tme values and dotted lines show estimated values. Figure 4 shows the altitude, figure 5 shows the total velocity (the absolute value of the velocity). Figures 6 through 8 show three Euler angles. These figures show that, these state variables are precisely estimated without any noticeable time delay and error. Figures 9 through 14 show time histories of three aerodynamic force coefficients Cx,t, Cy^t, Czj and three aerodynamic moment coefficients C/,/, Cm,t and Cn,t. The result shows far better estimation comparing with the result of Ref.4. The readers may think that there are still significant differences between true values and estimated values, however, the first integral of true values and estimated values are ahnost coincided with. As it is well known that the estimation of the second derivative terms, and these force and torque coefficients are directly proportional to them, we may say the estimated values are fairly well. Figures 15 through 26 show the results about a High-speed yo yo, and figures 27 through 38 show the results about a Low-speed yo yo maneuver. Figure 15, 27 show trajectories these maneuver, respectively. Figures 16, 28 show the altitude, figures 17, 29 show the total velocities. Figures 18 through 20 and 30 through 32 shows three Euler angles. Figures 21 through 26 and 33 trough 38 show three aerodynamic force coefficients Cx,t, Cy^t, Cz,t and three aerodynamic moment coefficients Cij , Cm,t and Cn,t. The overall results show the fine performance of the modified output error method developed by us. 5.CONCLUSION Some new results processed by a modffied output error method, which has been developed by authors are shown. The time histories of state variables such as altitude, total velocity, Euler angles have been very precisely estimated with the former version, however, as for the estimation of three axes aerodynamic force coefficientsCjc,/, C;;,f,Cz,/and aerodynamic moment coefficients C/,?, Cm,t and C„,/, there were some discrepancies between estimated values and tme values. W^th this improved algorithm, not only the former state variables, but also aerodynamic coefficients are well estimated. REFERENCES 1.W.R Gilbert, L.T. Nguyen and R. W. Von Gust," Simulator Study of tiie Effectiveness of An Automatic Control System Designed to Improve The High-Angle-of-Attack Characteristics of A Fighter Airplane", NASA TN D-9176, May 1976 2. S. Akishita and F. Imado," General Purpose Tactical Missile Simulation Program", Proc. of 13tii Annual Simulation Symposium, 257-270, March,1980 3 .F. Imado and Y. Koyama, "The Modified Output Error Method and Its AppHcation on Inverse Problems in Aeronautics", Proc. of ISIP'98,365-370,1998 4 .Y. Koyama and F. Imado, "The AppHcation of Modified Output Error Method and Its Verification (No.2)", Proc. of 37th Aircraft Symposium, 269-272,1999

434

Y. Koyama and F. Imado

10 Time (sec)

Fig3Maneui)er(ifiQentaiid]dl)

10 Time (sec)

20

Fi&4AUitiiie(a8Qa]taii(iiQD)

0

20

Eig5Vyuily(aBQe[itandBQlO

10 Time (sec)

Fii^7FitEh(a9oentandidD

10 Tune (sec)

10 Time (sec)

20

Ei|^ficll(aBoentaiidrQlO

O.Olf

10 Time (sec)

Fi&9 Cx,t (asoentaDdralO

10 lime (sec)

P!ig.lO Cy,t (afiOBntaniroD)

10 Time (sec)

20

Rg.llCz,/ (asoaitandid} 0.ooo3r-

10 Time (sec)

Eig.12 C/,/(i»QentandidO

10 Time (sec)

Fi&13 Cm,t (aaoentaididO

10 'nme(sec)

Fi&14 Cn,t (ascentandxblO

The application of modified output error method 160(V

Fig.l5 Maneuver (liigh-epeed yo yo)

10 20 Tmie(sec)

:»u

Fig.18 l ^ w Obigh-speed yo yo)

290-

10 20 30 Tine (sec) Fig.l6 Altitude (hi^-i^)eed yo yo)

0

435

10 20 Time (sec)

30

Fig.l9 Pitch Qii^-epeed y o yo)

0

10 20 30 Time (sec) Fig.l7 Veloci1y(Jugh-fipeed yo yo)

10 20 30 Time (sec) Fig.20 Roll Qiii^-^peed yo yo)

0-

10 20 Time (sec) Fig.21 Oc,r0iighH9peedyoyo)

10

20

Time (sec) Pig.24 C/,/ (hi^-fipeedyoyo)

10 20 Time (sec) Fig.22 C y , / ( h i ^ - q ) e e d y o y o )

10 20 Time (sec) Fig.23 Cz,/(high-speedyoyo)

10 20 Time (sec) Fig.25 Ciw,/(hiiJi-«peedyoyo)

Fig^

Time (sec) Oi,/(highHq)eedyoyo)

436

Y. Koyama and F. Imado 30Q-

F%27Maiieuver Oow-^peed yo yo)

10 20 Time (sec) Fig.d0 Yaw (km-Bpeed yo yo)

10 20 Time (sec) Fig.33 Cx,t Oow-speedyoyo)

10 20 Time fsec) F i g ^ Altitude Oaw-«peed yo yo)

10 20 Time (sec) Fig.3lPitdi (k)w-qpeedyo yo) 0.02r

10 20 Time (sec) flg.34 Cy, t (low-speed yo yo)

10 20 Time (sec) F i g ^ Velodly (bw-flpeed yo yo)

10 20 Time (sec) Fig.32 Ron (low-speed yo yo)

10 20 Time (sec) Pig.35 Cz, t (kiw-q)eed yo yo)

-0.02

-OM

10 20 30 Time (sec) Fig.36 C/,/ (bw-gpeedyoyo)

30 '" "'" •'v—^Q iQ 20 Time (sec) rmie(sec) Pig.37 Cm,t (low-speed yo yo) Pig.38 C/i,/(l0w-iq)eedyoyo)

""^^

Inverse Problems in Electromagnetics and Acoustics

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

439

OPTIMIZATION OF ELECTROPLATING ON SILICON WAFER K. AMAYA* S. AOKI* H. TAKAZAWA* and M. MIYASAKA** * Department of Mechanical and Environmental Informatics Tokyo Institute of Technology 0-okayama, Meguro-ku, Tokyo 152-8552 Japan ^^ Ehara Research Co., Ltd., 2-1, Honfujisawa 4-choume,Fujisawa 251-8502 Japan

ABSTRACT A boundary element method for simulating the electroplating of a silicon wafer with copper was developed. In this method, the Laplace equation is solved with three-dimensional or axisymmetric elements by representing the complicated phenomena near the anode and cathode surfaces as the polarization characteristics (PC) and using the PC as nonlinear boundary conditions. The electrical resistance of the thin copper and tantalum nitride (TaN) films on a silicon wafer, which is coated with a TaN spattering film before electroplating, is taken into account. To make a uniform copper film, the anode (copper plate) is divided into several pieces, and optimization of current to be supplied to each piece of the anode is performed by employing the Simplex method. KEYWORDS Electroplating Simulation, Boundary Element Method, Silicon Wafer, Polarization Curve, Laplace Equation, Electrical Resistance of Thin Films, Optimization of Current INTRODUCTION Recently, electroplating is tried to apply on manufacture of IC devices and printing circuits. [1] It is important to control the uniformity of macro thickness of plating in the plating process, in order to simplify the following grinding process. Therefore, a optimum designing of electroplating machine and reasonable selecting of plating condition is highly required. On the other hand, many researches has been done on application of numerical analysis on electroplating. Kohara and Kobayashi estimated the current density distribution in the plating bath by FEM and BEM, respectively. In this paper a boundary element method for simulating the electroplating of a silicon wafer with copper was developed. The electrical resistance of the thin copper and tantalum nitride (TaN) films on a silicon wafer, which is coated with a TaN spattering film before electroplating, is taken into account. To make a uniform copper film, the anode (copper plate) is divided into several pieces, and optimization of current to be supplied to each piece of the anode is performed by employing the Simplex method.

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K. Amaya et al.

MATHEMATICAL MODELING Plating Solution Domain Let us consider a electroplating system as shown in Fig. 1, where the electrical potential with in the domain is 0. We note that the electrical potential is defined with reference to the current supply point and the sign is opposite of that usually employed in corrosion science, where the potential is defined with reference to an electrode such as saturated calomel electrode (SCE). It may be assumed that electro-potential 0 within the electrolyte obeys the Laplace equation by macroscopic modeling. [2] V^0 = O

in Q

(1)

The complicated microscopic behavior near the electrodes are taken in account with a potential gap as polarization curve which is represented as boundary condition. Also, the microscopic geometry of the wafer surface is not explicitly considered. However this effect can also be taken into account as polarization curve implicitly, because polarization curve is measured experimentally on a real wafer which involves complicated microscopic behavior and microscopic geometry. Therefore boundary conditions are given as follow. (/) = 00

on Vd

(2)

i{= K-~-) = zo on

on r „

(3)

- ( 0 - 0 c ) = /cW

on Te

-{-

a) = fa{i)

on

Ta

(4)

(5)

where, ^ is surrounded by r^ -h F^ -h Fa + Fc(= F), potential and current density are prescribed with values of 0o and ZQ on F^ and F„, respectively. Fa and Fc are anode and cathode surface, n denotes the electrical conductivity of the electrolyte, d/dn is the outward normal derivative to F. fa(i) and fc{i) are nonlinear polarization curve on anode and cathode which are obtained experimently. Anode Since thick copper disc anode is used for copper plating system, the resistance within the anode can be ignored and (/)a, the potential within the anode, can be assumed to be constant. In case of considering that the whole current impressed on anode is prescribed, (j)a will be an unknown value, and the following constraint will be required:

L

idV - /o

(6)

Cathode Practically, a thin barrier layer is previously formed by spattering process in order to avoid the diffusion of copper into Si02 layer. The barrier material (TiN, Ta/TaN, Ta) shows high resistance characteristic. This characteristic curses the potential distribution with in the cathode. For the macroscopic analysis, the 3D micro-circuit pattern on the wafer can be ignored and the electrical field with in the cathode (wafer) Therefore, we define the current density within the cathode ic[A/m] as follows. *c = -{tst^s

+ tpi^p) V2(0c)

(7)

Optimization of electroplating on silicon wafer

^,— ^

441

- ^

^

^

^

V^(f) == 0 in Q

^^iSi^iS^^= i on Tc Fig. 1

Governing equations and B.C

where, ^^[m] and /^^[^"^m"^] are the thickness and electrical conductivity of spattering layer, respectively. tp[m] and Kp[Tt~^m~^] are the thickness and electrical conductivity of plating layer, respectively. The subscript 2 denotes the differential operator in 2D (in x — y surface). If we assume the electrical charge within the cathode obeys the consavation low, the following equation can be satisfied. div2(ic) +i = 0

(8)

Equation (7) and (8) yield the following governing equation in the cathode. {tsKs + tpKp)Vl{(l)c) = i

(9)

BEM FORMULATION The boundary integral equation for eq.(l) is: '^c{y)(t>{y) =

J

(t)*{x,y)i{x)dr /

-

i*{x,y)(l){x)dT

(10)

where, x and y are the locations of a source point and a observation point, respectively. The fundamental solutions * and z* are given by:

'(-'^) i\x,y)

=

4^

= K

d(t)*{x,y) dn

(11) (12)

where, r = \r\ = \x — y\, and n is the outward normal at the observation point x. substitution of (2) and (3) into (10) yields

mw = [Gm

(13)

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K. Amaya et al

where the detailed expressions of matrices, [H] and [G], are given in reference[4], and {(/>} and {%] are the vectors which component is (j) and i at each node, respectively. Discretizing eq.(4) and (6) yields {0}a = {<>«}»+ {-/„(J)}a = h

{A]l{i}a

(14) (15)

where, { }a shows that the value on the anode. v4 is a vector which components are area of elements. The boundary integral equation for eq.(9) is: c{tsK,s + tpKp)(l)c{y)

=

I {(t)\{x,y)ic{x) + /

-

i*^{x,y)(pc{x)}dj (16)

(l>;{x,y)i{x)dr

JTC

where, 7 is a curl around the cathode T^ Zc(unl)olded) is current density across 7 (= {tsK,s -h tpKp)d(t)c/dn2))- d/dn2 is an derivative in 2D along outward normal. The fundamental solutions for 2D Laplace equation are given by (17)

4>;{^,y)

=

~ln{-)

z;{x,y)

=

{t,K, + t , K , ) ^ ^ ^

ZTT

r

(18)

Discretizing eq.(15), yields [H2]{^ch = [G2]{lch^[B2]{l}c

(19)

where, [H2], [G2] and [B2] are the known matrices which depend on the geometry of 7 and elements[4]. {^CJT and {zc}^ are the vectors whose components are 0^. and ic on 7, respectively. { }c indicates the value at the nodes on the cathode (Fc). If the boundary condition on 7 and i on the cathode are given, eq.(19) can be solved, and using the boundary integral equation for internal point, the potential distribution on the cathode can be obtained as: {0c}c=,

!,

[glWc

(20)

where, [C] is a matrix which depends on the location of the internal point. Above equation and boundary condition (5) yield Wc

= {-/c(i)}c + ,

!,

[glWc

(21)

It is noted that {i}c is a part of {i} in eq.(13). Considering eq.(14) and (21) as the boundary conditions on the anode and the cathode, the system of nonlinear algebraic equations, eq.(13) eq.(15), is solved by the iterative procedure, e.g, the Newton-Raphson method.[3]

Optimization of electroplating on silicon wafer

443

Current 1=1.5A

Electorical Conductivity of catJiode K„=4.0x1cP ff^ mM

Fig. 2

Electroplating bath

terminal(Analysis domain

-«c=0

Fig. 3

Example model of cathode (wafer)

NUMERICAL EXAMPLE To examine the viability of the method, an example problem was solved, he example is shown in Fig. 2 in which a wafer is electrically plated with copper in a cylindrical bath. 8 terminals are attached on the edge of the cathode (wafer) as shown in Fig. 3. From the symmetry conditions, only a 1/16 of the domain is modeled in the boundary element analysis as shown in Fig. 4. Figure 5 shows the polarization curves which are employed in the analysis. The side surface of the bath is assumed to be insulated. The geometry of the bath is shown in Fig. 3. The current density distribution on the cathode, which is proportional to the plating thickness growth rate, is shown in Fig. 6. The potential distribution within the cathode is shown in Fig. 7. The distribution shows that the internal resistance of the cathode influences on it.

OPTIMIZATION Let us consider an optimization problem to uniform the thickness of the plating. In order to achieve this aim, the anode is divided into A^ number of multiple anodes, and currents for

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K. Amaya et al

Fig. 4

BEM mesh for the example

-0.5

-1

0

0.5

1

Current density i[mA/mrft ] Fig. 5

Polization Curve

x[mm]

Fig. 6 Current density distribution on cathode ( - / is proportional to the plating thickness growth rate)

optimization of electroplating on silicon wafer

Fig. 7

445

Potential distribution with in cathode (wafer)

these multiple anodes will be designed. The geometry of the multiple anodes are concentric rings. For simplicity, the current densities on these anodes are assumed homogeneous. The design valuables for this optimization problem are current densities ZQJ [j = 1, • • •, A^^) for the anodes. The objective function for this optimization problem is the deviation of current density on the cathode (wafer surface) as follows. (22) where, m is the number of boundary elements on cathode. Therefore, the optimization problem here is reduced to find the set of Zoj (;/ = 1, • • •, A^) which minimize the objective function (22). It is noted that the whole amount of impressed current is set to be constant (/Q), SO that the following constraint should be satisfied. 20,1^1 + 20,2^2 + •••• + ^0,7V^7V = ^0

(23)

where, Ak is a area of each /c-th anode. Example optimization was performed on the axisymmetric plating system. The condition is the same as the example analysis in the previous section. The anode is divided into 5 as shown in Fig. 8. The minimization is performed by Simplex method [3]. The current density distribution of anode and cathode is shown in Fig. 9. The dashed fine denotes the distribution of pre-optimization and the solid line denotes the one of postoptimization. It is seen that the distribution on the cathode of post-optimization is uniform compare with the one of pre-optimization.

CONCLUSION A new boundary element method for simulating the electroplating of a silicon wafer with copper was developed. In this method, the Laplace equation is solved with threedimensional or axisymmetric elements by representing the complicated phenomena near the anode and cathode surfaces as the polarization characteristics (PC) and using the PC as nonlinear boundary conditions. The electrical resistance of the thin copper and tantalum nitride (TaN) films on a silicon wafer, which is coated with a TaN spattering film before electroplating, is taken into account. To make a uniform copper film, the anode

446

K. Amaya et al.

12

Supply2

\ Supply 1

10 8

Supply4 BupplyS

Supply3

*

•

'

1

48elements 49nodes

N 4

•

2 0 -2,

20

Fig. 8

40

radius p [mm]

1

1

F 0.06 E ^ 0.04 ^

^^

100

1

1

r

1

1

1

1

1

ANODE

0.02

(Ti

0

•D

-0.02

C

*

80

BEM mesh and multiple anode for optimization

0.08

e

60

Optimal Initial

C

J J

CATHODE

g> -0.04 3

o -0.06 -0.08

0

1

1

L

_l

10

20

30

40

_ l _

50

1

1

1

1

1

60

70

80

90

100

Radius r [mm] Fig. 9

Current density distribution on anode and cathode of pre-post optimization

Optimization of electroplating on silicon wafer (copper plate) is divided into several pieces, and optimization of current to be supplied to each piece of the anode is performed by employing the Simplex method.

REFERENCES (1) Dubin V.M. (1997). Copper electroplating for USLI metalization. Proc. VIMC, 16, 192. (2) Aoki S., Amaya K., Miyasaka M. (1998). Boundary Element Analysis on Corrosion Problems. Syohabo, Tokyo. (3) Press W.H.(1996). Numerical Recipes in Fortran 77. Cambridge University Press. New York. (4) Brebbia C. A. (1978). The boundary element method for engineering. Pentech Press. London.

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

449

DEFECT SHAPE RECOVERING FOR ELECTROMAGNETIC PROBLEM USING HTS-SQUID GRADIOMETER ^ Fumio K O J I M A \ Ryosuke K A W A I ^ Naoko K A S A I ^ , and Yoshimi H A T S U K A D E ^

^ Graduate School of Science and Technology, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, JAPAN e-mail: kojimages. kohe~u. ac. jp ^ Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 805-8568, JAPAN ^ Graduate School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, JAPAN

ABSTRACT This paper is concerned with a quantitative nondestructive evaluation of conductors using high temperature superconducting quantum interference devices ( HTS-SQUIDs ). A mathematical model of the nondestructive evaluation system is described by electromagnetic potential problems. The forward analyses using finite element model are implemented for the conductor with a crack. A computational method using a genetic algorithm is proposed for recovering internal defect profiles with HTS-SQUID data.

KEYWORDS Nondestructive evaluation, parameter estimation, genetic algorithm, finite element method, Maxwell's equation.

INTRODUCTION Quantitative nondestructive evaluation is t h e critical issues for the structural integrity of material system used in airplanes and aerospace planes. Until recently, various types of alternative evaluation methods have been developed, including ultrasound analyses, thermal tomographies, and electrical impedance techniques, etc. These methods involve an attempt to characterize structural defects which may not be detectable by visual inspection. Although those methods are well-known practical techniques, there exists growing interest of developing the new and improved methods for evaluating the more accurate ^presented at ISIP 2000, International Symposium of Inverse Problems in Engineering, 6-10 march 2000, Nagano, Japan

450

F. Kojima et al

defect profiles of material systems. SQUIDs have the potential to detect material defects in conductors due to their extremely high magnetic flux sensitivity for magnetic fields. Some previous efforts on defect identification problem have been reported in [1][2J for the nondestructive system using direct current force. In this paper, we propose a computational method for recovering internal defect shapes with magnetic flux density data from HTS-SQUID gradiometer. Figure 1 illustrates a nondestructive evaluation (NDE) system using HTS-SQUID [3]. The system is composed of a HTS-SQUID gradiometer, a dewar, a scanning table, a lock-in ampUfier and a personal computer to gather data and control the scanning table. A sample material fixed on the scanning table is moved in the horizontal direction by compressed air and in the perpendicular and the vertical directions by mechanical gears. The gradiometer is a planer type one which pick-up coils are directly coupled to a SQUID arranged at the center of the pick-up coils. In the proposed nondestructive test, an alternative current density is applied to the conductor inspected [3]. As shown in Fig. 1, the return-current pad can be used for the purpose of detecting damage shape. The pad is set under the sample inserting a thin insulating film between the sample and itself. The sample and the pad are connected using the silver adhesive sheet at one end of these. The current within the sheet induces a magnetic field perpendicular to it and in turn produces a current within the sample, called an eddy current. When a flaw is present within the sample, the flaw disturbs the eddy current flow near the flaw and this disturbance is manifested in the magnetic field detected by the measuring device.The magnetic flux density can be measured from a HTS-SQUID gradiometer located near the conductor. The existence of a defect ( represented by a nonconducting volume ) disturbs the current flow inside the conductor and, as a result, this material defect can be detected as the perturbation of magnetic flux data. This paper is organized as follows. Section 2 is devoted to the mathematical modeUng of the eddy current analysis. In Section 3, the finite element model of HTS-SQUID system and the computational experiments are discussed for the conductor with defects. In Section 4, a computational method by using a simple genetic algorithm is proposed for recovering the internal defect shapes.

scanning line SQUID coil

insularer film sample material return-current pad

Fig. 1. The overall configuration of the NDE system

Defect shape recovering for electromagnetic problem

451

MATHEMATICAL MODELING OF NDE SYSTEM Let G be the bounded domain in E? such that G = {

I

x = {xuX2,Xi)

\xi\
i = l,2,3}

(1)

^-l,2}.

(2)

and let Gc be the conductor domain that is subset of G, a

= {

X G C I

|x,|
The NDE system can be described by the magnetic vector potential A = {Ai^A2^ A^) and the electrical scalar potential (j). From the Maxwell's equations, the governing equation of the NDE system can be expressed by 1 ~—V'^X

=

Jo

(3)

=

0

(4)

A -- 0 on dGo dA_ 0 on dG-dGo dn where cr^ plays a role of characteristic function of each region,

(5)

+ jujG^{A

+ V^)

-ja;Va^(A + V$) with boundary conditions

^

^ f CTc in Gc 1^ 0 otherwise.

(6)

/yN

In the above equation, ^ denotes the time integration of the electric scalar potential 0 , i. e.,

m=

(r)dT.

(8)

^—oo

The input function of the NDE system is the current density Jo in RHS of (3) . This can be obtained by ^

\

0 otherwise

^^

where the electrical scalar potential 0o is the solution of the following electrical potential problem: V-aV0o

=

0

in

Cc

(10)

^ + Xct>o = V on dGo (11) on ^ =:= 0 on dG,-dGo (12) on where dGc and dGo denote the boundary of Gc and its sub-boundary, respectively. The output from the HTS-SQUID can be made by (13) where Xp implies the center of the sensor location.

452

F. Kojima et al

FOWARD ANALYSES O F NDE SYSTEM Finite element model In this section, the finite element method is appUed in order to implement our NDE system numerically. Each finite element consists of rectangular parallelepiped and the number of nodes composing G and Gc are A^^ and A^^, respectively. Applying the Ritz-Galerkin method, The solution Ak{k = 1, 2,3) and $ is approximated as ~

A,

^H/,'=i/;,(x),

(14)

A: = 1,2,3

Nd

(15) where xj^i is a basis function in H'> = {^\

vlaG = 0 } .

^,VipeL\Gc),

(16)

Then, from Eqs.(3-6), the finite element model can be represented as

/ [''^ul

0

0 0

[Kn\ 0

V \KuV [Kt.V

0 [K,,\ \ (W,\ W2 0 [K2,] W3 [Kn] [KM] \Ki,Y [A-441 /

\wj

(

F,\ F2 \ F3 j

(17)

10;

where the component of each block matrix is denoted by [Ku]ij

=

—

[K44\ij =

ji^

[Kii],,

'^tj'^^idx

+ juj

/ / / a^Vii>j-V%l>i dx

a^xlJj^idx

(18) (19)

(/ = 1,2,3)

(20)

i^'h = jJJ Joi'idx (/ = 1,2,3).

(21)

= ju> fff

a^^xPidx

and the component of each vector is denoted by

The element vector F can be computed by solving the finite element model corresponding to Eqs. (10) and (12). Computational

experiments

The proposed algorithm is tested for a copper plate sample with an artificial defect of a stair-stepped hole, which shape varies depending on the depth. The size of the test sample is 200 x 100 x 20 mm. The sample is scanned in the X2 direction by a HTS-SQUID gradiometer along a fine over the center of the flaw as shown in Fig. 2. The hole has 3 steps and their diameters at each stage are 5, 10, and 20 mm. The depth ranges of the

Defect shape recovering for electromagnetic problem

453

stages are 0-8 mm, 8-14 mm and 14-20 mm from its surface, respectively. Figure 3-(a) illustrates the configuration of the hall. In order to solve Eq. (17), the number of nodes in G and Gc are Nd = 15984 and A^^ = 2900, respectively. By dividing the hole part smaller, the hole is approximated by rectangular parallelepiped as illustrated in Fig. 3-(b).

Fig. 2. The scanning direction in computational experiments

(a) Stair-stepped holes

(b) Finite element decomposition of holes

Fig. 3. The shape of stair-stepped holes in computational experiments

454

F. Kojima et al

The magnetic permeability /XQ and the conductivity GC are taken as ;^o = 47r X 10-^(///m)

GC = 5.8 x 10^(^-^771"^).

(22)

In computational experiments, the integration in the element matrices and the element vectors in the previous section can be computed numerically by Gauss-Legendre formula. Taking into account of the structure of the element matrix (17), the incomplete conjugate gradient method is used as the linear system solver. Figures 4 depicts the current force vector Jo = -GC V 00 around the top and the bottom of the stair-stepped holes. Figure 5 demonstrates the numerical results of magnetic flux density measurements by HTS-SQUID.

'^

y ^ ^

^

y.

>/ /

(a) The top level y'y

-y

y ,

,

_ _

xxxx^^~

xxxxx^xxxxxx-

y

/ / / / / /^

h ^ ^

y /

WXXXx\ W X X x \ WXX-.~

y / / / / / / / / / y////

• - X W W W X X v W W W W X

vww \ wx x^

'

/ / /A y/

\

/'

y/ n

^/A

W N

\\x \ \^ \ \"-

' X

w -XX -XX

W W

-f-f

x w \ w xxxxxx xxxxx^ xxxxx^

/ / / /

/ / / y ^ ^ . / / / yy^. / / / y ^^. /y^ y y ^^. y y^ y y^

y'^^

y y y y ^ ^^

(b) The middle level

f / y

/ /^ ///-

x\\l x\\\L xw\i xw\x

/ / / y

x\xww^ ,

TS^

N X W W W W

xxwwwx^

(b) The bottom level

Fig. 4. The current flow around the stair-stepped holes

// / /

455

Defect shape recovering for electromagnetic problem B3[nTI

B3 [Hz]

A

, . . ^ n o crack — -

x/'" -0.06

-0.04

-0.02

•O

0

0.02

0.04

0.06

-0,1

xA -0.08

-0.06

-0.04

-0.02

0

0.02

B3 [nT]

—^—s,

-0.06

-0.04

-0.02

0

0.08

0.1

f^\

B3 [nT]

-0.08

0.06

(c) / = 80 {Hz)

(a) / = 20 {Hz)

-0.1

0.04

X2 [m]

X2[m]

-0.06

0.02

-0.04

-0.02

0

0.02

0.04

X2[m]

X2[m]

(d) / = 110 [Hz)

(b) / = 50 {Hz)

Fig. 5. Numerical results of magnetic flux measurements

INVERSE ANALYSIS FOR DEPTH PROFILE IDENTIFICATION In this section, we propose a computational method for recovering internal defect shapes with magnetic flux density data from HTS-SQUID gradiometer. A simple genetic algorithm [4] is adopted for this inverse procedures. Our objective function is given by

E{q) =

-\Y[q)-Y'\'

(23)

456

F. Kojima et al

where Y{q) and Y^ denote the model output and the associated experimental data. The parameter q restricts the defect geometry and we will state it explicitly in the later discussions. From Eq. (13), the model output Y[q) is given by

(24)

Y(q) = {Y,{q;A)}7:v

Our focus on defect information is the radius size of cylindrical defect at each depth. To this end, we introduce the following stepwise function r(9,X3) = ^ ^ . B f ^ ( x 3 )

(25)

where {B^ "j-J^ denote the zero-order spline with knot A^" AN . . n ^ ^ ( 0 ) ^ ^ ( i ) ^ . . . < ^ W ^ ^ ^ where tc denotes the thickness of the conductor inspected here. In our approach, each component of the unknown parameter q takes a natural number. Then each parameter vector q can be transformed into one chromosome with the fixed length. Figures 6 and 7 illustrates the relation between each code and the corresponding defect geometrical shape. The admissible class of parameters Qad is defined as follows: Qad = {q--{qir--

. qN^

€ {1, 2, 3, • • • , TV,}, z = 1, 2, • • • , N^}

(27)

Thus the population {qiYi=\ can be chosen among Qad- In the computational experiments, test data were generated by using the procedures stated in the forward analyses.

ffi=|: Np

I)'

m.

