A unified approach for adjoint and direct method in shape design sensitivity analysis using boundary integral formulation

A unihed approach for adjoint and direct method in shape design sensitivity analysis using boundary integral formulation Joo Ho Choi and Byung Man Kwak

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Cheongryang P.O. Box 150, Seoul, KOREA A unified approach for shape design sensitivity analyses is presented in the Boundary Integral Equation (BIE) formulation, which covers both the adjoint variable and direct differentiation method. The effect of discontinuity in geometry and boundary conditions is included, and expressed as jump terms in the sensitivity formula. An analytic example is taken to show the validity of the jump terms by considering the change of a loaded length by a uniform pressure on a semi-infinite plate problem. The method is shown applicable to a wide class of boundary formulations such as the Trefftz formulation. In the direct differentiation method, the singularity problem arising in the new BIE is solved by utilizing boundary integral identities. Numerical accuracies are examined through an elliptic hole problem under a tensile stress at infinity.

1. INTRODUCTION In the problems of structural shape optimization, the determination of sensitivity coefficients is complicated and cannot easily be expressed explicitly in terms of design function. The accuracy problem in shape design sensitivity analysis (DSA) can be a crucial ingredient for successful shape optimization algorithms. Much literature is now available, but mostly in finite element formulations (see, e.g., Refs 1, 2). Shape DSA in the past was often based on the finite dimensional approach such as implicit differentiation3 or semi-analytical method 4 where the system equation is first discretized, and the derivative of the system matrix is next calculated analytically or by finite differences. Current research effort is focused on the continuum approach where the sensitivity formula is derived analytically based on the continuum basis, and the discretization is next conducted only for the numerical implementation. Some generalized formulas have been developed in this direction by Dems and Mroz 5, Choi and Haug 6, and others, using the variational formulation where the Finite Element Method (FEM) is mainly utilized for the numerical solution. In recent years, the Boundary Element Method (BEM) has emerged as an attractive and alternative technique in the shape DSA due to its reduced dimensionality and accuracy of the boundary solution. Moreover, regridding due to shape change is easier for the BEM than the FEM. Although there have been numerous applications of the BEM to the shape optimization Paper accepted April 1989. Discussioncloses August 1990

© ComputationalMechanicsPublications 1990

problems (see for a survey, l~ef. 7), few rigorous studies are made for the shape DSA based on the boundary element formulation. Recently, finite dimensional approaches for the shape DSA have been studied by Wu s and Kane 9, where the derivative of the boundary element matrix is calculated analytically or semianalytically. While those approaches provide an easy and straightforward process, they have shown some problems such as the complexities of new singular kernels in the boundary element matrices or arbitrariness in the finite differencing. On the other hand, there have been continuum approaches for the shape DSA using the boundary element formulations. Depending on the way how the design derivatives of the state variables are treated, there have been two methods; one is the adjoint variable method which employs an adjoint system to get an explicit sensitivity expression in terms of design change, and the other is the direct differentiation method where design derivatives are determined directly from the derivative of the state equation. Meric 1° has used the adjoint variable method to derive the sensitivity formula in the thermo-elastic problem using the Lagrange multiplier technique. While his work throughout the derivation was independent of the boundary element formulation, he proposed the BEM for the solution of the original and the resulting adjoint system. Choi and Kwak 1~ have developed a general procedure for the adjoint method of shape DSA using the formal Boundary Integral Equation (BIE) formulation. They showed that when the primal problem is formulated by the direct BIE, the adjoint system takes the form of an indirect BIE, which can also be solved using the same direct

