Computational Methods in Engineering & Science: Proceedings of Enhancement and Promotion of Computational Methods in Engineering and Science X -- Aug. 21-23, 2006 Sanya, China

In this problem statement A^,...,AM are weighting factors expressing the relative importance of the given stress tensors<J\...,<J M

. The second definition of Aijkl signifies a spatial average over a domain of interest for some

given stress field and the homogenized elasticity tensor EH definition is given in Eq. (3). The reader is referred to [11, 12] where a complete description of this model and its computational approximation is presented. — 103 —

2. Numerical Examples This first example considers two load cases corresponding to uniaxial equal stress fields in two perpendicular directions, respectively (see Fig. 3). The weighting factor X varies from zero to one. For A=0 we are at the single load situation cr22= 1 and for k - 1 we have a single load with ah =1. Table 1 presents the weighted energy values attained by the optimal microstructures and the respective optimal bound for different k values and /?=0.5 (50% volume fraction). The same results, comparing the energy value attained by optimal microstructures and the optimal analytical bound, are graphically shown in Fig. 4.

Figure 3: Load/stress cases Table 1: Weighted Energy Values Comparison X

Analytical Bound

Computational

Dif. (%)

o

1,000

1,000

0,0

0,1

1,300

1,369

5,3

0,2

1,407

1,479

5,1

0,3

1,465

1,530

4,5

0,4

1,490

1,579

6,0

0,5

1,500

1,584

5,6

0,6

1,490

1,579

6,0

0,7

1,460

1,530

4,8

0,8

1,408

1,479

5,1

0,9

1,301

1,369

5,3

1

1,000

1,000

0,0

In Fig. 5 some microstructure designs are shown for different /lvalues. Note that the single length scale design problem is non-convex with several local optimal solutions. Also there exist several optimal microstructures for the same volume fraction and load values. Fig. 5(c) illustrates this, showing an alternative design obtained with a different computational initial design point in the case of p=0.5, ^=0.5 (compare with Fig. 5(b)). Fig. 5(d) shows a 3D result. It is an optimal microstructure for a single case uniaxial load (for example with k=Q or 1). The second example considers nineteen load cases corresponding to equal shear stress fields each one rotated 6= 5° from the previous one (see Fig. 6(a)). The weighting factor is equal to all stress fields. The optimal microstructure is presented graphically in Figs. 6(b) and 6(c). Due to the high number of load cases with different orientation the optimal cellular material has an almost isotropic equivalent behavior as expected. The interested reader is referred to references [11-14] where this type of modeling and the respective computational approximation are discussed and demonstrated with several application examples. — 104 —

p-0.5 1.8 1.6 1.4 1.2 S? 1.0

£ °

■Bound •Comput.

8

0.6 0.4 0.2 0.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A, Figure 4: Graphical comparison of weighted energy values

(b)p=0.5,A=0.5

(a)p=0.5,A=03

(c)p=0.5,Z=0.5

(d) 3D microstructure for uniaxial load Figure 5: Optimal microstructures WWWrtHARTWWW

&&&•*:*•£;■£$•&•;:«*!

mmmmmm

*i«i©t©»©iQi©ij;*ii /m/m#i\Afwft/H\/m^f\ (a) Design Domain and loads

(b) Optimal Characteristic Cell

Figure 6: Design Domain and Optimal Material — 105 —

(c) Cellular Material

HIERARCHICAL MODEL FOR SIMULTANEOUS OPTIMIZATION OF MATERIAL AND STRUCTURE The hierarchical material design model can be illustrated based on Fig. 1. The design domain Q is defined for the macro level (global) where the goal is to find an optimal structure topology for given "load(s)" and "support" conditions. Conversely the design domain Y is defined for the micro level (local) where the aim is to find the optimal design of the material unit cell from which the structure is manufactured. Thus at each point x in i2, the hierarchical design model computes not only the volume fraction p of the cellular material but also the material characteristic cell geometry defined by the "micro" density y{x , y) function of y in Y. Thus a two scale material distribution problem (macro/global and micro/local) has been identified and a "density" field governs and defines each one, p(x) and y(x, y), respectively. The local problem solution is the topology for a single microstructure (material base cell) that is assumed to be periodically repeated inside a small neighborhood of each point jce/2. This periodicity is only local so the unit cell topology may vary with the macroscopic variable x although smooth transitions must be assumed. This feature is illustrated in Fig. 1 where two different unit cell designs are shown at different JC points in Q In the following we will describe two applications of this methodology. One in structural design with a compliance based objective and another in design for a thermal application. 1. Structural Compliance Optimization Assuming a multiple load measure (with weighing factors r) for structural compliance as objective function, the density fields p(x) and y(x, y), are solution of the optimization problem,

™n I X J*XdO+ J/XdT

with:

fy

=i

(9 )

Subjected to: jp(x)dQ
VxeQ

(10)

Q

p{x) = ^\y{x,y)AY,

VxeO

(11)

0
(12)

Where the displacement field u r solves the elastostatic problem, for the applied body and boundary forces pr and f, J £ ^ ( r K ( u ' M v r ) d i 2 - f z > ; v ; d £ - J r X d r = 0, a

a

Vv r admissible,r = l,...,M

(13)

r;

with E"u (r) given by Eqs. (2, 3). This optimization problem can be equivalently restated as (see e.g. [3, 15-16] for a full discussion of this aspect):

max

"l , mm - x J0(p,u\...,uM)dQ-^ar\

"

(, jpr-u'd£+ j Y u ' d s rkT

p(x) iTeC/ 0
0(p,u\...9up)=

max 0
Z«r[^W^(

u r

)^(

u r

)]

V

(14)

(15)

r_1

Y

The previous formulation clearly shows the global-local character of the design uncoupling it into a global problem for p(x) (14) and the local problem on y(y) (15), linked through constraint (11). — 106 —

Note that the local optimization problem (15) is simply a strain based formulation of the material optimization problem (8) presented in the previous section. Thus one is faced with a series of material optimization problems (one for each location "JC") where now the strain (stress) fields are not prescribed (as in Eq. (8)) but instead solution of the global equilibrium problem (13), i.e. induced by the respective boundary and body loads. Furthermore the uncoupling observed in the problem statement (14-15) is especially appropriate for the use of parallel processing techniques, where each of the local problems can be solved in a different processor, with important savings on computational time and efforts. A complete description of this hierarchical model and discussion of the respective computational implementation can be found in references [15, 16]. 2. Numerical Example This example considers a supported beam subjected to a multiple load condition as shown in Fig. 7. The problem was solved with 588 finite elements for the global problem while the local problem was solved with a 20x20x20 mesh. The volume constraint considered is 35% of the total volume of the structure. It is assumed a base material E°ijki linear and isotropic. The material properties are 210 GPa for Young's Modulus and 0.3 for Poisson ratio (see [16] for a complete description on the computational modeling for this example). The final global material distribution and some selected cellular material microstructures are presented in Fig. 8. In this figure, and the following ones, darker areas denote higher cellular material volume fraction p, lighter areas lower volume fraction values.

Figure 7: Hierarchical optimization-design domain and loads This type of hierarchical model as been studied also in the context of the remodeling behavior observed in natural cellular materials such as bone (see e.g. [17]). 3. Thermal Transient Problems A similar hierarchical modelling can be developed for thermal problems. Let us consider a structure occupying the domain Q (see Fig. 1) but now let us assume, not a cellular material but instead a composite material (see previous section "HOMOGENIZED PROPERTIES AND DESIGN PARAMETRIZATION"). The thermal transient problem is stated as,

qt(x,t) = K!Hx)

de_

8x:

(16)

dq (x

' ' 0 = (PC)H 0(x,t) dxi

in Q x (0, T]

with the initial and boundary conditions, 0(x,t) = 0(x,t)

on

qt (x, t) ni = q(x, t) q.(x9t)n.

on dQq x (0, T]

a

= h(0 (x,t)-O{x,t))

0(x,O) = 0o(x)

dQ0x(Oj] on

(17) 3Qhx(0j]

on Q — 107 —

Figure 8: Optimal solution In the previous problem statement, 6(x) is the temperature field, K" the homogenized (equivalent) conductivity tensor (5), (p C)H the homogenized heat capacity (7). The boundary and initial conditions are defined by; the prescribed temperature 0 on the dQ boundary, the prescribed heat flux q through dQq , natural or forced convection with the pair [//,^ a (jc,0] on dQ and initial temperature field 0o(x).

Vector n is the outward unit

normal to the domain boundary. Following the material description present earlier let us assume a locally periodic material whose characteristic cell is made of two base materials; one high-conducting (material 1) and the other low-conducting (material 2). The distribution of each material within the unit cell will be defined by the variable ^identifying the local volume fraction of material 1. In the limit in the unit cell regions where /equal to one, material 1 is present while in regions where ^equals zero one has material 2 (see interpolation formula (4)). The objective is to identify locally the composite material and its distribution that minimizes the difference between the temperature distribution 0{x, 7), at a prescribed time T, and a desired target distribution 0tg(x) i.e.

Mm\\(e{x,T)-ff>{xjfdX

(18)

such that,

p(*)=wMx'y)>dY 0
(19)

jp(x)dX
In this sense the present formulation is an extension of the one presented by S. Turteltaub [18] where a simpler law of mixtures was utilized. A complete derivation of the necessary conditions for the optimization problem (18-19), its separation into local and global problems and a discussion of the computational model developed and its implementation is presented in [19]. 4. Numerical Example The structural domain Q is defined as a 3 x 2 rectangular domain. The boundary and initial conditions are obtained from the following data: the initial temperature field is uniform 0° = 100, the flux is constant along part of the lower boundary for all t e(0,T], all the rest of the boundary has a convection condition (see Fig. 9). The objective is to have a uniform temperature 6tg = 75 °C in the domain, at time 7= 10. Note that for the problem conditions the target uniform temperature distribution is unattainable, so the objective is to get as close as possible in a L2 norm sense. The materials properties used are the following: For material 1 one has k\=l and (p\C\) =10, for material 2 the properties are k2=\0~2> k\ and (jhC2)=\ (see [18, 19] for further details). The volume upper bound is set equal to 83% and the initial design is uniform with initial p uniform and set to 0.83. The macro domain finite element model has a mesh with 60x40 four node isoparametric elements and the micro domain (unit cell) has a 30x30 mesh. In Fig. 10 the initial solution and its temperature distribution is shown. Fig. 11 depicts the "optimal" structure, as well as the material cell geometry at selected locations (macro elements), and the resultant temperature distribution. The difference between the target temperature and the initial structure and final structure temperatures is depicted in Figs. 12(a) and 12(b), respectively. Note here that at the optimal solution the "error" between actual temperature field and temperature target is concentrated close to the boundary as anticipated. Convection

[/? = 1,0' = 25 ]

DDDDDDDDDDDDDDD D □

□ □

D D

□ □

D D

□

D DDDD

D

wmi

□

DDDDDD

Figure 9: Thermal Problem

(a) Initial Design

(b) Temperature Distribution at Initial Design

Figure 10: Initial Design and Temperature Distribution — 109 —

j >^y^|^^g^|^1 *' *'' ^^^^^^^^^^^^^IMII . ' 65.588

69.303 67.445

(a) Final Design

73.01B 71.16

76.732 74.S75

wwiwi 80.447

76.59

(b) Temperature Distribution at Final Design

Figure 11: Final Design and Temperature Distribution

(a) Initial Design Ox, T)- 0tg (Fobj= 1637)

(b) Final Design Qx, T) - 0tg {Fobj = 197) Figure 12: Temperature Difference at Initial and Final Designs — 110 —

62.304

FINAL COMMENTS Some problems and the respective computational models were presented in the application of topology design techniques for the optimization of cellular/composite materials and the simultaneous optimization of material and structure. The proposed approaches for the simultaneous design of material and structure are based on the hierarchical design model concept, presented for elasto-static applications and thermal transient problems. The hierarchical model described, which uncouples the problem into local and global sub-problems, has several advantages. Formulating and solving separately the material optimization problem permits the easy introduction of material specific design constraints, such as manufacturing ones. Also even though computationally expensive the uncoupling in a local and global problem permits the use if parallel processing methodologies with significant gains in terms if computational cost and time (see e.g. [16, 20]). Lately it has been observed an even greater interest in this area of structural optimization with applications to design materials and structures for impact control, vibration and noise attenuation and interesting extensions to multidisciplinary engineering applications (see e.g. [4] and references therein). Acknowledgements The results presented in this communication derive from a very gratifying collaboration with Profs. M. P. Bendsoe (DTU, DK), S. Turteltaub (Univ. Delft, NL), Mr. P. Coelho (UNL, PT) and Profs. J. M. Guedes P.R. Fernandes, J. Folgado of IST-UTL. I would like to acknowledge these contributions and thank these researchers for the very interesting collaboration and discussions. I would like to acknowledge also the support given by Portuguese Science Foundation (FCT), through several programs, and by NATO-RTA to the research work reported here. REFERENCES 1. Bendsoe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenisation method. Comput. Methods Appl. Mech. Eng., 1988; 71: 197-224. 2. Allaire G. Shape Optimization by the Homogenization Method. Springer, New York, Berlin Heidelberg, 2002. 3. Bendsoe MP, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer, Berlin, Heidelberg, New York, 2003. 4. Bendsoe MP, Olhoff N, Sigmund O eds. Topological Design Optimization of Structures Machines and Materials, IUTAM Symposium. Springer, Dordrecht, 2006. 5. Bendsoe MP. Computational Challenges for multi-physics topology optimization, in Mota Soares CA, Martins JAC, Rodrigues HC, Ambrosio JAC eds. Computational Mechanics: Solids, Structures and Coupled Problems. Springer, 2006. 6. Lions JL. Some Methods in Mathematical Analysis of Systems and Their Control. Science Press, Beijing, China, Gordon Breach, NY, USA, 1981. 7. Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Springer, Berlin, Germany, 1980. 8. Guedes JM, Kikuchi N. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite elements methods. Comput. Meths. Appl. Mech. Engng., 1990; 83: 143-198. 9. Rozvany GIN, Zhou M, Birker T. Generalized shape optimization without homogenization. Struct Multidiscipl Optim., 1992; 4: 250-254. 10. Gibianski LV, Cherkaev, AV. Microstructures of composites of extremal rigidity and exact bounds on the associated energy density, in Cherkaev A, Kohn RU eds. Topics in Mathematical Modeling of Composite Materials, Birkhauser, New York, USA, 1997, pp. 273-317. 11. Guedes JM, Rodrigues H, Bendsoe MP. A material optimization model to approximate energy bounds for cellular materials under multiple load conditions. Structural and Multidisciplinary Optimization, 2003; 25: 446-452. 12. Bendsoe MP, Guedes JM, Neves MM, Rodrigues HC, Sigmund O. Aspects of the design of microstructures by computational Means. In Carbone L, Arcangelis R eds, Gakuto International Series Mathematical Sciences and Applications, vol. 18: Homogenization 2001, Gakkotosho, Tokyo, Japan, 2003. — Ill —

13. Sigmund O. Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. of Solids and Structures, 1994; 31(17): 2313-2329. 14. Diaz, AR, Benard A. Designing materials with prescribed elastic properties using polygonal cells. Int. J. Numer. Meth. Engng, 2003; 57: 301-314. 15. Rodrigues H, Guedes JM, Bendsoe MP. Hierarchical optimization of material and structure. Structural Optimization, 2002; 24(1): 1-10. 16. Coelho P, Fernandes PR, Guedes JM, Rodrigues HC. Hierarchical model for topology and material optimization of three-dimensional structures. Structural and Multidisciplinary Optimization, 2006 (submitted). 17. Rodrigues H, Jacobs C, Guedes JM, Bendsoe MP. Global and local material optimization models applied to anisotropic bone adaptation, in Pedersen P, Bendsoe MP eds. Synthesis in Bio Solid Mechanics, IUTAM Symposium, Kluwer, London, UK, 1999. 18. Turteltaub S. Optimal material properties for transient problems. Structural and Multidisciplinary Optimization, 2001; 22(2): 157-166. 19. Guedes JM, Lubrano E, Rodrigues HC, Turteltaub S. Hierarchical optimization of material and structure for transient thermal problems, in Bendsoe MP, Olhoff N, Sigmund O eds. Topological Design Optimization of Structures Machines and Materials, IUTAM Symposium. Springer, Dordrecht, 2006. 20. Coelho PG, Fernandes PR, Cardoso JB, Guedes JM, Rodrigues HC. A three-dimensional hierarchical model for topology optimization of structures, in Mota Soares CA et al eds. Ill European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, Lisbon, June 5-8, 2006. Book of Abstracts, Springer, 2006.

— 112 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Rigid Body Considerations for Geometric Nonlinear Analysis of Structures Based on the Updated Lagrangian Formulation Y. B. Yang *, S. P. Lin, C. S. Chen Department of Civil Engineering, Taiwan University, Taipei, 10617 Taiwan, China Email: [email protected] Abstract In the nonlinear analysis of elastic structures, the displacement increments generated at each incremental step can be divided into two parts as the rigid displacements and natural deformations. Based on the updated Lagrangian (UL) formulation, the geometric stiffness matrix [kg] is derived for a 3D rigid beam element from the virtual work equation using a rigid displacement field. Further, by treating the three-node triangular plate element (TPE) as the composition of three rigid beams lying along the three sides, the [kg] matrix for the TPE can be assembled from those of the rigid beams. The idea for the UL-type incremental-iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. The present approach is advantageous in that the formulation is simple, the expressions are explicit, and all kinds of actions are taken into account. The robustness of the procedure is demonstrated in the solution of several problems involving the postbuckling response. Key words: geometric nonlinearity, rigid element, rigid body, stiffness matrix, updated Lagrangian formulation INTRODUCTION The nonlinear analysis of elastic structures is usually conducted in an incremental-iterative way based on the three configurations: the initial deformed configuration Co, the last calculated configuration C\, and the current deformed configuration C2, as indicated in Fig. 1. In a step-by-step nonlinear analysis, we are interested in the behavior of the structure during the incremental step from C\ to C2. The deformations occurring within each incremental step are assumed to be small, but the displacements accumulated for all incremental steps can be very large. The concept to be presented herein is based on the updated Lagrangian (UL) formulation, in the sense that all quantities of the structure are expressed with reference to the last configuration C\.

Figure 1: Motion of body in 3D space The displacement increments generated at each incremental step of an elastic nonlinear analysis can be composed into two components as the rigid displacements and natural deformations [1,2]. For most structures encountered in practice, the rigid component constitutes a much larger portion of the displacement increments at each incremental step with respect to the deformational component. For a UL-type incremental-iterative analysis, the idea is that if the rigid rotation effects for elements with initial forces (or stresses) are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. — 113 —

Concerning the incremental-iterative procedure, distinction can be made between the predictor and corrector stages [3,4]. The predictor relates to solution of the displacement increments {U} of the structure for given load increments \P} based on the structural equation [A']{£/}= \P}. This stage determines the trial direction of iteration of the structure in the load-deflection space and thus affects the number of iterations or speed of convergence. For this reason, the stiffness matrix [K] used in the structural equation need not be exact, but should be rigid-body qualified to avoid convergence to incorrect directions. In the UL formulation, the corrector refers to recovery of the force increments {f} at C2 from the displacement increments {u} made available through the structural displacement increments {£/}, and the superimposition of these force increments with the initial nodal forces [Iff following the rigid body rule [5,6] to obtain the total element forces | 2 2 / } at C2. In this paper, a rigid-body qualified geometric stiffness matrix [kg] will be derived for the 3D beam element from the virtual work equation by assuming the displacement field to be of the rigid type. Such an element is referred to as the rigid element. To the knowledge of the authors, no similar elements were presented by others to explicitly accommodate the rigid behaviors of structures. For the 3D beam, the initial surface tractions may generate some moment terms upon 3D rotations during the incremental step from C\ to C2, commonly recognized as the moments induced by the semitangential torques and quasitangential bending moments [1,7]. Naturally, all such terms should be included in the virtual work formulation for the 3D beam. The other issue to be considered for the space frames is the equilibrium of angled joints in the rotated configuration C2, rather than in Q . Based on such a consideration, only the symmetric part of the geometric stiffness matrix of each element has to be retained in the structural stiffness matrix, as the antisymmetric parts of all the elements meeting at the same joint cancel out with each other [6,7]. As for the analysis of plate/shell problems, a triangular plate element (TPE) with three translational and three rotational degrees of freedom (DOFs) at each of the three tip nodes will be considered, for its compatibility with the 12-DOF beam element derived above. Since the rigid body behavior of each finite element is solely determined by its external shape or nodal DOFs, the geometric stiffness matrix for the TPE will be derived by treating it as the composition of three rigid beams lying along the three sides, which has the advantage of being explicitly given and capable of dealing with all kinds of in-plane and out-plane actions. For a review of related works on geometric nonlinear analysis of structure, Ref. [8] may be consulted, in which a total of 122 papers were cited. The purpose of this paper is not to review any related works. Rather, efforts will be focused on application of the rigid body concept and derivation of rigid-body qualified geometric stiffness matrices [kg] for the 3D beam element and TPE. The elastic stiffness matrices [ke] adopted are those readily available in the literature, namely, the elastic stiffness matrix [ke] adopted for the 3D beam element is the one commonly used [6,9], and the elastic stiffness matrix [ke] adopted for the TPE is constructed as the composition of Cook's plane hybrid element for membrane actions [10] and the hybrid stress model (HSM) of Batoz et al. for bending actions [11]. For the sake of brevity, the repetition of related derivations is kept to the minimum. \y

'UF

%, *>J

L

a

) b

^.~

'A/* x

,"F

Figure 2: Three-dimensional beam element THEORY OF THREE-DIMENSIONAL BEAMS Before we proceed to derive the rigid element for the 3D beam, a summary of the theory for the 3D beam with bisymmetric solid cross sections will be given first. The beam element considered has a total of 12 DOFs as shown in Fig. 2, in which x denotes the centroidal axis and (y, z) the two principal axes of the bisymmetric cross section. Based on the UL formulation, the virtual work equation for a 3D beam at C2, but with reference to C\, can be expressed in a linearized sense as [6]:

— 114 —

where E and G denote the elastic and shear modulus, respectively, Fis the volume of the element, and the factors of 4 and 2 are added to account for the symmetry of shear strains, i.e., , e^ =x e^, , exz =x e^, , rj^ =x rjyx, , rjxz =x rj^, ( l r x x , V , lTX2) are the initial (Cauchy) axial and shear stresses, and 8 denotes the variation of the quantity following. By allowing the surface tractions \tk and \tk to exist only at the two end sections, the external virtual works \R and at C\ and C2, respectively, can be written as \R=[^txdux+\tyduy+\t2duz)dS,

]R=[^txSux+\tyduy+\t2duz)dS

2

XR

(2a,b)

where S is the surface area of the element at C\. The linear and nonlinear components of the strain increments can be expressed with reference to C\ as :

_h dux ~i dy

dux ax9

x6xy 2

xrlxx=-

r du + 1 dx

du}

1 (duY

du7

dx

2\dz

dx

\2

y_

dx

i^ dx du z

Kdy ){ dx

l?lxz =

2\

(3)

' du V d u dz A dx

dx dz [dx J

{dz

du^ dx

(4)

du z dx

where (ux, uy, uz) are the displacements of a generic point N(y, z) at section x of the beam. Based on the Bernoulli-Euler hypothesis of plane sections remaining plane and normal to the centroidal axis after deformation, the displacements (WJC, uy, uz) of the generic point TV can be related to the displacements (w, v, w) of the centroid of the same section as ux = u-yv

-zw\

uy=v-z6x,

u2 =w + yOx

(5)

where 6X is the angle of twist. For the 3D beam, the displacement increments {u} are

{u}=(uo

va wa exa eya eza ub vb wb exb eyb ezb)

(6)

Correspondingly, the nodal forces \xf) and \xfj acting on the element at C\ and C2 are l

M„

{.'/}«<*-

l

M„.

{?/}-

'M„. Z

M„

V,. >M„.

l

F*

' xb 2

Fyb

2

F2b

M* 2

Mxb

l

M„,

(7)

X*

2

M2b

where the left subscript " 1 " is omitted from each term on the right hand side, i.e., {Fxa=\Fxa, xMxa=\Mxa, 2

2

(8) ...,

F„=?F,„, =\M , ..., etc. All the nodal forces and moments should be interpreted as the stress resultants, as xa 1 xa' Mxa xa 1 xaxa will be given below.

i



Figure 3: Coordinates of a generic point N at section x at: (a) d, (b) C2

— 115 —

1. Stress resultant definitions Let us cut a beam at section x and consider its left portion as in Fig. 3(a). Also, let us attach two sets of coordinates to the centroid C of the cross section. The first set is the rj - £ coordinates embedded at the cross section, with 77 and £ denoting the two principal axes. The second set has an origin fixed at centroid C, 2 2 2 referred to as the xx-y-z axes at d , with (ly, lz) coincident with (rj, £) (see Fig. 3(a)), and as the x- y- z 2 2 l l axes at C2, with ( y , z ) parallel to ( y and z) (see Fig. 3(b)). Based on the conditions of equilibrium, the forces acting on section x at Q , where k = 1,2, can be interpreted as stress resultants over the cross section of area A [6,7]:

*Fx = liS„dA,

k

Fy = [klSxydA, 'F^jJS^dA

k

Mx = [ ( f t , • ky - ^ ■ kz)dA, ? *A/, = [(ft,-*x-&.•*?)<«

k

(9a-c)

My = [ ( f t , ■ kz - ft • kx)dA **

(lOa-c)

where (*5'xx, J5 , ^S^) are the second Piola-Kirchhoff stresses acting at C* with reference to Ci. For the element at C\, these stresses reduce to the Cauchy stresses ( l r x x , V , lrxz), and the embedded coordinates ( 7 , ^ ) coincide with theC1)?, 1 z)axes, !

x = 0,

^^17=^

^ ^ ^ =2

(11)

Thus, the forces and moments in Eqs. (9) and (10) reduce to those commonly known for Cu e.g., the moments are ' ^ =i('^-V)^

l

My = llTazdA 'M^-l'r^ydA

(12)

For the element to deform from C\ to C2, the displacements (ux, uy, uz) of the generic point TV at C\ were given in Eq. (5). Consequently, the coordinates of point TV at C2 are (Fig. 3(b)) 2

x = -yOz+zOy,

2

2

y = y-z0x,

z = z + y0x

(13)

Meanwhile, the second Piola-Kirchhoff stresses can be expressed in an incremental form, \S,=\+St

(14) 2

2

2

for i = xx, xy, xz, where St are the stress increments. Since the forces ( FX, Fy, F2) at C2 are self-explanatory, we shall focus on derivation of the moments (2A4,2My, 2MZ) at C2. By substituting Eqs. (13) and (14) into Eq. (10), along with the stress resultant definitions in Eq. (12), the following can be obtained for the moments at C2: 2

MX =U\Sa-y-

X ■ z)dA,

2

My = J ft • zdA - lMz0x +1 *Mx9t 2

(15)

' ^ = " I' S ™ ' y d A + lMr&* ' 2 'Mx°y where products of displacement and stress increments are neglected and the factor 1/2 is adopted for bisymmetric solid cross sections. The expressions in Eq. (15) are derived based merely on the hypothesis of planar sections in Eq. (5) and the stress resultant definitions in Eq. (10). Evidently, the torque XMX should be interpreted as the semitangential moment, as indicated by the terms y?Mx0z and -)^lMx0y, and the bending moments lMy and lMz as the quasitangential moments, as indicated by the terms XMy0x and —MZ0X [1,6,7,12]. 2. Strain energy term With the linear strain components in Eq. (3) and the displacements in Eq. (5), one can obtain the strain energy 5U in variational form from thefirstintegral of Eq. (1) as follows [6]: SU= l(ElexxSlexx+4GlexySlexy+4GlexzSlexz)dV=

£(EAU'SU'+

EIywnSw"+ EiySv''+

GJ0'xS0x)dx

(16)

where A is the area, L is the length, ly and Iz are the moments of inertia, and J is the torsional constant. From the strain energy term SU, a 12x12 elastic matrix [kc] can be derived:

(17>

5U-=.W[*J«} — 116 —

as available elsewhere [6,9]. For the case of rigid displacements to be considered later on, the strain energy term SU vanishes. 3. Virtual work of initial stresses By substituting the nonlinear strain components in Eq. (4) into the second integral of Eq. (1), along with the displacements in Eq. (5) and the forces in Eq. (9) (for k = 1) and moments in Eq. (12), the virtual work SV of the initial stresses can be derived as [6]

SV=[(lTjlrlxx+2\Stfxy+2lTjxTlx2)dV

-H

,

FxS(u'2+v'2+w'2)+iFx

x

x

+ f [- MzS(w'0'x)+ £[lFyS(W6x

MyS{Vffx)-

-Sw"2+-±Sv" A A x

My8(u'w") +

- w ' v ' ) - lF2S(v'0x+u'W)]dx

{

FXI-2^±86'X2 A

+

dx (18)

x

MzS(uV)\dx -£[lMx8(v"w')-1MxS(w"v')]dx

+

where higher order terms have been neglected. 4. External virtual work terms By treating the surface tractions at C\ as Cauchy stresses, i.e., \tx=Ta,

\ty=T,

\tz=rX2, and by using the displacements in Eq. (5) and the forces in Eq. (9) (for k = 1) and moments in Eq. (12), the external virtual work ,'/? at C\, Eq. (2a), can be written as

\R = WM

(19)

where the displacements {«} and initial forces [If j were defined in Eqs. (6) and (7). Similarly, by treating the surface tractions at C2 as the second Piola-Kirchhoff stresses, i.e., ^t^S^,

\t =\S

, ^t^S^

, expressing the stresses as in

Eq. (14), and using the displacement field in Eq. (5) and the definitions for forces in Eq. (9) (for k = 2) and moments in Eq. (12), one can derive from Eq. (2b) the external virtual work \R at C2 as follows:

?* = W & / } H

1

MA ~lMx0z)soy 4-lMyox +±Mxey \sez

(20)

where {f/j was defined in Eq. (8). Clearly, the semi- and quasi-tangential properties of the torque lMx and bending moments lMy, lM2, respectively, are duly taken into account. GEOMETRIC STIFFNESS MATRIS FOR THE RIGID BEAM ELEMENT Now, we shall derive the geometric stiffness matrix for the 3D rigid beam. For an element subjected only to nodal loads, the bending moments can be related to those at the two ends as "A/, =-'M, a (l - X/Ly>Myb{x/L),

'M 2 =-'M 2 a (l - x/tyM^x/L)

(21a,b)

Further, based on the conditions of equilibrium, the following can be written, l

Fx = -lF„ = lFxb, % = -lFya = % = - i ( X , + XMzh)

X = -F*a = % = }("Mya + 'Myb),

X = -M„ = xMxb

(22a,b)

(23a,b)

For a rigid rotation, the axial and angular displacements can be expressed as u = {l-x/L)ua+(x/L)ub,

6x={l-x/L)0xa+{x/L)0xb

(24a,b)

subject to the constraints: u

a =ub>

(25a,b)

0xa=0xb 117

The lateral and rotational displacements can be expressed as v = (l - x/L)va + (x/L)vb, 0y ={l-x/L)Oya

w = (l- x/L)wa + {x/L)wb

(26a,b)

0Z ={\-x/L)0za+{xlL)92b

(27a,b)

+{x/L)0yb,

subject to the constraints: *„=4*.

em=6A

(28a,b)

For the rigid beam, the strain energy vanishes, i.e., SU = 0 . Consequently, the virtual work equation in Eq. (1), along with the aid of Eqs. (16), (18)-(20), reduces to the following: 2

H'F^ +v'W)^

+ J[ \-lMz8(w'ffx)-

l

My5(v'#x)-

L 2 -+Sw"2+^-<5v" A {

+ 'K x

MyS(u'w") +

L. + L -8ff} dx

MzS(uV)~\dx (29)

+ £[lFyS(w'0x

-u'v')-

]

FzS(v'0x +u'w')]dx + - £[]MxS(v"w')l

Mz0x -^Mx9Asey

+ ( ->My0x + ^MX0„

:

MxS(w"v')]dx \S0,

Since this is a problem with constraints, the following points must be considered in deriving each of the terms involved in Eq. (29) using the rigid displacement field: (a) The following variational terms are equal to zero: Su'2 = 0 , S0'x2 = 0, Sv"2 = 0 , dw"2 = 0 , Syu'w") = 0 , 5\u'v) = 0 , as a direct consequence of the rigid displacement field, (b) The variation of a quantity that is zero in the rigid displacement field need not be zero. For instance, u' = (ub-ua )/L = 0 according to Eq. (25a), but du' = (dub - dua )/L is not equal to zero since Sub * dua . (c) Even though some of the variational terms in Eq. (29) are known to be zero, such as u'Sv', u'Sw', 0'xdv', 0'xdw', they are kept in the derivation for the sake of symmetry of the related entries in the stiffness matrix, (d) The second derivative terms v" and wn should be treated as 0[ and - 0', respectively, and interpolated by Eqs. (27a,b) in order to capture the property of rotations. With the above considerations in mind, we may substitute the force expressions in Eq. (21-23) and the displacement fields in Eqs. (24), (26) and (27) into the virtual work equation for the rigid beam in Eq. (29). By taking into account the arbitrary nature of the variations, we can derive the stiffness equation for the rigid beam element as

<3°>

[ * . r {«}={?/}-{.•/} where the geometric stiffness matrix is

[g] [h.] - W [\] [K]T [L] -[»J [o] -Is) - W le] - W [K]T [o] - K f [k]

[*J

(31)

Here, 0] is a 3x3 null matrix containing all zero entries, a nd -lFyt/L

0

M= lh]=

x

l

- FyblL

Fxb/L

-lFzb/L 0

0 0

Myb

0 l

0

-xMzb x

-xFzbIL

l

- Mcb/2

FxblL

0

0

. w=

'MybIL l

Mb/L

x

- Mxb/2L 0

0 0 -lMxb/2L

0 l

(32a-c)

Mxbl 2 0

The submatrices [ha] and [ia] associated with end a can be obtained by replacing the terms ( ! M^, ]Myb, lMzb) in the submatrices [hb] and [ib] by the terms (lMxa, lMya, lMza), respectively. The matrix as presented in Eq. (31) is asymmetric, due to the asymmetry of the submatrices [ia] and [ib] originating from the boundary terms in Eq. (29). — 118 —

The geometric stiffness matrix derived in Eq. (31) is the same as that derived from the incrementally small-deformation theory [13]. It is qualified by the rigid body rule [5,6], in the sense that when an element with the initial nodal forces [Iff is subjected to a rigid rotation {u}n the total forces \xf\ obtained from Eq. (30) will have a magnitude equal to that of the initial nodal forces [Iff, but with the directions of the acting forces rotated by an angle equal to the rigid rotation. GEOMETRIC STIFFNESS MATRIX FOR THE TRIANGULAR PLATE ELEMENT (TPE) Let us consider the triangular plate element (TPE) shown in Fig. 4, which has 3 translational and 3 rotational DOFs at each of the three tip nodes [2]. This element is chosen for its compatibility with the 12-DOF beam element.

™i0

&

Oyl

■z' AJ

' YX.,Y,)

1 ~*x\ C(0,0)

\

x2,

c : CentriodI

(X„Y,)

% »'

Figure 4: Three-node triangular plate element Previously, the geometric stiffness matrix for a rigid beam element has been given in an explicit form, in terms of only the nodal forces and element length. It should be noted that as far as the rigid body behavior of an element is concerned, only the initial forces acting on the element and the external shape of the element need to be considered. The elastic properties that are essential to the deformation of the element, such as Young's modulus, Poisson's ratio, cross-sectional area and moments of inertia, can be totally ignored. Based on such an idea, the rigid behavior of the TPE can be simulated as if it is composed of three rigid beams lying along the three sides of the element, as in Fig. 5. It is in this sense that the geometric stiffness matrix for the rigid TPE will be derived.

Figure 5: TPE as the composition of three rigid beams

"Mil / " ' 3 ' F 3 1 'M31

3 . / r-?

,Mf

Figure 6: The forces and moments acting on each beam element — 119 —

Fig. 6 shows the nodal forces acting on each of the three beam elements, named as beam 12, beam 23 and beam 31. In this figure, ! F / and xMkij denote the nodal force and nodal moment, respectively, acting on beam if at the C\ configuration, with the right subscript k denotes the direction and the nodal point at which the force or moment is acting. In the following, we shall show how to determine the nodal forces acting on beam if: 1. Equations of equilibrium for each node of the TPE There are three forces and three moments acting at each node of the TPE. The following is the vector of forces {,*/ } for the element at the C\ configuration: ilf) = (lFxl 'F„ % 'A/,, %

' " „ ' ^ lFyi lFz2 'Mx2 >My2 'Mz2 'Fx3 % 'F2, <MxZ 'A/,3 >M,3)T (33)

Considering the equilibrium of the forces and moments acting at each node, we can write Model:

>F» + >F» = %,

"F» + 7 £ = ' F „ ,

' / £ ' + *F? = %,

"Ml + 'M- = *MA, ' < + 'M- = ' J / „ , ' < + M/» = >MZ1

(34a-0

' < 3 + 'Mil = >MX2, ' < ' + ' M - = >My2, 'A/- + 'MH = XM22

(35a-f)

Node3:>Fl> 'Mi:

+

>F» = 'Fxi,

'F»+ >F» = %,

' F z > ' F z ? ='Fz3,

+

>M?b = 'Mxi, *M» + *M$ = *Myi, 'A/i' + 'A/- = 'A/Z3

(36 a-f)

2. Equations of equilibrium of beam if All the forces and moments acting on beam if (with if = 12, 23 or 31) must satisfy the conditions of equilibrium for the beam, as given below: Equilibrium of forces: ^

+^ = 0 ,

l

F£ + lFZ=0

^ +^ = 0 ,

Equilibrium of moments: ( W + YJ ' F j ) + lMl + lMi = 0,

(37a-c)

-(x, xFi + Xj ' F j ) + ' < + XM% = 0

- f t ' F i + Yj ' F j ) + (X, ' F j + X, ' F ; ) + 'A/f, + 'A/* = 0

(38a-c)

for if = 12, 23 or 31, where (Xh Yj) are the coordinates of node /. 3. Equations of equilibrium for the triangular plate element The conditions of equilibrium must be obeyed by the TPE as a whole, as given below: Equilibrium of forces:

2 X = 0 , t'F M =0,

£'Fzt=0

(39a-c)

Equilibrium of moments: i{Yk>Flk + 'Mxk) = 0, k=l

±{-Xk'Flk

+

'Myk) = 0,

*=1

E ( - n ' ^ + Xk % + 'Mzk) = 0

(40a-c)

k=\

Now, we can solve Eqs. (34)-(40) to obtain the nodal forces for beam if as l

Fij =-lF + f xa

1

/j

xij

r

/rv - _ ! /rv xb~

r

xa->

l

FiJ =-F + f

J x '

ya !

~

7/7

F

Fij - - l Fij yb~

r

l

FiJ =-F + f

J y » l ij

ya>

zo

-5

zi/

^z '

£

F - -l Fij F

zb ~

(41 n-f) V*LdI)

za

and the nodal moments as 'Mf, = I ' ^ + I y / ^ + K - V , ) ,

l

l

M%

Ml ={'A/Z, - { f t % -Xtj%) + [m2 +(Yifx

=bM.j-±X0%+(my+XJz),

-X,fy)], — 120 —

ijLf'j =-lMu 1V1

xa

£

Ml=-lMl+Xs,

l

F,J

-Y

1V1

xb

F

ij

za 9

X

\rpij 1

za >

(42a-f)

Mi=^Ml^Fi-X^ l

l

l

where a variable with right subscript ij denotes the difference of two quantities, e.g., Mxij = Mxi- MXJ, Xij=Xi-Xj, and fx,fy,fz, &X, my and mz are the constants to be determined. One feature with the rigid element is that there are more unknowns than the equations that can be utilized. For instance, the total number of equations given in Eqs. (34)-(40) is 30, but the total number of unknown forces in Eqs. (41) and (42) is 36, with 12 for each of the three beam elements. This means that the force distribution of a rigid element is not unique, unlike that of an elastic element, and that six more conditions should be made in order to obtain a solution. In this paper, we simply let the six variables fx,fyifz, mx, my and mz equal to zero. All the nodal forces and moments solved and given in Eqs. (41) and (42) have been expressed with reference to the coordinates of the TPE, as indicated in Fig. 6. They have to be transformed into the local coordinates of each element in order to compute the geometric stiffness matrix [kg]beam of the element. Let I \fl}} denote the nodal forces of element ij in the global coordinates of the TPE and \lflJ\ in the local coordinates of element ij, {\f}

= (*Fl

{. , 7'}=( 1 ^

X

K

X

K

'Fl 'Pi

'Ml "Ml

*M% 'Ml l

M%

'Ml

'F| 'Fj

'F! l

F«,

'Fj l

n

'jl/» 'Ml

'M«b 'M$ 'M% Wjf

(43) (44)

The following is the transformation between the two sets of forces for element ij:

{lf'J}=[T]{lf'J}

(45)

where the transformation matrix [7] is

[T) =

\[R]

[0] [0] [0]

[0] [0] [0]

[R] [0] [0] [0] [R] [0] [0] [0] [R]

(46)

[*] = 0

0

with Ly denoting the length of element ij. By substituting \xfij ) into Eq. (31), the geometric stiffness matrix [kt-tbeamij for element ij can be obtained. Finally, by assembling the geometric stiffness matrices for the three elements following the standard finite element procedure, the geometric stiffness matrix [kg f

for the TPE can be obtained,

[kg]pE= iiT-rkh'-'M

(47)

iy=12,23,31

which has been explicitly given in Ref. [14]. It is easy to verify that the geometric stiffness matrix [kg f

derived above for the TPE can pass the rigid body test

[5,6], in the sense that the resulting forces \xf) computed from Eq. (30) for the TPE undergoing a rigid rotation maintain a magnitude equal to that of the initial forces {Iff, but with the directions of the acting forces rotated by an angle equal to the rigid rotation. Both the geometric stiffness matrix [kg]beam for the rigid beam, as given in Eq. (31), and the [kg]TPE matrix for the TPE [14], are asymmetric, as indicated by the submatrices [/*] and [/J, respectively, which relate to the behavior of nodal moments undergoing 3D rotations [6,7]. Such a property of asymmetry is restricted to the element level, but not the structure level, which adds little extra effort to nonlinear analysis. It has been demonstrated [6,7] that the anti-symmetric parts of the geometric stiffness matrices of all beam elements meeting at the same joint will cancel out, once the conditions of equilibrium for the joint in the rotated configuration C2 are enforced. As a result, the stiffness matrix assembled for the structure turns out to be symmetric. The same is also true for the plates and shells represented by the TPE, as was proved in [14]. PREDICTOR AND CORRECTOR FOR INCREMENTAL-ITERATIVE ANALYSIS Only incremental-iterative nonlinear analysis of the UL type is of concern in this study. The idea is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. In this regard, two major stages can be identified [3,4]. — 121 —

The first is the predictor or trial stage, which relates to solution of the structural displacements {U}, given the load increments {2P} — { ! P}, as indicated by the following equation:

< 48 >

[K]{V} = {2P}-M

where {lP} and {2P} denote the applied loads acting on the structure at C\ and C2, respectively. Once the structural displacement increments {U} are made available, the displacements increments {u} for each element can be computed accordingly. It is known that the predictor affects only the number of iterations or the speed of convergence [3,4]. Therefore, the structural stiffness matrix [K] is allowed to be approximate, but to the limit that the direction of iterations is not misguided. In addition to the elastic stiffness matrix, this will require the use of geometric stiffness matrices that are capable of simulating the rigid rotational behavior of the initially stressed elements, such as those presented herein for the 3D beam and TPE elements. For the 3D beam element, both the elastic stiffness matrix [ke] available in Refs. [6,9] and the geometric stiffness matrix [kg] in Eq. (31) will be used in constructing the structural stiffness matrix [K\. For the TPE, the elastic stiffness matrix [ke] is obtained as the composition of Cook's plane hybrid element for membrane actions [10] and the HSM element of Batoz et al. for bending actions [11]. The elastic stiffness matrices [ke] so obtained, plus the geometric stiffness matrices [kg]TPE given in the Appendix, will be used to assemble the structural stiffness matrix [K] for the plate/shell structures. The corrector stage is concerned with the recovery of the element forces { 2 /j at C2 with reference to C2. For incremental analysis of the UL type, two contributions need to be considered. The first is the initial nodal forces j \f\ existing at C\ and expressed with reference to C\. According to the rigid body rule, the initial nodal forces [Iff will be rotated by an angle equal to the rigid rotation (with no limit in magnitude) generated during the incremental step from C\ to C2, while their magnitude remains unchanged [5,6]. Thus, one can directly treat the initial nodal forces [Iff as the forces acting at C2 and expressed with respect to C2. So long as the rigid rotation effect has been taken into account, the force increments {2/} can be computed from the displacement increments {«} using the elastic stiffness matrix [ke] alone based on the small-deformation linearized theory, that is,

{2/}=fc]M

(49)

Summing the above two effects yields the total element forces {\f\

at C2 as

{22/} = {,'/} + {2/}

(50)

where the reference configuration is C2. As the structural and element displacements, i.e., {U} and {u}, are made available from the predictor stage in Eq. (48), one may update the nodal coordinates and element orientations to update the geometry of the structure at C2. Here, the nodal rotations at each incremental step are assumed to be so small that the law of commutativity applies. Then, by summing the element forces [2ff computed from Eq. (50) for each node of the structure at C2 and by comparing them with the total applied loads {2P}, the unbalanced forces for the structure at C2 can be obtained. Another iteration involving the predictor and corrector stages should be conducted to eliminate the unbalanced forces, should they be greater than preset tolerances. In a UL-type incremental-iterative nonlinear analysis, the accuracy of the solution is governed primarily by the corrector; while the predictor can only affect the speed of convergence or the number of iterations [3,6]. For the purpose of investigating the capacity of the geometric stiffness matrix derived in this paper, the following combinations of predictor and corrector will be tested in the analysis of the space frames: 1) Predictor: The structural stiffness matrix [K] in Eq. (48) is assembled using either eam

(PI) [ke J + [k \

beam

, where [kg]

of the following:

is the one derived in this paper for the rigid beam.

(P2) 1&J+ [kg\+ [&J, where \k \ is the (conventional) geometric stiffness matrix and [&J the induced moment matrix available on pp. 361-362 of Ref. [6]. (P3) [ke] matrix only. 2) Corrector: — 122 —

(CI) { 2 / } = [K]{u] as proposed in Eq. (49). (C2) \2f} = (IAJ + \kg J+ [^/]){w«}' where {w„} denotes the natural deformations of the element. As for the plates and shells, the TRIC element developed by Argyris et al. [15] will be used to yield solutions for comparison. The following are the combinations to be used: 1.

Predictor: The structural stiffness matrix [K] in Eq. (48) is assembled using either of the following: r

-\TPE

(PI) [ke\ r

|~

~\TPE

+\K \

-iTRJC

T

r

-iTPE

> where \ke\

is composed of the elastic stiffness matrices available in Refs. [10,11];

-\TRIC

(P2) \ke J

+1 kg J

, where both the stiffness matrices are given in Ref. [15].

(P3) \ke ]

matrix only.

2) Corrector: (CI) { 2 / } = [fce]

{w} , as proposed in Eq. (49) of this paper.

(C2){2f}=[ke]TR,C{u). NUMERICAL EXAMPLES The generalized displacement control (GDC) method [16] is adopted for tracing the nonlinear load-deflection response of the problems. With the aid of the generalized stiffness parameter (GSP), this method can automatically adjust the load increment sizes to reflect the variation in the stiffness of the structure, while reversing the direction of loading when approaching a limit point. It can easily deal with the limit points and snap-back points encountered in the postbuckling response. Since all the analyses are allowed to self adjust their load increments by the GDC method, so the total computation time spent by each scheme may not truly reflect the efficiency of the scheme. 1. Example 1 Fig. 7 shows a single beam restrained against the rotations about the x and y axes at the left end and against the rotations about the y and z axes at the other end. The following data are adopted: length L = 100 mm, cross-sectional area ,4 = 0.18 cm2, torsional constant J = 2.16 mm4, moments of inertia Iy= 0.54 mm4, Iz = 1,350 mm4, Young's modulus E = 71,240 N/mm2, and shear modulus G = 27,190 N/mm2. The beam is subjected to a moment of Mza at the left end, and a perturbation Mxb= 0.01 Mza at the right end. Ten elements are used for the beam. The theoretical critical moment for the beam under uniform moment is Mcr = ±7rJEIyGJ L = ±1,493.3 N-mm [6,7].

VYTT\

H""

/ \.

H'

2

Fig. 7 Single beam subjected to moment Mza The numerical results obtained are plotted in Fig. 8, along with the computation time of each analysis listed in Table 1. The peak load in Fig. 8 is 1463.0 N-mm. As can be seen, the proposed scheme PI CI can be used to yield solutions that are as accurate as the conventional one, though at the cost of longer computation time. It is interesting that the same problem can be solved using the elastic stiffness matrix [ke] alone in all stages of analysis, as indicated by P3C1 in Table 1. Of course, this is achieved at the cost of much longer computation time. Table 1 Running time (s) for analysis of Examples 1 and 2 Example

Combination P1C1

P2C2

P3C1

1

55.99

38.02

474.33

2

116.91

109.25

1081.13

— 123 —

0

20

40

m

80

100

120

-80

-60

-40

-20

?i)

0

10

60

80

Fig. 8 Moment-Displacement curves for single beam: (a) Uxa, (b) Uzb 2. Example 2 Fig. 9 shows an angled frame under uniform bending (i.e., with Mza = Mzc) and a shear load Fzb (N) of magnitude 5x10"5 Mza [1,6,7]. The length L is 240 mm and other properties are the same as in Example 1. Due to symmetry of the frame, only the left half modeled by 10 elements is analyzed. The member ab is restrained against rotations about the x, y axes and translations along the y, z axes. The translation along the x axis at node b is also restrained. The theoretical critical loads for the frame are: Mcr = ±nJEIyGJ L = ±622.2 N-mm.

h Fig. 9 Hinged angled frame in pure bending From the results plotted in Fig. 10, one observes that the proposed scheme PI CI can be reliably used to trace the postbuckling behavior of the frame, though at the expense of slightly longer computation time, compared with the conventional one (P2C2). Of interest is that using the [ke] matrix alone in both the predictor and corrector, as indicated by P3C1, can arrive exactly at the same solution, but with much longer computation time (see Table 1).

-20

0

20

-Jfl

M $0 m r^ u:v.rA

!20 !■«> 1A0 ISO 200

~~ o

" 60

120

!so

2'fo

xxs

.*«.)

Figure 10: Moment-displacement curves for hinged angled frame: (a) Uxa, (b) 9za 3. Example 3 The right-angled frame subjected to an in-plane load Fyb at the free end in Fig. 11(a) was studied [1], for which the critical load is Fybcr = 1.088 N. The same material and geometry data as those used in Example 2 are adopted. Each member of the frame is modeled by 10 beam elements. The entire frame was also modeled by 38 triangular elements as in Fig. 11(b). An imperfection load Fzb equal to one thousandth of Fyb is applied at the free end. From the results plotted in Fig. 12, it is seen that the proposed scheme PI CI using either the beam or TPE approach is capable of tracing the postbuckling response of the angled frame. Table 2 indicates that the time consumed by the present beam approach is slightly longer than the conventional approach. The same is also true for the TPE approach, compared with the TRIC approach. It should be noted that the number of iterations at each incremental step is about the same for both the TPE and TRIC approaches. However, less time is consumed by the TRIC approach in computing the stiffness matrices, since it considers only the in-plane actions, while the TPE considers all kinds of in-plane and out-of-plane actions. — 124 —

240 n

4'"

•!:

\MAAAAAM

240 mm

\\ h = 30 mm

255 mm

t = 0.6 mm

(b)

(a)

/////////

mr~

Figure 11: Right angled frame with fixed support: (a) beam elements, (b) plate elements

U

x * *<*P1C1 (present)

(a)

x P2C2

o

?o

m

H)

m

I ,/,

-P1C1 (present)

(b)

P2C2 (TRIQ

Figure 12: Load-displacement curves for angled frame: (a) beam approach, (b) plate approach Table 2 Running time for analysis of Example 3 Combination Approach P1C1

P2C2

P3C1

Beam

37.59

36.16

Divergent

Plate

1492.22

1211.25

Divergent

No convergent solution was obtained by the P3C1 scheme using only the [ke] matrix, due to the fact that the trial directions produced by the predictor based on the [ke] matrix alone deviates too much from the equilibrium path in the region near the bifurcation point. la

/ /la

R = 2540 mm a= 784.90 mm t = 99.45 mm E = 68.95 N/mm2

Figure 13: Hinged spherical shell subjected to a central load — 125 —

4. Example 4 Figure 13 shows a spherical shell subjected to the central point load P with all edges hinged and immovable [17]. The data adopted herein are: half of side length a = 784.9 mm, thickness t = 99.45 mm, Young's modulus radius of curvature R = 2,540 mm, E = 68.95 N/mm2, and Poisson's ratio v = 0.3. Because of symmetry, only one quarter of the shell is modeled as an 8x8 mesh and analyzed. 60

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— 126 —

The results obtained for the central deflection of the shell are plotted in Fig. 16, along with the computation time of each analysis in Table 3. Again, no distinction can be made between the results obtained by different approaches. The computation time consumed by the present TPE approach is slightly longer than the TRIC approach for the reasons stated in Example 3. The P3C1 approach with the [ke] matrix alone is able to trace the entire load-deflection response, though at the cost of extra computation time. Table 3 Running time (s) for analysis of Examples 4-6. Combination Example P1C1

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6. Example 6 This example is identical to Example 5 except that the thickness of the shell is halved to h = 6.35 mm [18]. The results obtained using an 8x8 mesh have been shown in Fig. 17, together with the computation time in Table 3. A comparison of Figs. 17 with 16 indicates that reducing the thickness of the shell by half has resulted not only in a drastic decline of the limit load capacity, but also in a shift of the response curve pattern to embrace a snap-back region. As can be seen from Table 3, slightly longer computation time is consumed by the present TPE approach, compared with the TRIC approach, for which the reason was given in Example 3. Again, this problem can be solved using exclusively the [ke] matrix in each stage of analysis, as indicated by P3C1, though much longer computation time is required.

0

5

10 15 20 25 30 Central deflection ( mm ) Fig. 17 Central deflection of hinged cylindrical shell with h = 6.35 mm CONCLUDING REMARKS The procedure presented in this paper is based on the idea that if the rigid rotation effects are fully taken into account at each stage of the incremental-iterative nonlinear analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. Based on the UL formulation, two stages are considered essential, i.e., the predictor stage for computing the structural displacements given the load increments, and the corrector stage for recovering the element forces. The former affects the number of iterations, but the latter determines the accuracy of the solution. As for use in the predictor, the geometric stiffness matrix [kg] is derived for a 3D rigid beam element from the virtual work equation by using a rigid displacement field, and the geometric stiffness matrix [kg] for the rigid triangular plate element (TPE) is assembled from those for the three rigid beams lying along the three sides of the element. One advantage with the present approach is that the geometric stiffness matrices [kg] for both the 3D rigid beam and rigid TPE are explicitly given and all kinds of in-plane and out-of-plane actions are taken into account. Using the rigid element concept, the effort of derivation is greatly reduced, as only rigid displacement fields are required. As for the corrector, the rigid body rule is used to update the initial nodal forces {\f\

existing at C\ with no limit on

the magnitude of rigid rotations. With this, the force increments generated at each incremental step are computed using only the elastic stiffness matrix [ke] based on the small-deformation linearized theory. — 127 —

The robustness of the proposed combination of predictor and corrector has been demonstrated in the solution of a number of frame and shell problems involving postbuckling responses. A slightly longer computation time is required for the TPE approach due to consideration of all kinds of actions, compared with the TRIC approach that considers only in-plane actions, based on the framework of the GDC method for adjustment of load increments. For problems with no abrupt change in the slope of the load-deflection curves, the entire postbuckling response can be traced by using only the elastic stiffness matrix [ke] in each stage of analysis, though at the cost of extra computation time. Acknowledgement The research works reported herein have been conducted as the result of a series of research projects granted by the National Science Council to the senior author, including NSC 81-0410-E002-05. REFERENCES 1. Argyris JH, Hilpert O, Malejannakis GA, Scharpf DW. On the geometrical stiffness matrix of a beam in space —a consistent v.w. approach. Comp. Meth. in Applied Mech. & Eng., 1979; 20: 105-131. 2. Yang YB, Chang JT, Yau J D. A simple nonlinear triangular plate element and strategies of computation for nonlinear analysis. Comp. Meth. Appl. Mech. Eng., 1999; 178: 307-321. 3. Yang YB, Leu LJ. Force recovery procedures in nonlinear analysis. Comp. & Struct, 1991; 41(6): 1255-1261. 4. Turkalj G, Brnic J, Prpic-Orsic J. ESA formulation for large displacement analysis of framed structures with elastic-plasticity. Comp. & Struct, 2004; 82: 2001-2013. 5. Yang YB, Chiou HT. Rigid body motion test for nonlinear analysis with beam elements. J. Eng. Mech., ASCE, 1987; 113(9): 1404-1419. 6. Yang YB, Kuo SR. Theory and Analysis of Nonlinear Framed Structures, Prentice Hall, Englewood Cliffs, N. J., 1994. 7. Yang YB, Kuo SR. Frame buckling analysis with full consideration of joint compatibilities. J. Struct. Eng., ASCE, 1992; 118: 871-889. 8. Yang YB, Yau JD, Leu LJ. Recent developments on geometrically nonlinear and postbuckling analysis of framed structures. Appl. Mech. Reviews, 2003; 56(4): 431-449. 9. McGuire W, Gallagher RH, Ziemian RD. Matrix Structural Analysis. 2nd ed., John Wiley, NY, USA, 2000. 10. Cook RD. A plate hybrid element with rotational d.o.f. and adjustable stiffness. Int. J. Numer. Meth. Eng., 1987; 24: 1499-1508. 11. Batoz JL, Bathe KJ, Ho LW. A study of three-node triangular plate bending elements. Int. J. Numer. Meth. Eng., 1980; 15: 1771-1812. 12. Ziegler H. Principles of Structural Stability. 2nd ed., Birkhauser, Stuttgart, Germany, 1977. 13. Yang YB, Kuo SR, Wu YS. Incrementally small-deformation theory for nonlinear analysis of structural frames. Eng. Struct, 2002; 24: 783-798. 14. Yang YB, Lin SP, Chen CS. Rigid body concept for geometric nonlinear analysis of 3D frames, plates and shells based on the updated Lagrangian formulation, submitted to Comp. Meth. in Appl. Mech. & Eng. for publication (2006). 15. Argyris J, Tenek L, Olofsson L. TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells. Comp. Meth. in Applied Mech. & Eng., 1997; 145: 11-85. 16. Yang YB, Shieh MS. Solution method for nonlinear problems with multiple critical points. AIAA J., 1990; 28(12): 2110-2116. 17. Horrigmoe G, Bergan PG. Numerical analysis of free-form shells by flat finite elements. Comp. Meth. in Applied Mech. & Eng., 1978; 16: 11-35. 18. Surana KS. Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements. Comp. & Struct, 1982; 15(5): 549-566.