3 Nr

Fig. 6. Admissible parameter class of stir-step holes

q={q..q2

qwp} —^

U'.JjX'.nZn.I^

mbitTn

Np X Nr bits

Fig. 7. The structure of one chromosome

457

Defect shape recoveringfor electromagnetic problem CONCLUDING REMARKS

The computational method for a quantitative nondestructive evaluation using HTS-SQUID was discussed. The mathematical model of the inspection process was derived from the Maxwell's equations. The finite element model was successfully tested for the detection of crack shapes. The computational method using a genetic algorithm was proposed for recovering internal defect shapes with HTS-SQUID data. We are currently pursuing the effectiveness and validity of the proposed algorithm with experimental data.

REFERENCES [1] KOJIMA, F. and KASAI, N . , 2000. In: Review of Progress in QNDE, 19, American Institute of Physics. [2] BANKS, H . T . and KOJIMA, F . , 1998. Technical Report CRSC, TR98-15, North Carohne State University: to appear in Journal of Inverse and Ill-posed Problems. [3] HATSUKADE, Y . , KASAI, N . , KOJIMA, F . , TAKASHIMA, H . , and

1999. In: Proc. of EUCAS99, Spain.

ISHIYAMA Y . ,

[4] FOGEL, D.B. ,1995. Evolutionary Computation. IEEE PRESS, pp. 89-97.

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459

DESIGN OPTIMIZATION OF ELECTROMAGNETIC DEVICES WITH NEURAL NETWORK AkioKimura^ and Yukio Kagawa^^ ^Hachinohe Institute of Technology, Hachinohe ,031-8501, JAPAN ^^Okayama University, Okayama, 700-8530, JAPAN

ABSTRACT The Neural Network (NN) provides mapping and pattern-matching capability, which can be used for optimization. The application to the shape optimization was made for the pole piece design of the electromagnetic devices to realize the uniform magnetic field distribution in which the NN approach was also proposed in association with the Kalman filtering. The one of the drawbacks of the NN is however a long period of time required for the training in the back propagation. It has been reported that the improvement could be achieved by introducing the momentum term for acceleration. In the present paper, this approach is examined to apply the shape optimization of the pole piece of the rotor in electric motors. It is found that the method gives the reasonable solution, in which a proper choice of the stability factor is essential for achieving acceleration with stable convergence. KEYWORDS Neural Network, design optimization, magnetic field analysis.

1. INTRODUCTION The neural network is to extract essential information from the signals contaminated with noise and has been used for recognition and classification. For optimization application, the method has been examined to achieve a uniform magnetic field distribution by the shape variation of magnetic poles. The application of the neural network together with Kalman filter has also been examined for the optimization [1]. The drawback of the back propagation method in NN is the time required for the training. This paper discusses the normal back propagation (BP) method and the momentum method, and also the modified momentum method. The normal back propagation (BP) method does not injure the threshold to the internal conditions of the NN unit, in which the coupling factor of the network is modified based on an

A. Kimura and Y. Kagawa

460

evaluation function and the output [2]. The momentum method and the modified momentum method both belong to the back propagation method [3], [4]. The momentum method includes the momentum term in the learning process for acceleration. The momentum term is modified by multiplying a factor so as to present the improper operation when the initial momentum is too large. The modified momentum method is the method in which above operation is performed in each step of modification [5]. Both momentum methods are known to reduce the time required for learning. In the present study, these three methods are applied to the shape optimization of the rotor in the electric motors. The examination shows that the momentum methods give reasonable solutions while the time of learning is greatly reduced. For the present study, the software code is developed referring to the books previously referred [2],[3],[4],[5]. Our simulation experience is given in the following.

2. MODEL UNDER EXZAMINATION The shape optimization of the magnetic pole is considered so that the magnetic flux distribution in the gap between the stator and rotor in the electric motors is desired to be sinusoidal. Fig. 1 shows a two-dimensional model of the object which is the magnetic pole of a synchronous motor [6]. The magnetic flux distribution depends on the gap due to the shape of the rotor. The influence of the slot in the stator is here ignored. 0.04 STATOR 0.03 &2 83 ^4 ^S ^6 h

^8 ^9 *10 *11

P2 P3 P4 Ps Ps P7 Pg P9 Pio

Pii

0.02

COIL 0.01

0.01

0

L

P 0

0.01

0.02

0.03

A-C distance [ m ]

ROTOR

0.02

0.03

Fig. 1 Magnetic pole configuration for the analysis

Fig.2 Comparison between the desired flux density in the gap and the flux density in the initial shape

Design optimization of electronic devices with neural work The A-B space is a region to be considered for which the discrete boundary points ai (i=l to 11) are chosen for the interest. The other conditions imposed for the optimization are as follows. The magnetic flux density distribution must be less than 0.5 T along A-C space. The boundary points pi (i=l to 11) over the surface of the rotor are supposed to be moveable only in the normal direction, and the radius of the rotor is limited not to exceed 0.028m. Boundaries a - i3, r - 5 , and 6 - a are the fixed boundary in which vector potential A is set to zero. On the boundary ^ - T set is the natural boundary condition.

3. SHAPE OPTIMIZATION The magnetic field is calculated on the rotor surface for a supposed shape using the two-dimensional finite element modeling to evaluate the field distribution along A-B. In the learning process with the NN, the convergence is determined to be achieved when the average error E became less than 1.0X10"^. The average error is defined by ^^)

^^i_P^y^ 2

^p

where Np is the number of the teaching data, No the number of the output, ypk are the number of the learning patterns, hpk the number of the units of the output layer. The number of the learning patterns is chosen to be 10. The learning times is as many as 5000 times for each learning pattern. The optimization is carried out using normal back propagation (BP), the momentum method, and the modified momentum method. The shape is considered to be converged when the average norm B between the calculated magnetic flux density and the desired magnetic flux density reaches the smallest at all the points of consideration. The average error E is thus defined by No B -B*

^ = I(\^

(2)

where 5^ and B* are the desired magnetic flux density and the magnetic flux density obtained referring to the points /. (a) The normal BP [2] In the learning process, the learning coefficient V used for normal BP learning was chosen to be 0.6,0.75 or 0.9 respectively. Table 1 shows the result of the learning error and the error of the achieved magnetic flux density.

461

462

A. Kimura and Y. Kagawa

(b) The momentum method [3], [4] The learning coefficient is chosen to be constant ( r] =0.75). The stabilization coefficient a is chosen to be 0.7, 0.8 or 0.9. Table 2 shows that lager the stability factor, faster the convergence. (c) The modified momentum method [5] As in the case of the momentum method, the learning coefficient is chosen to be constant ( V =0.75). The stabilization coefficient a is initially chosen to be smaller 0.2, 0.4 or 0.6, and the limit is 0.8. The increment of the stabilization coefficient A a is chosen to be 0.002. Table 3 shows the result obtained by the modified momentum method. When the initial value of the stabilization coefficient a is chosen to be 0.4, at the iteration number 840, the average error of the achieved magnetic flux density at the consideration points is as small as 1.45 X 10"'. Table 4 shows the result obtained in which the upper limit of the stabilization coefficient is slightly increased to 0.9. The number of the iteration is half to achieve the same result.

Tablel. Results obtained by normal BP Learning error

Error of achieved magnetic flux

Learning factor 7J

Iteration

0.60

5000

12.19

2.32

0.75

5000

14.13

0.90

5000

16.03

2.53 2.68

E

(10-')

density fi( 1 0 - ^ )

1 1

Table2. Results obtained by momentum method Stability factor value Iteration to

Learning error

Error of achieved magnetic flux density fi( 1 0 - ^ )

a

convergence

0.7

1474

9.996

1.49

0.8

763

9.993

1.50

0.9

281

9.815

1.50

E

(10-*)

Table3. Results by the modified momentum method Stability factor, initial Iteration to value

a

convergence

Learning error E

(10-*)

0.2

1049

9.999

0.4

840

9.997

0.6

825

9.995

Error of achieved magnetic flux density a[ 1 0 - ^ )

( When the stabilization coefficient is limited to 0.8. )

1.53 1.45 1.53

1 1 1

463

Design optimization of electronic devices with neural work

Table 4 . Results by the modified momentum method Stability factor, initial Iteration to value

a

convergence

0.2

552

Learning error E

Error of achieved magnetic flux density E

(10-^)

(10~^)

9.931

1.53

0.4

401

9.965

1.45

0.6

449

9.998

1.53

( When the stabilization coefficient is limited to 0.9.)

0.8 0.6 • LU

0.5

? 0.4

HI bO

c 03

V (D _J

0.2

h

-^

_b yt

• uL

0.7

V =0.6

--.fl-^ 77=0.75 —o— T) =0.9

1

"Of =0.8

^ 0.5

•Of =0.9

^0.4

c^^>

I 0.3 (0

s^L

^0.2

0.1

nn

• a =0.7

uj 0.6

0.1 1

10

1

1

20

30

,.

40

50

0.0

10

20

30

40

50

Iterations

Iterations

Fig.4 Learning error in the momentum Method (U =0.75)

Fig.3 Learning error in normal BP

-•--a=0.2

• o r =0.2

- Of =0.4

~ ^ H - a =0.4

•a? =0.6

-™~O~~Qf=0.6

!^U»i»vu»({({(((f{C(ff.f.{UUU.{.{{{C 20

30

40

50

Iterations

Fig.5

Learning error in the modified momentum method (M=0.75,a„,ax=0.8)

10

20

30

40

50

Iterations

Fig.6 Learning error in the modified momentum method (/x=0.75,amax=0.9)

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A. Kimura and Y. Kagawa

In the followings, some details of the process are also given. Fig.3 shows the convergence of the learning in the normal BP learning process. The learning error does not much depend on the learning coefficient. Fig.4 shows the convergence of the learning in the momentum method. The convergence in likely to occur at less iteration some vibration appeared in initial stage, when stabilization coefficient a is not properly chosen to 0.9. Fig.5 shows the convergence of the learning in the modified momentum method when the stabilization coefficient a max is limited to 0.8. With the modified momentum method the factor learning is about 10 times without exhibiting any vibration. Fig. 6 shows the convergence of the learning in which the stabilization coefficient is limited to 0.9 without regarding to the initial coefficient. Fig. 7 shows the magnetic flux density between A-C after the shape is optimized, in which the initial and desired magnetic flux density are also shown. At points, 1,2,3, and 11 of the consideration, the desired magnetic flux density is not perfectly achieved. Fig. 8 shows the optimized shape of the magnetic pole that satisfies the desired flux distribution. These are all achieved by the normal BP method, the momentum method, and the modified momentum method. There is not much difference in the shape of the magnetic pole and the magnetic field density distribution, except that the surface of the magnetic pole is slightly smoother and the gap between the stator and rotor is slightly narrower in the case of the modified momentum estimation. With our simulation experiments the conclusion similar to the references. The use of the modified momentum method is most promising to accelerate the leaning process.

X

10''

[m]

Fig. 7 The magnetic flux density distribution between A-C in shape optimization.

Design optimization of electronic devices with neural work

0.032 0.03 0.028 0.026 0.024 0.022 0.02

(a) Normal BP (/i=0.75)

0.032

0.03

0.028

0.026

0.024

0.022

0.02

(b) Momentum method (M =0.75, a =0.8)

0.032 0.03 0.028 0.026 0.024 0.022 0.02

(c) Modified momentum method (/^=0.75,ao=0.4,an,ax=0.8)

465

466

A. Kimura and Y. Kagawa

0.032 0.03 0.028 0.026 0024 0.022 002

(d) Modified momentum method (//=0.75,ao=0.4,«max=0.9) Fig.8 Shape and magnetic flux distribution after the optimization.

4 . CONCLUDING REMARKS The shape optimization of the magnetic pole in the electromagnetic motors was examined to achieve a desired magnetic field distribution using the neural network approach. The modified momentum method is capable of reducing the number of iteration and the time of learning about one truth to achieve the similar result.

REFARENCE 1. K. Kimura and Y. Kagawa: Electromagnetic Device Optimization Using Kalman Filtering Neuro Computing, Journal of the Japan Society for Simulation Technology, Vol.18 No.l, 39/46,1999 in Japan 2. Ichikawa: Hierarchy neural network, kyoritu, 1993 in Japan 3. Nishikawa: Neural network and Measuring control, 1995 in Japan 4. Yagawa: Neural network, Baifukan, 1995 in Japan 5. Kiritani:Neuro Computer,Gijutuhyouron,1994 in Japan 6. Levent Ovacik and Sheppard J. Salon: Shape optimization of Nonlinear Magnetostatic Problems using the Finite element Method Embedded in Optimizer, IEEE trans. On Magn., vol. 32, No. 5, pp4323-4325, Sep. 1996

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

467

ON THE RECONSTRUCTION OF MAGNETIC SOURCE IN CYLINDRICAL PERMANENT MAGNETS Hajime

IGARASHI

Faculty of Engineering, Kagawa University, Saiwai-cho 1-1, Takamatsu, 760-8526, Japan

e-mail: igaOeng.kagawa-u.ac.jp

ABSTRACT This paper describes reconstruction of magnetization distribution in a cylindrical permanent magnet from the magnetic fields measured outside the magnet. The inverse problem is solved by two approaches, one of which is the Tikhonov regularization with the L-curve method for the determination of the regularization parameter. Another approach is based on the conjugate gradient (CG) method which solves the ill-conditioned linear system without regularization. The magnetization distribution to be identified by the present methods is obtained by the forward analysis of magnetization process of the magnet taking its magnetic hysteresis characteristics into account. It is shown that both methods reconstruct similar magnetization fields which qualitatively agree well with the original ones. The Tikhonov method provides more accurate and smooth solutions than the CG method when the regularization parameter is appropriately determined by the L-curve method, while the latter is advantageously free of the regularization parameter. KEYWORDS Permanent magnet, magnetic roller, Tikhonov regularization, conjugate gradient method INTRODUCTION Cylindrical permanent magnets, called magnetic rollers, play a crucial role in copy machines and laser printers. The magnetic roller attracts toner particles by magnetic force and transfers them by its rotational motion to the development zone. It is known that the quality of printed images sensitively depends on the magnetization distribution in the magnetic roller. The numerical analysis of the magnetic fields around the magnetic roller would give useful information for understanding of the motion of toner particles. However the magnetization distribution inside the magnetic roller, which is necessary for the analysis of the magnetic fields around it, is unknown in general. In this paper the magnetization distribution inside the magnetic roller is reconstructed

468

H. Igarashi

from the magnetic field measured on a circle outside the roller. It can be shown that there exist non-trivial magnetization fields which produce no magnetic field outside the magnetic roller. This means that the problem does not have unique solution, and requires prescription for the ill-posed system. In this paper the magnetization distribution is reconstructed by two approaches; the Tikhonov regularization and the conjugate gradient (CG) method. The former regularizes the system matrix by introducing stabilization function with the regularization parameter, while the latter solves the ill-conditioned linear system without any regularization. This paper is organized as follows: The second section gives problem description with mathematical formulation. The third section describes the Tikhonov and CG methods, and the forward analysis of the magnetization process. The fourth section compares the reconstruction results obtained by the two approaches, and the last section provides some concluding remarks.

PROBLEM DEFINITION We consider a magnetic roller of radius a whose cross section Q. is shown in Fig.l. The length of the roller is much greater than a so that the magnetic field H around it can be regarded as a two dimensional field. The roller is inhomogeneously magnetized to generate appropriate magnetic fields for imaging. The magnetic field H generated by the magnetization M in the roller is given by

"«-i/„[

M{x') \x — x'?

M{x')

• (x \x — x'\

x'){x-x')

dS'.

(1)

The radial component Hr{0) of the magnetic field H is measured on the circle of radius h shown by the broken line in Fig.l, where a
measurement points

Figure 1: Cross section of a cylindrical permanent magnet

On the reconstruction of magnetic source in cylindrical permanent magnets

469

struction is expressed by the finite series with Nf terms, H,{e) = - y " \ cos{kO) /

^^

I

Jo

H^^'^ie') cos(kO')dO' + sm{kd) /

Jo

H'^^^'^'iO') sm{k9')de' (2)

to filter out noises with high spatial frequencies. Here we used the fact that a magnetic field in free space is a harmonic vector field. The region Cl is subdivided into triangular elements, in each of which M is assumed to be constant. Applying this assumption to (1), we have a system of equations of order n, r = Km-hr

= 0,

(3)

where K, m and hr denote the system matrix of n x 2m, magnetization vector with 2m columns and magnetic field vector with n columns, respectively, m and n are the numbers of the triangular elements and measurement points. The vectors m and hr represent the discrete counterparts of M and Hr, that is, (M^, M j , . . . , M ^ , M^, ...,M^y and {H^, H^,..., i^J?)*. We seek for the solution to (3) in a least square sense. To do so, by minimizing the squared residue (r, r ) , we have K * K m = K*/i^.

(4)

However (4) does not have unique solution since there exist M s like T

M = -BQ, a

for r < a,

(5)

which generates no magnetic fields outside the roller [1], where ee denotes the azimuthal unit vector. That is, (5) belongs to the kernel of the operator K. To remedy this difficulty, we introduce two approaches in the next section. TWO APPROACHES FOR INVERSE PROBLEM Tikhonov

regularization

Instead of minimizing (r, r ) , we consider the problem (r, r) + A(m, W m ) —)• min,

(6)

where the second term with Tikhonov's regularization parameter A represents the solution norm weighted by the diagonal matrix W. When A is sufficiently small, the first term representing residual norm becomes dominant in the minimization. In contrast the solution norm becomes dominant when A is enough large. Note here that (6) reduces to the original problem (r, r ) -> min when A = 0. The entries of W is set to the area of the corresponding triangular elements so that the smoothing function (m, W m ) corresponds to magnetization energy / ^ | M | 2 dS. The problem (6) leads to a system of equations as {K'K + XW)m

= K'hr.

(7)

Now this regularized equation would be solved with ease. The value of A will be determined by the L-curve method below.

470

H. Igarashi

Conjugate gradient method In this approach, (4) is solved by the CG method without regularization. The algorithm of the CG method for solution of a linear system kx = 6 of order N is given as follows: Choose the initial solution XQ and set p^^ rQ — h — KXQ. Then for A: = 0, 1,... repeat o^k = Xk+i

{Pk. rk)/{Pk^

=

Pk = Pk+l =

APk).

Xk+OtkPk,

-{rk+u

APk)/{Pk.

Apfc), W

Vk+l+PkPk'

As can be seen in (8), the approximate solution XkJ^i is searched in the space spanned by the residual vectors r^, i = 0, 1, ...,A:, which can be written in the form k

(k) rk = ro + sr^ Ta\'^AW

(9)

1=1

Now ro is expanded by the eigenvectors Wi of A as ro = J2iLi ^i'^i- ^^ assume that the nullity of A is u, and its non-zero eigenvalues are Aj^+i, A^+2,.-., Ajy. The residue Vk can then be written as

1=1

iz=U+l

\

1=1

J

We can see from (10) that in the solution process the search is done in the orthogonal complement of the space spanned by the eigenvectors txi,...,xx^ which correspond to the zero-eigenvalues. Consequently the solution process of the CG method is independent of the zero-eigenvalues, and hence one of the solutions can be found even if A is singular. The CG method is applied to the solution of (4) in the next section.

Forward analysis of

magnetization

Test samples of magnetization distribution are obtained by the forward analysis of magnetization process of the magnet. In this analysis the magnetic hysteresis of the magnetic roller is modeled by the vector Preisach model [2]. The external magnetic field, generated by magnetic dipoles located outside, is sinusoidally varied in a half period in time. After the external field vanishes, the magnetic field generated by M remains by the hysteretic property. At each time step of the forward analysis, we start from the state H = H^, where i f e is the external magnetic field. The magnetization M is then computed from H and the hysteretic characteristics by the vector Preisach model [2] as follows: p

/.7r/2

M =

J—K/2

BrdO

JT

P{a, p)Lc,0iH

• e,) dadff,

(11)

where e^. is the unit vector in the radial direction, T the triangular region satisfying a > /? in the a — P plane, L^fi the elementary hysteresis loop consisting of the step functions

On the reconstruction of magnetic source in cylindrical permanent magnets

All

-a N

a

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 H [normalized] Figure 2: analysis

Hysteresis loops for forward

-0.5

0 H [normalized]

0.5

Figure 3: Typical changes of \M\ in forward analysis

which jump up to 1 from -1 at a and jump down at ^, and P the distribution function of a and P which determines the hysteretic characteristics. Figure 2 shows the hysteresis loop used for the analysis. The self-magnetic field Hs generated by M is computed by (1) to obtain the total magnetic field H = H^ + Hs- The magnetization M is again computed by the Preisach model for the new H. This process is repeated until the self-consistent magnetic field is obtained to proceed to the next time step. Figure 3 shows the typical changes of | M | at two different points in the magnet during the analysis. The tips of the curves correspond to the final state at which the external field vanishes and only the magnetic field generated by M remains.

NUMERICAL RESULTS In the inverse analysis, we assume that 2m = n for simplicity although 2m can be different from n in principle. (Equation (2) can always provide 2m data oiHr if enough measurement data are available.) The parameters are set as follows: n = 260, a = 1, 6 = 1.05 and A^/-150. We have three test magnetization samples, called Ml, M2 and M3, which are obtained by the forward analysis described above. Equation (1) is used to compute the magnetic field H on the outer circle shown in Fig. 1, from which the magnetization M is reconstructed by the Tikhonov regularization and CG methods. We define the error norm Nerr, residual norm /V^-e and solution norm Nsoi as follows: N N

= y ( m - mp, W(m - mp)),

Nsol

=V(m, Wm),

where mp represents the discrete counterpart of the original distribution of M. Figure 4 shows the locus on the A^^e - Nsoi plane for Ml, which clearly indicates an L-curve. The state point moves from left-top to right-bottom on the L-curve as A becomes

472

H. Igarashi

100000

50000

10000

Solution norm N_sol Residual norm N_re Error norm N_err

20000

e

u O

%

1000 \

10000

G O

o

100 5000

C/5

2000 10 100 Redsidual Norm N re

Figure 4: L-curve of model Ml

1000

le-10 le-08 le-06 0.0001 0.01 1 Regularization parameter Lambda

100

Figure 5: Norms of model Ml

large. The appropriate value of A would be at the corner of L, since N^oi and A/re are simultaneously minimized there [3]. To evaluate this expectation, the norms A^err, ^soi and Nre are plotted against A in Fig. 5. Note that A^err cannot be known for real situations of reconstruction in contrast to Nsoi and A^^-e since m^ is unknown. The corner of L in Fig.4 corresponds to the region of A where both N^oi and A/re keep constant. We see from Fig.5 that the determination of A at the corner of the L-curve is reasonable. The value of A is safely determined by setting it as large as possible within the range where A gives the L-curve corner. This tendency is also observed for artificially created distribution of M [4]. Figure 6 shows the original magnetization M of model Ml. The results of reconstruction by the Tikhonov regularization and CG method are shown in Fig.7. The CG method converges after less than 10 iterative steps under the tolerance condition \r}^^ < 1.0"^. Both results quantitatively agree well with the original magnetization. Figure 8 plots the original values of | M | versus the reconstructed ones. We can see that both reconstruction methods have tendency to make overestimation of \M\ where they are relatively small. Moreover, the deviation for the Tikhonov regularization method is clearly smaller than that of the CG method. This would be due to the fact that regularization increases smoothness of solutions. The error norm Nerr for the Tikhonov regularization is also smaller than that for the CG method. The results for model M2 and M3 are shown from Fig.9 to Fig. 14. The overall characteristics of the reconstruction, such as the L-curve and norm curves shown in Figs.4 and 5, were found to be similar to those of M l . Both reconstruction methods provide similar errors especially in the region where \M\ takes small values, that is, they give too diffusive distribution there in comparison with the original M . This common tendency in the errors suggests that the original fields mo contain the components which belong to the kernel of K. It is rather surprising that the solutions of the CG method are very similar to those of the Tikhonov regularization since the former solutions depend on the initial solution by its definition. The mathematical study on this point is remained. Moreover, it would be effective, if possible, to utilize the hysteretic characteristics as a priori data for improvement of accuracy in the reconstruction. This is also a future work.

On the reconstruction of magnetic source in cylindrical permanent

473

magnets

Figure 6: Original magnetization M of model Ml

(a) Tikhonov method (A = 0.001)

(b) CG method

Figure 7: Reconstruction of M for model Ml

800 -

800

_J

700 -

700 i

600 -

600

500 -

500

400 -

400

300 200 • 100

300

,^.*'

* •*

0 -

100 200 300 400 500 600 700 800 IM-originall (arbitrary unit) (a) T i k h o n o v m e t h o d (A = 0.001, Nerr = 1-01 X 10^)

100

0

1

1

1

+

1

t*

U

\

j J

200 1 i

h

1

u

1 0

•

>

•*•

r

* -1

1

1

1

1

1

100 200 300 400 500 600 700 IM-originall (arbitrary unit) (b) C G m e t h o d {Nerr = 1-55 X 10^)

Figure 8: Original and reconstructed values of \M\ for model M l

r

474

H. Igarashi

Figure 9: Original magnetization M of model M2

(a) Tikhonov method (A = 0.001)

(b) CG method

Figure 10: Reconstruction of M for model M2

800 •

1

1

1

1

1

1

700 -

800 i

1

^

4-

4

600 •

3

1

1

1

1

1

700

400 •

o

3 u G 0

300 • 200 -

o

i

100 • n . U T 0

1

1

1

1

1

1

1

1

100 200 300 400 500 600 700 8(X) IM-originall (arbitrary unit)

(a) Tikhonov method = (A 0.001, Nerr = 1-30 X 10^)

1

+ *

+4.* X*

400 300

* +

*>++

1

r

>

t.

U

1

v.*'

600 500 -

500

1

...

L

r

X

200 100 1 A U

J -

0

t

1

1

1

1

1

1

100 200 300 400 500 600 700 80 IM-originall (arbitrary unit)

(b) CG method {Nerr = 1-35 X 10^)

Figure 11: Original and reconstructed values of |iVf| for model M2

On the reconstruction of magnetic source in cylindrical permanent magnets

Figure 12: Original magnetization M of model M3

(a) Tikhonov method (A = 0.001)

(b) CG method

Figure 13: Reconstruction of M for model M3

700 600 500 400 300

: '^

200 100

0

100 200 300 400 500 600 700 800 IM-originall (arbitrary unit)

(a) Tikhonov method (A = 0.001, Nerr = 1-17 X 10^)

0

0 100 200 300 400 500 600 700 IM-originall (arbitrary unit) (b) CG method {Nerr = 1-29 X 10^)

Figure 14: Original and reconstructed values of | M | for model M3

475

476

H. Igarashi

CONCLUSIONS This paper describes reconstruction of magnetization in a magnetic roller from the magnetic fields measured outside. The Tikhonov regularization and CO methods are applied to the reconstruction. The regularization parameter is determined on the basis of the L-curve method for the Tikhonov regularization method. The CO method directly solves the least square problem without regularization. Both methods provide the reconstructed results which quantitatively agree well with the original magnetization. The smoothness and accuracy for the Tikhonov regularization is superior to those for the CO method, while the latter is advantageously free of the regularization parameter. The discrepancy between the magnetizations reconstructed by both present methods and the original ones seems very similar. This suggests that the original magnetizations include the components which produce no magnetic fields outside the roller. It would be necessary to use the hysteresis characteristics of the magnet as a priori information in the reconstruction for the increase of its accuracy. This is a remained work for future.

REFERENCES [1] V.D.Barger and M.G.Olsson, Classical Electricity and Magnetism, Inc., Newton, 1987. [2] I.D. Mayergoyz Mathematical

Allyn and Bacon

Models of Hysteresis, Springer, New York 1991.

[3] RC.Hansen and D.RO'leary, SIAM J. ScL Comput,

14, 1993, pp.1487-1503.

