a boundary element formulation for design sensitivities in problems involving both geometric and...

hfathl Comput. Adodelling Vol. 15, NO. 3-5, pp. 245-255, 1991 0895-7177191 $3.00 + 0.00

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A BOUNDARY ELEMENT FORMULATION FOR DESIGN SENSITIVITIES IN PROBLEMS INVOLVING BOTH GEOMETRIC AND MATERIAL NONLINEARITIES

SUBRATA MUKHERJEE

Department of Theoretical and Applied Mechanics, Cornell University

Ithaca, NY 14853

ABHIJIT CHANDRA

Department of Aerospace and Mechanical Engineering, University of Arizona

Tucson, AZ 85721

Abstract-A BEM formulation for the determination of design sensitivities for shape optimization

in problems involving both geometric and material nonlinearities is presented in this paper. This approach in based on direct diflerentiation (DDA) of th e relevant BEM formnlat,ion of the problem.

It retains the advantage of the BEM regarding accuracy, while avoiding strongly singular kernels. This approach provides a new avenue toward efficient shape optimization of elastic-viscoplastic or elastic-plastic problems involving large strains and rotations.

1. INTRODUCTION

Optimal design of structural shape is typically carried out by nonlinear programming methods. Such algorithms require repeated iterations on the shape of a structure and the solution of the appropriate boundary value problem at each stage. Even for problems of linear mechanics, such as small strain elasticity, this process can be extremely computer intensive. Problems involving material and geometric nonlinearities are much more complicated. The rewards for success in solving such nonlinear problems, however, are also immense, since such a technique can be applied to very important manufacturing processes, such as metal forming or metal cutting.

This paper presents a formulation, based on the boundary element method, for accurate determination of design sensitivities in problems involving both geometric and material nonlinearities. Analytical expressions for design sensitivities are obtained, which makes the proposed approach amenable to incorporation within a design synthesis strategy and provides a new avenue toward rational design of bodies undergoing inelastic deformations involving large strains and rotations. Thus, for example, optimal design of die shape for extrusion or of a pre-form for forging becomes feasible.

A crucial ingredient for obtaining successful and economical solutions to such optimization problems is the accurate determination of design sensitivities, i.e., the rates of quantities such as displacements, stresses, nonelastic strains, or other state variables in a deforming body with respect to a shape design variable. Of concern here are problems where the shape of the structure itself, rather than the variable such as cross-sectional dimensions or thickness for a fixed shape, is the design variable of interest. Accurate determination of design sensitivities is complicated enough for linear elastic problems. For nonlinear problems, where the design sensitivities are themselves history dependent, the challenges are much more formidable.

A large amount of literature exists on this important subject area. In a paper such as this, it is very difficult to acknowledge all the worthwhile contributions in this field. Instead, the reader is

This research has been supported by Grant No. DMC-86573.45 of the U. S. N a ional t Science Foundation to the

University of Arizona and by the U. S. Army Research Olllce through the hlathematical Sciences Institute at Cornell Univemity. hlnkherjee also acknowledges the hospitality of the University of Arizona during his sabbatical

visit there.

Typeset by A,&-TEi\;

245

246 S. MUKHERJEE, A. CHANDRA

referred here to a. comprehensive recent book by Haug e2 al. [l]. This book presents various kinds of optimization problems with linear material response. Very recently, Tsay and Arora [2] have used FEM analysis to obtain design sensitivities in nonlinear structures with history-dependent effects, and Mukherjee and Chandra [3] have obtained a BEM formulation for design sensitivities in small strain elastic-viscoplastic problems.

In general, two methods emerge as the most powerful ones for the determination of shape design sensitivities. These are the Direct Differentiation Approach (DDA) and the Adjoint Structure Approach (ASA). The DDA typically starts from a variational equation like the principle of virtual work (e.g., [2]) or from boundary integral equations (e.g., [3-51). Such an equation (or its discretized counterpart) is differentiated with respect to the design variables, and the resulting equations are solved in order to obtain the sensitivities. The ASA, on the order hand, defines adjoint structures whose solutions permit explicit evaluation of the sensitivity coefficients. Again, variational equations, together with the finite element method (FEM, e.g., Haug e2 al. [l]) or

boundary integral equations (BIE, e.g., Mota Soares et al. [G]; Kwak and Choi [7]), have been used.