— 128 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Nano-Modeling Structure and Micromechanical Properties of Mesoscopic Composite Systems Yu. G. Yanovsky * Department of Mechanics of Structured and Heterogeneous Continua, Institute of Applied Mechanics of Russian Academy of Sciences, Moscow, 119991 Russia Email: [email protected] Abstract We discuss the results of molecular modeling (quantum mechanics and molecular dynamics methods) of mesoscopic composite systems based on thermoplastic (polyethylene, polypropylene, etc.) and elastomeric (polybutadiene, polyisoprene, etc.) polymer matrices and active nanofillers (commercial carbon in different structural modifications, silicate, fullerens, nanotubes). Consideration is given to the structure, energy and micromechanical properties of model particles of commercial carbon, including those with surfaces terminated with various chemical compounds, and to the adsorption of various polymer chains on them. Shear strain and uniaxial tension in different adsorption complexes as well as molecular friction are calculated. The reinforcement effect, i.e. the change of most important physico-mechanical characteristics, and energy parameters are analyzed. We make important inferences about the influence of filler properties on its activity during interaction with polymer matrix particles. It is noticed that the best adhesion of polymer chain fragments (largest force of microscopic molecular friction) is observed for the isoprene elastomer — silicate filler system. Such nanoparticles as fulleiene and carbon nanotubes exhibit weak (almost equal) molecular adhesion, which makes it impossible to consider them as promising reinforcing fillers without proper modification. Carbon fillers are found to exert a different effect on the structure of thermoplastics and elastomers. In the first case, the structure becomes amorphous, while in the second one it stabilizes. This explains the known effect of elastomer reinforcement with carbon fillers. Thus, we have developed molecular simulation algorithms and methods, which employ parallel computing technologies on a supercomputer, to study the structure and micromechanical characteristics of large molecular systems, namely, clusters of representative elements of polymer composites. Based on the investigation by quantum mechanics and molecular dynamics methods, conclusions have been made concerning the effect of the filler nature on its interaction with polymer molecules comprising the polymer composite matrix. Key words: quantum mechanics, modeling, supercomputer, reinforcement INTRODUCTION It is known that the majority of computational methods used in the mechanics of composites are based on the continuum approximation when an object is considered as continuum. In particular, the stress-strain state is simulated by the finite element method with a step-by-step numerical integration, in which an integration step has no relation to the atomic-molecular structure of the material. To solve necessary partial equations, a number of parameters such as physical and empirical constants must be introduced. Reasoning from the nature of any matter it is obvious, however, that there is a natural threshold of its numerical division into minor constituents. The threshold is provided by atoms and molecules that make up this matter and carry information about its chemical properties. It is therefore reasonable to consider microscopic mechanical properties of composites taking into account their atomic-molecular structure. In the general case, it makes sense to investigate mechanical and strength characteristics of solids in the framework of a complex multilevel hierarchical approach [1]. The latter comprises interrelated and complementary micro- and macroscopic methods ranging from molecular modeling (quantum mechanics, molecular dynamics, etc.) to methods for describing effective properties of a medium at the macrolevel. In our opinion, molecular modeling in this hierarchy is a first and important step in describing the structure and mechanical properties of composites [2]. — 129 —

Below we consider some results of molecular modeling as the first level in the hierarchical approach. In our opinion, the results are of significant interest from the standpoint of explaining the formation mechanisms of the structure and mechanical properties of a composite at the microlevel. Particular attention will be given to such molecular modeling methods as quantum mechanics and molecular dynamics. SUBMISSIONQUANTUM-MECHANICAL SIMULATION OF COMPOSITE M E S O S C O P I C SYSTEMS The NDDO semiempirical method with the PM3 parameterization scheme was taken as the major quantum-mechanical method for direct modeling of the structure and mechanical properties of composites with regard to physico-chemical interactions on boundaries of their constituent components [3]. This method is realized in the framework of the original CLUSTER 1 program designed to calculate basic structural and energy characteristics of nanosized polyatomic clusters in the electron ground state as well as to calculate force fields of clusters, oscillation frequencies and infrared spectrum intensities. Phase boundaries of real nanocomposite materials were simulated by relatively large clusters (up to a few tens of nm and several hundreds of atoms) of different chemical composition and spatial structure. In this case, we first constructed molecular models of separate nanocomposite components and then a cluster to simulate an adsorption complex of composites. The calculated binding enthalpy of components characterized the bond strength in a real nanocomposite at the microscopic level. Notice that the quantum-mechanical computing experiment was performed in analogy with mechanical tests under active loading. It involved a consistent, step-by-step deformation of a molecular system from the stable initial state to bond rupture. On every deformation step the spatial structure of the molecular system was fully optimized. The relation between the microscopic structure of the phase boundary in a composite and characteristics of its stress state was studied in the framework of the approach of the microscopic coordinate of friction. The computing experiment consisted in a successive, step-by-step displacement of a particle from the initial state along the microscopic coordinate of friction. The spatial structure of atoms, which do not govern the microscopic coordinate of friction, was fully optimized on each step. The microscopic friction force was determined as energy gradient of the system with respect to the microscopic coordinate of friction. The data given below were obtained through parallel computing with the use of the NANOPACK package and supercomputer MBC-1000 in the interdepartmental supercomputer center of RAS [4]. 1. Internal microscopic friction in the adsorption complex "polymer - carbon filler surface" Comparison of energy and force characteristics of microscopic "intermolecular" friction of polymer molecules in a pure polymer matrix and friction of a polymer molecule on the filler particle surface in the adsorption complex at the boundary of interphase layers obtained in quantum-mechanical numerical experiment can provide useful information about chemical (microscopic) causes of reinforcement of organic (improvement of certain physico-chemical properties) polymers when they are filled with an ultradispersed carbon filler. It is experimentally known that the surface of commercial carbon presents a disturbed structure of the graphite crystal lattice. Defects in the lattice and thermal treatment lead to the fact that even on atomic scales almost no short-range order is observed. At further thermal treatment there appear graphitized carbon black. Taking into account the above-mentioned, we have constructed and calculated four possible models of various carbon surfaces in the quantum-mechanical approach: (1) in model CI the carbon surface )has a graphite, sp3 structure, (2) in model C2 carbon is both sp2- and sp3-hybridized, this surface has a defect-free periodic structure, (3) in model C3 carbon is solely sp2-hybridized like graphite, this model has defects in the form of various "holes" in the structure, and (4) in model C4 carbon is both sp2- and sp3-hybridized and has the structure with defects in the form of various "holes" and five-terms rings. To analyze the "activity" of the described carbon surfaces, we calculated adsorption of organic polymer molecules H-(CH2)„-H on them for n = 25. Fig. 1 illustrates adsorption enthalpies calculated for one monomeric unit -CH2- and energies of shear strain during motion of an organic polymer segment by lA along the surface, or microscopic forces of friction of the polymer on carbon surface. The obtained results allow us to conclude that the binding enthalpy of one monomeric unit -CH2- decreases in the series C1>C2>C3>C4, i.e. the most energetically unfavorable contact is registered for the system "polymer - graphite surface (CI)", while the most energetically favorable contact is observed for the case of polymer adsorption on the sp2/sp3-hybridized surface with hole-shaped and five-terms defects (C4). Diagrams of forces of friction for the analyzed surfaces substantiate the obtained result. Really, forces of friction is minimum in the case of "molecular friction" of a polymer chain segment along a strictly periodic graphite surface (CI) and maximum in the case of the most defect surface (C4) being almost 2.5 times higher. — 130 —

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Carbon pailicle .surface model Figure 1: Adsorption enthalpies (a) for an organic polymer segment H-(CH2)„-H, n = 25, and its energy of shear strain (b) at a 1 A shift calculated for one monomeric unit -CH2- for models CI, C2, C3 and C4 of carbon surfaces 2. Shear strains in the adsorption complex "polymer - carbon filler surface" The investigation of microscopic force characteristics and energies of shear strain depending on the shear value seems to be rather important in interpreting the reinforcement mechanism. These characteristics were studied for the adsorption complex "polymer (H-(CH2)„-H for n = 25) - carbon surfaces (C1-C4)". The results have shown that force of shear friction for the polymer - polymer system is about 2-3 times lower than for the polymer - filler (carbon surface) system. Moreover, this parameter increases in the series CI < C2 < C3 < C4. It is minimum for the case of "molecular friction" of a polymer chain segment along a strictly periodic graphite surface (CI) and peaks in the case of surface with the maximum number of defects (C4). Thus, we may state that the system "polymer - graphite surface (CI)" has an energetically unfavorable contact, whereas in the case of polymer adsorption on the sp2/sp3-hybridized surface the contact is energetically favorable. The latter system has hole-shaped and five-terms defects (C4). These data allow us to conclude that polymer chain fragments are better immobilized on the surface C4 and consequently carbon in the form C4 will exhibit better reinforcing properties as active filler. The above-discussed structures of carbon surfaces (models C1-C4) have unsaturated or broken valences that are arranged along the boundary of terminal aromatic rings, over the surface of hole defects or 6n surface sp3-hybridized carbon atoms. When nonterminated, these parts of the surface have rather high-energy, i.e. increased chemical and adsorption activity. They can be terminated in air atmosphere and during annealing of commercial carbon by hydrogen atoms or hydroxy Is. The chemical activity of terminated surfaces changes in this case, but they are still adsorption active. Similarly to the above-described method we calculated adsorption on the surface of each hydrogen-terminated model of the organic polymer segment H-(CH2)„-H for n = 25. When comparing the obtained results, the following conclusions can be made. The binding enthalpy of one monomeric unit -CH2- with hydrogen-terminated carbon surfaces decreases in the same series as for nonterminated carbon — 131 —

surfaces, namely, Cl-H > C2-H > C3-H > C4-H. The least energetically favorable contact is thus observed for the system "polymer - H-terminated graphite surface", while the most energetically favorable contact for the case of polymer adsorption on the hydrogen-terminated surface containing defects with sp2/sp3-hybridized carbon atoms and having hole-shaped and five-membered defects C4-H. Diagrams of forces of friction for the studied surfaces also substantiate the obtained conclusions. Really, force of shear microscopic friction is minimum for the case of "molecular friction" of a polymer chain segment along a strictly periodic graphite surface (Cl-H). The same result has been obtained for models both nonterminated and terminated by hydrogen. The energy of shear strain for a polymer segment along the hydrogen-terminated surface is also maximal for the case of the most defect surface (C4-H). Comparison of results obtained for nonterminated and terminated surfaces shows that force of microscopic friction along the terminated surface is about 0.9-0.7 of microscopic friction along the nonterminated surface. 3. Structure and adsorption properties of the complex "polymer - 3D carbon particle" Let us now turn to the structure and surface adsorption characteristics of 3D carbon particles imbedded into a medium of polymer molecules of different chemical structure. Commercial carbon we simulated by two 3D clusters with 170 and 670 atoms. The first particle was ~ 15 A in diameter and the second one ~ 25 A. The core of the particles had the structure of carbon in sp3-configuration, while their surface had both C-sp2 and C-sp3 structure, i.e. the surface contained both purely graphite and diamond-like structures arranged without short-range order and presented amorphous carbon. Consider results for adsorption complexes with the basic model of an amorphous carbon-black particle in the form of a carbon cluster ~ 15 A in diameter with 170 atoms. Adsorption of polymer molecules of different chemical nature was investigated, namely, polyethylene, polypropylene, polyacrylic acid, polyurethane, and polyvinyl acetate (oligomers CH3-(R)„-CH3, where n = 5). Adsorption of these molecules on the surface of model carbon-black particles with hydrogen-nonterminated and terminated surface was calculated; corresponding structures of complexes are given in Table 1. Table 1 Optimized models of adsorption complexes Polymer fragment

Adsoiption on nonterminared carbon black surface

Adsoiption on H-terminated carbon black surface

Polyethylene (PE)

Polypropylene (PP)

Polyacrylic acid (PAA)

Polyurethane (PU)

Polyvinyl acetate ' (PVA)

For analyzed adsorption complexes we calculated binding enthalpy and energy of shear strain during motion of a polymer fragment by 1 A along the carbon-black surface, which are referred to one monomelic unit (Figs. 2 and 3). — 132 —

Based on the above-given data, the following conclusions can be made. Adsorption energy and force of a polymer fragment on the amorphous carbon-black surface depends on the hydrophobic contact area (groups CH2 and CH3) as well as on the presence of electronegative atoms (O, N) in the polymer structure. Besides, conformation of a polymer fragment, namely, complementary character of its geometry on the carbon-black surface, also influences its adsorption. For example, adsorption enthalpy and energy of shear strain of the polypropylene monomer is ~ 1.5 times higher than that of polyethylene, which is due to a larger hydrophobic contact area of polypropylene. Adsorption enthalpy and energy of shear strain of the polyurethane monomer is ~ 1.7 times higher than that of polyethylene. At a relatively similar contact area, electronegative atoms in the polyurethane structure play a crucial role in this case. Polyvinyl acetate has the most favorable contact of a polymer fragment (~ 2.0 times higher than for polyethylene), which can be due to a simultaneous influence of two factors such as hydrophobic contact area and electronegative oxygen atoms.

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For polyacrylic acid having electronegative atoms, conformation of a polymer chain fragment adsorbed on the carbon-black surface is not optimal for contact interaction. This is demonstrated by a relatively low adsorption enthalpy and energy of shear strain of its monomer on the carbon-black surface. For a hydrogen-terminated surface, all regularities revealed above are qualitatively similar. However, adsorption enthalpy and energy of shear strain on such a surface are ~ 0.7 times lower. 4. Strain characteristics of the adsorption complex "polymer - 3D carbon particle". Uniaxial deformation and molecular friction Let us now consider strain characteristics of adsorption complexes that contain a cluster model of an amorphous carbon-black particle with a hydrogen-terminated surface ~ 15 A in diameter with 170 atoms and polymer molecules, such as polyethylene, polybutadiene and polyisoprene. Model oligomers were individual molecules CH3-(R)„-CH3, where n = 10. In the quantum-mechanical approximation we calculated energetically optimal structures of the mentioned adsorption complexes and their geometrical, energy and strain characteristics (Table 2). Table 2 also gives minimum distances between the polymer molecule and carbon black particle R, binding enthalpy AHbind calculated for one monomeric unit of polymer, maximum uniaxial tensile force (critical force of rupture) Fdeform,max of a polymer chain, and maximum microscopic friction force FShift,maxFor each model adsorption complex two microscopic strain characteristics were calculated: (1) the dependence energy of strain of a polymer chain contacting with a carbon-black particle on the elongation value (uniaxial tension of the polymer chain up to molecular bond rupture) and (2) force of microscopic friction (change in the adhesive force of the polymer chain and particle surface when a polymer molecule moves along its surface). Corresponding force curves are given in Fig. 4. One can see from the obtained data that stress-strain curves for polyethylene, polybutadiene and isoprene molecules have a common regularity from the qualitative standpoint. In the first curve portion the force of strain is almost constant and corresponds to low tensions; this is an entropy region of torsional and conformational changes of the polymer chain. In the second portion the curve changes abruptly; this is Hook's or enthalpy region where deformation (elongation) of valence bonds takes place. The maximum in the curve corresponds to the critical rupture force of the polymer chain, after which the analyzed curve drops sharply. From the qualitative point of view, it can be stated that in the series from polyethylene to polybutadiene and isoprene the entropy region of uniaxial deformation increases and the critical rupture force of the chain decreases. This bears witness to the fact that at the microscopic level elastic properties of polymer molecules improve in the considered series, and they are most pronounced for polyisoprene. This is known to correspond to experimentally observed facts. Table 2 Model adsorption complexes of polymer molecules with a carbon black particle and their geometrical, energy and strain characteristics calculated in quantum-mechanical simulation

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Complex

Optimized structure of the complex

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Elongation, A

Figure 4: Dependences of strain energy on elongation of a polymer chain contacting with a carbon black particle for polyethylene (a), polybutadiene (b) and polyisoprene (c) Analysis of forces of molecular friction has revealed that the best adhesion with the surface of a carbon-black particle is observed in the case of a branched, flexible isoprene molecule, while the worst adhesion for polyethylene. It can be therefore predicted that the reinforcing effect would be more pronounced in the combination of isoprene and carbon black, and less pronounced in the combination of polyethylene and carbon black. The conclusions made are in complete agreement with experiments. 5. Structure and adsorption properties of complexes "polymer - nanodispersed particles" We further analyze features of adsorption complexes based on isoprene and various cluster models of particles of ultradispersed fillers, such as commercial carbon (amorphous carbon black), ultradispersed amorphous silicate (white soot), fullerene particles (model C60), and carbon tubes (model C200). An amorphous carbon particle was modeled and optimized at the quantum-mechanical level. It included 170 atoms (~ 15 A in diameter) and its surface was partially hydrogen-terminated. An ultradispersed amorphous silicate particle contained 220 atoms with a Si-O-Si backbone, its size was ~ 22 x 14 and the surface was completely hydroxy 1-terminated. Fullerene was represented by a C60 particle ~ 7 A in diameter and having 60 carbon atoms. A C200 carbon tube particle contained 200 atoms and was ~ 8 A in length and diameter. The surface of the two latter particles was nonterminated and had an aromatic nature with sp3-hybridized carbon. In the quantum-mechanical simulation a fragment of a rubber chain was represented by the oligomer CH3-(CH2-C(CH3) = CH-CH2V-CH3, where n=\0. Table 3 gives structures of corresponding adsorption complexes optimized in the quantum-mechanical approximation and certain geometrical, mechanical and energy characteristics of these complexes, namely, minimum distance between the rubber oligomer and filler particle, binding enthalpy calculated for one monomelic unit, and maximum force of microscopic friction Fshift,max calculated for a 1 A shift of the polymer along the filler surface. The performed investigation allows the following conclusions. Binding enthalpy and force of shear friction are mainly determined by hydrophobic contacts between molecules and by the conformational similarity of interacting surfaces. The best adhesion of polymer (polyisoprene) molecule chain fragments to the filler surface (the highest force of microscopic molecular friction) is revealed for the polyisoprene - silicate complex, and a worse adhesion for polyisoprene - carbon black. Such fillers as fullerene and carbon tubes demonstrate very close values of even weaker molecular adhesion (for tubes the adhesion is somewhat stronger). MOLECULAR DYNAMICS SIMULATION OF MESOSCOPIC COMPOSITE SYSTEMS Molecular dynamics simulation is a promising method for describing virtually the microscopic structure, some energy properties and temporal behavior of mesoscopic polymer - filler complexes. Omitting the details of the used approach, notice that the calculations were performed in the framework of the CHARMM29 package for molecular dynamics simulations. Currently the package is the most developed, fast and theoretically substantiated molecular dynamics — 135 —

package [5]. It includes the Gborn continuum model (generalized Born model) [6]. The molecular dynamics calculation scheme of a molecular system involves the following steps: (1) molecule minimization, 100 steps, (2) heating of the system up to 300 K (3 ps at one step), (3) establishment of equilibrium, 50 ps, and (iiii) calculation of the molecular dynamics trajectory, 600 ps; while the trajectory is covered the molecular system relaxes. Coordinates are recorded in every 10 steps, and the temporal behavior of the system is analyzed by the obtained trajectories. When equilibrium is established at the end of the trajectory, geometrical and energy parameters of the system are obtained. In calculation experiments below we analyze the interaction of filler particles and a carbon black particle with partially hydrogen-terminated surface with the polymer matrix consisting of several layers of polymer molecules. The latter are polyethylene and polyisoprene. The polyethylene model represents two parallel layers of 10 individual polyethylene molecules in each, which are spaced at a distance from each other. Each molecule consists of 50 monomeric units. The entire model includes about 6 000 atoms. The initial structure of a polymer layer is assumed to be flat. The structure of the adsorption complex obtained in simulation is illustrated in Fig. 5. It is seen that the initial flat configuration of the polyethylene layers has been modified, i.e. the polyethylene component presents a complex conglomerate of amorphous mass with a carbon black particle embedded in it. The adsorption complex of a carbon particle and flat layer of polyisoprene molecules is depicted in Fig. 6. In this case, addition of a carbon black particle is seen not to change the structural configuration of polyisoprene that remains almost flat not enveloping the carbon particle.

Figure 5: Structure of the adsorption complex "carbon particle - two layers of polyethylene molecules" (molecular dynamics simulation data)

Figure 6: Structure of the adsorption complex "carbon particle - one layer of polyisoprene molecules" (molecular dynamics simulation data) Fig. 7 illustrates the adsorption complex of a carbon black particle and three polyisoprene layers (the carbon black molecule is put on the surface of the layers). Fig. 8 depicts the adsorption complex consisting of two layers of polyisoprene molecules (each layer comprises 9 chains of 50 monomeric units) spaced at a distance and parallel to — 136 —

each other with a carbon particle in between. The polyisoprene layers almost retain their flat, parallel structure in the presence of the carbon particle.

Figure 7: Structure of the adsorption complex "carbon particle - three interacting polyisoprene layers" (molecular dynamics simulation data)

Figure 8: Molecular dynamics structure of the adsorption complex where the carbon particle is put between two polyisoprene layers (molecular dynamics simulation data) CONCLUSIONS We have developed molecular simulation algorithms and methods, which employ parallel computing technologies on a supercomputer, to study the structure and micromechanical characteristics of large molecular systems, namely, clusters of representative elements of polymer composites. Based on the investigation by quantum mechanics and molecular dynamics methods, conclusions have been made concerning the effect of the filler nature on its interaction with polymer molecules comprising the polymer composite matrix. Particularly, it is found that the binding enthalpy and force of shear friction are mainly determined by hydrophobic contacts and by the conformational similarity of interacting surfaces. The best adhesion of polymer chain fragments with the filler surface (largest force of microscopic molecular friction) is revealed for the polyisoprene - silicate case. The coefficient is lower for the polyisoprene carbon black complex. Such fillers as fullerene and carbon tubes exhibit weaker and almost identical molecular adhesion. In the case of carbon tubes, however, the adhesion is somewhat better than for fullerenes. The molecular dynamics simulation allows revealing differences in the behavior of adsorption composite complexes on the basis of thermoplastics and elastomers. With carbon fillers introduced the thermoplastic matrix partially changes its initial ordered structure and becomes amorphous, i.e. molecules aggregate around filler particles. The elastomeric matrix stabilizes its initial structure with the introduction of carbon fillers. This is attested by both visual structural observations and energy calculations. Based on the given results we may state that computational molecular modeling is an efficient modern method for virtual analysis of fine structural, micromechanical and energy peculiarities and behavior under deformation of adsorption complexes of nano- and meso-sized composite media. — 137 —

Acknowledgements The work has been performed at the financial support of the Russian Foundation for Basic Research, Grants Nos. 05-01-00879, 05-01-08038 and 05-08-65529. REFERENCES 1. Yanovskii YuG, Zgaevskii VE. Hierarchical modeling of the mechanical behavior and properties of heterogeneous media. Phys. Mesomech, 2001; 4(3): 63-71. 2. Yanovskii YuG, Nikitina EA, Teplukhin AV, Basistov YuA, Filipenkov PA, Karnet YuN. New computer technologies for simulating structure and micromechanical properties of heterogeneous viscoelastic polymer media. Phys. Mesomech., 2003; 6(4): 129-142. 3.

Stewart JJP. Optimization of parameters for semiempirical methods. J. Comput. Chem., 1989; 10(2): 209-264.

4. Nikitina E. Computational modeling of surfaces and interfaces of nanoobjects. Mechanics of Composite Materials and Constructions, 2001; 7(3): 288-309. 5. Neria E, Fisher S, Karplus M. Simulation of activation free energies in molecular system. J. Chem. Phys., 1996; 105: 1902-1921. 6. Still WC, Tempczyk A, Hawley RC, Hendrickson T. Generalized Born equation. J. Am. Chem. Soc, 1990; 112: 6127-6129.

— 138 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESCX,Awg. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

An Extremely Efficient Finite-Element Model Updating Methodology with Applications to Damage Detection Ka-Veng Yuen* Department of Civil and Environmental Engineering Engineering, University of Macau, Macao, China Email: [email protected] Abstract: This paper presents a finite-element model updating methodology using noisy incomplete measurement of the natural frequencies and mode shapes with applications to damage detection. The proposed method does not require matching between the measured and calculated modes from the finite-element model, which is in contrast to most of the existing methods. Furthermore, the proposed iterative scheme is computationally efficient without nonlinear optimization programming. A ten-story building and a three-dimensional braced frame are used to demonstrate the proposed approach with applications to damage detection. Key words: damage detection, modal testing, model correction, model updating, structural health monitoring INTRODUCTION Damage detection/structural health monitoring has been attracting much attention in the past two decades, including several workshops, e.g., [1-3]; and special issues of journals, e.g., Journal of Engineering Mechanics (July 2000 and January 2004) and Computer-Aided Civil and Infrastructure Engineering (January 2001 and May 2006). Many methods have been developed including the class of direct methods using pattern recognition techniques [4-6] and the class of structural model-based inverse methods [7-11]. In the existing methods, they usually involve an optimization problem of a measure of the difference between the measured quantities, modal frequencies and mode shapes, and those calculated from the finite-element model. A generic form of this objective function can be written as follows: m

m

/=Xa,(to?-ci)?)2 + l P , l t - ^ 2 1=1

(1)

1=1

where <x>, and (j), are the measured natural frequency and mode shape of the ith mode; co, and (|)t are the model natural frequency and mode shape of the ith mode; and cc, and p, are the weighting, where i = 1,2,..., m. Note that one of the major difficulties is on the mode matching, that is, it is necessary to determine which model mode matches which measured mode. In the case when only measurements of partial mode shapes are available, this is not an easy task. Another major difficulty is that the m measured modes might not necessary be the lowest m modes in practice. Furthermore, in the case when there is possible damage in the structure, the order of the modes might switch, which increases the difficulty of this problem. The first attempt for solving this model updating problem without mode matching was made in [10] and [12] with the Bayesian probabilistic framework. This paper presents a very efficient finite-element model updating method using incomplete measured frequencies and mode shapes with applications to damage detection. The frequencies and mode shapes are measured. The proposed method does not require any mode matching between the model modes and measured modes. Furthermore, the pro­ posed method operates in an efficient iterative procedure instead of nonlinear optimization/searching. First, let's introduce the key idea of the proposed approach with a simple ideal case. Consider the eigenvalue problem of a structure with d degrees-of-freedom (DOFs): K^

= CG^MC^

(2)

— 139 —

where M and K are the d x d mass and stiffness matrix of the structure, respectively; coi and (^ are the natural frequency and mode shape of the first mode. The stiffness matrix K can be parameterized as follows: K = K0+Xe,K;

(3)

7=1

where 0 = [0i, 02,..., 0 n ] r is the uncertain/unknown parameter vector that governs the stiffness matrix and the matrices Ky, j = 0 , 1 , . . . , n, are known. Then, the eigenvalue problem becomes: [K1^1,...,K^1]0-(co?M-KoH1

(4)

Therefore, the stiffness parameter vector 0 is readily obtained: 0 = [K 1 ^ 1 ,...,K^ 1 ]- 1 (cofM-K 0 )(t) 1

(5)

However, complete mode shape measurements are not available in practice. Furthermore, the measurements are con­ taminated by measurement noise. In the next section, the formulation of the problem is described. Then, an expanded mode shape technique is introduced following with a method for fine tuning the noisy measurement of the natural fre­ quencies. Then, the stiffness parameter updating methodology will be introduced. Finally, the proposed approach will be presented which is an iterative process, that updates the mode shapes, natural frequencies and stiffness parameters alternately. Examples with a ten-story building and a three-dimensional braced frame will be used to demonstrate the proposed method with applications to damage detection. PROBLEM DESCRIPTION Assume that m modes are measured. Note that they are not necessary the first m modes. Consider the eigen-equation of the ith measured mode of a structure with d DOFs: ftfc = fltfMfc, i = l,2,...,m

(6)

where M and K are the d x d mass and stiffness matrix of the structure; and CD; and ()), are the natural frequency and mode shape of the ith measured mode. Use d>, and (j), to denote the measurements of the natural frequency and mode shape for the ith measured mode, respec­ tively. These measurements are contaminated by measurement noise, that is, there is a difference between the measured and the actual frequency and mode shape. Let 8 denote the difference operator. For example, 8co? is the difference, or measurement noise, between the actual and measured natural frequency of the ith measured mode:

5co? = ($-&}

(7) th

Similarly, the difference between the actual and measured mode shape of the i measured mode is given by Sfc-= ♦,-♦,■

(8)

In the case when a component of the mode shape is unmeasured, one can simply assign zero to its measured value. Then, all these random differences are grouped into an error vector: Z={h
(9)

For an unbiased estimate of the modal quantities and the stiffness parameters, this error vector has zero mean with covariance matrix: E[ezT]=Zz

^O)2

= Z

co 2 ,<j>

^C02,((!

(10)

^

which can be obtained by Bayesian modal identification methods [13-16]. Note that the error of the modal frequen­ cies/mode shapes of different modes might be correlated. For the unobserved components of the mode shapes, one can simply assign a large value for their variance without correlation with other modal quantities. — 140 —

By using a substructuring approach, the stiffness matrix K can be parameterized as follows: (11) where 6 = [61,62,.. , 6 n ] r is the unknown parameter vector that governs the stiffness matrix and the substructural stiffness matrices K,, j = 0,1,...,«, are known up to a scaling 6 7 . Then, in the similar fashion as in Eq. (7) and (8), the difference between the identified values and the nominal values 6 for the stiffness parameter vector is denoted as 1 = 6-9

(12)

Therefore, the difference between the actual and identified stiffness matrix depends on the difference between the actual and identified stiffness parameters 86 as follows: 5K = K - K = £ (6, - Qj)Kj = £ 86,K,

(13)

MODE SHAPES EXPANSION If sensors are available at s DOFs, the mode shape components can be measured at these DOFs only. In order to re­ cover the unmeasured components, many existing mode shape expansion techniques were proposed. A comprehensive summary and comparison among them was reported in [17]. Note that in the existing methods, the measured mode shape components are fixed at their observed values. Therefore, the expanded unobserved mode shape components are sensitive to the measurement noise for the observed components. In the following, a 'flexible' and computationally efficient method is presented. It does not completely constrain the observed values. Instead, the measured components will be probabilistically constrained by the measurements. In other words, the measured components can be different from their observed values and the allowable difference is controlled by the covariance of the measurement uncertainty. This is expected to improve the accuracy of the expanded mode shapes. Now, the mode shape <(), is updated using Eq. (6) but the stiffness parameters 6, and hence the stiffness matrix K, and the natural frequency co2 are uncertain. By using Eq. (7) and (13), Eq. (6) can be rewritten as (K - ctfM)^- = 8co2M
(14)

where YJ)=\ 86/K^- can be rewritten in terms of 86: X 8 6 ; K ^ . = [K1^,...,K^]86

(15)

Define the equation error eeq$ = Sco^Mc^ — [Ki^-,... ,Kn(|)t]86, which is the right hand side of Eq. (14). Therefore, the covariance matrix of the equation error X ^ is given by t-'eqti

^ ^ z F ^

+ F^XeF^

(16)

where the matrix Ze is the covariance matrix of the stiffness parameter vector by prior information. In the case when there is no prior information, large values can be assigned to the variances. The matrices F^) and F^2) are given by 0

M4>, M(|) 2

]

F( ):

(17) Iv%

0

F<2) =

-K,4», -K,m

dmxn

-K 2 4>, -K 2 4> 2

K„<|>2

- K

K„
2

^

(18) dmxn

141

Note that even though the eigen-equations of different modes are uncoupled, the equation errors are correlated. Then, the expanded mode shapes are obtained by minimizing the following objective function: -l T

(K-fflfM)^

(K-fl^M)fc

/*> =

5T

(K-OBJM)^, (K-W£M)<|> 2

1

>l-fl' r +

L(K-o£MHmJ

(K-fl&M)4>mJ

fc-$2

"♦i-§i :

V

2_^2

(19)

An-ftd

. ♦ m - m_ th

where <j>,-, i — l,...,m is the mode shape measurement of the i measured mode. For those unmeasured components, zero value can be assigned. 1^ is the covariance matrix of all the mode shapes. Minimizing this objective function is equivalent to minimizing the equation error with probabilistic constraint for the measured components of the mode shape. It allows the measured components to vary from the measured values but the variation will be constrained by a scale of its standard deviation of the error in the measurement. It can be easily shown that the updated mode shapes (recovering the unmeasured components and fine tuning the measured components) is given by

$2

-(F^:/3^^-1)-1!,-1

$2

(20)

where the (symmetric) matrix F^3^ is given by I©JM - K

0 a> 2 M-K

3

F< > =

(21) 0

fi&M

- K

dmxdm

FINE TUNING OF THE MEASURED NATURAL FREQUENCIES Note that the frequencies are measured with noises so it is possible to further improve the prediction by the following procedure. Now, the squared natural frequency co2 is updated using Eq. (6) but the stiffness parameters 0 (and so the stiffness matrix K) and the mode shape §t are uncertain. By using Eq. (8) and (13), Eq. (6) can be rewritten by neglecting the second order error: (K - co2M)$( = (co2M - &)&|>, - X M y M ,

' (22)

where £"_, 807K/<j), can be written in terms of 88: (23)

2 8 e ; K A = [K l $,,...,K n t.]8e

Define the equation error zeq ^ = (u)2M — K)8,- — [Ki <j>;,..., K„^,]80, which is the right hand side of Eq. (22). There­ fore, the covariance matrix of the equation error~Leqlsf.is given by 2 ^

= GOS (o2 G(» r + G(2)ZeG(2)r

(24)

where the matrices G ^ and G^2^ are given by co2M-K G =

co^M-K

(25) 2

co M-K

dmxdm

142 —

G<2> =

-Ki$! —Ki<j>2

-K2$! — K2$2

-K,$m

-K2$m

Knf, Kn$2

(26) dmxn

Then, the natural frequencies can be updated by minimizing the following objective function: '(K-0)fM)$ 1 " , r (K-cofM)^ Jtf =

■(R-cofM)$," (K-coJM)^

y-l

e^co2

Cof-Cbj" 0^-0)2

+

r

"cof-fi)?" ©1-^

*v 9

(K-co^M)^

^9

(27)

«m-«>mj

where d),, i = l,,..,m is the measurement of the natural frequency of the ith measured mode. Z^ is the covariance matrix of all the measured frequencies. Minimizing this objective function is equivalent to minimizing the equation error with probabilistic constraint for the measured frequencies. It allows them to be different from the measured values but the variation will be constrained by a scale of the standard deviation of the error in the measurement. By minimizing J^p. with respect the co2, the natural frequencies can be fine tuned as follows:

rv[V

3

= (G( >+Z- 2 ')-

2

( \

"Kr 2"

K$ 2

1

(28)

+ *3

2

eq,
P>m.

<&m\

3

where the matrix G' ' is given by fvr£-i

„.

„rv-i

vrE_I

et

V

G<3> = y

tn^eq,<&

Vl

^

0

)

^

2^,(o2

y

™

(29)

eq,®2^"1.

and the vector v,-, i = 1,2,..., m, is given by Vi = [Oix(/-iM,(M^) r ,0 lx(m _ jV ] 7

(30)

that is, v/ is a zero vector except that the sub-vector that includes the (i—\)d+\

to id elements is equal to Mfy.

UPDATING OF THE STRUCTURAL PARAMETERS Consider the eigen-equation for the ith measured mode in Eq. (6). Note that the natural frequencies and mode shapes are uncertain. By using Eq. (7), (8) and (11), Eq. (6) can be rewritten by neglecting the second order error: [K,^-,... ,K„^] 9 = (cb 2 M- Ko)$f. + 8 a M + (co 2 M- KJ&fc

(31)

Here, the updated values are used for the natural frequencies and mode shapes instead of the measurements. If these equations for all measured modes are grouped together, one can obtain the following equation: (32)

A9 = b + 6b where the matrix A is given by

"K,$, A=

K 2 $! •

Knjh

Ki$2

K2^2

'

K„2

K!$m

K2$m

•

Kn$m

(33)

— 143 —

the vector b is given by (c^M-Ko)^,' (G)5M-Ko)2

L(co?wM-Ko)$ff/

(34) dm x 1

and the equation error vector 8b is given by 5b = [H (1 \H (2) ]£

(35)

where 8 is the error vector given in Eq. (9) and the matrices H

( l}

and H

(2)

are given by

0

M(t), 3VId>?

H0) =

(36)

mm cofM-K H (2)

(KM

_

-

K

(37) dm -dm

Therefore, the covariance matrix of the equation error is given by I s / , = [H ( ".H< 2, ]I fc .[H (1 ».H ,2, ] r

(38)

where the matrix I,, is the covariance matrix of e given in Eq. (10). It is shown in the Appendix that the optimal estimate of the stiffness parameter vector 0 is given by e=(A r X 6 / ,'A)

'A r I W ) 'b

(39)

where the matrix Z§£ is the same as I5/,, except that it replaces the zero eigenvalues of Its/?by a small value. The matrix A X&, A is invertible if enough constraints (number of sensors and measured modes) are available. Otherwise, more sensors or measured modes are needed. Note that the least squares solution is a special case of Eq. (39) under the condition when X^ = a 2 1: e=(A r A)~ 1 A 7 "b

(40)

Uncertainty Estimation The uncertainty of the estimation can be quantified by its covariance matrix which is given by U =

(xTiSblAyl

(41)

The diagonal elements give an estimation of the variance for each stiffness parameter.

PROPOSED ITERATIVE APPROACH The proposed methodology updates the natural frequencies, mode shapes and stiffness parameters in an alternating manner with the aforementioned methods. It is recommended to operate in the following order: 1) Assume an initial trial vector for 9 and calculate the stiffness matrix K with 8. 2) Expand and fine tune the mode shapes ,, / = 1, 2, . . . , w, using Eq. (20). Note that this mode shape expansion technique can be used for reducing measurement noise even when complete measurements of mode shapes are available. 3) Fine tune the squared natural frequencies ci)*-, / = 1, 2 , . . . , m, using Eq. (28). 4) Update the estimation of the stiffness parameter vector 9 by using Eq. (39). In the case if a component becomes negative, replace it by the value in the last iteration. — 144 —

5) Iterate the previous steps 2-4 until the stiffness parameter vector 9 converges. 6) Calculate the covariance matrix of the estimates of the stiffness parameters using Eq. (41). It is recommended to skip step 3 in the first few, say 10, iterations in order to have faster convergence. ILLUSTRATIVE EXAMPLES 1. Example 1: Ten-story Building In this example, a ten-story building is considered. It is assumed that this building has uniformly distributed floor mass and stiffness across all floors. The mass per floor is taken to be 100 tons, while the interstory stiffness is chosen to be kQ= 176.729 MN/m so that the first five modal frequencies are 1.00,2.98,4.89,6.69 and 8.34 Hz. The coefficient of variation of the measurement error of the squared natural frequencies and mode shapes are taken to be 1.0% for all modes. The initial trials of the stiffness parameters are taken to be uniformly distributed from 2#o to 3#o, where k0 is the actual interstory stiffness. In this case, the initial trial is significantly overestimated and the variation between different interstory stiffness is substantial. Table 1 shows the initial trial values and identified values of the stiffness parameters using different number of measured modes, where the actual value is 176.729 MN/m for all these parameters. The estimated standard deviations, calculated using Eq. (41), are shown in parenthesis. In these cases, complete measurements at all DOFs are utilized. It is not surprising that the uncertainty reduces when the number of measured modes increases. Table 1 Identification results using different number of measured modes (Example 1) Parameter 9l

°2

1 e3 e4 e. e6 e7 e8 e9

Oio

Initial trial 359.29 435.09 489.67 400.89 456.37 518.83 425.93 415.03 379.62 407.28

2 modes 177.01 (2.95) 175.59(4.73) 178.42(6.69) 187.88(11.5) 182.44(7.28) 173.69(4.70) 179.92(4.22) 171.88(3.85) 174.06(4.76) 183.02(6.71)

3 modes 173.96(1.81) 181.11 (3.56) 179.42(4.54) 177.44(2.83) 178.77(2.52) 177.22(2.63) 176.81 (3.45) 173.81 (2.17) 179.14(2.14) 174.05(2.09)

4 modes 174.05(1.45) 182.25(3.36) 175.66(1.83) 175.81 (1.62) 177.37(2.33) 176.66(1.49) 175.62(1.62) 175.74(1.81) 175.86(1.27) 174.93(1.21)

5 modes 174.53(1.27) 178.99(2.08) 176.06(1.25) 176.28(1.51) 177.19(1.32) 176.79(1.16) 175.42(1.17) 176.03(1.12) 176.16(1.04)

175.93(0.95) 1

Incomplete mode shape measurements: Consider only five sensors on the first, fourth, fifth, seventh and top floor. Table 2 shows the initial trial values, identified values, standard deviation and the coefficient of variation (C.O.V.) of the stiffness parameters. Fig. 1 shows the iterative history of the identification process. It converges with 50 iterations. The difference in 8, is smaller than l.OxlO-6 compared to those estimated after 1000 iterations. The CPU time is about one second with a normal personal computer with 2.4 GHz under the MATLAB environment [18]. It is seen that the iteration virtually converges after ten iterations. Table 2 Identification results with measurements of five sensors and five modes (Example 1) Parameter 01

02

e3 e4 85

e6 e7 e8 e9 Oio

Initial trial 359.29 435.09 489.67 400.89 456.37 518.83 425.93 415.03 379.62 407.28

Identified values 0 175.54 178.52 174.04 173.92 178.62 178.17 175.97 176.19 175.33 176.41

— 145 —

Standard deviation Ge 1.84 2.61 2.09 2.09 2.03 1.47 1.60 1.68 1.56 1.26

C.O.V. 0.010 0.015 0.012 0.012 0.011 0.008 0.009 0.009 0.009 0.007

600

|

l! !

l

fi \ I! I

i

jl ■

i

I

I

I

1

I

1

1

j| | ; ; j

550^ r

M■ M I

450 ■ - ■ ■ ■ ,

400

fi

i

;

~

I!

3501 300

~

250

\ i/\l

I'M-

-

200 L i / .

;' i , ,f

s^f-J™**"'*-"■-■"

J

—

■ ■ ■ - ' - - • ' ' - ■ ' ■ - " '•■•A,...™

....-,....

..-...■■.■...■...»..

150

1A ;W'' ;/ 100

It/

lV

v

' 1

1

10

1

15

1

I '

1

20

25

30

35

1

40

1

45

50

Number of iterations

Figure 1: Iteration history for the stiffness parameters with incomplete measurements of mode shapes (Example 1) 2. Example 2: Three-dimensional braced frame The proposed method is applied to update the finite-element model of a three-dimensional five-story braced frame. It has a square section with width a = 4 m. There are four columns for each floor, one at each corner. Each of them have interstory stiffness 10 MN/m and 15 MN/m in the x andy direction, respectively. Furthermore, each face in each floor is stiffened by a brace and its stiffness is taken to be 20 MN/m. As a result, the interstory stiffness is 80 MN/m and 100 MN/m in the x andy direction, respectively. The floor mass is taken to be 10 tons for each floor. As a result, the first five natural frequencies of the structure are 4.05,4.53, 7.44, 11.83 and 13.22 Hz.

(+y) (1)

L±ii_

h+*

^U

(-x)

i (2)

o* - p ^

^r>

Oi

(+x)

**.-» (3)

(-y) Figure 2: Floor plan for the 12-DOF model for identification (Example 2)

— 146 —

In order to locate the face(s) that sustain damage, four stiffness parameters are used for each story to give twenty stiffness parameters, ©4(/-i)+i = ^/.+.v, &4(i~i)+2 ~ ^/,+V' ^4(/--i)+3 ~ ty.~x a n d ®4(/-i)+4 — £/,->■•> / = 1, 2, . . . , 5, where the index / represents the story number and '+JC', '+y\ 6-x9 and '->>' represent the direction of the outward normal of a face. The actual values of these stiffness are A/.+.v = A/.-* = 50 MN/m and A/.+v = A7. _ v - 40 MN/m, / = 1, 2 , . . . , 5. In other words, 61 = 63 = • • • = 819 — 50 MN/m and 62 = O4 = • • • — 02o = 40 MN/m. The floor plan is shown in Fig. 2. The point Oj (xj, yj) is the stiffness center of story /, where xj and >7 , / = 1,2,. . . , 5, are given by a(ki+x-ki,

-x)

2(kx+x + klf.xy

yi ■

'"

«(A7, + v-/v7,-v)

(42)

2(*,. + r + */._,..)

The stiffness matrix for story / with respect to the O'j coordinates is iL+Y + ki,~Y 0 0

K; =

£/,+v - ^/.-i 0 0

0 ^/.+.r + ^7,-.r 0 0 -A7. + . v - kj^x 0

0 0 kit 0 0

-kn

-A7.+V - A7,-.i 0 0 A7i+v-hA7.-v 0 0

xifk,.+x + ( - -yi)2k,.+y + ( - +x,)2ki.~x + ( -

kit = (--

0 —ki,+x-ki.„x 0 0

fc/,+.r + fy-.r 0

+y,)2k,...y

0 0 -kn 0 0

(43)

ku (44)

Here, the first three DOFs and the last three DOFs correspond to the lower floor and the upper floor of the story, respectively. Each of these sets of three DOFs correspond to the x-translational, y-translational and torsional motion. The stiffness matrix for story / with respect to the (1), (2) snd (3) DOFs in Fig. 2 of the upper and lower floor is given by K/ = T r KjT

(45)

where T is given by 1

T

T =

0

0 T

1

1 0 - S + y/ T = 0 1 -$-xi 1 0 2 + v/

(46)

The stiffness matrix for the 12-DOF structural model is assembled from those of the floors. The DOFs for this stiffness matrix are the (1), (2) and (3) shown in Fig. 2 for each floor. However, this stiffness matrix is not linear to the stiffness parameters 9 ; . In this case, one can linearize the relationship between the stiffness matrix and the stiffness parameter: 20

K=Ko+XKA-

(47)

7=1

where

.20

(48)

and K0 = K - 2^ K7-0y

(49)

where the matrix Ko will need to be updated in every iteration. To increase the difficulty, it is assumed that only first three x-directional and ^-directional modes are measured but not any of the torsional modes. This is done deliberately to simulate the reality that some of the modes might not be excited so they are not able to be measured. In the identification, it is assumed that we do not know there are missing modes. These measured modes correspond to the 1 st (4.05 Hz), 2nd (4.53 Hz), 4th (11.83 Hz), 5th (13.22 Hz), 7th (20.84 Hz) and 9th (23.95 Hz) mode. Sensors are placed at the +y, -x and -y face of the 1st, 3rd and 4th floor to measure the — 147 —

natural frequencies and mode shapes. The coefficient of variation of the modal data is taken to be 0.5%. Initial trials are taken to be 100 MN/m, which overestimates by 100% and 150% for the ±x and ±y faces, respectively. The iteration history is shown in Fig. 3 and it virtually converges after 20 iterations. It took about 3.1 seconds after 50 iterations. These stiffness parameters converges into two branches, one for the ±x faces (approximately 50 MN/m) and the other for the ±y faces (approximately 40 MN/m). Table 3 shows the actual values, identified values, standard deviations and coefficients of variation (C.O.V.) of the stiffness parameters. The standard deviations are computed using Eq. (41) and they are up to about 1%. The estimates are all close to the actual values. The difference between the actual and identified values are of similar order to the corresponding calculated standard deviations. 100

20

25

30

50

Number of iterations

Figure 3: Iteration history for the stiffness parameters of the undamaged structure (Example 2) Application to damage detection: The structure is assumed to be damaged on the +y face of the first story and the +x face of the third story. One third and a quarter of the stiffness reduction of the corresponding brace is imposed to these two faces, respectively. It corresponds to 16.67% and 10% stiffness reduction of these faces, respectively. It is noted that there is no sensor at the +JC face. The first five natural frequencies of the damaged structures are 3.98, 4.50, 7.37, 11.67 and 13.19 Hz. Note that these damages alter the order of the modes. Furthermore, the translational and torsional modes are mixed, especially for the high modes. In this case, the first six translational modes become the 1st (3.98 Hz), 2nd(4.50 Hz), 4th(l 1.67 Hz), 5th(13.19 Hz), 6th(18.47 Hz), and 7th(20.67 Hz) mode. Initial trials are again taken to be 100 MN/m. Independent modal data is used to identify this damaged structure. The iteration history is shown in Fig. 4. It took about 3.3 seconds for 50 iterations. These stiffness parameters converges into four branches, one for the +y face of the first story, one for the other ±y faces, one for the +JC face of the third story and the last one for the other ±JC faces. Identification results are shown in Table 4, which is similar to the one in Table 3. It is clearly seen that the +y face of the first story and the +x face of the third story has substantial stiffness reduction compared to the undamaged structure. In order to visualize the damages, the means and standard deviations for the stiffness parameters are used to find the probability that a given stiffness parameter 9/has been reduced by certain fraction d, compared to the undamaged state of the structure. An asymptotic Gaussian approximation [19] is used for the integrals involved to give:

,

p}™(d) = P{&f < (1 -d)Vf)^ 4> ^'

(l-d)ef-ef l

J

-

^ ( l - d ) 2 ( < ) 2 + ( <^ )2 — 148 —

)

(50)

Table 3 Identification results of the undamaged structure (Example 2) Parameter | Actual values 8 | Identified values 6 | Standard deviation CQ 1 0.34 50 50.426 ei.+v 40 39.901 0.20 ei.+v 50 49.924 0.26 1 6l.-.v 40 39.677 0.20 e».-v 50 49.466 0.40 &2.+.Y 40 39.436 0.22 e2.+v 50 49.799 0.36 e2.-.v e 2 ._ v 40 40.199 0.22 50 50.018 0.34 63.+.T 40 40.030 0.43 1 ®3,+v 50 50.404 0.26 83,-x 40 40.088 0.43 e 3 ,-v 50 49.749 0.33 | fy.+T 40 40.202 0.24 e 4 .+ v 50 50.025 0.29 1 ®4,-x 40 39.481 0.23 fy.-v 50 50.361 0.16 fy>,+.v 40 39.805 0.17 05.4-y 50 49.839 0.14 85. - r 40 39.759 0.18 ! e*-, 1

| C.O.V. 0.0069 0.0050 0.0052 0.0051 ! 0.0080 | 0.0055 0.0072 0.0055 0.0068 0.0108 0.0053 0.0109 0.0067 0.0060 0.0058 0.0058 0.0033 0.0044 0.0029 0.0046 1

100

90 I

1

0

5

10

15

20

25

30

35

40

45

50

Number of iterations

Figure 4: Iteration history for the stiffness parameters of the damaged structure (Example 2) where 0( •) is the standard Gaussian cumulative distribution function; B'f and &li denote the most probable values of the stiffness parameters for the undamaged and (possibly) damaged structure, respectively; and &f and tf are the corresponding standard deviations of the stiffness parameters. — 149 —

Table 4 Identification results of the damaged structure (Example 2) Parameter 6i. + . v ©l.+v 6i._, 6,,_ v

e2.+,. 62,+v

e2._.v e2._,, 63.+,-

e3.+v

63._,. 63.-.V ^4,+.Y

e 4 . +v ®4,-x

e 4 .- v e5.+.v ^5,+v

e5.-v 05.-,

Actual values 0 50 33.33 50 40 50 40 50 40 45 40 50 40 50 40 50 40 50 40 50 40

Identified values 0 50.246 33.465 50.030 40.400 50.415 38.826 50.486 40.160 45.258 39.946 50.379 39.423 49.861 40.147 50.179 39.669 50.334 39.686 50.254 40.106

Standard deviation Ge 0.33 0.19 0.26 0.21 0.45 0.36 0.46 0.39 0.35 0.27 0.26 0.25 0.34 0.24 0.28 0.24 0.19 0.17 0.17 0.17

C.O.V. 0.0066 0.0057 0.0052 0.0053 0.0090 0.0090 0.0092 0.0099 0.0079 0.0069 0.0053 0.0064 0.0068 0.0060 0.0057 0.0060 0.0038 0.0043 0.0035 0.0044

M,+y)

:3,+x)

-0.05

0.1

0.15

0.2

0.25

Damage extent d

Figure 5: Probability of damage (Example 2) The probabilities of damage for the twenty 07 are shown in Fig. 5. It can be clearly seen that the +y face of the first story and the +x face of the third story certainly have damage with probability almost unity. The mean of the damages

— 150 —

are 16.1% and 9.5% and their target values are 16.61% and 10%. Furthermore, uncertainty of these estimates are 0.6% and 0.9%. The figure can be interpreted as follows. Consider the probability of damage curve of the +y face of the first story. Damage is 15% damage or more with probability 0.95. All the other eighteen faces are undamaged. The stiffness parameters difference between the undamaged and damaged structure is within the uncertainty level. This probability of damage can be used directly to set up the confidence interval for damage with a given percentage of confidence. CONCLUDING REMARKS A finite-element model updating methodology is presented with applications to structural damage detection. The proposed method does not require any matching between the measured and calculated modes from the finite-element model as it does not require any calculation of the natural frequencies and mode shapes of the finite-element model. Furthermore, it operates in a computationally efficient iterative manner without nonlinear optimization programming. The illustrative examples confirm the efficiency and effectiveness of the proposed approach. Acknowledgements This work was supported by University of Macau under research grants RG097/03-04S/YKV/FST and RG068/0405S/YKV/FST. These grants are gratefully acknowledged. APPENDIX Consider a set of linear algebraic equations: A9 = b + 5b

(51)

where 5b is a random vector that describes that the equation error with zero mean and covariance matrix !&,. Since X§& is symmetric, it is diagonalizable: X^VDV"1

(52)

where the diagonal matrix D contains the eigenvalues of I&, and the matrix V contains the corresponding eigenvectors. With a proper choice of the scaling factors of the eigenvectors, a matrix that satisfies V r = V" 1 can be chosen since Ega is symmetric. By pre-multiplying Eq. (51) with \ T , one obtains: V r A9 = V r b + 5b'

(53)

where 5b' = V r 5b has zero mean and covariance matrix V r I^,V = D, which is diagonal. This implies that the random components in 5b' are uncorrected. Note that the covariance matrix X&, is in general singular. Therefore, some of the diagonal elements of D are zero. The equations associated with these zero eigenvalues will have zero error, that is, they give exact constraints to the unknown parameters. One possible solution is to consider those constraints to eliminate some unknown parameters. Then, one can solve the remaining unknown parameters by considering the remaining equations weighted by the non-zero entities in the diagonal of the matrix D. However, it is suggested to solve this problem with the following approximated procedure. The values on the diagonal elements of D indicate the variance of the error of each equation. A larger value implies larger uncertainty. For those diagonal elements that are zero or close to zero, one can replace them by a small number, say 10 6 / V/v? , where Kn-g denotes the average of the eigenvalues of X&,. This means that the weighting for those equations are still large compared to the others. There are two reasons for this procedure: 1) The matrix D becomes invertible; 2) Instead of completely rely on those equations with zero eigenvalues, it just gives large weighting for them because the covariance matrix £&> might not be exact, e.g., due to the fact that the second order errors are neglected. Use D to denote that the diagonal matrix that replaces the zero diagonal elements by small values in D. Then, by pre-multiplying D 7, one obtains I ) H T 7 A e - D"i V r b + 5b"

(54)

where 5b" — D ^Sb' has zero mean and unity covariance matrix I. Then, all the equations have independent errors with the same variance. Therefore, the optimal solution of the unknown parameter vector 0 is the least squares solution: — 151 —

0 = [(i)-lvTA)T{D-iYTA)]-\D-?VTA)Tii-iVTb

(55)

Then, the unknown parameter vector 0 can be estimated as follows: e=(A%1Ar1ArS»b

(56)

where the matrix I&, is the same as Z&,, except that the zero eigenvalues are replaced by a small value. REFERENCES 1. Natke MA, Yao JTP eds. Proceedings of the Workshop on Structural Safety Evaluation Based on System Identification Approaches. Vieweg and Sons, Wiesbaden, Germany, 1988. 2. Agbabian MS, Masri SF eds. Proceedings of the International Workshop on Nondestructive Evaluation for Performance of Civil Structures. Department of Civil Engineering, University of Southern California, Los Angeles, CA, USA, 1988. 3. Chang FK ed. Proceedings of the 4th International Workshop on Structural Health Monitoring. Stanford University, USA, 2003. 4. Mazurek DF, De Wolf JT. Experimental study of bridge monitoring technique. ASCE J. Structural Engineering, 1990; 116(9): 2532-2549. 5. Hearn G, Testa RB. Modal analysis for damage detection in structures. ASCE J. Structural Engineering, 1991; 117(10): 3042-3063. 6. Lam HF, Ko JM, Wong CW. Localization of damaged structural connections based on experimental modal and sensitivity analysis. J. Sound and Vibration, 1998; 210(1): 91-115. 7. Hemez FM, Farhat C. Structural damage detection via a finite element model updating methodology. International J. Analytical and Experimental Modal Analysis, 1995; 10(3): 152-166. 8. Pandey AK, Biswas M. Damage detection in structures using changes in flexibility. J. Sound and Vibration, 1994; 169:3-17. 9. Beck JL, Katafygiotis LS. Updating models and their uncertainties. I: Bayesian statistical framework. ASCE J. Engineering Mechanics, 1998; 124(4): 455-461. 10. Beck JL, Au SK, Vanik MW. Monitoring structural health using a probabilistic measure. Computer-Aided Civil and Infrastructure Engineering, 2001; 16(1): 1-11. 11. Ching J, Beck JL. New Bayesian model updating algorithm applied to a structural health monitoring benchmark. Structural Health Monitoring, 2004; 3: 313-332. 12. Yuen KV, Beck JL, Katafygiotis LS. Efficient model updating and monitoring methodology using incomplete modal data without mode matching. Special issue in memory of Prof. T.K. Caughey, J. Structural Control and Monitoring, 2006; 13(1): 91-107. 13. Yuen KV, Katafygiotis LS. Bayesian time-domain approach for modal updating using ambient data. Probabilistic Engineering Mechanics, 2001; 16(3): 219-231. 14. Katafygiotis LS, Yuen KV. Bayesian spectral density approach for modal updating using ambient data. Earthquake Engineering and Structural Dynamics, 2001; 30(8): 1103-1123. 15. Yuen KV, Katafygiotis LS. Bayesian modal updating using complete input and incomplete response noisy measurements. ASCE J. Engineering Mechanics, 2002; 128(3): 340-350. 16. Yuen KV, Beck JL, Katafygiotis LS. Probabilistic approach for modal identification using non-stationary noisy response measurements only. Earthquake Engineering and Structural Dynamics, 2002; 31(4): 1007-1023. 17. Levine-West M, Milman M, Kissil A. Mode shape expansion techniques for prediction: experimental evaluation. AIAA J., 1996; 34(4): 821-829. 18. MATLAB. Matlab User's Guide. The MathWorks, Inc., Natick, MA, USA, 1994. 19. Papadimitriou C, Beck JL, Katafygiotis LS. Asymptotic expansions for reliability and moments of uncertain systems. ASCE J. Engineering Mechanics, 1997; 123(12): 1219-1229.