[4] H. Igarashi, A. Kost and T. Honma, Proc. COMPUMAG-Sapporo, 1999, to appear in IEEE Trans. Magn.

vol.2, 310-311,

INVERSE PROBLEMS IN ENGE^EERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

477

EXTRACTION OF TRANSFER CHARACTERISTICS OF VOCAL TRACT FROM SPEECH SIGNALS K. TSUTSUMI Kumamoto National College of Technology 2659-2 Suya, Nishigosi, Kumamoto, 861-1102,Japan

Y. KAGAWA Okayama University 1-1-1 Tsushimanaka, Okayama, 700-8530, Japan

ABSTRACT In order to estimate the vocal tract shape from speech signals, it is need to extract the transfer characteristics, i.e, impulse response, of the vocal tract from speech signals by removing the effects of glottal source wave. Here an inverse filtering technique used linear prediction is applied to remove the effects of the source wave. The simulation by the use of artificial voices shows the effectiveness of the proposed way. KEYWORDS Vocal tract shape, area function, speech signal, source wave, two-mass model, artificial voice, inverse filtering, linear prediction. INTRODUCTION The vocal organ (vocal tract) consists of glottis, pharynx, oral cavity and lips. As the speech is produced in the result of the convolution of the transfer characteristics of the vocal tract and the glottal source, the measurement of the vocal tract shape is interesting and also important in the area of practical application. The vocal tract shape uttered a speech has been examined by the X-Ray pictures or MRI [1,2]. However these methods are laborious and time consuming. In the present papers, more practical direct estimation of the vocal tract shape from speech signals is investigated [3,4]. In the direct estimation, the transfer characteristics of the vocal tract, i.e., the impulse response of the vocal tract, are to be extracted from the speech signals, in other words, the effects of the glottal source wave must be removed from the speech signals as a pre-processing. Removing the source wave characteristics from the speech signals is, however, not easy. In most case, a pre-emphasis that simply compensates the spectrum decay due to the glottal source is applied to the speech signals [5]. However this pre-emphasis is not enough because the spectrum decay depends on the shape of glottal source wave. Here we propose a new approach that takes the effects of the source wave into account. First the outline of the glottal source wave is estimated from the speech signals using an inverse

478

H. Tsutsumi and Y. Kagawa

filtering by linear prediction. Then the source wave characteristics are inverse-filtered from the speech signals by another inverse filtering. Thus the transfer characteristics of the vocal tract are extracted. INVERSE FILTERING AND EXTRACTION First the linear prediction is here outlined. s(n) is a speech signal sampled by time step T. If s{n) is causal, s(n) can be predicted by a linear combination of the last M sampled signals such as s{n - l),s(n - 2\...,s{n - hf), so that M

(1)

s(n) = ^aMjSin-i) The coefficients a^^ are selected for the squared some of error to be minimized ^s=i;W«)}'

(2)

where A^ is the number of data and (3)

ein) = sin)-sin)

This results in that a^j is the solution of the following equation (called as normal equation) ta^.R(\i-j\)^-R(j),

y=l,2,...,A/,

(4)

here R(J) is auto-correlation of s(n). From (1) and (3) E(z) = S(z) - S(z) = F(z)Siz)

(5)

where F(z) = I.^a^,z-',

a^,o=l

(6)

E(z), S(z), Siz) are the Z-transformations of e(n), s(n), s{n) respectively. Filtering is to extract some causal characteristics, i.e., to find linear prediction coefficients (LPC) OMJ from speech signals. Inverse filtering is to remove the causal characteristics from speech signals using F(z)as shown in (5). E(z) is the residue which consists of the non-causal signal, in other words, the input signal. Here we assume that this input signal corresponds to the glottal source wave. Defining G{z) as the Z-transformation of glottal source wave g(t) Giz) = F(z)S(z)

(7)

Thus we can estimate the glottal source wave by applying the inverse filtering on the speech signals as shown in (7). Finally the transfer characteristics of the vocal tract are extracted by inverse-filtering the estimated glottal source characteristics from the speech signals.

Extraction of transfer characteristics of vocal tract from speech signals

479

SIMULATION Artificial voice In order to verify the proposed method through the simulation, we employ an artificial voice instead of the real speech signals in which neither the true source wave nor the true transfer characteristics are known. For the glottal source, we employ two-mass model developed by Ishizaka et al [6], which can produce the fairly natural glottal source wave that is adapted to the vocal tract shape. In order to obtain the vocal tract transfer characteristics, first we calculate the power spectrum by the use of acoustic tube model of the vocal tract [7], in which the effects of loss, radiation impedance and wall impedance can be included. Then the auto-correlation is evaluated for the power spectrum by the inverse discrete Fourier transformation (IDFT). The LPC that represents the transfer characteristics of the vocal tract is calculated from the auto-correlation by (4). From (7) S{z)^G{z)IF{z)

(8)

The artificial voice is synthesized using the relation of (8) (so-called LPC synthesizer). Extraction methods The following two methods are proposed depending on the pitch period of the speech to extract the transfer characteristics of the vocal tract. Method 1 (for the speech with long pitch period) 1) Integrate the speech signals one time in order to compensate the effects of the radiation impedance at lips which works as a differentiator. 2) In the speech signals obtained at stepl), the source-free portion is found, which is not affected by the glottal source, i.e., the portion just after the glottis closure. 3) Calculate LPC^fromthe speech signals in the source-free portion. This LPC' represents the pure vocal tract characteristics which are not affected by the source wave. 4) Remove the vocal tract characteristics from the speech signals obtained at step 1) by the inverse filtering used the LPC'. 5) The residue of the inverse filtering at step 4) should include only the glottal source characteristics but it still includes some high frequency components at the low level. So the lower level than D.C., i.e., average value is cut, thus the upper level than D.C. regards as the estimate of source wave. 6) Calculate LPC^ from the estimated glottal source wave. This LPC ^represents the characteristics of the glottal source wave. 7) Remove the glottal source characteristics from the integrated speech signals obtained at step 1) by the inverse filtering used LPC^. Then the residue of the inverse filtering is the estimate of the transfer characteristics of the vocal tract.

H. Tsutsumi and Y. Kagawa

480

Method 2 (for the speech with short pitch period) When the pitch period of the speech is short, detecting the source-free portion seems to be difficult, therefor the step 2 to 4 in method 1 are changed as follows. 2) Filtered by the low-pass with the cut-off frequency of 1500-2000 Hz in order to remove the high frequency components. 3) Filtered by the high-pass with the cut-off frequency of about 2 times of pitch frequency in order to remove the lowfrequencycomponents of the source wave. 4) Calculate LPC' fj-om the speech signals obtained at step 3), then remove the first or the second formant characteristics from the speech signals obtained at step 2) by inverse filtering used ^PC RESULT The simulation is conducted for the 5 vowel vocal tract shapes which are obtained from the XRay pictures by Fant and others. Fig. 1 shows the variation of the source waves generated by two-mass model. The source waves of the vowel /a/, /e/ and /if are similar, while the source waves of the vowel /of and /u/ are rather different. These shapes vary depending on the glottal condition such as vocal fold tension and sub-glottal pressure.

3

1 5 10 Time in ms (d)/o/

5 10 Time in ms

10 Time in ms (b)/e/

Time in ms (a) /a/

(c)/i/

JWiiiljtil 1

1

1

1

1

1

1

1

1

1

1

1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 —'—1—i—•—1—'—1—1—1—1—•—1—t-

5 10 Time in ms (e)/u/

Fig.l Shapes of source wave generated by two-mass model for 5 vowels. Fig.2 shows the variation of source waves on the sub-glottal pressure p (expressed in water height). Fig.3 shows the power spectra estimated by 3 order of LPC for the same condition of Fig.2. From the Fig.3, it is clear that the spectrum decay varies much on the shape of source wave.

Extraction of transfer characteristics of vocal tract from speech signals

0

3 6 Time in ms (a) p=6

Time in ms (b)p=8

Time in ms (c)p=10

481

0 3 5 Time in ms (d) p=12

Fig.2 Variation of the source waves on the sub-glottal pressure p for the same vowel /a/.

3 4 Frequency in kHz Fig.3 Power spectra of the source waves for the four kind of sub-glottal pressures. 0

1

2

Fig.4 shows a synthesized voice by LPC synthesizer for vowel /a/ and its integrated wave form. The source-free portion is designated by the arrow in the figure. If the pitch period is long enough, the transfer characteristics of the vocal tract are well represented by LPC. Source

Source

Source-free

Time

in ms

. 5 . Tune m ms

Fig.4 Synthesized voice wave and its integrated wave.

10

482

H. Tsutsumi and Y. Kagawa

Fig.5 shows one example of the estimated source waves by method 1. It shows that in the case of speech with long pitch period, the source wave can be estimated rather well from the speech signals. The comparison between the true source wave (used in the synthesizer) and the estimated source waves is shown in Fig.6. The pitch period and wave width are almost the same. True

in

ms

lime

in

ms

Fig.5 Estimated source wave by method 1. True Estimated

5

10 "15 Time in ms Fig.6 Comparison between the true and the estimated source waves in Fig.5. Fig.7 shows one example of the estimated waves by method 2. Comparing with method 1, the estimated source wave by method 2 is a little narrow in width and smaller in the rising slope than the estimated one by method 1. This seems to be the effects of low-pass filter at step 2). So appropriate cut-off frequency must be used in the low-pass filter. Fig.8 shows the comparison of the spectra between the true source wave and the estimated source wave (by method 2). Both are almost the same in spectrum. Finally Fig.9 shows the comparison in spectrum between the estimated transfer characteristics (by method 2) and the true transfer characteristics (by the acoustic tube model). They also show good agreement in spectrum shape, but a little difference remains in formant frequency. This seems to be the difference between the spectrum of the synthesized voice and the spectrum of the transfer fimction by the acoustic tube model.

Extraction of transfer characteristics of vocal tract from speech signals True

True

Fig.7 Estimated source waves by method 2.

Estimated

1

2

3 Frequency in kHz

4

Fig.8 Spectra of the true and estimated source waves. GQ

50+

o

401 301 20 f 10 +

o Q.

O-h

H- -+2 3 Frequency in KHz

Fig.9 Transfer characteristics of the vocal tract in power spectrum

483

484

H. Tsutsumi and Y. Kagawa

CONCLUSION Two methods are proposed to extract the transfer characteristics of the vocal tract from the speech signals. hi the simulation by the use of artificial voice, it is shown that both the source wave and the transfer characteristics of the vocal tract are rather well extracted from the speech signals by the proposed way. The application of this method to the real speech is left as one of theftitureworks. REFERENCE 1. 2. 3. 4. 5. 6. 7.

Fant, G. (1960). Acoustic Theory of Speech Production. Mouton and co., 's- Gravenhage. Baer, J. and Gore,J. G. and Gracco, L. C. and Nye, P. W. (1991). In: Journal ofAcoustic Society ofAmerica, pp. 799-828. Wakita, H. (1970). hi: IEEE Trans. Vol ASSP-27No.3. pp.281-285. Tsutsumi, K. and Kagawa, Y. (2000). In: Journal ofAcoustic Society ofJapan Vol. 56-1. pp.21-31. Nakajima, T. and Omura, K. and Ishizaki, S. (1973). In: Bulletin of the Electrotechnical Laboratory Vol.37 No.4. pp.462-481. Ishizaka, K. and Flanagan, J. L. (1972). In: The Bell System Technical Journal JulyAugust. pp.\233-\26S. Flanagan, J. L. (1972). Speech Analysis Synthesis and Perception. Springer-Verlag, Berlin Heidelberg New York.

Uniqueness, lU-posedness, Regularization

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487

REGULARIZING PROCEDURES FOR SOLVING THE GENERAL INVERSE PROBLEM OF STRUCTURAL CHEMISTRY AND THEIR APPLICATIONS I.V. KOCHKOV, G.M. KURAMSHINA, V.P. SPIRIDONOV, Yu.I. TARASOV and A.G. YAGOLA Moscow State University, Moscow 119899, Russia ABSTRACT The general inverse problem of structural chemistry is to find the molecular force field and geometry parameters using experimental data (IR and Raman spectra, electron diffraction, etc.) and ab initio quantum mechanical calculations. We formulated the problem and proposed a new approach based on the theory of nonlinear ill-posed problems in finite-dimensional spaces. Numerical methods and sofi:ware package SPECTRUM were applied for data processing and investigation of different molecules. KEYWORDS Inverse problems, ill-posed problems, regularization, inverse vibrational problem, general inverse problem of structural chemistry INTRODUCTION Many mathematical problems of science, technology and engineering are formulated in the form of an operator equation of the first kind with operator and right side approximately known. Ofi;en such problems turn out to be ill-posed. It is possible to solve such problems only by using the special methods - regularizing algorithms. The theory of solving linear ill-posed problems is advanced greatly today [1]. It is not in the case of nonlinear ill-posed problems, since the direct transfer of linear theory to the nonlinear case is not possible in general. For the solution of nonlinear ill-posed problems, different variational approaches for constructing of the regularizing algorithms based on Tikhonov's fiinctional and generaUzed principles for the choice of regularization parameter and formulation of very general conditions for their implementation could be found in [2].

488

/. v^ Kochikov et al.

These methods were applied to many inverse problems of physics, chemistry, etc. Below we shall describe the general inverse problem of structural chemistry for free molecules as nonlinear inverse problem of molecular vibrations. The purpose of the study is molecular force field and geometry restoration using all available experimental data (vibrational (IR and Raman) spectroscopy, electron diffraction, etc.), a priori constraints and ab initio quantum mechanical calculations. Vibrational spectra are among the most important sources of our knowledge about molecular structure. From the viewpoint of analyzing experimental data (obtained by means of infrared and Raman spectroscopy) a number of inverse problems arise. One of the most important is the socalled inverse vibrational problem of determining parameters of the molecular force field from given experimental data (vibrational frequencies, isotope frequency shifts, CorioUs constants, centrifugal distortion constants, etc.). The accumulation of data on molecular constants helps us to predict spectra and other properties of compounds not yet investigated and assists the development of physical models in a theory of molecular structure. This problem in details as well as regularizing procedures for its solution was described in [3,4], Another problem is to find molecular geometry parameters using electron diffraction data. If we consider these two problems together then we get a general inverse problem of structural chemistry (GIPSC). The GIPSC is a nonlinear ill-posed problem in finite-dimensional spaces. It could have nonunique pseudosolution. So, it is necessary to select a rule how to choose an unique normal pseudosolution. We proposed to use results obtained from ab initio or quantum mechanical calculations for constructing molecular force field and geometry parameters such that they are most close to theoretical data, but exactly reproduce the available experimental data within chosen set of a priori limitations (£)) and specified experimental precision. A set D can include limitations in a form of fixing some constants or including intra- or inter-molecule-in-pair equalities for some parameters as well as Pulay's scaled force constant matrix constraints. A special regularizing procedure based on a theory of nonlinear ill-posed problems [2] was developed. As result, we receive a regularized quantum mechanical molecular force field and geometry. This approach is based on usage of data obtained from ab initio and quantum mechanical calculations on computers of University Wisconsin-Madison (USA) and Waseda University (Japan) + calculations on PC computers of Moscow State University within regularizing algorithms realized in the software package SPECTRUM. Details of this approach could be found in our recent pubHcations [5, 6, 7]. In these publications and also in [3,4], the structure of the software package SPECTRUM and results of numerical calculations of important practical objects, including benzene, fuUerenes and other molecules were described. This work was partially supported by the RFBR grants 98-03-33166, 96-15-97469 and 99-0100447. A.G.Y. is also grateful for the financial support that was granted by Ministry of Education, Science, Sports and Culture, Japan, and to the Russian Ministry of Education for a grant "Universities of Russia - Basic Researches" (project 4-5220).

MATHEMATICAL STATEMENT OF GIPSC AND REGULARIZING PROCEDURES To implement an integrated procedure of simultaneous refinement of the force field and equilibrium geometry, a dynamic model of a molecule has been created. Based on the general approximation of small vibrations, the model has been extended to include cubic and quartic

489

Regularizing procedures for solving the general inverse problem

anharmonic potential terms for proper description of the large-amplitude motion. Within this approach, a molecule is described using a set of equilibrium geometry parameters R, and a molecular force field F represented by the matrices of quadratic, cubic and possibly quartic force constants defined in the framework of a certain nonlinear system of internal coordinates. Both parameter sets R and F can be considered asfinite-dimensionalvectors. The model is used to predict experimentally measured values, such as vibrationalfrequenciesco, electron diflfraction intensity M{s\ rotational constants {A, B, C} obtained from microwave molecular spectra, etc. All of these values are Sanctions of geometric (R) and force field (F) parameters. With experimental data and parameters represented as elements of normalized finite-dimensional spaces, we can formulate the problem of simultaneous refinement of the force field and equilibrium geometry of the molecule as a system of non-linear equations {G)(F,R) = CO,^,

(1)

\Mis,F,R)=M,^(s), [{A,B,C}(R^F) = {AB,C},^ on a set of predefined constraints FGDF, R^DR. This system can be extended to include additional experimental evidence when available (for example, data for isotopic species of a molecule sharing the same force field and equilibrium geometry).

Due to experimental errors, lack of experimental data and model limitations, this system of equations (that can be also treated as afinite-dimensionalnon-linear operator equation) usually fails to define unique solution, often proves to be incompatible and does not provide stability with respect to the errors of input data. To avoid these unfavorable features characteristic to the ill-posed problems, it is necessary to implement a regularizing algorithm for its solution. We suggest using a regularizing algorithm based on optimization of the TikhonoVs fijnctional M\F,R) = \\co(F,R)-co\f ^]M(s,F,]^-M,^(s)f fn „2 „ ,,21 ^J||i7_i70|p+||^_;^0||2l

^{A,B,Q(F,R)-{A,B,Q,^f

+ <2)

where in the last ("stabilizer") term F^ and R^ represent parameters of ab initio force field and equilibrium geometry, respectively. With the appropriate choice of regularization parameter a (that depends on the experimental errors characterized by some numerical parameter S), it proves possible to obtain approximations converging to a normal pseudosolution of the system (1) when experimental errors tend to zero [3,4]. These approximations are obtained as extremals {/^a,i^«}offimctional(2). The following schematic picture shows the main features of data processing in the software package SPECTRUM used to process spectroscopic and electron diffraction data.

490

/. V. Kochikov et al.

Input

Model assumptions and results of a^ initio calculations

Spectroscopic frequencies of vibrations (Hk

1. Assumed initial equilibrium geometry (R^) 2. Geometry constraints (R e DR) 1. Initial approximation for the harmonic force field (matrix F^) 2. Additional constraints imposed on matrix elements (including force field model using scaling factors)

Electron diffraction intensity

Microwave rotational constants {A,B,C}

FGDF

Anharmonic terms in the force field 1. Kinematic anharmonicity 2. Dynamic anharmonicity

Optimization procedure minM"(F,7?) for FGDJ,,

RGDJ^

where

M\F,R) = = \\co(F,R)-co,j( ^\\^(s,F,l^-M,^(s)f

^{A,B,Q(F,R)-{A,B,Q,^f

+

M^

Output

1. Geometrical parameters (r^, r^, /•«), correlation coefficients and estimated errors 2. Mean square amplitudes. 3. Quadratic force field (matrix F), vibrational frequencies and eigenvectors of GF matrix. 4. Graphic representation of molecular geometry and vibrations

Regularizing procedures for solving the general inverse problem

491

REFERENCES 1. 2. 3. 4. 5. 6. 7.

Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V. and Yagola, A.G. (1995). Numerical methods for the solution of ill-posed problems. Kluwer Academic Press, Dordrecht. Tikhonov, A.N., Leonov, A.S. and Yagola, A.G. (1998). Nonlinear ill-posed problems. Chapman and Hall, London. Yagola, A.G., Kochikov, LV., Kuramshina, G.M. and Pentin, Yu.A. (1999). Inverse problems of vibrational spectroscopy. VSP, Utrecht, Tokyo. Kochikov, I.V., Kuramshina, G.M. and Yagola, A.G. (1998) Surveys on Mathematics in Industry 8, 63. Kochikov, I.V., Tarasov, Yu.L, Kuramshina, G.M., Spiridonov, V.P., Yagola, A.G. and Strand, T.G. (1998) J. Mol. Struct. 445,243. Kochikov, I.V., Tarasov, Yu.L, Spiridonov, V.P., Kuramshina, G.M., Yagola, A.G, Saakjan, AS., Popik, M.V. and Samdal, S. (1999) J. Mol. Struct. 485/486 (1), 421. Kuramshina, G.M., Pentin, Yu.A. and Yagola, A.G. (1999) J. Mol. Struct. 509,255.

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

^"^

N U M E R I C A L S O L U T I O N OF A C A U C H Y P R O B L E M FOR A N ELLIPTIC EQUATION J. C H E N G \ Y . C . H O N ^ , T . W E I ^

M. Y A M A M O T O ^

^ Department of Mathematics, Fudan University, Shanghai 200433, China. & Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376-8515, Japan. e-mail: j chengQmath. sci. gunma-u.ac.jp

^ Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, HongKong e-mail: maychonQcityu. edu. hk ^ Department of Mathematics, Lanzhou University, Lanzhou, Gansu Province 730000, China ^ Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaha, Meguro, Tokyo 153, Japan e-mail: myamaQms.u-tokyo . a c . j p ABSTRACT In this paper, we investigate the numerical solution of a Cauchy problem for an elliptic equation in two-dimension. By using Green's formula, the Cauchy problem is first transformed to a moment problem whose numerical approximation can be sought. Furthermore, a necessary and sufficient condition for the existence of the solution of the Cauchy problem for Laplace's equation in two-dimension is also given from solving the corresponding moment problem.

KEYWORDS Numerical computation, Cauchy problem, ill-posedness

INTRODUCTION Let Oj, j = 1, 2 be simply connected domains in IZ^ such that Hi C ^ 2 . We assume that dVlj^ j = 1,2 are C^-boundaries. Consider the following Cauchy problem: Au{x) u{x) du ^—(x)

= =

0, f{x),

xeQ.2\'^i X e dQ.2

(1) (2)

=

g{x),

xedn^

(3)

494

J. Cheng et al.

where A is an elliptic operator and ~- is the conormal derivative with respect to A on 9^2 (see [10]). This is the classical Cauchy problem for an elliptic equation. It arises in many real applications such as nondestructive testing [1], [3], [4], [7], geophysics [13] and cardiology [5]. It is well known that there exists at most one solution to (1) - (3) ([7]). The problem is ill-posed in the sense that any small changes of the initial data may induce large changes of the solutions. Under additional a-priori boundedness, a continuous dependence of the solution on / , p, and hence is called conditional stability, can be restored (e.g. [7], [11]). The ill-posedness of this problem makes the numerical computations very difficult. Usually, some regularization techniques have to be employed to obtain a stable solution. Several regularization methods were proposed: quasi-reversibility method [8], [9], Tikhonov regularization method [2], [13] and discretization [6], [12], etc. This paper proposes a new numerical method for an approximation of the solution of (1) - (3). The new ingredient is to transform the Cauchy problem to a moment problem. This transformation has a spirit similar to the boundary element method for solving elliptic equations. A numerical method for the moment problem gives the boundary values of |^ on 30i which yields u in Qi \ Q^i through the Neumann problem for (1). In this paper, the theoretical formulation of the proposed numerical method is given. Detailed numerical computations will be given in a forthcoming paper. It should be remarked here that the proposed method can also be extended to solve Cauchy problems for general elliptic equations of second order in three-dimension.

TRANSFORMATION OF THE CAUCHY PROBLEM TO A MOMENT PROBLEM For simplicity, consider the simpliest case when ^ = A. For a general elliptic operator A, the methodology is similar. Let V satisfy Av{x) /^v{x) dv{x) dv

== 0,

0,

X£VL2\Q^U

(4)

xedVii.

(5)

Here u is the unit outward normal vector to 9(^2 \ Hi) and ^ ^ =zVv -P. By using Green's formula, we have f

/A

A \ 7

/*

/ <9^

^^N ,

f

, du

dv. ^

(6) / (viXu-ullv)dG - / ivu-^)ds^- I {v~ u-^)ds Jn2\ni Jdn2 du dv JQ^^ dv dv' where da and ds are the area and curve elements respectively. Since u is the solution of (1) - (3) and v is the solution of (4) - (5), the identity (6) can be rewritten as

Denote n = {v laoi I V satisfies (4) - (5)} . We have the following result.

(8)

Numerical solution of a Cauchy problem for an elliptic equation

495

P r o p o s i t i o n 1. If the Cauchy problem (1) - (3) has a solution u such that |^|ani ^ L^(^ni), then (3 = fj|ai2i satisfies the following moment problem: /

vf^ds = /

Jani

{f-

JdU2

gv)ds =

fiv{f,g):

(9)

^^

where v E H. Conversely if ^^ G L'^idfti) is the solution of (9), then there exists a solution u of the Cauchy problem (1) - (3) such that f^\dn, E P{dQ.i). Proof: We can easily verify that the solution of the Cauchy problem (1) - (3) satisfies the moment problem (9). It is sufficient to show tha.t the solution of the moment problem (9) satisfies the Cauchy problem (1) - (3). Assume that /? is the solution of the moment problem. Consider the following Neumann boundary problem for Laplace's equation in 02 \ Hi: Aa(x) da{x) da{x)

xe^2\'^u

(10)

(3{x),

xed^i,

(11)

g{x),

xedn2,

(12)

=

0,

= =

/ ads =

f

(13)

fds.

7^2

For the solvability of (10) - (12), by Green's formula, again we have /

gds + /

f^ds = 0.

(14)

It is obvious that any constant function c is in H. Therefore, simply taking v ^ c in (9), the equality (14) will then follow. From a standard result (e.g. [10] p. 195) for elliptic equations, there exists a unique solution a G H^{rt) (Here /f 2 (0) is a Sobolev space). It is clear that if a satisfies (10) (12), then a -\- c also satisfies (10) - (12) for any constant c. Hence, with a suitable choice of c, we can obtain a solution which further satisfies (13). To show that a satisfies (1) (3), it is sufficient to verify (2). In fact, for any v £ K, the equations (9), (11) and (12) imply Jan2

9jy

dv

Je^^

that is

f r9v . f da . f / }-^ds= / v-^ds-^ I Jdn2 ^^ Jdn2 ^^ Jea, On the other hand, the use of the Green's formula yields

dv da ^ v-^ds. ^^

/* da , /• da , f dv ^ f dv ^ f / v-^ds + / v-T—ds = / a'T-a5 + / a—-ds = / a -^ds. du JQ^^ du JQ^^ Jd^2 ^^ Jan, ov Jea, du Therefore, we have /

^{f-a)ds

= ^.

(15)

496

J. Cheng et al

Now. consider Ai; =

0

1^

=

/ - a

=

0

dv

in

n2\ni, on

on

(16)

an2,

(17)

901.

(18)

From (13) and the standard result for the elliptic equa,tion, there exists v which satisfies (16)- (18) and v eU. Therefore (15) becomes

/

(19)

[f-a^ds^O,

and hence / = a'|c,a2- Thus, a is the solution of the Cauch}^ problem (1) - (3). The proof is complete. From the uniqueness of Cauchy problem for Laplace's equation and Proposition 1, we have Proposition 2. There is at most one solution to the moment problem (9). Furthermore, from Proposition 2, we ha,ve that H = L^{drti). {vj}%o C n such that L'{dn,). Span{v,}jLo =

Therefore, we can choose

The moment problem (9) becomes

L

v,pds = fij,

J = 0,1,---

(20)

where f^j = J^^ ( / " ^ ~ 9^j)(^^ ^i^d Vj are functions satisfying (4) - (6). It is noted here that fjj is determined uniquely by / , g and v^ e H. In the follow^ing, we shall show that the functions Vj (j = 0 , 1 . 2, • • • ) can be chosen suitably to give a numerical approximation of the solution.

MOMENT PROBLEM In this section, we show that a special choice of the functions Vj can give a very simple moment problem. This also gives the necessary and sufficient condition for the existence of the solution in the domain 02 \ Hi. For illustration, let n, = {{xuX2)\xl^xl
(21)

and ^2 be a simply connected domain such that Hi C 02Remark 1. It should be noted here that 02 can be an arbitrary simply connected domain. This makes the proposed method robust and effective for solving practical problems. Remark 2. In most of the cases Oi is assumed to be a unit disk. It is known that in two-dimension any simph^ connected domain can be transformed by a conformal mapping to a unit disk. Note that a conformal mapping does not change the original Laplace's equation.

Numerical solution of a Cauchy problem for an elliptic equation

497

For J = 0,1, 2, • • • , choose V2j + l{xi,X2) = (;7 + H)cOs(j(9), 1^2,(^1,^-2) = (;7 + H)sin(j/'^), where xi + ix2 = re^K It is easy to verify that Vk satisfies {^Vk{x) = ^,

.22)

xelV\

From (20), the Cauchy problem (1) - (3) is equivalent to the following moment problem, forj=0,l,2,..-.

where

^

1 [

and

dv2j+i

. ,

rdVO

From the expression (23), it is clear that the moments are just the Fourier coefficients of the function P{9). Based on the result on Fourier analysis, we have the following existence result for the solution of the Cauchy problem (1) - (3). Proposition 3. (Existence) A necessary and sufficient condition for the existence of the solution for the Cauchy problem (1) - (3) with f^la^i € L^{dfti) is that 00

E(I/^'I' + KI')<~-

(24)

J=0

Thus, in this case, it is sufficient to reconstruct /? from the the Fourier coefficients. However, by the ill-posedness of the original Cauchy problem, such reconstruction is unstable in general. For stable computation, we have to apply some regularization. The details will be given in a succeeding paper.

CONCLUSIONS In this paper, a new numerical method for solving the Cauchy problem for an elhptic, particularly the Laplace's, equation was proposed on the basis of solving a moment problem. The necessary and sufficient condition for the existence of the solution was also given. Although this paper investigates the Cauchy problem for Laplace's equation in a two dimensional plane domain, the method can be extended to study the Cauchy problem of general elliptic equations of second order in three-dimension.

498

J. Cheng et al

ACKNOWLEDGEMENT

The first author is partially supported by the NSF of China (No. 19971016). This work is partially supported by a grant from the Research Grants Council of the 'Hong Kong Special Administrative Region, China (Project No. 9040428), a Strategic Research Grant from the City University of Hong Kong (Project No. 7000943) and the Sanwa Systems Development Company Limited (Tokyo)

REFERENCES [1] ALESSANDRINI G . , (1993). Stable determination of a crack from boundary measurements. Proc. Roy. Soc. Edinburgh Sect. A 123. pp. 497-516. [2] A N G , D . D . , N G H I A . N . H . and T A M , N . C , (1998). Regularized solutions of a Cauchy problem for the Laplace equation in an irregular layer: a three-dimensional model. Acta Math. Vietnam. 23, pp. 65-74[3] BUKHGEIM, A.L., C H E N G , J. and Y A M A M O T O , M . , (1999). Stability for an inverse boundary problem of determining a part of a boundary. Inverse Problems 15, pp. 1021-1032. [4] C H E N G , J. and Y A M A M O T O , M . , (1998). Local stability of a hnearized inverse problem in detecting steel reinforcement bars. Proceedings of the International Conference on Inverse Problems and Applications (Quezon City, 1998). Matimyds Mat. 21, Special Issue, pp. 18-33. [5] C O L L I - F R A N Z O N E , P . ,

G U E R R I , L . , T E N T O N I , S., V I G A N O T T I , C . , B A R U F F I ,

S.,

SPAGGIARI, S . and T A C C A R D I , B . , (1985). A mathematical procedure for solving the inverse potential problem of electrocardiography. Analysis of the time-space accuracy from in vitro experimental data. Math. Biosci, 11, pp. 353-396. [6] FALK, R.S.I, and MONK, P . B . , (1986). Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson's equation. Math. Comp. 41, pp. 135-149. [7] ISAKOV, v . , (1998). Inverse Problems for Partial Differential Equations. Verlag, New York.