In their recent work on sensitivity analysis for nonlinear problems, Tsay and Arora [2], starting from the principle of virtual work, discuss the relative merits of the DDA and the ASA and suggest a hybrid method that combines the advantages of the two. They also present some analytical examples and present FEM discretizations of the general equations.

The DDA, in conjunction with the BEM, provides an extremely elegant approach toward the determination of design sensitivities. Barone and Yang [4] start with the usual BEM equations of elasticity of Rizzo [8] and differentiate them directly with respect to a design variable. What is remarkable here is that this differentiation process does not increase the singularity of the relevant kernels. Thus, while for two-dimensional problems one usually starts with kernels that are In r and l/r singular (r being t,he distance between a source and a field point), the differentiated kernels are regular and l/r singular, respectively. The displacement sensitivities can be evaluated directly by solving the same matrix equations as for the regular BEM, with a different right-hand side. The situation regarding stress sensitivities on the boundary, however, is not as simple, and one must deal with kernels that are l/v singular. These singularities cannot be removed with rigid body modes and other methods, such as the finite part integration of Kutt [9], must be employed here.

This paper presents a BEM formulation for obtaining design sensitivities for elastic-viscoplastic problems with large strains and rotations. Elastic-plastic problems with coventional rate- independent material models can also be solved by this approach. The BEhZ formulation of Mukherjee and Chandra [3] f or materially nonlinear problems is extended here to include geometric nonlinearities as well. The ideas presented here have two important features. The first is that this formulation starts from a recent BEhl elasticity formulation [lo-111 using tractions a.nd tangential derivatives of displacements (in two dimensions) or displacement gradients (in three dimensions) as primary boundary variables. Remarkably, this new approach allows the determination of bounda y stress sensifivities, for two-dimensional elasticity problems, from diflerentiated

kernels that are com,pletely regular. The other important idea is the inclusion of material and geometric noulinearities. It is shown that the BEh1 can be systematically applied to obtain sensitivities of quantities of interest, such as displacements, stresses, nonelastic strains, or other state variables, anywhere in the solid with respect to changes of initial shape. These sensitivities are now history dependent. The present formulation is based on earlier work on small strain elastic-

viscoplastic problems (e.g., [3, 12-131) and follows the formulation of Chandra and h,lukherjee

[14-161 for large strain problems of elasto-viscoplasticity. The sensitivity analysis uses many of the matrices of the original large strain elastic-viscoplastic problem. It is estimated that the SO-

lution of the complete sensitivity problem would require approximately twice the computa,tional effort of solving the original elastic-viscoplastic problem involving large strains and rotations.

2. IZ BEhI FORhIULATION \I’ITII DISPLXCERIENT DERIVATIVES

Ghosh et al. [lo] and Ghosh and Mukherjee [ll] 1 iave proposed a BEhI formulation for linear

elastic problems in which either the t.angentia.1 derivatives of displacements (for two-dimensional

A boundary element formulation 247

problems) or the displacement gradients (for three-dimensional problems), together with tractions, are the primary boundary variables. Once the boundary problem is solved, this formulation allows boundary stresses to be determined directly from Hooke’s law without further differentiation. Thus, it provides excellent accuracy for boundary stress calculations and is expected to be very useful for determining stress sensitivities. Mukherjee and Chandra [3] also present an extension of this formulation for small strain elastic-viscoplastic problems.

The two-dimensional plane strain formulation for elastic-viscoplastic problems involving small elastic strains but large inelastic strains and rotations is discussed in this paper. For metallic bodies, the components of elastic strain are generally limited to about 10e3, since the elastic moduli of metals are typically about three orders of magnitude larger than the yield stress. Thus, in metal forming or metal cutting operations, the nonelastic strain components, which can be of the order of unity, greatly dominate the elastic strains.

Bounday Equations

Using an updated Lagrangian frame [14-161, the rate form of the BEM formulation [3, 14-161 may be written as (i,j, L = 1,2),

0 = / [~ij(WW)~~L)(b.Q) - Nj(U=,Q)$,Q)] ds t3B

+ /, [XUij,i(b, P, q)d”)(h q) + 2GUij,k(h P, dd:;)(h q,] CM

+ J

Uij,m(b, P, q)gmi@, q)dA, B

where

d(“) = dp + dg + d$.