— 152 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Dynamic Infinite Elements for Soil-Structure Interaction Analysis in a Layered Soil Medium C.B.Yun^J.M.Kim2 1

2

Department of Civil & Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejon 305-701, Korea Department of Civil & Environmental Engineering, Chonnam National University, Yeosu, Chonnam 550-749, Korea

Email: [email protected] Abstract: This paper presents dynamic infinite element formulations which have been developed for soil-structure interaction analysis both in frequency and in time domains by the present authors and our colleagues during the past twenty years. Axisymmetric, 2D and 3D layered half-space soil media were considered in the developments. The displacement shape functions of the infinite elements were established using approximate expressions of analytical solutions in frequency domain to represent the characteristics of multiple waves propagating into the unbounded outer domain of the media. The shape functions were determined in terms of the excitation frequency as well as the spatial and material characteristics of the far-field soil region. Thereby the element mass and stiffness matrices become frequency dependent. As far as time domain analysis, the shape functions were further simplified to obtain closed-form frequency-dependent mass and stiffness matrices, which can analytically be transformed into time domain terms by a continuous Fourier transform. The proposed infinite elements were verified using benchmark examples, which showed that the present formulations are very effective for the soil-structure interaction analysis either in frequency or in time domain. Example applications to actual soil-structure interaction problems are also given to demonstrate the capability and versatility of the present methodology. Key Words: finite element, Infinite element, soil-structure interaction, frequency domain analysis, time domain analysis INTRODUCTION When an engineer analyzes a structure supported or surrounded by soil medium subjected to dynamic loads, it is used to consider the flexibility of the soil (kinematic interaction) and the energy radiation into outer far-field region (inertial interaction). These effects are noticeable particularly for stiff and/or massive structures resting on relatively soft soil. Thus the interaction phenomena have to be considered in dynamic analysis often called 'soil-structure interaction' (SSI) analysis. On the other hand, because the soil domain is practically much larger than the other dimension of the problem such as the size of the structure, it can be regarded as infinite. In order to model the effect of wave radiation into the infinite domain on structural behaviors, several kinds of modeling techniques have been developed including viscous boundary, transmitting boundary, boundary elements, infinite elements, and system identification method [1]. The infinite element method is one of the popular schemes, since it is a mean of extending the well-known finite element method to problems which extend to infinity, at the expense of requiring special shape functions and integration schemes [2-39]. In the infinite element method, the interior region is modeled with finite elements, while the exterior region is represented by infinite elements. Unlike the finite element solution, the behavior in the exterior highly depends on the problem considered, for example the solution of elastostatics problem is non periodic while that of elastodynamic periodic. Thus the shape functions of infinite element for approximating field variables were usually derived using analytical solutions in the exterior region. This may be the reason why lots of infinite element formulations are available and still under development. The idea of the infinite element was originally introduced by Ungless [2]; Bettess published the first paper on infinite element in mid 1970's [3]; and later Astley [4] and Zienkiewicz et al. [5] respectively proposed the wave — 153 —

envelope type and the mapped type infinite elements. Owing to these earliest works, the method then became popular and has successfully been extended to various fields. There is now a considerable literature on the topic of infinite elements extended to over 300 papers. Among the research papers, we note Chow & Smith [6], Beer & Meek [7], Lynn & Hadid [8], Rajapakse & Karasudhi [9], and Koh & Lee [10] for elastostatics; Schrefler & Simoni [11] and Karpurapu [12] for consolidation analysis; Honjo & Pokharel [13] and Zhao & Valliappan [14] for seepage flow; Bettess & Zienkiewicz [15], Lau & Ji [16], Park et al. [17, 18], and Astley et al. [19] for the Helmholtz equation including hydrodynamics and acoustics; and other fields [20, 21]. Most of the formulations are reviewed in Bettess' monograph [22], and more recently Astley presented a review of formulations for scalar wave problems [23]. Among the infinite elements, the static type is judged to be most successful, so that it is now available in some commercial finite element programs. Infinite elements for scalar wave problems governed by the Helmholtz equation have also been developed well enough to apply for industry. However, relatively limited work has been done for elastic wave propagation problems in layered media, such as the SSI analysis in which multiple wave components generated by vibrating structures coupled with neighboring soil travel simultaneously into the unbounded layered half-space medium. Medina carried out pioneering works with Penzien and Taylor in this field [24, 25]. Rajapakse and Karasudhi [26] suggested an infinite element which is capable of propagating multiple waves in a homogeneous half-space. While favorable results were obtained using these elements, the accuracy of the methods was found to deteriorate due to incompatible nature of the shape functions along interfaces with adjacent finite and infinite elements. Yang & Yun [27] and Karasudhi & Liu [28] eliminated this limitation by introducing extra unknowns and effective integration scheme on the infinite formulation. At this juncture it is noteworthy that only 'decay' type infinite elements have been proposed for the elastic wave propagation problem, while 'mapped' type ones are mostly developed for the scalar wave problem. This is because the 'decay' type can easily deal with multiple wave components in the infinite element formulation. Thus the infinite element for the elastodynamic problem necessitates integration over infinite interval, as follows: / = rG(x)e-ikxdx Jo

(1)

where / = v — 1 and k is wavenumber which is in general complex-valued and represents attenuation and frequency of displacement in space. Various schemes including the Gauss- Laguerre quadrature, Newton-Cotes formula and analytical integration can be employed for computation of the integral [22]. In the aforementioned infinite elements for the SSI analysis, the interior near-field consists of a upper cylindrical region and a hemisphere underneath. The exterior far-field of the cylindrical near-field is then modeled by horizontal infinite elements, while the outer domain of the hemispherical near-field is represented by radial infinite elements. Introducing the hemispherical near-field, the body waves which have hemispherical wavefronts can easily and exactly be formulated in the outer hemispherical region. Another way of modeling strategy is proposed by Chow & Smith [6] and Zhang & Zhao [29]. This approach is more natural and economical than the previous method, and its effectiveness is noticeable especially for wider near-field domain. However, in their formulations, the number of wave components included in an infinite element was restricted to one. Yun et al. [30] further developed to include any number of multiple waves in infinite element formulation for the layered half-space. In addition, we note work done by Yang et al. [31] who successfully solved dynamic problems in 3D layered half-space subjected to moving loads. In spite of these progresses, three dimensional infinite element technique for the SSI analysis has left us with a challenging topic, which studied by Zhao & Valliappan [32], Park [33], and recently Seo et al. [34]. It would also be interesting to note some recent works dealing with the wave propagation in two-phase soil media, in which interactions between solid skeleton and pore water are taken into account by considering compressibility and flow of the pore water [35-38]. All the elastodynamic infinite elements aforesaid have dealt exclusively with time-harmonic loading. For the sake of transient analysis, an indirect approach utilizing a special infinite element formulation resulting in closed-form frequency-dependent mass and stiffness matrices in conjunction with a continuous Fourier transform turned out to be very effective [39-41, 33]. Unfortunately a direct time domain elastodynamic infinite element whose shape functions may describe transient behavior has not been developed. This paper presents dynamic infinite element formulations which have been developed for the dynamic SSI analysis both in frequency and in time domains by the present authors and our colleagues during the past twenty years. Axisymmetric, 2D and 3D layered half-space media were considered in the developments. Most of efforts have been devoted to establish wave functions of the infinite elements for those dimensions by using approximate expressions of analytical solutions in frequency domain. They represent the characteristics of multiple waves propagating into the outer domain of multi-layered soil. The shape functions are then constructed based on the shape functions of adjacent finite elements as well as the wave functions which reflect the spatial and material properties of the far-field — 154 —

soil. For the sake of time domain analysis, a special infinite element formulation whose element matrices can analytically be transformed into time domain ones by utilizing a continuous Fourier transform. The proposed infinite elements were verified using various benchmark examples, which showed that the present formulations are very effective for SSI analysis either in frequency or in time domain. Example applications to actual SSI problems, including the Hualien LSST project [42-45] and a liquid storage tank on a compliant ground [46], were also given to demonstrate the capability and versatility of the present methodology. DYNAMIC INFINITE ELEMENT FORMULATIONS 1. Governing Equations The harmonic motion of a multi-layered exterior region QE in a SSI system can be represented by the Navier's equation as: x u + pco2u = 0 mClE

for a homogeneous linear elastic solid

(2)

where the displacement field is defined as u(x;co)e /to/ ; co is circular frequency; x denotes location vector; X and G are Lame's constants; and p is the mass density. Since the problem, except for special cases with a bedrock, remains unsolved, approximate solutions of the free-field problem have been utilized for the shape functions of infinite elements. 2. Geometric Mapping The structure and near-field region, in this study, are modeled by utilizing the standard finite elements, and the exterior is discretized by the proposed horizontal, vertical and corner infinite elements for 2D and axisymmetric bodies; i.e., HIE, VIE and CIE as in Fig. 1(a), while HIE, HCIE, VIE, VCIE, and VHCIE for a 3D body as in Fig. 1(b). The cylindrical coordinate system (r,Q,z) and Cartesian coordinate system (x,z) are respectively chosen for the axisymmetric and 2D bodies, while Cartesian coordinate system (x,y,z) for 3D problem. The Fourier series expansion method in the 0-direction is employed for the axisymmetric problem under 3D loads. The global and natural coordinate systems and a typical mesh for finite element incorporating the infinite elements are shown in Fig. 1. In the 2D plane as shown in Fig. 1 (a), geometric mappings of the infinite elements from the local coordinates to the global are defined for the HIE, CIE and VIE, as follows:

* = *o( 1 + S)>

Z = ^LJ(J\)ZJ

for

z=

for CIE

ZQ-C,

HIE

in 2D plane

(3)

N

for VIE y=i

where N is the number of nodes for horizontal and vertical infinite elements; Lj(r\) denotes Lagrange's interpolation function associated with nodey; (x 0 ,z 0 ) are the coordinates at the corner node of the near-field; and £,£ E [0,oo). The mapping in the 3D space is also defined in the same way [34].

K

H 2-D plane strain finite elements r

^ Horizontally Layered %?

Equivalent earthquake

><*/ ,\ rr(co)

force on

Y.

M:

Inhomogeneous Near-Field Soil

Homogeneous Half-Space

tmtmtfUi nu

flffffffffJJH Vertically Incident Plane Body Waves

j

u

M*7

(a) 2D or axisymmetric plane

< >< : . - t H -

(b) 3D space

Figure 1: Modeling of soil-structure systems with multi-layered half-space by FE-IE method — 155 —

Table 1

Approximate wave functions (see Fig. 1 to refer local coordinate systems) Horizontal layer Body waves

Exact wave functions

Underlying half-space

Surface waves

Body waves

Surface waves

2D

H™{kr)

e~ikx

H™{kr)

6r'*V | z |

Axi. 3D

h«\kR)

H«\kr)

h«\hR)

H™(kr)e-*A

g-iO-lS+ikoK

2D

e-**

e**e*

e-(a25tifab^-(a2$+*oK

e-C0.75+ifc)^

Approximate Axi. wave functions

e-<0.75Hkr0K

e-i025Hkr0K

^U&tfrM^ e -(0.75f*o)^ e - ( a 7 5 t * 5 ) ?

-ej(o.7s+ifa,.£

3D

e-8>(0.75+*)C

-^(025t*^.)r,

-^«X7Wty)r,

-^«X75+*c, % -G/ (tt75+iitK - ^ ( a 7 5 * ^ -aj(a?w^)n -a((a7$+*)?

3. Formulations for Frequency Domain Analysis The displacement field of an infinite element in the 2D plane, u(x; GO) , can be approximated as: N M

u(x;(o) = ^^Njm(x;(6)pjm((o)

or u(x;co) = N p (x;co)p(co)

(4)

j=\ m=l

where TV is the number of nodes or the number of wave functions depending on the type of infinite element; M is the number of wave functions included in the formulation; and pym(co) is a unknown vector in the generalized coordinate associated with Njm . The shape function Njm is expressed in the 2D plane as: iV,m(X;co) = L y (T 1 )/ m 'fe(o)

for HIE;

^ ( X ; < D ) = /;(£©)/;(&«>)

for CIE; and

A^(X;CO) = L,.(TI)/;(C;CO)

for VIE. In the shape functions, f^ and / J are wave functions in x- and z-axis respectively which can be obtained from the approximate expressions of analytical solution as given in Table 1. All the wave functions have unit value at FE-IE interface, i.e., t, = 0 or <^ = 0. In Table 1, wavenumber k associated with the body waves varies with layer and is a function of frequency. On the other hand, the values of k and ju associated with surface waves are in general nonlinear function of frequency, which is called 'dispersion'. In this study, the dispersion curves are determined by solving transcendental eigen equations prior to the SSI analysis. For the purpose of assembling element matrices, it is required to express the displacement field in terms of shape functions associated with nodal displacements, displacements along sides of the infinite element and internal displacements. Hence, Eq. (4) is rewritten as: u(x; co) = Nq (x; co)q(cv) in which N 9 (x;co) = [N^,N 5 ,NJ ; q = (dT,sT,bTY

(5) ; d is nodal displacement vector; while s and b are

amplitude vectors of side and bubble modes, respectively. Typical shape functions for HIE and CIE are shown in Fig. 2 and Fig. 3. Thus, a transformation matrix between two generalized coordinates, p(co) in Eq. (4) and q(co) in Eq. (5), can be introduced as: p(co) = TMq(a>),

N,(i;ffl) = N J ,(i;cD)T w

(6)

where T is the constant transformation matrix which is defined for each type of infinite elements [30]. For 3D infinite element formulation, the wave functions shown in Table 1 are established in conjunction with local coordinate systems depicted in Fig. 1. All the procedures are similar to those as for the 2D formulation given above, although the shape functions and the transformation are much more complicated as the dimension increase. Typical shape functions are shown in Fig. 4, and Fig. 5 demonstrate how the shape functions for the 3D infinite elements describe a displacement field. — 156 —

The element stiffness and mass matrices of the infinite element can be computed in a way similar to the finite element method, as follows:

K„(co) = k B / DB, d& , M„(
(7)

Thus the dynamic stiffness matrix of infinite element, S (co), is also constructed in the same manner of the finite element method as: S „ (co) = (1 + »'24) K w (co) - co2 M „ (co)

(8)

in which £ is the hysteretic damping ratio of the infinite element. An effective integration procedure is proposed by Yang & Yun [27] by devising wave-by-wave integration procedure and by modifying the Gauss-Laguerre quadrature, and later elaborated by Yun et al. [30] by employing the transformations as: r M T "~qq = T ~pq M "*-pp*-pq

K qq • *-pq T '■^pp'-pq* K T

1T

*w

P
l

(9)

K

ppKpq

where Tpq is the transformation matrix defined in Eq. (6). It should be noted that computations of the matrices K and Mpp

are much easier than those of K

and M

, since shape function N^ involves single wave

component only while N^ contains multiple wave functions. The integration including single wave functions in the infinite direction may be carried out by using the Gauss-Laguerre quadrature as [27]:

£' s

[G(x)esxdx=^G(^)Wk s

k=l

(10)

s

where s is in general a complex number; xk and Wk are the Gauss-Laguerre integration point and weight factor, respectively; and A^int is the number of required integration points. In general, this integration scheme gives good results for non-oscillating functions of G(x), while it may yield erroneous results for oscillating functions. In this study, the five-point Gauss-Laguerre rule has been used for all numerical examples, i.e., Afint = 5. The equation of motion for the SSI system including impedance matrix of the far-field soil, Sf 6 ((o), can be written as: S

s

»

Sis(co)

Ssi(co)

U,(ffl>)'

S6A(co) + S £ > ) U6(co)

(11)

F,(co)

where U(co) and F(co) are respectively displacement and force vectors, and S(co) is the dynamic stiffness matrix in frequency domain. Subscript b stands for the nodes along the interface between the near- and far-field and s for those of the structure and the near-field soil region.

Re{tfs.i2ft. Re{tf„($,Ti)}

(b) Side mode Re{Ww2ft,Ti)}

(a) Nodal modes (c) Bubble mode Figure 2: Example shape functions of a 3-node horizontal infinite element: Real parts only (Axisymmetric bodies) — 157 —

Figure 3: Typical shape functions for a corner infinite element: Real parts only (Axisymmetric bodies)

(a) Nodal mode Figure 4:

(b) Edge mode

(c) Face mode

Typical shape functions for a 3D horizontal infinite element: Real parts only (3D body)

(a) Plan view

(b) 3D view

Figure 5: An example of displacement field constructed using 3D infinite elements (3D body) 4. Formulations for Time Domain Analysis For the SSI analysis in frequency domain, the dynamic stiffness matrices of the infinite elements shall be calculated at each frequency step in accordance with wavenumbers which are nonlinear function of the frequency. The frequency-dependency of the dynamic stiffness is a distinct characteristic of the SSI system. However, it makes the time domain analysis of the system be extremely difficult or inefficient compared with the frequency domain analysis [1]. If we let the wave function to be as /(^;co) = exp(-K m ^), then the exponent of the wave function becomes implicit functions of the frequency. Therefore the dynamic stiffness matrix for the infinite domain can be only obtained numerically at each frequency point. To derive the dynamic stiffness matrix analytically, the exponent can be further approximated as linear function of frequency as: K m (co)s(a + Ko)C„,

(12)

where the newly introduced positive constant, a, which is related to the geometric attenuation, is taken to be the same for all wave components. The value of a in Eq. (12) is determined by an error minimization procedure [39]. Owing to the simplification, the element mass and stiffness matrices are expressed analytically in terms of frequency and constant matrices as: M , » =- ^ - M

0

+

*.

Ml3

K 9? (co) = K 0 + ( t f + / c o ) K 1 + — — K 2

— 158 —

(13)

where M 0 , M , , K 0 , K, and K 2 are complex-valued constant matrices. Finally, assembling the mass and stiffness matrices of the infinite elements, the impedance matrix of the far-field soil region, S*b (a>) , can be obtained as: S ? »

=Sg+f
(14)

where Sf, Sf, Sf and Sf are complex-valued constant matrices. Since the impedance matrix of the far-field region is obtained as an analytical function of frequency, Eq. (11) can be analytically transformed into time domain by taking Fourier transform as [40]: Mcc M,

"0 MLllu.rOl T O 0" 0 | f uk(0l .(0l [J 0 Sf M *». . J Ito(0J i i . M l 10 Sf||u f c (0l

IK,, IK,.

K, Kw+S*

■■•>!. f

„

W

1 (.5)

where u(t) and / ( / ) are displacement and force vectors in time domain respectively; Sf and Sf represent contributions of the exterior region to the static stiffness and viscous damping matrices; and Sf and Sf are related to the lingering responses on the interface between the near- and far-field. The SSI system may be analyzed in time domain by using Eq. (15). Furthermore, in case that the nonlinear behavior of the structure or the near-field soil is significant, the nonlinear effect may easily be included in Eq. (15) and the response can effectively be analyzed in time domain. 5. Earthquake Response Analysis The earthquake responses can be obtained by solving the wave radiation problem with equivalent earthquake force rbfree((o) applied along interface Tb as shown in Fig. 1(a). The force vector can be calculated using the free-field motions (displacement u{ree and traction t{ree) and the impedance matrix of the far-field regardless of the properties of near-field as [47]: »/"(«>) = S ^ u f

(©) - A t f - ( © )

(16)

where A is a constant transformation matrix. When a nonlinear analysis is carried out by changing its shear modulus and damping ratio which are equivalent to a reference strain in the region, the dynamic stiffness matrix for the near-field soil shall be modified at each iteration step. However, the earthquake force vector can be assumed unchanged during the iteration process, if the far-field region was selected so far from the structure that nonlinear behavior of soil may be negligible in the region. For the purpose of time domain earthquake response analysis, the equivalent earthquake input force, rbee (co), is firstly computed along the interface in the frequency domain. Then, the earthquake forces on the interface in the time domain, fb (t) = xbee (t), are computed through the discrete Fourier transform technique. Finally, the earthquake responses are computed by solving Eq. (15), in which force vector fs(t) is equal to zero [41]. VERIFICATION EXAMPLES 1. A Rigid Strip Foundation on an Elastic Layer with Rigid Bedrock (2D IE) A rigid strip foundation on an elastic soil with rigid bedrock is modeled as in Fig. 6. Two Rayleigh waves and two body waves are used in the infinite element formulation. The vertical compliance of the foundation was obtained using the numerical and closed-form frequency-dependent infinite elements, and compared with each other along with reference solution by Tassoulas [48] in Fig. 6. Very good agreements have been found between them. 2. A Rigid Circular Disk on Layered Hals-Space Medium (Axisymmetric IE) Horizontal impedance has been obtained for the case that the depth of the upper horizontal layer (h) is RQ . Both the horizontal layer and the halfspace are assumed to be elastic, homogeneous and isotropic but having different soil properties; i.e., shear wave velocities (C 5l / Cs2 = 0.8), densities (p 1 / p 2 = 0.85) and Poisson's ratio (v t = v2 = 0.25). Two body waves and three Rayleigh waves have been also used for infinite elements. The effect of the horizontal dimension of the nearfield has been also investigated using three different meshes shown in Fig. 6; i.e., r0 = 12RQ , \5RQ and 2RQ . The results shown in Fig. 7 indicate that reasonable solution can be obtained if the horizontal dimension of the near-field is taken to be larger than 15RQ . The results are also found to be in good agreement with the analytical solutions obtained by Luco [49]. — 159 —

Tassoulas o

Numerical E

*

Analytical E

H=2b\

a0

(a) FE & IE mesh Figure 6:

-cobIcs

(b) Compliance of vertical motion

Vertical compliance of a rigid strip foundation on a layer with fixed bedrock (2D analysis) **--

■■-■■H

r*

1 r™

V f h h TTT\y

-*M)

h«-

-■•»-j

y

|7\r\

■-H

^ 11 1 1 / \\\

^ 1 11 r' \1

^X&\ P\n WE

^V)

\1 /

Ar\\

f 1.4 | Luco

r0 / /2o = 15 r 0 //?o = 2.0

r

stu(

jy

J

=

Figure 7:

w^

Horizontal impedances of a rigid disk on a layered half-space using three different finite element meshes for the near-field (h IR^ = 1.0) (Axisymmetric analysis) Chow(1987) This study

Ox / /xO

*\

(a) Vertical impedance (h/B=4)

\

/ 0

(b) Vertical impedance (h/B=6)

Figure 8: Vertical impedance of rectangle footing on a soil layer with bedrock (L/B=2, v=0.33, \ =0.05) (3D analysis)

— 160 —

3. A Rigid Rectangular Disk on Layered Half-Space (3D IE) A rigid rectangular footing on a horizontal soil layer with an underlying rigid rock is analyzed. The impedance function of the rectangular footing is expressed as kz(a0) = ksz'a(k+ia0c)(l+2iZsd)^ w h e r e ks2'a ig t h e v e r t i c a l s t a t i c s t if m e ss. HIE and HCIE are used for the horizontal exterior region of the soil layer, while the boundary condition along the interface with the bed rock is taken as fixed. Two cases with different depth ratios (h = 4B and 6B ) are considered. Fig. 8 shows the results for the rectangular footing along with the results by Chow [50]. The present results are found to be in good agreement with the reference values. 4. A Strip Foundation on a Two-Phase Medium (2D Body) As a demonstrative extension of the present methodology, we present results of forced vibration of a strip foundation on a two-phase soil layer with fixed bedrock, as in Fig. 9. In the two-phase layers, the wave propagation mechanism is much more complicated than that in single-phase soil, e.g., three body wave components exist including P1-, P2- and S-waves. However, the infinite element formulation essentially remains the same and straightforward [37]. Fig. 9 shows the accuracy of the present method, and indicates that multiple wave components shall be considered in the analysis even though other researchers [35, 36, 38] obtained reasonable results by considering one or two wave components only. 20MPa —}

tyKMitt

iiimm HIE HIE

5m

m

HIE HIE HIE

\—1

<—

le of Footing Displacement (u_z)

XQ

-*

Real (u_z) at ground surface

(x0/B = 11)

(a> = 20 rad/sec)

Xo/B = 11

X-coordinate (m)

Frequency (rad/sec)

Figure 9: Frequency response of a 2D strip foundation on a two-phase soil layer upon fixed bedrock i(Exact:

.) (FE only: ~^~)

(FE-IE : ♦ = including 1 wave, ■ = 2 waves, • = 3 waves)

5. A Rigid Circular Disk on Flexible Hollow Caisson (Axisymmetric IE and 3D IE) A rigid circular footing on a RC caisson shown in Fig. 10 is analyzed using the FE-IE approach. The axisymmetric and 3D infinite elements are employed for modeling the far-field of soil. The material properties are given in the figure. The impedance functions are normalized by GSR0 for the horizontal and the vertical components, while they are normalized by GSR03 for the rocking and the torsional components. The results are shown in Fig. 10, and compared with those based on the indirect boundary element method by Chen & Penzien [51]. Excellent agreements have been achieved using both types of axisymmetric and 3D infinite elements. APPLICATION EXAMPLES 1. Simulation of Forced Vibration Test & Earthquake Responses (Axisymmetric Body) This example presents the result of an international cooperative research on the post-correlation analysis of forced vibration tests and the prediction of earthquake responses of a large-scale seismic test (LSST) structure. Through the system identification technique, the properties of the soil layers are revised so that the best correlation in the responses is obtained as shown Fig. 11. The revised values are shown in Table 2 [41]. Utilizing the revised soil properties as the initial linear value, the seismic responses are predicted for an earthquake using the equivalent linearlization technique. It has been found that the predicted responses by the equivalent nonlinear procedure are in excellent agreement with the observed responses as shown in Fig. 12. — 161 —

Ve"»'

,m

| Ec/&=500 1 R;/ft=1.5 \ v c =v,=025 \ ^=0.02

^=oo

-

£

* 30

1 1

I 0.6 m

\\

_

o

V

Presentstudy

CHH

ME ME

12m

Cats*n

\

Chen and Penzien

0)

impec

mmm

40

,.

"3

ME

N

ME

0

2°

-

~-^r—1>^::8

ME

Pi"

ME

a i

Ro=4m

ME

«-

(

^HH

vm

VIE VIE

CE

in

0.0

0.5

1.0

i

i

i

1.5

i

i

i

i

2.0

, (oR>

(a) Using axisymmetric FE-IE Chan & Penzlen(19B6) This study

I

120

i

k„M

•• •

(b) Using 3D FE-IE Figure 10: Impedances of a rigid disk on a hollow caisson embedded in homogeneous half-space 8.0 6.0

o Test (D^

H15

—Calculated

4.0"2.0 0.0-

n 5

1 ■ r10 15 Frequency (Hz) ■

Figure 11: Simulated and measured forced vibration test (Di-direction) — 162 —

20

1 10 Frequency (Hz)

1 10 Frequency (Hz)

Figure 12: Response spectra of simulated and measured accelerations in structure (Di-direction) Table 2

Identified values of the parameters for from the Hualien LSST structure using FVT data, and equivalent linear properties (Note: Values in parentheses are hysteretic damping ratios in percent.) FVT simulation & Linear Earthquake Response Analysis

Parameters (V s in m/sec, E in GPa)

133(2.0),

Sand-l,Sand-2(V s )

Nonlinear Earthquake Response Analysis (May 1, 1995) 113(4.7),

231(2.0)

199(4.5)

Backfill-1, SB-1 (Vs)

270 (2.0)

244 (2.4),

207 (5.3)

Backfill-2, SB-2 (Vs)

325 (2.0)

257 (4.8),

234 (6.4)

Gravel-1, SG-1 (Vs)

308 (2.0)

262 (5.0),

248 (6.0)

Gravel-2, SG-2 (Vs)

281 (2.0)

214(7.0),

203(7.9)

211(7.2),

340(4.4)

333 (2.0),

Gravel-3, Gravel-4 (Vs)

388 (2.0)

Roof & Base (E)

28.2 (2.0)

28.2 (2.0)

Shell (E): Upper, Middle, Lower

19.7 (2.0), 21.3 (2.0), 21.8 (2.0)

19.7 (2.0), 21.3 (2.0), 21.8 (2.0)

|

2. Time Domain SSI Analysis For verification of the proposed time-domain SSI analysis procedure, earthquake response analysis of a multi-layered free field half-space shown in Fig. 13 is carried out. The near-field soil region is discretized with plane strain finite elements and the remaining far-field soil region is modeled by the analytical frequency-dependent infinite elements [39]. The properties of the soil layers are shown in Table 2. A horizontal acceleration record is used as the input control motion on the ground surface, which is the NS-component of an earthquake measured at Hualien, Taiwan on January 20, 1994. The peak ground acceleration is 0.0318g, and the time history is shown in Fig. 13. The acceleration histories are compared with those of the free field analysis which are obtained based on the frequency domain method. The results in Fig. 13 show excellent agreements. (a) Control motion (Free surface)

9-node FE AI Sand 1

* * * *

2.0 m

V

> \

3.15 m

Sand 2

I Graved

i

7.0 m

J

' Half-space Boundary for

|

|

|

|

|

|

|

^earthquake force input

3-nodevlE

C 0.00 < -0.04

Vertically Incident P-wave and SV-wave

(a) Free field analysis problem

Time (sec)

(b) FE-IE mesh

(c) Time history responses

Figure 13: Earthquake response analysis of a layered soil medium — 163 —

3. Flexible RC tank on a layered half-space (Axisymmetric Body) In order to demonstrate the soil-structure interaction effect on the member forces of a liquid storage tank, a stress analysis is carried out for a structure depicted in Fig. 14 under various soil conditions. The structure is supported by a horizontal layer with the underlying bedrock. Three values of the shear wave velocity for the horizontal soil layer, i.e., 500m/s, 800m/s, and 5000m/s, are considered in this investigation. The other material properties for the structure and soil regions are given in Fig. 14. The frequency-dependent mass matrix associated with the DOF along wetted interface is included in the equation of motion in the frequency domain, in which the fluid motion is represented by Bessel functions [46]. An acceleration time history with PGA of 0.14g, which is compatible with a design response spectrum for a rock site is simulated for the earthquake input as in Fig. 12. In this analysis, the control acceleration is assigned at the top of bedrock as a horizontal outcrop motion. Thus, seismic motion can be amplified at the ground surface depending on the properties of the horizontal soil layer. Member forces are calculated on the vertical shell for three different soil conditions including both the fluid-structure interaction and soil-structure interaction, and their maximum values are plotted along the height of the structure in Fig. 14. For the purpose of comparison, the maximum member forces are also computed using ANSYS program for the same structure but on a rigid ground. A fully coupled fluid-strucrure-soil interaction analysis cannot be carried out by ANSYS program. In ANSYS analysis, the input ground acceleration at the fixed base is prepared for each soil condition by carrying out the free-field analysis using SHAKE91 program. Thus, the solution by ANSYS can be considered as the response for the same input motion but excluding the soilstructure interaction effect. Two sets of the results for a rigid soil condition by the present and ANSYS analysis (in Fig. 14) are found in good agreements, which confirms the accuracy of the present analysis. The results for the softer soil conditions indicate that the member forces on the shell reduce considerably as the soil stiffness decreases. This result re-confirms that accurate dynamic analysis of a large liquid storage tank considering the soil-structure interaction may yield cost-effective cross-section for the structure. C.L.

I H=40m

50m 3m Compliant Soil

/?=30m

! water pw=1.0Mg/m3

RC Shell pc=2.6Mg/m3, Es=30MPa, vc=0.2, h=2%

"f" 20m

(p s =2.5Mg/m\ v s =0.2, h=2%) Bedrock (ps=2.5Mg/m\ Vs=5000m/sec vs=0.2, h=2%)

104m

^ 1 SSI excluded ^ z | (ANSYS) SSI included (Present Study)

2000 4000 6000 Member force (KN/ m)

(a) Ks=5000m/s

2000 4000 6000 Member force (KN/ m)

(b) Ks=800m/s

2000 4000 6000 Member force (KN/m)

(c) Ks=500m/s

Figure 14: RC liquid storage tank and maximum member force profiles for a RC liquid storage tank (N t at 0 = 0°, N z at 0 = 0°, and Nte at 0 = 90°) — 164 —

CONCLUDING REMARKS This paper presented infinite element formulations for the dynamic SSI analysis of an axisymmetric body subject to 3-D loads, a 2D problem, and a 3D analysis both in frequency and in time domains. The displacement shape functions of the infinite elements were constructed using approximate expressions of analytical solutions in frequency domain to represent the characteristics of multiple waves propagating into the infinite multi-layered soil medium. For the sake of time domain analysis, the dispersive wavenumbers included in shape functions were linearized to obtain closed-form frequency-dependent element matrices, which can analytically be transformed into the time domain terms by a continuous Fourier transform. The proposed elements were verified using several benchmark examples. Comparisons with the results by other studies showed that the present formulations are very effective for the SSI analysis either in frequency or in time domain. Example applications to actual SSI problems were also given to demonstrate the versatility and effectiveness of the present methodology. Acknowledgement This study has been supported by SISTEC (KOSEF/MOST: Grant No. Rl 1-2002-101-03001-0). Their financial supports are greatly acknowledged. Also the second author would like to thank KOSEF (Grant No. R01-2003-00010635-0) for the financial support to this study. REFERENCES 1. Wolf JP. Dynamic Soil-Structure Interaction, Prentice-Hall, 1985. 2. Ungless RF. An infinite finite element. MS Thesis, University of British Columbia, Vancouver, Canada, 1973. 3. Bettess P. Infinite elements. Int. J. Num. Meth. Eng., 1977; 11: 54-64. 4. Astley RJ. Wave envelope and infinite elements for acoustic radiation. Int. J. Num. Meth. Fluids, 1983; 3: 507-526. 5. Zienkiewicz OC, Emson C, Bettess P. A novel boundary infinite element. Int. J. Num. Meth. Eng., 1983; 19: 393-404. 6. Chow YK, Smith IM. Static and periodic infinite solid elements. Int. J. Num. Meth. Eng., 1981; 17: 503-526. 7. Beer G, Meek JL. Infinite domain elements. Int. J. Num. Meth. Eng., 1981; 17: 43-52. 8. Lynn PP, Hadid HA. Infinite elements with 1/r" type decay. Int. J. Num. Meth. Eng., 1981; 17: 347-355. 9. Rajapakse RKND, Karasudhi P. Elastostatic infinite elements for layered half space. J. Eng. Mech., ASCE, 1985; 111: 1144-1158. 10. Koh KH, Lee SR. p-Version static infinite element for representing 1/r" type decay problems in unbounded media. Computers and Geomechanics, 1998; 22: 73-89. 11. Schrefler BA, Simoni L. Non-isothermal consolidation of unbounded porous media using mapped infinite elements. Comm. Num. Meth. Eng., 1987; 3: 445-452. 12. Karpurapu GR. Composite infinite element analysis of unbounded two-phase media. Adv. Eng. Software, 1988;10:202-209. 13. Honjo Y, Pokharel G. Parametric infinite elements for seepage analysis. Int. J. Num. Anal. Meth. Geomechanics, 1993; 17: 45-66. 14. Zhao C, Valliappan S. Transient infinite elements for seepage problems in infinite media. Int. J. Num. Meth. Eng., 1993;17:323-341. 15. Bettess P, Zienkiewicz OC. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Num. Meth. Eng., 1977; 11: 1271-1290. 16. Lau SL, Ji Z. An efficient 3-D infinite element for water wave diffraction problems. Int. J. Num. Meth. Eng., 1989; 28: 1371-1387. 17. Park WS, Yun CB, Pyun CK. Infinite elements for evaluation of hydrodynamic forces on offshore structures. Computers & Structures, 1991; 40: 837-847. 18. Park WS, Yun CB, Pyun CK. Infinite elements for 3-dimensional wave-structure interaction problems. Eng. Structures, 1992; 14: 335-346. — 165 —

19. Astley RJ, Macaulay GJ, Coyette JP. Mapped wave envelope elements for acoustic radiation and scattering. J. Sound and Vibration, 1994; 170: 97-118. 20. Bettess P, Abram RA. Finite and infinite elements for a simple problem in quantum mechanics. Comm. Num. Meth. Eng., 2002; 18: 325-334. 21. McDougall MJ, Webb JP. Infinite element for the analysis of open dielectric waveguide. IEEE Trans. Microwave Theory and Techniques, 1999; 37: 1724-1731. 22. Bettess P. Infinite elements. Penshaw Press, UK, 1992. 23. Astley, RJ. Infinite elements for wave problems: a review of current formulation and an assessment of accuracy. Int. J. Num. Meth. Eng., 2000; 49: 951-976. 24. Medina F, Penzien J. Infinite elements for elastodynamics. Earthq. Eng. & Str. Dyn., 1982; 10: 699-709. 25. Medina F, Taylor RL. Finite element techniques for problems of unbounded domains. Int. J. Num. Meth. Eng., 1983; 19: 1209-1226. 26. Rajapakse RKND, Karasudhi P. An efficient elastodynamic infinite element. Int. J. Solids & Structures, 1986; 22: 643-657. 27. Yang SC, Yun CB. Axisymmetric infinite elements for soil-structure interaction analysis. Eng. Structures, 1992; 14: 361-370. 28. Karasudhi P, Liu YC. Vibration of elastic and viscoelastic multi-layered spaces. Str. Eng. & Mech., 1993; 1: 103-118. 29. Zhang C, Zhao C. Coupling method of finite and infinite elements for strip foundation wave problems. Earthq. Eng. & Str. Dyn., 1987; 15: 839-851. 30. Yun CB, Kim JM, Hyun CH. Axisymmetric elastodynamic infinite elements for multi-layered half-space. Int. J. Numer. Meth. Eng., 1995; 38: 3723-3743. 31. Yang YB, Hung HH. A 2.5D finite/infinite element approach for modelling visco-elastic bodies subjected to moving loads. Int. J. Num. Meth. Eng., 2001; 51: 1317-1336. 32. Zhao C, Valliappan S. A dynamic infinite element for three-dimensional infinite-domain wave problems. Int. J. Numer. Meth. Eng., 1993; 36: 2567-2580. 33. Park KR. Development of three-dimensional frequency-dependent dynamic infinite elements for multi-layered elastic media. PhD Thesis, Kyoto University, Japan, 2004. 34. Seo CK, Yun CB, Kim JM. Three-dimensional cuboidal infinite elements for soil-structure interaction analysis in multi-layered half-space, (in preparation). 35. Khalili N, Valliappan S, Yazdi JT, Yazdchi M. ID infinite element for dynamic problems in saturated porous media. Comm. Num. Meth. Eng., 1997; 13: 727-738. 36. Khalili N, Yazdchi M, Valliappan S. Wave propagation of two-phase saturated porous media using coupled finite-infinite element method. Soil Dyn. & Earthq. Eng., 1999; 18: 533-553. 37. Kim JM. Two-dimensional infinite element for dynamic analysis of saturated two-phase soil. J. Earthq. Eng. Soc. Korea, 2005; 9: 67-74. 38. Wang G, Chen L, Song C. Finite-infinite element for dynamic analysis of axisymmetrically saturated composite foundations. Int. J. Num. Meth. Eng., 2006 (in press). 39. Yun CB, Kim DK, Kim JM. Analytical frequency-dependent infinite elements for soil-structure interaction analysis in two-dimensional medium. Eng. Structures, 2000; 22: 258-271. 40. Kim DK, Yun CB. Time domain soil-structure interaction in two-dimensional medium based on analytical frequency-dependent infinite elements. Int. J. Num. Meth. Eng., 2000; 47: 1241-1261. 41. Kim DK, Yun CB. Earthquake response analysis in time domain for 2-D soil-structure systems using analytical frequency-dependent infinite elements. Str. Eng. Mechanics, 2003; 15: 717-733. 42. Yun CB, Choi JS, Kim JM. Identification of the Hualien soil-structure interaction system. Soil Dyn. & Earthq. Eng., 1999; 18: 395-408. 43. Choi JS, Yun CB, Kim JM. Earthquake response analysis of the Hualien large scale seismic test structure using updated soil-structure properties based on vibration test data. Earthq. Eng. & Str. Dyn., 2001; 30: 1-26. — 166 —

44. Choi JS, Lee JS, Yun CB. Input and system identification of the Hualien soil-structure interaction system using earthquake response data. Earthq. Eng. & Str. Dyn., 2003; 32: 1955-1975. 45. Choi JS, Lee JS, Yun CB. Identification of the soil-structure interaction system using earthquake response data. J. Engrg. Mech., ASCE, 2004; 130: 753-761. 46. Kim JM, Jang SH, Yun CB. Fluid-structure-soil interaction analysis of cylindrical liquid storage tanks subjected to horizontal earthquake loading. Str. Eng. Mechanics, 2002; 13: 615-638. 47. Yun CB, Kim JM. KIESSI - A computer program for soil-structure interaction analysis using finite and infinite element techniques. Res. Rep., Dept. of Civil Eng., KAIST, Korea, 1996. 48. Tassoulas JL. Elements for the numerical analysis of wave motion in layered media. PhD Thesis, MIT, Cambridge, Mass., USA, 1981. 49. Luco JE. Impedance functions for a rigid foundation on a layered medium. Nucl. Eng. Des., 1974; 31: 204217. 50. Chow YK. Vertical vibration of three-dimensional rigid foundation on layered media. Earthq. Eng. & Str. Dyn., 1987; 15: 585-594. 51. Chen CH, Penzien J. Dynamic modeling of axisymmetric foundation. Earthq. Eng. and Str. Dyn., 1986; 14: 823-840. 52. ANSYS Inc., ANSYS 5.5, User's Reference Manual, 1999. 53. Schnabel PB, Lysmer J, Seed HB. SHAKE91 —A computer program for earthquake response analysis of horizontally layered sites. Earthquake Engineering Research Center, UCB, USA, 1991.

— 167 —

LIST OF OTHER SEMI-PLENARY LECTURES Computational methods at different observation scales - from material characterization to performance-based optimization of structures Roman Lackner (Austria) Towards a deeper understanding of the Wittrick-Williams algorithm for solving eigenproblems Fred W. Williams (UK) Enabling technology for large-scale multidisciplinary simulations Yao Zheng (China)

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Numerical Simulation of Cavitation Generation in Tandem Cascades C. Kang*, D. Liu, M. G. Yang School of Energy and Power Engineering, Jiangsu University, Zhenjiang, 212013 China Email: [email protected] Abstract Theory and numerical prediction remain the most difficult and worthwhile subjects in investigation of cavitation, whether in mechanism of cavitation or in practical cavitation in fluid machinery. Cavitation in flows passing through combined hydrofoils often leads to serious effects on hydrofoils' performance. As the flow structure in hydrofoil cascade is relatively complex, generation of cavitation is difficult to define and the interaction between flow and cavitation keeps even secret. To discover the cavitation around hydrofoil, twin hydrofoils were arranged with several configurations. A new model was set up to simulate the flows passing through the cascades. The results show that different cascade solidity and transverse distance have deep influence on cavitation bubble generation condition. The cavitation number can be changed accordingly. Additionally, the distributions of pressure and velocity are given to support the obtained cavitation characteristics. It is proved that the increase of vertical distance of two hydrofoils enlarge the cavitation zone and the low-pressure zone's diffusion tendency gets more obvious. However, the cavitation generation position does not vary with transverse distance between the two hydrofoils. The conclusions obtained here can be applied in design practice of fluid machinery.

Figure 1: Cavitation around hydrofoil cascade (t=0.45/) Firure 2: Cavitation around hydrofoil cascade (t=0.55/) REFERENCES 1. Singhal AK, Athavale MM, Li Huiying, Jiang Yu. Mathematical basis and validation of the full cavitation model. Journal of Fluid Engineering, 2002; 124: 617-624. 2. Horiguchi H, Arai S, Fukutomi J, Nakase Y, Tsujimoto Y. Quasi-three-dimensional analysis of cavitation in an inducer. Journal of Fluid Engineering, 2004; 126: 709-715. 3. Ventikos Y, Tzabiras G. A numerical method for the simulation of steady and unsteady cavitating flows. Computers & Fluids, 2000; 29: 63-88. — 169 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Simulation of Atrium Fire using Two CFD Tools V. K. Sin1, L. M. Tarn1'2, S. K. Lao1, H. F. Choi1* 1

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macau, China 2 Institute for the Development and Quality, Macau, China Email: [email protected] Abstract Building fires lead to casualties which are mainly caused by smoke. Therefore, smoke management system is one of the most important factors to be considered in Fire Safety Engineering (FSE). Its major aim is to ensure that lives are protected from heat and smoke [1]. Applying full scale experimental study on fire dynamics research is relatively expensive and time-consuming. Thus, computational fluid dynamics (CFD) has been widely used in assessments of fire hazards. The most popular models for fire simulation using CFD are zone model and the field model. According to [2], atria, covered shopping malls, convention centers, airport terminals, sport arenas, and warehouses are examples of large spaces for which these conventional zone-model approaches are not always effective. In this paper, numerical simulation of fire-produced smoke movement in a large complex space proposed in [3] has been carried out in field model with two CFD tools, StarCD and FDS. StarCD is a general-purpose, unstructured CFD code that allows extremely complex scenarios often encountered in many situations to be simulated. Fire Dynamics Simulator (FDS) is a fire model based on large-eddy simulation CFD code developed by the National Institute of Standards and Technology (NIST). The data obtained from calculations based on these two field models are being compared. The results, including the transient development of the smoke profile, temperature distribution and velocity field, etc., are presented and analyzed in this study. It is found that the trend of the results obtained from both of these two CFD tools are in good agreement.

\I Figure: Temperature (K) field at center section and time = 360 sec. (left - FDS and right - StarCD) REFERENCES 1. NFPA 92B, Guide for Smoke Management Systems in Malls, Atria and Large areas, National Fire Protection Association, Quincy, MA, 2000. 2. Chang CH, Banks D, Meroney RN. Computational fluid dynamics simulation of the progress of fire smoke in large space, Building Atria. Tamkang J. Science and Engineering, 2003; 6(3): 151-157. 3. Yang CS. Design Analysis and Experimental Investigation of Smoke Management and Egress System of a Large Shopping Mall. PhD dissertation, National Sun Yat - Sen University, Taiwan, China, 2003. — 170 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Investigation of the Particulate Matters with the Aid of CFD L. M. Tarn \ V. K. Sin \ H. I. Sun2*, K. I. Wong2 1

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macau, China 2 Institute for the Development and Quality, Macau, China Email: [email protected] Abstract Particulate matters, being one of the major indoor contaminates, can lead to serious effect to our health, especially for those having the aerosol diameter less than lOum (PM10), since those are the particles that generally pass through the throat and nose and enter the lungs. Once inhaled, these particles can affect the heart and lungs and cause serious health effects. Therefore, concern should be paid to the indoor pollution from PM10. In this study, the particulate matters of an industrial workshop are suspected to be at a high concentration, therefore, an investigation has been performed to check the level of the particulate matters and the air flow path of them. The investigation is conducted by two means: experimental measurement for the particulate matter and computational simulation for the air flow field of the room. Experimental data showed that the particulate matters are much higher than the Hong Kong indoor air quality objective value which is 0.180mg/m3 [1] and simulation results showed that the particulates generated are spreading everywhere within the room before extracted away by the HVAC system. In order to protect the occupants inside the room, the HVAC system is proposed to be modified into a three-level negative pressure system. Performance of the three-level negative pressure system is verified by CFD tool and result showed that the new HVAC system can immediately extracted the contaminants away without contaminating the occupants inside.

Figure: Flow path of particulate matter: (left) existing HVAC system, (right) three-level negative pressure system) REFERENCES 1. The Government of the Hong Kong Special Administrative Region Indoor Air Quality Management Group. Guidance Notes for the Management of Indoor Air Quality in Offices and Public Places. September 2003.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X.Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

CFD Analysis of Fire in a Forced Ventilated Enclosure L. M. Tarn1,2, V. K. Sin1, S. K. Lao1*, H. F. Choi1 1

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macau, China 2 Institute for the Development and Quality, Macau, China Email: [email protected] Abstract Due to the rapid development of computer technology and numerical technique, many complex thermalfluid phenomena such as fire dynamics can now be simulated by computational fluid dynamics (CFD). StarCD, a general-purpose CFD package with finite volume method, is used to simulate the experimental data [1] obtained at the Lawrence Livermore National Laboratory (LLNL). The experiment consists of a fire in a room with a ventilated exit and an air inlet. An enclosure of 6 m long, 4 m wide and 4.5 m high, with an air intake of 2 m wide and 0.12 m high, has been investigated. Air was ventilated at a constant rate of 0.5 m3/s at the outlet of 0.65 m x 0.65 m. A heat plate model was implemented to represent the natural pool fire at the center of the floor. Combustion chemical reactions were excluded [2]. This study employed k-e model together with Discrete Ordinates Method (DOM) radiation model to predict the turbulent flow and radiative heat transfer in the enclosure. The energy absorbed by soot particles as well as the conjugate heat transfer to the wall have been taken into consideration. The simulation results show good agreement with the experimental data. Moreover, the data indicated that 70% of heat generated by fire was absorbed by the walls, the floor, and the ceiling. It has also been concluded that radiation is the dominant mode of heat transfer. This study demonstrates that StarCD is capable of modeling fire generated heat transfer in a forced ventilated enclosure.

Figure: Temperature (K) field at one center section at time 20 minute REFERENCES 1. Alvares NJ, Foote KL, Pagni PJ. Forced ventilated enclosure fires. Combustion Science and Technology, 1984;39:55-81. 2. Liu Y, Moser A, Sinai Y. Comparison of a CFD fire model against a ventilated fire experiment in an enclosure. International Journal of Ventilation, 2004; 3(2): 169-181 — 172 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESCX,^wg. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Development of the Diffusion Control System using Air Shutter T. Asami *, Y. Nakabayashi Department of Computational Science and Engineering, Toyo University, 2100 Kujirai, Kawagoe, 3508585 Japan Email: [email protected] Abstract It often seems that the people are harmed by the smoke when a fire breaks out in the building. In such situation, the fire wall is used for blocking the fire and smoke. But, it becomes difficult for the people to run away through it. Applying the air curtain system to the Air Shutter proposed by this study, it will be possible both to block the smoke and to escape from fire accident. In order to develop Air Shutter, some numerical flow analyses are performed using ADVENTURE system.

Figure 1: The model of air shutter

Figure 2: Streamline of air shutter REFERENCES 1. National Air Curtain: http://www.hat.hi-ho.ne.jp/jun_matsui/top.htm 2. ADVENTURE Project: http:// adventure.q.t.u-tokyo.ac.jp/

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

A High Order Compact Difference Scheme for Solving the Unsteady Convection-Diffusion Equation Zhihua Xie *, Jianguo Lin, Juntao Zhou College of Environmental Science and Engineering, Dalian Maritime University, Dalian, 116026 China Email: [email protected] Abstract In practical engineering applications, convection diffusion equations are generally used to describe the transport processes involving fluid motion, heat transfer, astrophysics, oceanography, meteorology, semiconductors, hydraulics, pollutant & sediment transport and chemical engineering. In this paper, a high order compact difference scheme based on the fourth order compact difference scheme in spatial discretization and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of the unsteady convection-diffusion equation. The validity and effectiveness of the proposed method is firstly tested by a two-dimensional convection-diffusion equation with a Gaussian pulse type concentration. The L2 error norms are used to measure differences between the exact and numerical solutions and compared to those obtained by other methods. It is shown that the results obtained by proposed method agree very well with the analytical solutions and is more accurate than other methods. Then, a two-dimensional non-linear Burgers equation is used to validate the effectiveness of the proposed method used to solve the non-linear convection-diffusion equation, which also models well. Finally, the Taylor's vortex problem is investigated by the proposed method and good agreement is obtained with the exact solutions. From the three test problems, it is shown that the proposed high order compact difference scheme is an efficient and accurate method to simulate the transport problems and also can be applied to many engineering problems. REFERENCES 1. Lele SK. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 1992,103: 16-42. 2. Kalita JC, Dalai DC, Dass AK. A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients. International Journal for Numerical Methods in Fluids, 2002, 38: 1111-1131. 3. Karaa S, Zhang J. High order ADI method for solving unsteady convection-diffusion problems. Journal of Computational Physics, 2004, 198: 1-9.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Computation of One-Dimensional Dam-Break Flow Using ENO Scheme Yuling Liu u , Yongtao Guo, W. B. Fan 2 1

Institute of Water Conservancy and Hydraulic Engineering, Xi 'an University of Technology, Xi 'an, 710048 China 2 Economics and Management School, Northwest University, XVan, 710069 China Email: [email protected] Abstract This paper is concerned with a high-resolution mathematical model for one-dimensional (ID) dam-break flows by solving the ID shallow water equations of Saint-Venant with the essentially non-oscillatory (ENO) scheme, the finite volume method (FVM) and the total variation diminishing (TVD) Runge-Kutta-type method. And this model is used to predict the flood evolution process caused by the instantaneous ID dam-break flow, and the simulating results are compared with the analytical solutions, which shows that the numerical resolutions accord the exact solutions well, and indicates that this model is fairly effective for simulating dam-break flood waves. REFERENCES 1. Harten A, Engquist B, Osher S, Chakravarthy S. Uniformly high order essentially non-oscillatory schemes. Journal of Computational Physics, 1987; 71: 231-303. 2. Vukovic Senka, Sopta Luka. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. Journal of Computational Physics, 2002;179: 593-621. 3. Harten A. High resolution schemes for hyperbolic conservation laws. J. Comp. Phys., 1983; 49: 357- 393. 4. Stoker JJ. Water Waves. The Mathematical Theory with Applications, Interscience, London, 1957.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Numerical Simulation of 2D Flood Waves Using TVD Scheme WenliWei^W.Y.He 2 1

Institute of Water Conservancy and Hydraulic Engineering, Xi 'an University of Technology, Xi 'an, Shanxi, 710048 China 2 Science school, Xi 'an University of Technology, Xi 'an, 710048 China Email: [email protected] Abstract In recent studies, the type of shock-capturing scheme called total variation-diminishing (TVD) scheme, was proposed by Harten, and was applied widely in gas dynamics. The TVD scheme has second-order accuracy, is oscillation-free across discontinuities, and does not require additional artificial viscosity. Therefore it was introduced to solve hydrodynamics for free-surface flows, in particular, for recent complex dam-break flows. Delis and Skeels made a comparison with several different TVD schemes (i.e., symmetric, upwind, TVD-MacCormack, and MUSCL scheme) to predict one-dimensional (ID) dam-break flows. Based on the above research results, a high-resolution algorithm model is proposed for the solution of the 2D shallow water equations by adopting the numerical flux for the TVD scheme, the Strang type operator splitting method and the finite volume method (FVM).And this model was used to predict the flood evolution process caused by the instantaneous partial dam-break, and the simulating results indicate that this model is fairly effective for simulating dam-break flood waves. REFERENCES 1. Harten A. High resolution schemes for hyperbolic conservation laws. J. Comp. Phys., 1983; 49: 357-393. 2. Delis AI, Skeels CP. TVD schemes for open channel flow. Int. J. Numer. Methods in Fluids, 1998; 26: 791-809. 3. Strang G. On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 1968; 5: 506.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Motion Analysis of the Elevating Ball by the Effect of Buoyant Force Y. Matsuo *, Y. Nakabayashi Department of Computational Science and Engineering, Toyo University, 2100 Kujirai, Kawagoe, 3508585 Japan Email: [email protected] Abstract The multidisciplinary analysis of incompressible fluid flow and rigid body motion is performed to investigate the behavior of the ball elevating by the effect of buoyant force. In the two dimensional analysis, the ball vibrates periodically by the effect of swing force generated by Karman vortex street behind the ball. Our final target is to investigate the reason the ball does not jump perpendicularly when it comes on the surface.

Figure 1: Mesh configurations of elevating ball

Figure 2: Velocity field around elevating ball (left: 7te=100, right Re=10000) REFERENCES 1. Adventure Project: http://adventure.q.t.u-tokyo.ac.jp/

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Development of an Educational Flow Simulation System Tameo Nakanishi*, Hironori Shibata, Mutsumi Sato Department of Mechanical Systems Engineering, Yamagata University, Yonezawa, 9928510 Japan Email: [email protected] Abstract In late years computational fluid dynamics (CFD) progresses drastically and becomes an important tool of engineering design. Various flow simulation systems have been developed and gained big success. Effective use of these systems, however, requires background of fluid mechanics and numerical analysis. These systems are still expensive to be used for college education. To meet the demand of engineering education, we developed an interactive two-dimensional flow simulation system. The system solves incompressible laminar and turbulent flows around multiple bodies of complex geometry. The core technology of the system is based on an overset grid method, a semi-Lagrangian scheme of third-order accuracy, and a one-equation turbulence model newly developed by the first author. The system integrates all necessary functions of flow simulation, and is tough enough that anyone could easily handle it. The system enables qualitative real-time simulation by use of large time step and quantitative simulation by use of adequate time step of a specified flow problem. The operation of the flow system is similar to that of using the Microsoft PowerPoint to draw a picture. The control points of an arbitrarily body shape are sequentially inputted by mouse. The system allows addition, deletion and moving of control points. Dragging, enlarging, rotating and copying of bodies are supported. As sample shapes, a circle, a square, NACA 4-digit airfoils, a car shape are prepared. These shapes can be modified to make new shapes. The main-grid covering the whole computational domain and the sub-grid around each body are automatically generated. The flow is visualized by velocity vectors, pressure and velocity contours, and stream lines. The pressure and shear stress plots of body surface, the time-plot of aerodynamic coefficients are supported.

Figure: Outline of the flow system — 178 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Using Finite Element Program Generator to Solve N-S Equation S. Wan1*, M.P. Nielsen2, G. Chai * 1

Transportation College, Southeast University, Nanjing 210096, China Department of the Civil Engineering Technical University of Denmark Email: [email protected] 2

Abstract The operator splitting method is used to deal with the N-S equation, in which the physical process described by the equation is decomposed into 2 processes: a diffusion process and a convection one. They are analyzed separately. Diffusion equation is solved first, and then the result is used in the solution of the convection equations. The virtual work equation of the diffusion equation is made. According to the standard FE assembling process, its FE equation is obtained. The mixed interpolation method in which the velocity field in the element is describe by shape function of the isoparametric element with 9 nodes and the pressure field is described by the interpolate function of the 4 nodes at the vertex of the isoparametric element with 9 nodes. The subroutine of the element and the integrated FE code are generated by the Finite Element Program Generator (FEPG) successfully. The lid-driven cavity problem is analysed. The physical parameters of the liquid in the cavity are: p = 1.0kg/m3 , ju = 10"3 kg/m • s .The boundary conditions in the lid-driven cavity problem are veloity on the surface on the cavity is zero excepte the top surface.The velocity along the top surface is given as a contant value. The grid is generated by software Gid. Mesh is also graded near the wall using a geometrical progression.The multi-frontal solver is used to solve the FE equation.Time increment is At = 0.01 sec and total time is 30 sec.The solution is used to compare with results of reference paper. The component in x direction of velocity distribution in the cavity along vertical centre-line,stream-function contour and the pressure contour calculated for Reynolds number Re=1000 are given.The solutions follow the results of reference paper very well.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Experimental and CFD Study of the Effects of Design Parameters on Reynolds Number in a Short Duration Hypersonic Test Facility Amir Al-Falahil*9 T. Yusaf2, M. Z. Yusoff2 1

INTI College Malaysia, School of Engineering and Technology, Jalan BBN 12/1, Bandar Bam Nilai, 71800 Negeri Sembilan, Malaysia 2 University Tenaga Nasional UNITEN, College of Engineering, jalan puchong-kajang, 43009 Kajang Selangor Darul Ehsan, Malaysia Email: [email protected] Abstract The first phase of this paper is aimed to develop a standard test procedure to check the effects of the design parameters (e.g. diaphragm pressure ratio, speed of sound ratio, temperature ratios, and geometry) in a short-duration hypersonic test facility that build at the Universiti Tenaga Nasional "UNITEN" in Malaysia on Reynolds number. The facility has been designed, built, and commissioned for different values of diaphragm pressure ratios and Mach number. A theoretical model was developed to evaluate the Mach number values as a function of diaphragm pressure ratio for different working fluids. The second phase is to run experimental tests for different operating conditions. The calculated parameters which are pressure, temperature and velocity were very comparable to the practical results. A high precision in house made thermocouple was used to measure the temperature profile during the facility operation. A numerical transient heat transfer mathematical model was developed to evaluate the heat flux from the surface temperature history. Comparing these results with a CFD model using commercial software Fluent was found to be much matched. The principle of operation and the reasoning behind building such a facility are explained, and the governing equations for the shock tube are presented. The selection of the shock tube parameters is explained. REFERENCES 1. Anderson JD. Modern Compressible Flow with Historical Perspective. McGraw Hill, New York, 1990. 2. Buttsworth DR, Jacobs PA, Jones TV. Simulation of Oxford University Gun Tunnel performance using a quasi-One-dimensional model. Shock Waves, 2002; 11: 377-383. 3. Buttsworth DR. Heat transfer during transient compression: measurements and simulation. Shock Waves, 2002; 12: 87-91. 4. Jacobs PA. Quasi-one-dimensional modeling of a free-piston shock tunnel. AIAA Jornal, 1994; 32(1):

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Investigation of Multiphase Flows near Walls with Textures by the Lattice Boltzmann Method Y. T. Chew*, J. J. Huang, C. Shu, H. W. Zheng Department of Mechanical Engineering, National University of Singapore, Singapore 119260 Email: [email protected] Abstract During the last decade interest in multiphase flows near textured surface has revived due to its importance in microfluidics and the study of the self-cleaning property of some biological surfaces (often called "Lotus Effect"). Here we investigate the three dimensional multiphase flows near solid walls with textures, e.g., array of small pillars, with the free energy based lattice Boltzmann method (LBM). The model uses two sets of distribution functions along some fixed directions which evolve at every discrete time step. It incorporates the physics of surface tension and wetting on a solid wall using a prescribed free energy function which effects are reflected through the equilibrium distribution functions and the collision step of LBM. Using the Chapman-Enskog multiscale expansion, the Navier-Stokes equations including the surface tension effect and the Cahn-Hilliard equation for the order parameter can be obtained at the long wavelength and long time limits. In dealing with the wetting boundary conditions, the method in [2] is adopted. Drop behaviors near a single pillar and multiple pillars have been studied. It is found that the model can simulate the static equilibrium and dynamic evolution of drops to a reasonably satisfactory level. It is noted that similar work has been performed (e.g., [1]) with the single component model for which only one set of distribution function is used and the density ratio is limited to be small. In some of the present simulations, with the newly improved model [3], the density ratio can be much larger.