Springer-

[8] KLIBANOV, M . V . and SANTOSA, F . , (1991). A computational quasi-reversibility method for Cauchy problems for Laplace's equation. SIAM J. Appl. Math. 51, pp. 1653-1615. [9] L A T T E S , R . and LlONS, J.L., (1986). The Method of Quasi-Reversibihty: Applications to Partial Differential Equations. Elsevier, New York. [10] LIONS, J . L . and M A G E N E S , E . , (1972). Non-homo ereneous Boundarv Value Problems and Applications. Vol. I. Springer-Verlag, New York-Heidelberg. [11] PAYNE, L.E., (1960). Bounds in the Cauchy problem for the Laplace equation. Arch. Rational Mech. Anal. 5, pp. 35-45.

Numerical solution of a Cauchy problem for an elliptic equation

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[12] R E I N H A R D T , H . -J., H A N , H . and HAo, D. N., (1999). Stability and regularization of a discrete approximation to the Cauchy problem for Laplace's equation. SIAM J. Numer. Anal. 36, pp. 890-905. [13] TiKHONOV, A.N. and A R S E N I N , V.Y., (1977). Solutions of Ill-posed Problems. Winston and Sons, Washington.

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

501

VARIATIONAL A N D D-N A P P R O A C H E S FOR A MAGNETOSTATIC CAUCHY PROBLEM TAKEMI SHIGETA^ ^ Graduate School of Mathematical Sciences, Ibaraki University 2-1-1, Bunkyo, Mito Ibaraki 310-8512, Japan e-mail: nd7404@mcs. i b a r a k i . a c . j p

ABSTRACT The purpose of this paper is to propose two numerical methods for a magnetostatic Cauchy problem. In this problem, the Cauchy data are prescribed on a part of the boundary of a bounded domain. Based on the variational method, the proper boundary value on the rest of the boundary and the corresponding magnetic field in the domain are determined. Another method is based on the Dirichlet-Neumann (D-N) alternating method. After computations by using these two numerical methods, it is concluded that our variational algorithm is stable, and the numerical solution is in good agreement with the exact one. The D-N algorithm is better than the variational algorithm in the cost of computations. KEYWORDS Cauchy problem, Dirichlet-Neumann alternating method, inverse boundary value problem, magnetostatic problem, steepest descent method, variational formulation INTRODUCTION We consider a magnetostatic problem in which two kinds of boundary data are simultaneously imposed on a part of the boundary of a bounded domain. This is a variant of the Cauchy problem of the Laplace equation, in which Dirichlet and Neumann data are simultaneously prescribed on a part of the boundary [4]. In this paper, we call the magnetostatic problems of this kind magnetostatic Cauchy problem or simply Cauchy problem. Let r^ be a bounded domain enclosed by the smooth boundary F in B? with the rectangular coordinates (x,7/), let the vector n be the unit outward normal to F, and let the vector r be the unit tangent in the direction along F. As the Cauchy data, we prescribe simultaneously the normal component of magnetic flux density B and the tangential component of magnetic field ii" on a part of the boundary Fa C F. We note that any boundary conditions are not imposed on the rest of the boundary Fid '•=" T \ Td- The problem aims at identifying the proper boundary condition on Fjd and also aims at finding the magnetic field H in Q. Namely, the problem can be regarded as an inverse boundary value problem.

502

T. Shigeta

We give the electric current density J in the 2:-direction perpendicularly to the xy-plane. The magnetic permeability JJL is assumed to be a positive constant for simplicity. In the xi/'plane, we recall that dvy dvx (d(l) d(j)Y 1. xotv = —^- —-, rot ox oy for a vector function v = [v^, VyY ^nd a scalar function 0. Then, our problem can be written as follows: Problem 1 Find H e H\Qf

d^x , dvy

such that

rot i f = J

and

div B = 0 in Q

with the boundary conditions B n = Bn

and

on Fd,

H - T = Hr

where Bn.Hr G i/^/^(Fd) are given boundary data. Let A e H^{Q) be a scalar function that satisfies B = rot* A = {dA/dy,-dA/dx){e if ^(Q)^). If Bn = 0 on Fd, Problem 1 is equivalent to the following Cauchy problem of the Poisson equation:

^ = 0

and

1 —AA = J M IdA -7 - - ^ - = Hr /i dn

in

Q,

on

Pd.

That is the reason why we call Problem 1 magnetostatic Cauchy problem, and Bn and Hr the Cauchy data in this paper. The numerical method for the Cauchy problem of the Laplace equation studied in [3] is extended to the numerical method for our problem. VARIATIONAL FORMULATION We assume that a solution of Problem 1 can be extended to Fid through ft. In order to identify the proper boundary value a; = ff • r on the boundary Fid, we transform Problem 1 to the following variational problem: Problem 2 Find u e /f ^/^(Fid) that minimizes the following objective functional: F{uj)'= f

\Hiu;)-T-Hr\^dr-\-rj

- / \H{uj)'T-Hr\^dr-\-rj Jr^

f

H{uj)'B{Lj)dn

[ fi\H{uj)\'^dn Jn

for some constant 77 > 0 called regularization parameter. The second term of the functional denotes the total energy, which plays a role of regularization. Since the solution u makes the energy small, we can expect to obtain a stable solution, even if errors are contained in

Variational and D-N approaches for a magnetostatic Cauchy problem

5 03

the Cauchy data. Here H = H{co) is the solution of the boundary value problem (called primary problem): and

TotH = J

div B == 0

in

Q

(1)

with the boundary conditions B-n H

= Bn T = u;

on

Td,

(2)

on

Fid-

(3)

Our strategy is to find the minimum of F{UJ) by generating a minimizing sequence {^^^^}£o t>y the steepest descent method: starting with an initial guess u^^\ with a suitably chosen numerical sequence {pk}kLo- The derivative F'{UJ) is the first variation of F(a;), defined by F{u; + Su) - F{LV) = {F'{uj), SLO) + o{\\Su;\\), where (•, •) and || • || denote the inner product and the norm of L^(rid), respectively:

\\f\\--=[l l/Nr) .

{L9):=£f9dT,

For the implementation of the steepest descent method, we require an explicit form of F'{uj), which will be calculated in the next section.

DERIVATION OF F'{co) To seek a concrete expression for F'iu),

we note that

F{ijo -f duo) - F{uj) = [ {\H{u; + Scj) ' T -Hr\^ -hr] [ Jn

- \H{LO) ' T -Hr\^}

fi{\H{u~h6uj)\^~\H{uj)\^}dn

= f {[H{uj + Su) + i f (a;)] • r - 2Hr}{[H{u; -hV [ fi{H{u

dr

+ Su) - H{u)] - r} dT

-h Scu) + H{cu)} • {H{uj + Su) - H{u;)} dQ

= [ {[2H{u) -f SH{uj)] . T - 2Hr}{SH{u;)

• r } dT

+ rj [ ii{2H{uj) + 6H{uj)} • 6H{ij) dVt Jn = [

2{H{u;) ' T - 'Hr){6H{uo) -TJdT-hr] + [ \6H{uj)'T\^dr-^r] Jva

= [

2{H{uj) • r - Hr}{SH{uj)

f Jn

[ 2/xif (a;) • SH{u;) dQ

ix\6H[u))\^d^

- r} dT + rj f 2/iif (cj) • SH{uj) dQ + o{\\Scj\\).

504

T. Shigeta

Here we put 6H{uj) := H(UJ + 6uj) The following equations hold:

H{u).

rot(^i/(a;)-0

where SB{u) := B{u-\-Su) A{cj), we can see from

in

Q,

(4)

div(5B(a;) = 0

in

Q,

(5)

JB(a;)-n = 0

on

Td,

(6)

6H{uj)'

on

Fid,

(7)

- B{u).

T = SLJ

Putting B{u) = rotM(a;) and SA{UJ) := A{U;-\-SLJ)

-,{SHi.).r}

=

-

(8)

'-^,

(9)

6 B i . ) . n = ' ^ , that equations (4)-(7) is equivalent to A6A{u;) = 0

in

Q,

(10)

SA{uj) = 0

on

Td,

(11)

on

Fid-

(12)

dSAju) dn

= —/JLSLJ

Using the Stokes formula: [ rot H{uj) 6A{u;) dQ = ({H[uj)

Jn fQ

Jr Jr

• T)6A{U)

dV-\- f H{u)

Jn

• rot* 5A{u)) dQ,

we know from (1), (3), and (11) that f JSA{uj)dn=

f

u;6A{uj)dr+

(13)

f H{UJ)-rot*SA{uj)dn.

Let A G H^[Q) be a solution of the following boundary value problem (called adjoint problem): P r o b l e m 3 Find A e H\n)

such that -AA

= 2r]nJ

in

Q

(14)

on

Fd,

(15)

on

Fid-

(16)

subject to the boundary conditions: A = 2{H{u)'T-Hr} —- = -2r//ia; on

Substitution of (10)-(12) into the Green's formula: f ASA{uj) Adn

= f ^^^^AdT

-

f SA{u;)^dr

-h /

SA{uj)AAdn

505

Variational and D-N approaches for a magnetostatic Cauchy problem yields 0= f

^ ^ ^ l c / r + I (-^i6uj)AdT-

f

6A{uj)^dr+

[ 6A{u;)AAdn.

Consequently, we know that F{uj -f Su) - F{u) = f A (--^^4^) Jr^ \ ^ dn

J

dr + ri [ 2Hiuj)'rot* SA{uj)dQ + o{\\Su;\\) J^

(using (8), (15), and ^i6H{uj) = rot* SA{uj)) =-[[

{-fiSuj)AdT-

f

SA{u;)^dr+

-^2r]( f JSA{u)dn-

f

f SA{u;)AAdQ]

ujSA{uj)dr]+o{\\6uj\\)

(using (17) and (13)) = / {-A)Sujdr+ / - /

•/rid

f

\-^-h2r]u;]sA{uj)dr

(-AA-\-2r]JjSA{uj)dQ-^o{\\Suj\\) (using (14) and (16))

{-A)Sujdr-^o{\\6uj\\)

= ( - A M + o(Pa;||). Therefore, we know that the first variation

F'{UJ)

F'iu) = -A\r,, e

is given by the following form: H'/\r,a).

NUMERICAL ALGORITHM Our algorithm can be summarized as follows: ALGORITHM Step 1. Pick an initial guess a;^^) G if V2(rjd). Set k := 0. Step 2. Solve rot i f (^) = J

and

to find i f (^)(a;W) • r | r , e if^/"(Ld).

divB^^^ - 0

in

0,

B^^^-n = Bn

on Td,

(17)

506

T. Shigeta

S t e p 3 . Solve - A ^ ( ^ ) = 2r//xJ

in

Q,

on

Td,

— — = -2r]^iuj^^^ on an

Fid

A^''^^2{H^^\uj^^^)'T-Tir]

to find F'(a;(^)) = -l^^^lr,, G //^/'(Fid). S t e p 4. Choose /o^. S t e p 5. Update

S t e p 6. Set A: := A; + 1. Go to Step 2. Problems in Steps 2 and 3 are solved by using the edge and the nodal finite elements, respectively [1, 2]. In order to choose suitably the numerical sequence {pfcj^o ^^ ^^^P 4' we adopt the Armijo criterion [5], which guarantees for the sequence {uo^^"^) to satisfy F(a;W - /9,F'(a;(*))) < F(a;W) - ip,\\F'{J''^)f

(18)

with a constant 0 < ^ < 1/2.

D-N APPROACH We propose a new numerical approach based on the Dirichlet-Neumann (D-N) alternating method [6]: S t e p 1. Give an initial guess a;(^) G H^l'^{Ti^). Set k := 0. S t e p 2. Solve the magnetostatic problem: rot i f i*^^ = J,

div(/xHf^) = 0 {fiH[^^)'n Hf'>-T

to find {fiH[^^)

in

Q,

on

Fd,

= uj^^^ on

Fid

= Bn

"n\r,^.

S t e p 3 . Solve the magnetostatic problem: rot i f ^*^^ = J,

div(/iif^^^) = 0 H[^).r

= T{^

(/iif ^^^) • n = ifiH^^) tofindH^^^-rlr^a-

•n

in

n,

on

Fd,

on

Fid

Variational and D-N approaches for a magnetostatic Cauchy problem

507

Step 4. Update the boundary value: (^C^+i) = ak{Hf^

• r ) + (1 - ak)J>'^ on Ty,

where the relaxation factor ak is a suitably chosen real number. Step 5. Set A; := A: + 1. Go to Step 2. If this algorithm is convergent, then H^ (or H2) converges to a solution to the Cauchy problem as k tends to infinity. Suppose that Hy converges to Hi {i = 1,2), then we note that {iiHij-n — (/iH2)-n(= A) and Hi T = H2 • T{= LO) on Fid- Hence, we can see that Hi and H2 are solutions to the common Cauchy problem with the Cauchy data A and u on the boundary Fid. By uniqueness of solution to the Cauchy problem, we obtain Hi = H2{= HQ) in Q. We can see that (//if0) • ^ = i/^Hi) •n = Bn and H o • r = H2 -T = Hr on Td- Therefore, i f 0 is a solution to our Cauchy problem.

NUMERICAL EXAMPLES In this section, we consider three examples. We will confirm effectiveness of our numerical method based on the variational method through numerical experiments, and observe influence of the size of the boundary Fd on the accuracy of the numerical solutions. We notice that the function i f (a:, y) = {x ~ y, —y)^ is a solution of the magnetostatic equations in the case J = 1 and /i = 1. We suppose that the function H is unknown in the square domain Q, := {{x,y)] 0 < r r < l , 0 < ^ < l } , and impose two kinds of boundary data Bji and Hr on the boundary Fd- The boundary data in the three examples are: (a) Fd: three sides of the square Bn = 0, Hr = X on the side y ~ 0, Bn = I — y^ HT- = —y on the side x = 1, Bn — —1, HT — —X + 1 on the side y ~\. (b) Fd: one side of the square Bn — —1,

B.^ — —X + 1 on the side y = 1.

(c) Fd: a part of one side of the square 'Bn = - 1 , Jlr = -x + l

on the side y = 1 {0 < x < 0.4).

The domain is divided into triangular finite elements as shown in Fig. . As an initial guess, we took uo{x,y) = 0 along Fid- Fig. shows the exact magnetic field H{x,y), and Fig. 3 shows the estimated magnetic field i f ^^^(x, y) after k steps for each example, where we set 77 = 0, ^ = 0.1, r = 0.5, and £ = 10"^. Through the examples, we can confirm that the estimated magnetic fields are precise in the neighborhood of the boundary Fd, where boundary data are prescribed. From Fig. 4, we also know that the shorter the boundary Fd, the worse the accuracy of the estimated magnetic field.

508

T. Shigeta

V V V 7 -/ i' h H ^V N 4 ^ , * ; , it, !«•, >^, .i-, Y r^

.,

^

.1

J.

1,

.^^

Fig. 1. Finite element mesh

Fig. 2. Exact magnetic field H

4 VY V V > > ^^ V H H^ V V V^'^^ ^^

f / fe^/ / , V-, >!', >V> -4|

ii

^

H\ ^ ^ ^

"^i. --

V ^^ U h h h ^^ ^ ^-^ ^^ ^N^ ^v^ h U \ ^ ''^ ""-^ • •

(a) A: - 30

-

^

^

^

-

^

^

(b) A: = 50

(c) A; = 50

Fig. 3. Estimated magnetic field H^^^

gO.5

•s °

1 (a)

>

'" fi; f-t

(b)

y coitponent

«

(c)

Fig. 4. Exact solutions v.s. estimated solutions Next, we will observe if the regularization term stabilizes the numerical solutions when errors are contained in the boundary data. We notice that the function i f (x, y) = {x/2 -\- 2y/3, 5x/3 - y/2)'^ is a solution of the magnetostatic equations in the case J = 1 and /x = 1. We suppose that the function H is unknown in the same square domain Q as one in the previous examples, and impose two kinds of boundary data Bn and Hr with the relative errors 10% on two sides Fd of the

Variational and D-N approaches for a magnetostatic Cauchy problem

509

square as follows: 2 1 1 5 h 10% errors on the side x — 1^ ^n = -y -\- --\-10% errors, Hr — — y -\ 5 \ 1 2 h 10% errors on the side y = 1. ^n = -X — - + 10% errors, Hr = — x The domain is divided into triangular finite elements as shown in Fig. . As an initial guess, we took u)Q{x,y) = 10 along Fid- Fig. 5 shows the errors \\H - H^^^Wi^^n) for each rj. From this figure, we can observe that the optimal rj is 0.1. We can also see that the numerical solution for 77 = 0.1 is better than one for 77 = 0 as shown in Figs. 6 and 7. Hence, we know that the regularization term stabilizes the numerical solution. Through the numerical results, we can confirm that the regularization term with the suitably chosen rj > 0 yields good approximate solutions even if errors are contained in the boundary data.

Fig. 5.

Errors for each 77

Fig. 6.

H v.s. ff^^^ for r] = 0

Fig. 7.

H v.s. i f ^^^ for 77 = 0.1

Moreover, Fig. 8 shows the numerical results for the exact Cauchy data by the D-N and the variational algorithms, where we set a^ = 0.5. From this figure, we can see that accuracy of the solution by the D-N algorithm is fairly good.

,X^x>yy

^^^^ ^x

- ^

-^_

- ^ ^

% -^^ ^/ % -"^ -> ' . ^^ ^f

/ ^ ;

A

', 't ft n /t\

1.666

fl/i

>

(a) exact H

(b) i f (^^^ (D-N algorithm) Fig. 8.

(c) i/^^^^ (variational algorithm)

D-N algorithm v.s. variational algorithm

510

T.Shigeta

CONCLUDING REMARKS We considered an approximate method for the magnetostatic Cauchy problem. The problem is recast to a variational problem to identify the proper boundary condition. Minimization of a functional using the steepest descent method leads our variational problem to an iterative process in which two boundary value problems are involved. These problems are called primary and adjoint problems respectively. We apply the mixed finite element method using the edge element and the conventional finite element method using the nodal element for approximate solutions of the primary and the adjoint problems, respectively. Based on the result of the numerical examples, it is concluded that our numerical process is stable, the estimated solutions agree with the exact solution in the neighborhood of the boundary where Cauchy data are prescribed, and the regularization term with the suitably chosen regularization parameter yields good approximate solutions for the noisy boundary data. Another approach based on the Dirichlet-Neumann alternating method was proposed. Computational cost of the method by this approach is low. However, accuracy of the solution by the method was not so good.

REFERENCES 1. Kikuchi, F. (1989). Mixed formulations for finite element analysis of magnetostatic and electrostatic problems, Japan J. Appl. Math., 6, pp. 209-221. 2. Nedelec, J. C. (1980). Mixed finite elements in /^^ Numer. Math., 35, pp. 315-341. 3. Onishi, K., Kobayashi, K., and Ohura, Y. (1996). Numerical solution of a boundary inverse problem for the Laplace equation. Theoretical and Applied Mechanics, 45, pp. 257-264. 4. Tosaka, N., Onishi, K., and Yamamoto, M. (1999). Mathematical Approach and Solution Methods for Inverse Problems: Inverse Analysis of Partial Differential Equations {in Japanese), University of Tokyo Press. 5. Suzuki, S., Yabe, H., lida, Y., and Nakayama, T. (1994). Modern Numerical Methods {in Japanese), Ohm Corporation. 6. Yu, D., Xue, W., and Huang, H. (1995). A Dirichlet-Neumann alternating method in infinite domain: Algorithm and convergence analysis. Research Report ICM-9531, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences.

Numerical and Computational Algorithms

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INVERSE PROBLEMS IN ENGINEERESTG MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

513

I D E N T I F I C A T I O N C O E F F I C I E N T P R O B L E M S F O R ELLIPTIC HEMIVARIATIONAL INEQUALITIES A N D APPLICATIONS

S. MIGORSKI Jagiellonian University, Faculty of Mathematics and Physics Institute of Computer Science uL Nawojki 11, 30072 Cracow, Poland

ABSTRACT The inverse problems for a class of elliptic hemivariational inequalities are considered. The nonlinear elhptic operator is supposed to be monotone and the multivalued nonlinearity is given by the Clarke subdifFerential of a nonconvex superpotential. The unknown coefficient to be identified appears in the operator and depends on the gradient of the solution. Two results are delivered. First one is on the existence and uniqueness of solutions to hemivariational inequality, and the dependence of solutions on the data. The second one establishes the existence of solutions to the inverse problem. An application of our results to an inverse problem of determination of elastoplastic properties of a bending cylindrical bar is presented. KEYWORDS Hemivariational inequality, Clarke's subdifFerential, nonmonotone nonlinearities, nonconvex superpotential, elastoplastic. INTRODUCTION This note reports on a part of a recent progress done by the author during last years on the development of the hemivariational inequalities as models used for the description of mechanical systems and structures (cf. [1, 2, 3, 4, 5, 6]). We study the inverse problem of identification of a coefficient in the nonlinear boundary value problem for elliptic hemivariational inequality from measurements performed on a part of the boundary or in the interior of a domain. The problem of determination of coefficients in differential equations has been studied earlier in some particular cases: for elliptic equations by Cannon and Rundell [7], Mamedov [8], Hasanov [9], Acar [10] and for elliptic variational inequalities by Hasanov [11, 12].

514

S. Migorski

The goal of the present paper is to generalize the results of the above papers and to consider systems governed by hemivariational inequalities. The notion of hemivariational inequality was introduced by Panagiotopoulos, see [13] and [14] who defined also nonconvex superpotentials. The hemivariational inequalities express a physical principle (the principle of virtual work or power) and are not equivalent to minimum problems but they give rise to substationarity problems. We stress the fact that since the nonhnearities in the direct problem are multivalued and nonmonotone in general, the direct problem can not be formulated as a usual variational inequality. An overview of the theory of hemivariational inequalities is given in a recent monograph by Naniewicz and Panagiotopoulos [15]. The optimal shape design problems for a class of hemivariational inequalities have been studied by Denkowski and Migorski in [3, 4]. Furthermore, there is still a need to model processes involving nonmonotone laws and to identify parameters numerically. We remark that the identification problems for the hemivariational inequality appear in various situations in physics and engineering, for instance, in the determination of conductivity or permittivity of a medium occupying a given region (cf. [16, 15]). Working with a model based on deformation theory of plasticity, we give the application of our results to the inverse problem for determining the elastoplastic properties of a bending cylindrical bar (cf. [11, 8]).

SETTING OF THE PROBLEM The direct problem. Let H be an open bounded subset of IR^, occupied by a body under consideration, with Lipschitz boundary dVt = F. Let n — {rii} be the outward unit vector normal to F and let F = 17 U F^ with Fi fl F2 = 0. Consider the following boundary value problem: -div(5r(|Vu|^) Vw)+aiz = / i / 1 - 7 + 7, u = 0

on

-l{x)edj[u{x))

in

0

a.e. X G H (1)

Fi

Here j : IR —> IR is a nonconvex locally Lipschitz function and dj denotes the generalized Clarke gradient of j given by dj{i) = {( G IR : j\i',

v) > (v for all v e U}

(see Clarke [17]) while the generalized directional derivative of j at (f in the direction v^ denoted by j°(^; i'), is defined by J {(,;v) = hmsup

.

In order to study the direct problem (1) we use the variational approach. Introducing V = {veH\n)

: i ; - O o n Fi}

Identification coefficient problems

515

and the operator A\V —^ V* by div [g (|Vu|^) Vw) + m,

Au^-

(2)

the weak formulation of (1) is as follows find u ^V

such that there is x G ^^(0) satisfying

{Au,v) + {x,v) = U,v), x{^) ^ dj{u{x))

(3)

^veV

a.e. in H,

where (/, t') ==

fvdx -\Lpvd(j{x). The variational formulation (3) is called the Jo. JT2 ^ ^ hemivariational inequality. It is well known (see [15]) that dj{^) = /^(O? where /?: IR -^ 2 is a multivalued function obtained from a given function /3 G LJ^^l^) ^Y "filing in jumps" procedure (see e.g. [16, 15]). In the case j is a convex function, (3) reduces to a variational inequality (considered e.g. by Duvaut and Lions [18]) and dj is understood in the sense of convex analysis. The inverse problem. The inverse problem consists in the determination of the function g = g{^) in a set Q of admissible coefficients which solves the equation u{x]g) = h{x),

(4)

X eT2

for a given data h = /i(x), where u{g) = u[-]g) denotes the solution of the problem (3). In practice instead of solving (4), one associates the functional

/(<;,«) = |K.;ff)-/*(•)!U^r.)

(5)

and considers the double minimization problem m:m{J{g,u{g))

: geS,

uE % ) }

(6)

where S{g) denotes the set of solutions to (3) corresponding to the coefficient g. In applications, instead of the measured data (4) on the part of boundary, one may give the nonlocal measured data / u{x) dx — $. In such a case the functional (5) is replaced in JO.

(6) by

J{9^^) = I / u{x;g)dx-

$|.

Concerning the admissible set of coefficients we consider the class Q of functions (see [8]) which satisly hypothesis H{g) below: (a)

0 < Co < g{0 < ci,

(b)

g'iO > 0,

(c)

5(0 + 2<7'(e)e>7o>0on(0,r),

(d)

3 Co e ( 0 , D : g{0 + 2g'{0(o > 70 > 0 on ( 0 , 6 ) -

516

S.Migorski

MAIN RESULTS We present results on the existence of solutions to the direct and inverse problems. First using the technique of Hasanov (see [9], Lemma 4.1) we can prove the following Lemma 1.

The set Q of adm,issible coefficients is a com,pact subset of H^{0^^*).

We make the following hypotheses concerning the data of the direct problem: H{f3) :

function /3 G LJ^^(IR) satisfies the conditions (i) (ii)

lim /?(() exists for each <^ G IR; the graph of /3 ultimately increases, i.e. there is ^ > 0 such that ess sup 13(0 < 0 < ess inf f3{(); (-co,-0

(^,+oo)

(iii) \0(i)\ < c(l + lei) for ^ G IR with c > 0. (Ho) :

a G L^n), 0 < a{x) < C2J e 1^(0), (^ G L'{T2)-

Let us observe that if ^ satisfies condition H[g)a) and a satisfies (i/^o), then the operator A defined by (2) is coercive with constant CQ (recall that J^ \Vu\^ dx is an equivalent norm on \\ cf. p. 1033 of Zeidler [19]). Theorem 1. (i) Under hypotheses H{g)a)^b),c), (HQ) and H{P), the direct problem (3) has at least one solution. Moreover, the selection x appearing in (3) belongs to L'^(yi). (ii) Under the assumptions of (i), every solution u G V of (3) satisfies the estimate \\u\\v-^\\x\\L^
L2

(Hi) If (5 satisfies additionally the growth condition

essM^-M^iM>_k,

(7)

with k such that Q < k < co/d, where d > 0 is the constant of the embedding V —> L'^{^), then the problem (3) has at most one solution. The existence part of the theorem can be proved by using the methods of Chapter 3.4 of Naniewicz and Panagiotopoulos [15]. The condition (7) has appeared for the first time in a paper by Miettinen [20]. The a pn'on estimate in (ii) follows directly from hypotheses. We are now in a position to present an existence result for the inverse problem (6). Theorem 2. solution.

If hypotheses 11(g), II[fi), (HQ) and (7) hold, then the problem (6) has a

Proof. We give only the main steps of the proof which is carried out by using the direct method of the calculus of variations and the method of successive approximation. Let u^^^

517

Identification coefficient problems

be known in the A;th approximation. We show that there exists a solution to the following minimization problem mm{Ffc(5) = \\u^'*'\g{u('^))

- A|U.(r.)}

gEQ

(8)

where g{u^^^) is written instead of gdVu^^^"^). Taking into account Lemma 1, to obtain existence of solutions to (8), it suffices to prove that the map

is continuous, where u^^"^-^^ = u^^'^^\g{u^^^)) is the unique solution to the linearized problem

< JQ

JQ

[ x^^^^\x)

G dj{u^^+^\x))

a.e.

xe^

for all V e V. Let {gn{u^^^)}n C 5 , gn{u^^^) -^ go{u^^^) in 5 , as n -^ oo. Let us assume k to be fixed. By Theorem 1 (ii), we have ||ul^+'^||y < const\\f\\L2. Hence ^(^^+1) _ ^(^+1) weakly in y.

(9)

Moreover, by H{(3){ii), we have ||X1^"^^^I|L2 < c (l + ||4^+^^||y) < const, and we may assume xl^"^'^ -> Xo^"^'^ weakly in 1^(0), as n -> oo.

(10)

We show that UQ "^ ^ solves the direct problem with go{u^^^) in place of g{\Vu^^^'^). To this end, let us denote ai{ul'^'\v)

= [

gi{\\/u^'^\')Vui'^'^Vvdx.