(14

(lb) Here, the cross section B of the body has the boundary dB in the 21 - 12 plane in a refer-

ence configuration; vi, dvi/ds, and d$) are components of velocity, its tangential derivative, and nonelastic deformation rate respectively; and b denotes the design variable. The quantity gnai in the last domain integral may be expressed as [14-161

where

gnai = Umkwki + dmk~ki - Umidkkr (lc)

dij = i(vi,j + vj,i), (14 and

Wij = $(Ui,j - Vj,i).

dL) denotes the components of Lagrange traction rate and is denoted as [14-161, 1

(le)

with

(‘I ri = 7lnBmd = Tic) - ?2,,,(bnakWki + dmkUki - Umidkk), I W)

where rjc) denotes the components of Cauchy traction rate. Bere, sji and uji represent the components of Lagrange stress and Cauchy stress, respectively. A superscribed dot (.) denotes a material rate and a superscribed hat ( - ) d enotes a corotational or Jaumann rate.

It should be noted that, for some problems, the physics of the problem may require prescrip- tion of velocities on a portion 8Br of dB. Conversion of these velocity boundary conditions to

MCM 15:3/5-Q


prescriptions of tangential derivatives of the velocity on dBr may lead to loss of information on the velocity itself. This may lead to loss of uniqueness of the solution as obtained from this formulation. In such cases, this difficulty may be resolved by using an additional constraint equation of type

(lh)

where A and B are suitably chosen points on the boundary dB.

The kernels, written in terms of a source point p (or P) and a field point q (or Q) (where upper case letters denote points on the boundary dB and lower case letters denote points inside the

body B) are [lo]

Uij = -

1

8n(l- v)G [(3 - 4Y)ln TSij - r,ir,j],

CVij = 4,(1’_~)[~(~ - v)4Sij +YikT,jT,k + (1 -2v)yjilnr],

(24

where v and G are, respectively, the Poisson’s ratio and the elastic shear modulus; r is the distance between a source and a field point; and 4 is the angle, measured in an anticlockwise sense, between the line joining P and Q and the line through P parallel to the zi-axis. Finally, 6ij is the Kronecker delta function, 711 = ~2s = 0,712 = --y21 = 1, and a comma denotes a derivative with respect to a field point. It is important to note that both of these kernels are only lnr singular. It is also interesting to note at this point that equation (1) contains velocity gradients (instead of velocities) and traction rates on dB and in B.

Stress Rates on the Boundary

Once the boundary integral equation is solved in the usual way [lo, 14-161 and ~1~) and &i/as are obtained on dB, the Jaumann rates of stresses on dB may be calculated from the following algebraic equations (i, j, Ic = 1,2):

dvi ds = Vi,jtjr

eji = X(Vk,k - d(“))bij + C(Vi,j + Vj,i) - 2GdiT).

(3b)

There are seven equations for the seven unknowns ui,j and dij [d$’ in assumed to be known

at any time from an appropriate constitutive model]. While this so-called boundary algorithm is well known [17], the important difference here is that &i/as is now a primary variable and need not be computed by tangential differentiation of the boundary velocity field. Thus, no further inaccuracy is introduced by numerical differentiation and the boundary stress rates are as accurate as the boundary traction rates. This fact has very important consequences for sensitivity analysis, as will be discussed later.

The material rate of Cauchy stress can be obtained as

oij = 6ij - aikwkj + Wikakj. (34


Internal Equations

Velocity gradients and stress rates are also needed at the internal points. To this end, the version of equation (la) at an internal point [3, lo-111 is differentiated at a source point xc(p) to

yield

vj,.db, P> - vj,f(b, ii) = J

t3B

a + h(P)

- J [XUij,i(b,P, q)d(“)(b, q> + ‘JGUij,k(b,p, q)di)(b, q)] dA (4)

B d

- + WP) J

Uij,m(b, Pt q)grni(b, q)dA, B

where ,j z a/ax,(p) and p is the image of point P. The point p is chosen to be the point of intersection of dB with the line parallel to the global xi-axis through p,