Figure: Drop on a hydrophobic wall with (left) One pillar (right) Pillar arrays REFERENCES 1. Dupuis A, Yeomans JM. Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions. Langmuir, 2005; 21: 2624-2629. 2. Briant A J, Papatzacos P, Yeomans JM. Lattice Boltzmann simulations of contact line motion in a liquid-gas system. Phil. Trans. R. Soc. Lond., A, 2002; 360: 485-495. — 181 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Numerical Simulation of the Microstructure of Magnetorheological Fluids in Magnetic Fields X. H. Peng *, H. T. Li Institute of Intelligent Structures, Chongqing University, Chongqing, 400044 China Email: [email protected] Abstract A magnetorheological fluid (MRF) is a suspension consisting of ferromagnetic particles suspended in some kind of liquid. When an MRF is set in a parallel static magnetic field, the particles will be polarized and interact with each other, leading to an anisotropic chain structure. The apparent viscosity of the MRF may increase remarkably, and its fluidity may decrease and even disappear, which is so called MR effect. The currently widely used constitutive relationship for the behavior of MRFs is the Bingham's model, which phenomenologically describes the effect of viscosity, shear strain rate and the initial yield shear stress on stress response. It is known that the macroscopic behavior of an MRF is determined by the interaction between many factors, which cannot be simply described with the Bingham's model. In this paper, a micro-to-macro approach is proposed for the constitutive behavior of MRFs, based on static magnetics and the 2nd Newtonian Law. Three kinds of forces: magnetic force, repelling force and viscous resistance, on each particle in an MRF subjected to a parallel static magnetic field are introduced, and the process of chain formation is analyzed with the viewpoint of micromechanics and making use of some concepts of molecular dynamics and discrete element methods. Considering the difference with the conventional molecular dynamics in the aspect of the size of particles, the distance of two particles at equilibrium and the forces between particles, which lead to additional difficulties in the stability of numerical simulation, a combined fixed and variable time step method is adopted. Different from the conventional discrete element method, long distance interaction exists in MRFs. In order to reduce computational time, the searching region is reduced to a minimum by defining a cut-off distance, provided the distance between two particles is sufficiently large so that the long distance interaction is negligible. A combined link-cell and Verlet list method is employed in the simulation, so that the algorithm has a time complexity of O(N). It proves that the developed algorithm is of satisfactory convergence and efficiency, and is capable of computing the response of an MRF with sufficiently large number of ferromagnetic particles suspended in viscous liquid. The chain structure and its variation of an MRF under a static magnetic field and/or external shear displacement as well as the corresponding macroscopic response, are simulated with the proposed model and the corresponding numerical approach. The parameters used for the simulation are listed. The figures in the paper have shown the simulation of mieroetrueture in MRF under an applied magnetic field and applied shear strain, and the response of the shear stress versus shear rate. The results are in satisfactory agreement with experimental results. It should be noticed that, the effects of the influencing factors, such as the intensity of the magnetic field, the size, property and volume fraction of particles, the viscosity of fluid, temperature and even the surface roughness of particles, on the macroscopic responses, can be taken into account. This is advantageous for the design of new and high-performance MRFs.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Implementation of a 3D Multilaminated Hydromechanical Model for Analysis of an Unlined High Pressure Tunnel N. S. Leitao, L. N. Lamas * LNEC - National Laboratory for Civil Engineering, Av. do Brasil 101, 1700-066 Lisboa, Portugal Email: [email protected] Abstract In fractured rock masses in the presence of water, modelling of the excavation of an underground opening and of its infilling with water under high pressure needs to take into consideration the interaction between the hydraulic and the mechanical problems as an integrated process. The purpose of this paper is to report the first stage of the implementation within FLAC30 of a coupled hydromechanical model using an iterative procedure between two independent hydraulic and mechanical sub-models. The fracture network of the rock mass is considered using the multilaminated medium concept. The implemented model is illustrated through its application to the hydraulic circuit of the Venda Nova II hydroelectric power scheme, in Portugal [1]. Hydromechanical interaction is of particular relevance in this project, owing to the high water pressures involved and to the fact that the pressure tunnel is unlined. The paper starts with a short description of the main characteristics of the Venda Nova II hydraulic circuit. The main assumptions and the conceptual model are then presented, namely as regards the geometry of the openings, the mechanical and hydraulic behaviour of the rock mass and the hydromechanical interaction [2]. The main characteristics of the numerical model and the implementation of the hydromechanical interaction mechanisms are described. Finally, some results corresponding to the first infilling of the unlined pressure tunnel are presented.

Figure: Geometry of the FLAC30 numerical model and grid at tunnels level and main powerhouse cavern REFERENCES 1. Oliveira MA, Ribeiro V, Apolinario V, Costa JA. The Venda Nova II pumped storage scheme. Int. J. on Hydropower & Dams, 2004; 11(5): 88-92. 2. Lamas LN. Contributions to Understanding the Hydromechanical Behaviour of Pressure Tunnels. PhD thesis, University of London, UK, 1993. — 183 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Prediction of Mixing and Reacting Flow Inside a Combustor A. L. De Bortoli * UFRGS/IM- DMPA, Bento Goncalves 9500 -P.O. Box 15080, Porto Alegre, RS - Brazil Email: [email protected] Abstract The improvement of combustor devices is very important especially for aeronautical applications; combustors or burners are mechanical equipments where the combustion takes place. New technologies, including mixture and reaction, instability, low cost and emission are helping the development of numerical codes employed in combustion problems. But, when the burners are improved, unexpected problems can appear. The field of combustion can be divided into non-premixed and premixed combustion. The majority applications of technical interest are classified as non premixed and turbulent. To model non premixed flames it is necessary a good understanding of the combustion process and of turbulent mixing because the reaction take place when the fuel and the oxidizer mix in a molecular level. In combustion problems there is a strong coupling among transport (heat transfer, molecular diffusion, convection, turbulent transport,....) chemistry and fluid dynamics: it is a multidisciplinary topic of research. Moreover, combustion models may turn very complex: the complete mechanism of methane combustion has been identified as having more than 300 elementary reactions and over 30 species. The complete reaction mechanism of iso-octane oxidation includes 3600 elementary reactions among 860 chemical species. Due to computational resource limitations to perform Direct Numerical Simulation (DNS), for moderate to high Reynolds values, Large-Eddy Simulation (LES) seems to be a good alternative. LES turns attractive since only the sub-grid scales and chemical reactions need to be modeled; the large scales are fully resolved. The LES equations are closed by modeling the stress tensor with an eddy viscosity. Therefore, the application of LES requires appropriate sub-grid scale models, fast computers and accurate and robust methods. The aim of this work is the numerical solution of mixing and reacting flows inside a combustor. The model considers the overall, single step, irreversible reaction between two species, being the reaction rate governed by the Arrhenius law. Numerical tests, based on the finite difference explicit Runge-Kutta five-stage scheme, for third order time and fourth order space approximations, are carried out for Reynolds 10000, Damkohler 300, Zel'dovich 10, Heat Release 10 and Schmidt and Prandtl both equal to 0.9 and the results contribute to obtain a better understanding of the mixture-reaction behavior for the reacting flow inside a combustor. REFERENCES 1. Boersma B J, Lele SK. Large Eddy Simulation of Compressible Turbulent Jets, Annual Research Briefs, Center for Turbulence Research, 1999, pp. 365-377. 2. Poinsot T, Veynante D. Theoretical and Numerical Combustion, 2nd Edition, Edwards, 2005. 3. Sone K, Patel N, Menon S. Large-Eddy Simulation of Fuel-Air Mixing in an Internal Combustion Engine, AIAA paper 2001-0635, 2001.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Large Eddy Simulation of Turbulent Reactive Flow Y. Liu l - 2 *, Y. Q. Chen3, J. G Chen l>2, R. Yang l>2 1

Department of Engineering Physics, Tsinghua University, Beijing, 100084 China Center for Public Safety Research, Tsinghua University, Beijing, 100084 China Department of Mechanics and Engineering Science, Peking University, Beijing, 100871 China Email: [email protected] 2

Abstract The wide range of temporal scales involved in the turbulent reactive flow brings great requirements to the numerical simulation processes. The modeling of reactive flows with multi-step detailed chemical kinetics calls for the small enough time step size, which will result directly in a huge computational cost. Different from the fixed time step methods adopted in most numerical simulations of the turbulent reactive flow currently, a method of self-adaptive time step is developed in this paper, in which the size of the time step in computation is automatically determined according to the time scales of both the chemical reaction and the turbulent fluctuation. Thus, compared with the fixed, very small time step throughout the whole calculation, the current self-adaptive time step method can reduce computational expense greatly. In order to validate this method, large eddy simulations of a methane-air turbulent reactive planar jet flow is carried out in this paper with the self-adaptive time step method. The mechanism of a simplified 4-step chemical kinetics is applied for the methane-air reaction. The dynamic model is adopted for the turbulent motion of sub-grid scale. The dynamic similarity model is used as the SGS model for the reaction rate. The LES results with the self-adaptive time step method depict satisfactorily the ignition process of the turbulent jet flame and give clear and detailed coherent structures of the fully developed turbulent reactive jet flow. The self-adaptive time step method helps to match the balance between saving computational efforts and catching more flow details.

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(a) LES result with self-adaptive time step (b) LES result with fixed time step Figure: Comparison between results of fixed and self-adaptive time step REFERENCES 1. Chen YQ. Theoretical statistical solution and numerical simulation to heterogeneous brittle materials. Acta Mechanica Sinica, 2003; 19(3): 16-25. 2

T in Y. Lau KS, Chan CK, Guo YC, Lin WY. Structures of scalar transport in 2D transitional jet diffusion flames by LES. Int. J Heat and Mass Transfer, 2003; 46(20): 3841-3851.

3. James S, Jaberi FA. Large scale simulations of two-dimensional nonpremixed methane jet flames. Combustion and Flame, 2000; 123: 465-487. — 185 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Computational Modeling of Coal Water Slurry Combustion Processes in Industrial Heating Boiler L. J. Zhu, B. Q. Gu * College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, 210009 China Email: [email protected]; [email protected] Abstract Coal water slurry (CWS) is typically composed of 60-70% coal, 30-40% water, and 1% chemical additives. It has been developed over the last 20 years as an alternative to fuel oil mainly in industrial and utility boilers. In this paper, a commercial computational fluid dynamics (CFD) software Fluent was utilized to investigate numerically the flow field, the heat transfer and the combustion processes in a CWS-fired industrial heating boiler. The simulation results indicated that there existed swirl flow field in the furnace; the high-temperature gases expanded outward to form the recirculation zones near the burner areas; the temperature dropped with the increase of the furnace length, and the highest temperature appeared in the area of the burners; from the front wall to the back wall of the furnace, the swirl intensity firstly strengthened and then weakened; the distribution of CO, O2 and CO2 concentration was strongly related to the temperature field, that is, the higher the temperature, the higher the CO concentration is and the lower the O2 and CO2 concentration; the average exit temperature, the maximum furnace temperature and the burn-off rate were affected by the boiler load. The distribution of the velocity, the temperature and the particles trajectories in the furnace was also analyzed. The obtained results provided a helpful reference for the design, the running and the reconstruction of the CWS-fired industrial heating boiler. A v e r a g e * x » M n x i n u u u ftinwce rdinjxnture fenijxtr.inire

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Figure: Change of some parameters with boiler load while firing Datong CWS REFERENCES 1. Miller BG. Coal-water slurry fuel utilization in utility and industrial boilers. Chemical Engineering Progress, 1989; 85: 329-338. 2. Shih TH, Liou WW, Shabbir A et al. A new k-e eddy-viscosity model for high Reynolds number turbulent flows-model development and validation. Computers & Fluids, 1995; 24: 227-238. 3. Philip JS. Recent applications of CFD modeling in the power generation and combustion industries. Applied mathematical modeling, 2002; 26: 351-374. — 186 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Large Eddy Simulation of Unsteady Turbulent Flow and Pressure Fluctuation in an Axial-Flow Pump with Various SGS Models Guohui Cong *, Fujun Wang, Ling Zhang College of Water Conservancy & Civil Engineering, China Agriculture University, Beijing, 100083 China Email: [email protected] Abstract Three dimensional Large Eddy Simulation (LES) is performed to predict the unsteady turbulent flow and pressure fluctuation in an axial-flow pump with two different Subgrid-Scale (SGS) models. The specific speed of the axial-flow pump is 518. The calculated flow rates are Q/Qd =0.7, 0.8, 0.9 and 1.0, where Qd is the flow design point. Both the classical Smagorinsky-Lily SGS model [1] and the dynamic SmagorinskyLily SGS model [2] are used as the turbulent eddy viscosity model of LES with finite volume method and slide-mesh methodology, and the constant Cs in the later model is dynamically computed based on the information provided by the resolved scales of motion. The computational domain includes the rotator, the fixed guide-vane and the diffuser of the axial-flow pump (See Figure 1). The total head, shaft power and efficiency of the pump are obtained by the simulation and compared with the experimental measurements. The prediction results of the two SGS models are also compared. Both the SGS models successfully resolve the turbulent flow in the pump, and have good agreements with the experimental data. The pressure fluctuation of the flow in pump is predicted too, and the results show that the pressure fluctuation may be the leading role in unsteady hydraulic force generation. The simulation indicates that the numerical method presented in this paper has its potential capability to predict 3D unsteady turbulent flow and pressure fluctuation in axial-flow pumps, and the method can extend to other similar hydraulic machine too.

Figure: 3D model of the axial-flow pump REFERENCES 1. Smagorinsky J. General circulation experiments with the primitive equations. I. The basic experiment. Month. Wea. Rev., 1963; 91: 99-164,. 2. Kim S-E. Large eddy simulation using unstructured meshes and dynamic subgrid-scale turbulence models. Technical Report AIAA 2004 2548, American Institute of Aeronautics and Astronautics, 34th Fluid Dynamics Conference and Exhibit, June 2004.

— 187 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Computation of Turbulent Flows in Natural Gas Pipes with Different Rectifiers Z. L. Li *, Y. X. Zhang Department of Mechanical and Electronic Engineering, China University of Petroleum-Beijing, Beijing, 102249 China Email: [email protected] Abstract With the development of natural gas industry, improving the accuracy of the flowmeter systems is becoming the common concerned problem both corporations and customers. Many experiments demonstrated that the different flow field structures of pipes with diverse baffle installations had great influence on the performance of flowmeters. In order to measure the flux in pipes more accurate, the flowmeters must be installed downstream the baffles far away so that the irregular flows induced by baffles had less influence on the flowmeters. For example, CFD was used to compute the flow fields of a diffusive pipe with different rectifiers in this paper. The lengths of the irregular flows induced by the diffuser and the rectifiers were shown in flow fields. By analysis and comparison the numerical results, the rectification of the different rectifier were confirmed. Furthermore, the experiments will be done soon with PIV to verify the numerical results. If the numerical results are in good agreement with the experiments, the conclusions derived from CFD will become the leading guidance to deal with the local flowmeter selection and installation in the future.

— 188 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Computation of Unsteady Incompressible Turbulent Flow by Using an Implicit SMAC Method Y. X. Zhang1*, Z. L. Li1, B. S. Zhu2 1

Department of Mechanical and Electronic Engineering, China University of Petroleum-Beijing, Beijing, 102249 China 2 Department of Thermal Engineering, Tsinghua University, Beijing, 100084 China Email: [email protected] Abstract An implicit numerical method is developed based on the simplified marker and cell (SMAC) method to solve the 3-D unsteady incompressible turbulence Reynolds-averaging equations in general curvilinear coordinates. The governing equations include the Reynolds averaged momentum equations with contravariant velocities as the unknown variables, pressure Poisson equation and k-s turbulent model equations. The governing equations are discretized in 3-D MAC staggered grid. To improve the numerical stability of the implicit SMAC scheme, the higher-order high resolution Chakravarthy-Osher TVD scheme is used to discretize the convective terms in momentum equations and k-s equations. The algebraic equations of momentum equations and k-s equations are solved by CTDMA method. The algebraic Poisson equations are solved by the Tschebyscheff SLOR method with alternating computational directions. The unsteady flows in a simplified hollow-jet valve are simulated with the code written by the first author. The computational results are in satisfactory agreement with the experimental photographs.

— 189 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Simulation on Vortex Effect for Superconducing Devices Siu-Long Lei, Man I Lao *, Iat-Neng Chan Department of Electrical and Electronics Engineering, University of Macau, Macau, China Email: [email protected] Abstract The simulation of vortex dynamics motion in superconductor is based on the time-dependent Ginzberg-Landa equations in the non-equilibrium state. Runge-Kutta's method is used for time stepping and finite difference method is for space derivative during the simulation for the effects of applied magnetic field, vortex motion and material defects. In the research, simulation for several special electronic devices are accomplished for functional test. At least one design has been compared with measurement data in a real circuit. Major characters of simulation result are analyzed and reflected the reality of the notable phenomena as Meissner effect, penetration depth, coherent length and pinning force of superconductivity. There are two major objects have been done in the simulations. One is for the modification on the electric field and the magnetic field. The other is for the material properties of superconductivity. One interested subject is about the field slightly lower than the critical field He. The vortex movement during non-equilibrium state of the surface superconductivity is studied. The primary result show that the superconductivity near the surface is fluctuated and the central part is forming small vortex clusters. The relation between vortex fluctuation with material property and applied field is also investigated. REFERENCES 1. Gaitan F. Microscopical analysis of the nondissipative force on a line vortex in a superconductor. J. Phys.: Condens. Matter, 1995; 7(12): L165-L170. 2. Du Q, Gunzburger MD, Peterson JS. Analysis and approximation of the Ginzbrug-Landau model of superconductivity. SIAM Review, 1992; 34(1): 54-81. 3. Lele Sanjiva K. Compact finite-difference schemes with spectral-like resolution. Journal of Computational Physics, 1992; 103: 16-42.

— 190 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Simulation of Corrosion Rate of Carbon Steel Subjected to Elastic/ Plastic Strain M.Ridha 1 *, S.Aoki 2 1

Department of Mechanical Engineering,Syiah Kuala University, Banda Aceh, NAD, Indonesia Department of Computational Science and Engineering, Toyo University, Kawagoe, Saitama, Japan Email: [email protected]

2

Abstract In this research, the effects of elastric and plastic strain to the corrosion rate of carbon steel were studied. The corrosion rate of the steel was represented by its polarization curve. Experimental works using carbon steel plate specimen were carried out. The carbon steel plate was bended into U shape to initiate the elastic or plastic strain on the specimen. The U-bend type specimen was immersed into the artificial seawater and the polarization curves were measured on the surface of the steel specimen which has the highest and lowest stress. The MSC Nastran code was used to simulate the stress distribution on the U-bend steel specimen subjected to elastic/plastic strain. Boundary element method was applied to obtain the potential and current distribution on the surface of the U-bend type steel specimen due to the galvanic effect. The result shows that corrosion rate of the carbon steel was affected by the magnitude of the elastic and plastic strain working on the steel. In addition, the bigger magnitude of the elastic or plastic strain working on the steel the higher the corrosion rate. In corrosive environment, the effect of galvanic will occur on the steel surface with different magnitude of the elastic or plastic strain. The area on the steel surface which has biggest magnitude of the elastic or plastic strain becomes the most anodic area and becomes the easiest area to be attacked by corrosion. REFERENCES 1. Aoki S, Amaya K, Miyasaka M. Boundary Element Analysis on Corrosion Problems. Shokabo, Tokyo, 1998 (in Japanese). 2. Adey RA, Niku SM. Boundary element simulation of galvanic corrosion - the story of a major success for boundary element method. Boundary Element X, Vol. 2: Fluid Flow and Electrical Application, Computational Mechanics Publications, Springer-Verlag, 1988. 3. Fontana MG, Greene ND. Corrosion Engineering, 2nd edition, McGraw-Hill International, 1983.

— 191 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Effect of Surface Traction on the Shakedown Limit under Moving Surface Loads Jim Shiau * Faculty of Engineering and Surveying, University of Southern Queensland, 4350, QLD, Australia Email: [email protected] Abstract The effect of repeated surface friction on the shakedown limits for an isotropic, homogeneous Tresca material is examined in this paper using a lower bound finite element and mathematical programming approach. Based on Bleich-Melan's static shakedown theorem, numerical solutions are obtained for this problem and compared with the analytical solution proposed by Johnson (1985). A shakedown map for various surface friction is prepared for design purposes. 4.5-r Subsurface Failure 4.04 3.54 C

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Figure: A shakedown map indicating the effect of surface friction upon dimensionless shakedown limits REFERENCES 1. Shiau JS. Numerical Methods for Shakedown Analysis of Pavements under Moving Surface Loads. Ph.D. Thesis. The University of Newcastle, NSW, Australia, 2001. 2. Johnson KL. Contact Mechanics. Cambridge University Press, 1985.

— 192 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Elasto - Plastic Finite Element Analysis of Tapered Steel Silo C. L. Xu *, Y. F. Luo, H. J. Song College ofcivi Engineering, Tongji University, Shanghai, 200092 China Email: [email protected] Abstract The tapered steel silos are used in storage and accumulation of bulk granular material. Their usage in different branches of industry is numbered about one hundred already, but they are one of the worst studied building constructions. The reason is that steel silo design is a very complicated subject covered by the thin shells and stiffened plate structures. The load distribution of the steel silos is uncertain. The stresses of the members and shells is difficult to be accurately calculated during the design of the steel silos. In addition, there are no special analysis techniques and assesing methods for the design of the tapered steel silos in current national design codes of our country, which results in creating uneconomic and unreliable silo and supporing system. The cases of accidents and damages of the steel silos are well known to us in engineering practice. It is necessary to carry on the large displacement elasto-plastic finite element analysis for the tapered steel silo structure to obtain the ultimate loading of the members and shells during design. According to the design command, the large displacement elasto-plastic analysis of the tapered steel silo structure of Taiyuan Mining Vertical Preheater is conducted in the paper. The silo structure is divided into simple elements and calculations were carried out by means of ANSYS software. The reliable numerical data are provided to the design. The valuable ideas about the loading behavior of the tapered silo are concluded. It is a useful reference for design of similar structure.

Figure 1: Finite element model of tapered steel silo

— 193 —

Figure 2: Sress distribution of shell elements

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Simulation of Failure Process in Heterogeneous Elastoplastic Materials Yazhe Li, Yongqiang Chen* Department of Mechanics and Engineering Science, Peking University, Beijing, 100871 China Email: [email protected], [email protected] Abstract A general numerical approach is developed to simulate the failure process of structure or heterogeneous material, in which different local failure models are defined for different structures and materials to describe the failure of structure joint or material element. A two-parameter Weibull distribution is employed to produce the initial heterogeneity of material, i.e., the Young modulus and the failure strain. An appropriate elastic-plastic constitutive relation with failure strain value is used to describe the material behavior and failure in elements. During the displacement-controlled loading, the different elements would be in different stages of deformation, i.e. linear elastic stage and plastic stage. At each load step, an overall sweep is first carried out to determine deformation statue of each element, and find the one which is most probable to fail to determine the size of load step, so that a specific material element will enter the plastic stage from elastic stage or reach failure strain if it is already in plastic stage. Thus when this adaptive load incremental is applied, it would occur that only element changes its behavior status, i.e., from elastic stage to plastic stage or strain failure. The failed element would be assigned with a very small modulus to make it broken rather than remove it from network, which keep the geometric continuity. Applications of this proposed method to 3D and plane stress problems are presented. The numerical results have a good agreement with the test data. The numerical results show that it is suitable to simulate the failure process of materials with local elastoplastic relations of stresses vs. strains (e.g. concrete at higher temperature) under the quasi-static loading and to model the complicated nonlinearity of realistic material. It also provides a method to evaluate the effective properties and statistic measurements of practical engineering material and some composite materials. REFERENCES 1. Lilliu G, van Mier JGM. 3D Lattice Type Fracture Model for Concrete. Engineering Fracture Mechanics, 2003;70:927-941. 2. Schlangen E, Garboczi EJ. Fracture Simulations of Concrete Using Lattice Models: Computational Aspects. Engineering Fracture Mechanics, 1997; 57: 319-332. 3. Chen YQ, Yao ZH, Zheng XP. Theoretical Statistical Solution and Numerical Simulation of Heterogeneous Brittle Materials. Acta Mechanica Sinica, 2003; 19(3): 276-284.

— 194 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

Fatigue Damage Analysis of Reactor Vessel Model Under Repeated Thermal Loading M. Takagaki1*, Y. Toi1, T. Asayama2 1

Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan O-arai Research and Development Center, Japan Atomic Energy Agency, 4002 Narita-cho, O-arai-machi, Higashi-ibaraki-gun, Ibaraki 311-1393, Japan Email: [email protected] 2

Abstract The local approach to fracture based on continuum damage mechanics and the finite element method has been applied to the low-cycle fatigue damage and fracture behavior of a nuclear reactor vessel model subjected to repeated thermal loading due to the alternate inflow of high-temperature and low-temperature liquid sodium. The viscoplastic strain based on the creep plasticity isotropic hardening theory with damage evolution is used in the analysis. The calculated results for the neibourhood of intake nozzle and the conical body have been compared with the experimental results. The calculated results considering the difference of mechanical properties of parent metal, weld metal and heat-affected zone have corresponded well with the experimental results for the initiation and propagation of thermal fatigue cracks. 0.99 079 059 040 020 0C0

Figure: Damage distribution in conical body of nuclear reactor vessel model REFERENCES 1. Lemaitre J. A Course on Damage Mechanics. 2nd Ed., Springer, 1996. 2. Skrzypek J, Canczarski A. Modeling of Material Damage and Failure of Structures (Theory and Applications). Springer, 1999. 3. Toi Y, Lee JM. Thermal Elasto-Viscoplastic Behavior of Structural Members in Hot-Dip Galvanization, Int. J. of Damage Mechanics, 2002; 11(2): 171-185.

— 195 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Interaction of Moving Interfacial Cracks between Bonded Dissimilar Elastic Strips under Antiplane Shear Subir Das * Department of Mathematics, B. P. Poddar Institute of Management & Technology, Poddar Vihar, 137, V.I. P. Road, Kolkata, 700052 India Email: [email protected] Abstract In this paper the Interaction between three moving collinear Griffith cracks under antiplane shear stress situated at the interface of two bonded dissimilar fixed elastic strips has been studied. Fourier Transform and Finite Hilbert transform techniques have been employed to solve the problem. Analytical expressions for stress intensity factors at the crack tips have been derived. Numerical results connected to the interaction effect have also been obtained. Depending on the spacing of the cracks and their common velocity of propagation, occurrence of shielding and amplification phenomena of the cracks have been noticed.

— 196 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X.Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Stress Intensity Factor of a Wide Range of Semi-Elliptical Partly Through-Wall Crack in a Finite-Thickness Plate K.P.Kou* Department of Civil and Environmental Engineering, University of Macau, Macao, China Email: [email protected] Abstract In practice, cracks of any awkward shape initiated accidentally, e.g. fabrication errors, exist in structures such as offshore steel structure. Existing of these flaws could lead to disastrous accidence due to fatigue effect. Therefore, a safe interval of maintenances is substantial in order to keep a reliable residual life and strength of the structure. To assess the fatigue effect, one approach is to make use of the stress intensity factor(SIF) of the cracks in facture mechanics. These flaws are usually treated as semi-elliptical surface cracks(with a and c as the minor and major axis) on the member of the structure e.g. BS7910. For plates with semi-elliptical surface crack, Newman, Raju and Kou have reported the stress intensity factor. However, for plates with partly through-wall cracks, namely an semi-elliptical crack with an imaginary crack depth a which is greater than the thickness (T) of the plate, the stress intensity factor was not reported. In the current work, semi-elliptical partly through-wall cracks (of sizes 1.0
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Figure: A 3D model with partly through-wall crack REFERENCES 1. Raju IS, Newman JCJr. Stress-intensity factor for a wide range of semi-elliptical surface cracks in finite-thickness plates. Engineering Fracture Mechanics, 1979; 11: 817-829. 2. Kou KP. The SIF for long-deep semi-elliptical surface crack in finite thickness plates. Proc.ABAQUS User Conference, Stockhom, 2005.

— 197 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Numerical Simulation of the Fracture Spacing in Two-Layered Material Subjected to Thermal and Mechanical Loading Lianchong Li **, Chun'an Tang2, Zhengzhao Liang2 1

Centre for Rock Instability and Seismicity Research, Northeastern University, Shenyang, 110006 China School of Civil Engineering, Dalian University of Technology, Dalian, 116622 China Email: [email protected]

2

Abstract With the knowledge of heterogeneous characteristics of materials at mesoscopic level, a thermomechanical-damage (TMD Model) coupled model, based on elastic damage mechanics, thermal mechanics theory and FEM technology, is imported into Material Failure Process Analysis (MFPA) code and the MFPA-TMD analysis method for the failure process analysis of materials subjected to coupled thermo-mechanical loading is proposed. Using MFPA-TMD, analyses have been carried out to investigate the fracture behavior in ceramic coatings subjected to thermal loading. Hypothetical material properties have been considered as material data for coupled (thermal and mechanical) finite element analyses. These properties were chosen by assumed changes in some functional properties of ZrO coatings. The aim was to evaluate the bearing capacity in different coatings. Furthermore to demonstrate the influence of crack length and coating geometry on the bearing capacity of coating, numerical analyses were carried out for various cases. Numerical results reproduced the fracturing process of coating materials subject to thermal and mechanical loading. The bearing capacity of coating was measured. The results showed that the shorter the crack length and the thinner the coating, the sounder the coating. Mathematically, MFPA-TMD successfully gives an effective solution to the nonlinear and discontinuum mechanics problems in the failure analysis of heterogeneous materials using continuum mechanics method. It provides us with a more sensible physical intuition and a more accurate mathematical for optimizing the design and the processing of ceramic coatings subjected to coupled thermal-mechanical loading. The following is one of the numerical results for case study. The coating fractured and became warped.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

VCCM Rule-Based Meshing Algorithm for an Automatic 3D Analysis of Crack Propagation of Mixed Mode Kohei Murotani**, Kanda Yasuyoki2, Toshimitsu Fujisawa2, Genki Yagawa * 1

Center for Computational Mechanics Research (CCMR), Toyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo, 112-0001, Japan Prometech Software, Inc., Tokyo, Japan Email: [email protected] Abstract An analysis of crack propagation using FEM need to perform re-meshing for an updated shape of crack with each cycle. In an analysis of crack propagation of 3D mixed mode, the shape of crack seems to be complex. In addition, we must effectively calculate parameters of fracture mechanics and exactly forecast a direction of crack propagation. Because of this situation, we can analyze an automatic 3D analysis of crack propagation of mixed mode, if we can perform meshing satisfied the two basic conditions of VCCM (Virtual Crack Closure Method) for an arbitrary shape of crack. In this paper, we show an algorithm of a VCCM rule-based meshing for the arbitrary shape of crack and calculation results.

Figure: A crack propagation using FEM.

REFERENCES 1. Okada Hiroshi, Higashi Mayumi, Kikuchi Masanori, Fukui Yasuyoshi, Kumazawa Noriyoshi. Three dimensional virtual crack closure-integral method (VCCM) with skewed and non-symmetric mesh arrangement at the crack front. Engineering Fracture Mechanics, 2005; 72: 1717-1737. 2. Okada H, Kamibeppu T. A virtual crack closure-integral method (VCCM) for three-dimensional crack problems using linear tetrahedral finite elements. CMES: Computer Modeling in Engineering & Sciences, (accepted for the publication).

— 199 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

The State-of-the-Art Methodology to Compute the 3-D Stress Intensity Factors for Arbitrary Shaped Cracks in Complex Shaped Structures H. Okada1*, G. Yagawa2, H. Kawai3, K. Shijo4, D. Fujita4,Y. Kanda5, T. Fujisawa5, T. Iribe 5 Department of Nano-structure and Advanced Materials, Kagoshima University, 1-21-40 Korimoto, Kagoshima, 890-0065 Japan Center for Computational Mechanics Research, Toyo University, 2-36-5 Hakusan, Bunkyoku, Tokyo 112-8611, Japan Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohokuku, Yokohama 233-0051, Japan 4 Technostar Co., Ltd., M21 Bldg., 2-2-5, Roppongi, Minatoku, Tokyo 106-0032, Japan Prometech Software, Inc., Sangakurenkei-Plaza, 4F, 7-3-1 Hongo, Bunkyoku, Tokyo 113-0033, Japan Email: [email protected] Abstract In present research, engineering softwares that can perform highly efficient and accurate fracture analyses based on the finite element method (FEM) is being developed. The developed softwares will drastically shorten time and labor cost to perform the fracture mechanics analyses. Structural integrity problems have been recognized as the major concerns by engineers in power plant, civil engineering, aerospace and ship-architectural industries. One of the scenario leading to structural failure is that (1) cracks from in the structure, (2) they propagate through the structure under the applied stress and (3) unstable crack propagation lading to catastrophic structural failure occurs. In this scenario, it is very important to detect the structural defects by non-destructive inspection (NDI) techniques and to predict how the existing cracks propagate through the structure. Thus, the remaining service life of damaged structure can be predicted. The major tool to perform the residual life prediction analysis is the finite element method (FEM). Once FEM analysis is performed for the damaged structure, fracture mechanics parameter such as the stress intensity factor is evaluated and it characterizes the severity of crack tip deformation field. Rate and direction of crack propagation are predicted based on the stress intensity factor. In this research we aim at developing softwares that can automate the fracture mechanics analysis by coupling the (1) automatic mesh generation software, (2) finite element program for large scale analysis and (3) a new virtual crack closure-integral method (VCCM) [1] for evaluating the stress intensity factor based on tetrahedral finite elements. Among them the VCCM technique is the key technology. In the presentation, the VCCM for quadratic tetrahedral finite element is described first and the-state-of-the-art mesh generation software called VENUS [2] and the use of free mesh method (FMM) [3] are discussed. REFERENCES 1. Okada H, Araki K, Kawai H. Stress intensity factor evaluation for large scale finite element analyses (virtual crack closure-integral method (VCCM) for tetrahedral finite element), submitted to Transactions of Japan Society of Mechanical Engineers (JSME), 2006. 2. Yagawa G. Node-by-node parallel finite elements: a virtually meshless method. Int. J. Numer. Meth. Engng., 2004; 60: 69-102. 3. WWW page of Technostar Co., Ltd., Japan, http://www.e-technostar.com/english/ — 200 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

3D Crack Propagation Analysis Using Free Mesh Method Hiroaki Osaki**, Hitoshi Matsubara2, Genki Yagawa x 1

Center for Computational Mechanics Research (CCMR), Toyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo, 112-0001, Japan 2 Japan Atomic Energy Agency Email: [email protected] Abstract The Free Mesh Method (FMM) is a kind of meshless method in the sense that the user can get the solution without the consciousness of meshing process. Also, the solution obtained by using the FMM is equivalent to that of the FEM. On the other hand, the computational fracture mechanics requires frequent re-meshing as the crack propagates. Therefore, the Free Mesh Method (FMM) is considered to be useful in the crack propagation analysis. In this paper, in order to show the effectiveness of the FMM in crack propagation analysis, we study the 2D and 3D crack propagation problems with the FMM . The crack propagation is simulated here according to the Erdogan-Sih Criterion and the Paris law.

REFERENCES 1. Yagawa G, Yamada T. Free mesh method: a new meshless finite element method. Computational Mechanics, 1996; 18: 383-386.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Method for Analyzing Stress Intensity Factor of a Surface Crack in Tubular Joints Yongbo Shao*, Zhifu Du, Weidong Hu School of Civil Engineering, Yantai University, Yantai, 264005 China Email: [email protected] Abstract Tubular joints are widely encountered in offshore engineering as supporting structures. Fatigue failure is very common for these structures because they are frequently subjected to cyclic loading caused by seawater and wind. In the fatigue failure process of tubular structures, surface crack will initiate at the weld toe and propagate along the chord/brace intersecting curve. The residual life of a tubular joint is dependent much on the accurate estimation of the stress intensity factor of the surface crack. As the chord/brace intersecting curve is a complicated 3-D curve, it is critically important to present a reasonable method to produce high-quality mesh in finite element analysis of the stress intensity factor of the surface crack. In this paper, a sub-zone method for generating the FE mesh of a surface crack in a tubular joint is proposed. In this method, the surface crack can be located at any position along the weld toe and be of any length. To avoid the element around the surface crack badly distorted, five element types are used to model the crack region. Thereafter, two methods, namely interaction J-integral method and displacement extrapolation method, are used to compute the stress intensity factors along the crack front. The numerical results have been verified from some reported experimental results on tubular T- and K-joints. It is found that the presented finite element method for analyzing the stress intensity factor of a surface crack in a tubular joint is effective, accurate and reliable.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

The Solutions of Stress and Displacement Fields of Orthogonal Anisotropic Plate with Edge-Cracks Z.R.Tian*,Z. Sun, Z.Y.Li Department of Mechanics, Northwestern Polytechnical University, XVan, Shaanxi, 710072 China Email: [email protected] Abstract To solve the stress-field {o^} and the displacement-field {«,} in the orthotropic plate with edge-crack, the authors established a particular method based on the concept of equivalent-space. The singularity of Kelvin solution of orthogonal anisotropic plate with crack has been using the virtual displacement method of Boundary Element Method. The Fourier transform and the basic solution of Laplace equation on the variables x and y, we could get the expressions of the stress-field {cr^} and displacement-field {w;} in the 1/4 plane(jt>0,j>>0), which are with the coefficients C(£)and D(n). In order to obtain the coefficients C(£)and D(TJ) in the integral equations which satisfy the boundary conditions, it need to solve Weber-Schafheitlins discontinuous integral equation, Bessel's and Abel style integral equations, the main problem is solving the Abel integral equation, and it can be done by transforming the Abel integral equation to the second kind Fredholm integral equation. From the solving of the second kind Fredholm integral equations, by using the third order B-Spline BjA(t), we can get the correlative coefficients a(t) and b(t) of the C(£)and D(TJ), then we have the C(£)and D(rj) to get the solutions of the {cr } and the {w,}. Thus, stress field and displacement field have also been solved. Final numerical solutions are obtained by C programming. The entire solving method is intricate, but precise in logic.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESCX,^wg. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Nonlinear FE Model for RC Shear Walls Based on Multi-Layer Shell Element and Micro-Plane Constitutive Model Z. W. Miao, X. Z. Lu *, J. J. Jiang, L. P. Ye Department of Civil Engineering, Tsinghua University, Beijing, 100084 China Email: [email protected] Abstract Nonlinear simulations for structures under disasters have been widely focused in recent years. However, precise modeling for the nonlinear behavior of reinforced concrete (RC) shear walls, which are the major lateral-force-resistant structural member in high-rise buildings, still has not been successfully solved. In this paper, based on the principles of composite material mechanics, a multi-layer shell element model is proposed to simulate the coupled in-plane/out-plane bending or the coupled bending/shear nonlinear behaviors of RC shear wall. The multi-layer shell element is made up of many layers with different thickness. And different material models (concrete or steel) are assigned to various layers so that the structural performance of the shear wall can be directly connected with the material constitutive law. And besides the traditional elasto-plastic-fracture constitutive model for concrete, which is efficient but does not give satisfying performance for concrete under complicated stress condition, a novel concrete constitutive model, referred as micro-plane model, which is originally proposed by Bazant et al., is developed to provide a better simulation for concrete in shear wall under complicated stress conditions and stress histories. Three 10-story buildings under static push-over load and dynamic seismic load, with various shear wall arrangements, were analyzed with the shear wall model proposed in this study. The simulation results show that the multi-layer shell elements can correctly simulate the coupled in-plane/out-plane bending failure for tall walls and the coupled in-plane bending-shear failure for short walls. And with micro-plane concrete constitutive law, the cycle behavior and the damage accumulation of shear wall can be precisely modeled, which is very important for the performance-based design of structures under disaster loads. REFERENCES 1. Jiang JJ, Lu XZ, Ye LP. Finite Elelement Analysis of Concrete Structures. Tsinghua University Press, Beijing, China, 2005 (in Chinese). 2. Lu XZ, Ye LP, Teng JG, Jiang JJ. Meso-scale finite element model for FRP sheets/ plates bonded to concrete. Engineering Structures, 2005; 27(4): 564-575. 3. Bazant ZP, Ferhun CC, Carol I, Adley MD, Akers SA. Microplane model M4 for concrete. Journal of Engineering Mechanics, 2000; 126(9): 944-953.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Numerical Modelling and Simulation of an Internal Combustion Engine Piston with a Surface Coating Tadeusz Niezgoda, Zdzislaw Kurowski, Jerzy Malachowski * Faculty of Mechanical Engineering, Military University of Technology, 00-908 Warsaw, Poland Email: [email protected], [email protected], [email protected] Abstract In recent years, one can observe an ever growing interest in application of special kinds of surface coatings called Graded Materials - GM and Functionally Graded Materials - FGM. They are used in various kinds of industry and have a wide range of applications, one of which are thermal barriers. The aim of applying such coatings, e.g. on elements of an internal combustion engine, may be multiple: on the one hand - upgrading the efficiency of combustion, better performance of the engine, less pollution, and on the other - obtaining higher strength and durability, as well as better tribological properties of its parts. Application of coatings for thermal barriers causes however considerable problems to overcome, such as residual stresses resulting from the difference in material properties, above all the thermal conductivity, as well as their brittleness. A universal and versatile numerical tool for the analysis of these coatings is the Finite Element Method. The aim of works presented herein was to elaborate an original methodology of Finite Element Method numerical analysis for determining the surface coatings' influence on the strength of elements of internal combustion engines, and then to perform modelling and simulation of a piston with a zirconia coating, subjected to thermal and thermo-mechanical loadings. Several FEM models of the piston with the coating were elaborated, starting with simple axisymmetric ones, and ending with a 3D model of 30° "cake slice" of the piston. The models were subjected to thermal and thermo-mechanical loadings and the analysis included calculations of temperature and heat fluxes fields, distribution of residual stresses and stresses in the working conditions of the piston. The results of computations proved high efficiency of the coating as a thermal barrier. The working temperature in some areas inside the piston with a coating was 40% lower as compared with the temperature for the same piston without a coating. An analysis of the influence of the coating's parameters on the strength of the piston was carried out.

Figure: A 3D model of the coated piston (left), and comparison of the temperature distribution in the piston without (middle) and with a coating (right) — 205 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Reliability analysis using saddlepoint approximation Jia Wang'*, Ka-Veng Yuen \ Siu-Kui Au 2 1

Faculty of Science and Technology, University of Macau, Macao, China Department of Building and Construction, City University of Hong Kong, Hong Kong, China Email: [email protected]

2

Abstract Reliability assessment has been attracting much attention for several decades in different areas of research, including aerospace, civil engineering and finance. The problem of reliability analysis is essentially the integration of the probability density function in the failure domain. Although straight-forward in principle, it is computationally prohibitive when the integration dimension (number of uncertain parameters) is large. Monte Carlo simulation (MCS) was then developed for this purpose. Its efficiency and applicability do not depend on the problem dimension but the coefficient of variation of the estimated Pp is given by COV = yj{\-PF)l{nPF) where PF is the failure probability and n is the number of samples. In practical circumstances, this failure probability is expected to be small, e.g., 10-6, because one does not anticipate to construct an engineering system to fail easily. In this case, MCS is inefficient for small PF. The saddlepoint method is a choice that does not depend on the dimension of the problem. It aims at estimating the cumulative distribution function (CDF) given a set of extreme samples of the concerned quantity. In the context of reliability, this CDF gives the reliability of the system if the argument of the function is chosen to be the prescribed bound defining failure. A commonly used saddlepoint approximation to the cumulative distribution function P(Y >)*(D(w) + ^(w)

w v where O and <j) are the CDF and PDF of the standard normal random variable w and the internal variables w and v are given by w = sign ( £ J J 2 [ £ ^ - # ( £ ) ] and v = ^yJK'^). The saddlepoint £ is found by solving K'(%s) = y. The cumulant generating function, which is defined by the natural logarithm of the moment N

generating function, is expanded as a finite series K(%s) = ln(m(£)) = ^K£}

j)'\ • In practice, Nis chosen to

be 4. One can show that the monotonicity of the y-% curve is not maintained. In other words, the relationship between the threshold value y and the saddlepoint % is not one-to-one so given a value of the threshold y, there exist more than one possible values for £. As a result, there might exist more than one possible results of the saddlepoint formulation. In this paper, we will focus on solving this crucial problem of the saddlepoint approximation. A new formulation for estimating the "saddlepoint-threshold" relationship will be developed. Numerical examples are presented to demonstrate the efficiency of the proposed methodology. REFERENCES 1. Lugannani R, Rice SO. Saddlepoint approximation for the distribution of the sum of independent random variables. Advances in Applied Probability, 1980; 12: 475-490.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Advanced Computational Method for Reliability Analysis of Concrete-Faced Rockfill Dam Qingxi Wu1*, KuiZhi Zhao2, Mingzhu Yang1 1

Department of Engineering Mechanics, Hohai University, Nanjing, 210098 China Department of Geotechnical Engineering, Nanjing Hydraulic Research Institute, Nanjing, 210024 China Email: [email protected] 2

Abstract Concrete-faced Rockfill Dam (CFRD) has the advantages of less engineering workload, short construction period, low cost, convenient construction and safe operation. It becomes one of the most chosen dam types, and has the trend of large-scale popularization. The rockfill materials have complicated physical and mechanical properties. The parameters (such as elastic modulus, friction angles, etc.) have strong dispersed characteristics, and the variability is large in different zones. The load on the faced rockfill dam (such as water pressure, seepage pressure, deadweight, etc.) also has some uncertainties. Factors mentioned above give rise to the uncertainties of stress and deformation of CFRD. Therefore, analyzing reliability of CFRD and measuring structural reliability using reliability index is correct and reasonable. Because of the complexities of the structure of CFRD and the nature of rockfill, it is very difficult to carry through the reliability analysis on this kind of structures. Presently, the satisfactory results cannot be obtained using the conventional methods in structural reliability analysis for large-scale, nonlinear structures. Advanced computational method was adopted to calculate the reliability of CFRD and the corresponding calculation formula were derived. The method has the following characteristics: (1) Original 3D nonlinear finite element code can be used directly; (2) Nonlinear response surface function was adopted to substitute the actual complicated limit state function, and the reliability computation can be done briefly, efficiently with better precision. According to a real-life engineering example, the calculation and analyses of anti-crack reliability of the face slab were conducted and better results were obtained. REFERENCES 1. Simo JC, Taylor RL. A return mapping algorithm for plane stress elastopasticity. Int. J. Num. Meth. Eng., 1986; 22: 649-670. Duncan JM et al. Strength, stress-strain and bulk modulus parameters for finite element analysis of stresses and movements in soil masses. Report No.UCB/GT/80-01, University of California, Berkerly, 1980. 2. Wong FS. Slope reliability and response surface method. J. of Geotech. Engrg., ASCE, 1985; 111 (1): 32-53.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Structural Dynamic Reliability of Solid Rocket Motor Grains Shujun Zhang*, Junguo Ren College of Aerospace and Material Engineering, National University of Defense Technology, Changs ha, 410073 China Email: [email protected] Abstract Analytical methods to predict the stochastic response of MDOF-system are quite limited [1]. In fact numerical procedures, such as Monte Carlo simulation (MCS) techniques are the only available methods to treat practical reliability problems. Direct Monte Carlo simulation, however, requires a substantial amount of computer time and resources. Variance reduction techniques such as importance, directional, etc, are utilized to increase the efficiency of Monte Carlo simulation for analysis of complex dynamical system. This paper proposes an efficient simulation procedure for stochastic analysis of nonlinear dynamical systems with multi-degree freedom, namely rocket solid motor grains, based on Monte Carlo simulation. A simple criterion is established for indicating the importance of each dynamical response sample. According to this criterion, Russian Roulette & Split (RRS) method is applied to deal with the selected samples. The efficiency of this algorithm is much higher than that of direct MCS while the number of response samples in the low probability regions is increased. Then, structure dynamic reliability of grains was analyzed based on Monte Carlo viscoelastic stochastic finite element method. On the basis of nearly incompressible viscoelastic deterministic finite element method, this efficient simulation procedure was applied to increase the efficiency on account of the inefficiency of direct Monte Carlo simulation. Considering the randomness of internal press when motor starts, reliability and its variational trend of the solid rocket motor grain under internal pressure was analyzed in combination with failure criterion and minimal transform. In one numerical example, this procedure is compared with direct MCS demonstrating comparable accuracy and universal, thus suitable for engineering application. REFERENCES 1. Pradlwarter HJ, Schueller GI. On advanced Monte Carlo simulation procedures in stochastic structural dynamics. International Journal of Non-Linear Mechanics, 1997; 32(4): 735-744. 2. Pradlwarter HJ, Schueller GI, Melnikov PG. Reliability of MDOF systems. Probabilistic Engineering Mechanics, 1994; 14: 235-243. 3. Harnpornchai N, Pradlwarter HJ, Schueller GI. Stochastic analysis of dynamical systems by phasespace-controlled Monte Carlo simulation. Computer Methods in Applied Mechanics and Engineering, 1999;168:273-283.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Wave Propagation in Orthotropic Elastic Shells: Theoretical and Numerical Modeling B. Tie*, D. Aubry Laboratoire de Mecanique des Sols, Structures, et Materiaux (CNRS UMR 8579), Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France Email: [email protected], [email protected] Abstract Shell type structures are used in a wide variety of industrial components, especially in automotive or aeronautic industries. A full understanding and predictive modeling of elastic waves propagation through those structures can be of great importance at their design stage for several reasons, such as the noise reduction in a vehicle or the control of high energy shock waves triggered by pyrotechnic cut in a space launcher. This paper deals with the theoretical and numerical modeling of elastic wave propagation in shells. The aim is to discuss the relevance of theoretical models and to develop appropriate and efficient numerical tools. Orthotropic elastic shells are considered in this paper and the classical Mindlin shell kinematics is used to model them. First theoretical analyses developing the acoustic tensor are presented, in order to highlight the role played by the shell curvature in the transmission, reflection and conversion phenomena of membrane and flexure waves. Dispersion equations are discussed in the case of cylinder or cone shells. Then in the case where high frequencies are involved and the shortest wavelength is smaller than the shell thickness or the local curvature radius, the relevance of the Mindlin kinematics and its enriched variants is discussed. Besides, the theoretical analyses are always compared to and enhanced by the numerical modeling performed using a space-time Galerkin discontinuous solver. Estimates of model errors are also numerically evaluated to quantify differences between the considered shell kinematical models. Finally, several numerical results on shell type industrial structures are presented. REFERENCES 1. Batra RC. Plane wave solutions and moda analysis in higher order shear and normal deformable plate theories. Journal of Sound and Vibration, 2002; 257(1): 63-88,. 2. Tie B, Aubry D, Boullard A. Adaptative computation for elastic wave propagation in plate/shell structures under moving loads. European Revue of FEM, 2003; 12(6): 717-736. 3. Tie B, Aubry D, Grede A. Numerical modeling of elastic wave propagation in honeycomb sandwich panels under moving loads: application to space launcher. 6th International Symposium on Launcher Technologies, Munich 8-11, Nov. 2005.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Study on the Criterion of In-plane Instability of Non-reinforced U-shaped Bellows Ye Chen*, Boqin Gu College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, 210009 China Email: [email protected] Abstract In-plane instability is a type of plastic failure of non-reinforced U-shaped bellows. The instability will not only reduce the load-carrying ability and the thermal compensation capacity but also decrease the fatigue lifespan of bellows expansion joints. A study on the in-plane instability mechanism of non-reinforced U-shaped bellows under internal pressure or internal pressure-axial displacement is carried out by theoretical analysis, nonlinear finite element method (FEM) and experiments. The results indicate that whether bellows are stable or distorted mostly depends on the generation and propagation of the plastic regions. The in-plane instability critical loads can be obtained by calculating the stress distribution of the plastic regions and analyzing the propagation of the plastic regions both in the roots and the circular plates of bellows. The results obtained by theoretical analysis are in good agreement with experimental data. The now frequently used criterion of in-plane instability is based on the hypothesis of the generation of the plastic hinges both in the roots and crowns, and it may not be applicable to the bellows under the combined loads of internal pressure and axial compression. Based on the results obtained in this paper, a new criterion of in-plane instability of the U-shaped bellows is put forward. a 8000 Cl 7500 ? 7000' £ 6500 & 6000 tn 5500 . g 5000 g 4500 "£j 4 0 0 0 C 3500 ~ 3000 > 2500 g . 2000 §1500

- RZ10-300BI - RZ10-150BI

"

13 1000 jg

500

P

°

.5 CT,

-500

> ^ + 3

5

8

m

l.&X.

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10 13 15 18 20 23 25 28 30

The node number along longitude from root to crown of convolution Figure: Distribution of equivalent elastic stress in some bellows obtained by FEM REFERENCES 1. EJMA. Standards of the Expansion Joint Manufacture Association. New York, EJMA, 1993. 2. Broyles RK. Bellows instability. ASME PVP, 1989; 168: 37-43. 3. Hu Jian, Guo Bingliang et al. In-plane instability of U-shaped bellows subjected to pressure loadings. ASME PVP, 1992; 51: 229-240. — 210 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Coupled Thermal-Dynamic Stability Analysis of Large-Scale Space Structures by FEM Wei Li *, Zhihai Xiang, Mingde Xue Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China Email: [email protected] Abstract Thermally induced vibrations of flexible spacecraft appendages, which are subjected to incident solar heat flux during the orbital eclipse transitions, have been typical failure of modern spacecrafts since 1960's. For example, this failure was observed on the Voyager space explorer [1], the Hubble Space Telescope [1] and the UARS satellite [2] etc. When the thermally-induced vibration is unstable, it is called thermal flutter, which is due to the coupling effect between structural deformations and the incident normal solar heat flux. Owing to the existence of the coupling effect and the transient heat conduction problem involving heat transfer by radiation, it is really a highly nonlinear problem. In this paper, the thermally induced nonlinear vibration of practical thin-walled large-scale space structures subjected to suddenly applied thermal loading is analyzed. By using the finite element method, the governing equations of the coupled system are established which contain the temperature field and dynamic response accurately for actual complex space structures and heating conditions. By using a kind of Fourier-FEM technique, the transient heat conduction problem involving heat transfer by radiation is solved, in which the average temperature and temperature differences in the structure's cross-section are evaluated simultaneously [3]. In addition, the dynamic response of structures is also simulated by FEM. In the coupled thermal-dynamic system, the stability property of the system depends on only the heat incident angle vector v and damping ratios^ . Finally the criterion of thermal flutter is investigated under the frame of nonlinear vibration theories and the necessary condition of thermally-induced vibration is verified.

REFERENCES 1. Thornton EA. Thermal Structures for Aerospace Applications. AIAA Education Series. AIAA: Washington DC, 1996. 2. Johnston JD, Thornton EA. Thermal snap of satellite solar panels. In Proceedings of the 1999 Flight Mechanics Symposium, NASA CP 209235, 1999, pp. 215-229. 3. Xue MD, Ding Y. Two kinds of tube elements for transient thermal-structural analysis of large space structures. International Journal for Numerical Methods in Engineering, 2004; 59: 1335-1353.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Modelling of the Lateral-Torsional Buckling of Stainless Steel I-Beams: Comparison with Eurocode 3 N. Lopes1*, P. M. M. Vila Real1, L. Simoes da Silva2, E. Mirambell3 1

University ofAveiro, Dep. Civil Engineering, 3810 Aveiro, Portugal University ofCoimbra, Dep. Civil Engineering, 3030 Coimbra, Portugal 3 Universitat Politecnica de Catalunya, Dep. Enginyeria de la Construccio, Spain Email: [email protected] 2

Abstract This work presents a numerical study of the behaviour of stainless steel I-beams subjected to lateral torsional buckling and compares the obtained results with the beam design curves of Eurocode 3. The EN version of part 1-1 of Eurocode 3 [1] introduces significant changes in the evaluation of the lateral torsional buckling resistance of unrestrained beams, that improves the too conservative approach of 1992 version [2] of Eurocode 3 in case of non-uniform bending through the introduction of a correction factor. In fact there is an additional beneficial effect resulting from reduced plastic zones connected with the variable bending diagram along the member that was not considered in the earlier versions of the Eurocode 3. This beneficial effect (see Figure) still remains to be taken into account in Part 1.4 of Eurocode 3 - 'Supplementary rules for stainless steels' [3]. The present study aims at evaluating the influence of the loading type on the lateral-torsional buckling resistance of stainless steel I-beams and investigates the need of using a similar correction factor as the one used for carbon steel. XJ/-1 MMpl

-prENl993-l-4 Safir

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Figure: Comparison of the numerical results with the Eurocode 3 REFERENCES 1. European Committee for Standardisation. EN 1993-1-1, Eurocode 3, Design of Steel Structures - part 1-1. General rules and rules for buildings. Brussels, Belgium, 2005. 2. European Committee for Standardisation. ENV1993-1-1, Eurocode 3, Design of Steel Structures - part 1-1. General rules and rules for buildings. Brussels, Belgium, 1992. 3. European Committee for Standardisation. prEN 1993-1-4, Eurocode 3, Design of Steel Structures part 1-4. Suplementary rules for stainless steels Belgium, 2005. — 212 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Research on Simulation Analysis for Stability Problem of PressurePenstock with Imperfection Wenyuan Meng **, Xiuqin Li \ Jiashou Zhuo 2 1

North China Institute of Water Conservancy and Hydroelectric Power, Personnel Department, Zhengzhou, Henan, 450008 China 2 College of Civil Engineering, Hohai University, Nanjing, Jiangsu, 210098 China Email: [email protected] Abstract The construction of large-scale hydropower stations and large-capacity pumped-storage power stations make penstocks construction be large-scale and huge-scale. The use of high quality steel makes penstocks become more and more flexible and thin. The stability of penstock has become the key factor in penstock design. In this paper the simulating technology of Element-Free Method and experiment is applied to the research on the stability analysis of penstock under external pressure. After a brief review of the element-free method (EFM) and the stability analysis of pressure penstock, a numerical model is established for the stability analysis of geometric nonlinear ribbed shell with initial imperfections. At the same time, the seams between penstock and concrete were turned into initial imperfections. Some improvement on the EFM is presented, including a method for constructing the field function of EFM. The thick curved-beam geometry equation and the nonlinearity geometry equation of thin shell with initial geometric imperfections are derived. This geometry equation can be conveniently reduced to the cases of perfect shell and linear shell. The model of ribbed shell is regarded as the combination of thick curved-beam and thin shell. This model can reflect the real stress and strain state of penstock. Finally, as one of the key works in this paper, the experimental scheme for the stability analysis of penstock under negative-pressure is presented. With the help of experiments, the buckling mechanism of perfect penstock and imperfect penstock are investigated. The reliability of new model is verified. The sensitivity of critical load to imperfection and geometric size is analyzed.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Nonlinear Finite Element Buckling Analysis of Square Reinforced Concrete long Columns Confined with Carbon Fiber Reinforced Plastic (CFRP) Sheets under Uniaxial Compression Q. X. Ren ] *, T. G. Chen \ C. K. Huang \ Y. H. Liu 2 State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024 China School of Hydraulic Engineering, Shenyang University of Agriculture, Shenyang, 110161 China Email: [email protected] Abstract The general software ANSYS is adopted to establish the finite element models of square reinforced concrete long columns confined with CFRP sheets under uniaxial compression and used to analyze the influence of CFRP sheets to these structural buckling loads. Height of columns, layers of CFRP sheets, concrete grade, longitudinal rebar diameter, stirrup diameter and stirrup spacing are some factors considered. Based on the nonlinear finite element buckling analysis of these columns, some major conclusions are summarized as follows: (1) The enhanced level of buckling load is descending with the increasing columns' height; (2) The buckling load is rising along with the increasing layers on a nonlinear manner; (3) The buckling load increment is independent of concrete grade, longitudinal rebar diameter, stirrup diameter and stirrup spacing; (4) The improvement is not very obvious to use CFRP sheets to strengthen the structure when it's stability goes wrong. Influences of CFRP sheets to square reinforced concrete long columns can be found by analyzing the stability of these columns, and the analysis results are available for the strengthening of some structures.

Figure: A 3D model of reinforced concrete column strengthened with CFRP sheets REFERENCES 1. Taljsten Bjorn. Strengthening of beams by plate bonding. Mater Civil Engineering, 1997; 9(4): 206-212. 2.