JQ

We have \a4ul'+'\v)-ao{ut'\v)\< < K{ui;'+'\v)

- ao{ui'+'\v)\

+ \aoiui'^'\v)

- ao{ui'-''\v)\

= h + h.

Estimating I^ and /2, we get h < I k d V w ^ n - 5o(|Vu('=))p)||c(o.e) \Wi'^'%

\Hv

- 0

and l2<

I / [v4'=+>) - VuJ,*+')] i^odVuC^^nVixisI ^ 0,

From (10), we may suppose that ul^+''\x) -^ 4 * + ' ^ ( K ) for a.e. x G Q, as n -^ CO. Invoking the Convergence Theorem (see Aubin and Cellina [21]), we obtain Xt'\x)€dj{ui'+'\x))

a.e. r r e a

(11)

So we have shown that (ug '^ ,Xo ^ ) satisfies / ( 5 o ( | V « W n v 4 * + ' V i ; + aui'+'K

+ xi'^'^v)

dx = f

fvdx

518

S.Migorski

and (11) holds. By the trace theorem (see e.g. Zeidler [19]), from (9) we infer

This completes the proof that (8) admits a solution which we denote by gk^i{u^^'^). The corresponding solution to the direct problem is denoted by u^^^^\ In this way we get the sequences {gn{u^''~^^)]^, {u^'^^]n- Since {u^'^'^]n is bounded in V, there exists {n^} C {n] and u* e V such that u^'^''^ -^ u* weakly in V. Chosing the sequence {gnumi^*)) ^ idnuiu")} such that gn^^{u*) -^ g*{u*) in C{Q,C)i as m -> oo, we have F{gn^^{u*)) —> F{g^{u*))^ as m -^ oo. This completes the proof.

APPLICATIONS Let us consider the inverse problem of determining the elastoplastic properties of a bending cylindrical bar by using the experimentally given value of the torque A4. Let the lower end of the bar be fixed and a torque is applied to the upper end of the bar. The quasi-statical elastoplastic torsion of the bar is governed by the following equation dxi

»<^''l;) + s;(»<^''l;)-^<'+fl"^"^"'"

Now the parameter 0 is the angle of twist per unit length, —f{x) G /S{u{x))^ u = u{xi^X2) is the Prandtl's stress function and T = I (-—)^ I + ( {-—)^ ) ^dxi J \ dx2 The theoretical value of the torque is given by M{u{x;g))

is the stress intensity.

= 2 / u{x]g)dx

and M is the

torque given from the experiment. Using the deformation theory of plasticity (cf. e.g. [8, 9]), we know that between the intensities of shift strain F and tangential stress T the following relation holds

T = /(r^)r.

The function / = /(P^) is the modulus of plasticity and the inverse function P = g{T^)T exists and the function g satisfies the following bounds 1)

0
2)

0 < as < giT') +

3) 4)

g{T^) < 02, 2g'{T')T'

g'(T^)>0,T^eiO,Tl,) 3 T^ e (0, Tit) •• 9{T^) = const, for T € (0, To).

Similarly to Lemma 1, the following property holds: Lemma 2.

The set Qi of coefficients which satisfy properties 1) - 4) is compact in

The direct problem can be written in the form (3) with a = 0 and V = HQ{^). Other boundary conditions can be considered in an analogous way. The inverse problem consists

Identification coefficient problems

519

now in the determination of the coefficient g G Gi using the additional information on the "measured" value Ai. Theorem 3.

Under assumptions

1) - 4), H{f3), f e L'^{^) and (1), the inverse problem iin|I / u{q) u{g)idx — M.\ min =^1

JQ

has a solution. Remark. It is possible to consider the elliptic problem with a multivalued law appearing on a part of the boundary. In this case we get the following hemivariational inequality: find u e V such that {Au,v) + {x^v)^,^^^^ = {f,v) I

x(^) ^ dj{u{x))

, yveV

(12)

a.e. on r2.

The existence of solutions to (12) and the existence of solutions to the inverse problem for (12) can be obtained by exploiting the same method we present above.

ACKNOWLEDGEMENT This study was supported in part by the State Committee for Scientific Research of the RepubHc of Poland (KBN) under Research Grants no. 2P03A04015 and 7T07A04718.

REFERENCES 1.

2. 3. 4. 5.

6. 7. 8. 9. 10.

Migorski, S. (1998). In: Proceedings of International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP'98), pp.27-37, M. Tanaka and G. S. DuHkravich (Eds.). Denkowski, Z. and Migorski, S. (1998). J. Global Optimiz., 12, 37. Migorski, S. (1999). Dynamic Systems and AppL, 8, 73. Denkowski, Z. and Migorski, S. (1998). In: International Series of Numerical Mathematics, 124, 31. Migorski, S. (1999). In: Proceedings of the International Conference on Nonlinear Programming and Variational Inequalities, Hong Kong, December 15-18, 1998, accepted. Migorski, S. (1999). In: Proceedings of the International Congress on Dynamics and Control of Systems: DYCONS99, Ottawa, Ontario, A. Guran et al (Eds). Cannon, J. R. and Rundell, W. (1987). J. Math. Anal AppL, 126, 329. Mamedov, A. (1995). Int. J. Non-Linear Mech., bf 30, 23. Hasanov, A. (1997). Inverse Problems, 13, 1265. Acar, R. (1995). SI AM J. Control Optim., 3 1 , 1221.

520 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

S. Migorski Hasanov, A. (1995). SIAM J. AppL Math., 55, 1736. Hasanov, A. (1998). Inverse Problems, 14, 1151. Panagiotopoulos, P. D. (1985). Inequality Problems in Mechanics and Applications. Convex and Nonconvex energy functions. Birkhauser, Basel. Panagiotopoulos, P.D. (1993). Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer-Verlag, New York. Naniewicz, Z. and Panagiotopoulos, P. D. (1995). Mathematical Theory of Hemivariational Inequalities. Dekker, New York. Panagiotopoulos, P.D. (1985). Z. Angew. Math. Mech., 6 5 , 29. Clarke, F. H. (1983). Optimization and Nonsmooth Analysis. John Wiley & Sons, New York. Duvaut, G. and Lions J. L. (1972). Les inequations en Mecanique et en Physique. Dunod, Paris. Zeidler, E. (1990). Nonlinear Functional Analysis and Applications II, Springer, New York. Miettinen, M. (1993). PhD Thesis, University of Jyvaskyla, Finland. Aubin, J.-P. and Cellina, A. (1984). Differential Inclusions, Springer-Verlag, Berlin - Heidelberg - New York - Tokyo, 1984.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

521

VARIATIONAL APPROACH FOR THE PROBLEM OF COEFFICIENT IDENTIFICATION OF THE WAVE EQUATION Kenji

SHIROTA^

^ Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan e-mail: shirotaQmito. ipc. ibaraki .ac.jp ABSTRACT The purpose of this paper is to present a new numerical method for the inverse problem of coefficient identification of the wave equation: Determine the unknown coefficient function with the knowledge of the Dirichlet boundary value and the Neumann boundary value. We recast our inverse problem to the variational problem. In order to find the minimizing function, we make use of the projected gradient method. The first variation of the functional is given by solving two initial-boundary value problems. We confirm the effectiveness of our algorithm by two numerical experiments. KEYWORDS Coefficient identification, wave equation, variational problem, projected gradient method, first variation INTRODUCTION In this paper, we consider the problem of coefficient identification of the wave equation. This problem is to determine the unknown coefficient function with the knowledge of the boundary measurements. This problem has been considered by several people (see Isakov[l]). Let Q C RP' {n = 2, 3) be a bounded domain with smooth boundary. A conventional problem is to find the function u such that ^

- V • {K{x)Vu) - 0 u=^

=0 ot _ u=u

in O X (0, T], on fix {0},

^^^

on d^ X (0,T),

where w G C MO, T]; H^dD.)j. Here we assume that the coefficient function K belongs to L°°(fi) and satisfies the condition 0 < Ci < K{x) < C2 for all x e Q,

(2)

522

K. Shirota

where Ci and C2 are given positive constants. Under these assumptions, the initialboundary value problem (1) has a unique solution w G ^ ^ ( Q x (0, T)). Our inverse problem is to determine the unknown coefficient function K{x) with the knowledge of the Dirichlet boundary value u and the Neumann boundary value q as shown in Figure 1. The uniqueness and stability of this problem were guaranteed under the appropriate assumptions [2, 3].

anx(o,T] = "w anx(o,T]

= 9

Given

Fig.l. The problem of coefficient identification The purpose of this paper is to present an algorithm for the numerical resolution of our inverse problem. We introduce an object functional to be minimized, then our problem is recast to a variational problem. We adopt the direct variational method using the projected gradient method. We confirm the effectiveness of our algorithm by two numerical experiments. VARIATIONAL METHOD AND NUMERICAL ALGORITHM Let F be a convex subset of L°°(Q) characterized by V ={Ke

L°°(Q) I Ci < K{x) < C2 for all x e Q} .

The unknown coefficient function K is determined by minimizing the functional J :V

R^ := [0, 00), defined by

J{K}

fl Jo

dsdt, Jd\

where the function u[K] is the solution of the initial-boundary value problem (1) with the coefficient function K. To minimize the functional J, we make use of the projected gradient method: For j = 0,1,2,..., Kj^i{x) = Kj{x) — ajSj{x). The search direction Sj{x) is obtained by s,{x) = K^ix) - P{Kj -

J'{Kj)){x),

where

f Ci {PK){x) = \ K{x) [ C2

(K{x) < Ci), (Ci < K{x) < C2) (C2 < K{x))

(3)

523

Variational approach for the problem of coefficient identification Here J'{K) is the first variation of the functional J, defined by J{K + 6K) - J{K) = {f{K),

8K) + o {\\6K\\),

where (/, 9) := / f{x)9{x) dx, Jet

11/11 :=( f {fix)]' dx \JQ

We can guarantee that the updated coefficient function Kj generated by (3) belongs to V if the initial coefficient function KQ belongs to V and the step size 0 < a^ < 1. To implement the projected gradient method, we must know concrete expression of the first variation J'. du[K] I For d.ny K e V and SK G L^{Q) such that We denote q[K] := K dn lafix(o,T] K -\- SK G F, we notice that J{K + 6K)-J{K)=

[ [ {{q[K + Jo Jdn = f f Jo Jdn

SK]-qf-{q[K]-qf}dsdt

{q[K-\-SK]-^q[K]-2q)Sqdsdt

= [ f 2 {q[K] - q) Sq dsdt + J ( \H^ dsdt. Jo Jdn Jo Jdn

(4)

Let V be the solution of the initial-boundary value problem ^^^""^ " ^ Ov v = w, — = 0

in 0 X [0, T),

^ W~^'

v = 2 (xp-

o n Q x {T},

(5)

on dQ x (0,r),

- q\

where the function w is the solution of the boundary value problem in r^,

V • {KVw) = 0 nj = Since

(6)

on (90.

2(^K^{^,T)-q{-,T)

du[K] (•, 0) = 0 and v{', T) = w, we notice that dt

/7

Jo Jn

du[K] dv dxdt = dt dt

dt fdu[K]^

Jo

0

^,

^

r


vdt^ dx

f ^u[K]

,

..

(7)

Moreover we can get the following relation: / / KVu[K] . Vvdxdt = [ I [ vq[K] ds - / V • {K\/u[K])vdx\ Jo Jn Jo UdQ Jn J

dt.

(8)

524

K. Shirota

Prom (7) and (8), JQ JQ

\ at at

= f^i;T)wdxr I vq[K]dsdt. Jo Jn /n ^ <^^ Jo Jdn Jdn In similar way, we can get the following relation:

= f^^^^^^±^{.^T)wdxJQ ^

r f vq[K^SK]dsdt Jo Jdn

(9)

(10)

Prom (9) and (10), we notice /

/ 1 ^ ^

~ ^^^ "^ ^i^)Vit[/^ + SK] - KS/u[K]) . V ^ l dxdt (11)

= I ^{',T)wdxf f vSqdsdt, JQ ^ Jo Jdn where 6u := u[K + SK] - u[K]. dv Since 6u{', 0) = 0, —(•, T) = 0 and (^ix|ar)x(o,r] = 0,

/7„^l'^'"-/„{hl],-/

-/7

and

Su-—7r dxdt

= - / / (^i/V • (KVv) Jo Jn Therefore we can get the following relation:

dxdt

£l(^^^-KV6u.Vv)d.dt = 0.

(12)

Prom (11) and (12), we have -f

f ^{',T)wdxJn In ^ Since f |af2x(o,r) = 2(g'[A'] - g), it follows that pT

f SKVu[K-\-SK]-Vvdxdt= Jo Jn r

f f vSqdsdt Jo Jdn

(13)

f ^{•,T)wdx. Jn ot

(14)

rT

I 2 {q[K] - q) dsdt = I j SKVu • V^ dxdt Jo Jdn Jo Jn f f SKVSu'Vvdxdt+ Jo Jn

Variational approach for the problem of coefficient identification Accordingly, from (4) and (14), we obtain J{K + SK) - J{K) = [ [ SKVu . Vv dxdt + /* / SKVSu - Vv dxdt Jo Jn Jo Jn + f^{',T)wdx+ Jn crt

[ [ \Sq\'dsdt. Jo JdQ

We notice that the solution w of the problem (6) is identically equal to zero when the coefficient K is equal to the exact K*. As the variation SK in K* + SK tends to zero, we can expect that the solution w satisfies lim \\w\\ = 0. This implies that l^^{.,T)wdx

= oi\\6K\\)

for K* + SK in the neighborhood of the exact K*. Hence the first variation J' is given by f{K)=

f Jo

Vu{',t)-Vv{',t)dt

We employ the Armijo criterion for choosing the step size aj in (3). Armijo criterion Given the parameters ^ (0 < ^ < 0.5), r (0 < r < 1) and e > 0. (i) If \\sj\\ < £, then stop; else go to next step. (a) Prn '-= 1. (Hi) Do m = 0,1,2,

; // J{Kj - pmSj) < J{Kj) - ^pm I f{Kj)sj Jn

dx,

then aj := Pmi else Prn+i -^ rprnIn order to solve our inverse problem, we summarize the algorithm as follows: Numerical algorithm 1. Pick an initial coefficient function Ko EV. 2. Forj =

0,1,2,..., du •

(a) Solve the initial-boundary value problem (1) to find Vuj and K-—— anx(o,T) (b) Solve the boundary value problem (6) to find Wj. (c) Solve the initial-boundary value problem (5) to find Vvj.

525

526

K. Shirota (d) Calculate the first

variation J\Kj)=

I Jo

Vuj'Vvjdt

(e) Set the search direction: Sj := Kj — P{Kj — (f) Choose the step size aj by using the Armijo (g) Update the coefficient function:

J'[Kj)). criterion.

Aj+i = Kj — ajSj.

NUMERICAL EXPERIMENTS In this section, we show two numerical examples about our algorithm. We take some examples as bench mark tests within the following framework: 1. Let Q be a unit disk and T = 4.0. 2. The corresponding Dirichlet and Neumann data are generated by solving numerically the initial-boundary value problem with the following data: The domain is a circle with radius 2.75. The initial and Neumann boundary values are set to |sinl.257r(y^fTx|-2.6) lo.O

(1 < v ' x f T x f < 2.6, X2 > 0) {otherwise)

and 0.0, respectively. 3. The initial-boundary value problems in our algorithm are solved numerically by using Newmark method for time integration with linear triangular finite elements in space. 4. The domain is divided into triangular mesh as shown in Figure 2.

Fig.2. Finite elements (1400 elements) First, we consider the following two-dimensional example: Suppose that the exact coefficient function K is set by

i^(..x.) J^-^ (^^^/^xl+ <xl<0.25)1.0) ^

^

\LO

(0.25

< V ^

The initial estimate is set as KQ = 1.5. We suppose that Ci = 0.9 and C2 = 2.1. After 50 times of iterations, we have J = 0.55 x 10~^. Figure 3 shows the distribution of the

527

Variational approach for the problem of coefficient identification

exact coefficient function. The calculated coefficient function is shown in Figure 4. This calculated coefficient function is in good agreement with the exact one. T—r-i—I—I—\ v^'y;—i—r-i—i—i—r-

'•^c i5^'>'-i^^-l

-1

-0.5

0

0.5

Fig.3. Exact coefficient function

-0.5

1

0.5

0

1

Fig.4. Calculated coefficient function

In the previous experiment, a priori information about Ci and C2 was good enough. A priori information is set anew as Ci = 0.5 and C2 = 3.0. After 50 times of iterations, we have J = 0.87 x 10"^. Figure 5 shows the calculated coefficient function. The calculated coefficient function becomes worse.

—— ' ""

•

1 1 • 1 1

1 1 ^ ^ : — > _ « _ jL.I

1 1 1 1 1

^ ^

j^^^^i

i^Pi

1 1 1

-0.5

-1

0

0.5

1

Fig.5. Calculated coefficient function {C\ — 0.5, C2 = 3.0) To investigate the sensitivity of a priori information Ci and C2, we try to change values of Ci and C2. Figure 6 shows the calculated coefficient function when we set Ci = 0.5 and C2 = 2.1. This calculated result does not improve. The calculated coefficient function when we set Ci = 0.9 and C2 = 3.0 is shown in Figure 7. This calculated result improves, which is comparable with the case in Figure 4.

I'

- ' • ' ' '

1

'

I

'

1 1 1 1 1 ^

-_

10

d

11

0

lO

]

0

1

•

»-4

-1

-0.5

0

Fig.6. Calculated coefficient ( C i = 0 . 5 , C2 = 2.1)

0.5

1

function

-1

. 1 1 1 1 1 1 l-J

-0.5

0

0.5

Fig. 7. Calculated coefficient (Ci = 0.9, C2 - 3.0)

1

function

528

K. Shirota

In the previous example, the domain consists of two materials. Second, we consider three materials case. We suppose that the exact coefficient function K is set by 2.0

(0 < v ^ x f + x f < 0.25)

K[xi, X2) = < 1.5

(0.25 < y/xl + xl < 0.5)

1.0

(0.5 < v ^ f T x f < 1.0)

The initial estimate is set as KQ = 1.75. We suppose that Ci = 0.9 and C2 = 2.1. Figure 8 shows the distribution of the exact coefficient function. The calculated coefficient function is shown in Figure 9. The calculated coefficient function is in good agreement with the exact one.

-1

-0.5

0

0.5

Fig.8. Exact coefficient function

1

-1

-0.5

0

0.5

1

Fig.9. Calculated coefficient function

CONCLUDING REMARKS We presented a numerical algorithm for the problem of coefficient identification of the wave equation. To identify the unknown coefficient function, we make use of the direct variational method. Our numerical algorithm is based on minimizing the functional by the projected gradient method. By numerical experiments, we confirm that our algorithm is effective if we have good a priori information for the coefficient function. In particular, our algorithm is sensitive for the lower bound of the coefficient function. Our future work contains the following points: 1) We will show that our minimizing problem has a unique minimum for any boundary data in an appropriate function space. 2) The convergence of our algorithm will be demonstrated theoretically. 3) Because the initial guess should be chosen near exact coefficient, we will try to find a direct method for our inverse problem to get good initial guess. REFERENCES [1] ISAKOV, v . , 1998. Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin. [2] SUN, Z . , 1990. On Continuous Dependence for an Inverse Initial Boundary Value Problem for the Wave Equation. J. Math. Anal Appl. 150, pp. 188-204. [3] Y A M A M O T O , M . , 1999. Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 78, pp. 65-98.

INVERSE PROBLEMS EST ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

529

ON AN INVERSE PHASE CHANGE PROBLEM

A. EL BADIA

University of Technology of Compiegne. Equipe Mathematiques Appliquees. B.R 20529, 60205 Compiegne cedex, France. ABSTRACT This paper deals with an inverse one Stefan problem in ID and 2D cases. Namely identification of moving solid/liquid of a melting solid in ID case and a rectangular domain, from temperature measurements performed on the cold solid side of the solid only. KEKWORDS One phase inverse Stefan problem, moving boundary, identification.

INTRODUCTION Problems with free boundaries concern a large number of physical phenomena of which many can be encountered in thermal industrial processes, such as : casting, welding, purification by metal beams or laser machining by beams. These phenomena are based on fusion and solidification. These transformations are accompanied by absorption (for fusion) or by release (solidification) of the latent heat on solid/liquid interface, which characterizes the process of phase change. The optimization of the industrial process requires the control of the moving interface, which makes its identification essential. Many studies of inverse problems, in phase change, have been devoted to the control. It consists of searching for the boundary conditions in order to generate a prescribed interface. But, fewer results are available for the interface identification problem considered here [1, 2, 3, 4, 5, 7]. In this paper, we consider an identification problem of a particular moving boundary problem (Stefan problem): the isothermal interface, between the solid phase and the liquid phase the phase. Basically, two approaches leading to this identification are possible. The first one consists of measuring directly the position of the interface and thereby, avoids system modelling. However, it is difficult to apply because of the inaccuracy and the difficulty of installation in measuring instruments. The second approach is based on the system modelling including both liquid and solid phases. It consists in connecting the physical sizes observed and the position of the interface. This approach presents a difficulty related to the modelling of the liquid p a r t : many physical phenomena occur (natural convection, thermal conduction, surface effects,...). Moreover, the boundary conditions on the liquid side are hard to measure

530

A. El-Badia

in practice. For example, in the case of welding, in addition to the thermal conduction, the electro-magnetic forces and surface tension can intervene. To overcome this difficulty one usually uses simplified models in the liquid part. Given these facts, to solve our problem, only the solid part is considered, it is governed by the heat equation. Therefore, the domain of study is restricted to the solid part, and measurements (temperature and flux) are taken on the fixed part of the boundary. The difficulty is that the moving part of the boundary is unknown and there is no explicit relation to reach it, contrary to the Stefan problem. So, we have to solve an inverse problem of the heat equation, where the unknown is the moving part of the boundary. It is a non-linear and ill-posed problem, so a regularization method must be applied to obtain its numerical solution. We constructed an identification algorithm based on the Tikhonov regularization method of the zero order, and a minimization of the smoothing functional by using a constrained minimization method. In order to calculate the gradient of this functional, the corresponding sensitivity and adjoint problems are introduced. Similar inverse Stefan problems are developed in [5]. There are many theorical and applied studies of a wide class of such problems. At first the author considers a class of one-phase boundary inverse Stefan problems with the given phase boundary. These problems involve determining the boundary conditions on some boundary of the domain in the presence of phase transition in the other boundary. Moreover, the movement of the phase transition boundary in known. Also, some one-phase boundary inverse Stefan problems have been considered with unknown phase boundary. The unknown boundary conditions and the phase boundary ^(^),0 < t < T have been determined from the additional information u[l{t)^t) on the solution u{x^t) of the direct Stefan problem, where 0 < l{t) < ((t). Another problem has been considered with the additional information u(a:,T),0 < x < {(T). In all these problems the Stefan condition on the phase transition has been considered, but it is not the case in our study. Moreover, our additional information is given only on the fixed part of the boundary. Another similar problem, called geometric inverse heat conduction problem, has been considered in [3]. It concerns heat equation with constant coefficient, defined on a semi-infinite solid body with a moving unknown boundary ^{t). The initial position and the flux on the moving boundary at infinity are given. Then the author considers the identification problem of the moving boundary from additional data at a certain point in ]0,oo[. Its determination is reduced to the resolution of two nonlinear equations of Volterra type. Our approach is direct and systematic. It could be extended to varying coefficient problem in multidimensional space. Moreover, our problem is considered in a bounded domain with additional data given on the fixed solid part only.

STATEMENT OF THE PROBLEM IN ID CASE Let Q = {{x,t);a{t) <x 0 the Holder space of functions u{t) continuous on the closet set [0, T] together with derivatives u^'^ for 0 < 5 < [/] and having the derivatives of the order [/] Holder continuous with the exponent /? = / - [ / ] . Here [/] is the largest integer less than / cf. [5].

On an inverse phase change problem

531

Let C^'^/^(0) be the Holder space of functions 0{x,t) continuous on the closet set Q together with their derivatives D[ DJ <9 for 2 r + 5 < [/] cf. [5]. Let us consider the one phase Stefan problem, in one space dimensional which is a particular moving boundary problem: the isothermal interface between the solid phase and the liquid phase is driven by the diffusive heat in two connected phases. The only modelled part is the solid one. The interface sohd/liquid is characterized by a positive function aGC'[0,T]. The solid p a r t :

I,{t) = {x : a{t) < ;r < 1}

is gouverned by the heat equation ^ . - ^ . . = 0 in

/,(t)x(0,T),

(1)

satisfying the initial condition 0{x,0) = 0 in the boundary condition and the interface equation

]<7(0),1[,

(2)

e,{i,t) = u{t) in (o,r),

(3)

= Tf in

(4)

0iait),t)

(0,r),

where the function u and the constant T/, respectively the temperature and the melting point, are assumed to be known. Without loss of generality, we will suppose that T/ = 0. If the interface a is given and Hue C^'^'^[0, T], then the problem ( (l)-(4) ) admits an unique solution 0 G C^"*"^' 2 (Q) cf. [5]. One can calculate 0{l^t) and then define the observation operator

C{a) = e{l,t). This is the direct problem. The inverse problem we are concerned with, is as follows: Inverse Problem: Given the observation ^, can we find a function a such t h a t : C{a) = <j> ? Resolution

Method.

Criterion. The identification of the parameter cr is a nonlinear problem that is solved by the minimization of a penalized least square criterion J cf. (5). The minimization is performed with a constrained optimization method, which requires the calculation of the directional derivative of J , using the sensitivity state given by theorem 1 and the adjoint state given by equations (10)-(13). Where e is a regularization parameter.

JW) = \ J\^(^M - mfdt + ^ r a'dt.

(5)

532

A. El-Badia

Sensitivity Equation. To obtain The Frechet derivative of J it is necessary to determine the derivative of C To calculate the Frechet derivative of C, it suffices to look for its Gateaux derivative and to verify that is the same in each directio. So, this section aims to calculate the Gateaux derivative of the operator C at the point a in the direction /i, defined by:

A

A-)-0

where /i is a given direction. That is is given by the following theorem: T h e o r e m 1 Let 6 he the solution of the problem (1) — (4). Let ip be the solution of the problem: i^t-i^..

=

rP(x,0)

^ 0 , , + ( - ^ ) [{l-x)e,] 1 — (T \ 1 — (T/ = Oin ]a(0),l[,

i^{a(t),t)

=

Mht)

=

Om

in

(6)

]a(t),l[x]0,Tl

(7)

]0,T[,

i — (7

(8) (9)

u{t) in]0,T[,

th en {C'{a),h) = 4'{l,t)Proof [see 4] Adjoint

State

The aim of this section is to express the Frechet derivative {J'{cr), h) of the criterion J at point a in the direction /i, by means of the state associated to the system ((6) — (9)). Let p be the adjoint state defined by the equations: =

0, in

]
p{x,T)

=

0, in

](T(0),1[,

p,{l,t)

= C{a)(t)-
-Pt-P..

p{(Tit),t)

= 0, in

(10) (11)

in

(0,7),

(12)

(0,r).

(13)

Let / be the function defined by f{x,h,a,e)

= ^ 0 , , +

( - ^ ]

in

[{l-x)0,]

then the expression of the Frechet derivative ^J'[G\

]a{t),l[x]0,Tl

h) is given by the following result:

P r o p e r t y 1 If the function a is strictly increasing, then {J'{a),h)=

I Jo

u{t)p{l,t)dt^e ^ ~ ^

/

ahdt^ Jo

Jo

Ja{t)

f{h,a,0)p{x,t)dxdt.

533

On an inverse phase change problem DEFINITION OF THE PROBLEM IN 2D CASE

Let us consider now the problem in 2D case. As in ID case the only modelled part is the solid one. The interface sohd/liquid is characterized by a positive function s. The solid part is Qsit) = {{x, y):0


s{t) < x < 1}

with boundary d^s = To U Fi U Fs U F(t), where Fo = { ( l , y ) : 0 < 2 / < 1 } Fi = {(5(0,t),0)} F2 = { ( . ( ! , t ) , l ) } V{t) =

{{s{y,t),y):0
Using dimensionless variables, one gets the heat conduction equation in the time dependent solid domain: ut - u^x -Uyy = ^ in n , ( t ) X (0, T), (14) satisfying the initial condition u(:.,y,0) = 0 in

n,(0),

(15)

the boundary conditions w^ = 0 on ( F i U F 2 ) x ( 0 , r ) , u, = ip on F o x ( 0 , T ) , and the interface condition

it = 0 on

(16) (17)

T{t) X (0,T).