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 1 39

A unified approach for adjoint and direct method: Joo Ho Choi and Byung Man Kwak BIE. Some numerical studies have followed for the potentialS2 and 2-D elasticity problems~3. In computational aspects, it was found that the adjoint solutions in most cases correspond to the cases with a concentrated force or moment, which is not suitable for the usual boundary element analysis because they give rise to unbounded integrals. In practice, they were represented by a statically equivalent distributed tractions ~3 In contrast to the adjoint method, the direct differentiation method has a computational advantage in the sense that it avoids the problem of solving for the singular adjoint load. Very recently, an analytic procedure was developed in this direction .by Barone and Yang ~4, by directly differentiating the conventional BIE. The new BIE for the design derivatives was efficiently solved using an identity to avoid singular integrations arising in the new BIE. Choi and Choi ~s have also proposed a direct method in their recent work, utilizing the existing expression that was developed for the adjoint method in Ref. 13. In this paper, a new BIE was developed too, in which a regular BIE technique ~6 was used by locating the collocation point outside the boundary to avoid the singular integrations. While both papers are for the direct method, the method of Barone and Yang is conceptually straightforward since the original BIE itself is differentiated. Also, it is limited in application to the problems which are formulated by only the conventional direct BIE. In this paper, a unified approach for the shape DSA is presented in the BIE formulation for the 2-D elasticity, which covers both the adjoint and direct differentiation method. The basic procedure is the same as those in Refs 13, 15, where the boundary integral identity, derived from the original BIE, is utilized to develop the sensitivity formula. For a practical consideration, the effect of a corner and the discontinuity of boundary conditions is included here, which results in jump terms in the sensitivity formula. An analytic example is presented for the adjoint method to show the validity of the jump terms by considering uniform pressure over a finite length on a semi-infinite plate. Although the method is developed in the BIE formulation, it can also be extended to a wide class of boundary formulations such as the Trefftz formulation, since the boundary integral identity holds for any set of functions satisfying the governing equation. In the direct differentiation method, a new BIE similar to that of Ref. 15 is developed. In Ref. 15, singularities of the integrals and jump terms arising in the new BIE were avoided by the regular BIE technique ~6. Hence, it has left some arbitrariness as to the best location of the collocation point outside the boundary. In this paper, a conventional BIE technique is again employed to eliminate the arbitrariness, removing the singular integrals and jump terms by utilizing suitable identities. The numerical accuracies of the adjoint and direct methods are studied with an elliptic hole problem subjected to a tensile stress at infinity. The analytic solutions are utilized for comparison.

FT(T = TO)

Boundary F

Corner Point

Fig. 1. Elastic body in 2-D with piecewise smooth boundary placements u0i and tractions poi on the boundary F, where the index runs 1 and 2. The body forces are not considered for simplicity (they are discussed in Ref. 13). The boundary F is assumed piecewise smooth as shown in Fig. 1, which has some corners and discontinuous boundary conditions. The equilibrium equation and the associated boundary conditions are written as

(Yij,j(X) :

O,

X ~ [~

(1)

ui(x) = u0i(x),

x ~ r.]

Pi (X) = Poi(X),

X E rp)

(2)

The above boundary value problem can be transformed to the direct Boundary Integral Equation by Somigliana's identity 17 as follows

Oiki(Xo)Ui(Xo) "4- IF { Pki(Xo, X)R/(X) -- Uki(Xo, X)pi(x)}

ds(x)=0, xo~r

(3)

where Xo denotes the load point, and ds(x) means the integration with respect to the field point x along r'. The coefficient OLk~is a function of geometry of F. It is not necessary to be known if the identity of rigid body movement is utilized ~7. The kernels Uki and Pki in (3) are the fundamental Kelvin solutions for displacement and traction, respectively, which are given in Ref. 17:

gki(xo, x)=87rp(l_~) [(3-4~')1n(1) tSki+r, kr, i1 -

1

[Or

P k i ( x ° ' x ) - 4 7 r ( 1 - ~)r Onn [(1 - 2 ~ ) 8ki+2r, kr, i} --(1 -- 2~)(r, kn, i - r, in,k)]

(4) where r = I x 0 - x I, ~ and ~ denote material constants, and n is the outward unit normal. While equation (3) is for the direct BIE, one can consider an arbitrary system of displacement u* and traction p * which can be expressed by the indirect BIE aT.

u*(x) = ~Ir pk(Xo)Uki(Xo,X) ds(x0), 2. B O U N D A R Y I N T E G R A L F O R M U L A T I O N IN P L A N E ELASTICITY

Consider a 2-D elastic body fl subject to prescribed dis40

p ? ( x ) : pk(X)Olki(X)

+ I r pk(Xo)Pk,(Xo,x)

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 1

ds(xo),

x ~r

(5)

A unified approach for adjoint and direct method: Joo Ho Choi and Byung Man Kwak where the function Pk is a fictitious source density distribution over I'. If we multiply Pk to the direct BIE (3), integrate over I' with respect to x0, and apply the indirect BIE expressions o f (5), a boundary integral identity is obtained as follows

Iv (uip* - piu*)

ds = 0

derivation, the jump term o f ( q i V , - p~Vs) has been removed due to the following relation qiVn -- p i Vs = aij Vk (Sjnk -- njSk) = aij Vk ejk

where

(6)

This identity corresponds to Betti's reciprocal theorem xs for two arbitrary states: one with dis,placement u and traction p, and the other with u* and p . From this, one can observe an important relationship that the two systems formulated by direct and indirect BIE are mutually adjoint. Although the identity (6) is derived using the BIE formulations, it holds in general for any two systems if they satisfy the governing equation. Hence, one can also utilize the identity in the Trefftz formulation.

Since ouVkejk is continuous at a corner, the jump term (qiVn - piVs) vanishes.