Liu Hao, Zhao Yinghua. A finite element study on the load carrying capacity of CFRP strengthened concrete columns. Journal of Shenyang Architecture and Civil Engineering University (Natural Science), 2002; 18(3): 161-163 (in Chinese). — 214 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

A Kind of Channel-Section Beam Element for Transient Coupled Thermal-Structural Dynamic Analysis J. Duan *, M. D. Xue, Z. H. Xiang Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China Email: [email protected] Abstract Thermally induced vibrations of spacecraft appendages with frame- and truss-type structures have been a recurrent problem for over 40 years. When an appendage is subjected to sudden heating exchanges, the deformation and vibration will be induced by the time-dependent bending moments and bimoments due to the rapid temperature gradients. The solution sequence of the thermal-structural analysis is as follows: (1) transient temperature field analysis of the structure; (2) thermal-structural static and dynamic analysis. In this paper, a kind of thin-walled Channel-Section beam element is presented for transient coupled thermal-structural dynamic analysis by the finite element method (FEM). The temperature gradient in the longitudinal direction and the circumferential direction of the beam are taken into account in the transient temperature analysis. The average temperature and the perturbation temperature along the circumference are decoupled in the deduction of the one dimensional element and the nonlinear analysis caused by the radiation heat transfer is restricted to solving the equations of the average temperature. A kind of straight Euler-Bernoulli beam element including the effect of warping deformation is adopted and the equations of thermally induced coupled bending and torsional vibration are deduced and solved. The comparison of the thermal and deflection analysis of a cantilever with the conventional two dimensional FEM results demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Furthermore, the coupled thermal-structural analysis which includes the effects of beam deformations on heating and temperature gradients is developed by the presented identical thermal-structural element. y

■

>' a

u

\ 1

\ L

2

♦

b

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\

a

S

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A

a

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r

I^

a

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Figure: A cantilever model with channel-section

REFERENCES 1. Xue Mingde, Ding Yong. Two kinds of tube elements for transient thermal-structural analysis of large space structures. Int. J. Numer. Meth. Engng, 2004; 59: 1335-1353. 2. Thornton EA, Kim YA. Thermally induced bending vibrations of a flexible rolled-up solar array. Journal of Spacecraft and Rockets, 1993; 30(4): 438-448.

— 215 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Semi-Analytical Analysis of Super Tall Building Bundled-Tube Structures Yaoqing Gong *, Ke Li Civil Engineering School, Henan Polytechnic University, Jiaozuo, Henan, 454000 China Email: [email protected] Abstract A new semi-analytical method is developed for the analysis of interactions between the subgrade and the foundation and the superstructure of the super tall building bundled-tube structure by three-dimensional model which is a combination of stiffened-thin-wall tubes on semi-infinite elastic body. The subgrade is idealized as a semi-infinite elastic body, and the rigidities of the elastic body pertinent to various deformations of the foundation have been expressed as analytical equations [1], with which the reactions between the foundation and subgrade can be quantified expediently. The foundation is taken as a part of the superstructure. In fact, the foundation is the extension of the superstructure toward the underground. The only difference is the size, since in most cases the foundation must be large enough to make the soil stable. The superstructure and its foundation of the super tall building bundled-tube structure are simplified equivalently and continuously to a combination of stiffened-thin-wall tubes on semi-infinite elastic subgrade. Then discretization is made by some nodal lines, the unknown functions defined on the lines are used as primary unknowns, and interpolating functions are implemented between the lines. So the displacement field of the computing model can be expressed by the unknown functions. After using the principle of minimum potential energy, the governing equations will then be obtained, which is actually a group of ordinary differential equations. Therefore, analysis of a tall building structure will be changed into the solution of the boundary problem of a group of ordinary differential equations that can be solved by the precise and powerful Ordinary Differential Equation Solver — COLSYS [2], a kind of computational software. The interactions between the subgrade and the foundation and the superstructure of a super tall building bundled-tube structure due to static loadings are analyzed by the method based on the model. The numerical results show that the analytical model is reasonable and feasible. Therefore, a practicable method for the global analysis of the super tall building bundled-tube structure is obtained, and some valuable conclusions are acquired through analyzing the computing results as well. REFERENCES 1. Gong Yaoqing. Tall building structures on elastic subgrade and research of semi-analytical method. Tsinghua University Press, Beijing, China, 1999 (in Chinese). 2. Yuan Si. Introduction of a common compute program for boundary problems of ordinary differential equation—COLSYS. Computational Mechanics and Application, 1990; 2: 104-105, (in Chinese).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Computational Design of Beam Sections under Impact Loading Shujuan Hou1*, Qing Li1,2 , Shuyao Long1, Xujing Yang1 1

College of Mechanics and Aerospace, Hunan University, Changsha, Hunan 410082, China School of Engineering, James Cook University, Townsville, QLD 4811, Australia Email: [email protected]; [email protected]; [email protected] 2

Abstract Computational crashworthiness design is of special interest in automotive engineering to ensure the vehicle structural integrity and more importantly the occupant safety in the even of the crash. Nevertheless, most of the collision energy is absorbed by various types of structural components during the impact. For this purpose, hollow thin-walled beam sections have been extensively adopted as an important element of energy absorption in various vehicles. It is highly desirable to have such beam structure to absorb as much energy as possible once impact occurs. To the authors' best knowledge it is found that, unfortunately, there have been very limited publications on the design optimization involving crashworthiness criteria. Computationally, the impact involves modeling fhctional contact, large deformation and rate dependent material constitutive relationship. The high nonlinearity and computing cost in the sensitivity analysis make conventional gradient-based optimization methods very difficult if not impossible. As an effective alternative, the response surface method (RSM) exhibits excellent simplicity and suitability to such a complex problems as crashworthiness design. For this reason, RSM is adopted in this study. This paper seeks for the optimal shape of the tapered cylindrical beam cross-section under a crashworthiness criterion. The effect of selection of quartic polynomial basis functions on the optimization results is investigated, To reflect the crashworthiness criterion, the total strain energy is set as the objective function. The explicit Finite Element (FE) simulation, response surface techniques together with nonlinear constrained multivariable mathematical programming algorithm are involved in the optimization procedure. Two examples of cylindrical beam sections with and without taper are considered in this paper. The optimization results demonstrate reasonable computational efficiency and algorithmic suitability in different design cases.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X.Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Pretension Control of the Long-span Roof Structure of South Shanghai Railway Station Y. Huang*, Y. F. Luo, R. Yu College of Civil Engineering, Tongji University, Shanghai, 200092 China Email: [email protected], [email protected] Abstract The roof of South Shanghai railway station is a long-span beam-cable hybrid steel structure. The essential role of cables pretension between and under steel beams is to keep whole roof stable under possible load conditions. Using finite element analysis computer program ANSYS, the time-history analysis of seismic response and the stability behavior of the steel roof is calculated. Based on the numerical results, the optimal tension force of cables and reasonable pretension sequence are obtained. Combined with a time-history analysis and a buckling analysis, the tension forces of prestressed steel rods are optimized on the basis of guaranteeing no cable slack. Then four kinds of steel rod pretension schemes are proposed. An optimal one is chosen to enssuring cable stress value uniformity. Subsequently, beginning with the ideal state of completed roof, the counter-pretension procedure, in which cables are disassembled in the opposite direction of pretension, is modeled stepwise, and the initial cable force values are obtained to insure a successful pretension process. AN MB

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Figure 1: First buckling mode of steel roof

Figure 2: Finite model for Shanghai railway station

REFERENCES 1. Simo JC, Taylor RL. A return mapping algorithm for plane stress elastopasticity. Int. J. Num. Meth. Eng., 1986; 22: 649-670. Feng Jian, Zhang Yaokang. Static optimizing analysis of prestressed cable-stayed space truss structure. Journal of Southeast University, 2003; 33(5): 583-587. 2. Huang Y, Luo YF. Optimal design and buckling analysis of a long-span steel truss. ICASS 05, 2: 1329-1333.

— 218 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Damage Detection and Sensor Placement Design for Two Highway Bridges Y. Q. Li1, Z. H. Xiang1*, M. S. Zhou1, G. Swoboda2, Z. Z. Cen1 1

Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China Department of Structural Engineering, University of Innsbruck, Innsbruck, A- 6020 Austria Email: [email protected] Abstract Xinxingtang and Beixingtang Bridge were located on the highway connecting Nanjing and Shanghai, China. Because of the fast growing traffic flows, observable deterioration was recorded during the routine manual inspection. To further investigate the health condition of these two bridges, extensive examinations were carried out. The following work is how to detect the damages in these bridges from the onsite measurements. To this end there are two aspects should be considered: the sensitivity of measurements to damage and the selection of measurement locations. In this paper, a damage detection algorithm is developed for bridges based on the static deflection, which is sensitive to structural stiffness reduction. In addition, how to choose an optimal measurement set is discussed by evaluating the influence of measuring errors. The performance of the damage identification algorithm and the criterion of selecting optimal measurement set is evaluated by an academic example. Then the proposed method is implemented to the damage detection work of Xinxingtang and Beixingtang Bridges. The identified results are in good accordance with onsite observations.

Figure: Elevation of Xinxingtang bridge (left) and Beixingtang bridge (right) REFERENCES 1. Banan MR, Hjelmstad KD. Parameter estimation of structures from static response, I and II. Journal of Structural Engineering, 1994; 120: 3243-3283. 2. Sanayei M, Sletnik MJ. Parameter estimation of structures from static strain measurements, I and II. Journal of Structural Engineering, 1996; 122: 555-572. 3. Xiang ZH, Swoboda G, Cen ZZ. On the optimal layout of displacement measurements for parameter identification process in geomechanics. ASCE The International Journal of Geomechanics, 2003, 3: 205-216.

— 219 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

A new iterative method for solution of rectangular elastic structure Fuyong Lin* Department of Engineering Mechanical, Huaqiao University, Fujian, 362021 China Email: [email protected] Abstract Kernel solution of elastic problem in periodic zone is obtained by orthogonal element method, the solution of elastic problem in other area can be gotten by expanding the area to periodic zone with acting the forces in the boundary. By using shift operator one can get the kernel solution of displacement of elastic problem with forces acting at any point in periodic zone. By satisfying the boundary conditions one can get the solution of given problem. A new iterative algorithm to solution rectangular elastic structure problem is proposed by orthogonalizing the boundary conditions. By the proposed method the calculation of inverse of large matrix is transformed to calculate a serial inverses of Matrix of 4x4 for plate system, 2^2 three dimension problem. By proposed method, for two dimension problem the calculation to get displacement in periodic system without consider boundary condition be reduced to 0(NxMx/og(NxM)). Where N, M are the division in x dimension and y dimension, the calculation every step also be 0(NxM), therefore the computation of the proposed method is 0(NxMx/og(NxM)), compared to the calculation of boundary element method (0(N+M)3), it is a great reduction in computation. It is potential to apply proposed method to construct new domain decomposition and iterative algorithms. REFERENCES 1. Lin Fuyong. Orthogonal continuous segmentation function, Applied Mathematics and Computation. 2004; 154: 599-607. 2. Lin Fuyong. Multi-solution theory of signal processing based on finite element method. Applied Mathematics and Computation, 2005; 161: 293-310. 3. Lin Fuyong, Orthogonal bases of 3-B-Spline and its application in bending problem of plate and beam system. Applied Mathematics and Computation, 2005; 162: 723:733.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Overall Stability of the Long Span Steel Roof Structure of Hangzhou International Conference Center Y. F. Luo*, X. Liu, Z. B. Wang College of Civil Engineering, Tongji University, Shanghai, 200092 China Email: [email protected], [email protected] Abstract Large span structures are widely used in modern buildings because of their novel forms or shapes, light self-weight and lower construction cost. The buckling behavior of the large span structures under loads are much complex and hard to understand. The non-linear stability analysis and the incremental-interation tracing methods are necessary for design complementory during design. The geometry imperfection influences on structure stability are also needed to be considered .There are no special analysis technics and judging methods for overall stability of the large span structures in current national design codes of our country. It is necessary to carry on non-linear overall stability analysis for every specific structure to obtain the overall stability behavior of structures during design. According to the design demand, the finite element model of the steel roof structure of Hangzhou international conferernce center (shown as in Fig.l) are set up in the paper. The large displacement elastic and elasto-plastic overall stability analysis of the perfect and imperfect steel roof structure are conducted. The critical loads, member stresses and maximum displacements under different load cases are obtained. The reliable numerical data are provided to the design. The valuable ideas about the overall stability behavior of the large span structure are concluded. It is a useful reference for design of similar structures.

Figure: Finite element model of Large Span Roof Structure Hangzhou International Conference Center

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Regionwise Modeling Approach for the Analysis of Layered Structures P. M. Mohite, C. S. Upadhyay * Department ofAerospace Engineering, Indian Institute of Technology Kanpur, 208016 India Email: [email protected] Abstract Laminated composites are widely used in aerospace, automobile and marine applications for shell like structures due to their excellent strength properties over conventional materials. There is always a need to optimize the structures made from these composites with constrains on response quantities like state of the stress at a point, maximum transverse deflection, buckling, first-ply failure load etc. An accurate computation of these constraints is essential for the final optimal solution. A large number of dimensionally reduced plate theories are available in literature for the analysis of the laminated plate structures. These are popularly known as 'plate models' or 'equivalent single layer models'. Another class of plate theory is zig-zag theories. Some of the zig-zag has also shown to be convergent to the three-dimensional elasticity solution with respect to the strain energy (energy norm). The basic idea of these models is to give a higher order approximation of displacement field in the transverse direction to accurately represent the transverse shear effects. These models are quite effective in predicting the global response like maximum transverse deflection, buckling of the laminates. The attractive feature of these theories is that they are independent of layers in the laminate. The major drawback with these theories is that the transverse stresses, obtained using these models, are not accurate. For this reason the equilibrium based postprocessing approach has to be employed to extract the better through thickness transverse stresses. However, for the domains with corners, damaged zones, and unsymmetrical lay-ups these models are ineffective. These situations can be handled more efficiently using more refined analysis. Layerwise models are often used to alleviate these problems. In these models a higher order displacement field is applied layer-by-layer. Hence, the cost of computation increases with number of layers in the laminate. The three dimensional effects are localized in the vicinity of corners, boundaries, parts of damaged lamina/ interfaces. In such regions a more refined analysis (layerwise) is imperative and beyond this region of unsmoothness a dimensionally reduced model of intermediate (e.g. lump all laminae above and below a delamination separately) models can be used effectively. This is achieved in this paper by the proposed region-by region approach where any model can be used in any region of the laminate. As shown in this paper, this approach leads to tremendous savings in computational cost and gives accurate representation of the state of stress everywhere in the domain. This approach is a generalization of the planar constrained approximation approach and the h-d approach in literature, given for homogeneous materials. As an example, the stress component computed using three-dimensional solution is compared with those computed by equivalent single layer model, layerwise model and regionwise model. It is seen that the stress component computed using regionwise approach is accurate and almost overlaps with the three-dimensional solution. Further the layerwise theory is very accurate. The equivalent single layer model is very far from three-dimensional solution both qualitatively and quantitatively. The number of unknowns for the layerwise model is 11875, for the equivalent model 6875 and for regionwise approach it is just 4863.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Computational and Experimental Study of Energy Absorption Metter by Composite Structures T. Niezgoda *, W. Barnat Military University of Technology, Kaliskiego Street 2, 00-908 Warsaw, Poland Email: [email protected] Abstract The main goal of this research is to evaluate and compare the ability to absorb energy due to impact by two types of basic composite structures (conic and cylinder). Structures made of glass-epoxy composite with the properties taken from tests were analyzed using classic laminate theory. Most of the design process will base on Finite Element Method (FEM) calculations conducted with commercial explicit code. Numeric results were compared with experimental results. Using results from the tests the energy absorbed by the basic composite structures was estimated. The preliminary results of the dynamic simulations compared with the experimental readings shows the excellent coincidence. At the next stages of calculations authors decided to pay more attention on the problem of delamination modeling. This way better correctness of failure phenomena in composite structure will be possible to describe using numerical methods. In this new approach, which is currently during the testing phase, the delamination model with special finite elements is developed. In conclusion one must admit that the designing of the structures from the glass-based composites with an energy absorption comparable or better than that of metal constructions is possible. However, because of a still insufficient knowledge about composite mechanics and the quality of changeable parameters concerning the designing of composite structures, it is much more complicated to find an optimal structural solution than in the case of metals.

Figure: Numerical (a) and experimental (b) study of composite conic deformation (so called "brush effect")

REFERENCES 1. Fleming DC. Delamination modelling of composites for improved crash analysis, Journal of Composite Materials, 2001; 35(19): 1777-1792. 2. McCarthy MA, Harte CG, Wiggenraad JFM, Michielsen ALP J, Kohlgruber D, Kamoulakos A. Finite element modelling of crash response of composite aerospace sub-floor structures. Comput. Mech., 2000; 26: 250-258. 3. McCarthy MA, Wiggenraad JFM. Numerical investigation of a crash test of a composite helicopter subfloor structure. Composite Structures, 2001; 51: 345-359. — 223 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

The Dynamic Analysis of Main Building of Hangzhou International Conference Center H. J. Song*, Y. F. Luo, C. L. Xu,, M. W. Yang College of Civil Engineering, Tongji University, Shanghai, 200092 China Email: [email protected] Abstract Hangzhou International Conference Center is a high-rise steel building. Its structure is much complex. The main structure of the building is a sun-shaped braced-frame structure. Lots of curved beams, slanting columns and struts are adopted in the structure. Plane arrangement of the members is extremely complex. Horizontal and vertical stiffness vary greatly. Since the irregularity of the structure, the spectrums are much dense. The calculation of dynamic responses is much complicated. The coupled Reaction of different modes is much apparent. The response spectrum method and time-history method are both used to analyze the dynamic characters and seismic response of the complicated main building in this paper. The dynamic characters of the structure are obtained. The capability of seismic resistance is also calculated. It is a valuable reference for design of this structure and similar structures.

Figure 1: Finite element model of main building

Figure 2: Deformation shape under earthquake action

REFERENCES 1. ATC. Seismic evaluation and retrofit of concrete buildings. Report No. ATC-40, Applied Tschnology Council, Redwood City, California, USA, 1996. 2. Leger P, Ide IM, Paultre P. Multiple-support seismic analusis of large structures. Computer and Structures, 1990; 36 (6): 1153-1158.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Second-Order Analysis for Steel Frame Structures with a Distributed Plasticity Numerical Model Ke Wang1*, L. W. Tong \ Tian Li 2 College of Civil Engineering, Tongji University, Shanghai, 200092 China School of Civil Engineering, Zhengzhou University, Henan Zhengzhou, 450000 China Email: [email protected] Abstract This paper provides an efficient numeric model of steel frames using second-order distributed plasticity FEM (finite element method). Details of the modeling including element type, mesh discretization, material model, residual stresses, initial geometric imperfections, and boundary conditions are presented. Its validity is verified by several numerical examples. Case studies of Vogel's portal frame and space steel frames are performed. The ultimate loads obtained from the proposed analysis and Vogel agree well within 2% error. Therefore , the model is accurate and provides benchmark solutions of steel frames using second-order distributed plasticity analysis. The benchmark solutions of the steel frames are useful for the verification of various simplified second-order inelastic analyses. It is observed that the load carrying capacities calculated by the Code for design of Steel Structures (GB50017-2003) method are 20%~30% conservative when compared with those of the proposed analysis.and using this model the study was carried out on the effects of the nonlinear factors on the behavior of steel frames.

b

1

f

\f

HEA340

w

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cc UL X

I

7T*rr

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Horizontal displacement (mm)

Figure: Dimension and loading condition, and Comparison of horizontal load-displacement curves of Vogel's portal frame

REFERENCES 1. King WS, Chen WF. Practical second-order inelastic analysis of semirigid frames. Journal of Structural Engineering (ASCE), 1994; 120 (7): 2156-2175. 2. Seung-Eock Kim, Dong-Ho Lee. Second-order distributed plasticity analysis of space steel frames. Engineering Structures, 2002; 24: 735-744. — 225 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Pretension Simulation of the Long Span Truss String Supported by the Temporary Structures R.Yu*,Y.F.Luo, Y.Huang College of civil Engineering, Tongji University,Shanghai, 200092 China Email: [email protected],[email protected] Abstract The truss string structure, composed of lower string, upper truss and struts, is one kind of the long span pre-stressed steel structure. During the pretension procedure, the struts are compressed since the tension of the string. At the same time the inner force and deformations, which are opposite to the effect of outside load, appear in the truss. The structure stiffness and integrity are improved. Before pretension, lots of falseworks are set up in order to supply the temporary support for the upper truss. The reasonable pretension of the string commonly conforms a principle that the upper truss is just moved up from the falseworks. And meanwhile the predefined tension force is reached. Therefore, for getting a reasonable pretension method and pretension force, it is important to precisely simulate the coaction of the elastic falseworks and the truss. A actual truss string of a roof structure is taken as a comparison model for pretension simulation and ANSYS program is used in this paper. Detail analysis models are showed as Fig 1 and Fig 2. Different simplifications are used to simulate the coaction of the falseworks and the truss. After comparison of the results, an reasonable simulation model and a tension control method are obtained. The reliable pretension method and optimal pretension force are provided. It is valuable for future similar engineering.

^ ^

Figure 1: Single truss string model

Figure 2: Whole model

— 226 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Accurate Form-Finding Method for Cable-Dome Structues Based on Catenary Element X. Z. Zhao1*, R. W. Tang2, Z. Y. Shen1 1

Department of Structural Engineering, Tongji University, Shanghai, 200092 China China Academy of Building Research, Beijing, 100013 China Email: [email protected] 2

Abstract Cable dome, comprised of a large amount of prestressed cables and a few compressive stubs, is a type of structure with highly geometrical nonlinearity. This nonlinearity makes it difficult to carry out the numerical analysis on form-finding and constructional control of cable domes. In this paper, an iterative algorithm in form-finding of cable dome structures was proposed to determine precisely the prestress of the cable dome and the original length of the cables which will be definitely balanced in the initial geometrical condition taking into account the effect of self-weight. The most important issues in analyzing cable structures using catenary elements are how to efficiently calculate the element stiffness matrix and the original length of cables. In this study, a globally convergent method was proposed first to calculate the element stiffness matrix and a direct method to obtain the original cable length was then employed. These methods are able to solve the inadequacy of the iterate algorithms presented in the literature. Based on these methods this paper finally presents a form-finding method using catenary elements. The results obtained from a benchmark problem analyzed using this method demonstrate that the form-finding method proposed in this paper is efficient and accurate.

REFERENCES 1. Jayaraman HB, Knudson WCA. A curved element for the analysis of cable structures. Computters and Structures, 1981; 14: 325-330. 2. Zhang QL. Cable Nets and Membrane Structures. Tongji University Press, Shanghai, China, 2002 (in Chinese).

— 227 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Higher Order Modes in Thin-Walled Beam Analysis R. Vieira *, F. Virtuoso, E. Pereira Departamento de Engenharia Civil e Arquitectura, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Av Rovisco Pais 1049-001, Lisboa, Portugal Email: [email protected]{ricardo.vieira, francisco.virtuoso, eduardo.pereira}@civil.ist.utl.pt Abstract The analysis of a thin-walled beam taking in account higher order deformation modes (i.e. warping, shear lag) is presented. The analysis is performed using a beam model that considers an interpolation of the displacements over the beam cross section by a set of pre-selected functions {z)}Considering a set of functions only dependent of the beam axis coordinate {u(x); v(x); w(x)} as the corresponding amplitudes of the previous cross section interpolations, the beam displacement field is described as u(x,y,z) = u(x) (p(y,z); v(x,y,z) = v(x) Y(y,z); w(x,y,z) = w(x) x(y,z). Admitting the cross section to be constituted by a set of elements, assembled in such a way that the compatibility between them is satisfied, the set of the differential equilibrium equations can then be written in a compact form: [K 1 ]8 ff +[K 2 ]6'+[K 3 ]8=0 where: 8 = {u(x), v(x), w(x)}; Ki and K3 are symmetric and K2 skew-symmetric. The solution procedure for the equilibrium equations is related to that of a generalized eigenvalue problem, which permits to obtain a set of orthogonal deformation modes. The generalized eigenvalue problem is achieved by an assumption of the solution form, suggested by the theorem of Toupin concerning the quantification of Saint-Venant Principle. A procedure based on a quadratic eigenvalue problem is also considered for the solution of the differential equations, which is verified to lead to identical results. The governing differential equilibrium equations are solved numerically by the conventional finite element method over the beam axis, adopting Hermite functions for the amplitudes {u(x), v(x), w(x)}. Numerical examples are presented for the analysis of a thin walled beam. Some important conclusions can be drawn from the analysis of results, namely: i) the set of deformation modes of most relevance for a specific cross section subjected to a specific load can be clearly identified; ii) the solution decay behaviour of higher order modes is observed and iii) the solutions obtained from the generalized eigenvalue problem are in excellent agreement with those obtained trough the quadratic eigenvalue procedure.

— 228 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Research on Rigidity Limits of Bridge with Conventional Spans for Chinese High-speed Railway Mangmang Gao *, Jiaying Pan, Yiqian Yang China Academy of Railway Science, No. 2 Daliushu Road, Haidian district, Beijing, 100081 China Email: [email protected]; [email protected]; [email protected] Abstract In this paper, an efficient dynamic model of coupling vibration analysis for train-bridge system is given. Aiming at whole train passing through the multi-span bridge with ballasted track, a model of bridge with various finite elements is established. The motion of a four-axle car with two stage suspension is modeled based on multibody dynamic theory with linear springs and dampers. According to the assumptions that both wheel and rail are rigid bodies and keep contact to each other in vertical direction, the vehicle special vibration model has 27 degree of freedom. Meanwhile, proceeding from microcosmic analysis of wheel/rail relationship, describing wheel/rail interaction and displacement coordination with wheel/rail contact creep theory in horizontal direction, regarding track irregularity as the excitation source, and adopting time domain method, an overall analysis of the procedure from train's entering into to exiting from the bridge has been made. To the train- bridge coupling vibration, implicit integration method is used, and thus the limitation of time step can be avoided in explicit integration method which is caused by high frequency of bridge, at the same time the accuracy and rapidity of analysis can both be achieved and well effects are obtained.. A computer program TYCHE has been developed for the dynamic analysis of train-bridge system. The above procedure has been used to make train-bridge coupling vibration analysis for single track prestressed concrete box girder bridges with 20m,24m,32m and 40m span respectively. In order to reflect the actual situation, the homemade Pioneer within 160~220km/h, Chinese Star within 160~250km/h and German ICE3 within 250-350km/h has been considered respectively. By analyzing the disciplinarians between beam rigidity and the car body acceleration and Sperling index, the vertical rigidity limits of bridge with Conventional Spans aimed at high-speed are proposed. REFERENCES 1. Gao Mangmang. Studies on Train-Track-Bridge Coupling Vibratiion And Runnablility of Train on High-speed Railway Bridges, Doctor Thesis of China Academy of Railway Science, 2001 (in Chinese). 2. Zhai WM, Cai CB, Guo SZ. Coupling model of vertical and lateral vehicle/track interactions. Vehicle System Dynamics, 1996; 26 (1): 61-79. 3. Shen ZY, Hedrick JK, Elkins JA. A comparison of alternative creep force models for rail vehicle dynamic analysis, in Proc. 8th IAVSD Symposium, MIT, Cambridge, 1983, pp. 591-605.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Implementation and Calibration of a Hysteretic Model for Cyclic Response of End-Plate Beam-to-Column Steel Joints under Arbitrary Cyclic Loading Pedro Nogueiro **, Luis Simoes da Silva2, Rita Bento 3 1

Instituto Politecnico de Braganca, Braganca, Portugal Civil Engineering Department, University ofCoimbra, Coimbra, Portugal 3 Civil Engineering Department, Technical University Lisbon, Lisbon, Portugal Email: [email protected], [email protected], [email protected] 2

Abstract The maintenance of the competitiveness of steel construction in seismic areas is the aim of our studies. Several earthquakes in many countries have shown some steel design errors done in the last years and which must be avoid. In other hand and from the economical point of view, especially the increase of the steel price reinforces the need of the standardised steel construction. It is the purpose of the present paper (1) to show the results of some laboratories tests in endplate beam-to-column steel joints; (2) to present a numerical hysteretic model for cyclic response of these laboratorial tests implemented in the SeismoStruct program; (3) to calibrate the hysteretic model for arbitrary cyclic loading. The numerical model was made in Delphi platform, and it was integrated into numerical software SeismoStruct by means of a spring element. The laboratorial tests are for external steel bare joints. These joints were chosen and analytically design to have a ductile behaviour. They are first loaded monotonically and cyclically, and then characterized in terms of strength, stiffness and energy dissipation.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Analysis of Singular Inplane Stress Field around an Inclusion's Corner Tip* M.C. Chen1*, X.C. Ping2 1

School of Civil Engineering, East China Jiaotong University, Nanchang Jiangxi 330013, China School of Mechatronics Engineering, East China Jiaotong University, Nanchang Jiangxi 330013, China Email: [email protected] Abstract Many engineering applications involve a junction of multiple materials to enhance the mechanical performance of engineering materials. However, according to linear elasticity, the stresses at the bonded interface are often unbonded and present high stress singularities, which provides a source of crack initiation. In order to study the fracture starting at the inclusion corner tip by the method of fracture mechanics, the singular stress field has to be taken into account. It is well known that singular stresses are often assumed to have the asymptotic form: cr = Krk fi}(9), in which (r,0) are the polar coordinates originated at the singular stress point, X is the singularity order, fj{0) gives the angular variation and K is the generalized stress intensity factors. There are a lot of studies on the singular order and the singular stress fields, as far as the inclusion problems, the work of Chen [1] based on complex function method is representative. It should be noted that analytical technique can be only applied to some typical problems of simple geometrical discontinuity and/or particular combination of materials, therefore, numerical methods are powerful alternative, in this paper, based on the principle of a novel hybrid finite element technique introduced in [2], a kind of corner-tip element (see Figure) is established to study singular stress fields around the inclusion corner tip. As applications, the influences of the ratio of lengh and width and the distance of two inclusions on the generalized stress intensity factors are investigated. 8,

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Figure: Definition of an eight-node super corner-tip element

REFERENCES 1. Chen DH. Analysis of singular stress field around the inclusion corner tip. Eng. Fract. Mech., 1994; 49: 533-546. 2. Chen MC, Sze KY. A novel hybrid finite element analysis of bimaterial wedge problems. Eng. Fract. AfecA.,2001,68: 1463-1476. * This project is supported by the National Natural Science Foundation of China (No. 10362002) — 231 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Analysis for the Metallic Gasket Effective Width X. Feng*, B. Q. Gu, R.Liu College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, 210009 China Email: [email protected], [email protected] Abstract The leakage behavior of bolted flanged joints is related to the gasket contact pressure. In particular, the non-uniform distribution of this pressure in the radial direction caused by the flange rotational flexibility has a major influence on the tightness of bolted flanged connections. The current ASME flange design rules and the new ASME proposed design rules addresses this effect by introducing the concept of gasket effective width. In this paper, the distribution of the contact pressure of metallic flat gaskets is investigated by Finite Element Analysis (FEA). The gasket effective width values can be obtained when the contact pressure distribution, the gasket factor m and the internal pressure are given. The results indicate that the gasket effective width is mainly influenced by the flange material and the gasket dimension. The gasket effective width obtained in this paper is smaller than that according to ASME design rules. The obtained results provided a helpful reference for the design of bolted flange joints. 160-

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4 6 8 10 Gasket width mm Figure: Radial distribution of the gasket contact pressure REFERENCES 1. Sawa T, Higurashi N, Akagawa H. A stress analysis of pipe flanged connections. ASME J, Pressure Vessel Technol., 1991; 113: 497-503. 2. Bouzid A, Derenne M. Distribution of the gasket contact stress in bolted flanged connections. Pressure Vessel and Piping, 1997; 354: 185-193. 3. Sawa T, Ogata N. Stress analysis and evaluation of the sealing performance in pipe flange connections with spiral would gaskets under internal pressure. Analysis of Bolted Joins, ASME PVP. 2001; 416: 27-34. — 232 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Analysis of Electrochemical-Poroelastic Behaviors of Conducting Polymer (PPy) Films W. S.Jung *,Y.Toi Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505 Japan Email: [email protected] Abstract Computational modeling has been formulated for the electrochemical-poroelastic behaviors of conducting polymers. The three-dimensional continuum modeling based on poroelastic theory for the passive poroelastic behavior of polypyrrole films given by Delia Santa et al. has been extended to the analysis of active electrochemical-poroelastic behavior. The system of governing equations has been established by combining the above modeling with the one-dimensional ionic transportation equations given by Tadokoro et al. The established equations have been discretized by the finite element method. The validity of the proposed computational modeling has been illustrated by comparing the calculated solutions for the passive and active behaviors of polypyrrole films with the theoretical and experimental results.

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Figure: Pressure distribution in polypyrrole film REFERENCES 1. Delia Santa A et al. Performance and work capacity of a polypyrrole conducting polymer linear actuator. Synthetic metals, 1997; 90: 93-100. 2. Tadokoro S et al. An actuator model of ICPF for robotic applications on the basis of physicochemical hypotheses, in Proc. 2000 IEEE International Conference on Robotics and Actuation, 2000, pp. 1340-1346. 3. Toi Y, Kang SS. Finite element analysis of two-dimensional electrochemical-mechanical response of ionic conducting polymer-metal composite beam. Computers and Structures, 2005; 83: 2573-2583.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Analyses of Multi-Material Wedges and Junctions with Singular Antiplane Stress Field X. C. Ping**, M. C. Chen \ J. L. Xie 2 School of Mechatronics Engineering, East China Jiaotong University, Nanchang, Jiangxi, 330013 China School of Mechanical and Electronic control Engineering, Beijing Jiaotong University, Beijing, 100044 China Email: [email protected] Abstract Some analytical methods can be used to solve singular antiplane stress fields of typical geometrcal discontinuity (e.g., [1]), but they can not be used directly to various kinds of multi-material wedges and junctions. Conventional finite element models with refined meshes near the singular point can be used to predict singualr stresses, but it has long been recognized that the results is not accurate. For this reason, special elements have been created over the years which are especially suited to evaluate the stress state present around the singular point. Among them, singular elements that incorporate the angular variation of the singular fields are more appropriate for dealing with singular stress field problems. However, derivation of analytical angular variations of is either limited to very simple configurations or requiring high level of mathematical competence. Naturally, numerical generated fields is resorted to. It is recommendable that the finite element eigenanalysis method can predict numerical angular variaitions of muti-material wedges and junctions [2], and recently, a kind of ad doc finite element eigenanalysis method has been established by the authors and used to solve the singulrity orders and the angular variations of multi-material wedges and junctions [3]. In this paper a kind of wedge-tip element based on the eigenanalyisis solutions [3] is established and incorperated with the assumed hybrid-stress finite element model to study singular problems in the multi-material wedges and junctions. The new model is validated firstly by the well known crack problem, then, as applications, some other examples are given. It is proved that present method gives converged solution with fewer elements, and more imortantly, it can be used to study various kinds of wedges and juctions only by changing the shape and node number of the wedge-tip element. REFERENCES 1. Li XF, Tang GJ. Closed-form solution for two collinear mode-Ill cracks in an orthotropic elastic strip of finite width. Mechanics Research Communications, 2003; 30: 365-370. 2. Pageau SS, Biggers SB. Enrichment of finite elements with numerical solutions for singular stress fields. Int. J. Num. Meth. Eng., 1997; 40: 2693-2713. 3. Ping XC, Chen MC, Xie JL. Singular stress states in tips of anisotropic multi-material wedges and junctions subjected to antiplane shear. Acta Mechanica Solida Sinica, 2005; 26: 193-198.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Plane Strain Finite Element Analysis of a Piled Bridge Abutment on Soft Ground H. T. Wang '*, Z. P. Chen2, L. J. Xiao2 1

Department of Civil Engineering, South China University of Technology, Guangzhou City, 510640 China Guangdong Guanshen Civil Engineering New Tech Corporation, Guangzhou, 510000 China Email: [email protected] Abstract In order to predict the behaviour of a piled bridge abutment constructed on soft ground, a plane strain finite element model is developed by using commercial finite element code ABAQUS. With the help of the finite element model, the loading and displacement of the structure and embankment at any time after the construction are readily obtained. In the present model, the consolidation of soil is realized by employing 'couple pore fluid diffusion and stress procedure' in ABAQUS which models soil as saturated porous media and uses pore pressure element at the same time. Critcal state plasticity model provided by ABAQUS, which is an extention of the modified Cam-clay model, is used to describe the inelastic behavior of clay. Vertical, horizontal sand-drain and major construction process are all taken into accouts. For better conforming the intrinsic 3D nature of the structure (especially piles) and soil-structure interaction, some measures are taken when constructing the plane strain model. The success of the present model is verified by data from onsite experiments. REFERENCES 1. Hara T, Yu Y, Ugai K. Behaviour of piled bridge abutments on soft ground: A design method proposal based on 2D elasto-plastic-consolidation coupled FEM. Computers and Geotechnics, 2004; 31: 339-355. 2. Ellis EA, Springman SM. Modelling of soil-structure interaction for a piled bridge abutment in plane strain FEM analyses. Computers and Geotechnics, 2001; 28: 79-98.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Finite Element Analysis of a Coal Liquefaction Reactor during Lifting Z. B. Wang*, Y. F. Luo, X. Liu College of Civil Engineering, Tongji University, Shanghai, 200092 China Email: [email protected], [email protected] Abstract The lift project of shenhua direct-coal-liquefaction reactor ranks the largest one among the pressure vessels in china at present. The stresses of the reactor during lifting are analyzed using finite element method, based upon which the safety assessment, reactor and lifting additional structure members, is conducted for lifting scheme. The FEA investigates several important positions in the equipment including the manway neck of the reactor, the lifting cover, the skirt and the base ring. The analysis is performed using finite element analysis computer program ANSYS. After symmetric simplification, half 3D model is founded to simulate the whole equipment. Considering the calculating time and the computer memory capacity, the several elements, with brick elements for lifting cover and manway neck, shell elements for skirt and base ring, beam elements for bolts, braces and tailing lugs, are adopted. During judgment, since yielding-destroy characterizes the steel material, the intensity stresses are taken as the controlling item, and the safety factors are introduced according to the "Unified standard for reliability design of building structures". Typical situations are selected as to cover the entire lift procedure, which include horizontal lifting position, 45 degree decline lifting position and vertical lifting position. The correspond numerical results are gained for assessment. Compared with each other, it can be concluded that the horizontal lifting position is the unfavorable and critical position and the manway neck is the controlling part. Following above ways, the equivalent stresses and the allowable stresses can be gained for this part. The equivalent stresses are beyond the allowable ones. So the conclusion can be made that the lifting scheme adopted currently is unsafe for the equipment and it should be revised.

Figure 1: Finite element mesh for the manway neck

Figure 2: Intensity stress of the manway neck — 236 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Anaylsis of Welded Cruciform Joint A. H. Wu **, S. Syngellakis2, B. G Mellor2 1

Department of Structural Engineering, Tongji University, Shanghai, 200092 China School of Engineering Science, University of Southampton, Southampton, SOI 7 IBJ UK Email: [email protected]; [email protected]; [email protected] 2

Abstract The behaviour of a welded cruciform joint under quasi-static loads was simulated using the finite element method (FEM). The reliability of the FEM models was examined by solving a benchmark problem, carrying out a mesh sensitivity study and comparing the strain-moment relation with experimental measurements. Particular attention was given to the accurate representation of material property variations from the base metal (BM), across the heat-affected zone (HAZ) to the weld metal (WM), which is present in a real weldment but has not been addressed in the literature. The limited material information from the earlier experimental work was supplemented with data on strain hardening and maximum elongation for all three material zones. Material parameters were selected within the ranges specified for the grades of steel used. The sensitivity of strain distribution to strain hardening model was examined. Based on hardness data, a property gradient approach was applied and it has been found from this modelling process that the strain distribution within and adjacent to the HAZ is very sensitive to the accurate representation of the gradual change of material properties in this area. This information is very useful particularly for studies on failure mechanisms under low applied loads, such as fatigue fracture. The effect of varying weld geometry and the HAZ size on the strain and stress distributions may be studied using the generated model.

Figure: Effect of gradual change of material properties on principal strain difference distribution REFERENCES 1. Masubuchi K. Analaysis of welded Structures. MIT, Pergamon Press, 1980. 2. Fessler H, Pappalettere G. Fillet welds under bending and shear, in Proc. 5th Int. Conf. of Pressure Vessel Technology, San Francisco, 1984, pp. 47-61. 3. Kamtekar AG. A new analysis of the strength of some simple fillet welded connections. J. Constr. Steel Res., 1982;2:33-45. — 237 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Static and Dynamic Testing of the SATUOeiras Viaducts M. Xu *, L. O. Santos, J. Rodrigues Department of Structures, National Laboratory for Civil Engineering, Lisbon, Portugal Email: [email protected] Abstract The first phase of SATUOeiras (Automated System of Urban Transport of Oeiras) was inaugurated in May 2004 (fig.l). The system consists of a single rail track that runs above ground level, on specially built viaducts, with fully automated vehicles moving along the track. At the 1st phase the system is 1260 m long, with three stations and two connecting viaducts between them. The viaducts VI and V2, with 33 m spans, have a length of 360 m and 460 m, respectively. A prefabricated solution was adopted for their construction[l]. Before the opening for traffic, static and dynamic tests were carried out on the viadutcs. The static test was performed with the train which was loaded up to a total of 197.9 kN. This train was placed in accordance to the load plan to obtain the maximum moments at the middle spans and check the behaviour of the joints between the prefabricated beams. Vertical displacements and rotations were measured during the static tests. The structural behaviour of the viaducts was also analysed with three dimensional finite element models (fig.2) [2]. A good agreement between experimental values and analytical results has been obtained. The dynamic tests consisted in ambient vibration measurements and tests with the train crossing the viaducts with different speeds. The ambient vibration tests were carried out in nine set-ups. During these tests 15 accelerometers were used. Sampling rate was selected as 1000 Hz. The software ARTeMIS was used for the modal identification [4]. The enhanced frequency domain decomposition technique (EFDD) was applied to estimate natural frequencies, mode shapes and the modal damping. The vibration modes obtained by the tests were compared with the calculated ones.

Figure 1: SATUOeiras

Figure 2: Finite element model of the viaduct VI

REFERENCES 1. Camara J, Rocha J, Cardoso D. SATUOeiras cable railway system. Struct. Eng. Int., 2005; 15(2): 82-84. 2. CSI. SAP2000 - Integrated Finite Element Analysis and Design of Structures, 2000. 3. Structural Vibration Solution - ARTeMIS Extractor Handy, Release 3.1, Denmark, 2002. — 238 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Finite Element Approach to Resin Flow during the Resin Film Infusion Process M. Yang, S. L. Yan* School of Science, Wuhan University of Technology, Wuhan, 430070 China Email: [email protected], [email protected] Abstract Resin film infusion (RFI) is a new cost-effective fabrication technique. The success of RFI process depends largely on the successful impregnation of the preform by the resin. Prediction of the location of the flow front and pressure distribution during the mold filling is useful in addressing several critical issues of the process. Mold filling of resin film infusion process involves a moving flow front, and so presents a moving boundary problem. The major method currently being used to solve this problem is the finite element/control volume (FE/CV) method. The FE/CV approach makes the resin impregnation process is regarded as a quasi-steady process by assuming a steady state condition at each time step even though it is a transient process. The selection of the time step increment for each of the quasi-steady state is based on the consideration that the time increment used allows only one control volume region to be completely filled, which may become cumbersome and time-consuming when higher-order nodes are involved. A physically accurate and computationally effective finite element method (FEM) is presented to simulate the isothermal resin infusing process. The FEM is based on conservation of resin mass at any instant of time and is objective of resin film infusion (RFI) fiber impregnation and mold filling. Darcy's low is invoked for the resin flow velocity field, thereby forming a transient governing equation involving the pressure field and the resin saturation fill factor which tracks the location of the resin front surface. Finite element approximations are then introduced for both the fill factor and the pressure field, and the resulting transient discrete equations are solved in an iterative manner for both the pressures and the fill factors for tracking the progression of the resin front in an Eulerian mold cavity. The FEM primarily works with the finite element mesh geometry, and the control volume regions associated with nodes need not be specified, so it is superior to the traditional finite element/control-volume (FE/CV) method. Based on the FEM method, a computer code is developed to simulate the resin infusing visually during the resin film infusion process. A numerical example presented here also demonstrated that compared with FE/CV method, FEM is physically accurate and computationally efficient. REFERENCES 1. Bruschke MV, Adrvuni SG. A finite element/control volume approach to mold filling in antistrophic porous media. Polymer Composites, 1990; 11(6): 398-405. 2. Mohan RV, Ngo ND, Tamma KK. On a pure finite-element-based methodology for resin transfer mold filling simulations. Polymer Engineering and Science, 1999; 39(1): 26-43. 3. Park J, Kang MK. A numerical simulation of the resin film infusion process. Composite Structures, 2003;60:431-437.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Simulation of a New Complex FRP Pipe Culvert by FEA M.W.Yang * College of Civil Engineering, Tongji University, Shanghai, 200092 China Email: [email protected] Abstract A new type of complex FRP pipe culvert is mainly composed of FRP fibre, liquid of resin and concrete. It is manufactured as following steps. Firstly, soak the liquid of resin into FRP fibre. Secondly, twin the FRP fibre around the core model orderly under the pre-defined tension. Finally, concrete pipe culvert under the normal or high temperature environment. It has three layers in pipe culvert. The interior layer is FRP fibre added material of antiseptic, permeation resistance, temperature resistance. The middle layer is structure layer composed of concrete and FRP fibre. The exterior layer is also FRP fibre, which is added material of permeation resistance and temperature resistance. Considering mutual reaction of soil and pipe culvert, the mechanic performance of FRP pipe culvert is simulated by FEA in this paper . The loading behavior of the pipe with different ratios of diameter to thickness and different depth in the soil is concluded. Some important conclusions are drawn. The comparisons between the new type of FRP pipe culvert and traditional steel concrete pipe culvert are given. The results show that the FRP pipe culverts have good performance than others. It provides some useful evidences for us to the further study of complex FRP material pipe culvert.

Figure 1: FRP pipe culvert in soil, sketch of material and analysis model

Figure 2: Result of analysis

REFERENCES 1. Wu YY, Yin YG. The research of new material and technics of pipe culvert in road. The Material of Engineering, 2005; 19 (2): 118-120. 2. Kuai HC, Ren BJ. The finite element analysis and reinforcing measures of pipe culvert. Hunan Communication Science and Technology, 2000; 26 (4): 50-52 (in Chinese).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Study of Two-Dimensional Transient Heat Conduction Using Finite Element Method Lee Yuk Choi **, Wong Sau Keong \ Ooi Hooi Woon ! , Sia Chee Kiong2 1

Faculty of Mechanical Engineering, Kolej Universiti Teknikal Kebangsaan Malaysia (KUTKM), Locked Bag 1200, Ayer Keroh, Melaka, Malaysia Department of Material and Design Engineering. Kolej Universiti Teknologi Tun Hussein Onn (KUiTTHO), Locked Bag 101, Batu Pahat, Johor, Malaysia Email: yukchoi@ kutkm.edu.my Abstract A mathematical formulation applied to a numerically study of two dimensional transient heat conduction in a building slab to determine the temperature distribution as a function of time. The finite element method is a method of choice and it is a numerical procedure for analyzing structures and continua. Usually the problem addressed is too complicated to be solved satisfactorily by classical analytical methods. The Galerkin method is the method of choice in the formulation, which involves transient heat conduction by using the bilinear quadrilateral element discretization; the integration of weighted residual of the differential equation and also the boundary condition was perform using finite element method. A computer program is developed using Matlab to run the analysis. The finite element analysis software MSC Nastran has also been used to analyze the temperature distribution and for the comparison purposes. It was found that temperature declined in the early stage for delta time, At =1 second which is more efficient. For comparison with MSC Nastran, different delta time for 32 and 200 elements is taking into consideration. The result of the analysis shows that the error is less than 1% by using Matlab and MSC Nastran, which was deemed satisfactory. There are 3 nodes selected for comparison, which is Node 5-ML, Node 23-ML and Node 41-ML for Matlab simulation tools and Node 5-MSC, Node 23-MSC and Node 41 -MSC is from MSC Nastran simulation tools. It can be said that with 32 elements, At =0.4 s shown not much difference as most of the point situated at the same exponential curve line for both MSC Nastran and Matlab simulation tools. It achieves saturation at 213°C for Node 5-ML and Node 5-MSC, 175°C for Node 23-ML and Node 23-MSC and 67°C for Node 41-MLandNode41-MSC.

REFERENCES 1. Krarti M. Steady state heat ransfer from horizontally insulated slabs. Int. J. Heat & Mass Transfer, 1993; 36:2135-2145. 2. Mendes N, Oliveira GHC, Humberto X de Araujo. Building thermal performance analysis by using MATLAB/ Simulink. Proc. 7th Int. IBPSA Conf., Rio de Janeiro, Brazil, 2001, pp. 473-480. 3. Huebner HK, Dewhirst DL, Smith DE, Byrom TG. The Finite Element Method For Engineers, Fourth Edition. Canada: A Wily-Inc, 2001.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Chemical Reaction, Heat and Mass Transfer on Nonlinear MHD Boundary Layer Flow through a Vertical Porous Surface with Thermal Stratification in the Presence of Suction R. Kandasamy l*, K. Periasamy2, K. K. Sivagnana Prabhu 3 1

Department of Mathematics, Institute of Road and Transport Technology, Erode -638 316, India Department of Chemistry, Institute of Road and Transport Technology, Erode -638 316, India 3 Department of Chemical Engineering, R.M.K. Engineering College, Chennai, India Email: [email protected] Abstract In this paper a similarity analysis is made for the forced and free convection boundary layer flow in a semi infinite expanse of an electrically conducting viscous incompressible fluid past a semi-infinite non conducting porous plate with chemical reaction, heat and mass transfer in the presence of suction. A magnetic field is applied transversely to the direction of the flow. Adopting the similarity transformation, governing nonlinear partial differential equations of the problem are transformed to nonlinear ordinary differential equations. Then the numerical solution of the problem is derived using Gill method, for different values of the dimensionless parameters. The results obtained show that the flow field is influenced appreciably by the presence of chemical reaction, buoyancy effect and magnetic field.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Development of a Mathematical Model for Heat and Mass Transfer Inside a Granular Medium V. J. Petty1*, A. L. De Bortoli1, O. Khatchatourian2 1

Department of Pure andApplied Mathematics, UFRG, P O. Box 15080, Porto Alegre, RS - Brazil Departament of Physics, Statistics and Mathematics, UNIJUI, P. O. Box 560, Ijui, RS - Brazil Email: [email protected], [email protected], [email protected] 2

Abstract In many situations of engineering and applied mathematics interest we are confronted with heat and mass transfer problems in granular medium, as in the grains drying processes and in small load nuclear reactors safety, where the spheres are fuel elements. In the literature we have found several models that try to describe these two processes. Most of the models found in the literature are empiric or semi-empiric and valid, therefore, for a quite limited range of the involved parameters. In this work we present a mathematical model based on the Navier-Stokes equations in three dimensions that describe the hot air flow going through the granular middle. We consider the grains in the form of small spheres inside a chamber. During the air passage in the grains middle occurs heat and water mass transfer between the spheres and the air. The heat transfer in the air occurs by convection and conduction processes, while the water mass (as vapor) in the air, is transferred by the convection and the diffusion processes. The heat and mass transfer between the air and the grains is considered in the sources terms of the energy and mass conservation equations for the grains and the air. Those terms are modeled obeying the Fourier's and Fick's laws, respectively. Our objective is to evaluate the dimensionless parameters influence in the heat and mass transfer processes, as well as to establish the values of that parameters for which the model is valid. Among the dimensionless parameters analyzed, we have the Reynolds, the Schmidt, the Prandtl and the Eckert numbers. Numerical tests, using the Gauss-Seidel central finite difference scheme, are carried out for non dimensional parameters ranging from Re = 200 - 2000, Sc = 0.7 - 0.9, Pr = 0.7 - 0.9, Ec = 10"5 -10" 4 . The results will contribute to obtain a better understanding of the parameters influence during the drying process of a granular medium. REFERENCES 1. Mhimid A, Ben Nasrallah S, Forh JP. Heat and mass transfer during drying of granular products simulations whit convective and conductive boundary conditions. Int. J. of Heat and Mass Transfer, 2000;43:2779-2791. 2. Petry VJ, De Bortoli AL, Sefidvash F. Passive cooling of a fixed bed nuclear reactor, in Proceedings of 4th ICCHMT, Paris-Cachan, France, 2005, pp. 305-308. 3. Srivastava VK, John J. Deep bed grain drying modelling, Energy Conversion and Management, 2002; 43: 1689-1708.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Factor Analysis for Convective Heat Transfer Problem by Using the ANN Method H. K. Tarn l *, S. C. Tam \ A. J. Ghajar2, L. M. Tarnl 1

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macau, China 2 Department of Mechanical and Aerospace Engineering, College of Engineering, Oklahoma State University, USA Email: [email protected] Abstract Artificial neural network (ANN) has been shown its superior predictive power compared with the conventional approaches in many studies. However, they always be treated as 'black box' because it provides little explanatory insight into the relative influence of the independent variables in the prediction process. Ghajar et. al. (2002) used the ANN method to develop an empirical correlation for the heat transfer data for a horizontal tube with a reentrant inlet under uniform wall heat flux boundary condition. In their work, the least and the most important variables were examined using an index of importance obtained from the weight and bias matrices based on a single training. However, the method is only applied to one set of experimental data and whether this method can be applied to other sets of data is still not known. Therefore, in this study, additional experimental data for different inlet configurations, namely, the square-edge and bell-mouth inlets from Ghajar and Tam (1994) in the turbulent region are used to further verify this method. Unlike the previous research, the index of importance in this study is from the average of multiple trainings. Furthermore, the number of neurons and the computation iteration are also changed. Using the methodology, the Reynolds number (Re) can be observed as the most important parameters and the length-to-diameter ratio (X/D) is the least important parameter except for the bellmouth inlet. The findings agree well with the physical meanings for convective heat transfer in the turbulent flow region.

REFERENCES 1. Ghajar AJ, Tam LM, Tam SC. A new heat transfer correlation in the transition region for a horizontal pipe with a reentrant inlet—using neural network, in Heat Transfer 2002, Proc. 12th International Heat Transfer Conference, Elsevier, 2002, pp. 189-194. 2. Ghajar AJ, Tam L M. Heat transfer measurements and correlations in the transition region for a circular tube with three different inlet configurations. Experimental Thermal and Fluid Science, 1994; 8(1): 79-90.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Thermally Induced Mechanical Changes around a Potential Nuclear Waste Repository in China Y. M. Liu lj2 *, J. Wang2, M. F. Cai \ S. R. Wang3 1

University of Science and Technology Beijing, Beijing, 100083 China Beijing Research Institute of Uranium Geology, Beijing, 100029 China China University of Geosciences Beijing, Beijing, 100083 China Email: [email protected] 2

Abstract Deep geological disposal is considered to be the best option to isolate the high-level radioactive waste (HLW) from the biosphere. Heat generated from the waste in the repository will result in an increase in rock temperature and substantially change the stress state of the host rock surrounding a nuclear waste repository. For the safe design of a deep underground HLW repository, there has to be a systematic consideration of all thermal, hydrological and mechanical effects that could prejudice the integrity of the repository and its man-made and natural barriers in the short and long terms [1]. The Beishan area in Gobi desert in Gansu province is considered as the most potential candidate area for HLW geological repository in China [2]. A numerical investigation is conducted on the impacts of the thermal loading history on the evolution of mechanical response of a fractured rock mass containing a hypothetical HLW repository in Beishan. The geological data are extracted from the site investigation results at Jiujing section, Beishan area. The large-scale far-field analysis is conducted using FLAC2D for coupled thermo-mechanical process. The results show that the temperature distributions at different time scales and the thermal stresses of fractured rock masses vary significantly with mechanical properties determined at the representative scale. Vertical heaving and horizontal tensile displacement are observed above the repository. This study provides an important base for site selection and repository design of HLW repository in China. REFERENCES 1. Hudsona JA, Stephansson O, Andersson J, Tsang CF, Jing L. Coupled T-H-M issues relating to radioactive waste repository design and performance. International Journal of Rock Mechanics & Mining Sciences, 2001; 38: 143-161. 2. Wang J, Chen W, Su R, Fan H. Studies on Geological Disposal of Radioactive Waste in China. Atomic Energy Science and Technoloty, 2004; 38(4): 339-342.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Research on Thermo Quality Transmission Problems for Large-Scale Slab with Creep* J. X.Wang 1 ' 2 *, X.C.Wang 2 1

Key Lab for the Prevention and Control of Explosion Disaster, Beijing Institute of Technology, Beijing, 100081 China 2 College of Architecture and Civil Engineering, North China University of Technology, Beijing, 100041 China Email: [email protected] Abstract Based on Biot theory, the basic equations for large-scale slab are presented and the transient thermal or thermo-mechanical loading are considered, four types of material surface of large-scale slab forced by transient thermal or thermo-mechanical loading are discussed. The characteristics and regularity of heat quality transmission in large-scale slab were investigated. The displacement fields and pore pressure is obtained by using the formulation of the corresponding transient thermal loading. There is significance for the research on thermo quality transmission problems of large-scale slab with permeates flow and creep. Biot's work was nonlinear and 3-demensional, it marked that the general idea of fluid permeates in large-scale slab had been set up. A series of achievement has been given about the question for study or discussion of fluid permeation, for instance, in geomechanics and biomechanics and so on. But the results for the thermal quality transmission of large-scale slab with permeate flow and creep are less presented in the literature. Wang has investigated its theory and application considering thermal elasticity and thermal loading, some important conclusion has been presented. Based on this method, the displacement fields and pore pressure are obtained in this paper. REFERENCES 1. Wang JX, Wang XC. Constitutive relation of thermo-elasto-plastic impact problems for frozen soil porous medium, in: Mechanical properties of advanced engineering materials, Tsinghua University Press & Springer-Verlag, 2003, pp. 69-75. 2. Wang JX, Wang XC. Numerical analysis of contact thermo-elastic-plasticity problems with creep for frozen soil, in: Mechanical properties of advanced engineering materials, Tsinghua University Press & Springer-Verlag, 2003, pp. 63-68.

* The investigation was supported by the grant of the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KZCX1-SW-04); Scientific Research Common Program of Beijing Municipal Commission of Education (No.:KM200610009010), Beijing Natural Science Foundation Program and Scientific Research Key Program of Beijing Municipal Commission of Education (No. :KZ200610009005). — 246 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Current Developments on the Coupled Thermomechanical Computational Modeling of Metal Casting Processes C. Agelet de Saracibar*, M. Chiumenti, M. Cervera International Center for Numerical Methods in Engineering, Barcelona, Spain ETS Ingenieros de Caminos, Canales y Puertos, UPC, Barcelona, Spain Email: [email protected] Abstract The coupled thermomechanical computational modeling of metal casting processes has been one of the research topics of great interest over the last years. However, despite the considerable advances achieved in computational mechanics, the large-scale numerical simulation of these processes continues to be nowadays a very complex task. This is mainly due to the highly nonlinear nature of the problem, involving nonlinear material models, liquid-solid phase change, nonlinear thermomechanical boundary conditions and thermomechanical contact, among others. In this paper, current developments to deal with an accurate, efficient and robust coupled thermomechanical computational simulation of metal casting processes is presented. A thermodynamically consistent constitutive material model is derived from a thermoviscoplastic free energy function. A continuous transition between the initial fluid-like and the final solid-like is modeled by considering a J2 thermoviscoplastic model. Thus, an thermoelastoviscoplastic model, suitable for the solid-like phase, degenerates into a pure thermoviscous model, suitable for the liquid-like phase, according to the evolution of the solid fraction function [1-2]. A thermomechanical contact model, taking into account the insulated effects of the air-gap due to thermal shrinkage of the part during solidification and cooling, is introduced [1-2]. A fractional step method, arising from an operator split of the governing differential equations, is considered to solve the coupled problem using a staggered scheme [1-2]. Within a finite element setting, using low-order interpolation elements, a multiscale stabilization technique is introduced as a convenient framework to overcome the Babuska-Brezzi condition and avoid volumetric locking and pressure instabilities arising in incompressible or quasi-incompressible problems [3-4]. Computational simulation of industrial castings show the good performance of the model.