(18)

If the interface 5 is given then one can calculate W|rox(o,T) and then define the observation operator C{s) = U|rox(o,T). This is the direct problem. The inverse problem we are concerned with, is as follows: Inverse Problem: Given the observation (j)^ can we find a function s such t h a t : C{s) = l As in ID case the problem is solved by the minimization of a penalized last square criterion:

A^) = \ l

^ JTO

I {C{s){y,t)-4>{y,t)fdtdy+'-f I s'dtdy. ^ JFO

JO

JO

The minimization is performed with a constrained optimization method, which requires the calculation of the Gateaux derivative of J . To obtain it, it is necessary to determine the derivative of C. The latter is defined at the point 5, in the direction /i, by

(c'(3U) = i i m ^ ( i ± M i : M A-^0

A

534

A. El-Badia

where /i is a given direction. That is given by the following theorem: T h e o r e m 2 . Let u he the solution of problem (14) — (18). Let W he the solution of the prohlem: '^t-'^XX-'^yy

=

W^,; + 2 ( X - 1 ) ( -

1—5

+(1 - x)(-^)tu^

in n,{t) X (0, T)

1—5

^

= 0 onr{t)

^

= Q in n,(0)

)yU:,y -i- (X - l){)yyU:, 1 — 5 i — 5

X (0,T)

^y

= 0 oLn ( r i U r 2 ) X (0,T)

*x

= :;

^onTox{0,T)

i —5

then

{C'{s),h) = ^\romn Proof Let /i be a given function and A such that 5 + A/i < 1. Let 5+ = 5 + A/z, we consider the function: \,y,t :]5,1[—y]s'^,l[

defined by: ,

/ X

. /

X

A,t/,Ha:) = ^{x,y,t)

=

1 - "S"^

a n d we set

w{x,y,t) where u{(^y^t\s'^)

=

A/l

X+ 1—5 1—5

u{^,y,t]s'^),

is the solution of the problem (14) — (18) with s = 5"^.

Let a = and

P = It is easy to see that

a=

1-5+ 1—5 h l-s'

l-\p

and

(^ =

\p(i-x)^x.

By derivation of the functions ^ and w;, we obtain

6

=

ix

= a

^y =

\{i-x)(it X{l-X)(3y

535

On an inverse phase change problem So

In addition

Wy

=

Wyy

=

Uy-\-(y.U^

Uyy+2^y.Uy^-\-{Cyf.U^^+^yyU^.

Thus

Assuming that s satisfies

one obtains:

.

'^ Wy = 0 on

( r i U T s ) X (0,T).

Thus the function w satisfies the system a"^

\a =

Wx =

0 on

a

a

X (0,r)

mQs{t) 10^

J

( F i U r s ) X (0,T)

a(f on To x (0, T)

w

= Oon r ( t ) X ( 0 , r )

w

=

0 in 0,(0).

As a^ —> 1 when A —y 0 and 6 , ^y, iyy —> 0 when A —> 0, on deduces that the function w —> u, solution of the problem (14)-(18). Now let us consider the function A

•

It follows from (14) — (18) that it satisfies the system:

^*^-*i-^\ = ' ~ ^tr.""'--- - 2^v (^»'.)^ - ^..--^^+^^.6 i n n , ( i ) X (0,T)

$^ = Oon r(t) X (o,r) #^

=

Oinft,(0)

$^

=

0 on

^t

=

(riUr2)x(0,T)

^^^^c^onrox(0,T)

536

A. El'Badia

Now it is easy to see that the function ^^ —> ^ , the solution given by theorem 2. CONCLUSION This method was tested, in ID case, by F. Moutazaim [4] and compared with the one proposed in [1]. The results are very satisfactory. It would be very interesting to generalize it for n dimensional case (n=2,3) with any domain Qg • This will be done in a coming work.

REFERENCES 1. Afshari, A. (1990). Thesis, Pris-Orsay University. 2. Afshari, A., Benard, C , Duhamel, C. and Guerrie, B. (1989). In: Proceeding of 5th IFAC symposium on control of distributed systems, Perpignan (France), 209. 3. Alifanov, O.M. (1994). Inverse Heat Transfer Problems, Springer-Verlag, Berlin, Heidelberg, New York. 4. El Badia, A. and F. Moutazaim, F. (1999). In: Inverse Problems, 15, 1507. 5. Gold'man, N. L., (1997). In: Inverse Stefan Problems, Kluver Academic Pubhshers, Dordrecht, Boston, London. 6. Katz, M. A., and Reemtsen, R.,(1984). In: Num. Heat Transfer, vol.7, 269. 7. Wang, X., Rosset-Louerat, M.M., and C. Benard, B. (1992), Int. Series of Num. Math., 107, 99.

INVERSE PROBLEMS IN ENGESTEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

537

ON SOME INVERSE EEC PROBLEMS CHAFIK M., EL BADIA A., and HA-DUONG T. Universite de Technologic de Compiegne B.P. 529, 60205 Compiegne cedex, France ABSTRACT The main purpose of this paper is the inverse source problem when the source terms are dipolar, i.e. of the form given in (4) below, the equation is elliptic and the available information is the Cauchy data from the boundary of a bounded domain 0 in IR"^, n = 2, 3. The problem comes from usual models in Electroencephalographie. After resolving the question of identifiability, we give an algebraic method to identify the number of dipoles, their locations and moments. Some numerical tests are presented. KEYWORDS Inverse Source Problems, Electro-Encephalo-Graphic, Non Destructive Testing, Elliptic Equations. INTRODUCTION We consider in this paper the problem of determining a source term F in an elliptic equation in a bounded domain 0 -\/.{aVu)

= F

(1)

from the Cauchy data of u on d^t. The main application (but not the only one) we are interested in is the Electro-EncephaloGraphy (EEG), which is a noninvasive means of measuring some of the electrical activity of the brain. See the review paper of Hamalainen et al [4]. In the quasi-static approximation of the Maxwell equations, which is justified by the very low frequencies involved, the relevant equations of EEG are actually : -V.(crVu)

=

V.J^inO

(2)

a ^ ou

=

(^ on r = aO

(3)

where 0 is the head and <j the conductivity. The inverse problem consists in determining the unknown primary cerebral current J^ from the input data Lp and the measurement / = ^|r* of the electric potential w at a subset with non void interior of the surface F of the head.

538

M. Chafik, A. El-Badia and T. Ha-Duong

Usually, the head is assumed to be made out of disjoint coated regions Vti.i = 0 , . . . , / (the scalp, skull, cerebrospinal fluid, gray matter and white matter) in each of them the conductivity is constant. The primary current is some distribution which is localized in the cortex OQ. These assumptions will be also ours in this work. More precisely, we suppose that 0 is a bounded domain of R"^ (n = 2, 3), with sufficiently regular boundary, and partitionned into subdomains Vti^i = 0 , . . . , / such that 0 , , i = 1 , . . . , / are connected and coated in layers around HQ. Their boundaries are dQo = To

r,nr,_i

= 0; j - i , . . . , / ,

where Tj is the interface between Hj and Hj+i, and F/ = F = dft. Finally, we consider general equation (1) rather than (2), with piecewise constant a = CTJ in Dj and with a source F supported in fto. One of the first studies in inverse source problem for elliptic equation is the work of Novikov [7], where F is the characteristic function of a domain a;, and the available observation consists in the Newtonian potential created by this function source in a vicinity of infinity. Generalizing the method of Novikov, Isakov proved in [5] that subdomains of ft which are star-shaped or convex in a direction can be identified from the Cauchy data at ^ 0 . On the opposite side, we proved in [3] that Cauchy data permit only to identify the harmonic part of a square integrable source term F. Naturally, this negative result doesn't contradict the Isakov's result, and more generally doesn't prevent us from looking to specific classes of sources that could be identified from the Cauchy data. In particular, with respect to the EEG problem where the sources are dipolar, our main results in this paper are twofold : first, we prove an identifiability theorem for any finite distribution of dipoles - with unknown moment (or strength) for each of them ; on the other hand, we give an algorithm to calculate the locations and moments of these dipoles. THE IDENTIFIABILITY QUESTION We consider a linear combination of dipolar sources : m

F='£p,.V5s,

(4)

A:=l

with distinct centers Sk localized in QQ. The number m of dipoles, their centers S'k and vector strength pk are all unknown. Then our first result is: T h e o r e m 1 Let ui^i = 1,2 he the solutions of the problems mi

-V.{aVui)

= ^pi^lV(^^(.) m n

(5)

k=i

-g^

= ^ onTi

such that ui — U2 on F*,

(6)

On some inverse EEGproblems

539

then : mi = m 2 ( = m )

Pi'' = p f

Vfc=l,...,m

5(') = 5 f

Vfc = l , . . . , m

Proof First, we use Holmgren theorem to prove that w = U2 ~ ui \s identically null in 0 / . Then, the transmission conditions included in the equation (5) permit to go through 0 , , i — / , / — 1 , . . . , 1 until Ho, where w satisfies the following problem (one can always set CTQ to bel) :

k-l

w

k=l

= 0 on To

^^ -^r- =

n r 0 on i o .

From that, one can easily extend w out of Ho by 0, and obtain its explicit expression :

and w{x) = -—(> ^ TTT-^ - > ^ r - — ) for n = 3. k=l F *^A; I A;=l 1^ ~ *-^/i; I Since w is analytic in the connected domain R^\ U {S^} and null out of OQ, the theorem follows. We notice that the above argument is no more valid when the source contains an infinity of dipoles. Counter-examples can be constructed when F is a surfacic distribution of dipoles. AN ALGEBRAIC ALGORITHM OF IDENTIFYING DIPOLAR SOURCES From the proof of the theorem 1, one sees that the identification process can be decomposed into two dictinct parts : The first part consists in the resolution of Cauchy problems respectively in H^, z = i-,i — 1 , . . . , 1, to obtain the Cauchy data of u on the ultimate domain OQ. We refer to the paper of Kozlov et al [6] for the numerical methods for Cauchy problems for elliptic equations. Once this first step is done, we are led to the identification problem in a homogeneous domain, i.e. to the case where 0 = OQ in previous notations : to identify the dipoles (Pk, 5'A;) such that m

Au = X^Pit-V^S;, in n

(7)

A;=l

du d^

= (^ on r /onr

u =

(8) (9)

540

M. Chafik, A. El-Badia and T. Ha-Duong

with known (^ and / . We shall use the concept of reciprocity gap functional introduced by S. Andrieux in [1] : R{v) = < ^, V >^^^^^j^_^^^^ -
dv

>Hhr).H-hry

(^°)

where i; is a harmonic function in Q :

V e H{n) = {ve H\n)/Av = 0} By Green's formula, it is easy to see that m

R{v) = -Y^

Pk-^v{Sk),

Vt; e H{n).

(11)

Now, suppose that m < M where the integer M is known (this is an a priori knowledge on the dipoles). Let us consider first the case of M? where polynomials Vj{x,y) = {x-\-iyy;

j € N

are harmonic. We then compute the numbers (ij = ^ ^ ^ := E ( p . , i + ipk,2)Si-\ •^

1 < J < 2M - 1,

(12)

k=l

where Sk stands for the complex affix of the point. It is then a simple algebraic matter to verify that the familly of complex vectors {rjj} defined by

(

P.

\

Vj =

G C^,

1 < j < M,

(13)

has the rank m, and that the subfamilly (^1,7725 • • •, Vm) is independent. Thus, the number m of dipoles is determined. We then restraint the above vectors to their m first components, and define the matrix T by ^j+i = TrjjJ = l , . . . , m . We can then prove that the complex numbers Sk are eigenvalues of T. Once the S'k determined, the pk can be computed from the equations (11, 12). Stability The main difficulty we met in numerical experiments described below concerns in fact the determination of the number m of dipoles, since the determination of the rank of a matrix (using singular values) is rather unstable. Once this number m is known, however, the stability of the following steps in our algorithm can be assured by the classical theory of eigenvalues calculations. More precisely, if the norms of the perturbations {9 = f~f^h = (p — (p) satisfied certains conditions, we have proved that exist a and b two positive numbers depending only on m, ft, f and (^ such that for all A; = 1 , . . . , m: II S, - S, h<

mil — R^) ^ ^ : i ( ^ _ jj) rnax {Ci_,W;

0 < j < m - l] {a \\ g | | ^ j ^ ^ ^ +b || h

\\^^^^^)

541

On some inverse EEG problems

where d is the minimal distance between the Sk^ Sk the centers of dipoles (5^, PA;) associated to ( / , (p) and i? 7^ 1 is a real number superior to the norm of any point on F. We have an analogous result for the strength vectors p^;, p^. Numerical experimentations

for 2D

We have tested this algorithm for several situations. We present here some examples, when n is a disk, for m equal to 3 (Fig.l) and 5 (Fig.2) respectively. The Cauchy data used in the experiments are generated from exact (analytical) formula given in [2]. The determination of the centers and moments of dipoles are here made with known value of m. In each figure, we present the calculated dipoles with either the exact Cauchy data or the perturbed ones using a random perturbation with maximum variation equal to 10"^, 10~^ and 10"^ respectively. Perturbation = 0.001

Without perturbation

-0.5

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Perturbation = 0.1

Perturbation = 0.01

-0.5

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Fig. 1. An example in 2D with m = 3 dipoles

The 3D case In this case we suppose that the points Sk have distinct projections Qk on the plan xOy. Using harmonic polynomials independent of z^ Vj{x,y,z) = [x -\- iyY I < j < 2M, one sees that, if the projections {pk,i',Pk,2) of the vectors' strength pk are not null, one can apply the above algorithm to determine m, then Qk and {pk,i-,Pka) for 1 < A; < m. Now, to determine the projections Zk of Sk and the 3rd component pk^2, of PA; we have several

M Chafik, A. El-Badia and T. Ha-Duong

542

Perturbation = 0.001

Without perturbation

-0.5

-1

Perturbation = 0.1

Perturbation = 0.01

1 0.5

o*

0

1

O- y^Qf^

\

J

•0.5

-0.5

^

-1

-1

^ -0.5

—

^

0

0.5

-1

1

-0.5

0

0.5

1

Fig. 2. An example in 2D with m = 5 dipoles choices. One of them is to consider the harmonic functions: Wk{x,y,z)

=

z{x-\-iy-Qk)Yli^-^^y-Qjy

and to deduce Zk by the formula

zj^ =

<(^,iy,>- QI^

{Pk,i + ipk,2)l[j:^kiQk

- QJY

The 3rd component of vector p^ can then be obtained by:

< ^, t/. > - < /,dUk^ > -2zk{pk,i+iPk,2){ii{Qk - QMYI n (Q^ - oo) 3^k

Pk,3 =

Uj^kiQ^

- QJY

where Uk is the harmonic polynomial functions defined by: Uk{x,y,z)

=

zJ\{x^iy-Qjf. 3+h

Jl^k l^k,j

543

On some inverse EEG problems Numerical experimentations

for 3D

We present here an example of calculations for a sphere and m = 3 dipoles. In Fig. 3, the projections of the dipoles on the plane xOy are displayed as in Fig. 1 and Fig. 2, while their projections on the axis Oz are given in the table 1.

O

O

Without perturbation

perturbation = 0.001

perturbation = 0.01

perturbation = 0.1

0

0

O

0 -0.5

Fig. 3. Projections of the dipoles on xOy

Zl

true value Without perturbation perturbation=0.001 perturbation=:0.01 perturbation=0.1

0.6549 0.6549 0.6549 0.6549 0.6605

Pi,3

-0.1524 -0.1524 -0.1523 -0.1517 -0.1403

Z2

0.0046 0.0046 0.0031 -0.0088 -0.0599

P2,3

0.2398 0.2398 0.2478 0.3172 0.8049

zs -0.1159 -0.1159 -0.1178 -0.1355 -0.3288

P3,3

-0.0721 -0.0721 -0.0802 -0.1502 -0.6484

Table 1. Projections of the dipoles on Oz CONCLUSION We have presented some new results in the inverse source problem when the source is a sum of gradient of the Dirac distributions. The singularity of the source terms, in contrast to the case when it is regular, permits its identifiability. On the other hand, the particular structure of harmonic polynomials gives rise to an algebraic algorithm to numerically

544

M. Chafik, A. El-Badia and T. Ha-Duong

identify the source. Our method is applicable to the Magneto-encephalo-graphy (MEG) problem. It will be reported in another paper. On the other hand, more numerical works are in progress, including comparison with other methods as well as consideration of more realistic geometries and measures.

References [1] Andrieux S. (1995) C.RA.S.,

Paris, Serie I, 320, 1553.

[2] Dautray R., Lions J.L. (1987) Analyse mathematique sciences et techniques Vol. 2. Masson, Paris.

et calcul numerique pour les

[3] EL Badia A., Ha Duong T., (1998) Inverse Problems, 14, 883. [4] Hamalainen M., Hari R., Ilmoniemi R. J. , Knuutila J. and Lounasmaa O.V. (1993) Rev. Mod. Phys. 65, 413. [5] Isakov V. (1989) Inverse source problems. AMS, Providence, Rhode Island. [6] Kozlov V.A., Maz'ya V.G. and Fomin A.V. (1991) Comput. Math. Phys. 31, 45. [7] Novikoif P. (1938) Doki Akad. Nauk SSSR 18, 165.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

545

N U M E R I C A L S O L U T I O N OF U N D E R - D E T E R M I N E D 2 D L A P L A C E EQUATION WITH INTERNAL INFORMATION Qingchuan WANG^), Yoko OHURA^) and Kazuei ONISHI^) 1) Graduate School of Mathematical Sciences, Ibaraki University, Bunkyo 2-1-1, Mito 310-8512, Email: [email protected] 2) Dept. of Management Information, Kyushu Institute of Information Sciences, Saifu 6-3-1, Dazaifu, Fukuoka 818-0117, Email: [email protected] 3) Dept. of Mathematical Sciences, Ibaraki University, Bunkyo 2-1-1, Mito 310-8512, Email: [email protected]. ibaraki. ac.jp

ABSTRACT Concerning an under-determined problem of the Laplace equation in two spatial dimensions, a numerical treatment is presented in this paper. The Dirichlet and Neumann data are respectively imposed on two parts of the boundary of the domain. Besides, the values of the unknown function are specified at n distinct internal points. This new problem can be regarded as a boundary inverse problem, because the proper boundary conditions are to be identified for the rest of the boundary. The treatment is based on the direct variational method. A functional is minimized by the method of the steepest descent. The minimization problem is transformed into iterative primary and dual boundary value problems of the Laplace equation. In simple numerical examples, we compared our numerical solution containing internal information with the numerical solution not containing internal information. It is concluded that adding internal information can significantly improve our numerical solution. KEYWORDS Boundary inverse problem, Laplace equation, direct variational method, numerical optimization, boundary element method

UNDER-DETERMINED PROBLEM OF THE LAPLACE EQUATION Let Q C H^ be a domain bounded by a smooth curve F with the rectangular coordinates X = (xi,X2). We consider the Laplace equation -Aw(x) - 0

m Q..

(1)

546

g. Wang, Y. Ohura and K. Onishi

du Let n denote the outward unit normal to the boundary F, and let g' = 7— on the _ on boundary. Assume that the Dirichlet data u are prescribed on a non-zero measure part r^^ of the boundary F, and the Neumann data 'q are prescribed on another non-zero measure part Vq of the boundary F. We notice here that F^ and F^ are taken arbitrarily. Moreover we assume that the values it(x(j)) — u^^^ are given at n distinct internal points x^^\ j = 1,2, • • • , n. We consider the case in which the data contain measurement errors. As a matter of fact, we also notice that, even if the data u and q are exact, the solution u{x) of the Laplace equation is not uniquely determined provided that F-^ fl F^ = 0 and Fu U Fq 7«^ F. In this case the problem is under-determined. We consider such new problem, reflecting real situations in practice of the measurement, because it is often difficult to measure u and q in such a way that F^^ fl F^ = 0 and F^^ U F^ = F. We call such problem as boundary inverse problem, because the problem falls essentially into identification of the proper boundary values on F\(Fw U F^). To solve the under-determined problem the basic idea contains two aspects. One is to identify a proper boundary value u = cu for the rest of the boundary F^ = F\F-u so that the solution u{x) of the Laplace equation also satisfies the boundary condition given on Fq. Another is to identify a proper boundary value g' := r for the rest of the boundary F^ = F\Fg so that the solution u{x) of the Laplace equation also satisfies the boundary condition given on F^. Meanwhile the solution should satisfy the internal information. Ohura, et al. [1] discussed under-determined problem for the Laplace equation. Kobayashi, et al. [2] investigated a numerical aspect of the Cauchy problem for the Laplace equation. Onishi, et al. [3] considered a boundary inverse problem for the Laplace equation through a theoretical point of view. Based on those works, we discuss the under-determined problem with internal information. DIRECT VARIATIONAL APPROACH Identification of u: We write u[x) = u{x',uo) and q{x) = q[x\uo) respectively to stress the dependence of the solution of the Laplace equation (1) on the boundary value uj to be identified on F^ = F\F^^. By the words proper boundary values, we mean the boundary values uj which minimize the object functional: J{uj)

=

f

\q{x;u;)-q{x)\^dr

subject to

+ "^\u{x^^^;uj)-u^^^\^

+ rj [ \^^^

U\Y^ = u.

(2) (3)

Here we assume J: if^/^(F^) 3 a; 1—> R^ = [0,+00). The Dirichlet integral added in Eq.(2) is a regularizer with the regularization parameter 77 > 0 to guarantee the unique existence of the minimum of the functional J{uj). With a suitable choice of positive real numbers a^ for A: = 0 , 1 , 2, • • •, we consider the minimizing process: uJk+\{x) = u)k{x) - akJ'{(jJk)^

(4)

where the gradient J'{uj) G i^~^/^(F^) is defined by the first variation: J[ijj -h 6u) - J{uj) = < J'{uj), Suj > +o(||^cc;||)

(5)

547

Numerical solution of under-determined 2D Laplace equation

with real-valued function o(||(5cc;||) of higher order than || Ja;||, as it tends to zero with respect to the L^(r^)-norm. We also assume J'{UJ) G i7^/^(r^). We consider v{x) as a solution of the Poisson equation n

/\v{x)

in

= Y^2[u{x]uj)-u{x)]8{x-x^^^)

Q,

(6)

3=1

subject to the boundary conditions: i; | r e - 0 .

v\r^=2{q{x',uj)-q{x)),

(7)

Then it can be shown that the exphcit form of J ' ( ^ ) is given by: rin

J'{co)^~{x;uj)

fill

on

+ 2v^ix;uj)

T^.

(8)

Our algorithm can be summarized as follows: Algorithm Given uJo\r^. For A; = 0,1,2, • • • ^ until convergence, do: Solve the primary problem: —Auk{x) = 0 in Q., with Uk\ru = ^^ ^k\r^ = ^k, to find qk{x;(jJk) on Fq. Solve the dual problem: n

Avk{x)

= y j 2[uk{x; ujk) — u{x)]5{x — x^^"^) in

^,

with Vk\Tq = 2{qk{x;ujk) - q{x)}, Vk\r- = 0, to find J\uJk) = ~^

~^ ^^~d~ ^^ ^^'

Update ujk+i = ^k - (y-kJ'{ujk) on F^. To choose a suitable sequence of positive real numbers {ak}-, we employ the Armijo criterion [4] in mathematical programming. This guarantees that the sequence {a;^} satisfies J{uJk - akJ'{iOk)) < J{^k) - ioLk II J\(^k) IP with a constant ^ (0 < <^ < 1/2). Controlling the step size ak Given parameters 0 < ( ^ < | , 0 < r < l , and 0 < 6: < 1. / / 1 | J'(uJk) \\< £, then stop, else For m = 0,1,2, •••, do: IfJ{^k - PmJ'ioJk)) < J{0Jk) - ^Pm II J^OJk) then ak := Pm, else Pm+i '•= r/3m-

f,

Identification of q: We write u{x) — u[x\ r) to explicitly stress the dependence of the solution of the Laplace equation (1) on the normal derivative r to be identified on F^ = r \ F g . This time, by the words proper boundary values^ we mean the boundary values r which minimize the object functional K{T)

=

[

\u{x;T)-u{x)\^dr

subject to

^|r, = q.

+ y2\u{x^'\r)-u^^^\^

+ r] f \Vu{x^^

(9) (10)

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Q. Wang, Y. Ohura and K. Onishi

Here we assume K: /^^/^(r^) 3 r i—> i^+ = [0, H-oo). The Dirichlet integral added in Eq.(8) is a regularizer with the regularization parameter 7/ > 0 to guarantee unique existence of the minimum of the functional K[T). Then the first variation is given in the form: K'{T)=v[x\T)

+ 2r]u{x]T)

on

T\.

(11)

Our algorithm can be summarized as follows: Algorithm Given ro|rc. For A; = 0 , 1 , 2, • • •, until convergence, do: Solve the primary problem: —Aujfc(x) = ^inVt, with qk\Tq = q, Qk\r<^ = ^k, to find Uk{x;Tk) on TuSolve the dual problem: n

Avk{x) with

= ^^2[uk{x]Tk)

dvk dn

= 2{uk{x;Tk)

Update Tfc+i =Tk-

- u{x)]S{x - x^^^) in Q, dvk dn

-u{x)}.

= 0, to find K'{Tk) ^Vk + 2rjUk on H .

akK'{Tk) on T^.

The Armijo criterion is employed again to choose the sequence {offc}.

NUMERICAL EXAMPLES The boundary element method is used for the numerical solution of the primary and dual boundary value problems of the Laplace equation and the Poisson equation. We note that the function in polar coordinates u*{xi,X2) = x\ — x\ = r^cos(2i?) is harmonic. Under-determined problem with internal information Suppose that the function u* is unknown, and consider the Laplace equation in the unit circle Vt = {(r, d)\^
yjx\ + xl < 1,

0 < t? < 27r}

with the Dirichlet data; u = cos (2^) o n r ^ = {(l,??) | 0 < i ? < | } and the Neumann data; q = 2cos(27^) on Tg = {(l,i9) | TT < t? < |7r} as shown in Fig.L

Figure 1. Boundaries F^ and F^

Figure 2.

Boundary and internal points

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Numerical solution of under-determined 2D Laplace equation

The boundary V is uniformly divided into 24 boundary elements, and the referred internal points are indicated in Fig.2. The black points on the boundary denote the nodes on P. The five small circles in Q. denote the internal points. We take ry = 0 to examine how our algorithm behaves without the regularizer. We take an initial guess: (Jo = - c o s f - ( < 9 - - ; This initial guess is smooth on F^ and has the same slope as tx* at points (r,6) = {1,7r/2) and (r,6>) = (l,27r). After 10 times of iterations, J(a;fc) converged to a local minimum. That is, || J'(^io) ||< e {E = 10~^). We can regard UIQ as our numerical solution. The initial guess CJQ, calculated values of Uk, and the exact it* on F are presented in Fig.3. The numerical process was stable. The convergent solution UIQ is in good agreement with our target u* in the vicinity of the boundary Tq. 1.5

0.5

(^k

0

-0.5 -!3!-k=

-15

Figure 3.

J

1

-^(«-k =

5

=

10

The initial CJQ, calculated Uk, and the exact u* on the boundary

Under-determined problem without internal information Consider the same example as in the subsection above, but the internal information is not available. We take the same initial guess UJQ and regularization parameter rj. Then we get II J\^2o) \\< £ {e = 10~^). This means that the functional J{(jJk) reached a local minimum at UJ20. We regard U20 as our numerical solution, which does not coincide with u*. On the boundary, the initial guess UJQ, calculated 0;^, and the exact solution u* are shown in Figure 4.

550

Q. Wang, Y. Ohura and K. Onishi

^k

e Figure 4. The initial CJQ, calculated uJk, and the exact u* on the boundary

CONCLUSIONS We considered an under-determined problem of the Laplace equation with internal information. Using the method of the steepest descent for a functional to be minimized, our problem is led to an iterative process consisting of the solution of primary and dual boundary value problems of the Laplace equation. The boundary element method is applied for numerical solution of the primary and dual problems. In simple numerical examples, we compared our numerical solution containing internal information with the numerical solution not containing internal information. We conclude that adding internal information can significantly improve our numerical solution as long as we choose suitable distribution and number of the internal points.

REFERENCES 1. Ohura, Y., Kobayashi, K. k Onishi, K. (1999), Numerical solution of an underdetermined problem of the Laplace equation, Journal of Applied Mechanics, Japan Society of Civil Engineers, 2, pp. 185-189. 2. Kobayashi, K., Onishi, K., and Ohura, Y. (1996), On identifying Dirichlet condition for 2D Laplace equation by BEM, Engineering Analysis with Boundary Elements, Vol.17, pp. 223-230. 3. K. Onishi, K. Kobayashi, k Y. Ohura (1996), Numerical solution of a boundary inverse problem for the Laplace equation, Theoretical and Applied Mechanics, 45, pp. 257-264. 4. Armijo, L. (1966), Minimization of functions having Lipschitz-continuous derivatives, Pacific Journal of Mathematics, Vol. 16, pp. 1-3.