Adjoint variable method Let the adjoint system for (u*, p*) satisfy the boundary condition U* = - ffpi

Consider a performance functional arising in shape design problems in the boundary integral form

,I, = ~ ¢,(u. p~) ds

(7)

dF

(8)

[~/u,l~iq- ~p,bi -1" ~/Vj, sSj] ds

or

P .) F

-( ,) 1"

(~HV~- ff, sV~) ds+ ~ (ffVD

(9)

c

where ff~, and ~bp, denote partial derivatives, H and sj are curvature and unit tangent o f F, and Vn and V~ are the normal and tangential components of design velocity field 1~, respectively. Note that l~.s in (8) has been removed in (9) by integration by parts on the boundary, resulting in the jump terms denoted by ( . ) at points with discontinuity. The first integral in (9) contains implicit derivatives of the boundary variables, which cannot be expressed explicitly by V. In the adjoint method, these derivatives are eliminated by introducing an adjoint system, whereas in the direct differentiation method, they are determined directly. Whichever method is used, the boundary integral identity (6) can be utilized. Take the material derivative of (6), using various formulas given in Ref. 13, to obtain

I = (

Fu

ds

(~bu,-p*)(dtoi- Uoi,sVs) ds

tJ

+ ~

d Up

(¢,9, + u * ) ( i , oi - poi, svs) ds

+ ~] ((piu*+ ~)VD

(13)

c

In this expression, the adjoint system is in the form of the indirect BIE, which can also be solved using the same direct BIE as the original due to the equivalence between the direct and indirect BIE 19. It is interesting that if the prescribed values Uo~ and p0i are constants and if all quantities are smooth on the boundary, the resulting expression contains only the normal component Vn of the boundary velocity. However in the general case, one should consider also the tangential component V~ in the sensitivity of the boundary functional q, as shown above. An analytic example is given here to show the importance of the jump term which includes the tangential velocity V~ in (13). Consider a uniform pressure a0 applied over a finite length 2a on a semi-infinite plate as shown in Fig. 2. Due to the symmetry, one half of the problem is considered. The objective now is to get the exact sensitivity expression for the vertical displacement uy at the origin O, due to the change of the loaded length a. Note that the discontinuity of the traction is observed at point A. Assuming uy = 0 at y = - R where R ~, a to impose uniqueness of the solution, the analytic solution for Uy at the origin is given in Ref. 20

Uy =

2o0(I - v) 4ao a rE a + ~--~ a In

(14)

where E denotes Young's modulus. Hence, the sensitivity of Uy with respect to the semi-length a becomes

r ( u i p * -- p i u * ) ds

o I"

(12)

[ - - U i, sqi * + p i ( U i ,* n + u ~ H ) + ~ H } V n

F

+ f

J I"

Since the boundary is moving, the material derivative concept ~3 is utilized for the variation of ,I, with respect to the shape change. Then,

x fi 1-'u/

Substituting (12) into (9), and utilizing (10), thedesired sensitivity formula for • is obtained in terms o f V as cb' = f

3. SHAPE DESIGN SENSITIVITY ANALYSIS

¢~t = ~

(11)

d

-~a Uy =

[[-ui,~q*+pi(ui,*+u*H)lV,

+ (u~,sp~ - p~,,u*)Vd ds + ~ (piu*VD

(10)

c

where qi is the tangential component of the stress tensor, i.e., q~ = (Tijsj. It should be noted that during the

(1 + v)ao

rE

2a0

a

+ - ~ In ~

(15)

If the developed method is used, the functional to be considered is d I"

Uy(X) 6 ( x - xo) ds

(16)

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 1 41

A unified approach for adjoint and direct method: Joo Ho Choi and Byung Man Kwak y

derivative of the velocity Vj,s in (21). Recalling that (u*, pi*) are in the form of the indirect BIE, u*,.s in (21) can be expressed as

t

HiTs(X) = Pk(X)[3ki(X) i

+

lF

Or

R

/

(22)

where the coefficient/3ki appears as a result of the singularity of Ugi,~ at x0 = x, and is also a function of geometry of I'. If we substitute the indirect BIE expressions for u*, p * and ui*s of (5) and (22) into (21), and enforce that the resulting equation hold for arbitrary source density pk, then a new BIE is obtained for tii and ,hi:

i i l t

/

I Pk(Xo)Uki, s(Xo, X) ds(xo),xE F

i

~ °itJ° + fr ( P k i t J i - UkiiJi) ds

=-f3°~(°iYle~t)°- f r

Uki, s(oijVlejl) ds + ~ °i(ui, j ~ ) 0

+ f r Pki(Ui, j V j ) ds

Fig. 2. Uniform pressure applied over finite length of semi-infinite plate where Xo denotes the origin. In this problem, the entire velocity field is zero except the tangential movement at point A, which corresponds to ~Sa. Hence, the sensitivity formula becomes simply the following jump term ~' = (piU*Vs)

(17)

at point A

where the adjoint variable u* is determined by the condition (12) which corresponds to a unit concentrated force applied in the vertical direction at the origin. The analytic solution of Uy* for point A in Ref. 20

, ( l + v ) + 2 In a uy---if- ~E

(18)

Substituting (18) into (17), and noting that the discontinuity of py at point A is oo, one obtains the sensitivity of cI>, the same as the exact expression (15).