The boundary kernels are regular as long as p is an internal point and Q is a boundary point. This, however, is not the case for the domain integral, where a kernel which is l/r singular must be differentiated again. It appears best to treat the domain integral using the technique of Huang and Du [18]. The final form of the resulting equation, which is used instead of (4), may be written as (i,j,k,C = 1,2),

vj,i(b, P) - vj,i(b, p) = J[

Uij,i(b,P, Q)T~~) (by Q) - Wij,i(b,P, Q)%(b, Q)] ds 8B

- ~dn)(b, PI J Uij,i(b, P, Q)nt(b, Q>ds 8B

- 2G&)(b,P) J

Uij,k(bj P, Q)nt(b, &Ids

aB

- gmi(b,p) J

Uij,m(b,P, Q)nt(b, &Ids BB

(5)

+ lim 17-O J X&&P, d[d(“)@, q> - ~“)(b,p)ldA B-B(v)

+ lim fI+O J Uij,mdb, P, q)bni(b, q> - Smi(b,P)ldA* B-B(v) Since [d(“)(b, q) - d(“)(b,p)], [d$‘(b, q) - d$)(b,p)], and [g,i(b, q) - g,i(b,p)] are of the order of r, the domain integrals are now only l/y+ singular. Rajiyah and Mukherjee [19] also present

alternate ways of treating these differentiated domain integrals. Finally, the stress rate components at an internal point may be easily obtained from the

velocity gradients and nonelastic deformation rates using the assumption that the elastic field in the problem obeys the law of hypoelasticity given in equation (3~).

3. DESIGN SENSITIVITIES FOR SHAPE OPTIMIZATION

Boundary Equations

Following Mukherjee and Chandra [3] and Barone and Yang [4], the first step is the differentiation of equation (la) with respect to a design variable b. Let a superscribed asterisk (*)

250 S. MIJKHERJEE, A. CHANDRA

denote the design derivative (w.r.t. 6) of a variable of interest and a superscribed circle (0) and a superscribed square (Cl) denote design derivatives of its material rate and of its Jaumann rate,

respectively, in the original configuration (at zero time) [i.e., (;ij = d/db(aij),$ij = d/db(irij),

and iij = d/db(dij)]. N ow, one obtains the equation (i,j = 1,2),

o= Uij(6, P,Q);iL'(6, Q) - Wij(6, P, Q)$(b, Q) 1

ds

+ J [iTij(a,P,Q)~~")(6,Q)-~ij(6,P,Q)~(6,Q)] ds i3B

+ J[ Uij (6, P, &)~i(~) (6,Q)-W,j(6,P,Q)~(6.Q)] dg BB

+ J[ AUij,i(b, P, q)i(“)(b, n) + 2GUij,k(b, P, *)4;)(6, n) I

dA

B

+ J[

Abij,i(b, P, q)d”)(b, q) + fJGcii,t(6, P, q)d$,“)(b, q) 1 dA

B

+ J [xuij,i(b, J’, q)d(“)(b, ‘1) + 2GUij,r(b, PI q)d;)(b, q)] di

B

+ I[

Uij,m(b, P, q)Gmi(b, ‘I) + cij,rn(b, P, q)grni(b, q) 1 dA

B

+ J Uij,m(b, J’, Q)Smi(b, q)di*

B

Here, 6 is defined in the original configuration (at t = 0) only. Hence,

(64

Gij(69 P,Q) = Uij,k(b, P,Q) Fkm(Q)im(Q) - Fkm(P);m(P) I

7 (6b)

kij(6, P, Q) = Wij,k(b, P, Q) [

Fkm(Q)Jim(Q) - Fkm(P)im(P) I

7 (6~)

cij,k(b, P, Q) = Uij,kn(b, P, Q) [

Fnm(Q);m(Q) - F,m(P)im(P) I

1 (64

and

kij,s(6, P, Q) = Wij,kn(b, P, 0) [ Fnm(Q)im(Q) - F,m(P);m(P) I 9 VW

where X, and x, are coordinates of a material particle in the original configuration (at t =

0) and at a reference configuration (at any time t, after updating), respectively. Fk,,, is the

deformation gradient at any time t and , k E a/axk(Q). The design derivative of the Lagrange traction rate may be expressed as