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Applications of Computational Algorithms

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

553

EVOLUTIONARY METHODS IN INVERSE PROBLEMS OF ENGINEERING MECHANICS T.BURCZYNSKI^'^ W.BELUCH'\ A.DLUGOSZ'\ P . 0 R A N T E K ' \ M . N O W A K O W S K I ' ^ ^''Department for Strength of Materials and Computational Mechanics Silesian University of Technology, Konarskiego 18a, 44-100 Gliwice, Poland, ^''Institute of Computer Modelling, Cracow University of Technology Warszawska 24, 31-155 Krakow, Poland ABSTRACT The paper deals with applications of evolutionary algorithms to inverse problems of engineering mechanics. Geometrical inverse problems are considered for identification of material defects as voids and cracks. Detection of voids is based on measured natural frequencies and measured dynamical displacements at boundary points. Identification of cracks is performed for prescribed number of cracks, unknown number of cracks and for stochastic disturbance of measured displacements. An inverse problem for thermoelasticity is also considered. The optimal boundary condition for temperature is searched for minimisation of a boundary displacement functional. KEYWORDS Genetic algorithms, evolutionary algorithms, boundary element method, geometrical inverse problems, identification of voids, identification of cracks. INTRODUCTION The aim of the paper is to develop an application of the evolutionary algorithms to inverse problems in applied mechanics. This work is an extension of previous papers in which the coupling of genetic/evolutionary algorithms and the boundary element method (BEM) has been used in shape optimization [4], generalized shape optimization (shape + topology optimization) [3], optimization of cracked structures [1] and identification of voids and cracks [2,7,8]. Evolutionary algorithms can be considered as modified and generalized classical genetic algorithms [6] in which populations of chromosomes are coded hy floating point representation and the new modified crossover and mutation operations are introduced. The evolutionary algorithm starts with a population of randomly generated chromosomes from a feasible solution domain. These chromosomes, which have the vector structure, evolve towards better solutions by applying genetic operators such as selection, mutation and crossover. After applying genetic operators, the new population has a better fitness. The probability of crossover and mutation does not have to be constant as in classical genetic algorithms and can change during the

554

T. Burczynski et al.

evolutionary process. An objective function (fitness function) with constraints plays the role of the environment to distinguish between good and bad chromosomes and select the better solution. In the present paper the new approaches of the evolutionary BEM computation are proposed to inverse problems: (i) identification of voids, (ii) identification of cracks, (iii) inverse problem of finding an optimal distribution of temperature on the boundary (boundary condition) for minimum of a displacement functional. FORMULATION OF THE EVOLUTIONARY IDENTIFICATION PROBLEM It is assumed that an elastic body, which occupies a domain Q bounded by a boundary F, contains internal defects in the form of voids or cracks. One should find a position and shape of defects while having additional information about boundary state fields such as displacements, stresses or natural frequencies. Defects are specified by a set of shape parameters x=(Xi), i=l,2,N. The problem can be solved by minimising an objective function with respect to x=(x^:

where q^ are measured values of a state field such as e.g. displacements, stresses or natural frequencies, ^ are computed values of the same state fields. The problem can be solved by using conventional optimization procedures, based on sensitivity information, but such approaches have some drawbacks: an objective function must be continuous, shape variation of the boundary defect should be regular, a hessian of the objective function should be positive definite, there is a large probability of convergence to a local optimum, computation starts from a single point narrowing the search domain and a choice of the starting point may influence the convergence. Because of this difficulties there is a demand for checking other methods, free from the restrictions mentioned above. The genetic algorithms belong to these methods. Genetic algorithms are stochastic algorithms whose search methods model natural phenomena: genetic inheritance and Darvinian strife for survival. Classical genetic algorithms are based only on the fitness value information and coded chromosomes. They work on populations of solutions and use binary operators of crossover and mutation and the probability of operators is constant. Evolutionary algorithms are modified and generalized classical genetic algorithms in which populations of chromosomes are not coded binary and floating point representation is used. They use modified crossover {simple, arithmetical and heuristic) and mutation {uniform, boundary and non-uniform) operations. The selection is performed in the form of the ranking selection or the tournament selection and the probability of operators can be variable [6]. A chromosome, which represents the design vector or matrix, can be expressed in the form jc=<xy, X2, ....x„ ...x,v> or

x=[x,k\ l
(2)

where restrictions on genes Xj and x,k are introduced in the form: X,L<X, <X,R

X,kL<X,k <X,kR

(3)

Genes Xj and Xik represent geometry of the defect and a number of defects. Values of genes Xj and Xik belong to the space of real or natural numbers.

555

Evolutionary methods in inverse problems of engineering mechanics The crossover operation swaps some chromosomes of the selected parents in order to create offspring. Simple, arithmetical and heuristic crossover operators are applied. Simple crossover This operator needs two parents and produces two offsprings. The simple crossover may produce an offspring outside the design space. To avoid this, a parameter aG[0,l] is applied. For randomly generated crossing parameter i it works as follows (chromosomes Xi, X2 are parents in the vector form): xi=<xi, X2, ....,xu

and

...XN>

X2='/, ---^A^^

'xi^<xi,...,,Xuyi^ia^x^^i(l-a),...,y}^a+X}^(l-a)>

and

(4) (5)

'x2= Arithmetical crossover This operator produces two offsprings which are a linear combination of two parents 'xi=axi +(l-a)x2

'xj=0x2 ^(l-a)xi

Heuristic crossover This operator produces a single offspring from two parents: 'xs=r(x2 - Xi) + X2

(6)

(7)

where r is a random value from the range [0,1] and J(x2)^(xi). Mutation operators such as uniform mutation, boundary mutation, non-uniform mutation and special gradient mutation are used: Xi=<Xi,

X2, ....,Xi,

...XN>

'XJ=<XJ,

X2, . . . . , 'X/, ...XM>

(8)

Uniform mutation Offspring are allowed to move freely within the feasible domain and the gene 'x, takes any arbitrary value from the range [x,^, X,R]. Boundary mutation The chromosome can take only boundary values of the design space, 'x, =X,L or 'x,= XJR. The boundary mutation works very well when the solution lies either on or near the boundary of the feasible search space. Non-uniform mutation This mutation operator depends on generation number / and is used to tune of the system ^

fx. + A(/, x^^ - x^) if a random digit is 0 [Xj - A(/, X. - x^^ // a random digit is 1

where the function A takes value from the range [0,y]. Gradient mutation This single-argument operator changes any chromosome on the ground of fitness function gradient: %=x,+/NJ, (10)

556

T. Burczynski et al.

where /? is a coefficient determining a step increment in the search direction given by fitness function gradient VxJ. The objective function given by (1) plays the role of o fitness function. Evolutionary identification is performed by using an evolutionary algorithm which minimizes the fitness function (1) with respect to shape parameters. In order to solve this problem one should introduce additional constraints which take into account the fact that the defect must be placed into the domain Q.. The fitness function is calculated for each chromosome in each generation by the boundary element method (Fig. 1). START •

1

Preparing the model Analysis of the structure

Evolutionary identification 1 EVOLUTIONARY ALGORITHM NO

? /^'^^Optwnai geometry ^ V4nd number of defects2^

a new geometry of the defect

h k

i

T

BEM a fitness value

]

IYES

END 1 Fig. 1. Evolutionary identification EVOLUTIONARY IDENTIFICATION OF VOIDS Detection of voids based on measured natural frequencies The identification problem is formulated as a minimization of the sum of differences between calculated natural frequencies cq (for a given position of the void) and measured natural frequencies d)j:

^ = Z(^;-^;)'

(H)

;=i

The chromosomes contain genes which are specified by geometrical parameters of the void. The direct problem for the eigenvalue problem is solved by using the dual reciprocity boundary element method (DRBEM). The vector boundary integral equation of DRBEM has the form cu + J p * u d r - J u * p d r = 6 ; V ] - c ^ ' + JU * I M F - J P * ^ M r

(12)

where U* and P* are fundamental solutions of elastostatics, ^^ and l!^ are pseudodisplacement and -traction fields, respectively, generated b y / , where \x{z)=a'f{z), ZGQ. Numerical example I. The problem of identification of a triangular void in an elastic rectangular plate under plane strain was considered for given M=3 natural frequencies d)j J=J,2,3. Solution

Evolutionary methods in inverse problems of engineering mechanics 557 for initial and 18th generation is given in Fig.2. ^

I

generation 1

generation 18

Fig. 2. Identification of a triangular void: (a) initial generation, (b) 18th generations Detection of voids based on measured dynamical boundary displacements The problem of a void identification in elastic structures being under dynamical loads is considered. The fitness function has the form: (13) j=\ r=\

where ii^(/Jis a measured total displacement at a sensor point 7=7,2,..M, at the time tr, r=J,2,..,R, and Uj{tr) is a computed total displacement at the same sensor point and time. The displacement field u(y,/) is computed by solving a boundary-initial value problem of elastodynamics by DRBEM. The vector boundary integral equation of DRBEM used in the paper has the form:

cu+jp*udr-Ju*pdr= J-cT'+ju*sMr-Jp*T'drU' where

a)

(14)

ii(z, t) = a^ (t)/^ (z), z G H, p is a mass density.

• ^ — ^

U.p(y)=H{t)po CUL

- sensor point • ^ — i ^

Fig.3a. Identification of two voids: rectangular plate with voids

Numerical example 2. The plate with two circular voids was loaded dynamically (Fig.3a). Heaving measured displacements at sensor points one should determine the positions and circular radii. The problem was solved for deterministic experimental data of displacements (no noise case) and for stochastic variation with normal distribution {noise case) (Fig. 3b).

558

T. Burczynski et al b)

9

O NO NOISE

Q)

o

Population 1

NOISE

Population I

1

O

Population 40

generation 40

generation 5

O

(D

O

©

o

o 1 Population 5

generation 1

O

(S) 1 Population 40

(S) 1 Population 100

generation 1 generation 40 generation 100 Fig.3b. Identification of two voids: solutions for no noise case and noise cases (with standard deviation D(u .) = E(u ^) / 30; where E(ii •) = expected value) a)

t^iii^^ik^4 p{t)=Asinco t

Numerical example 3. The problem of identification of two defects was considered for the harmonic excitation. The evolutionary algorithm decided what kind of defect is detected (ellipse void or crack). Solution of the problem was achieved in 50 generations (Fig.4)

- sensor point b)

generation 1

generation 15

generation 50

Fig.4. Identification of two different defects, a) plate with defects, b) solution in different generations.

Evolutionary methods in inverse problems of engineering mechanics

559

EVOLUTIONARY IDENTIFICATION OF CRACKS The problem of crack identification is formulated as a minimisation of the sum of differences between theoretical boundary displacements (or stresses) and measured ones.

M

M ,

^

1

or ;=i

.

(15)

The direct problem is solved by using the dual boundary element method (DBEM). Two vector boundary integral equations of DBEM are formulated: - for displacements:

and for tractions:

cu + j P * u d r = J u * p d r r r

(16)

- p + nJs*udr = nJD*pdr

(17)

where S* and D* are fundamental solutions, n unit normal vector. The identification problem is formulated both for single and multiple cracks. The genes in chromosomes are described by coordinates of tips of cracks for a simple shape of cracks or by other shape parameters in the case of more complicated shapes of cracks. The situation when information about measured displacements (or stresses) is not deterministic is also considered. One assumes that measured state fields have stochastic nature and are characterized by the mean value and the standard deviation. The problem of identification of unknown number and positions of cracks is also considered. In this case the chromosomes contain genes which describe the number of cracks as well as and their geometry. Numerical example 4. The problem of identification of a crack in the structural element loaded statically was considered. Actual and final positions of the crack are given in Fig. 5.

NNSSX

^ - sensor point Fig. 5. Identification of the crack

560

T. Burczynski et al

Numerical example 5. The problem of identification of unknown number of cracks was considered. One assumed that the number of cracks was from the range <1, 5>. The evolutionary algorithm selected two cracks and found the final position very close to the actual positions (Fig.6).

NVSNS

^ - sensor point Fig. 6. Identification of unknown number of cracks Numerical example 6. The problem of identification of a crack when the measured displacements were burdened by random fluctuations was considered. The final and actual position of the crack is presented in Fig.7.

VS\S\

« - sensor point Fig.7. Identification of a crack for the case of random variation of boundary displacements (with standard deviation D(ii.) - E(u ^) / 3 0; where E(ii •) - expected value)

INVERSE PROBLEMS IN THERMOELASTICITY The problem of finding optimal distribution of temperature on the part of the boundary F* is solved for minimum of displacements on another part of the boundary Fp! dF„

where u is total displacements at the boundary Fp, u„ is an admissible displacement.

(18)

561

Evolutionary methods in inverse problems of engineering mechanics

In order to evaluate the fitness fiinction (18) the direct boundary-value problem of linear, homogeneous, isotropic, steady-state thermoelasticity theory is solved. The boundary only integral equation for thermoelastic problem is obtained

cu+jp*udr=::Ju*pdr + J(pr-Q^)dr

(19)

in which needed boundary temperature and its normal derivative are calculated by solving the boundary integral equation for the heat conduction problem:

cr+J2*rdr = Jr*^dr

(20)

where Q* and T are fundamental solutions for the Laplace equation, P and Q are known functions. In this case the distribution of temperature is represented by genes, which are described by control points of Bezier curves. Numerical example 7. The problem of finding the optimal boundary condition for temperature T* for minimum displacements of the loaded boundary was considered. The optimal distribution of temperature for two traction fields p** are presented if Fig. 8. ///////y

1

U"= 500

S!^—

V

'••^=:--

. y ^

•% /

"''^Cv

140 Tl-C)

120

100

80

120

100

80

40

20

' ^ «

0

Fig.8.Solution for the thermoelasfic solid, a) a plate with boundary conditions, b) optimal distribution of temperature

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T. Burczynski et al

CONCLUDING REMARKS Several numerical examples of identification of voids and cracks and inverse themioelastic problems in 2-D structures have been presented. The evolutionary methods appear to be very convenient and useful in inverse problems. The solution is very stable and it is not necessary to modify the fitness function by adding regularization terms. Errors in the experimentally measured displacements (or stresses) do not cause a significant difference in the computed defect geometry. One disadvantage of the evolutionary methods is the time consuming calculation because in order to achieve a satisfactory solution one should produce many generations, hi order to omit this drawback, a hybrid approach is proposed. In this approach a special gradient mutation operator is introduced. The gradient mutation operator uses information based on sensitivity of the fitness function. There are several possibilities and strategies of using such mutation. In the paper one proposes a strategy which is controlled by neural nets. Numerical tests for detecting voids show that this approach is very effective and promising. The original feature of the paper is the formulation and solution of the evolutionary identification for structures being under mechanical, thermomechanical and dynamical loading.

ACKNOWLEDGEMENT This research was carried out in the framework of the KBN grant 8T11F00316.

REFERENCES \.

2.

3.

4.

5. 6. 7. 8.

Burczynski T., Beluch W., Kokot G. (1999). Optimization of cracked structures using boundary elements and evolutionary computation. In: Boundary Element Techniques (ed. M.H.Aliabadi), London. Burczynski T., Beluch W., Kokot G., Nowakowski M., Orantek P. (1999). Evolutionary BEM computation in optimization and identification, Proc. lUTAM Symposium, Cracow, 1999. Burczynski T., Kokot G. (1999). The evolutionary optimization using genetic algorithms and boundary elements. Proc. 3rd World Congress on Structural and Multidisciplinary Opimization, Buffalo, New York, USA. KitaE., TanieH. (1977). Shape optimization of continuum structures by genetic algorithm and boundary element method. Engineering Analysis with Boundary Elements, Vol. 19. Kleiber M. (ed.) (1998). Handbook of Computational Mechanics. Springer. Michalewicz Z. (1992). Genetic Algorithms + Data Structures = Evolutionary Programs. Springer-Verlag, AI Series, New York. Stavroulakis G.E., Antes H.(1998). Flaw identification in elastomechanics: BEM simulation with local and genetic optimization. Structural Optimization, Vol.16. TanakaM., NakamuraM. (1994). Application of genetic algorithms to plural defect identification in inverse problems. In: Inverse in Engineering Mechanics (eds. Bui H.D., Tanaka M., et al.), Balkema, Rotterdam.

INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

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SCREEN DETECTION WITH NEAR FIELD MEASUREMENTS

C. J. S. ALVES (^) and G. E. PIRES (2) Departamento de Matemdtica, Instituto Superior Tecnico Av.Rovisco Pais 1, 1049-001 Lishoa, Portugal ^^^ Centro de Matemdtica Aplicada; Email: [email protected] ^^^ Centro de Andlise Matemdtica, Geometria e Sistemas Dindmicos; Email: [email protected]

ABSTRACT We consider the inverse obstacle problem of identifying sound-soft plane screens. For spherical incident waves and for quite general scatterers we use the integral representation of the scattered field and a theoretical result is established in order to distinguish plane sound-soft screens from other obstacles. The scattered field is the solution of the Helmholtz equation satisfying Dirichlet boundary conditions and the Sommerfeld radiation condition. A consequence of this result is the presentation of a criterium for the location and identification of such screens. Numerical simulations are presented to illustrate these results. KEYWORDS Acoustic scattering, screens, inverse problems.

INTRODUCTION Identifying the shape of an object by means of the field pattern of acoustic waves, scattered by the object, has been studied from different points of view. The book of Colton and Kress [5] gives a good overview of the subject and applications in industry as well. In this text we derive a theoretical result for distinguishing plane sound soft screens from other obstacles, a counterpart of the results obtained in [4] (for sound hard cracks) and in [2] (for plane incident waves and far field measurements). The main purpose is the inverse obstacle problem of identifying sound-soft plane screens and to present criteria for their location, allowing several possibilities of sets of local measurements and of incident point-sources. In the usual framework, this problem is solved considering the measurement of the far field amplitude generated by incident plane waves (e.g. [2], [5], [7]).

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C.J.S. Alves and G.E. Pires

ACOUSTIC SCATTERING IN THE NEAR FIELD We consider acoustic scattering by an obstacle F with regular boundary in a three dimensional space. F can be the boundary of a round shape Q. or an admissible screen. We define admissible screens to be open subsets of the Lipschitz boundary of an open set in R^. The problem consists of determining the solution u of the Helmholtz equation (with wavenumber A; > 0) satisfying Dirichlet boundary conditions and the Sommerfeld radiation condition, i.e.

in R^\n. on r, drU — iku = o(^), when r =

{A + k'^)u = 0 u = -'^/^"^

(1) |:z:| —>• o o ,

(when the obstacle is a screen, Cl should be replaced by F). The well posedness of this problem has been established for round shapes Q (e.g.[5]) and for admissible screens F (cf. [8]). The asymptotic expansion of the scattered wave u is given by ^ik\x\

i

^W = -|^^oo(:r)-fO(|^) where ^ = jfi and U^Q is an analytic function, called far field amplitude, defined on the unit sphere. The usual framework of this problem is to consider incident plane waves with direction d, u^^^{x) = e^^^-^ and to measure Uoo- In [2], Alves and Ha-Duong present an overview of the developments of this framework and establish criteria to distinguish the far-field of plane screens from others. Here we will be interested in incident point sources, given by the fundamental solution of the Helmholtz equation, $(x) = f ^ ^ - e . we take u^^^{x) = ^{x — y), which we will also call spherical incident wave centered on y, and, at the same time, we will consider local measurements of the scattered field. We will consider point-sources located (centered) on an admissible set E (also called source set) defined by one of the following conditions (cf. [4]): i) E is the border of an open set S that contains F. (see Fig. 1). ii) E is the border of an open set S in R ^ \ F (A;^ is not an eigenvalue of the interior Dirichlet problem), (see Fig. 1). iii) E is an open set of dS, with S as defined in i) or ii), assuming that dS is an analytic surface, (see Fig. 1). iv) E is an open set of dS, where dS is an analytic surface that cuts R^ in two unbounded domains, S and S^ D F. We will be mainly interested in the case when dS is a plane, (see Fig. 2).

565

Screen detection with near field measurements

Fig. 1: The source set and the scatterer.

^i

Fig. 2: Plane source set . If we consider a source set E around the obstacle, for instance a sphere with sufficiently large radius, each spherical incident wave centered in y can be viewed asymptotically as an incident plane wave with direction —y. Therefore, the asymptotic approach (e.g. [2]) assumes that both the source set and the observation set (the set of points where the measurements take place) are located far away from the obstacle. As a consequence other possibilities we want to consider here are excluded (see also [4]). SOFT SCREEN DETECTION In the following, u{x,y) will denote the amplitude (measured at a point x) of the wave scattered by the object (with boundary F) after the incidence of a spherical wave centered on y. The measurements will be taken on an admissible set O which may coincide with E. In the case of sound-hard obstacles we can identify a plane crack from other shapes, using spherical incident waves and near field measurements, since the scattered field vanishes on the plane of the crack (see [4]). This is the case since the scattered field can be expressed by the double layer potential

5^.

dsz 1

Jr^

^

\x - z\^\x-

z\

— ik)^{x

— z)dsz

with -0 = [u], the jump of the field across F. For x on the plane of the crack we have n • (x — z) = 0. In [2], by considering incident plane waves and far field measurements, a condition on

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CJX Alves and G.E. Pires

the tangential derivative of the far field was obtained which was used to distinguish plane screens from other obstacles. In the case of sound-soft obstacles the scattered field is given by the single layer potential, (2)

u{x, y) = J ip{z)^{x - z) ds,, with ip = [dnu] e H~^^^{r),

the jump of the normal derivative across F.

In this work, using spherical incident waves, we establish a condition on the derivatives of the near field from which we derive criteria for identification and location of sound-soft plane screens. In fact, using (2) we obtain, for some fixed vector n, n • Viu{x, y) = J '^{z) n. Vx<3E>(x - z) ds,,

(3)

where Vi stands for the gradient with respect to the first three variables. Note that n • Vx^(x -z)

=n-

r(^ - ik)^{x \x — z\ \x — z\

- z).

(4)

Thus if n ± {x — z) we get n • Viu{x, y) = 0. This is the case of plane screens if we assume that the observation point x is taken on the plane of the crack. We will now prove that this situation occurs only for plane screens. We begin by stating a density lemma which can be proved in a similar way as for density lemmas presented in [1], [4] where such results were used to prove uniqueness for the inverse obstacle problem for spherical incident waves. Lemma. Let B be an open set with Lipschitz boundary such that k^ is not an eigenvalue of the interior Dirichlet problem in B. Consider any 7 C dB^ and take E an admissible source-set. Then F(7) = span{(x - y)\^ : 2/ e E} is dense in i7~^/^(7). Theorem. Consider E an admissible source set and F an admissible boundary (of a round object or a screen), such that T D S = 0. There exists x* and a vector n such that n-Viu{x*,y) = 0 for all y e T>, if and only if T is a screen on 11, the orthogonal plane to n defined by x*. Proof The necessary condition is clear from (3) and (4). Suppose now there is a point w; G F, It; ^ n , then there exists a neighborhood V^ of w that does not intersect 11. By the previous lemma F(F) is dense in i/~^/^(F) and by the well-posedness of the Dirichlet problem (if /c^ is not an eigenvalue of the interior Dirichlet problem in Q), we can produce an incident field u'^^^ on F such that there is a function ip with support equal to V^, and then n • VM^)

= / ipiz) n • V^$(x - z) ds, = f

i/jiz) n • V:,^{x - z) ds, y^ 0 (x G n \ F ) (5)

choosing a small V^, because '0 ^ 0 in K; and x G H. When /c^ is an eigenfunction of the interior Dirichlet problem in O, we can follow the same steps of [2] using a mixed potential representation.

Screen detection with near field measurements

567

Corollary. IfT is a sound-soft screen on the plane H then there exist three points ^1,^25^3 in E such that det M{x) = det(Viu(x, 2/1), Viu(x, 7/2), Viu{x, ys)) = 0, {x e

U\r).

Moreover, for a such x* G n \ r we have dim(ker(M(a;*))) = 1 and therefore the plane U can be identified by x* and by the normal vector n solution of M.{x*)n = 0. Proof The first part is a direct consequence of the necessary condition of the theorem, because we have n'Viu{x*,y) = 0 for all y G E which implies that the vectors Viu{x'',yi),'Viu{x*,y2),Viu{x*,ys) lie on the plane E and therefore they can not be linear independent. Suppose now that the dimension of the kernel is never equal to one. Then for arbitrary yi, 2/25 2/3 ^ E there exist two linear independent vectors n, m such that n-\/iu{x,yi) = 0, in-Viu(a;, y^) = 0, and this implies by the sufficient condition of the theorem that F should be a screen on a plane H orthogonal to n and to m which is impossible. Remark. The reciprocity relation u{x^ y) = u{y, x) for all x,y E R ^ \ r implies that Vi?i(x,y) = \/2u{y,x), and this allows to establish a criterium where the roles in the previous results are interchanged. For instance, in the corollary, 2/1,2/2,2/3 will be measurement points and x* will be a point-source. NUMERICAL SIMULATIONS To derive a numerical approximation of the wave scattered by a plane screen, we use a boundary finite element method (as in [3] for the Neumann case), applying the variational formulation of the single layer potential on H~^^'^{r) (see [6]), J J ^{x-

y)il){x)4)[y)ds:,dsy = J -u'''%x)^{x)

ds^.

(6)

The density ij; is approximated using a finite element method on the boundary F C R^ x {0}. The scattered field is then given by the single layer potential (2), using the approximation

^. In Fig. 3 we present the scattered field generated by a plane screen T={(x,y,Q):x'

+

y''
We remark that this is an axisymmetric problem and therefore we neglect the y axis. An incident spherical wave centered on (3,0,3) and wavenumber A: = 6 is considered. To check the necessary criterium established for finding the location of the plane of the screen, we considered several situations. Only two source points are considered (instead of three as stated in the corollary) because of the axisymmetric feature of F. We measure det(M(a;)) for x G 6 , whereO is a line that connects the two points Xi and X2, and M{x) — {\/u{x,yi),Vu{x,y2))We notice that if no axisymmetric feature occurs we should consider 6 to be a part of a plane. The following experiments were considered:

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C.J.S. Alves and G.E. Pires

Fig. 3: Plot of \u[x,y)\ for fixed y = (3,0,3), A; = 6. i) Fig. 4 Source points: yi — ( - 3 , 0 , 3 ) and 7/2 = (3,0,3). Measurement line: Xi — (5,0, —5) and x^ — (3,0,3) ii) Fig. 6 Source points: yi — (—3,0,3) and y^ — (3,0,3). Measurement line: Xx = (—6,0,2) and x^ = (4,0,2) iii) Fig. 8 Source points: yi = (3,0,3) and ?/2 = (4,0,4). Measurement line: Xi = (2,0, 4) and X2 = (2, 0, —4) iv) Fig. 10 Source points: yi = (3,0,4) and 2/2 = (4,0,4). Measurement line: Xi = (2,0,4) and X2 = (2,0, —4) In Fig. 5, on the left, we plot the graph of (t, | det M{xi + t{x2 - ^i))|) for case (i) and we confirm that det M(x) = 0 for the only x on the plane of the screen. Moreover, in Fig.5, on the right, we plot the angles made by the vectors Vu{x,yi),Vu{x,y2) and we confirm that they coincide for an angle of zero degrees which corresponds to the orientation of the crack. The same experiment was made in the other cases. We now notice that in case (ii) (see Fig. 6) the measurement line does not intersect the line of the crack, and therefore the determinant is not null (Fig. 7, on the left), as predicted. However, the determinant can have small values near the edges of the line, since the amplitude decreases fast. When we look at the angle variation (Fig. 7, on the right), we see that the graphs are quite different and never give the angle orientation of the crack.

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Screen detection with nearfieldmeasurements

Fig. 4: Case (i). Location of the screen and of the measurement set.

0.2

0.4

0.6

Fig. 5: Case (i). Plot of the determinant and angle variation. In figures Fig. 8 and Fig. 10 we consider two similar experiments (cases (iii) and (iv), respectively), where the only change was in the position of a source point. In case (iii) the extension of the line that connects the two source points crosses the crack and the values of the determinant (see Fig. 9, on the left) are ten times lower than in case (iv) (see Fig. 11, on the left), and therefore the two lines of the angle variation plot almost coincide (see Fig. 9, on the right), if we compare it with the angle variation on case (iv) (see Fig. 11, on the right). This high difference can be partially justified, since in case (iv) this line extension would verify the criteria in the definition of an admissible source set location, which does not happens in case (iii). Of course, this two points do not mean much of the all source set E, and if E is not a line, other points could be chosen. However, this gives a good example on the choice of the source points. All this experiments were made without noisy data. There is a difficulty with noisy data. 3 2.5

1.5 1 0.5 -6 Fig. 6: Case (ii). Location of the screen and of the measurement set.

570

CJX Alves and G.E. Pires

0.2

0.4

0.6

0.

Fig. 7: Case (ii). Plot of the determinant and angle variation, 4

(^

1

2

3

r

-2l

-4|

Fig. 8: Case (iii). Location of the screen and of the measurement set. on the calculation of the derivatives, since no regularity should be expected, that can be dealt with filter techniques. In fact, a very simple approach it to use the relation

Vu{x) =< 6^,Vu > since we can approach the Dirac delta on x by a mollifier, or simply by a hat function, fix, such that Vu{x) ~ — < V/ix, u > • However, extra problems are expected with small values of the determinant. This analysis will be the subject of a future work.

Fig. 9: Case (iii). Plot of the determinant and angle variation.

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Screen detection with near field measurements

-2

-4

Fig. 10: Case (iv). Location of the screen and of the measurement set.

Fig. 11: Case (iv). Plot of the determinant and angle variation. REFERENCES 1. Alves, C.J.S. (1998). In: Mathematical pp. 502-504; J. DeSanto (Ed.). SIAM.

and Numerical Aspects of Wave

Propagation,

2. Alves, C.J.S. and Ha Duong, T. (1997). Inverse Problems, 13, 1161. 3. Alves, C.J.S. and Ha Duong, T. (1995). Int. J. Num. Meth. in Eng. 38, 2347. 4. Alves, C.J.S. and Ribeiro, P.M.C. (1999). In: Boundary Elements XXI, Advances in Boundary Elements, vol.6, pp. 355-364; C. Brebbia and H. Power (Eds.). WIT Press, Southampton. 5. Colton, D. and Kress, R. (1992). Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences 93, Springer-Verlag, Berlin, Heidelberg. 6. Ha Duong, T. (1992). Integral Eq. Op. Theory , 15, 427. 7. Kress, R. (1995). Math. Meth. in Appl. Sc, 18, 267. 8. Stephan, E. P. (1987). Integral Eq. and Oper. Theory, 10, 236.