REFERENCES 1. Agelet de Saracibar C, Cervera M, Chiumenti M. On the constitutive modeling of coupled thermomechanical phase change problems. Int. J. Plasticity, 2001; 17: 1565-1622. 2. Chiumenti M, Agelet de Saracibar C, Cervera M. Constitutive modeling and numerical analysis of thermomechanical phase-change systems, Monograph M48, CIMNE, Barcelona 1999. 3. Agelet de Saracibar C, Chiumenti M, Cervera M, Valverde Q. On the orthogonal subgrid scale pressure stabilization of small and finite deformation J2 plasticity. Monograph CMFP 2, CIMNE, Barcelona 2004. 4. Chiumenti M, Valverde Q, Agelet de Saracibar C, Cervera M. A stabilized formulation for incompressible plasticity using linear triangles and tetrahedral. Int. J. Plasticity, 2004; 20: 1487-1504.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Application of the Mushy Cell Tracking Method for Gallium Melting S.-J. Liang **, Y.-J. Jan 2, M.-S. Chung

3

1

Department of Water Resources Engineering, Feng Chia University, Taichung, 407 Taiwan, China Department of Marine Engineering, Kaohsiung Marine University, Kaohsiung, 805 Taiwan, China Civil, Hydraulics, and Informatics Research Center, Sinotech Engineering Consultants, Inc., Taipei, 105 Taiwan, China Email: [email protected] 2

Abstract A thermally driven mushy cell tracking algorithm has been proposed and successfully applied to phase-change problems [3]. The interface tracking equation is based on the energy balance at the mushy cells which separate the liquid phase and solid phase. The mushy cell tracking method is applied to Gallium melting problem which has been studied experimentally [1] and numerically [2]. Previous study of this problem is not conclusive. It is found that prediction of the interface dynamics and thermo-fluid field with coarse meshes matches better than that with fine meshes. In this study, the mushy cell tracking method is applied to simulate Gallium melting systematically. Both unstructured meshes and structured meshes are used in the computations. Prediction of the moving interface and thermo-fluid field agrees well with the experiment data and other numerical solutions.

(a) Interface and streamlines of Hannoun, et al. [2] at various time instances

(b) Interface and streamlines of the present method at various time instances Figure: Comparison of numerical solution of Gallium melting of Hannoun, et al. [2] with the present method REFERENCES 1. Gau C, Viskanta R. Melting and solidification of a pure metal on a vertical wall. Journal of Heat Transfer Transactions of the ASME, 1986; 108: 174-181. 2. Hannoun N, Alexiades V, Mai TZ. Resolving the controversy over tin and Gallium melting in a rectangular cavity Heated from the side. Numerical Heat Transfer, Part B, 2003; 22: 253-276. 3. Jan YJ. A cell-by-cell thermally driven mushy cell tracking algorithm for phase change problem, Part II: Phase-change with natural convection. Computational Mechanics, 2006 (in press). — 248 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

New FDM for Plane Elasticity in Polar Coordinate B. Q. Zhu1*, J. S. Zhuo2, J. R Zhou1 1

College of Mechanical & Electrical Engineering, Hohai University, Changzhou, 213022 China College of Civil Engineering, Hohai University, Nanjing, 210098 China Email: [email protected] 2

Abstract Circular sector domain is the typical area of plane elasticity in polar coordinate for solution. The area can be changed to rectangle area with suitable transformation. With radial coordinate or circumfluent coordinate treated as "time" and dual variables introduced, two symplectic systems of different form can be established for plane elasticity in polar coordinate by using Hellinger-Reissner generalized variational principle. In this paper, FDM is introduced into radial symplectic system of elasticity. The idea of FDM in Lagrange system is adopted. After meshes and nodes being set, the surrounding area of each node is used as integrating area. From Hamiltonian dual equations, difference equations for each node are established by using integral interpolation method. Combining the difference equations of all nodes, whole difference format is constructed. The problems of thick-walled cylinder and curved beam are calculated by programming. The numerical results show that the proposed FDM is effective. Another numerical method is provided for plane elasticity in polar coordinate. Elasticity problems are always solved with one kind of variables in Lagrange system. Traditional FDM includes stress difference method and displacement difference method. These difference methods have some particular limit because they are restricted by the system itself. With dual variables for solution, the new FDM can treat complex boundary conditions better. Displacements and stresses can be obtained directly. The format of FDM with stress boundary condition is discussed in this paper. The problem with displacement boundary condition can be treated similarly. REFERENCES 1. Yao Weian, Zhong Wanxie. Symplictic Elasticity. Higher Education Press, Beijing, 2002 (in Chinese). 2. Zhou Jianfang, Zhuo Jiashou. A new solution of elasticity in polar coordinate. Acta Mechanica Sinica, 2001; 33: 839-846 (in Chinese).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Computational Strategies for Curved-side Elements Formulated by Quadrilateral Area Coordinates (QAC)* Song Cen1'3*, Depo Song1, Xiaoming Chen1, Yuqiu Long2 1

School ofAerospace, Tsinghua University, Beijing, 100084 China Department of Civil Engineering, Tsinghua University, Beijing, 100084 China 3 Failure Mechanics Laboratory, Tsinghua University, Beijing 100084, China Email: [email protected] 1

Abstract The sensitivity problem to mesh distortion often occurs for a quadrilateral finite element model. In order to avoiding this trouble, a new Quadrilateral Area Coordinate (QAC) method has been systematically established by generalizing the area coordinate system from triangle to quadrangle [1]. Based on the QAC method, some membrane, plate/shell elements have already been successfully developed. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion [2]. But for a quadrilateral element with curved sides, special difficulty may be encountered. This is because that the present QAC system can not handle curved-side cases directly. So in this paper, two computational strategies are proposed for dealing with such situation. The first pattern comes from isoparametric transformation: i) divide each quadrilateral element along its diagonal into two triangular regions; ii) establish the relationships between the QAC and local triangular area coordinates within each sub-triangle; iii) perform isoparametric transformation and evaluate stiffness matrices by Hammer numerical integration (full or reduced) in each sub-triangle; iv) obtain the final element stiffness matrix by sum of the results from two sub-triangles. The second pattern is an approximate one, which makes some simplification to the Jacobian determinant during the isoparametric transformation. Then, these two schemes are introduced into the eight-node quadrilateral, QAC element AQ8 [3]. Two new eight-node membrane QAC elements AQ8-M1 and AQ8-M2, which can be used for both straight and curved meshes, are successfully constructed. Numerical examples show the new elements are insensitive to straight and curved mesh distortion, and possess excellent performance in most cases. It also demonstrates that the present strategies are effective complements of QAC system. REFERENCES 1. Long YQ, Li JX, Long ZF, Cen S. Area coordinates used in quadrilateral elements. Commun. Numer. Meth. Engng., 1999; 15(8): 533-545. 2. Cen S, Long YQ, Yao ZH, Chiew SP. Application of the Quadrilateral Area Coordinate method: A New Element for Mindlin-Reissner plate. Int. J. Num. Meth. Eng., 2006; 66(1): 1-45. 3. Soh AK, Long YQ, Cen S. Development of eight-node quadrilateral membrane elements using the area coordinates method. Computational Mechanics, 2000; 25(4): 376-384. * This study was supported by the Natural Science Foundation of China (10502028), the Special Foundation for the Authors of the Nationwide (China) Excellent Doctoral Dissertation (200242).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Studies of 4-node Membrane Element with Analytical StiffnessMatrix Based on the Quadrilateral Area Coordinates* YuDu 1 *, SongCen 1 ' 2 1

School ofAerospace, Tsinghua University, Beijing, 100084 China Failure Mechanics Laboratory, Tsinghua University, Beijing, 100084 China Email: [email protected], [email protected] 2

Abstract The new Quadrilateral Area Coordinate (QAC) method is a powerful tool to construct 2D finite element models [1]. Compared with the traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. Various elements based on QAC method have been successfully developed, among which the 4-node quadrilateral membrane element AGQ6-I is a typical one [2]. Theoretically, by using the area integral formulae [3] the analytical stiffness-matrix could be obtained, which may greatly benefit the computation procedure. However, as the derivation of the analytical expression is relatively complicated, all the papers by far still adopt the numerical integration method for computer coding, which indeed impedes the advantages of the QAC method. So in this paper, by introducing the basic QAC formulae into two famous symbolic computation softwares, Maple and Mathematica, the analytical expression of the stiffness-matrix of AGQ6-I is obtained for the first time. Then a corresponding FORTRAN subroutine is compiled. Numerical examples show that the present scheme exhibits excellent performances in computation efficiency when compared with the QAC element using numerical integration and the isoparametric element. Furthermore, some general remarks for simplification are also concluded from the derivation process, which may provide significant reference for other researches. REFERENCES 1. Long YQ, Li JX, Long ZF, Cen S. Area coordinates used in quadrilateral elements. Commun. Numer. Meth. Engng., 1999; 15(8): 533-545. 2. Chen XM, Cen S, Long YQ, Yao ZH. Membrane elements insensitive to distortion using the quadrilateral area coordinate method. Computers & Structures, 2004; 82(1): 35-54. 3. Long ZF, Li JX, Cen S, Long YQ. Some basic formulae for area coordinates used in quadrilateral elements. Commun. Numer. Meth. Engng., 1999; 15(12): 841-852.

* This study was supported by the Natural Science Foundation of China (10502028), the Special Foundation for the Authors of the Nationwide (China) Excellent Doctoral Dissertation (200242).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Special Hybrid Multilayer Finite Elements for 3-D Stress Analyses Around Quasi-Elliptic Hole in Laminated Composites Z. S. Tian**, Q. P. Yang \ X. Q. Zhang 2 1

Department of Mechanics, Graduate School, Chinese Academy of Sciences, P.O.Box 2706, Beijing, 100080 China Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, 100080 China Email: [email protected] Abstract Two kinds of 3-dimensional assumed stress hybrid multilayer finite elements — one contains a traction-free planar surface and the other contains a traction-free cylindrical surface — have been developed based on the Hellinger-Reissner principle and a modified complementary energy principle respectively. The combination of these two kinds of special elements can be used for efficient and accurate analysis of stress around quasi-elliptic hole in thin to thick laminated composites. The expressions of fully 3-dimensional stresses of the special elements are derived, such that the homogeneous equilibrium equations in each layer, the continuity of the transverse stresses on the interlayer surface and the traction-free boundary conditions over the designated surfaces are satisfied a prior, while the interelement reciprocity traction condition is satisfied a posterior in a variational sense. All components of displacement are included since bending/stretching coupling may occur. The transverse-shear deformation effects are incorporated in each layer with displacement continuity enforced along interlayer surface. Example solutions have indicated that the combination of these special multilayer finite elements is far superior in predicting the distributions of circumferential stresses and transverse stresses for laminated composites with quasi-elliptic hole when very coarse element meshed are used. Meanwhile these special elements possess much more efficiency than the ordinary assumed stress finite elements and the conventional assumed displacement elements. REFERENCES 1. Tian ZS, Liu JS, Ye L, Pian THH. Studies of stress concentration by using special hybrid stress elements. Int. J. Num. Meth. Engng., 1997; 40(8): 1399-1412. 2. Tian ZS, Zhao FD. Stress concentration in a solid with symmetric U-shaped grooves. Strain Analysis, 2001; 36(2): 211-217. 3. Tian ZS, Zhao FD, Tian Z. Special hybrid stress element for stress analyses around circular cutouts in laminated composite. Scienc in China (Series E), 2001; 45(5): 531-541.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A 3-Dimensional Assumed Stress Hybrid Element with Drilling Degrees of Freedom A. P. Wang **, Z. S. Tian \ X. Q. Zhang2 1

Department of Mechanics, Graduate School, Chinese Academy of Sciences, P.O. Box 2706, Beijing, 100080 China Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, 100080 China Email: [email protected] Abstract A new 3-dimensional 8-node assumed stress hybrid finite element with drilling degrees of freedom — which contains a traction-free cylindrical surface — has been developed based on a modified complementary energy principle. The assumed stress field satisfies the homogeneous equilibrium equations in the element and the traction-free conditions over the cylindrical boundary. Meanwhile, it also satisfies the compatibility equations when the element is degenerated into 2-dimensional case. The combination of the special elements with the ordinary assumed displacements elements can be efficiently used for analyzing the stress concentrations around some cutouts. The stress distributions of solid with symmetric fillets, semi-circular notches and U-shaped notches — in different ratios of the height to the width — under tension or bending are analyzed to evaluate the following three cases: 1) Combining the special assumed stress hybrid elements with the ordinary isoparametric elements which are derived by the conventional assumed displacement approach. 2) Using the conventional assumed displacement elements with drilling degrees of freedom everywhere. 3) Using a numerical analytic method which is based on the superposition principle. Numerical results have demonstrated that the special element can provide much more accurate stress concentration factors and the distributions of circumferential stress along the rim of the grooves than those obtained by using other methods. REFERENCES 1. Tian ZS, Liu JS, Ye L, Pian THH. Studies of stress concentration by using special hybrid stress elements. Int. J. Num. Meth. Engng., 1997; 40(8): 1399-1412. 2. Yunus SM. A study of different hybrid elements with and without rotational d.o.f. for plane stress and plane stain. Comput.& Stuct., 1988; 30(5): 1127-1133. 3. Lee TY, Kang YS, Choi CK. Non-conforming membrane elements with drilling d.o.f. CD-ROM of WCCM'V, Vienna, Austria, 2002.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Suppression of Zero-Energy Modes in Hybrid Finite Elements via Assumed Stress Fields Canhui Zhang*, Dongdong Wang, Jianlin Zhang Department of Civil Engineering, Xiamen University, Xiamen, 361005 China Email: [email protected], [email protected], [email protected] Abstract For hybrid finite elements, if an assumed stress field does not contain enough appropriate stress modes, the resulting element will include zero-energy deformation modes and thus can not be used for pratical applications [1-3]. On the other hand, adding extra stress modes will require more computational effort. For a finite element including n degrees of freedom and r rigid body modes, generally m-n-r stress modes are considered to be the optimal choice from the computational point of view. However, so far there still does not exist a universal and rational way to derive the m optimal assumed stress modes that can be used to generate a hybrid element free of zero-energy or kinematic deformation modes. This paper present a methodology to suppress the zero-energy mode in hybrid element using assumed stress fields. The m basic deformation modes of hybrid element are derived straightforward from the displacement field, which are linearly independent to each other so that they can represent any deformation modes of given element. Their corresponding stress modes are employed to determine the zero-energy modes in the hybrid element. It is shown that a basic deformation mode is a spurious kinematic mode if its corresponding stress mode is orthogonal to all modes in the assumed stress field, and a assumed stress mode is zero-energy mode when it is orthogonal to all basic deformation modes. Based upon the orthogonality relationship between the initial stress modes and the basic stress modes that are corresponding to the basic deformation modes, one can find the necessary m stress modes to formulate a hybrid element free of hour-glass modes. The iso-function method is adopted here to derive the initial stress modes and a related systematic procedure is discussed. Furthermore, it is found that the zero-energy stress modes cannot suppress the zero-energy deformation modes, instead they increase the stiffness associated with the nonzero-energy deformation modes of the element. Thus it is not appropriate to include the zero-energy stress modes into the assumed stress field. The examples of 2-D, 4-node plane element and 3-D, 8-node solid element are provided to illustrate the proposed method. REFERENCES 1. Wu CC, Pian THH. Incompatible Numerical Analysis and Hybrid Finite Element Method. Science Press, Beijing (1997). 2. Pian THH, Chen DP. On the suppression of zero-energy deformation models. Int. J. Numer. Meth. Eng., 1983; 19: 1741-1753. 3. Zhang C, Feng W, Huang Q. The stress subspace of hybrid stress element and the diagonalization method for flexibility matrix H. Applied Mathematics and Mechanics, 2002; 23: 1263-1273.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Analysis of Concentrated Boundary Loads in the Scaled Boundary Finite Element Method T. H.Vulj,e,A. J. Deeks2 1 2

School of Civil and Resource Engineering, University of Western Australia, Perth, Australia School of Civil and Resource Engineering, University of Western Australia, Perth, Australia

Email: [email protected], [email protected] Abstract The scaled boundary finite element method models linear elastostatics problems excellently and out-performs the finite element method (FEM) when solving problems involving unbounded domains or stress singularities. This paper introduces a technique for solving concentrated boundary load problems using the scaled boundary finite element method. By employing fundamental solutions for the displacements and the stresses, the solution is computed as summation of a fundamental solution part and a regular part. The singularity of the solution is modelled exactly by the fundamental solution, and only the regular part, which enforces the boundary conditions of the domain onto the fundamental solution, needs to be approximated in the solution space of the basic scaled boundary finite element method. In the case of a half-plane where the source point is still in the interior but close to the boundary, a dipole image solution technique can be employed to obtain the fundamental solution. In this technique an extra image source, which is an image of the main source mirrored on the opposite side of the boundary, is included to cancel the boundary condition violations caused by the load singularity. The half plane fundamental solution is consists of the summation of the Kevin solution with a complementary part, which is the contribution of the image source to the main solution. When the source point is on the half-plane boundary, the fundamental solution is the called the Flamant solution when the point load is normal to the boundary, and the Cerruti solution when the point load is tangential to the boundary. Although the fundamental solutions are evaluated by different sets of formulae depending on the location of the concentrated load in the domain, the basic procedures for formulating and solving the boundary value problem for the regular part remain unchanged. Using the new technique, a solution with high accuracy is recovered when solving problems which have not previously been solved effectively, such as the problem with a concentrated load on the boundary, and problems which have not previously been solved with the scaled boundary finite element method, such as the problem with a concentrated load acting on a side face. For the sake of simplicity, only the case of a straight boundary is considered in this study Examples provided illustrates that the new approach is simple to implement and accurate for solving the concentrated load problems. The proposed technique is applied to 2D in this paper but an extension to any linear problem, for which fundamental solutions exist, is straightforward.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

A Frequency-Domain Approach for Transient Dynamic Analysis Using Scaled Boundary Finite Element Method (I): Approach and Validation Z. J. Yang1'2*, A. J. Deeks2, H. Hong2 1

College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310027 China School of Civil and Resource Engineering, the University of Western Australia, WA 6009 Australia Email: [email protected] 2

Abstract Solutions of the scaled boundary finite element method (SBFEM) are approximate in the circumferential direction in the finite element sense, and analytical in the radial direction. This semi-analytical nature makes the SBFEM particularly suitable for certain situations such as unbounded media and solids with stress singularities (e.g., cracks or corners). This method has been very successfully applied in elastostatics. Its application to elastodynamic problems, however, has lagged behind, primarily due to the lack of an effective solution procedure to the governing equilibrium equation system in the frequency domain, which consists of second-order nonhomogeneous differential equations with respect to the radial coordinate. The authors recently developed an easy-to-follow Frobenius solution procedure to this equation system and conducted initial validations against simple problems subjected to harmonic loadings [1]. This study further develops a frequency-domain approach for the general transient dynamic analysis, through combining the Frobenius solution procedure and the fast Fourier transform (FFT) technique. The complex frequency-response functions (CFRFs) are first computed using the Frobenius solution procedure. This is followed by a FFT of the transient load and a subsequent inverse FFT of the CFRFs to obtain the time history of responses. Two wave propagation problems, subjected to Heaviside step load and triangular blast load, are modeled with a small number of degrees of freedom using the new approach. The numerical results agree very well with the analytical solutions and those from FEM. Further applications of this approach to transient dynamic fracture problems are presented in the accompanying paper [2]. REFERENCES 1. Yang ZJ, Deeks AJ, Hao H. A Frobenius solution to the scaled boundary finite element equations in frequency domain. Int. J. Num. Meth. Eng., (under review). 2. Yang ZJ, Deeks AJ, Hao H. A frequency-domain approach for transient dynamic analysis using scaled boundary finite element method (II): application to fracture problems. The 10th Int. Conf. Comp. Meth. in Engng. & Sci (EPMESC X). Aug. 21-23, 2006, Sanya, Hainan, China.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A Frequency-Domain Approach for Transient Dynamic Analysis Using Scaled Boundary Finite Element Method (II): Application to Fracture Problems Z. J. Yang 12 , A. J. Deeks2*, H. Hong 2 1

College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310027 China School of Civil and Resource Engineering, the University of Western Australia, WA 6009 Australia Email: [email protected] Abstract The scaled boundary finite element method (SBFEM) allows mixed-mode stress intensity factors (SIFs) to be accurately determined directly from the definition because the stress solutions in the radial direction are analytical. This makes the SBFEM superior to the traditional finite element method and boundary element method when calculating SIFs, since the other approaches generally need special crack-tip treatment, such as refining the crack-tip mesh or using singular elements. In addition, anisotropic material behaviour can be handled with ease by the SBFEM. These advantages are exploited in this study, in which the new frequency-domain approach combining the Frobenius solution procedure [1] and the fast Fourier transform technique, as developed and validated in the accompanying paper [2], is applied to model transient dynamic fracture problems. Two benchmark problems with isotropic and anisotropic material behaviour are modeled with a small number of degrees of freedom. Excellent agreement is observed between the results of this study and those in published literature. The new frequency-domain approach thus provides a competitive alternative to model dynamic fracture problems with high accuracy and efficiency. REFERENCES 1. Yang ZJ, Deeks AJ, Hao H. A Frobenius solution to the scaled boundary finite element equations in frequency domain. Int. J. Num. Meth. Eng. (under review). 2. Yang ZJ, Deeks AJ, Hao H. A frequency-domain approach for transient dynamic analysis using scaled boundary finite element method (I): the approach and validation. The 10th Int. Conf. Comp. Meth. in Engng.&Sci (EPMESC X). Aug. 21-23, 2006, Sanya, Hainan, China.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan ,China ©2006 Tsinghua University Press & Springer

A Novel Fuzzy expert Diagnosis System of Inner-Faults for ThreePhase Squirrel Cage Induction Motors Tak Son Cheang*, Si Leong Chan, Booma Devi Sekar, Ming Chui Dong Faculty of Science and Technology, University of Macau, P.O. Box 3001, Macau, China Email: [email protected], [email protected] Abstract Three phase induction motors are widely used in industrial production all over the world. Due to high utilization rate, frequent starting and changing load condition, the appearance of failures in induction motors is relatively high. A sudden fault will cause large economic losses. Therefore, the early fault detection becomes very important. During practical operation of three-phase induction motors in industry, the inner faults may occur-in their rotor and stator windings. These kinds of faults will make serious influence on the operation of the motor. There is insufficient R&D in the diagnosis system of inner-faults of squirrel cage induction motor. Thus a new research on the fuzzyexpert diagnosis system of inner-faults for three-phase induction motors is presented. The purpose of fault diagnosis is to isolate the cause of a system malfunction in a timely manner. This paper presents a new sequence performing diagnosis of the three-phase induction motors as: sample the fault symptoms, carry the fuzzyexpert forward & backward inference, deduce the fault hypotheses and conclude the fault discrimination. Among them, a novel technique of how to define the various membership functions and the relevant fuzzy sets concerning the uncertainty issues based on the symptoms of motor faults (experimental, simulation & calculation data), and how to construct the proper inference nets for producing the production rules are explained. In summary, the following issues will be presented in this paper: 1. List of all faults of motor with related symptoms and hypotheses; 2. Expert's knowledge about the priori probabilities of each fault and the combinational faults of the motor; 3. Definition of fuzzy sets concerning the uncertainty symptoms & hypotheses; The inference nets with all necessary parameters for further calculating the probability propagations. Finally, a series of experiments of diagnosing the stator and rotor faults are shown.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Updating Noise Parameters of Kalman Filter Using Bayesian Approach K. I. Hoi *, K. V. Yuen, K. M. Mok Department of Civil and Environmental Engineering, University of Macau, Macau SAR, China Email: [email protected] Abstract Kalman filter [ 1 ] is known to be a robust tool in state estimation for linear or slightly nonlinear systems in diverse disciplines of science and engineering. In the area of structural engineering, Kalman filter has also received enormous attention over the years due to its importance for model updating, response prediction, structural control and health monitoring. The algorithm becomes popular since it provides not only the state estimation but also the associated uncertainty of the estimation. In addition, the algorithm is online so that the state vector is immediately updated once a new data point is obtained. However, the accuracy of Kalman filter depends on the prior knowledge of the process noise and measurement noise parameters, which is difficult to be obtained in practice. In the present study, the Bayesian propabilistic approach [2] is proposed to estimate these noise parameters in the Kalman filter for the case when the input is a zero-mean Gaussian white noise process and limited output measurements are available. The optimal estimates of the noise parameters are chosen by the maximum likelihood criterion. Through the two illustrative examples, the estimated noise parameters are close to the actual values in the sense the actual parameters are located in the region with significant probability density. Therefore, it is concluded that the Bayesian approach is able to provide accurate estimates of the noise parameters, and hence the state estimation, for Kalman filter. REFERENCES 1. Kalman RE. A new approach to linear filtering and prediction problems. Transactions of ASME, Journal of Basic Engineering, 1960; 82: 35-45. 2. Beck JL, Katafygiotis LS. Updating models and their uncertainties, I: Bayesian statistical framework. Journal of Engineering Mechanics, 1998; 124: 455-461.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Damage Detection of Vibrating Structure from Limited Natural Frequencies X. L. Li l*, H. Okuda \ G Yagawa2 1

Research into Artifacts, Center for Engineering, The University of Tokyo, Chiba, 277-8568 Japan Center for Computational Mechanics Research, Toyo University, Tokyo, 112-8611 Japan Email: [email protected]

2

Abstract Conventional non-destructive structural damage detecting approaches, such as eddy current, ultrasonic, and radiographic testing, the location of the damage must be known a priori or the whole structure needs to be tested. The vibration-based method uses changes in natural frequencies to detect damage and can avoid the above limitations. In mathematics it is proven that if all natural frequencies are known the stiffness matrix of structural system can be reconstructed fully, but in practice only limited natural frequencies are available, so we only can pursue the least squares solution of it. Accurate Jacobian matrix in each iterative step is necessary to solve this ill-posed underdetermined inverse eigenvalue problem by the Newton's method. In this paper, the Jacobian matrix of reconstruction equation is evaluated accurately and efficiently by the proposed symbolic-numerical hybrid approach. The numerical results of fixed-free spring-mass system show that good results can be obtained even only limited measured natural frequencies are used with different damage scenarios.

VW\_{

p V W q

]

AAAA_ .

^/v'VVV.^

]

Figure: Fixed-free spring-mass system

REFERENCES 1. Friedland S, Nocedal J, Overton ML. The formulation and analysis of numerical methods for inverse eigenvalue problems. SIAM J. Num. Anal., 1987; 24(3): 634-667. 2. Chu MT. Inverse eigenvalue problems. SIAM Review, 1998; 40(1): 1-39. 3. Chen Xuzhou, Chu MT. On the least squares solution of inverse eigenvalue problems. SIAM J. Num. Anal., 1996; 33(6): 2417-2430.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Error Estimations in LBIEM and Other Meshless Methods H. B. Chen1+, D. J. Fu \ X. F. Guo 2 , P. Q. Zhang ! CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230027 China 2 Teaching and Research Section of Mechanics, Dalian Jiaotong University, Dalian, 116028 China Email: [email protected] Abstract: Error estimation in numerical computations relates to the reliability of algorithms and improvement of computational efficiencies. Large amounts of relevant research have been developed and the most classes of error estimators are residue-based and recovery-based in FEM [1]. In meshless methods, nodes can be easily moved, added and deleted; therefore it leads to more convenient and attractive to implement adaptivity processes [2, 3]. In the current paper, the developments of error estimation in meshless methods are firstly reviewed. Some schemes to define the a-posteriori error estimators are quoted in detail. For instance, In EFGM, the stress projection technology, the strain gradient method, the cell energy method, recovery based estimation technology, etc, are respectively adapted to obtain the referenced solutions. Then, based on the relevant ideas of FEM and the other meshless methods, the dual error indicators defined by the differences of three sets of solution are introduced for the local boundary integral equation method (LBIEM) [4]. The three sets of potential solution are: "Original" denotes the potential calculated by MLS through the fictitious values at original nodes; "Taylor" denotes the potential obtained by MLS using Taylor series expansion, for the newly added nodes; "Refresh" denotes the potential calculated by MLS through the fictitious values at the original and newly added nodes, after a reanalysis of the LBIEs has been performed. The following figures show that, for the newly added nodes, the "Taylor" errors are always lower than their "Original" ones; in the well-distributed arrangements, the "Refresh" errors are similar to the "Taylor" ones at the newly added nodes. As a result, when these two defined error indicators are consistent with each other, the nodal arrangement with new additional nodes is well-distributed; otherwise, modulate the location and number of the new additional nodes. REFERENCES 1. Zienkiewicz OC. The background of error estimation and adaptivity in finite element computations. Comput. Meth. Appl. Mech. Eng., 2006; 195: 207-213. 2. Gavete L, Cuesta JL, Ruiz A. A procedure for approximation of the error in the EFG method. Int. J. Numer. Methods Eng., 2002; 53(3): 677-690. 3. Liu GR, Tu ZH. An adaptive procedure based on background cells for meshless methods. Comput. Meth. Appl. Mech. Eng., 2002; 191: 1923-1943. 4. Guo XF, Chen HB. Dual error indicators for the local boundary integral equation method in 2D potential problems. Eng. Anal. Bound. Elem., (in press).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Parallel Computing for Enriched Free Mesh Method (EFMM) Yosuke Kobayashi *, Genki Yagawa Center for Computational Mechanics Research (CCMR), Toyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo, 112-0001 Japan Email: [email protected] Abstract Performed in this paper is a structural analysis by the Parallel Free Mesh Method, which is a kind of mesh-free method. In order to improve the accuracy of the solution by the Free Mesh Method, we have developed the Enriched Free Mesh Method, where the displacement field and the strain field are independently assumed. The unknown parameters of these fields are determined by employing the Hellinger-Reissner principle. Some parallel performance study is made with a cluster computer. REFERENCES 1. Aoyama Y. A Guide to parallel programming: MPI version. RIKEN Information Base Center, 2004. 2. Yagawa G, Yamada T. Free mesh method: a new meshless finite element method. Int. J. Computational Mechanics, 1996; 18: 383-386.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

3D Animation for Free Mesh Method Shinsuke Nagaoka!*, Masakazu Inaba2, Genki Yagawal 1

Center for Computational Mechanics Research (CCMR), Toyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo, 112-0001, Japan 2 Techno Star, Inc., Tokyo, Japan Email: [email protected] Abstract In recent years, the performance of computer has been improved dramatically. Numerical or computation technology is also developing greatly with the above improvement in the performance of computer. On the other hand, the objects for the analysis become more and more complicated. To deal with the large-scale and complex problems, new approach called the meshless method has been actively studied. The Free Mesh Method is one of the meshless methods, which can solve the complicated phenomenon that the body is dynamically fragmented. In order to animate such an analysis result, this paper describes the 3D animation system for the free mesh method, which is intended to be a post-processor for dynamic phenomena.

REFERENCES 1. Yagawa G, Yamada T. Free mesh method: a new meshless finite element method. Comp. Mechanics.l8:383-386,1996.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A Stabilized Conforming Integartion Procedure for Galerkin Meshfree Analysis of Thin Beam and Plate Dongdong Wang* Department of Civil Engineering, Xiamen University, Xiamen, 361005 China Email: [email protected] Abstract This paper aims to develop an efficient Galerkin meshfree formuation to analyze thin beam and palte. First, through a reformulation of the reproducing conditions (or consistency conditions), a meshfree double-variable approximation based on both the deflection and slopes is constructed. It is shown that this approximation can approximate a given function with better accuracy than that purely based on the deflection variables. Then with this meshfree approximation function, the Galerkin framework for thin beam and plate is formulated. To treat the essential boundary conditions in the Galerkin meshfree formulation, the transformation method [ 1 ] is generalized herein to relate the nodal deflection and rotation coefficients and their corresponding physical values of deflection and rotations. Since the weak form of thin beam and plate involves the 2nd order derivatives, accurate numrical integration for the discrete stiffness matrix requires intensive computational effort. In this study, numerical tests show that low order Gauss quadrature as well as the previous stabilized conforming nodal integration [2-4] which was developed to integrate the stiffness with 1st order derivative, can not fully stabilize the mnumerical solutions. To obtain the pure bending solution and remain spatial stability, the srain smoothing idea in the stabilized conforming nodal integration is extended here. Rather than diectly employing the nodal strain smoothing and nodal integration, the problem domain is partitioned into integration cells and the numerical integration is carried out on several sampling integration points defined in each integration cell. At the smapling points, the curvature smoothing is performed within conforming sub-cells which comprise of the integration cell. Due to the conforming nature of curvature smoothing, the resulted meshfree scheme can exactly reproduce bending solutions. Numerical examples demonstrate the present strategy has superior convergence rates, accuracy and efficiency, compared with that using higher order Gauss integration. REFERENCES 1. Chen JS, Pan C, Wu CT, Liu WK. Reproducing kernel particle methods for large deformation analysis of nonlinear structures. Comput. Meth. Appl. Mech. Engng., 1996; 139: 195-227. 2. Chen JS, Wu CT, Yoon S, You Y. A stabilized conforming nodal integration for Galerkin meshfree methods. Int. J. Numer. Meth. Eng., 2001; 50: 435-466. 3. Wang D, Chen JS. Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput. Methods Appli. Mech. Eng., 2004; 193: 1065-1083. 4. Wang D, Dong SB, Chen JS. Extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry. Int. J. Solids Struct., 2006; 43: 144-171.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Element-Free Galerkin method with wavelet basis Y. H. Liii *, Y. N. Liu, Z. Z. Cen Department of Engineering Mechanics, Tsinghua University, Beijing, J 00084 China Email: [email protected] Abstract Many numerical methods have been developed and used to solve problems of computational mechanics. Recently, one of the hottest topics in computational mechanics is the meshless or meshfree method. Some meshless methods have been proposed and achieved remarkable progress, such as the element-free Galerkin (EFG) method [1], the meshless local Petrov-Galerkin (MLPG) methods [2],and so on. In above-mentioned meshless methods, it is key and necessary to construct the so-called shape function, which is complicated, time-consuming and even hard to realize in some special conditions. Furthermore, the complexity of shape function will increase computational cost in total process. It is desirable to find a new method, which is simple and reasonable to construct shape functions in meshless method. However, it seems to be a difficult task. So we should resort to some other mathematical tools. Wavelet is a powerful tool in solving many problems in science and engineering. In recent years, there has been an increasing interest in the wavelet-based approach due to the success in wavelet-based methods in several applications. Some of recent investigations on the wavelet method include papers by many scholars such as Amaratunga and Williams [3]. This paper is aimed at introducing a new meshless method by using DB wavelet. In the construction of shape function, scaling function in wavelet analysis is directly employed as shape function to approximate the unknown field functions. Because of the compact support and orthogonality, DB wavelets can describe the details of the problem conveniently and accurately. The other advantage brought by compact support is that the DB wavelet-based meshless method has relatively small degrees of freedom in solving practical problems. Compared with traditional meshless methods, this method is concise in formulations, and flexible in applications. The numerical examples illustrate that the DB wavelet-based meshless method has high computational accuracy. So DB wavelet-based meshless method has enormous potential in the analysis of other complicated problems. REFERENCES 1. Belytschko T, Lu YY, Gu L. Element free Galerkin methods. Int. J. Numer. Methods Engrg., 1994; 37: 229-256. 2. Alturi S N, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput.Mech.Engrg.,1998; 22: 117-127. 3. Amaratunga K, Williams JR. Wavelet-Galerkin solutions of boundary value problems. Comput. Mech. Engrg., 1997; 4: 243-285.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

The Numerical Prediction of Effective Properties of Non-Continuous Carbon Nano-Reinforced Composites by the Macro-Microscopic Homogenization Method Dongmei Luo **, Wenxue Wang2, Yoshihiro Takao2, Koichi Kakimoto2 1

Environment and Civil Engineering School, Foshan University, Guangdong, 528000, China Research Institute for Applied Mechanics, Kyushu University, 816-8580, Japan Email: [email protected]

1

Abstract In this paper, the homogenization method with exact periodic boundary conditions is adopted to predict the effective stiffness of composites reinforced with straight and sinusoidal wavy Carbon Nano-Tubes (CNTs) as shown in Figure (a) and (b), which are considered as the effective discontinuous transverse isotropic short fiber. Numerical calculations for regular and staggered array models (Figure (c)) are performed by using the commercial finite element software ABAQUS. Comparing with those from Mori-Tanaka and Halpin-Tsai theories, it is found that the stiffness of CNTs composites are not only related to aspect ratio and CNTs volume fraction as described in the above theories, but also to depend on CNTs arrays and distribution within the selected Representative Volume Element (RVE). The results from the two empirical approaches are included in the present results with special spacing ratios of horizontal and vertical fiber ends (Tf lHf ), and staggered arrays are significant to predict the effective stiffness with a high degree of axial overlap of fibers. For wavy CNTs reinforced composites, it is found that the CNTs waviness results in the reduction of effective stiffness of composites compared with straight CNTs reinforcement. The degree of reduction is dependent on the waviness amplitude, CNTs arrays method and wavy CNTs distributions. The results prove that the macro-micro homogenization method is efficient to predict the effective stiffness of a CNTs-composite by considering more microscopic parameters such as the CNTs properties, geometry characteristic and CNTs dispersion in the matrix, simultaneously.

^ m m m ^ ^ ^ ^ a ^ ^ ^ . ^
v

'

.. .. ,,...„ ; +* «*> * * m l * » t pmnm at *»«v»*r»«* array lor

mF^

(a) (b) (c) Figure: Straight and sinusoidal wavy CNTs, and regular and staggered array models REFERENCES 1. Luo DM, Wang WX, Takao Y, Kakimoto K. Prediction of the stiffness and stresses for carbon nano-tube composites based on homogenization analysis. Reports of Research Institute for Applied Mechanics, Kyushu University, Japan, 2005, No. 129, pp. 37-45. 2. Liu YJ, Chen XL. Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element. Mechanics of Materials, 2003; 35: 69-81. — 266 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Molecular Dynamics Simulation of Length Size Effect on Mechanical Properties of Nano-Metal D. Huang*, J. S. Zhuo Department of Engineering Mechanics, Hohai University, Nanjing, 210098 China Email: [email protected] Abstract Based on embedded atom potential and molecular dynamics method, a numerical simulation model suitable for analysis of mechanical properties and behavior of nano-scale metal was proposed. With this model the failure process and mechanical properties of nanocrystalline nickel wires with different cross section sizes were studied. Simulation reveals the motion of atoms, appearance and growth of nano-scale voids, changes of atomic arrangements and relationships between average atomic stress, energy and crystal lattice strain in nano-scale metal wires under static external load. Numerical results show that initial unstable energy, as a result of free surfaces and surface reconstruction, takes large effects on the deformation and failure mechanism of nano wires. Energy in surface atoms is much higher than that in the inner ones, so surface atoms depart from standard crystal lattice positions and nano voids come into being along the surfaces firstly. The deformation process of nano wire behaves as the expansion and connection of nano cavities from surface into inner lattices, which agrees well with experimental phenomena. What's more, the elastic modulus, yield and fracture strength on definite lattice directions of nickel nano wires with different cross section sizes were obtained, and the length size effect of nanocrystalline metal on mechanical properties was analyzed further. According to simulation results we put forward two calculation formulae on elastic modulus, yield strength and fracture strength of nickel nano wire. In the first one, elastic modulus and yield strength of metal nano wire are both linear to the logarithm of cross section length size, but fracture strength has an inverse relation to exponential cross section length size. In the other one, the elastic modulus and yield strength are divided into three parts: the contribution from corner lattices, surface lattices and inner ones, with different volume ratio and effect factor. Two formulae can predict the elastic modulus, yield strength and fracture strength of a nanocrystalline nickel nano wire according to the cross section size or surface lattice ratio respectively, and keep remarkable accordance with simulation results.

REFERENCES 1. Broughton JQ, Meli CA, Vashishta P, et al. Direct atomistic simulation of quartz crystal oscillators: bulk properties and nanoscale devices. Physical Review B, 1997; 56: 611-618. 2. Liu WK, Karpov EG, Zhang S, et al. An introduction to computational nanomechanics and materials. Computer Methods in Applied Mechanics and Engineering, 2004; 193: 1529-1578. 3. Voter AF, Chen SP. Accurate interatomic potentials for Ni, Al, and NisAl. Materials Research Society Symposium Proceeding, 1987; 82: 175-182.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Molecular Dynamic Simulations of CNT-Water Nanostructures J. Zou 1*, X. Q. Feng \ B. Ji \ H. Gao 2 1

Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China Division of Engineering, Brown University, 182 Hope Street, Providence 02912, USA Email: [email protected] 2

Abstract We report discoveries from molecular dynamics simulations that single-walled carbon nanotubes (CNT), with different diameters, lengths, and chiralities, can coaxially self-assemble into multiwalled carbon nanotubes in water via a set of consecutive spontaneous insertion of smaller tubes into larger ones. The assembly process is tube-size dependent and the driving force is primarily the inter-CNT van der Waals (vdW) interactions. Such coaxial self-assembly suggests possible "bottom-up" routes for fabrication of novel nanodevices. After the assembly, typical water structures besides bulk phase include a one-dimensional water chain inside the small tube, a uniform or non-uniform water shell between the two tubes and a "boundary layer" of water near the exterior wall of the large tube. We found that a concentric water shell consisting of one to three layers of water molecules can form between the two selected CNTs, which lead to a class of carbon-water-carbon composite nanotubes with distinct lifetimes. Analysis of the potential energy of the CNT-water system indicates that the carbon-water-carbon composite nanotubes are stabilized by both the tube-water and intertube van der Waals interactions, and a minute change in the CNT-water attraction will dramatically affect the stability of such composite nanotubes. Our results have general implications on filling nanoporous materials with nanoparticles or tubular nanostructures via the nonspecific, nonbonding vdW interactions. REFERENCES 1. Gao H, Kong Y, Cui D, Ozkan CS. Spontaneous insertion of DNA oligonucleotides into carbon nanotubes. Nano Lett., 2003; 3: 471-473. 2. Zou J, Ji B, Feng XQ, Gao H. Self-assembly of single-walled carbon nanotubes into multiwalled carbon nanotubes in water: molecular dynamics simulations. Nano Lett, 2006; 6: 430-434. 3. Zou J, Ji B, Feng XQ, Gao H. Molecular dynamics studies of carbon-water-carbon composite nanotubes. Small, (in press).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Computer Simulation of Quantum Dot Surface under Stress Xiaoming Liu, Zhuo Zhuang *, Tong Zhang School ofAerospace, Tsinghua University, Beijing, 100084 China Email: [email protected] Abstract In thin film structure, a traction-free quantum dot surface is usually under large stress for the presence of misfit strain between substrate and film. Such kind of material surface becomes unstable and roughens. The surface perturbation will rapidly lead to the breaking up of the coat, and the films eventually become damaged. In this paper, a variational formulation is proposed to study the evolution of the stressed quantum dot structure. Surface diffusion is considered as the dominant mass transport mechanism, while mass transport will be significant when process has the features of severe working condition such as relatively high stress, high temperature, electric current, small size scale. The elastic field was obtained by the first-order perturbation solutions. In our approach, the surface shape is expanded in terms of a complete set of basis functions, and surface governing equations can be obtained by the principle of minimization of energy. Our simulations reveal that surface profile grows due to the destabilizing influence of the large stress effects and shrinks due to the stabilizing influence of small stress effects. The competition between elastic strain energy and surface energy can lead to the growth of surface profile perturbations at large stress, small surface tension. The misfit strain between film and substrate can significantly influence the surface evolution. 006 Surttc* tvotulton profltM

005

(A 0j

t =0.0001)00

004

t'=0.000442 t"«0.002908

003

t*=0.009X28 ^3 02 001

-001 "VV*Q

~ " " ~ 0.2

04

08

0.8

1

y/X

Figure: Evolution of quantum dot surface

REFERENCES 1. Liu P, Zhang YW, Liu C. Computer simulations of the Stranski Krastanov growth of heteroepitaxial films with elastic anisotropy. Surface Science, 2003; 526: 375-382. 2. Yang WH, Srolovitz DJ. Cracklike surface instabilities in stressed solids. Physical Review Letters, 1993; 71: 1593-1596. 3. Shenoy VB, Ramasubramaniam A, Freund LB. A variational approach to nonlinear dynamics of nanoscale surface modulations. Surface Science, 2003; 529: 365-383. — 269 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

3D BEM for Piezoelectric Solids of General Anisotropy M. Denda1*, C.-Y Wang 2 ' 3 1

Mechanical and Aerospace Engineering Department, Rutgers University, NJ 08854-8058, USA Mathematics and Modelling Department, Schlumberger-Doll Research, CT 06877-4108, USA 3 {currently on secondment to) Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China Email: [email protected], [email protected] 2

Abstract In this paper we present a new scheme of three-dimensional boundary element method for the general anisotropic piezoelectric solids. We use the Radon transform representation of the three-dimensional fundamental solutions of piezoelectricity and integrate them analytically over the triangular boundary element with the linear interpolation. This reduces the computation for the system matrices G and H from the standard singular surface integrations to the simple regular line integrations and enables a drastic reduction of the computation time. The integrand of the line integral consists of the product of a function dependent and another function independent on the location vectors representing the source and observation points. The latter function depends only on the material and element properties and thus need be calculated only once for each element and saved for a repeated use in the calculation of G and H matrices and in the post-processing. Exploitation of this favorable structure results in the further reduction of the computation time for very large systems. The implementation of the proposed method with numerical examples will be presented. REFERENCES 1. Denda M, Wang CY, Yong YK. 2-D time-harmonic BEM for solids of general anisotropy with application to eigenvalue problems. J. Sound Vibrat., 2003; 261: 247-276. 2. Wang CY, Achenbach JD, Hirose S. Two-dimensional time domain bem for scatteringof elastic waves in solids of general anisotropy. Int. J. Solids Structures, 1996; 33(26): 3843-3864. 3. Denda M, Araki Y, Yong YK. Time-harmonic BEM for 2-D piezoelectricity applied toeigenvalue problems. Int. J. Solids Struct., 2004; 41: 7241- 7265. 4. Wang CY. Elastic fieldelds produced by a point source in solids of general anisotropy. J. Engng. Math., 1997; 32(1): 41-52. 5. Chen T, Lin FZ. Boundary integral formulations for three-dimensional anisotropic piezo-electric solids. Computational Mechanics, 1995; 15: 485-496. 6. Pan E, Yuan FG. Boundary element analysis of three-dimensional cracks in anisotropicsolids. Int. J. Numer. Meth. Engng., 2000; 48: 211-237.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Analysis of Quantum Dots Induced Strain and Electric-Field in Piezoelectric Semiconductor Substrate of General Anisotropy C.-Y. Wang **, M. Denda2, E. Pan 3 1

Mathematics and Modelling Department, Schlumberger-Doll Research, CT 06877-4108, USA, {currently on secondment to) Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China 2 Mechanical and Aerospace Engineering Department, Rutgers University, NJ 08854-8058, USA 3 College of Engineering, University of Akron, Akron, OH 44325-3905, USA Email: [email protected], [email protected], [email protected] Abstract Characteristics of the self-organized quantum dots (QDs) such as electron and hole energy levels and wave functions are dependent to the state of strain and electric field produced during the growing process of QDs in a semiconductor substrate. The calculation of the strain and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full three-dimensional and usually intricate shapes of the QDs. Numerical simulations are often performed by finite difference, finite element, or atomistic techniques, all require substantial computational time and memory. In this paper, we present a new Green's function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity. Following the literature of micromechanics, the problem is formulated as an Eshelby inclusion problem of which the solution can be expressed by a volume-integral equation that involves the Green's functions and the equivalent bodyforce of eigenstrain. The volume integral is subsequently reduced to a line integral based on exploring a unique structure of the Green's functions. The final equations are cast in a form that most of the computation results can be repeatedly used for QDs at different locations - a very attractive feature for simulating large systems of QD arrays. The proposed algorithm has been implemented and validated by comparison with analytical solutions. Numerical simulations are presented for pyramidal QDs in the substrates of Gallium Arsenite (GaAs) [001] andGaAs[lll].

Figure: Micrograph of pyramid-shaped quantum dots grown from indium, gallium, and arsenic. Each dot is about 20 nanometers wide and 8 nanometers in height. Image courtesy NIST.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Adaptive Under-Frequency Load Shedding Scheme by Genetic Algorithm Chinwang Lou *, Mingchui Dong, Chikong Wong Faculty of Science and Technology, University of Macau, P.O. Box 3001, Macau, China Email: [email protected], [email protected], [email protected] Abstract Under-Frequency Load Shedding (UFLS) is one technique used in the power industry for many years. The UFLS function is to rescue the system under extreme disturbances and avoid the system collapse. Traditionally, a scheme is designed based on the projected scenarios (considering the summer and winter conditions) as well as the worst generation deficiency that could occur in the system. Usually, when the old UFLS settings are no longer appropriate to the new operating mode, they must be re-designed. I.e. based on the past experience in design and operation, the new scheme and settings are assumed firstly. Then, off-line computer simulation is adopted to verify the UFLS performance. Settings are adjusted according to the simulation result. Afterwards, the simulation and performance are verified again under the adjusted settings. This procedure is repeated until the settings are sought such that the UFLS demonstrates the best performance on the selected scenarios. Many efforts were devoted to simplify this procedure. In this paper, a novel adaptive load shedding method based on Genetic Algorithm (GA) is proposed to automate the findings of optimal settings such that the repetitive trial-error can be minimized greatly. The simulation results on figures show that the new scheme based on GA has superior performance, compared with old scheme. Average Load shed Amount

Avg. Frequency Performance

I Old Dg60-4

I Old Dg60-4

4.50

|—^

100.0

s? 2.50 1.50 0.50

rlTrMl.1T.B-,. T :

%

Scenario

r

-

^

i° m\ 11 11

^

III1

Scenario

Figure: The performance comparison in two load shedding schemes (g60-4=settings optimized by GA) REFERENCES 1. Prasetijo D, Lachs WR, Sutanto D. A new load shedding scheme for limiting underfrequency, IEEE Transactions on Power Systems, 1994; 9(3): 1371-1377. 2. User guide of MTSP (mid-term stability program), Tsinghua University, Beijing, China. 3. Al-Hasawi WM, El Naggar KM. Optimum steady-state load-shedding scheme using genetic based algorithm. IEEE Melecon 2002, Cairo, Egypt, May 7-9, 2002. — 272 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

An Effective Computer Generation Method for the Domain with Random Distribution of Large Numbers of Heterogeneous Grains Yan Yu l *, Junzhi Cui 2 , Fei Han l Department of Applied Mathematics, School of Science, Northwestern Poly technical University, XVan, 710072 China 2 Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100080 China Email: [email protected], [email protected] Abstract In this paper in order to perform the prediction of the physics and mechanics properties for the composite materials with random distribution of large numbers of heterogeneous grains/cavities an effective computer generation method is presented to more perfectly and rapidly generate the samples of the random distribution domain with large numbers of grains/cavities. At first the heterogeneous geometries of grains/cavities are represented and the stationary random distribution with large numbers of grains/cavities is defined. In order to construct the algorithm for generating the samples of inhomogeneous stationary random distribution the selection algorithm with one stochastic variable is extended to the situation with several stochastic variables, and the selection theorem with several stochastic variables is stated. And then the detailed procedure of new algorithm is described. Finally several illustrations for 2 and 3 dimensional random distribution domains with the grains of different geometries are demonstrated respectively. The computer method in this paper is able to generate the sample with a higher percentage of grain volume and better stochastic property, and suitable to arbitrary stationary probability distribution model. The practice shows that the computer method in this paper is effective. REFERENCES 1. Li YY, Cui JZ. The computer simulated method for the domain with large numbers of random ellipse grains/cavities and the improved automatic generation triangle mesh algorithm. Chinese J. Comput. Mech., 2004; 20(3): 290-295 (in Chinese). 2. Wang ZK. Foundations and Application of Probability. Beijing Normal University Press, 1996 (in Chinese).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Three-Dimensional Mesh Generation Using the Crossed Circle Method H. Suzuki *, Y. Ezawa Department of Computational Science & Engineering, Toyo University, Kawagoe, 850-8585 Japan Email: [email protected] Abstract Delaunay method is a representative method in FEM. The quality of mesh generation by this method, however, depends on the allocation of nodes. Crossed Circle Method (CCM) was developed for solving this problem. CCM is the method of two-dimensional mesh generation that allocates nodes in advance only on a domain boundary and adds nodes automatically in the inside of domain. In this research, we developed the program that generates mesh automatically with three-dimensional tetrahedron elements using CCM. In addition, we checked our program with some mesh generation. CCM is based on Delaunay method. In Delaunay method, the circumscribed circle of the triangle does not contain any other nodes. When nodes except vertexes are contained in the circumscribed circle, the diagonal line of the quadrangle made from two adjoining triangles is swapped. In this research, we developed three-dimensional CCM, for tetrahedron elements and we proposed the following algorithm. First, when nodes except vertexes are contained inside the circumscribed sphere of a tetrahedron in the three-dimensional Delaunay method, the nodes are swapped, as shown in Figure 1. Tetrahedron elements will increase in number from two to three. Second, circumscribed sphere is calculated for three tetrahedrons after the swap, respectively. Only when the center of the circumscribed sphere is included in both of remaining two circumscribed spheres, it is investigated whether the existing node is in the near. When there is no existing node, the center-of-gravity point of the triangle of which is the center of three circumscribed spheres is added. Figure 2 is the example of mesh created by three-dimensional CCM. Some numerical examples showed the validity of this method using developed program.

Figure 1: Swapping of tetrahedron elements

Figure 2: Mesh created by Crossed circle method

REFERENCES 1. Ezawa Y. Triangulation with internal points generated by crossed circles. WCCM VI in conjunction with APCOM'04, Sept. 5-10, 2004, Beijing, China. 2. Taniguti T. Auto Mesh Generation for FEM. Morikita, 1992 (in Japanese).

— 274 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Study on Displacement Prediction of Landslide Based on Grey System and Evolutionary Neural Network W.Gao* Department of Civil Engineering, Wuhan Polytechnic University, Wuhan, 430023 China Email: [email protected] Abstract Landslide is a very serious geological disaster. So, how to control the landslide has become a very important work. To control landslide disaster, the forecasting of landslide is a very powerful method. But the development of landslide is a very complicated dynamic system. To describe this system very accurately is very hard. But the measured displacement series can describe the general laws of landslide development. So, some methods for displacement prediction of landslide are proposed. From analysis of those methods, we can find that, the neural network is a good method. Generally, the displacement time series of the landslide can be divided into some sections, such as, even section, periodic section and fluctuant section, et al. For different section, the different method can be taken. In this paper, considering the monotonously increasing character of the displacement time series of the landslide, this time series is divided into two sections, such as, the trend section and the deviation section. Here, based on the principles of displacement decomposition, the trend of displacement time series is extracted by Grey System and the deviation of Grey System is approximated by the new ENN model. In this new ENN model, the architecture and algorithm parameters of neural network are evolved simultaneously through combining modified BP algorithm and Immunized Evolutionary Programming proposed by author. The Immunized Evolutionary Programming is the combination of traditional evolutionary programming with artificial immune system principles. Using the above analysis, a new intelligent prediction method is proposed here. At last, one engineering examples is used to verify calculating effect of the proposed method. Xintan landslide in China is a very famous landslide for its successful prediction. To control this landslide, a lot of measured displacement data are gotten. With those data, our new method is used to forecast the landslide. The results show that the generalization of the new method is good and it can predict the displacement of landslide very well. So, this new method can be used in real engineering practice very well.

-Measured diaplacement -Computing displacement

,11 16 + time step

Figure: Measured displacement and computing displacement of Xintan landslide — 275 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Prediction of Ambient PMio Concentration with Artificial Neural Network L. H. Lam *, K. M. Mok Department of Civil and Environmental Engineering, Faculty of Science and Technology, University of Macau, Macao SAR, China Email: [email protected] Abstract Previous studies with artificial neural network (ANN) application on air quality prediction show success even most models developed use only temporal data of air concentrations; hence mainly time series analyses are performed. It is known that the concentrations of air pollutants are highly related to the variations of the local and regional meteorology conditions which dictate the dispersion and transport routes of them. Many correlation studies and models simulating the fate and impacts of the released pollutants are these bases. Meanwhile, ambient air pollutants may also affect the concentrations of each other therefore making prediction or modelling of their behaviours a very complex problem. This study aims at designing economic and flexible ANN models for the 24-hour-ahead predictions on concentrations of respiratory suspended particulates (PMio) taking into account the effects from local meteorological conditions and related pollutants. The ANN applied in this study is a three-layer feed-forward network (TLFN) of the back-propagation type. To achieve computation efficiency, the size of the input data is limited to six parameters per input set. In addition, variation of the predicted hourly PMio concentration is assumed to depend only on meteorological and air quality conditions within the last 72 hours. With these as the main criteria, models for PMio concentration prediction are developed and tested with one year of hourly measured data in a small coastal city, Macao. The measured data include seven meteorological parameters (dew point, wind direction, wind speed, relative humidity, precipitation, sunshine hour fraction, and temperature) and concentrations of five criteria pollutants (PMio, SO2, NO/NO2/NOX, O3, and CO). Based on previous study, these data are divided into two seasons namely summer and winter due to the prevailing monsoon climate; hence the development of the PMio models for these two seasons are treated independently. Individual season segment of data is further divided into three sub-sets for model training, testing and validation purposes. To select the proper six input parameters for each seasonal model, correlation coefficients among the hourly concentrations of the listed pollutants and meteorological parameters are calculated. Six input parameters with the highest absolute values of correlation coefficients are selected to form the model input pattern with the minimum correlation coefficient value for cut-off being 0.73. The number of neurons used in the hidden layer for each model with its selected six-parameter input pattern is determined by systematic trials of minimum root mean square error. Five and six neurons are determined for the summer and winter models. The trained models are then used to predict PMio concentrations for seven days and compared with actual measurements. Absolute relative percentage error (ARPE) and mean absolute percentage error (MAPE) are used as comparison criteria. Results show that 75% of the predictions in both models achieve an accuracy of ARPE less than 30%, while 100% of the predictions in the winter model could achieve an accuracy of ARPE less than 40%. The overall MAPE for the two models are 49% and 77%, respectively. Overall, the developed ANN models can capture the general trends of the real measurement. Larger errors occur at those hours with high and sharp concentration peaks which may be considered as noises caused by instantaneous, individual incidents occurred in the vicinity of the monitoring station. These abnormal individual incidents may not be easily captured based on knowledge of the past. — 276 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A Note on the Complexity of the PCG Algorithm for Solving Toeplitz Systems with a Fisher-Hartwig Singularity Seak-Weng Vong *, Wei Wang, Xiao-Qing Jin Department of Mathematics, University of Macau, Macao, China Email: [email protected], [email protected] Abstract Recently, Y. Lu and C. Hurvich showed that the complexity of T. Chan's preconditioned conjugate gradient algorithm for solving the Toeplitz system Tn (f)x = b is o(«log 3 «) where the generating function/is given by f(co) = \l-e^[2d

h(co)

with de(-^9 ^)\{0} and h(a>) being positive continuous on [~n, x] and differentiable on [-TZ, 7r]l{0}. Although their results are interesting, there exist some improper expressions in their proofs needed to be corrected. In this paper, we try to improve those improper expressions and demonstrate these important results by some numerical tests.

REFERENCES 1. Chan R, Ng M. Conjugate gradient methods for Toeplitz systems. SIAM Review, 1996; 38: 427-482. 2. Jin X. Developments and Applications of Block Toeplitz Iterative Solvers. Kluwer Academic Publishers, Dordrecht, and Science Press, Beijing, 2002. 3. Lu Y, Hurvich C. On the complexity of the preconditioned conjugate gradient algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity, SIAM J. Matrix Anal. Appl., 2005; 27: 638-653.