‘CL) ‘i =Tlm?mi + Gm&mi - (;l,,,b,,~Wki+n,(;,kWki+ %a~rnk~ki)

-(kndmkuki +nmZn~~ +nmdmtcGki) (W

+(&numid~!f +nm~midak + nrnurni ;i u),

where

I

nm = nm,t(Q)F~t(Q)k~(Q)j (64


(69

d;l = ;z’k,k(q)dA, (6.i)

* xi = -xk = Fjkik, 8Xi *

axk (6k)

and

li,j = Fik,j;k. (60

At the start of the time step, half of the sensitivities aGi/as and gi”’ are to be determined, while

all the rest of the quantities in equation (6a) are known. r,!L’

known from a solution of the regular large strain BEM problem at dvi/ds, d(v) and gmi are

this time, anl3d ihe sensitivity

of the nonelastic deformation rates are known from differentiating a constitutive model with state variables, such as [3, 121

(7a)

to yield

Qij -(~) = ham’ (bij , *~~‘), (7b)

and

y(k) Bh$) * ah!;) ecp)

ij = -fl,nt + -q-pntr &Tat

(7c)

(74

in terms of the sensitivities of the stress and state variables. It should be noted here that derivatives of the coordinates of the boundary, as well as internal

points in the original configuration, with respect to a design variable b must be evaluated in order to solve equation (6). Solving equation (6) also requires determination of n,,k(Q) and

Fik,j, which will have to be determined numerically. Finally, for large strain problems, r,!“’

and gmi contain velocity gradients, whose design sensitivities are not known a priori. Thus, like regular large strain problems, iterations will be needed in solving equation (6).

Sensitivities of Stress Rates on the Boundary

The boundary algorithm for sensitivities corresponding to equation (3) is obtained by differ-

entiating these equations with respect to the design variable b. The resulting equations are (i, j, k = 1,2; equations valid for P E aI?)

a;i = 1;. .t +v. .;.

8S 8,~ i ZJ I,

and

(8’3)


Yji = A(Ck,k - &“))6ij + G(Gij + ;j,i) - 2G3;‘. (8~)

There are seven scalar linear algebraic equations for seven unknowns ;i,j and ?ij on 8B. The sensitivity of material rate of Cauchy stress may be obtained as

r

0 0 * aji = aji - ffi&‘kj - ai$‘kj + ;ikbkj + WikGkj e

(84

Internal Equations

A sensitivity equation from (5) for the sensitivities of velocity gradients (and hence stress rates) at an internal point may also be written as (all indices 1,2)

;j,i(b,P) - ;rj,i(b, P) = Uij,i(b,P, Q);{L’(b, 0) - wij,i(b,P, Q)%(b, Q) 1 ds

+ i: [ Uij,i(b, P, Q>rjL'(b, 0) - Wij,i(b, P, Q)s(b, Q)] dH

- A”)(b,p) s Uij,i(b, P, Q)nt(a, Q)ds aB

- Xd(“)(b,p) J

cij,i(b,p, Q)nt(b, Q)ds BB

- Ad(“)@, p) J

Uij,i(b,p, Q)Gt(br Q)ds 8B

- Ad(“)@, p) I

*

uij,i(b, P, Q)nt(b, Q>ds

- 2G&b,p)~B Uij,k(b, P, QM4 QP

- PG~;)(~,P) J,, fiij,k(b,Pr Q)nt(b, Q>dS

- 334;)@, PI J,, Uij,k(b,p, Q)ct(br Q)ds

- 2Gd$)(b, p) J

uij,k(b,P, Q)nt(b, Q)d H BB

- imi Cb7 P> J Uij,m(b, P, Q>nt(a, QWS BB

- gmi(b,p) J

aB cij,rn(b,p, Q)nt(b, Q)ds

- Smi(b, P) J uij,m(b,P, Q);L(~, &Ids BB

- gtni (b, PI I

uij,nz(b,~, Qh(4 Q)dB ao

lim J Afiij ii(b,P, q)[d(")(b,q)- d(“)(b,p)ldA B-B(?)) ’

J q+' B--B(q) ;(")(',P)ldA

+ lim J ‘t-O B-B(q) AUij,,(a, p, q)[dcn)(b> Q) - Jn)(‘, P)]d’

and on 33

upon Integrate conver~mcr

-D I I to t+At

A boundary element formulation

_ ~I_~1

_E]_F] Figure 1. Solution algorithm.