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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

573

BAYESIAN ESTIMATION FOR NONLINEAR INVERSE PROBLEMS Makoto SUZUKI^ and Akira MURAKAMI^ ^ Senior Researcher, Shimizu, Co.

e-mail: suzuki@ori . shimz .co.jp ^ Professor, Okayama

University

ABSTRACT The framework of the numerical procedure for parameter identification is to minimize the objective function which includes both the square of the observational error and prior information on the parameters. This leads to a well-posed numerical solution for linear inverse problems. However, there are few numerical techniques which are effective in dealing with nonlinear inverse problems where the observation is expressed as a nonlinear function of an unknown vector. This paper presents the Bayesian estimation for overcoming difficulties in solving such nonlinear inverse problems through the discretization of the probabilistic density function for estimation. The Kalman filtering technique, an alternative tool, provides solutions without maximum likelihood under the assumption of linear approximation for nonlinear observation equations. The numerical strategy is described in conjunction with the finite element method for identifying the elastic parameters of a two-layered soil stratum based on settlement observation. KEYWORDS Bayesian estimation, Kalman filtering, nonlinear inverse problem, FEM INTRODUCTION In the field of geotechnical engineering, the inverse analysis procedure is a promising tool for predicting groifnd performance and for carrying out safety assessments of the ground within a limited number of soil experiments. Among the several numerical strategies for solving inverse problems, both the Bayesian estimation [3] and Kalman filtering [1] have benefits in numerically regularizing ill-posedness based on prior information, e.g., the statistical characteristics of unknown parameters. Within the framework of Kalman filtering, the observation update scheme for a set of linear system equations with Gaussian noise is performed using the Bayesian estimation and the time update scheme based on the state equation. For the parameter identification of linear observational equation, both the Bayesian estimation and Kalmanfilteringprovide the same presumption through the noisy observation. This is because the time update scheme does not work under the stationary condition of parameters in time. In cases where the nonlinear system dominates a field equation, the Taylor series expansion is

574

M Suzuki and A. Murakami

utilized to avoid nonlineaiity. Such a technique is referred to as 'extended Kalmanfiltering.'The Bayesian estimation is independent of the nonlinearity of the observational equation, while the Kalman filtering provides solutions dependent on the nonlinearity of system equation. The aim of the current paper is to discriminate the difference between two such numerical strategies for a set of nonlinear system equations which often appear in geotechnical engineering. Section 2 is devoted to examining the difference between Bayesian and Kalman filter estimates. Section 3 deals with a simple example of two connected springs and compares the different numerical solutions for nonlinear observational equation given by both strategies. Section 4 describes a further example for identifying the elastic moduli of a two-layered soil deposit beneath embankment loading. In Section 5, the numerical results are summarized and further development stemming from the present procedure is discussed. BAYESIAN ESTIMATION AND KALMAN FILTERING Scheme for observation update It is assumed that the following equation is obtained for the observation: (1)

y = Hx-\-v

where y is an observational vector, H is an observational matrix, x is an unknown vector, and v is an observational noise with zero mean and a covariance matrix of R. Prior distribution on x is also given as the following probabilistic density function (p.d.f.) of x, namely. ^("^^ = (27r)n/2^|M|V2 >< ^^P {-^(^ - ^fM-\x

- x)|

(2)

which follows to the normal distribution with the known average of x, x, and the variance matrix of X, M . They are given a priori. The problem is given as: To find X which gives the maximal value of the conditional p.d.f. of x under the observation y,p{x/y). To obtain the conditional p.d.f. of x under the observation of y, p{x/y), Bayes' theorem can be used as : (3)

p{x/y) = p{y/x)p{x)/p{y) where p{y/x) is given as follows through the use of equation (1) Piy/"") = (27r)J^\R\^/^ "" exp | - i ( y - HxfR-\y assuming that v has a Gaussian distribution with zero mean.

- Hx)^

(4)

575

Bayesian estimation for nonlinear inverse problems

Based on equation (1), the p.d.f. of y is also denoted as follows: (5)

E[y] = E[Hx ^v]=Hx E[{y-y){y-yf]

= H E[{x - x){x - xY]H^ = HMH^ + R

+ E[vv^] (6)

Hence, v{y)

= (27r)W2|ijrjvfjf^ + M | V 2 exp | - i ( y - HxY{HMH^

+ R)-\y

-

Hx)\

(7)

Finally,

exp | - ^ ( t / - HxfR-\y --{x

- XYM~^{X

-

Hx)

x)

+i(2/ - HxY{HMH^

+ R)-\y

-

Hx)\

(8)

Thus, the problem is rewritten as: To find X which minimizes the following function J: J=\{y-

HxfR-\y

- Hx) + ^{x - xfM'^x

- x)

(9)

The following set of equations can be obtained as the estimation of cc, x, by differentiating the above function: X = X -i- K{y — Hx)

- 1 +. H^R-^H) ^To-l.x^ - M .^ P = (M-^

r KHM .-n.

dO)

where K is the Kalman gain and P is the error estimation matrix. An estimation through the above scheme for observation update coincides with that of the Bayesian estimation.

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M. Suzuki and A. Murakami

Scheme for time update If the state variable has the time evolution denoted as follows, it should be incorporated into the scheme which appeared in the preceding section. In other words, (11)

Xt+i = FtXt + Wt

where Ft is a transition matrix at time t and Wt is a system noise with the following characteristics: E[wt-^] = 0

(12)

E[{wt-i - wt~i){wt-i - wt-i)'^] = Qt-i

(13)

E[xt-i] = xt-i

(14)

Also,

E[{xt-i - xt-i){xt~i

(15)

- xt-i)'^] = Pt-i

(16)

xt = E[xt] = Ft-iXt-i

(17)

Mt = Ft.,Pt.^Fti + Qt~i Based on the last equation, the Kalman gain at time t, Ku can be denoted as Kt = PtMjR;' = MtHjiHtMtHj + Rt)''

(18)

A set of resultant equations can be summarized as listed below: Filter equation:

Xt+i = FtXt

(19)

Xt = Xt + Ktivt - HtXt)

(20)

Kalman gain: + Rt)-'

Kt = MtHj{HtMtHj

(21)

Error estimate matrix: Mt+, = FtPtFj Pt = Mt-

+ Qt

(22)

KtHtMt

(23)

M o = So

(24)

Initial condition: Xo = Xo,

Bayesian estimation for nonlinear inverse problems

Sll

In the case of a nonlinear observation, for example, y^ = ht{xt)

(25)

The Taylor series expansion is used to linearize such nonlinear observational equation. As a result, the following matrix is adopted within the preceding algorithm. Ht=-^\,=^, dxt

(26)

The scheme which includes such a linearization is called 'extended Kalman filter.' Example of two connected springs To demonstrate the difference in the estimated values of the unknowns in nonlinear inverse problems between the Bayesian estimation and extended Kalman filtering, the following example is used to identify the spring constants, ki and k2, based on the observation of the displacement of spring #2, U2, as seen in Figure 1. U M2 [

/l. "l I"

i^////////////////^m Figure 1: Spring model The relation between force and displacement is obtained based on the force equilibrium, in other words.

This leads to the observational equation U2 = h{ki, k2) +vt

(28)

where h is the nonlinear function of the unknowns and Vt is the observational noise at time t. For the parameter identification problem to be dealt with here, x = {ki k2Y is adopted as the unknown variable. This quantity is constant in time and the following equation is given as the state equation.

These two equations correspond to the following set of equations, respectively, Xt+i — FtXt (State equation)

(30)

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M. Suzuki and A. Murakami

Table 1: Cases to be analyzed Bayesian estimate Simultaneous Successive

Case #1 Case #4

Extended Kalman filter With local iteration Without local iteration Case #2 Case #3 Case #5 Case #6

y^ = h{xt) + Vt (Observational equation)

(31)

where F is a state transition matrix (unit matrix in equation (29)) and h is given by equation (27). It is assumed that there are three different measurements, namely,

Thefirstand the third measurements are contaminated with noise under the true values of ki=OJ and A:2=1.0. Two cases are treated to examine the difference in solutions between the Bayesian estimation and Kalman filtering for nonlinear inverse problems; one is a case where all three measurements are observed simultaneously; the other is the case in which the three measured displacements are observed successively. In the extended Kalman filtering, the *locar iteration within a time step is often adopted to find a good convergency of the solution citeJazwinski:1970. Cases to be analyzed herein are listed in Table 1. The numerical results are summarized in Table 2. The following discussions on numerical solutions are obtained: 1. Cases #1 and #4 have the same posterior distribution of estimate errors through the Bayesian estimation procedure instead of simultaneous or successive observations. 2. Cases #1 and #2 provide similar solutions. 3. The solution of Case #3, without the local' iteration, is different from those of Cases #1 and #2. 4. The difference between Cases #2 and #5 tells us that the solution by the extended Kalman filtering depends on the order of the measurements.

Bayesian estimation for nonlinear inverse problems

579

Table 2: Numerical results and probabilistic densities

h

Case #1 0.460 Case #2 0.456 Case #3 0.443 0.700 Case #4 0.520 0.460 0.697 Case #5 0.471 0.399 0.593 Case #6 0.526 0.476

k2

1.093 1.116 1.163 1.393 1.393 1.093 1.407 1.556 1.398 1.351 1.326 0.919

p{x\y) 2.453 2.470 3.247 0.397 1.356 2.453 0.373 1.595 2.727 1.011 1.360 1.897

IDENTEFICATION OF THE MATERIAL PROPERTIES OF AN EMBANKMENT FOUNDATION Formulation The extended Kalman filtering in conjunction with finite elements is adopted here to pursue the elastic moduli of soft soil deposits under embankment loading [4, 5]. To begin with, the stiffness equation of solids such as Ku = f

(32)

is rewritten as the observational equation in the following manner: y^ = u-\-Vt = K~'^{xt) -\-Vt = ht{xt) + Vt

(33)

where Xt is the vector of unknown elastic moduli. The observation matrix can be calculated at time t based on OXt

(34)

where the derivative of load / with respect to Xt vanishes, because / is assumed to be independent of Xt. On the other hand, the following stationary condition of parameters in time describes the state equation, namely. Xt+l =

IXt

(35)

580

M. Suzuki and A. Murakami

Problem description Figure 2 depicts the problem description, where Co indicates the differential settlement gauge and Li (z = 1 '^ 4) draw settlement gauges on the ground surface. A forward analysis is carried out to obtain hypothetical measurements under the assumption of Young's moduli of the foundation layers and Poisson's ratio of both layers. Figure 3 shows the construction sequence of the embankment and the measured settlements based on the forward analysis. 9.80 7.00 7.00

16.25

in meter

L4

U

L2

Ml A 90.00

Figure 2: Problem description

IV\^

"

•

^

^ r V x<..

]

L4

Co

/

L J

he ght

0

100

/

/

0 L3 L2 Li

200 300 400 500 600 elapsed time (day)

Figure 3: Loading history Initial guesses of the unknown variable and the diagonal term of the estimate error covariance matrix are 1500 MPa and (150 MPa)^, respectively. The covariance matrix of the observational noise, R, has a diagonal value of 1.0 x 10~^. Numerical results Young's moduli of the two layers are identified using extended Kalman filtering based on the measured settlements. Figure 4 shows the identified results via embankment height. Solid lines indicate the identified Young's moduli with estimation errors expressed by dotted lines. The identified moduli gradually converge from the initial value of 1500 MPa to the true values of 2000 MPa and 1000 MPa, respectively. The converged values are not in good agreement with the true values. The reasons for such a disagreement are listed below:

Bayesian estimation for nonlinear inverse problems

581

1. Extended Kalman filtering adopts the approximated Gaussian distribution of estimate errors in the linearization of each observational update and its approximation errors may have an influence on the numerical solutions. 2. The initial guesses of unknowns, which correspond to the prior information of the Bayesian estimation, have a significant influence on the numerical solutions. 3. Since each element of the estimation error covariance matrix shrinks during the update scheme, the Kalman gain decreases in value and it is hard to refine the estimation.

H iiii

10-

5

0-

s

J

L

100

/

[

r

F

h

rr /

200

2000

r h500

300

400

elapsed time (day)

500

600

Figure 4: Numerical results Although extended Kalman filtering includes errors in its solutions, influenced by the hnearization of the formulation, such errors are not significant in the current example. From the viewpoint of computational effort, extended Kalman filtering with finite elements is quite a practical tool compared to the Bayesian estimation which computes under the non-Gaussian distribution of unknowns. CONCLUSIONS Numerical analyses with the Bayesian estimation and extended Kalman filtering are presented for nonlinear inverse problems. Comparing the numerical results of both strategies, several conclusions can be made: 1. The numerical solutions are different, because the Bayesian estimation maximizes the probabilistic density oi'p[x\y) and extended Kalman filtering obtains the average of that probabilistic density, namely, £^[p(cc|y)]. 2. The Bayesian estimation is advantageous in the sense that it provides a solution instead of the order of the measurements. 3. Extended Kalman filtering is a more practical technique than the Bayesian estimation in light of the feasibihty of the numerical method and its applicability to nonlinear inverse problems.

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M. Suzuki and A. Murakami

REFERENCES 1. Kalman, R.E. and Bucy R.S.(1961) / Basic Eng. (Trans. ofASME, Sen D), 83, pp.95-108. 2. Jazwinski, A.H.(1970) Stochastic process andfilteringtheory. Academic Press Inc. 3. Cividini, A., Maier, G. and Nappi, A.(1983) Int. J. RockMech. & Mining ScL, 20(5), pp.215226. 4. Murakami, A. and Hasegawa T.(1988) Proc. 6th Int. Conf. on Num. Methods in Geomech., pp.2051-2058. 5. Suzuki, M. and Ishii K.(1993) Proc. 6th Int. Conf. of Strucural Safety & Reliability, pp.843850.

INVERSE PROBLEMS EST ENGESTEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.

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STRUCTURAL OPTIMIZATION USING CELLULAR AUTOMATA T. TOYODA Department of Mechano-Infomatics and Systems^ Nagoya University, Nagoya 464-8603, JAPAN E. KITA School of Infomatics and Siences, Nagoya University, Nagoya 464-8601, JAPAN ABSTRACT In this paper, we propose a new structural optimization method using cellular automata. The design domain is divided into small square cells, which are regarded as finite elements when stress analysis is performed by the finite element method. The thickness of each cell is taken as a design variable, which is updated according to a local rule. In this study, the rule is derived as follows. Two objectives are considered in structural optimization, one is to minimize total weight of the structure and the other is to minimize the deviation between the equivalent stress and the reference stress. Besides, in order to formulate the local rule for an updated cell, the special constraint condition is introduced. The local rule is derived from the above optimization problem. The present method is applied to a simple example in order to confirm the vaUdity.

KEYWORDS Structural Optimization, Shape and Topology Design, Cellular Automata(CA), Local Rule, 2D Elastic Problem.

INTRODUCTION Cellular automata(CA) is one of numerical simulation methods for various phenomena, which has widely been studied since von Neumann [4,5] first introduced the concept of finite automata first. In CA simulation, the design domain is divided into many small elements called 'cells'. Each cell has a discretized quantity of state, which is renewed in every time step by applying a local rule. The rule is simply expressed as a function of the states of the updated cell and its neighbors at current and previous time steps. This updating of the quantity of state of the updated cell is carried out in the whole design domain at the same time step. In this way, the state of the whole region is renewed in order. As a local rule is derived as a function with respect to the state quantities of

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T. Toyoda and E. Kita

Moor neighborhood

Von Neumann neighborhood

Fig. 1: Neighboring cells the updated cell and its neighbors alone, the differential equation describing the whole domain uniformly is not required in the CA simulation. Therefore, it is apphed to various problems that have a difficulty with derivation of the equation as a model of the problem, for example, traffic simulation, diffusion simulation of the particle, behavior analysis of the coarse grain, flow analysis, etc. [1,2,6] Recently the CA simulation has been also used for the structural optimization problem. Inou et a/. [3] proposed a shape and topology optimization method of two- and threedimensional continuum structures. In this method, Young's modulus of each cell is taken as the design variable and they are updated using the nonlinear equation representing the relationship of Young's modulus and equivalent stress of each cell. As this equation is experientially derived, its theoretical background is not clear. Besides, many of the structural optimization using CA approach proposed so far had some diflSculties incorporating a constraint on stress to a local rule. So, in this paper, we propose a new structural optimization method based on the concept of CA to deal with these problems. The design domain is divided into small square cells and the thickness of each cell is taken as the design variable. The optimization problem for one cell and its neighboring cells is defined, which consists of two objectives and the special constraint condition. The local rule is derived from the optimization problem. Finally, the proposed method is applied to the shape optimization of two-dimensional structures.

DERIVATION OF LOCAL RULE AND ALGORITHM Cellular representation for two-dimensional continuum structure We consider two-dimensional continuum structure. The design domain is divided into small square cells and then, each of them is considered as finite element in finite element analysis. A cell has some neighboring cells around itself. In CA simulation, there are mainly two types of neighboring cells, Moor neighborhood and Von Neumann neighborhood(Fig.l). In this paper. Moor neighborhood is used as neighboring cells. The thickness of each cell is taken as the quantity of state, which is updated using a local rule defined as the relationship of the quantities of state of the updated cell and its neighboring cells.

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585

Defimtion of optimization problem In this study, two design objectives are considered. One is to minimize the total weight of the structure and the other is to make the equivalent stress of all cells to agree with the reference stress. Now we consider the optimization problem for one cell, the design objectives are to decreasing the thickness of the cell and to make the equivalent stress at the cell to agree with the reference stress. The objective functions representing them are shown below.

W,

=

( ^ - l ) ' s ( » - l ) '

(2)

where h and h^ are the thickness and the initial thickness of the updated cell, respectively. a and GC are the equivalent stress of the updated cell and the reference stress, respectively. Since these objective functions are minimized on the condition that the states of neighboring cells are fixed, the constraint condition is necessary so that the quantities of state of neighboring cells are fixed during the updating process. This condition is defined as follows. (i - 1, • • • , TV,)

g,^^-l=CJi-l=Q

(3)

where di and a^ are equivalent stresses of neighboring cell i in present and previous time steps, respectively. TV, is the total number of neighboring cells of the updated cell. Derivation of penalty function and stationarizing Introducing the weighting factors a, fi and penalty coefiicient p, the penalty function for the optimization problem of the updated cell is defined from the two objective functions (Eq.(l) and (2)) and constraint condition (Eq.(3)) as follows.

W

=

aWi^pW2+pJ2gi'' i=l

=

(4)

ah' + 0(a-lf+p^{ai-l)' i=l

where penalty coefficient p is specified by users. The weighting factors a and /5 are taken as follows.

«-^=''

" = { 1 !:>;>

(^'

When the equivalent stress of the updated cell exceeds the reference stress, the objective function Wi is disregarded and only the objective function W2 is considered. On the contrary, if the equivalent stress of the updated cell is much smaller than the reference stress, it can be considered that the objective on stress is satisfied, and thus only the objective function W2 on weight minimization is considered. When the penalty function (Eq.(4)) is expanded around h, we have W{h -h Sh) == a{h + Shf + f3{a + aSh - if

-h p ^ ^ ( a , + aiSh - 1)'

(6)

586

T. Toyoda and E. Kita

where ( ' ) stands for d/dh. Stationarizing Eq.(6), we have

^ ^

^ =

2a{h + 6h) 4- 2^(cr + aSh +2p^{ai

and then,

_

+ aiSh-l)&i

l)a

= 0

ah-{-P{a-l)a+pZ{(Ji-l)(yi

(7)

,^.

Calculation of approximate sensitivity If the finite element method can be formulated at each cell, we have the stiffness equation hK'^Ue = fe

(9)

where subscript e means the updated cell. Differentiating both sides of Eq.(9) with h, we have the approximate sensitivity of the displacement K > e + hK'^Ue = fe = 0

^e = 4K-KX = - ^ Using the coefficient matrix B, displacement and stress are related as a- = Bu. the approximate sensitivity of equivalent stress is calculated as follows.

(lO)

(11) Thus,

&e = BUe = —-r (1^) a This means that the stress sensitivity is in proportion to (Te and in inverse proportion to h, So, a in Eq(8) can be as Now we consider the approximate sensitivity of neighboring cells. The stiffness equation on the neighboring cells is expressed as KnUr^ = fn

(14)

where subscript n means the neighboring cell. Considering the relation between thickness of the updated cell and stress of the neighboring cell, it is said that the one is qualitatively in inverse proportion to the other. So we shall assume the following relation. hfn ~ const.

(15)

/; = - ^ .

(16)

So, we have

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587

By differentiating Eq.(14) with h and substituting it to Eq.(16), the approximate sensitivity of displacement of neighboring cell is derived as follows.

Urt =

{Kn)

^fn

=

- ^ (17) h Using the relation between displacement and stress, the approximate sensitivity of stress of neighboring cell is calculated as
(18)

Thus, the approximate sensitivity of equivalent stress of neighboring cell i is ai = - ^ .

(19)

Derivation of local rule Substituting Eq.(13) and (19) into Eq.(8), the change of thickness of the updated cell is expressed as + (5{a -l)a + pZhjcTj - l)a. Sh = -ah'' -^^^ -^PK^-ya^P2.i=^KOJ--^)u^^

(20)

Using Eq.(20), the thickness h is updated as h'+' = h^ + 6h

(21)

where superscript k is the time step. Algorithm The algorithm of the present method is as follows. 1. Input the initial data. 2. Perform the stress analysis by the finite element method. 3. Check the convergence criterion. If the criterion is satisfied, the process is terminated. If not so, the process goes to the next step. 4. The cell thickness is updated by the local rule Eq.(21). 5. Go to Step 2.

NUMERICAL EXAMPLE In this section, we shall show the numerical example of structural optimization. The initial

588

T. Toyoda and E. Kita

i

40 cm

MtM 60 cm Fig. 2: Initial structure

Table 1: Design parameters Number of cells

60x40

Young's modulus

E = l.Ox 10^° (Pa)

Poisson's ratio Initial cell thickness

ho = 1 (cm)

Force

P = 1000 {N)

Penalty coefficient

p=l{)

Refernce stress

(Jc = 0.8 X a^''''

Structural optimization using cellular automata

589

Fig. 3: Final structure structure of the object is shown in Fig.2. Table 1 indicates design parameters. The object is rectangle region with 60cm x 40cm. Both sides are fixed and the load P is applied on the center of the structure. In the Table 1, the initial max stress G^°'^ represents the maximum stress at the initial profile. The thickness distribution after 3000 iteration is shown in Fig.3. We notice four sides of structure are eliminated and a X-shaped structure is remained. Figure 4 represents the convergence histories of the maximum and mean stresses and the total weight. In this figure, cTmax c^-nd Gave are the maximum and mean stress of the whole structure, respectively, and CTQ is the maximum stress at initial analysis. W and WQ are the total weight and its initial value, respectively. Total weight is decreasing steadily as the thickness is updated, reaching 15% of the initial weight. Therefore we can say the objective on the weight minimizing is satisfied. On the other hand, the maximum and mean stresses are approaching the reference stress which is 0.8 x GQ and is shown as a broken fine in Fig.4. This means that because the stresses arising in all cells are closing in the reference stress, the objective on the stress is almost satisfied. Therefore the validity of the local rule proposed in this paper is shown.

CONCLUSION This paper described the structural optimization method using cellular automata. In the present method, the design domain is divided into small square cells and their thickness is updated by the local rule at every time step. The local rule was derived from the local optimization problem consisting of two objectives and the special constraint condition. One of the objectives is to reduce the thickness of the updated cell and the other is to make the equivalent stress of the updated cell to agree with the reference stress. The constraint condition is that the equivalent stress of neighboring cells does not change as the thickness of the updated cell changes, which is introduced to treat the optimization problem as local. The local rule was apphed to the shape optimization problem of two-dimensional structure in order to confirm the validity. The object to optimize was a rectangular plate, fixed at left and right side and applied a downward force at center. The design domain was di-

590

T. Toyoda and E. Kita

1

'

r

1000 2000 Number of iteration Fig. 4: Convergence history vided into 40 x 60 cells. The reference stress was set to 0.8 times as high as the maximum stress at initial stress analysis. The result obtained after 3000 iterations was X-shaped structure, which weight was reduced to 15% of initial one. As updating by local rule was carried out, the maximum stress and the mean stress of whole domain was getting closer to each other. This means that the stresses arising at all cells in the entire structure were becoming around the reference stress. Therefore, we may say that the objectives of the optimization problem were almost satisfied and the validity of this method was confirmed.

REFERENCES 1.

G. D. Doolen, U. Frisch, B. Hasslacher, S. Orszag and S. Wolfram(1987). Lattice Gas Methods for Partial Differential Equations, Addison-Wesley.

2.

W. Eissler, P. Drtina and A. Frohn(1992). ASME, 61, 1109.

3.

N. Inou, T.Uesugi, A. Iwasaki and S. Ujihashi(1998). Fracture and strength of solids Pt 2; Behavior of materials and structure (Proc. 3rd International conference, Hong Kong, 1997).

4.

J. von Neumann(1951). The General and Logical Theory of Automata - Cerebral Mechanisms in Behavior -, John Wilet & Sons.

5.

J. von Neumann(1966). Theory of Self-Reproducting Automata, IlUnoi University Press.

6.

S. Wolfram(1994). Cellular Automata and Complexity, AWPC.

591

Author Index Abboudi, S. 23 Adachi,T. 129 Alves,CJ.S. 213,563 Amaya, K. 439 Ando, S. 119 Aoki, S. 439 Arai,M. 129 Arimitsu, Y. 285 Artioukhine, E. 23, 33 Azegami, H. 277, 285, 295 Bailly,Y. 33 Beluch, W. 553 Bemtsson,F. 91,99 Bui, H.Y. 203 Burczyriski, T. 553 Chafik, M. 537 Chen, A. 343 Chen, W. 333 Cheng, J. 493 Chouaki, A. 13 Cola9o, M J. 409 Collard, E. 367 Constantinescu, A. 181, 203 Contro,R. 171 Cotta,R.M. 53 Dantas, L.B. 53 Darbord, L 13 Dennis, B.H. 139 Deraemaeker, A. 367 Divo,E. 63 Dlugosz, A. 553 Dulikravich, G.S. 43, 139 Eggers, Th. 389 El-Badia, A. 529, 537 Elden,L. 91,99 Furukawa, A. 149 Furukawa, T. 307 Ha-Duong, T. Hatsukade, Y.

213,537

449

Hayabusa,K. 109 Hay ami, K. 119 Herve,P. 13 Hon, Y.C. 493 Hori, M. 349 Hosokawa, T. 359 Hu,N. 239 Igarashi, H. 467 Ikeda, N. 267 Imado, F. 429 Inoue,H. 109 Ito, M. 267 Itoh,M. 191 Kagawa, Y. 459, 477 Kanbayashi, T. 327 Kanevce, G.H. 43 Kanevce, L.P. 43 Kanoh, M. 359 Kapat,J.S. 63 Kasai, N. 449 Kassab, A J . 63 Katamine, E. 295 Kawai, R. 449 Kawai, T. 267 Kimura, A. 459 Kishimoto, K. 109 Kita, E. 583 Kitahara, M. 219 Kobayashi, A.S. 249 Kobayashi, S. 327 Kochikov, I.V. 487 Kojima, F. 229, 449 Kokaly, M.T. 249 Koyama, Y. 429 Kubo, S. 149,257 Kuramshina, G.M. 487 Kuroki, T. 359 Kyogoku,H. 191 Laban,M. 419 Labrujere, Th.E. 419 Lacour, F. 33 Ladeveze, P. 367 Leconte, P. 367

592 Liu, D. 343 Lobo, P.D.C. 53 Maigre, H. 203 Matsumoto, T. 3 Matsushima, K. 399 Matsuura, Y. 295 Migorski, S. 513 Miyasaka, M. 439 Murakami, A. 573 Nakahata,K. 219 Nakajima, M. 119 Nishida,T. 129 Nishimura, N. 327 Nowak,AJ. 71 Nowak, I. 71 Nowakowski, M. 553 Ohura, Y. 545 Okajima, N. 229 Onishi, K. 545 Orantek, P. 553 Orlande, H.R.B. 53,409 Penzel,F. 213 Person, V. 317 Pires, G.E. 563 Qian,Y. 389 Quaglini,V. 171 Rikards,R.

161

Sakagami, T. 257 Sakuma, T. 192 Shibuya, T. 109 Shigeta,T. 501 Shirota,K. 521 Singh, K.M. 81 Sobieczky, H. 389 Soemarwoto, B.I. 419 Sogabe, Y. 285 Spiridonov, V.P. 487 Suzuki, M. 573 Takazawa, H. 439 Tanaka,M. 3,81,333 Tanaka, N. 257 Tapley, J. 63 Tarasov, Yu.I. 487 Tardieu,N. 181

Terao,J. 119 Toropov,V.V. 191 Toyoda, T. 583 Iran, D.K. 249 Trendafilova, I. 377 Tsutsumi, K. 477 Van Brussel, H. 377 Vannucci, P. 317 Vena,?. 171 Verchery, G. 317 Wang, Q. 545 Wang, X. 239 Watanabe, S. 119 Wei, T. 493 Wrobel,L.C. 71 Wu, Z.Q. 285 Yagawa, G. 307 Yagola, A.G. 487 Yamamoto, M. 493 Yano,T. 3 Yansyah,H. 419 Yao, Z.H. 239 Yoshida,F. 191 Yoshikawa, H. 327 Yoshimura, S. 307 Zhu, P.

343