— 277 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

One-Point Integration that Handles Shear-Locking in Cubic Splines Simon Wang **, Yingshun Zhang 2 Department ofAeronautical and Automotive Engineering, Loughborough University, Loughborough LE11 3TU, UK 2 School of Engineering, Coventry University, Coventry CV1 5FB, UK Email: [email protected] Abstract A layerwise cubic B-spline finite strip method is developed for free vibration analysis of laminated plates. Numerical tests show that the method can give accurate predictions for natural frequencies of moderately and truly thick plates. However, it encounters shear-locking difficulty when thin plates are considered. Studies reveal that conventional selectively reduced integration technique as used in FEM fails to remove the shear-locking, but discover that it can be completely eliminated by using an integration scheme such that one-point Gauss integration is used per spline section in the strip direction whilst the conventional selectively reduced integration scheme in the crosswise direction. With this approach, introduction of an extra constraint on transverse deflection is needed at the both ends of the strip. This approach is termed as LWB33-sFSM(SRI-lP) which has the capability to produce accurate predictions for natural frequencies of truly thin and thick isotropic and composite laminated plates. REFERENCES 1. Wang S, Dawe DJ. Vibration of shear-deformable rectangular plates using a spline function Rayleigh-Ritz approach. International Journal for Numerical Methods in Engineering, 1993; 36: 695-711. 2. Wang S, Dawe DJ. Vibration of shear-deformable beams and plates using spline representations of deflection and shear strains. International Journal of Mechanical Sciences, 1994; 36: 469-481. 3. Wang S. A unified Timoshenko beam B-spline Rayleigh-Ritz method for vibration and buckling analysis of thick and thin beams and plates. International Journal for Numerical Methods in Engineering, 1997; 40: 473-491.

— 278 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

An Improved ICCG Method of Large-Scale Sparse Linear Equation Group* Y.J. Zhang*, Q. Sun School ofAeronautics, Northwestern Poly technical University, XVan, 710072 China Email: [email protected] Abstract In the field of engineering computation, many problems finally go to solve a large-scale linear equation group. Computational efficiency greatly relies on solving methodology and storing way of the equation group. This paper employs an improved ICCG (incomplete Cholesky conjugate gradient) method for improving the solving efficiency, and takes the fully sparse strategy for the storage of the sparse and symmetrical matrix of the equation group such that can save storing requirement to computer. Coefficient matrix of the linear equations is sparse and symmetrical. For the sake of CPU operational time saving to accesses data in matrix decomposition and reducing storing requirement to computer, we introduce fully sparse strategy structure that stores only nonzero elements of symmetrical part using chain pattern management. Apparently, its storing quantities are small, and its structure is convenient for research, insertion and deletion. When condition number of coefficient matrix is very big, coefficient matrix is morbid, and then iterative solution methods for sparse linear equations may be poorly convergent. Fortunately, appropriate preconditioning techniques can reduce condition number. In 1992, Saad introduced incomplete Cholesky decomposition with two thresholds One-threshold r controls magnitude of decomposed elements in every row. The other threshold p controls quantity of decomposed elements of every row. This method is denoted as ICCG (p9 r ) . Although ICCG (/?, r) is effective for symmetrical and positive definite matrix, some diagonal elements may be negative or quite small during incomplete decomposition. If diagonal elements are negative, preconditioned matrix may not be positive definite or approximate A well. If diagonal elements are quite small, they must bring big rounding errors, and then influence quality of decomposition and efficiency of solution. Therefore, we bring an improved diagonal method: Supposed that T is 1-norm or 2-norm of the / th row of preconditioning matrix, if diagonal element dn < T , then dti = T; otherwise don't modify the diagonal element. The improved ICCG (p,r) method can ensure quality of decomposition, thus improve the solving efficiency. Numerical examples show that the method is reliable and the high efficient.

REFERENCES 1. Wu JP, Wang ZH. Problems and improvements to the incomplete Cholesky decomposition with thresholds. Journal on Numerical Methods and Computer Applications, 2003; 3: 207-214. 2. St Georges P, Warzee G. High-performance PCG solves for FEM structural analysis. International Journal for Numerical Methods in Engineeing. 1996; 39:1313-1340 3. Beaumens R. Iterative solution methods. Applied Numerical Mathemetics, 2004; 51: 437-450. * This work is supported by the National Natural Science Foundation of China (No. 10477018). — 279 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A Parallel Computing Method of Object-Oriented FEM Based on Substructure* H. M. Zhao *, K. Zhang, Z. Z. Dong Department of Mechanics, College of Science, China University of Mining & Technology, Xuzhou, Jiangsu, 221008 China Email: [email protected] Abstract The method of object-oriented programmer is combined with the parallel computation of FEM in this paper. Based on the usual substructure method in FEM, a substructure class is established. Its objects automatically generate their meshes with given precision because it consists of the member function which is able to generate meshes automatically according to the data of boundary nodes. This is really significant in practice since people may require different precisions in different substructures. With the realization of the parallel class of MPI in C++ program environment, the previous serial object-oriented tools, such as development, debug and other software, can still be used. It is more important that the method of class libraries has the better expansibility and transplantation. So the design of parallel programmer is easier. Having designed the parallel computation steps of the object-oriented FEM based on substructure, the author calculates the problem of rectangular beam as an example and analyzes the efficiency of these programs. It shows that the method has more flexibility and the efficiency will be higher with the increase of the number of substructures. REFERENCES 1. Mackie RI. Object-oriented programming of the finite element method. International Journal for Numerical Methods in Engineering, 1992; 35: 425-436. 2. Yuan ZQ, Zhong SS. Sub-structural method of FEM by the object-oriented program. Journal of Chongqing University (Natural Science Edition). 2001; 24(3): 35-37 (in Chinese). 3. Zhang JF, Jiang HD. Performance analysis in FEM substructure parallel algorithm. Mechanics in Engineering, 2002; 24(5): 35-37 (in Chinese).

* This work is supported by the Science and Technology Foundation of China University of Mining & Technology. — 280 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Promotion of Frontier Science Research by Aid of Automatic Program Generation Technology B. X. Wu u * , H. S. Qian2, S. Wan3 1

Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, 100080 China Beijing Fegen Software Company, Beijing, 100086 China 3 Southeast University, Nanjing, 210096 China Email: [email protected] 2

Abstract Based on a new DIY concept for software development [1], an automatic program-generating technology attached on a software system called as Finite Element Program generator (FEPG) provides a platform of developing software, through which a scientific researcher can submit his special physico-mathematical problem to the system in a more direct and convenient way for solution. The software execution manner is completely different with usual: In FEPG mode, a couple of files in a special language for describing both the partial differential equations to be solved and the solving algorithm and procedures should be prepared, in stead of importing parameters to solve partial differential equations in normal commercial software mode (see Figure 1). The calculating program is automatically generated by the system based on these files and then used to solve the problem. As an open platform, its flexibility greatly expand users' problem-solving ability, e.g. in integrating with new calculating algorithms and schemes, and in solving multiphysics problems. Two examples of numerical simulation are shown in this paper to illustrate the usage and superiority of automatic program generation technnology. Both are solved by using finite element method. For a thermo-elastic problem with non-Fourier heat conduction, implicit Newmark algorithm is adopted to solve the wave equations. For the simulation of the flow and heat transfer in a heat pipe, the solution of a typical multiphase, multi-zone and multi-physics problem is easily realized in a uniform manner. A lot of unique features of FEPG make it a powerful tool of numerical analysis for frontier sciences associated often with new physical models including novel constitutive equations and/or covering interactive multiphysics phenomena. Calculating scheme FEPG mode import

solution

Commercial software mode

parameters

" W

bfa'Ckbox

solution

Figure 1: Different modes of software execution REFERENCES 1. Bangxian WU. How to cope with unknowingly coming software revolution? Software World, 2004; 5: 33 (in Chinese) — 281 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Uniformed NURBS Surface Deformation Subject to Boundary Conditions Kin Man Lo *, Zhixin Yang Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Taipa, Macau, China Email: [email protected] Abstract To allow mass custermization in competitive industries, it is important to conduct new product design and development with concentration on few key features while reusing the existed portion of design in terms of geometric shape and boundary condition. Partition methods will be firstly applied to separate a product model into a base shape and working feature. Customized design will be implemented by replacing the working feature with a new pattern, which shall be deformed such that the boundary is matched and the finallized feature is closed to the original one. This paper presents a new surface deforming method with an attempt to solve mentioned design problem. The control net of a surface is treated as a space truss so that the control points are considered as the nodes of the structure. When any of the nodes is moved, the entire structure is deformed accordingly. The techniques on the construction of the control net, determination of displacement of boundary node and unconstrainted node, and surface morphing will discussed in detail. Due to the localization property of NURBS surface, the number of control points could be adjusted is often restricted in traditional method. The proposed deforming method enables the control net being deformed gloabally and in a uniformed manner, which result a 'similar' deformed surface to the original pattern the designer intended to apply in the new product. It is believed the proposed method could be applied in related industries including shoe and car design. REFERENCES 1. Sanchez-Reyes J. A simple technique for NURBS shape modification. Computer Graphics and Applications, IEEE, 1997; 17: 52-59. 2. Hu S-M, Li Y-F, Ju T, Zhu X. Modifying the shape of NURBS surfaces with geometric constraints. Computer-Aided Design, 2001; 33: 903-912. 3. Dachille IX F, Qin H, Kaufman A. A novel haptics-based interface and sculpting system for physics-based geometric design. Computer-Aided Design, 2001; 33: 403-420.

— 282 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

The Pseudo-spectral method and Matlab Implement* Songling Wang, Zhengren Wu*, Youliang Cheng School of Energy & Power Engineering, North china electric power university, Baoding,, 071003, China Email: [email protected] Abstract The pseudo-spectral method based on function approach is good for solving nonlinear equation. Because the surplus term in difference scheme sometimes can affect the computing result, thereby, the function approach method becomes a more popular one in the numerical simulation. Now, the pseudo-spectral method is the major one, for its small aliasing error. There are some kinds of language in solving the nonlinear equation, Matlab has been a good one in numerical computation and simulation, and it has been widely used in hydrodynamics and applied mathematics, etc. many complicated engineering problem can be solved by Matlab, and the numerical results can be showed by its excellent graphics. As a common spectral method, the pseudo-spectral method used spectral disperse (Fourier transform) treatment of the space dependence together with a difference scheme in time, then the derivatives (or other operations) are algebraic in the transformed variable. In the paper, for the leapfrog, the traditional Lax-Wendroff form was discarded, and the above pseudo-spectral method was adopted. As for the possibility of separation of the solutions between successive time levels, we adopted the averages of adjacent levels and restart the scheme from the new levels, so the precision can also be assured. At one time, the complex-valued problem in Fourier transform was treated by aliasing. For example, we solve the fKdV equation with pseudo-spectral method and draw the waterfall with Matlab, the results are shown in the Figure.

Figure: The numerical simulation results for fKdV equation REFERENCES 1. B.Fornberg, G. B. Whitham, A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena, J. Fluid. Mech, Trans R.Soc. London, 289, (1978), 333-404. 2. Yong Zhu, Resonant of Nonlinear Capillary-Gravity Waves, Phys. Fluids, 7, (1995), 2294-2296. 3. Ying Long Zhang,Song Ping Zhu. Subcritical, transctitical and superctitical flows over a step. J. Fluid Mech, 1997,333,257-271. * The project is supported by the National Natural Science Foundation of China (Grant No. 10272044), Research Fund for the Doctoral Program of the ministry of Education of China (20040079004) — 283 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Composite Construction in Reinforced Concrete Taking into Consideration the Non-Rigid Bond of Interfaces in Joints V. Lindig * Bonhoefferstr. 43, Weimar, 99427 Germany Email: [email protected] Abstract The meaningfully combination of various building materials is a fundamental component of modern con­ structions, the so-called composite constructions. As such, the recording of bond behavior in joints is a crucial factor to be considered in the design of load-bearing structures in respect of new building and retrofit. The behavior of the plain or reinforced concrete-concrete interfaces in joints taking into consideration potential effects such as various rough contact surfaces have a significant influence on structural behavior of complex reinforced concrete structures and composite constructions respectively. In contrast to the case of physically localized behavior, the behavior of the bond interfaces between old concrete and new concrete in joints under aspects of multi-parametered influential factors and how this influences the global structural behavior of composite construction raises issues which are still open contrary to research. A three-dimensional, physically non-linear continuum model has been developed on the basis of the finite element method. This enables the multi-parametered influence of the various surface structures and the reinforcement positioning of the concrete-concrete bond interface to be investigated discretely as a component part of complex composite constructions exemplary represented in figure. On the basis of the extensive simulation results [1], it can be proved, that the relevant European standard EC 2 do not reflect the realistic structural behavior. A novel differentiated design concept for concrete- concrete interface joints has been formulated and recommended therefore for the first time under consideration of relative displacements of the contact surfaces and with reference to the bearing structure [1].

t) flwjjrtnomwsiiufcf VWtaft&rw

t

Huff-J « Mrtwtfieli «nvt* t A AstmtiAqte Oct**WXOWMHI V\xnt

r) NwoukpannBiifUJintMl in VMwiidfeffwrhninff

() Akniwiw ichafckfi":»iVf YiMtiaflifiwwrltinifag

Figure: An exemplary rigid composite framework principle at the 3-d finite element analysis. REFERENCES 1. Lindig V. Numerical Simulation, Structural Analysis and Design of Composite Constructions in Reinforced Concrete taking into Consideration the Non-rigid Bond of Interfaces in Joints. Doctoral Thesis, Bauhaus-Universitat Weimar, Germany, Shaker Verlag Aachen, www.shaker.de (printed and online version), 2005 (in German). — 284 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X.Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Optimization of Observation Condition on Inverse Analysis for Identifying Corrosion of Steel in Concrete Kazuhiro Suga l*9 M. Ridha2, Shigeru Aoki 3 1

Center for Computational Mechanics Research, Toyo University, Hakusan2-36-5, Bunkyo-ku, Tokyo 112-8611, Japan 2 Department of Mechanical Engineering, Syiah Kuala University, BandaAceh, Indonesia 3 Department ofComputational Science andEngineering, Toyo University, Toyo University, Japan Email: [email protected], [email protected], [email protected] Abstract The purpose of this study is to optimize the observation condition on the inverse problem for estimating the real and imaginary parts of the concrete conductivity and the impedance of the steel-concrete interface. The optimization is achieved by minimizing the average of eigen values of a posteriori estimate error covariance matrix. We performed a numerical identification to demonstrate the effectiveness of the optimized observation condition. The estimation is carried out by using the Kalman Filter algorithm. The simulation result shows that the real part of the impedance can be estimated with a high accuracy while the others cannot be well estimated. To overcome the above difficulty, a priori information and other kinds of observation conditions are considered. REFERENCES 1. Ridha M, Amaya K, Aoki S. Improvement of AC impedance method for monitoring corrosion of rebar structure by boundary element inverse analusis. Zairyo-to-Kankyo, 1999; 48: 654-659. 2. Amaya K, Aoki S. Optimization of measurements for inverse problem. Proc. 3rd Int. Conf. on Inverse Problems in Analysis, Port Ludlow, WA, USA, June 13-18,1999, pp. 1-6.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

A Study on Temperature Distribution in a Cross Section of Concrete Box Girder Bridge Yiping Tan *, Dajian Han Department of Civil Engineering, South China University of Technology, Guangzhou, 510640 China Email: [email protected] Abstract It is known that nonlinear temperature variation over a cross section of long-span continuous box-girder beam bridges or continuous rigid-frame box girder bridges can cause longitudinal stresses and that in some cases such kind of stresses may reaches or even exceeds those induced by the live loads and that temperature cracking may occur in the structure component. However, the mechanism of such phenomenon is not very clear yet. Therefore, to predict the stresses caused by temperature distribution is important for a correct design of the bridge structures. The purpose of the present paper is to present a practical computation method to simulate the temperature distribution over cross sections of a concrete box girder bridge. The relative factors considered include the bridge geometry, the geographical location, the bridge orientation, the material properties and ambient climatic conditions. In this paper heat flow equations over a concrete bridge cross section and heat boundary conditions of outside and inside the box-girder are first derived. And then numerical solution for daily time variation of air temperature is presented. The relationships between inside concrete temperature and ambient climatic conditions that include solar radiation, relative humidity, and wind speed etc. are established. A prediction for variation of concrete temperature is obtained. The results are compared to the observed values measured from The Guangzhou Guanyinsha Bridge that is a single-cell box-girder bridge. Finally, numerical method and computer programming are given to predict the temperature distribute in concrete bridge cross sections. This can be used to generate the thermal loads for finite element analysis.

r / -^-i '

-horizonal ovehang providing shade

surface exposed to sotar radiation 3

surface

Figure: Geometry defining incidence angle of solar radiation REFERENCES 1. Elbadry MM, Ghall A. Temperature variations in concrete bridges. Journal of Structural Engineering, ASCE, 1983; 109(10): 2355-2374. 2. Roberts-Wollman CL, Breen JE, Cawrse J. Measurements of thermal gradients and their effects on segmental concrete bridge. Journal of Bridge Engineering, 2002; 7(3): 166-174. — 286 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Stress-Based Effective Space Anisotropic Damage Model for Concrete Jian-YingWu^JieLi2 1

Department of Civil Engineering, South China University of Technology, Guangzhou, 510640 China Department of Build Engineering, Tongji University, Shanghai, 200092 China Email: [email protected]

2

Abstract Damage induced anisotropy is crucial for those initially isotropic materials, e.g., quasi-brittle materials such as concrete, geomaterials, ceramics, etc. Just as already pointed out by other researcher, at least second-order tensor should be adopted as damage variable to describe the damage induced anisotropy. Despite the substantial research efforts and the noteworthy recent contributions, the modeling of anisotropic damage is not a straight-forward task as that of isotropic one and still remains a challenging issue, among which the key point is the establishment of thermodynamically consistent damage evolution law. In this paper a stress-based effective space anisotropic damage model is proposed and applied to concrete modeling. The general stress-based formulations of modeling anisotropic damage with a second-order tensorial damage variable are first discussed, and then the principle of damage dissipation equivalence is introduced to define the effective damage rate tensor by transforming the nominal damage rate tensor. Therefore obtained by the corresponding inverse transformation, the conjugated effective damage energy release rate is completely and simply expressed in the effective space and exhibits some convenient properties. The damage criteria can thus be thermodynamically consistently postulated in terms of the obtained effective damage energy release rates, after which the damage evolution law in the effective space is then established in accordance with the principal of maximum damage dissipation. By incorporating the positive and negative projection operators, the presented framework is generalized to take the unilateral effect into account. The proposed model is applied to concrete modeling, leading to an anisotropic damage model which is capable of describingmost of the nonlinear anisotropic behavior of concrete evident in the experiments. It has been demonstrated that, only six material properties are required in the presented orthotropic damage model, i.e. the Young's modulus £ 0 , the Poisson's ratio vo, the uniaxial tensile and compressive strength f and^c, and the Mode-I and Mode-II fracture energy G\ and Gf , to describe most of the nonlinear behavior evident in the experiments, such as the stiffness degradation, the strength softening, the enhancement of strength and ductility under compressive confinement, the strength decay induced by orthogonal tensile cracking, the unilateral effect under cyclic loading and the damage induced anisotropy, etc. Most importantly, all the parameters are physically meaningful and can be easily identified by conventional experiment procedures, which constitutes the main merit of the presented model over others existing in the literature.

REFERENCES 1. Cordebois JP, Sidoroff F. Damage-induced elastic anisotropy. in Jean-Paul Boehler ed. Mechanics of Behavior of Anisotropic Solids, No. 295, Martinus Nijhoff Publisher, 1979, pp. 19-22. 2. Carol I, Rizzi E, Willam K. On the formulation of anisotropic elastic degradation. I: theory based on a pseudo-logarithmic damage tensor rate; II: generalized pseudo-Rankine model for tensile damage. International Journal of Solids and Structures, 2001; 38(4): 491-546. 3. Ortiz M. A constitutive theory for inelastic behaviour of concrete. Mechanics of Materials, 1985; 4: 67-93. — 287 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Identification of Electric Conductivity and Impedance of Reinforced Concrete by Boundary Element Inverse Analysis Masato Yoshida **, Kazuhiro Suga 2 , M Ridha3, Shigeru Aoki 4 , Kenji Amaya5 1

GraduateSchool of Engineering, Toyo University, Japan Centerfor Computational Mechanics Research, Toyo University, 2-3 6-5 Hakusan, Bunkyo-ku Tokyo, Japan 3 Department of Mechanical Engineering, Syiah Kuala University, BandaAceh, Indonesia 4 Department of Computational science and Engineering, Toyo University, Japan 5 Department ofMechanical andEnviromental Informatics Tokyo Institute of Technology, 2-12-1 Ookayama, Meguroku Tokyo 152-8552, Japan Email: [email protected], {ksuga, ridha, saoki}@eng.toyo.ac.jp, [email protected] 2

Abstract Corrosion of steel bars in concrete is a major cause of failure of the reinforced concrete structures. Numerical inverse analysis is expected to become a useful tool for the corrosion detection. In this research, a new method is introduced for identifying the electrical conductivity of concrete and the impedance of steel-concrete interface by using some potential data which are measured on the concrete surface. The identification is based on the boundary element inverse analysis. The inverse analysis was carried out by minimizing the cost function, which is the different between measured and calculated potential in the concrete surface. Laplace equation, which is solved by boundary element method, is used to describe the potential in the concrete domain. An experiment using a concrete block with an embedded rebar is used to demonstrate the effectiveness of the proposed method. The alternative current is impressed into the concrete domain at one point on the concrete surface and the potential is measured at several points on the concrete surface. REFERENCES 1. Aoki S, Amaya K, Miyasaka M. Boundary Element Analysis on Corrosion Problems. Shokabo, Tokyo, 1998. 2. Press WH, Teukolsky SA,Vetterling WT, Flannery BP. Numerical Recipes in C\ Gijutsu Hyoron Co. Ltd., 1994 (in Japanese). 3. Ridha M, Amaya K, Aoki S. Improvement of AC impedance method for monitoring corrosion of rebar in concrete structure by boundary element method. Zairyo-to-Kankyo, 1999; 48(10): 654-659.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Stresses and Cracking Caused by Hydration Heat in Massive Concrete Structures Ziming Zhang1*, Zhitong Song1, Yan Zhang2 1

Department of Engineering Mechanics, Hohai University, Nanjing, 210098 China Department of Material Science, Hohai University, Hanjing, 210098 China Email: [email protected] 2

Abstract After development of the theory and analysis of the temperature [1], this paper analyzes the stresses and cracking caused by hydration heat in massive concrete wall. The analysis takes into account the fact that the adiabatic temperature rise and the creep model are based on Arrhenius' theory which lead to nonlinear differential equations, and the cracks are modeled by Bazant's crack band model [2]. The crack is divided into equal strips, and a numerical algorithm with step-by-step loading is developed to calculate the growth of the crack in each step. The interactions of temperature, stresses and cracking are computed by half-analytical iteration method. The systems of up to 360 nonlinear equations are solved in this paper. The following conclusions are drawn. A temperature increase has two mutually competing effects. Firstly, it accelerates creep, i.e. increase the creep rate. This indicates that the retardation or relaxation times should be reduced as temperature increases. Secondly, a temperature increase further causes an acceleration of hydration or aging, thereby indirectly also reducing creep. The placement temperature has a severe effect on cracking in massive concrete wall, the more the placement temperature, the earlier the crack happen, and the longer the crack length. REFERENCES 1. Bazant ZP, Oh BH. Crack Band Model Theory for Tracture of Concrete. Materials and Structures (RILEM. Paris), 1983; 16: 155-177. 2. RILEM Committee TC-69 (Bazant ZP et al.). Material Models for Structural Creep Analysis, Chapter 2. in Bazant ZP ed. Mathematical Modeling of Creep and Shrinkage of Concrete. John Wiley, Chichester and New York, 1988, pp, 99-200. 3. Bazant ZP, Cedolin L. Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. Oxford University Press, New York, 1991.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Modeling of Consolidation of Marine Clay under Vacuum Preloading Incorporating Prefabricated Vertical Drains Sao Man Ho *, Thomas Man Hoi Lok Department of Civil and Environmental Engineering, University of Macau, Macau SAR, China Email: [email protected] Abstract This study is to investigate by numerical modeling the consolidation of marine clay at Macao under vacuum preloading. A two dimensional finite element program CRISP was used for the numerical modeling of consolidation of marine clay under vacuum preloading. A unit cylinder model with three soil layers, sand, marine and alluvium, with thickness 6m, 13m and 11m, respectively, was first established to examine the one-dimensional consolidation behavior of soils under vacuum preloading. Vacuum pressure generated nearly identical effects compared to a conventional surcharge pressure of the same magnitude under one-dimensional condition, as evidenced by the responses of the settlement. The major difference between surcharge and vacuum preloading was the pore water pressure change. When vacuum pressure was applied to soil, negative pore water pressure was generated. The pore water pressure reduced from its initial hydrostatic state by the amount of the vacuum pressure applied, whereas in the case of surcharge with the excess pore water pressure first built up from its initial hydrostatic state by the same amount as the surcharge and then gradually dissipated. Based on the results obtained from unit cylinder model, a two-dimensional model was constructed for a case study of vacuum preloading at Macao. Vacuum pressure of 80kPa was applied at the top of the soil layer, with negative pore pressure generated along the soil surface and the length of the vertical drains, which was different from past studies by simply fixing the negative pore pressure along the top boundary only (e.g., Park et al. 1997). Slurry wall with 8m of depth and 0.7 m of diameter, used to prevent vacuum loss around the treatment area, was also included to the soil model. In order to model the unlimited soil boundary in real situation, 30m of soil around the treated area was found to be sufficient. The lateral displacement at the border of the treated area was also investigated. It was noted that the direction of the lateral displacement was inward under vacuum preloading instead of outward in the case of surcharge preloading. The degree of consolidation was used as the main criteria for assessing the effectiveness of soil improvement. Factors affecting the degree of consolidation measurements based on settlement and pore water pressure were also discussed. REFERENCES 1. SAGE Engineering Ltd. SAGE CRISP: User Manual SAGE Engineering Ltd., 1999. 2. Park CL, Jeong HJ, Park JB, Lee SW, Kim YS, Kim SJ. A case study of vacuum preloading with vertical drains, in Davies MCR ed. Ground Improvement Geosystems: Densification and Rein­ forcement, Thomas Telford, London, UK, 1997, pp. 69-74.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Drag Forces Applied on Rock Matrix by Fluid Flow through Fracture Network in Rock Mass Junrui Chai 1 ' 2 * 1

College of Civil and Hydroelectric Engineering, China Three Gorges University, Yichang, 443002 China College of Hydroelectric Engineering, XVan University of Technology, Xi 'an,/T10048 China Email: [email protected] 2

Abstract It is very important to investigate the forces applied on the rock mass by the fluid flow through fractured rock mass, which is one of the issues for analyzing the stability of rock mass. The forces applied on the fissure walls by the fluid flow include the hydrostatic seepage pressure and the dynamic seepage pressure, i.e., the drag force. Based on the cubic law of the single fissure flow, the equations of the drag forces applied on the fissure walls by the single fissure flow are deduced by means of the momentum law of fluid mechanics. The equations are designated for the cases of no-filled fissure, filled fissure and the combined flow of fluid and the fillings, respectively. The equations have important values for analyzing the effect of fluid flow on the behavior of deformation and strength of fractured rock mass, and are the foundation of the interaction mechanism between stress and fluid flow. Based on the equations, the two kinds of forces applied on the fracture wall of fracture network in rock mass are studied, which include the normal hydrostatic seepage pressure and the tangent drag force. The equivalent node force of joint element is deduced under the two-dimensional and three-dimensional conditions. A numerical computation example is also given to indicate the effect of seepage through fracture network on the stress of rock mass. REFERENCES 1. Lewis RW, Schrefler BA. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2ndedn. John Wiley & Sons Ltd., Chichester, England, 1998. 2. Chai Junrui, Li Shouyi, Wu Yanqing. Multi-level fracture network model for coupled seepage and stress fields in rock mass. Communications in Numerical Methods in Engineering, 2004; 19(1): 63-74. 3. Chai Junrui, Li Kanghong, Wu Yanqing, Li Shouyi. Coupled Seepage and Stress Fields in Roller Compacted Concrete Dam. Communications in Numerical Methods in Engineering, 2005; 21(1): 13-21.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

A 2-D Natural Element Model for Jointed Rock Masses Tiantang Yu*, Qingwen Ren Department of Engineering Mechanics, Hohai University, Nanjing, 210098 China Email: [email protected] Abstract According to the characteristic structural features of jointed rock masses, a natural element method (NEM) model is proposed for the mechanics analysis of jointed rock masses based on the Sibsonian interpolation or Laplace interpolation. In the model, the jointed rock mass is regarded as a system of intact rock blocks connected by discontinuous joints modeled by interfaces. The displacement field of each block is constructed by the natural neighbor interpolation. To deal with the discontinuities of rock masses, the displacement fields are constructed to be discontinuous between blocks. The displacement fields and their gradients are continuous in each block. Numerical simulations illustrate NEM has advantages of both element free method and finite element method, and does not have their disadvantages. The present method is developed for two-dimensional linear elastic analysis for jointed rock structures, and can be extended to three-dimensional and non-linear analysis.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Implementation of a Bounding Surface Bubble Model for Structured Soils J. R. Maranha*, A. Vieira Department ofGeotechnics, Laboratorio Nacional de Engenharia Civil Av. do Brasil, 101 1700-066 Lisboa, Portugal Email: [email protected] Abstract In this work, the numerical implementation of a bounding surface model for structured soils formulated by Kawadas and Belokasfl] in the explicit finite difference code FLAC is described. This model incorporates capabilities to simulate soil destructuring, inherent and induced anisotropy, cyclic loading and small strain response. In this work, the numerical implementation of a bounding surface model for structured soils formulated by Kawadas and Belokas[l] in the explicit finite difference code FLAC is described. This model incorporates capabilities to simulate soil destructuring, inherent and induced anisotropy, cyclic loading and small strain response.

Figure: Bounding surface and yield "bubble" surface with conjugate and image stress points (left), and Ko compression stress path and bounding and bubble surfaces evolution with 2 unloading-reloading loops (right)

REFERENCES 1. Kawadas M, Belokas G. An anisotropic elastoplastic constitutive model for natural soils, in Desai et al. eds. Computer Methods and Advances in Geomechanics, Balkema, Rotterdam, 2001.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Simulation of Nonlinear Interaction of Soil, Superstructure and Thick Raft with Irregular Plan Yongfeng Du *, Shengkui Di, Hui Li, Yu Song, Xinghai Dang Institute of Earthquake Protection and Disaster Mitigation, Lanzhou University of Technology, Lanzhou, 730050 China Email: [email protected] Abstract Of all types of foundation for tall buildings, thick raft is used most frequently nowadays in China. With the rapid progress of economy, tall buildings with various shapes have been constructed in recent years, thus promoted the developing of thick rafts with irregular plans. This paper presents a finite element method (FEM) based analysis for the interaction of soil, tall building superstructure, and raft foundation with a irregular plan. The irregular boundary is simulated as four nodes isoparametric element, Mindlin model for moderate thickness in which transverse shear is considered is used in order to improve the accuracy of calculation. The ground stiffness matrix is calculated by Boussinesq model and the nonlinear analysis is proposed to consider the foundation detach from soil. It is found from the investigation that the calculation accuracy of elastic half-space ground model can be improved by adopting the load test results and this make the results of calculation close to the experimental value. A finite element method program IRF is worked out to calculate the interaction of raft foundation and soil using FORTRAN90. The program SUPER is worked out using substructure can be used for calculating reduced superstructure stiffness matrix and reduced load vector. Four different types of irregular foundations are used as numerical examples, and the results of distribution of settlement and principle stresses of the foundation are briefly illustrated.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Soil Additionally Affected by Non Force Loading and Its Influence on Upper Structure P. Kuklik *, M. Broucek Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thdkurova 7, 166 29 Prague 6, Czech Republic Email: [email protected] [email protected]; [email protected] Abstract It is an experimentally confirmed fact that a soil substantially changes its material properties when subjected to external loading. Apart from that, the soil, when subjected to a certain loading history, has the ability to memorize the highest level of loading mathematically represented by over-consolidation ratio, and so called initial void ratio. In virgin state the soil deformability is relatively high. On the contrary, following the unloading/reloading path shows almost negligible deformation until the highest stress state the soil has experienced ever before is reached [1,2]. The paper presents basic resources for the evaluation of the numerical codes for soils affected by floods. One of them is research project FLAMIS describing the 2002 year flood in the region of the South Bohemia. All substantial influences, such as suction, piping etc., can change the overconsolidation and porosity inside the subsoil. This affects the upper structure due to collapses of the soil skeleton following by the stress redistribution. Time dependent progress of influence zone can be used for the fast estimation of this phenomenon in the subsoil [3]. Several numerical examples are carried out using professional code GEO 5 FEM from FINE Inc. Ltd. The results are compared with real damages in situ.

REFERENCES 1. Bowles JE. Foundation analysis and design. McGraw-Hill, New York, 1966. 2. Kuklik P, Sejnoha M, Mares J. The structural strength of soil from the isotropic consolidation point of view. APCOM 99, Singapore, 1999, pp. 797-802. 3. Kuklik P et al. Calculation and verification of the influence zone depth inside the subsoil. ICCM 2004, Singapore, 2004.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Advances in Unsaturated Soil Mechanics Zhankuan Mi *, Zhujiang Shen Nanjing Hydraulic Research Institute, Nanjing, 210024 China Email: [email protected]; [email protected] Abstract Most of soils above the groundwater level encountered in the engineering were in the state of non-stauration. The special soils such as loess, expansive soil and artificial fill were all typical unsaturated soils. Advances in unsaturated soil mechanics were lower than those in saturated soil mechanics. The restricted factors were as follows: 1. Testing technique was relatively not perfect. The test difficulties of unsaturated soil were as follows: (1) It was composed of solid phase, fluid phase, gas phase and contraction membrane. It was very difficult to measure the stress and deformation of the above four phases individually. (2) The suction value ranges from 0 kPa to 106 kPa. But it was very difficult to measure the matrix suction when its value was greater than 80 kPa. (3) As its permeability coefficient was small, the test duration was long. 2. Although it has been presented many constitutive relations such as elastic models, elastic-plastic models, damages models and thermo-mechanical models, the theoretical system of unsaturated soil was far immature. The new constitutive model should describe the reversible and irreversible change process along with the stress and suction change process. Advances in the measuring technology and constitutive model for unsaturated soil were comprehensively summed up. Some suggestions were proposed on the further research towards the application of unsaturated soil theories in engineering practice.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Simulations of the Behavior of Foundations on Reinforced Soil C.M.Tou,T. M. H. Lok* Department of Civil and Environmental Engineering, University of Macau, Macau SAR, China Email: [email protected], [email protected] Abstract Since after the discovery of the improvement of the ultimate bearing capacity of foundations with geosynthetics such as geotextiles, geogrid, etc., a lot of experimental studies have been carried out to investigate the optimium configurations of the geosynthetics. In this study, finite element plain strain numerical model was created to investigate several parameters that would probably affect the behavior of reinforced foundations. The model used Duncan Hyperbolic model to simulate the foundation soil. Failure was defined when the settlement of the foundation reached S= 0.30B. The parametric studies of the single-layer soil system include the depth to the first reinforcement layer, the length of the reinforcement, the number of reinforcement layers, spacing between the reinforcement, tensile strength of the reinforcement and types of soils such as cohesive and cohesionless soil. Based on the numerial simulation, it is shown that the optimum depth to the first reinforcement layer is at about 0.15B and the result implies that the reinforcement does not need to be anchored in any way prior to placing the fill, as was previously thought essential to achieve a completely reinforced soil system. Also, the increase in bearing capacity in cohesionless soil is much higher than in cohesive soil due to the different mechanisms of interaction between the reinforcement and the soil.

Figure: A FE plain strain model with reinforcement for single-layer soil system & optimum depth to the first reinforcement layer REFERENCES 1. Kotake N, Tatsuoka F, Tanaka T, Siddiquee MSA, Huang CC. FEM simulation of bearing capacity of level reinforced sand ground subjected to footing laod. Geosynthetics International, 8(6): 501-549. 2. Shin EC, Das BM. Experimental study of bearing capacity of a strip foundation on geogrid-reinforced sand. Geosynthetics International, 7(1): 59-71. — 297 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Stress-Strain Modeling of Tire Chip-Sand Mixture H. J. Yu**, Thomas M. H. Lok 2 1

Faculty of Science and Technology, University of Macau, Macao SAR, China University of Macau, Av. Padre Tomds Pereira S.J., Taipa, Macao SAR, China Email: [email protected], [email protected] Abstract The mechanical behavior of scrap tire chip-sand mixture was previously examined in laboratory for the use of scrap tire as a potential lightweight fill material. The purpose of this study is to numerically model the stress-strain relationship and volumetric behavior of tire chip-sand mixture with varying chip/mix ratio, by using two different modeling methods. One is the hyperbolic model, the other one is the artificial neural networks (ANNs). A series of triaxial consolidated-undrained (CU) and consolidated-drained (CD) compression tests for the tire chip-sand mixture of six different chip/mix ratio is performed under controlled conditions to develop the database. The test results were used for the hyperbolic modeling parameters determination and neural network training and testing. Comparisons between the laboratory testing results and the numerical modeling results indicate that the simple hyperbolic model has certain limitations in modeling the behavior of tire chip-sand mixture such as dilation and strain softening. On the other hand, ANN is a powerful mathematical tool for non-linear multi-variable analysis. As shown in the figure, the ANN based models have great capacity to simulate the effects of chip/mix ratio (R) on the stress-strain relations and volumetric behavior of the mixture. Details on training and learning algorithms to develop the models are discussed.

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Figure: ANN simulation of stress-strain relationship and volumetric behavior during CD triaxial testing REFERENCES 1. Duncan JM, Byrne P, Wong KS, Mabry P. Strength, stress-strain and bulk modulus parameters for finite element analyses of stresses and movements in soil masses. Report No. UCB/GT/80-01, UCB, 1980. 2. Ellis GW, Yao C, Zhao R, Penumadu D. Stress-strain modeling of sands using artificial neural networks. Journal of the Geotechnical Engineering Division, ASCE, 1995; 121(5): 429-435.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Multi-Objective Optimization for Shape Design of Arch Dams Linsong Sun1*, Weihua Zhang1, Nenggang Xie2 1

College of Hydraulic Science & Engineering, Yangzhou University, Yangzhou, 225009 China College of Mechanical Engineering, Anhui University of Technology, Maanshan, 243002 China Email: [email protected] 2

Abstract A multi-objective optimization model is established for the shape design of arch dams. In this model, the geometric parameters describing dam shape is taken as design variables; four objectives are considered of dam volume, maximal tensile stress, maximal pressure stress and relative depth of the zone with tensile stress that is larger than l.OMPa; and the constraints include geometric constraints, dam volume constraint and mechanical property constrains, i.e., stress constraints, stability constraints, etc. Traditionally, multi-objective optimization schemes transform multiple objective functions into a single objective function in some manner, weighted sum method and Utopia point method for examples, and the resulting problem is solved as a single objective optimization problem. In this paper the cooperative game model analogy to the multi-objective optimization is proposed with each player correspond to one of the objective functions. After defining the utility of each player, the Nash arbitrary scheme is used to solve the cooperative game. The optimization of Baihetan arch dam, which is located in Sichuan province of China, is calculated as an engineering example. The results are compared to those obtained by weighted sum method and Utopia point method and indicate that the cooperative game method is superior to traditional methods. The optimal design saves 9.77x104m3 of dam volume, compared to initial design, along with the decrease of maximal tensile stress with 38.22%, the decrease of maximal pressure stress with 28.83%, and the decrease of relative depth of large tensile stress zone with 10.15%. REFERENCES 1. Marlers RT, Arora, JS. Survey of multi-objective optimization methods for engineering. Struct. Multidisc Optim, 2004; 25: 369-395. 2. Sun Wenjun, Sun Linsong, Wang Dexin, Li Chunguang. Bin-objective shape optimization of arch dams. Journal of Hohai University (Natural Sciences Edition), 2000; 28(3): 39-43 (in Chinese). 3. Spallino R, Rizzo S. Multi-objective discrete optimization of laminated structures. Mechanics Research Communications, 2002; 29:17-25.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Optimal Shape Control of Multilayered Piezoelectric Smart Plate Structure Jianguo Wang *, Genfang Ding, Yan Qin School of Civil Engineering, Hefei University of Technology, Hefei, 230009 China Email: [email protected] Abstract The finite element equations are derived by application of the extended Hamilton principle based on the classic electroelastic theory. Mechanical-electric coupled program has been developed by use of ANSYS/APDL language. A sandwich plate with embedded piezo actuators and/or surface bonded piezo actuators are modeled. The piezoelectric patch size and voltage are considered as design parameters. Zero order and first order optimal algorithms are also applied in order to maximize the piezoelectric actuator efficiency, improve the structural performance. To show the performance of the proposed models, several simple examples are presented. Results show that the presented model is effective. The some conclusions are presented in optimal design and shape control for smart structures.

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REFERENCES 1. Crawley EF, Deluis J. Use of piezoelectric actuators as elements of intelligent structures. AIAA J, 1987; 25(10): 1373-1385. 2. Adali SJ, Bruch JC, Sadek IS et al. Robust shape control of beams with load uncertanties by optimally placed piezo actuators. Structural and Multidisciplinary Optimization, 2000; 19: 274-281. 3. Barboni R, Mannini A, Fantini E et al. Optimal placement of PZT actuators for the control of beam dynamics. Smart Materials and Structures, 2000; 9: 110-120.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Engineering Structural Optimization with an Improved Ant Colony Algorithm Y.B. Gong*, Q.Y.Li Department of Electronic Machinery and Traffic Engineering, Guilin University of Electronic Technology, Guilin, 541004 China Email: [email protected], [email protected] Abstract Ant colony optimization (ACO) algorithm is a newly developed bionics method, which has been successfully applied to several kinds of optimization problems as the traveling salesman problem, sequential ordering, and management of communications networks and so on. In this paper the ACO algorithms for optimization problems of engineering structure as the antenna structure was presented. It was shown how to construct the responding relations between the ACO algorithms and structural optimization problems. For improving the performance of ACO algorithms, several parameters of the algorithm, particularly the pheromone evaluation, had been improved here. Three examples on structural optimization were presented and solved by the improved ACO algorithms, Genetic Algorithm, Simulated Annealing Algorithm and so on. The comparisons between the improved ACO algorithm and other algorithm for the three examples have been obtained in terms of efficiency and effectiveness. The comparisons show the improved ACO algorithm is a very effective approach for solving structural optimization problems. And the improved ACO algorithm can greatly enforce the robustness of the optimal result. REFERENCES 1. Dorigo M. Optimization, learning and natural algorithms. PhD. Thesis, Politecnico di Milano, Italy, 1992. 2. Li QY, Li WY. Parallel genetic optimization design of antenna structure with simulated annealing mechanism, in Cheng GD, Olhoff N,Gu YX, Haftka RT eds. Structural and Multidisciplinary optimization, Proc. 4th World Congress WCSM04, Dalian, China, 2001: 241-242. 3. Gong YB, Li QY, Xiao ZX. The Application of ant algorithm in structural optimization, in Herskovits J, Mazorche S, Canelas A eds. Structural and Multidisciplinary optimization, Proc. 6th World Congress WCSM06, Rio de Janeiro, Brazil, 2005, No.4101.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Optimization Studies for Crashworthiness Design using Response Surface Method Xingtao Liao1*, Qing L i 1 2 , Weigang Zhang1 1

State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082 China 2 School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia Email: [email protected], [email protected] Abstract Structural optimization related to crashworthiness and impact energy absorption capability is of particular importance to the automotive industry, which involves highly nonlinear computational analysis and design with many material and structure parameters. Unfortunately, conventional design analysis techniques can only improve the structural crashworthiness to a limited extent, which frequently undergo significant difficulties to achieve a global optimized performance in a nonlinear analysis and design framework. This paper developed an innovative Response Surface Method (RSM) to tackle the crashworthiness design problems, where the number of FE analyses are significantly reduced. Objective and constraint functions in crashworthiness design problems are often non-smooth and highly nonlinear in terms of design variables and require considerably expensive computational effort. In this study, the approximations of objective and constraints with respect to the specific design variables are constructed by using a statistical modeling technique for the response surface method. These statistical approximations, or metamodels, are used to improve high cost simulations for facilitating crashworthiness optimization. This paper presents two novel examples with a certain practical implication, one thin plate with a square hole and one simplified front rail structure of vehicle. Both models were subjected to an impact into a rigid wall and the simulations were carried out in LSDYNA. The first problem aims to maximize the energy absorbed with the position and the shape of the hole as variables. The second problem focuses on minimizing the mass of the front rail structure meanwhile meet the requirement of crashworthiness performance using the gage thickness and the material properties as variables. In the latter, variable screening technique and the sequential approximate optimization method are used to improve the performance of Response Surface Method, which has more design variables than in the former and thus appreas more challenging in computational analysis and optimization. The results demonstrate that the new computational design method is efficient and effective in solving crashworthiness design optimization problems. REFERENCES 1. Stander N, Craig KJ. On the robustness of the successive response surface method for simulationbased optimization. Engineering Computations, 2002; 16: 431-450.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Path Optimization of Large-Scale Automated Three-Dimensional Garage Based on Ant Colony Algorithm Jianjun Meng **, Zeqing Yang \ Zhenrui Peng 2 1

Institute of Mech-Electronic Technology, Lanzhou Jiaotong University, Lanzhou, 730070 China National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, 310027 China Email: [email protected], [email protected] 2

Abstract The basic principle of the ant colony algorithm was introduced firstly, and mathematical model was established taking the TSP (Traveling Salesman Problem) as the example. In view of computational complexity, lower efficiency and stagnation of this model, the max-min ant colony algorithm (MMACA) was proposed. Only the best ant that constructed the shortest tour is allowed to deposit the pheromone utilizing the historical information, which can effectively avoid the pheromone of one way is bigger than other ways leading to all ants concentrate to the identical way. But when it is applied in the actual scene of big disturbance, the search ability is unsatisfactory. To settle the contradiction among convergence speed and precocity and stagnation in the other ACAs, a new-type intellectual ant colony algorithm (NIACA) was developed. The proposed intellectual ACA adopts a novel dynamic pheromone updating rule to guarantee the algorithm has the better convergence speed and the stability; it uses a particular variation scheme in genetics to optimize each searched result, a new individual is produced using single-parent exchange operator and inversion operator and is applied to by inverse variation method to change the variation conditions. The performance of iteration will be improved and the time will be greatly reduced owing to this mutation operation. Meanwhile it introduces artificial interference to adjust the chosen path dynamically and consider the energy consumption. NIACA is applied to the path optimization of the large-scale automated three-dimensional garage. We have carried on the test on Personal Computer (PC) with the higher-level language, the results show that the improved algorithm has higher convergence speed and stability, and it can get ideal searching result. The time for taking and parking vehicles in garage is greatly reduced and the efficiency is greatly improved. Meanwhile, this algorithm has good robustness and can be carried out in parallel mode. Moreover, we also have experimented with the artificial interference to this algorithm, the simulation results indicate that the strategy leads to the dynamic adjustment of the selected path, and the feasible path that blocks the entrance task will be discarded and the optimized path with fewer energy consuming will be saved. It can reduce the computational complexity greatly, and it can get ideal searching results in the fewer iterations. The research results show that the prospect of this algorithm in large-scale optimization problems is promising. REFERENCES 1. M. Dorigo, V. Maniezzo, A. Colorni.The Ant System: Optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics, Part-B, 1996; 26: 1-13. 2. M. Dorigo, G. Di.Caro, L. M. Gambardella. Ant algorithms for discrete optimization. Artificial Life, 1999; 5: 137-172. 3. M. Dorigo, L. M. Gambardella.Ant colonies for the traveling salesman problem. BioSystems, 1997; 43: 73-81. — 303 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A Continuous Approach to Discrete Structural Optimization T. Tan*, X.S.Li State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024 China Email: [email protected], [email protected] Abstract The mathematical model of the discrete optimization is generally defined as (PI): min f(x) s. t. gj(x)<0, j = \, 2,..., m xt £ S, = [sn, ..., siRj}, i = 1, 2,..., n where x is the design vector, S. stands for a specified discrete-valued set for x.. In this paper we introduce a continuous approach to solving discrete optimization problem (PI), which is both mathematically rigorous and easily implemented in engineering practice. With this approach, the variable x. is denoted by xi — ^2sirSir, and (PI) is converted into an equivalent 0-1 model (P2): r=\

min f(6) s.t. gj(6)<0,

y = l, 2,..., m

X X = 1, i = 1, 2,..., n\ 8ir e {0, 1}, i = 1, 2,..., *, r = 1, 2,..., R. r=\

It is interesting to note that this reformulation for (/?,) coincides with the segmental model [1]. However, a very different solution strategy is proposed with our continuous algorithm in which the constraint 6ir €{0, l } , is replaced by a continuous one -Sir\og6ir — (l — 5f>)log(l — ^.r) = 0 by means of so-called binary entropy function [2]. As such, we finally establish an equivalent continuous model (P3): min f(6) s.t. gj{6)<09 7 = 1,2,..., in X X = U

= 1,2,...,H;

-8ir\og6ir-{\-6ir)log{\-6ir)

= 0, i = l, 2,...,«, r = l, 2,...,/?,.

r=\

Numerical experiments give promising results for proposed algorithm. REFERENCES 1. Templeman AB, Yates DF. A segmental method for the discrete optimum design of structures. Eng. Opt., 1983; 6: 145-155. 2. Shannon CE. A mathematical theory of communication. Bell System Tech. J., 1948; 27: 379-423, 623-656. — 304 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Parametrical Analysis and Optimization of Partial Double-Layer Reticulated Shells Using Uniform Design Method and Second Order Rotation Method J. C. Xiao1*, T. Liang1, Y. Liu2 1

School of Civil Engineering & Architecture, Guizhou University, Guiyang, 550003 China Department of Civil Engineering, Dalhousie University, Halifax NS B3J1Z1, Canada Email: [email protected] Abstract Partial double-layer reticulated shells are of the capability to cover large span space. To fulfill the requirements of preliminary design of the structures, uniform design method and rotational second order method are adopted to make parametrical analysis and optimization. Compared with other response surface methods, this approach needs fewer experiment observation times for the same precision. In the analysis, just existing software packages for structural analysis and design are used. By means of regression of the calculation results, approximate functional relations between the objective values (steel weight, deflections, basic frequency and minimal geometrically nonlinear buckling load) and the design variables (rise of shell, single-layer area, and design stress ratio for single-layer members), and the significant order of factors, are obtained. In these functions, the interactions of design variables are considered. The optimum level combination of the factors that meet the requirements of design specifications is obtained by solving a simple programming problem. Design verification of a multipoint-supported hexagonal partial double-layer reticulated shell shows the reliability of the results, which precision is high enough for preliminary design.

REFERENCES 1. Chen WJ, He YL, Fu GY. Stability analysis of the partial double-layer forms derived from the complete double-layer reticulated dome. Journal of the International Association for Shell and Spatial Structures, 2001; 42 (12): 117-127. 2. Cochran WG, Cox GM. Experimetal Design (second edition), John Wiley & Sons Inc, New York, 1957. 3. Mbakogua FC, Pavlovic MN. The prestressing of shell roofs. Engineering Structures, 1999; 21: 16-33.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Optimum Design of Spiral Grooved Mechanical Seal Based on Thermo-Hydrodynamics J. F.Zhou*,B. Q. Gu College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, 210009 China Email: [email protected], [email protected] Abstract The spiral grooved mechanical seal is a new type of non-contacting mechanical seal, and is widely used as shaft sealing on pumps, compressors and agitators. A finite element program is developed to calculate the bearing force and the leakage rate of the fluid film in the spiral grooved mechanical seal. Based on the frictional heat transfer analysis of the sealing members and the coupling analysis of their thermal deformation and the frictional heat of the fluid film, the shape of the gap between the two deformed end faces and the viscosity of the fluid film can be determined when the bearing force of the fluid film is given. The optimum design of the spiral grooved mechanical seal is desired to obtain the optimal configuration of the sealing members which is corresponding to both the largest ratio of the stiffness of the fluid film to the leakage rate and the minimum temperature of the sealing members. By means of this technique, the optimal geometrical parameters of the sealing members are obtained, such as the length L, the inner radius R[, the outer radius R0, the end radius of the grooves Rg, the spiral angle a, the ratio of the groove width to the weir width S and the number of the grooves JVg, and the sealing members designed possess good heat transfer performance and sealing capability. Dam

„T .

Rotation

Figure: Model of the optimal spiral grooved seal ring REFERENCES 1. Pascovici MD, Etison I. Thermo-hydrodynamic analysis of a mechanical face seal. Journal of Triboligy, 1992; 114: 639-645. 2. Gabriel, Ralph P. Fundamentals of spiral groove noncontacting face seals. Lubrication Engineering, 1994; 50(3): 215-224. 3. Cicone T, Pascovici MD, Tournerie B. Non-isothermal performance characteristics of fluid film mechanical face seals. Journal of Engineering Tribology, 2001; 215(1): 35-44. — 306 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Evolutionary Topological Design of Frame for Impact Loads Xianyan Chen ! *, Qing Li *'2, Shuyao Long ! , Xujing Yang3 1

Department of Engineering Mechanics, Hunan University, Changsha, Hunan, 410082 China School of Engineering, James Cook University, Townsville, QLD 4811, Australia Key Laboratory of Advanced Technology for Vehicle Body Design & Manufacture of Ministry of Education, Hunan University, Changsha, 410082 China Email: [email protected]; [email protected]; [email protected]

2

Abstract Over the past decade remarkable progress has been made in the development of software used in crash simulation and analysis. A number of commercial codes have been available with accepted accuracy. However, there has been limited report in its inverse problem yet, i.e. topological and shape optimization for crashworthiness. The difficulty is primarily raised from two folders on (1) the nonlinear sensitivity analysis and computational cost, which become particularly challenging in various topological designs; (2) dynamic multi-modals and non-convex design space, which do not lend the crash problems themselves well to classical gradient techniques. This paper aims at developing an alternative approach to the crashworthiness design problems. The Evolutionary Structural Optimization (ESO) will be applied to reduce the dependence on the continuum sensitivities. In a crash simulation, it is frequently found that plastic deformation in some locations may be much higher than others. This implies that the material of the crashing elements may have different contributions to the crashworthiness goal. Ideally, the energy absorption levels in all location are near identical. To represent the relative performance of element's material, a dimensionless factor is formulated by dividing the crash energy absorption by each element to the highest one as, a* =U.IU ^

11

max

where Ut computes the strain energy absorbed by the candidate element (zth). It is worth pointing out that the total internal energy should contain the elastic and plastic components, i.e. U = Ue+Up. The factor a1 can be used to determine the relative efficiency of material usage. By gradually removing the inefficient material of the elements from the structure, the remaining structure evolves toward an optimum. The non-linear explicit finite element code LS-DYNA is employed to simulate the deformation and stain energy of the structure under impact load. The ESO method is implemented in an environment of LS-DYNA. An example of frame topology designs are also shown in this study to demonstrate the capabilities of the present method.

REFERENCES 1. Pedersen CBW. Topology optimization design of crushed 2D-frames for desired energy absorption history. Structural and Multidisciplinary Optimization, 2003; 25: 368-382. 2. Steven GP, Li Q, Xie YM. Evolutionary topology and shape design for general physical field problems. Computational Mechanics, 2000; 26: 129-139.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Topology Optimization of Space Vehicle Structures Considering Attitude Control Effort Z. Kang*, C. Zhang State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024 China Email: [email protected] Abstract This paper addresses the topology optimization of space vehicle components for reducing the attitude control efforts. The driving torque requirements for attitude control of a space vehicle are typically closely related to the mass moment inertia of the system. Particularly, as shown by the present study, the air consumption for attitude control actuators using compressed cool air is proportional to the mass moment of inertia. In such circumstances, it is meaningful to reduce the mass moment of inertia of the structural components in order to minimize the attitude control efforts. While structural topology optimization problems are conventionally formulated as to minimize the structural weight (or material volume) or to optimize the structural performance (compliance, eigenfrequencies, etc.) under material volume constraint, the problem for minimum compliance under constraints of mass moment of inertia is investigated in this paper. The SIMP approach (Bendsoe M.P. 1989) is used in the mathematical statement of the problem, where the material density of the artificial material for each element is taken as the design variable. For the solution of the problem with a single constraint on the moment of inertia, the Optimality Criteria method is employed. To this end, a design variable updating scheme is derived based on the Kuhn-Tucker optimality condition. Therein, a sensitivity filtering technique proposed by O. Sigmund (1997) is employed to alleviate numerical instabilities such as checkerboard and mesh-dependency problems. For problems with multiple constraints, the gradient-based Mathematical Programming approach is employed. Numerical examples will be given for demonstration of the validity of the present method. The different optimal structural layouts obtained with the proposed formulation and those with the conventional topology optimization will be also discussed. REFERENCES 1. Bendsoe MP. Optimal shape design as a material distribution problem. Struct. Optim., 1989; 1: 193-202. 2. Sigmund O. On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach., 1997; 25: 495-526.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Application and Research of Structure Topology Optimization of Scraper Conveyer with MSC.Nastran.optishape J. B. Sang*, B. Liu, S. F. Xing, L. C.Yang, Y. H. Qie Department of Mechanical Engineering, Hebei University of Technology, Tianjin, 300130 China Email: [email protected] Abstract Within the last 30 years, a number of methods have been proposed such as boundary-based shape optimization, homogenization-based method, evolutionary structural optimization etc [1]. Design optimization is used to produce a design that possesses some optimal character, such as minimum weight. However, these optimization capabilities are basically used to improve the design of structures or parts in the detailed design process. In the beginning of conceptual design process, another method of optimization what is called topology optimization is necessary. Based on the homogenization method, which is introduced by Bendsoe and Kikuchi [2] in 1998, topology optimization has become an important and well-recognized sub-area of structure optimization. This theory has been extended and applied to several of kinds of problems such as static problems and mode problems [3]. In this paper, topology optimization of scraper conveyer is based on homogenization method and carried out with MSC.Nastran.optishape program. On the basis of this theory of homogenization method, the design domain is assumed composing of infinitely periodic microstructures [4]. Starting with a discussion on several of optimization methods and their merit and shortage, homogenization method is discussed completely [5]. In the first phase of topology optimization, the model of scraper conveyer is analyzed with MSC.Nastran. The results shows that the maximal stress value in original design is much less than the yield stress value, which make material wasted. Then the model is submitted to MSC.Nastran.optishape to access the topology optimization results. We find that the maximal stress value in scraper conveyer doesn't still approaches its breaking point. Therefore this design makes material sufficient used, the weigh of scraper conveyer effectively lightened and the stress distribution is more reasonable. It is found that optimization result is according to the theory and it is also testified that topology optimization with MSC.Nastran.optishape can be applied in the engineering scope practically. REFERENCES 1. Boltyanski V. Application of topology in optimization theory. Topology and Its Application, 2005; 617-628. 2. Bendsoe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Comp. Meth. in App. Mesh. & Eng., 1988; 71: 197-224. 3. Bruns TE, Tortorelli DA. Topology optimization of nonlinear elastic structures and complaint mechaniasm. Computer Method in Applied Mechanics and Engineering, 2001; 3343-3359. 4. Fernandes P, Guedes JM, Rodrigues H. Topology optimization of three-dimensional linear elastic structure with a constraint on perimeter. Computers & Structures, 1999; 583-594. 5. Suzuki K, Kikuchi N. A homogenization method for shape and topology optimization. Comp. Meth. in Appl. Mech. & Eng, 1991; 93: 291-318.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Structural Topology Optimization Using Level Set Method Michael Yu Wang * Department of Automation & Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China Email: [email protected] Abstract Structural optimization, in particular the shape and topology optimization, plays an important role in structural design and analysis. Optimization techniques could deliver a design with optimal performance requirements such as strength, stiffness, weight, natural frequency, or buckling. Particularly, topology optimization has the greatest potential for the improvement in structural performance, since topology optimization procedures permit changes in the connectivity of the geometry of the structure during the design process. Over the past decade, there have been some extensive developments of various approaches to topology optimization. Here, we specifically put forth the level set method, in which the shape of a structure is employed and modified directly such that certain design objectives are obtained. The level-set method is a versatile and efficient technique for problems involving the motion of curves and surfaces. It is by now a classical tool in many fields of applications such as fluid mechanics and image processing. For structural topology optimization, the technique makes use of the classical shape sensitivity in an Eulerian framework. One attractive attribute of the method is that it gives a natural way of describing closed boundaries (curves or surfaces) and allows for automatic changes of topology, such as merging and breaking of boundaries, with calculations easily made on a fixed rectilinear grid. Over the recent years, the technique has been developed for many topology optimization problems, including solid structures [1] materials [2], and compliant mechanisms [3]. A bridge type structure is presented in this paper to illustrate the technique. It can be concluded that the topology optimization using level-set method is a novel technique for structure optimization. It yields a complete solution space and allows for the application to complex engineering problems, including material, boundary and geometric behavior, linear or nonlinear. It is suitable for complex problems and has been successfully used in industry for engineering designs.