253

f lim J q-0 B-L?(q)

2Giiij,kr(brP! *)[d,(k'(bYq)- d,(;'(b> PIIdA

+ lim J q-0 E-B(a)

2GLl,j,kr(b,p,q)[~S,"'(',q)- 'i:)(',~)l'A

f lim J 7-O B-B(q)

2G&,&, p, q)[&@, 4) - &(4 ~16

+ lim J

r;

69

7-o B-B(t)) ij,mZ(b,P, P>[SmiCb> q> - kaiCb,P)ldA

+ lim J 7-O B-D(q)

Uij,&Cb, P, q)GTTli(b, 9) - ?7nitb,P)ldA

+ lim J v-0 u-l3(1/)

Uij,mr(b,P, Q)[gmi(‘, q> - gmi(‘,P)I”*

Although this equation (9) is long, it may be easily evaluated. The boundary kernels are

regular and the domain integrands are l/r singular. These domain integrals can be accurately

evaluated by standard means (e.g., [12]). Tl ie entire right-hand side of equation (9) is known at

this stage except for the integrals involving zt,li, which depend on ti,j. Accordingly, iterations are needed over equations (6) through (9). These iterations are similar to those needed over velocity gradients for BEi\ analyses of large strain problems.

Finally, the stress rate sensitivities at an internal point may be obtained from equation (8~).

4. SOLUTION RLCORITIIM

The solution algorithm for large strain elastic-viscoplastic problems, which involves solutions

of appropriate equations at the beginning of each time step and then marching forward in time, is discussed in detail in several previous papers by Chandra and Mukherjee [14-161. Iterations

are needed, since the velocity gradients appearing in a domain integral and through (L) Ti in the

boundary integral in equation (1) are not known a priori. Then sensitivity calculations must also


be carried out using an analogous procedure, in parallel with the regular calculations. Iterations over sensitivities of velocity gradients are now needed. The algorithm for the sensitivity problem is described below.

The solution for sensitivities at the initial time is obtained by solving the appropriate elasticity equations. Figure 1 illustrates the procedure for moving to t + At when the solution up to time t is known. The REM equation (6) is first solved in order to obtain the unknown values of

ati/as and ri *CL) on dB using an estimated value for sensitivities of velocity gradients. Now, the

boundary algebraic equations (8) are used to determine ii,j and ti,j on dB. Next, the internal

BIE equation (9) is used to obtain Gi,j at selected internal points, and the law of hypoelasticity

(3~) is used to obtain the stress rate sensitivities at these internal points. The new values of ;i,j on dB and in B are now incorporated into equation (6), and the system of equations (6-9) is solved with the new right-hand side and iterated until convergence. It is important to note here that the kernels and the coefficient matrices need not be altered during iterations. The rates are integrated to obtain the sensitivities of the relevant quantities at time t + At. The constitutive

*(n) equations (7) are now used to calculate dij at t + At from the values and sensitivities of the

*(n) stresses and state variables at this time. For classical elastic-plastic material models, dij depends

on ?ij, as well as on the stress components and their sensitivities. This requires iterations over

$T) in each time step. The sensitivity problem, however, still has approximately the same level of complexity as the original elastic-plastic problem. For large strain problems, iterations over

i$) may be carried out within the iterations scheme for ii,j. Thus, the large strain sensitivity problems of elasto-plasticity and elasto-viscoplasticity are

expected to require approximately twice the computational effort needed for the regular BEM analysis including both geometric and material nonlinearities, when sensitivity with respect to one design variable is needed. In a typical design environment, however, sensitivities with respect to a large number of design variables are desired. It is interesting to note here that the determination of sensitivities with respect to additional design variables does not require solutions of new matrix systems. The coefficient matrices remain the same for all cases. Only the right-hand side changes. Hence, for the slight increase in additional costs due to additional evaluations of the right-hand side, it is possible to simultaneously track the sensitivities with respect to several design variables.