REFERENCES 1. Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003; 192(1): 227-246. 2. Wang MY, Zhou SW. Synthesis of shape and topology of multi-material structures with a phase-field method. Journal of Compute-Aided Materials Design, 2004; 11(2-3): 117-138. 3. Wang MY, Chen SK, Wang X, Mei Y. Design of multimaterial compliant mechanisms using level-set methods. Journal of Mechanical Design, Trans, of ASME, 2005; 127(5): 941-956.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Topological Optimization Analysis of 3-D Continuum Structure with Stress and Displacement Constraints* H. L. Ye*, Y. K. Sui Numerical Simulation Center for Engineering, Beijing University of Technology, Beijing, 100022 China Email: [email protected], [email protected] Abstract It is well known that topology optimization of 3D continuum structure is one of the major challenges because of difficulty to establish a good geometric model which comprising a large number of design variables, and complexity of optimization algorithm. On the other hand, the problem under multiple load case is not easy to be approached than one under single load case, because the former becomes a multiple objective problem based on compliance objective function. In order to overcome these difficulties, the optimal topology model of 3D continuum structure is established based on ICM (Independent Continuous Mapping) method, which refers to weight as objective and subjected to stress constraints and displacement constraints with multi-load-cases. A globalization of stress constraints is proposed by virtue of the von Mises' yield criteria in theory of elastic failure. Thus, transformation of all elements' stress constraints into a structural energy constraint is achieved, namely, a global constraint substitutes for lots of local constraints. As a result, the numbers of constraints is reduced, and the complexity of the sensitivity analysis is decreased. For global displacement constraints, an explicit expression of displacement with respect to the topological variables is formulated by using of unit virtual load method. In order to decrease the error of numerical calculation generated by the order magnitude between different physical quantities, the optimal model that normalizes with two types of dimensionless constraints is further derived for continuum structure with stress constraints and displacement constraints. Furthermore, the best path transmitted force in the multiple load cases is selected successfully. The dual quadratic programming is applied for to solve the optimal model of continuum. Consequently, the number of design variables is dramatically decreased; the efficiency of computation is improved. In addition, the present optimal model and its algorithm have been implemented by means of the MSC/Patran software platform using PCL. Several numerical examples indicate that the method is effective and efficient. As an example, Figure 1 illustrates the background structure of an elastic body with size 10X0. 6X2m 3 . The two corner points on the bottom side are fixed. Four central forces P\ = P2 = P3 = PA = 450 kN are located in the middle of top of beam. The allowable stress is 50 Mpa, and the displacement constraint value of the four nodal points where the forces are located is less than 0.8 mm along with the up-to-down direction. The optimal topology configuration of the structures is illustrated in Figure 2.

Figure 1: Finite element model

Figure 2: Optimal topology configuration

* Supported by the National Natural Science Foundation of China (10472003), Beijing Natural Science Foundation (3002002), Beijing Educational Committee (KM200410005019) and the Foundation of Beijing University of Technology for PhD. — 311 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Cross-Level Sentence Alignment Anna Ho *, Francisco Oliveira, Fai Wong Department of Computer Science, Faculty of Science and Technology of University of Macao, PO Box 3001 Macao SAR, China Email: [email protected] Abstract This paper describes a new model for sentence alignment system of structurally different languages such as Chinese and Portuguese. The alignment may be one-to-one, one-to-many, many-to-one and many-to-many. It is also not surprised that the first word or sentence of Chinese is a translation of the last word or sentence of Portuguese. In this proposed method, we try to combine the statistical approach and lexical approach in order to achieve the efficiency and accuracy. In our current research, we first complete the word level alignment by making use of the Chinese-Portuguese dictionary to get the basic translation rate between the two texts. However, the system cannot make a good decision in processing the word alignment by just only concerning the information achieved from the Chinese-Portuguese dictionary. This is because most often the named entities cannot be discovered from the dictionary and this causes the system make a wrong decision. In order to make the system more adaptive, we apply the maximum entropy model to align the named entities without perform the word segmentation for Chinese. Secondly, from the word level alignment, we achieve anchor point and process the sentence level alignment. We use the Hidden Markov Model (HMM) and Singular Value Decomposition (SVD) to get the statistical information of the sentences. For SVD model, we first set up a matrix which consists of the word level alignment statistic information. Then performs the two dimensional reconstruction of the original matrix. By comparing Figure 1 and Figure 2 (sample fragments of data), we can observe some relationships among the sentences and this can give an approximation of sentence alignment. Table: Original matrix (left) and 2D reconstruction of the original matrix (right) PI P2 P3 P4 P5 P6 P7 P8 P9 P10

CI 6 5 1 1 2 1 1 0 11 3

C2 2 4 0 1 1 1 0 0 7 3

C3 2 4 0 1 1 1 0 0 7 3

C4 6 4 1 1 2 1 1 0 12 3

C5 2 4 0 1 1 1 0 0 7 3

C6 2 4 0 1 1 1 0 0 7 3

C7 2 4 0 1 1 1 0 0 7 3

Pi P2 P3 P4 P5 P6 P7 P8 P9 P10

CI

C2

C3

C4

C5

C6

C7

5.74 4.51 0.92 1.02 1.94 1.02 0.92

2.09 3.92 0.04 0.98 1.01 0.98 0.04

2.09 3.92 0.04 0.98 1.01 0.98 0.04

6.19 4.48 1.05

2.09 3.92 0.04 0.98 1.01 0.98 0.04

2.09 3.92 0.04 0.98 1.01 0.98 0.04

1.73 4.31 -0.11 1.09 0.98 1.09 -0.11

1 2.05

1 1.05

0

0

0

0

0

0

0

11.32 3.06

6.96 2.92

6.96 2.92

11.73 2.99

6.96 2.92

6.96 2.92

7.09 3.26

REFERENCES 1. Fai Wong DCH, Mao Yu Hang, Dong Ming Chui. A flexible example annotation schema: translation corresponding tree representation. Proc. 20th Int. Conf. on Computational Linguistics (COLING-2004), Switzerland, Geneva, 2004, pp. 1079-1085. 2. Li Yiping, Pun Chiman, Wu Fei. Portuguese-Chinese machine translation in Macao. Proc. of Machine Translation, SUMMIT VIT'99, Singapore, 1999, pp. 236-243. — 312 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

CSAT: A Chinese Segmentation and Tagging Module Based on the Interpolated Probabilistic Model Ka Seng Leong1*, Fai Wong1,2, Chi Wai Tang2, Ming Chui Dong1'2 1

Faculty of Science and Technology of University of Macau, PO Box 3001, Macau SAR, China INESC Macau, Macau SAR, China Email: {sejohnny, derek, kevin, dmc}@inesc-macau.org.mo 2

Abstract Chinese is a challenging language in natural language processing. Unlike other languages like English, Portuguese, the first step in Chinese text processing is the segmentation because there are no delimiters in a Chinese sentence for identifying the words boundaries in it. And there are many ambiguity problems during Chinese processing like segmentation ambiguities, unknown words problem, part-of-speech ambiguities, etc. In segmentation and tagging, one of the main tasks is to identify unknown words and recognize proper nouns. In the research, efforts are being paid on this particular problem. In this paper, an integrated application with segmentation and tagging ability has been studied and implemented. In the segmentation, a line of Chinese text is first split up into a sequence of atomic characters. Like this Chinese statement, IfefP^iJT^ (we go to play ball) is split up into S, ff5 i JT i^c with spaces and then for every atomic character in this statement, we are going to search through a knowledge base (necessary statistics about a segmented and tagged Chinese corpus, PFR corpus provided at http://www.icl.pku.edu.cn) to find all the words beginning with those atomic characters and keep this data in an appropriate structure. After that, a depth first search is performed on the data got to generate all the possible segmentations for the Chinese statement based on the words bi-gram model. Upon getting the results, all the candidates are evaluated and the N-best candidate segmentations are selected. In the second phase, an interpolated probabilistic tagging model proposed in [2] with proper nouns recognition and tagging is applied for tagging the N-best candidate segmented Chinese statements. Experiments based on different parameters (e.g., the parameter N used for getting N-best candidate segmentations) were taken for comparison so as to improve the performance of the application. This application is the first step for Chinese processing and it is used as a preprocessing module in the Chinese to Portuguese machine translation system. REFERENCES 1. Zhang Hua-Ping et al. Chinese lexical analysis using hierarchical hidden Markov model. Second SIGHAN workshop affiliated with 41th ACL, Sapporo, Japan, July, 2003, pp. 63-70. 2. Wong Fai, Chao Sam, Hu Dong Cheng, Mao Yu Hang. Interpolated probabilistic tagging model optimized with genetic algorithm. Proceedings of the Third International Conference on Machine Learning and Cybernetics, Shanghai, China, 2004, 2569-2574.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Overcoming Data Sparseness Problem in Statistical Corpus Based Sense Disambiguation Francisco Oliveira*, Fai Wong, Anna Ho, Yiping Li, Mingchui Dong Faculty of Science and Technology, University of Macau, Macao, China Email: {olifran, derekfw, ma36560, ypli, mcdong}@umac.mo Abstract The problem of data sparseness is considered as a common problem for Statistical Corpus based Sense Disambiguation approaches [1]. Usually large amounts of data are required in the corpus to guarantee that all senses of an ambiguous word are presented. However, this is not easily achieved, especially for words that do not occur frequently in the training corpus. On the other hand, for languages that do not have large amount of digitized resources, the sparseness problem is even worse. In this paper, we present an unsupervised framework that first learns the relationships between the ambiguous and related words to reveal their most suitable senses based on a proposed mathematical model and a set of bilingual resources, including a non-aligned Portuguese-Chinese training corpus, a dictionary, and a sense inventory. For senses not found in the learning phase, bilingual examples from the dictionary and Singular Value Decomposition [2] techniques are applied to overcome the sparseness problem. It is believed that the examples in the bilingual dictionary can reflect the most suitable meaning and sense of each word. Thus, these can be taken into consideration for learning extra senses that do not occur frequently in the training corpus. For each example related to the ambiguous word, the proposed method will identify their corresponding related words based on a defined context window, relative distance and their mutual information. These words in the source language example are then aligned with the ones extracted from the target language for identifying new senses. Words related to an ambiguous word often have a strong relationship between each other. If these are ambiguous words, then extra relationships can be found. This is modelled by a vector space, a matrix of related words by the senses of the ambiguous word. A vector for a word v is derived from the set of words related to the ambiguous word w. Each entry in the vector represents the number of times that v occurs with one of the senses of w in the corpus. The closeness between vectors is determined by the cosine of the angle between the vectors. Singular Value Decomposition is applied in this vector space to overcome the sparseness problem and to further explore the relationships between words in the matrix. All the senses found are converted into a set of rules and stored in the Word Sense database for later use in disambiguation and translation process. Preliminary experiment results show an improvement of learning more senses in the sparseness environment with the use of the mentioned strategies.

REFERENCES 1. Ide N, Veronis J. Introduction to the special Issue on word sense disambiguation: the state of the art. Computational Linguistics, 1998; 22(1): 1-40. 2. Gene H Golub, van Loan Charles F. Matrix Computations. The Johns Hopkins University Press, Baltimore and London, 1989.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Application of Translation Corresponding Tree (TCT) Annotation Schema for Chinese to Portuguese Machine Translation Chi-Wai Tang , Fai Wong, Ka-Seng Leong, Ming-Chui Dong, Yi-Ping Li Faculty of Science and Technology, University of Macau, Macau SAR, China Email: {kevin, derek, sejohnny, dmc}@inesc-macau.org.mo, [email protected] Abstract In Example Based Machine Translation (EBMT) research, there are three main approaches: Surface Based, Pattern Based and Structure Based approach. In Structure Based EBMT system, such as SSTC approach [1], it has a problem that it relies on two syntax parsers to analyze the translation examples, but robust syntax parsers are not always available. On the other hand, Chinese and Portuguese belong to two different language families and there exist grammatical deviation problem between them. In order to resolve the weakness of the Structure Based EBMT system and linguistic problems between Chinese and Portuguese, Tang and Wong [2] propose a new Portuguese to Chinese Machine Translation method and this method is based on a novel technology called Translation Corresponding Tree (TCT) which is an example based knowledge annotation method for Portuguese to Chinese translation. In this paper, it adopts the TCT annotation scheme and introduces the knowledge based construction and translation for Chinese to Portuguese MT. In this parper, it presents the additional problems during Chinese analysis and the corresponding solutions. In this research, it also proposes a conversion algorithm to reuse the existing translation knowledge of Portuguese to Chinese MT system, which represented in terms of TCT trees. Based on the transformation algorithm, the knowledge trees of Portuguese to Chinese translation is converted into that of the translation knowledge which can be used to facilitate the Chinese to Portuguese . By this conversion method, existing knowledge can be easily reused without having to re-construct the knowledge from scratch. Based on the research result of this paper, a Chinese to Portuguese prototyping machine translation system is implemented and the empirical results show that the MT system can achieve the translation accuracy of 85% in the domain of Macau Law statements. REFERENCES 1. AI-Adhaileh MH, Tang EK, Zaharin Y. A synchronization structure of SSTC and its applications in machine translation. The COLING 2002 Post-Conference Workshop on Machine Translation in Asia, 2002. 2. Tang CW, Wong F, Li YP. TCT Schema in EBMT and its application. In Proc. the Symposium on Applied Science and Technology in Macau 2004, 2004, pp. 19-27.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Development of a Knowledge Based System for the Portuguese Code for Building Acoustics Joao Mariz Graca **, Jorge Patricio 2, Luis Santos Lopes 3 1

Lusiada University, Lisboa, Portugal LNEC, Av. do Brasil, 101, 1700-066, Lisboa, Portugal Acustiprojecto Lda. [email protected], Lisboa, Portugal Email: [email protected] 2

Abstract A knowledge based system for the Portuguese code for Building Acoustics is under development. The system is being developed using XPCE/SWIPROLOG language. This specific PROLOG interpreter was choosen because it provides both: An object oriented architecture linked to PROLOG and a good interface to grapical objects. The XPCE Architecture is not PROLOG, however it is linked to PROLOG and providing a very complete set of libraries for (graphical) user interfacing (GUI). It also provides object oriented (OO) techniques, which are very well suited to handle the complexity of GUI components. Since we are dealing with buildings and architectural shapes, GUI libraries were found very useful to implement the 3D routines that can provide friendly 3D images so that the user can check if he is doing well. The system provides calculations for the following acoustic parameters: 1) airborne sound transmission through adjacent rooms and flanking contributions; calculation of Dn,w parameter, according to EN ISO 12354 standard; 2) impact sound transmission through adjacent rooms and flanking contribution;calculation of L'n,w parameter, according to EN ISO - 12354 standard; 3) sound insulation of facades; calculation of D2m,n,w parameter; 4) reverberation time according to Sabine and Eyring Formulas. The system provides a 3D interface to the user so that he can watch the rooms in which the acoustic field is being studied. The system uses object oriented techniques to provide construction elements that can be used in several different situations. This option was choosen because it seemed similar to designing procedures for buildings. In fact, building designers aim to rationalise projects, simplifying their options by choosing as few types of construction elements as possible. Dispite the fact that the same construction element can be used for different room configurations, only solutions that comply with regulations can be used. Artificial Inteligence (AI) techniques are used at several levels inside the system, such as: 1) to recognise the shapes of the adjacent rooms and choose the correspondent appropriate dimensions and algorithms for calculation; 2) to impose some constraints to the 3D editor so that transmission between two rooms only can exist if they are adjacent and separated by a wall or floor; 3) to ensure that the user selected options are as much consistent as possible. For instance if a flexible element is selected only types of joints for flexible walls can be selected. Since the system is developed under the Prolog language, many other AI techniques and routines where largely applied within the code. The system provide a good interface to the user since he can see 3D images of the parts of the building under acoustic simulation. He also can be confident of the results obtained since AI techniques are used to ensure consistency of the simulations The link between AI and object-oriented techniques was found very usefull to implement this system particularly when dealing with 3D objects. — 316 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Interfacing Vision System with Robot for Pick and Place Operation R. Lalitha * Department of Mechanical Engineering, College of Engineering Guindy, Anna University, Chennai, India Email: [email protected] Abstract It is very expensive for a company to have to set up a factory line so that parts are presented in the exact same position to a robot over time. A non-vision -enabled robot will crash or miss if it goes for a part and comes up empty. This can result in a costly and time-consuming line shutdown or manual intervention. To over come this draw back, amore sophisticated system is required. This paper deals with vision -enabled robots. Vision-enabled robot system has a camera for capturing images of components as they travel, software for identifying and processing those images, and lighting to make sure the camera captures the best possible image. The software runs on a Windows-based computer connected to an Ethernet network. Instead of locking a part's position in advance to suit a blind robot, the camera in a vision-enabled robot lets the robot see the position of a loose part and adjust itself accordingly to manipulate it. The main objective of this work is to develop a system to see the component position, calculate the component's position, grasp it, inspect it, and move it to where it needs to go. Robot vision may be defined as a process of extracting, characterizing and interpreting the information from the images. The system comprises of a 5-axis robot driven by its own controller. In this work we use CCD (charge coupled device) camera for capturing the two-dimensional image of the scene where the objects to be picked up lie. CCD camera discretizes the scene into a 752 * 552 format, which can be stored in the RAM of a computer. The information from the CCD camera is processed by an image processing card mounted within the computer. The view area is digitized using VB 6.0 so that the coordinate values of any object in this area are found out. NI Vision Builder for Automated Inspection software is used for processing the image (geometric matching, Shape and colour analysis) and to find out the center of gravity of the object and its orientation. It can communicate triggering and inspection results directly to NI M Series DAQ devices or to industrial devices over serial and Ethernet protocols. It also includes the ability to set up complex pass/fail decisions to control digital I/O devices and to communicate with serial device like robot. Interfacing card is used for connecting the computer to the robot. Values are send to the robot via I/O card, which makes the robot to position its gripper accordingly in order to pick up the object and move it to the required destination.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan,China ©2006 Tsinghua University Press & Springer

A Web-Based Data Management and Decision Support System for Slope Safety Inspection and Evaluation J. Wang1*, M. C. Hung2 1

Department of Civil Engineering, Tamkang University, Tamsui, Taipei County, Taiwan, China MIS Division, Yosun Industrial Corp., Taipei, Taiwan, China Email: [email protected]

2

Abstract This paper examines how artificial intelligence and WWW technologies can be applied to disaster management to improve the current practice. A web-empowered database and decision support system was built for slope safety inspection and evaluation of residential communities on hillsides. Users can perform data management directly through the browser interface without any knowledge of database and query language. Furthermore, the system is capable of adding communities and instruments, as well as other text and image information of the communities on-line. As for safety evaluation, rule-based reasoning is used to decide the safety degree of every monitoring instrument. The system was implemented with the three-tier software architecture. The web-based environment let users use standard WWW browsers to enter, inquire and manage monitoring data without installing additional programs. The inference engine of the system is WebExpert [1] installed on a MS 2000 server with IIS. Its programming language is based on CLIPS [2], a lisp like and rule-based language. In addition, the system incorporates dynamic graph generation together with a friendly user interface implemented in HTML and JavaScript, etc. The system performs as an efficient tool for the safety inspection and evaluation of hillside residential communities. It acts as a collaboration portal that collects monitoring data, text and picture information from field safety inspectors at different locations. Also, it acts as an information and data visualization portal with various graph generation and decision support functions for government officials to evaluate the safety of the communities.

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Figure: Overview of the slope monitoring system REFERENCES 1. WebExpert Technical Reference Manual, Wise Web Ware, Inc., 2001. 2. Riley G. CLIPS: A Tool for Building Expert Systems, at http://www.ghg.net/clips/CLIPS.html, accessed February 2006. — 318 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Simplified Doubly Asymptotic Approximation Boundary for Foundations Dynamic Analysis Wenjun Lei, Demin Wei * College ofArchitecture and Civil Engineering, South China Univ. of Tech., Guangzhou, 510640 China Email: [email protected] Abstract The paper presents a simplified doubly asymptotic approximation boundary (spring-viscous boundary) for infinite medium used in finite element method, which can be applied to both static and dynamic foundation problems. The boundary can be realized by using special boundary elements. The spring part of this boundary can be determined by Mindlin Equations, and frequency dependent viscous material (resistance proportional to velocity) is introduced into the boundary elements to simulate dashpots of the viscous boundary. When choosing appropriate parameters of Young's modulus, Poisson's ratio and material damping ratio, these boundary elements can simulate not only the elasticity recovery capacity of the far field media, but also the radiation damping. Two case studies justify the validity and practicability of this simplified boundary, which proves that it is applicable to the dynamic soil-structure interaction analysis.

Figure: Flexibility functions F\ and F2 REFERENCES 1. Veletsos AS, Verbic B. Vibration of viscoelastic foundations. Earthquake Engineering and Structural Dynamics, 1973; 2(1): 87-102. 2. Underwood P, Geers TL. Doubly asymptotic, boundary-element analysis of Dynamic soil-structure interaction. Journal of Solid and Structures, 1981; 17(8): 687-697. 3. Engquist B, Majda A. Absorbing boundary conditions for the numerical simulation of wave. Math. Comput, 1977;31:629-651. — 319 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Limit State Analysis of Seismically Excited 3D R/C Beam Bearing Structures N. Kaufmann * Bauhausstrafie 4, 99423 Weimar, Germany Email: [email protected] Abstract Based on the capacity design method in [1] a calculation model is presented which enables the analysis of plane r/c frameworks using linear and quadratic optimization algorithms. In this connection, as criterion for evaluation the ultimate limit (shakedown) state, the adaptive load factor p A is defined. An extended and modified calculation model using non-linear optimization algorithms will be introduced, which allows the ULS analysis of 3D frameworks considering realistic non-linear plasticity conditions for the first time [2]. The plasticity conditions describe the interaction between the axial force and the bending moments of a 3d beam element. It is important to consider the additive division of the internal forces into a linear elastic part SE and into an irreversible residual-stress part SR. The elastic part is calculable using linear-elastic dy­ namic models, separated from the non-linear analysis. The reducing of the elastic solution SE to extreme combinations se is recommendable to ensure a restricted number of plasticity conditions. The determina­ tion of the combinations se results in dependence of the developed criterion of minimal load intensity p. This procedure allows the inclusion both linearly as well as non-linearly defined plasticity conditions. With an example of a three-dimensional framework subjected to TAFT-earthquake will be quantified the influ­ ence both of the formulation of the plasticity restrictions and of the determination of the linear elastic com­ binations SE on the results of the adaptive load state. Table 1: Calculation model for limit state analysis Determination of residual forces p < p A

Determination of adaptive load factor p A

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REFERENCES 1. Schiiler H. Zur Analyse und zur Bemessung adaptiver Tragwerke aus Stahlbeton unter dynamischen Einwirkungen. Dissertation, Bauhaus-Universitat Weimar, Germany, 1997. 2. Kaufmann N. Physikalisch nichtlineare Analyse dreidimensionaler Stabtragwerke aus Stahlbeton mit der Methode der mathematischen Optimierung. Dissertation, Bauhaus-Universitat Weimar, Shaker Verlag, Aachen, Germany, 2003.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Earthquake Response Analysis and Energy Calculation Based on Wavelet Transform Chen Wu *, Ruizhong Zhou College of civil engineering and architecture, Fuzhou University, Fujian, Fuzhou,3 50002 China Email: [email protected] Abstract In earthquake engineering, Fourier transform is a main way for spectrum analysis in the past. However, signal is requested to be stationary and the result couldn't offer time-frequency local character. Therefore, Fourier transform isn't suitable for dealing with seismic signal. Wavelet transform is a perfect way to analyze non-stationary signal for its multi-resolution nature. Any seismic signal can be decomposed to some frequency channels by multi-resolution analysis. Using this method, formulas for earthquake responses of MDOF system are deduced in this paper. Seismic signal and displacement energy in distinct frequency channel are discussed. With examples on MDOF system, an argument that in elastic system total earthquake responses can be obtained by adding all of the dynamic response contributions in distinct frequency channel is demonstrated and the reconstruction signal after restraining high frequency is applied to an approximate model. With energy distribution, dynamics response contributions of distinct frequency channel are analyzed. At the end of this paper, some problems about wavelet transform are discussed. tmmpvditq

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Figure 1: Components wavelet of distinct frequency Figure 2: Energy distributions of seismic signal and channel and corresponding displacements displacement signal in distinctfrequencychannel REFERENCES 1. Eysa Salajegheh, Ali Heidari. Time history dynamic analysis of structures using filter banks and wavelet transforms. Computers and Structures, 2005; 83: 53-68. 2. Wu Delun, Zhao Mingjie. A preliminary research on wavelet analysis applied to the response of structures. Acta Mechanica Solida Sinica, 1999; 20(s): 174-177 (in Chinese). 3. Jiang Jianguo, Zhou XH, Zou YS, Duan SW. Application of wavelet analysis to degree of single freedom dynamic system. Journal of Hunan University, 2002; 29 (5): 104-109 (in Chinese). — 321 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Elastoplastic Impact of the Sphere upon the Uflyand-Mindlin Plate I. A. Lokteva**, A. A. Loktev2 1

Voronezh State University, Voronezh, Moiseeva 47-134, 394055 Russia OOO "Konmark", Voronezh, 394000 Russia Email: [email protected]

2

Abstract The present paper is devoted to mathematical modeling of the low velocity impact of a sphere on a thin elastic isotropic plate, whose dynamic behaviour is described by the Uflyand-Mindlin equations taking the rotary inertia and shear deformations into account and, therefore, these equations are wave equations. They allow to assume that in a plate the transient wave of transverse shear, because of which there is a deformation of a plate material outside of the contact area, is generated with final velocity. As a method of the decision the ray method and method of splicing asymptotic expansion received for small times in the contact area and outside of it are used. In the present work, the procedure similar to the one proposed in [1] for the analysis of transverse impact of a solid sphere upon an elastic buffer positioned on an elastic orthotropic plate, is used to the case of shock interaction of a solid body in view of a rod with a elastic isotropic plate. The local deformations in the contact region are taken into account in terms of two elastoplastic models and the Hertz's theory. During the interaction of the indenter with the plate, a quasitransverse waves representing the surfaces of strong discontinuity begin to propagate, and the reflected wave has not had time to return to the boundary of the contact disk before the impact process completion. In the plate of the surface of strong discontinuity represent cylindrical surfaces - strip, whose generators are parallel to the normal to the median surfaces and guides locating in the median surface are circumferences extending with the normal velocities. Behind the wave fronts, the solution is constructed in terms of one-term ray expansions [2] whose coefficients are the discontinuities in time-derivatives of the required functions. To determine the desired values inside contact area the equations of motion the impactor and the region of bar's contact with the plate are used. The dynamic characteristics are determined at splicing on border of the contact area of the solution for required function inside a contact disk and outside of it. As a result, we are led to a system of two linear differential equations in the dynamic displacements of plate and the quasi-static strain of the plate material in the contact region. The solution of these equations is found numerically by iteration Timoshenko scheme on computer, and the time-dependence of the contact force is obtained. The carried out numerical researches allow to make the conclusion about influence of parameters of a construction, including plate's elastoplastic properties in the contact region, on dynamic characteristics of interaction. One-term ray series for the desired functions have allowed one to calculate with the given accuracy the stresses in the contact area of the plate and the dynamic safety margin of the thin plate. REFERENCES 1. Loktev AA. Elastic Transverse Impact on an Orthotropic Plate. Technical Physics Letters, 2005; 31 (9): 767-769. 2. Rossikhin YuA, Shitikova MV. A ray method of solving problems connected with a shock interaction. Acta Mechanica, 1994; 102: 103-121. — 322 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X,Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Numerical Analysis of Impact between Cue and Ball in Billiard (Effect of Tip Structure) Shinsuke Shimamura**, Shigeru Aoki 2 1

Graduate School of Engineering. Sci. Eng., Toyo University, Kujirai 2100, Kawagoe, Saitama 350-8585, Japan 2 Dept. ofComp. Sci. Eng., Toyo University, Kujirai 2100, Kawagoe, Saitama350-8585, Japan Email: [email protected], [email protected] Abstract A finite element analysis and an experiment on the impact between a cue and a ball in billiard are carried out. It is assumed that the cue is elastic and the ball is rigid. The displacement of the cue is calculated by FEM and that of the ball is estimated by solving the equation of motion. The contact force is determined by equating the sum of these displacements to the impact velocity times duration. An experiment, in which a freely supported cue is hit with a swinging ball, is performed to examine the accuracy of the analysis. It is shown that (1) the computational results on the time variation of contact force agree well with the experimental ones, (2) the contact period is reduced by mounting the plastic collar near the tip of the cue, and (3) the contact force rises steeply with the mounted plastic collar. REFERENCES 1. Aoki S et al. Transactions of the Japan Society of Mecanical Engineers, A 2002; 68(670): 81-83. 2. Toda S. Mechanics. Iwanami Shoten, Tokyo, 1994 (in Japanese).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A New Multi-Harmonic Method for Predicting the Forced Response of Mistimed Bladed Disks with Dry Friction Damping* Erming He, Hongjian Wang * College ofAeronautics, Northwestern Poly technical University, XVan, 710072 China Email: [email protected], [email protected] Abstract Mistiming refers to the variations of blades properties due to procession and wear etc. Though these variations is small, they can confine the vibration energy in a few blades, and increase the resonant response of those blades. Therefore, dry friction dampers are often used to reduce the vibration of blades. In order to investigate the effects of mistuning and the performance of dry friction dampers, it is necessary to develop an efficient numerical method to predict the resonant features of the bladed disks with dry friction dampers. We propose an efficient multi-harmonic (MH) method which is based on the harmonic balance method. The formulation of the method can take the advantage of fast Fourier transformation (FFT) which can facilitate the analysis efficiency significantly. The iterative Newton-Raphson procedure are used to solve the nonlinear vibration equation. The MH method has been used in a mass-string model of bladed disk, and validated by numerical integration method. The analysis of forced response features is performed by the MH method for both tuned and mistuned systems with various coupling strength, viscous damping, dry friction damping and blade stiffness mistuning etc. The results indicate that the MH method can efficiently predict the resonant response of the bladed disks with nonlinear dry friction damping. It is found that, for the weakly coupled bladed disk with nonlinear blade-to-blade dry friction damping, mistuning can reduce resonant amplitudes of the system significantly when dry friction and viscous damping are all smaller. This is not common for mistuning effects. In other situation, Mistuning can increase the resonant amplitudes, broaden the resonant frequency region, and produce multiple resonant peaks. REFERENCES 1. Griffin JH, Sinha A. The interaction between mistuning and friction in the forced response of bladed disk assemblies. Journal of Engineering for Gas Turbines and Power, 1985; 107: 205-211. 2. Wei ST, Pierre C. Effects of dry friction damping on the occurrence of localized forced vibrations in nearly cyclic structures. Journal of Sound and Vibration, 1989; 129(3): 397-416. 3. Berthillier M, Dupont C et al. Blades forced response analysis with friction dampers. ASME Journal of Vibration and Acoustics, 1998; 120: 468-474. 4. Cardona A. A multiharmonic method for non-linear vibration analysis. International Journal for Numerical Methods in Engineering, 1994; 37: 1593-1608.

* This work is supported by the National Natural Science Foundation of China (No.50275121)

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Coupled Vibration Analysis of Multiple Launch Rocket System by Finite Element Method Bingshang Li, Xinqi Xu * Naval Aeronautical Engineering Institute, Yantai, Shandong, 264001 China Email: [email protected] Abstract In multiple launch rocket system, initial disturbing is one important factor to influence the firing precision. It is meaningful theoretically and valuable practically to Study and to take right theory and effective calculating methods to analyze the vibration of the multiple launch rocket system . Based on CSG (Constructive Solid Geometry) and FEM (Finite Element Method), the elastic multiple launch rocket system modal is built and the vibration-coupling response of multiple launch rocket system when firing constantly is simulated. In order to verify the availability of researching methods, the conclusions are compared to practical situation.The result provides scientific basis for multiple launch rocket system design. REFERENCES 1. Xiaoting Rui, Yuqi Lu. Simulation and Test Methods of Launch Dynamics of multiple launch rocket system. National Defense Industry Press, Beijing, China, 2003 (in Chinese). 2. Guoping Wang, Xiaoting Rui, Weidong Chen. Dispersion Simulation Technique of Multiple Tube Weapon. Journal of system simulation, 2004; 16 (5): 963-966 (in Chinese).

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Three-Dimensional Vibration Analysis of Functionally Graded Material Rectangular Plates by Chebyshev Polynomials Q.Li*,V.P.Iu,K.P.Kou Department of Civil and Environmental Engineering, University of Macau, Av. Padre Tom 'as Pereira S. J., Taipa, Macao SAR, China Email: [email protected] Abstract Functionally graded materials (FGMs) are generally two-phase composites with continuously varying volume fractions. This advantage of continuousness can eliminate interface problems of composite materials. In this paper, threedimensional vibration analysis of FGM rectangular plates has been investigated. Consider the case of a uniform thickness, FGM rectangular plate simply supported along the four edges. The volume fraction of the FGM properties is assumed to obey a power-law function along the thickness direction. However, the Poisson's ratio is assumed to be constant for the effect of Poisson's ratio on the deformation is much less than that of Young's modulus. The three mechanical displacements components of the plate are expanded in triplicate series of Chebyshev polynomials multiplied by the boundary R-function which makes expansions satisfy the essential boundary conditions along the edges. Chebyshev polynomial series are chosen as admissible functions for they are a set of complete and orthogonal series in the interval [-1/1], The Lagrangian function L of the plate is expressed in terms of Chebyshev polynomial series. By Ritz method, the partial differential of Lagrangian function L with respect to independent coefficients leads to a set of linear equations in form of eigenvalue matrix for natural vibration frequencies. The mode shapes corresponding to each eigenvalue may be obtained by back substitution of the eigenvalues one by one in the usual manner. In present study, rectangular FGM plates with four simply supported edges are taken as examples for the convergence study. The trial plates are of different volume fraction exponents and thickness-side ratios. The convergence studies reveal the rapid convergence rate and high efficiency. The frequencies monotonically decrease and approach certain values with the increase in the number of terms of admissible functions. Three terms are sufficient for the requirement of expansion in the thickness direction to obtain the first twenty natural frequencies. It is also found that the convergence rate is independent of volume fraction exponents k values of FGMs. In the last part of this paper, two comparisons have been carried out to validate the present method. The numerical results of the first eight natural frequencies of simply supported square FGM plates are compared with the solutions of the higher-order shear and normal deformable plate theory for two special cases of isotropy; the natural fundamental frequency parameters of simply supported square FGM plates with different thickness-side ratios are compared with the solutions of finite element method. In the modeling of finite element analysis, there is an simplification of material definition for the material properties variation along the thickness of the plate. The relative thick plate and thin plates are meshed by solid and shell element types according to their thickness-side ratios, respectively. It is seen that both comparisons show good agreements. The method presented in this paper is a good and valid approach in analyzing the rectangular FGM plates.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

A hybrid elasticity method for bending and free vibration of composite laminates C. F. Lu, W. Q. Chen* Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China Email: [email protected] Abstract Semi-analytical solutions for bending and free vibration of composite laminated beams and plates have been derived based on 2D/3D elasticity theory using a newly developed hybrid analysis, which perfectly combines the state space approach (SSA) and the differential quadrature (DQ) technique. The thickness direction of laminates is selected to be the transfer direction in SSA, and the DQ technique is employed to discretize the domain(s) normal to the transfer domain. This enables us to transfer the original partial differential equations into a state equation consisting of first-order ordinary differential equations. In particular, the use of DQ technique makes ease of the treatment of various boundary conditions, which can not be considered in the conventional exact state-space approach. A transfer relation between the state vectors at the top and bottom surfaces of the laminates using matrix theory is established, from which the bending and free vibration problems can be solved eventually. Comprehensive numerical results have been obtained, which converge rapidly and agree well with that in the literature as well as finite element calculations. It is concluded that: (1) the DQ technique significantly widens the application of SSA; and (2) the proposed method is very efficient for analyzing layered structures. REFERENCES 1. Bahar LY. A sate space approach to elasticity. Journal of the Franklin Institute, 1975; 299: 33-41. 2. Bellman R, Casti J. Differential quadrature and long-term integration. Journal of Mathematical Analysis and Applications, 1971; 34: 235-238. 3. Shu C, Richards BE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 1992; 15:791-798.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China ©2006 Tsinghua University Press & Springer

Vibration Assessment of Railway Viaducts Under Real Traffic Using Bridge-Track Models Constan9a Rigueiro1, Carlos Rebelo2*, Luis Simoes da Silva2 1

Department of Civil Engineering, Institute Polytechnic ofCastelo Bronco, Portugal Department of Civil Engineering, University ofCoimbra, Portugal Email: [email protected]

2

Abstract Among the several safety checks that must be performed during the design of small to medium single span railway viaducts, the one concerning vertical accelerations is often crucial for high speed railways, since under these conditions resonance is much likely to occur. Recent results [1-3] of several field measurements carried out on simply supported medium span concrete railway viaducts showed that the non-linear behaviour of the superstructure composed by ballast and rails can play an important role in the dynamic behaviour of the structure and consequent evaluation of its maximum vertical accelerations. The conclusions drawn from those measurement results pointed out that the superstructure must be taken into account in the numeric analysis of small to medium span bridges, mainly what global ballast-structure damping is concerned. This revealed to be particularly important when decisions are to be taken, related to the strengthening of existing structures in order to increase traffic speeds and when comfort of passengers is an important issue. In order to evaluate the vertical vibrations of the bridges when the dynamic behaviour of the track is taken into account, same authors have already proposed different models including vertical springs, dampers and masses, which are interpose between loads and structure in order to simulate the railway track behaviour. These models include the rail with flexural stiffness in vertical and lateral direction and axial stiffness in longitudinal directions, the rail pads as vertical springs and dampers, the sleepers as rigid bodies with masses, and finally the ballast as springs and dampers. The main purpose of this paper is to compare the dynamic behaviour of ballasted simple supported viaducts with and without the referred track models, taking into account the structural dynamic response obtained from the field measurements, in order to evaluate the influence of the track in the safety check. Furthermore, the response acceleration of the ballasted simple supported viaducts is also compared with the results obtained from the simplified methods proposed by the ERRI D214 Committee used to estimated the maximum acceleration at mid-span of simply supported bridges decks, namely the DER Method (Decomposition of Excitation at Resonance Method) and the Virtual Influence Line Method.

REFERENCES 1. Rebelo C, Heiden M, Pircher M, Simoes da Silva L. Vibrations measurements on existing single-span concrete railway viaducts in Austria. EURODYN 2005, Paris, 2005, pp. 1637-1642. 2. Rigueiro C, Rebelo C. Simoes da Silva L, Pircher M, Heiden M. Dynamic behaviour of ballasted single railway bridges. Metodos Numericos en Ingenieria 2005, Granada, 2005. 3. Rebelo C, Simoes da Silva L, Pircher M, Rigueiro C, Heiden M. Vibrations measurements on small to medium single-span railway bridges. Experimental Vibration Analysis for Civil Engineering Stuctures - EVACES, Bordeaux, 2005. — 328 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

In-Plane Vibrations of Rectangular Plates with Rectangular Cutouts I. Shufrinl, M. Eisenberger!'2* 1

Faculty of Civil Engineering, Technion - Israel Institute ofTechnology, Haifa, 32000 Israel Department of Building and Construction, City University of Hong Kong, Hong Kong, China Email: [email protected], [email protected] Abstract This work presents highly accurate numerical calculations of the natural frequencies and modes for the in plane vibrations of rectangular plates with rectangular cutouts. The solutions are obtained using the multi term extended Kantorovich method. The solution is sought as the sum of multiplications of two one dimensional functions. In this method a solution is assumed in one direction of the plate, and this enables to transform the coupled partial differential equations of the plate equilibrium into an a set of ordinary differential equation. These equation are solved exactly by the exact element method [1], and an approximate natural frequency is obtained. In the second step, the derived solution is now taken as the assumed solution in one direction, and the process is repeated to find an improved approximation of the natural frequency. This process converges with a small number of solution cycles. For plates with cutouts this process yields very accurate values even with 1 term expansion, and in some cases one can improve these values by adding additional functions in the expansion. As an example the natural frequencies of a square plate fully restrained along all four edges, with centrally located square cutout are given in the Table. The size of the cutout is 0.4 of the edge size. The first 5 frequencies of vibration are given together with the results from a finite element analysis using ANSYS (with 8520 DOF). Many more results will be given. Table: Comparison of results with FEM analysis by ANSYS Present

ANSYS 8520 DOF

1 term

2 terms

3 terms

4.369

4.609

4.403

4.378

4.467

4.545

4.480

4.471

4.467

4.545

4.481

4.472

4.546

4.541

...

___

5.426

5.443

5.426

""'

REFERENCES 1. Shufrin I, Eisenberger M. Stability and vibration of shear deformable plates - first order and higher order analyses. Int. J. Solids Structures, 2005; 42: 1225-1251.

— 329 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Optimal Control of Temperature Gradient in a Large Size Magnetic Czochralski Silicon Crystal Growth by Response Surface Methodology H. P. Yu l*9 Y. K. Sui \ J. Wang2, X. L. Dai 2 , G P.Anl 1

Numerical Simulation Center of Engineering, Beijing University of Technology, Beijing, 100022 China General Research Institute for Nonferrous Metals, Beijing, 100088 China Email: [email protected] Abstract Numerical simulation of a Czochralski crystal growth is a very helpful tool in optimizing the real experiment to produce high-quality single crystals in microelectronic industry. With the increase of silicon crystal size by Czochralski method, the melt volumes and the corresponding crucible diameters have to increase as well, which make the melt flows become more and more turbulent, laminar model is no longer suitable, so a low-Reynolds number k-e model of Jones and Launder considered Lorentz force were used to describe a large-scale crystal growth process in this paper. The SIMPLE algorithm was employed to couple velocity and pressure field. A finite volume method was employed and a staggered grid arrangement was used to avoid zigzag pressure field. A third order QUICK (Quadratic Upwind Interpolation of Convective Kinematics) was used to discretize the convection term in the governing equations. In order to get low defects crystal ingot, the growth interface should be kept flat or slightly concave. This means that the isotherm at the growth interface is approximately perpendicular to the growth direction, the isotherm should be flatter, and the small temperature gradient at the interface is suitable. While the temperature gradient is strongly influenced by the flow pattern in the crucible, the strength of the magnetic field, the crystal and crucible rotations are the three major factors in determining the melt flow pattern, in this study, we aim at optimizing this three design variable to control the temperature gradient at the interface. Concepts and techniques of response surface methodology (RSM) have been widely applied in many branches of engineering, especially in the chemical and manufacturing areas. The basic idea of RSM is to approximate the actual state function, which may be implicit and/or very time-consuming to evaluate, with the so-called response surface function that is easier to deal with complex problem. This paper presents an application of the methodology with a turbulence model to optimize the temperature gradient. The simulation demonstrates that the response surface methodology is a feasible algorithm for the optimization of the Czochralski crystal growth process. REFERENCES 1. Lipchin A, Brown RA. Comparison of three turbulence models for simulation of melt convection in Czochralski crystal growth of silicon. J. Cryst. Growth, 1999; 205: 71-91. 2. Jones WP, Launder BE. The prediction of laminarization with a two-equation model of turbulence. Inter. J. Heat and Mass Transfer, 1972; 15: 301-311. 3. Redhe M, Forsberg J, Jansson T, et al. Using the response surface methodology and the D-optimality criterion in crashworthiness related problems. Struct. Multidisc. Optim., 2002; 24: 185-194.

— 330 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Active Vibration Control Analysis of Piezoelectric Intelligent Beam Tao Wang, Rong Qin *, Guirong Li Department of Architecture and Civil Engineering, Guangxi University, Nanning, 530004 China Email: [email protected], [email protected], [email protected] Abstract The new theory and new method of the analysis of intelligent structures are studied in the paper. With piezoelectric intelligent beam as a example: the spline dispersing dynamical equations of the piezoelectric intelligent beam is given by the spline finite-point method, from the displacement modal, basing on intelligent constitutive relation theory, instantaneous variational principle and spline dispersing. Then the control equation of the active vibration control analysis of the piezoelectric intelligent beam is get. In the paper, considering the influence of transverse shear deformation in the geometric equations, using the intelligent linear elastic constitutive relation of piezoelectric materials and the end loop feedback control. Modeling process include: spline dispersing, establishing the intelligent spline dispersing functional, establishing the intelligent spline dispersing dynamic equations and establishing the active control equations of the intelligent beam. Two typical examples are studied: one is the electro-mechanical coupling analysis of piezoelectric bimorph beam and the other is the active vibration control analysis of the intelligent beam with a single piezoelectric patch. The results are well closed to the theoretical solution and the experimental result. Using spline finite-point method could make the physical conception be definitude and the logic clear. Simultaneity, could deal with the stiffness matrix, load matrix and electro-piezoelectric matrix easily. The results indicate that spline finite point-point method used in the paper, with simply calculation and high precision, is a powerful and economical method. REFERENCES 1. Tzou HS. Development of a light-weight robot end-effect using polymeric piezoelectric bimorph. Proceeding of the 1989 IEEE International Conference on Robotics and Automation, 1989, pp. 17041709. 2. Qin Rong. Intelligent Sttructure Mechanics. Science Press, Beijing, 2005 (in Chinese). 3. Wang Shuhe. Electric-piezoelectric Analysis and Active Vibration Control Research of Piezoelectric Laminated Plate/Shell Structures. PhD Thesis, Tianjin University, Tianjin, China, 1997 (in Chinese).

— 331 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Interval Dynamic Analysis Using Interval Factor Method Wei Gao* School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, NSW 2052, Australia Email: [email protected] Abstract The dynamic analysis of structures with uncertain parameters is a very significant research field in engineering [1-3]. Recently, the interval analysis method has been used to deal with the static response, natural frequencies and dynamic response of structures with bounded uncertain (interval) parameters [4-5]. In their studies, several important results have been obtained by using interval analysis and matrix perturbation techniques. In perturbation methods, the uncertainty of all the structural parameters are expressed as small parameters in the structural mass and stiffness matrices. These small parameters are not interval variables, but simply small values. Therefore, their methods can not reflect the effect of the uncertainty of the individual parameters on the structural analytical results. A new method called the interval factor method (IFM) for the natural frequency and mode shape analysis of truss structures with interval parameters is presented in this paper. Using the IFM, the structural physical parameters and geometric dimensions can be considered as interval variables. The structural stiffness and mass matrices can then be respectively divided into the product of two parts corresponding to the interval factors and the deterministic matrix. The computational expressions for the midpoint value (mean value), lower bound, upper bound and interval change ratio of the natural frequency and mode shape are derived from Rayleigh quotient by means of the interval operations. The influences of the change of the structural parameters on the natural frequency and mode shape are demonstrated by using truss structures. REFERENCES 1. Stefanou G, Papadrakakis M. Stochastic finite element analysis of shells with combined random material and geometric properties. Computer Methods in Applied Mechanics and Engineering, 2004; 193: 139-160. 2. Liu Q, Rao SS. Fuzzy finite element approach for analysis of fiber-reinforced laminated composite beams. AIAA Journal, 2005; 43: 651-661. 3. Moens D, Vandepitte D. A survey of non-probabilistic uncertainty treatment in finite element analysis. Computer Methods in Applied Mechanics and Engineering, 2005; 194: 1527-1555. 4. Chen SH, Yang XW. Interval finite element method for beam structures. Finite Elements in Analysis and Design, 2000; 34: 75-88. 5. Qiu ZP, Wang XJ. Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis. International Journal of Solids and Structures, 2005; 42: 4958-4970.

— 332 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Sensibility Analysis of Violin Plates Mario Razeto*, Claudia Staforelli, Gabriel Barrientos Departamento de Ingenieria Mecdnica, Universidad de Conception, Casilla 160 C- Conception, Chile Email: [email protected] Abstract The study of dynamic and vibratory behavior of violins considers the effect of the different parts that compose it. The top and back plates have great importance in the transmission of vibrations and the characteristic tone of the violin; therefore, the study is centered in their vibratory behavior, since the ideal form of the vibrational modes and frequency of violins of greater prestige is known, according to bibliography. Modal Analysis Experimental tests determine the vibrational modes with their resonant frequencies of the violin plates. A numerical model of the top plate is made by means of the finite elements method, to determine the influence of the different mechanical and geometric properties on its main vibrational modes and natural frequencies; with the objective of obtaining the expected acoustic response, making precise changes in their structure. The height in the arching of the plates is modified to obtain different effects on the frequencies and the vibrational modes. A sensibility analysis is applied on a finite elements model of the top plate, to determine the influence of the thickness in different zones on its natural frequencies, and to study the effect produced by the thickness variations of the bass bar. With an optimisation study the thickness distribution is modified to obtain the ideal natural frequencies and achieve the right vibratory behaviour, with a similar dynamic response as violins of greater prestige.

— 333 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Effect of Pier and Abutment Non-Uniform Settlement on Train Running Behavior Jianzhen Xiong *, Hanbin Yu, Mangmang Gao China Academy of Railway Science, No.2 daliushu Road, Haidian district, Beijing, 100081 China Email: [email protected], [email protected], [email protected] Abstract Dynamic behavior of running vehicle on bridge is influenced by the attached irregularity of track caused by pier and abutment non-uniform settlement. In order to study the relationship between them, vehicle-track-bridge vertical coupling system dynamic model is established, in which the four-axle car with two suspension systems on independent bogies is adopted, the bridge is simulated with various finite elements. The vehicle components are treated as rigid bodies, assembled with linear springs and dampers. There are totally 10 degrees of freedom, with car body and each bogie having 2 degrees of freedom of vertical and pitching movements, and each wheel-set having 1 degrees of freedom of vertical movement. As one part of track irregularity, the attached deflection is determined by static mechanics. Dynamic analysis uses time domain method, and nonlinear elastic hertz contact theory is adopted in wheel/rail vertical direction, at the same time the wheel/rail deviations can be considered. Based on ballast track and prestressed concrete box girder bridges with 40m span, the homemade Pioneer within 160~220km/h, pier and abutment non-uniform settlement with 5mm, 10mm and 15mm taking into account respectively, the car body acceleration and Sperling index corresponding to different settlement at various running speed are obtained, which can provide a reference to the practice engineering design of railroad bridge. REFERENCES 1. Zhang Geming. Vehicle-Track-Bridge System Dynamic Analysis Model and Track Irregularities Control on Quasi & High-Speed Railway. Doctor Thesis of China Academy of Railway Science, 2001 (in Chinese). 2. Gao Mangmang. Studies on Train-Track-Bridge Coupling Vibratiion and Runnablility of Train on High-Speed Railway Bridges. Doctor Thesis of China Academy of Railway Science, 2001 (in Chinese). 3. Zhai Wanming. Vehicle-Track Coupling Dynamics. China Railway Publishing House, 2002.

— 334 —

Author Index * denotes the presenter of the paper Agelet de Saracibar C

*247

Dai XL

330

Al-Falahi A

♦180

Dang XH

294

Amaya K

288

DasS

AnGP

330

De Bortoli AL

AokiS

191,285,288,323

Asami T

♦173

♦196 ♦184,243

Deeks AJ

255, 256, + 257

Denda M

♦270,271

Asayama T

195

DiSK

294

AuSK

206

Ding GF

300

Aubry D

209

Doltsinis I

♦35

BarnatW

223

Dong MC

258,272,313,314,315

Barrientos G

333

Dong ZZ

Bento R

230

DuY

♦251

Brou5ek M

295

DuYF

♦294

CaiMF

245

DuZF

202

♦250,251

Duan J

♦215

Eisenberger M

♦329

CenS CenZZ

219,265

Cervera M

247

Cescotto S

*1

FanWB

179

FengX

ChaiG Chai JR

♦291

Ezawa Y

Feng XQ

Chan IN

190

FuDJ

Chan SL

258

Fujisawa T

Cheang TS

♦258

Chen CS

113

Chen HB

*261

Chen JG

185

280

274 175 ♦232 268 261 199,200

Fujita D

200

GaoH

268

GaoMM GaoSX

♦229, 334 1

Chen MC

♦231,234

Gao W ( 1 )

♦275

Chen TG

214

Gao W ( 2 )

♦332

Chen WQ

♦327

Ghajar A J

244

Chen XM

250

Gong YB

♦301

Chen XY

♦307

Gong YQ

♦216

ChenY

♦210

Graca JM

Chen YQ

185,*194

GuBQ

♦316 ♦186,210,232,306

Chen ZP

235

GuoXF

261

Cheng YL

283

GuoYT

175

ChewYT

♦181

Chiumenti M

247

Habte MA HanDJ

Choi HF

♦170,172

HanF

Choi LY

♦241

HanX

59 ♦51,286 273 71

Chung MS

248

He EM,

Cong GH

♦187

HeWY

176

273

Ho A N

♦312,314

CuiJZ

— 335 —

324

HoSM HoiKI HongH HouSJ HuWD Huang CK Huang D Huang JJ Huang Y HungMC Ikeda T Inaba M IribeT IuVP JanYJ JiB Jiang JJ JinXQ JungWS Kakimoto K Kanda Y Kandasamy R KangC KangZ Kaufmann N Kawai H Kee BBT Keong WS Khalili N Khatchatourian O KimJM Kiong SC Kobayashi Y Kou KP Kuklik P Kurowski Z Lackner R Lalitha R LamLH Lamas LN Lao MI LaoSK LeiSL LeiT LeiWJ Leitao NS Leong KS

♦290 *259 256, 257 ♦217 202 214 ♦267 181 *218,226 318 88 263 200 326 248 268 204 277 *233 266 200 ♦242 ♦169 35, *308 *320 200 71 241 ♦59 243 153 241 *262 ♦197,326 *295 205 *168 *317 *276 183* *190 170,* 172 190 19 319 183 *313,315

LiBS LiGR LiGY LiH LiHT Li J LiK LiLC LiQ ( 1 ) LiQ ( 2 ) LiQY LiT LiW Li XL LiXQ LiXS LiYP LiYQ LiYZ LiZL LiZY Liang SJ Liang T Liang ZZ Liao XT Lin FY LinJG LinSP Lindig V LiuB LiuD LiuGR LiuR LiuWK LiuX LiuXM LiuY ( 1 ) LiuY (2) LiuYH (1) LiuYH (2) LiuYL LiuYM LiuYN LoKM Lobo Ferreira JP Lok TMH Loktev AA

— 336 —

325 331 71 294 182 287 216 *198 217,302,307 *326 301 225 *211 *260 213 304 314,315 219 194 *188, 189 203 *248 305 198 *302 *220 174 113 *284 309 169 *71 232 *87 221,236 269 *185 305 214, *265 *175 *245 265 *282 1 290, *297, 298 322

Lokteva IA Long SY Long YQ Lopes LS Lopes N LouCW Lourenco N LiiCF, LuXZ LuoDM Luo JJ LuoYF Malachowski J Maranha JR Matsubara H Matsuo Y Mellor BG Meng JJ MengWY MiZK Miao ZW Mirambell E Miyazaki N Mohite PM MokKM Murotani K Nagaoka S Nakabayashi Y Nakanishi T Nielsen MP Niezgoda T Nogueiro P Okada H OkudaH Oliveira F Osaki H PanE PanJY Patricio J Peng XH Peng ZR Pereira E Periasamy K Petry VJ Ping XC Qian HS QieYH

*322 217,307 250 316 *212 ♦272 1 327 *204 *266 51 193, 218, *221, 224, 226, 236 *205 *293 15,201 *177 237 *303 ♦213 *296 204 212 *88 222 259, 276 *199 *263 173, 177 ♦178 179 205, *223 230* *200 260 312,*314 *201 271 229 316 *182 303 228 242 ♦243 231,*234 281 309

QinR QinY Razeto M Rebelo C RenJG RenQW RenQX RidhaM RigoN Rigueiro C Rodrigues HC Rodrigues J Roubens M SangJB Santos LO SatoM Sekar BD Shao YB Shen ZJ Shen ZY Shiau J Shibata H Shijo K Shimamura S ShuC Shufrin I Simoes da Silva L SinVK Sivagnana Prabhu KK Song DP Song HJ SongY Song ZT Staforelli C SugaK SuiYK Sun HI SunLS SunQ SunZ Suzuki H Swoboda G Syngellakis S Takagaki M Takao Y TamHK TamLM

— 337 —

♦331 300 *333 ♦328 208 292 *214 ♦191 ,285,288 1 328 ♦101 238 1 ♦309 238 178 258 *202 296 227 ♦192 178 200 *323 181 329 212,230,328 170,171, 172 242 250 193,*224 294 289 333 *285,288 311,330 *171 ♦299 279 203 *274 219 237 *195 266 ♦244 170, 171 ,172,244

WeiWL

*176

TanT

♦304

Williams FW

*168

TanYP

♦286

Wong CK

244

TamSC

Tang CA

198

Tang CW

313,*315

Tang RW

227

WongF

272 312,313,314,315

WongKI

171

Woon OH

241

Tian ZR

*203

WuAH

♦237

Tian ZS

♦252,253

WuBX

*281

WuC

♦321

WuJY

♦287

TieB

*209

ToiY

195,233

Tong LW

225

WuQX

*207

TouCM

297

WuZR

*283

Upadhyay CS Valliappan S

♦222 59

1

Xiang XM Xiang ZH

2 1 1 , 2 1 5 , *219

Vieira A

293

Xiao JC

♦305

Vieira R

*228

Xiao LJ

235

212

XieJL

234

228 ♦277

XieNG

299

XieZH

*174

VuTH

*255

Xing SF

309

WanS

♦179,281

Xiong JZ

*334

Vila Real PMM Virtuoso F Vong SW

XuCL

Wang AP

*253

Wang CY

270, *271

XuM

Wang DD

254, *264

XuXQ

187

Wang FJ

♦324

Wang HJ

XueMD Yagawa G

♦193,224 ♦238 ♦325 211,215 ♦15, 199, 200, 201, 260, 262, 263

(,)

19

WangHT ( 2 )

♦235

Yang LC

309

WangJ ( 1 )

*206

Yang M

239

WangHT

Wang J

(2)

(3)

245

YanSL

♦239

169

Yang MG +

224, 240

♦318

Yang MW

Wang J ( 4 )

330

Yang MZ

207

Wang JG

*300

Yang QP

252

Wang JX

♦246

YangR

185

WangK

♦225

Yang XJ

Wang MY

♦310

Yang YB

Wang PB

19

Yang YQ

Wang SL

283

Yang ZJ

Wang SM

*278

Yang ZQ

Wang SR

245

Yang ZX

WangT

331

Yanovsky YuG

WangW

277

YaoZH

♦19

Wang WX

266

Yasuyoki K

199

Wang XC

246

YeHL

Wang XD

1

Ye LP

Wang ZB

221, *236

WangJ

Wei DM

*319

Yoshida M YuHB

— 338 —

217,307 ♦113 229 ♦256, 257 303 282 ♦129

♦311 204 ♦288 334

YuHJ YuHP YuR YuTT YuY Yuen KV YunCB YusafT YusoffMZ Zhang AQ Zhang C Zhang CH Zhang JL Zhang K Zhang L Zhang PQ Zhang SJ Zhang T Zhang WG Zhang WH Zhang XQ Zhang Y

*298 *330 218,*226 *292 *273 ♦139,206,259 *153 180 180 1 308 *254 254 280 187 261 *208 269 302 299 252, 253 289

Zhang YJ Zhang YS Zhang YX Zhang ZM Zhao HM ZhaoKZ ZhaoXZ Zheng HW Zheng Y Zhong ZH ZhouJF(1) ZhouJF(2) ZhouJT(1) ZhouJT(2) Zhou MS ZhouRZ ZhuBQ ZhuBS ZhuLJ Zhuang Z Zhuo JS Zou J

*279 278 188, *189 *289 *280 207 *227 181 *168 71 249 *306 1 174 219 321 *249 189 186 ♦269 213,249,267 *268