5. DISCUSSION

This pa.per presents a BEM formulation for the determination of design sensitivities for problems involving both geometric and material nonlinearities. The material model can be elastic- plastic or elastic-viscoplastic. The sensitivity formulation for two-dimensional problems is based on boundary kernels which are completely regular and domain kernels (even for the strain rate sensitivity at internal points) which lead to integrads that are only l/r singular. Also, the equations for the sensitivities are exact mathematical expressions obtained by analytical differentiation of the relevant integral equations. Thus, this approach combines the usual accuracy advantages of

the BEM without the difficulties of dealing with strongly singular integrads. Numerical result,s using this formulation are currently being determined and will be reported in the future.

REFERENCES

E.J. Ilaug, K.K. Choi and V. Komkov, Design Sensitivity Analysis of StractuLral Systems, Academic Press,

New York, (1986). J-J. Tsay and J.S. Arora, Design sensitivity analysis of nonlinear structures with history dependent effects,

Technical Report No. OLD-88.4, Optimal Design Laboratory, College of Engineering, The University of Iowa City, Iowa, (1988). S. Mnkherjce and A. Chandra, A boundary element formulation for design sensitivities in materially nonlinear problems, Acta Afechanica 78, 243-253 (1989). MB. Barone and R-J. Yang, Boundary integral eclnations for recovery of design sensitivities in shape optimization, AIAri J. 26, 589-594 (1988).


5. J.H. Kane and S. Saigal, Design sensitivity analysis of solids using BEM, ASCE J. Engrg. Mech. 14,

1703-1722 (1988).

6. C.A. Mota Soares, R.P. Leal and K.K. Choi, Boundary elements in shape optimal design of structural

components, In Computer Aided Oplimal Design: Slructural and Mechanical Systems (Edited by C.A.

Mota Soares), NATO AS1 Series, Springer Verlag, Berlin, (1987).

7. B.M. Kwak and J.H. Choi, Shape design sensitivity analysis using boundary integral equation for potential

problems, In Computer Aided Optimal Design: Structural and Mechanical Systems (Edited by C.A. Mota

Soar&), NATO AS1 series, Springer Verlag, Berlin, (1987).

8. F.J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Q. Appl.

Math. 25, 83-95 (1967).

9. HR. Kutt, Quadrature formulae for finite part integrals, The National Research Institute for Mathematical

Sciences, Pretoria, South Africa, (1975).

10. N. Ghosh, H. Rajiyah, S. Ghosh and S. Mukherjee, A new boundary element method formulation for linear

elasticity, ASME J. Appl. Mech. 53, 69-76 (1986).

11. N. Ghosh and S. Mukherjee, A new boundary element method formulation for three-dimensional problems

in linear elasticity, Acta Mechanics 67, 107-119 (1987).

12. S. Mukherjee, Boundary Element Methods in Creep and FTac2ure, Elsevier Applied Science, London, (1982).

13. T.A. Cruse, A.B. Pifko and H. Armen (eds.), Advanced Topics in Boundary Element Analysis, ASME

AMD-72, (1985).

14. A. Chandra, A. and S. Mukherjee, Boundary element formulations for large strain large deformation problems

of viscoplasticity, Inl. J. Sol. S2nLcl. 20, 41-53 (1984).

15. A. Chandra and S. Mukherjee, A boundary element analysis of metal extrusion processes, ASME J. Appl.

Mech. 54, 335-340 (1987).

16. S. Mukherjee and A. Chandra, Nonlinear solid mechanics, In Boundary Element Methods in Mechanics

(Edited by D. Beskos), North Holland Pub., Amsterdam, pp. 285-331, (1987).

17. F.J. Rizzo and D.J. Shippy, A formulation and solution procedure for the general nonhomogeneous elastic

inclusion problem, Znt. J. Sol. Slruct. 4, 1161-1179 (1968).

18. Q. Huang and Q. Du, An improved formulation for domain stress evaluation by boundary element method in

elastoplastic problems, Presented at the China-U.S. Seminar on Boundary Integral Equations and Boundary

Finite Element Methods in Physica and Engineering, Xian, People’s Republic of China, (1988).

19. H. Rajiyah and S. Mukherjee, Boundary element analysis of inelastic axisymmetric problems with large

strains and rotations, Int. J. Sol. Struct. 23, 1679-1698 (1987).

a boundary element formulation for design sensitivities in problems involving both geometric and...

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