Direct differentiation method The direct differentiation method is to solve for the material derivatives ~ and b~ directly. To this end, a new BIE is developed, which is derived from the derivative of the boundary integral identity. Equation (10) can be rewritten by utilizing the following relations on the boundary * = piUi,* n + qi U i,s * tYijU i,j

= U i, joij*

(piu *Vs) = I (piui *V~),~ ds F

= Ui, npi * + ui, sqi *

(19) (20)

Then, equation (10) becomes

fr (flip* -

dF

Xo E F

(23)

where the arguments Xo and x have been omitted for simplicity of the expression, and the superscript 0 means the value at x0. Although this equation looks somewhat complicated, it is nothing but a simple substitution of (Uki, Pki) in place of (u*, p*) in (21). The coefficients et°i and/3°i came from the singularities of the kernels. In (23), it looks that the singular integration for Uki, s and Pki is necessary. However, these singularities can be removed by utilizing the following identities f

I"

u ,,s ' ds - f

P

du*=O

f p* ds - S"oij,j F

(24)

dfl = 0

(25)

fl

Equation (24) holds because of the single-valuedness of u i. Substitute Uk~ and Pki into (24) and (25) to obtain

~°i + ~ Uki,~ ds = 0

(26)

Ot°i + ~ Pki ds dF

(27)

,J F

= 0

which hold for arbitrary Xo E 1-'. Multiply (aijVtejt) ° and (ui, fi~) ° to (26) and (27), respectively, and subtract the resulting equations from (23). Then the new BIE becomes

°l°i[4i° + fr (Pkiiti- UkiiJi) ds = ~ [ -- Uki, s(OuVlejt) + Pki(Ui, jVj) + Uki(piVj, sSj)] ds dF

iJiuT) ds

Xo ~ F

= ~,J [- (o~Vkej,~)ui*,+ (ui,YAp*+ (piV),,sAu*] ds I"

(21) Note in this equation that the jump terms are reexpressed using (20). This has produced the tangential 42

+ ~ Uki(piVj, ssj) ds

(28)

In this equation, (~) means the difference ( * ) - ( . ) o between the value at x and Xo, which becomes O(s) when x approaches x0. Hence, the first two integrals in the right hand side of (28) are regular which can be evaluated using the standard Gauss quadrature. Also, the coefficients c~k,'and/3ki are not necessary since they have

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 1

A unified approach f o r adjoint and direct method: Joo Ho Choi and Byung Man Kwak 0

0

TIIIIT

F1

x u~

v]

e

I

cL

Fig. 3. Elliptic hole under tensile stress at infinity disappeared in this equation. The integral operator in the left hand side of (28) is identical to that of the original BIE (3), which means that the same boundary element equations as the original can be used to determine tJi and ,hi.

4. N U M E R I C A L PROBLEM

EXAMPLE

-

ELLIPTIC HOLE

The adjoint and direct method are applied to an elliptic hole problem under a tensile stress at infinity as shown in Fig. 3. The equation for the elliptic hole is given by x = a cos 0 and y = b sin 0. Only a quadrant of the problem is considered due to the symmetry. The analytic solution is available for this problem (see for example, Ref. 18), and the detailed expressions are given in Ref. 15, where the same example has been considered to study the direct method. The shape of the hole boundary I'~ is varied in the y-direction by changing b, hence,

Vx = 0

and

Vy =

~b sin 0

Fe, respectively. Hence in this case, the adjoint traction includes the concentrated forces in the tangential direction at x i and x e. For a practical boundary element implementation, they are approximated by an equivalent traction over I~c using appropriate shape functions. In the direct method, substituting (29) into (28), and calculating the right hand side of the resulting equation using the standard Gauss quadrature, the derivatives tJi and ,bi are directly determined. Two cases are considered with different ratios for a[b: first one is a circular hole with a unit radius, and the other is an elliptic hole with a[b = 4. The applied stress is oo = 1000 psi, and the material constants are # = 1.0(10) 4 psi and v = 0.3. The infinite plate is approximated by a finite but sufficiently large square plate, and linear elements are used for the analysis as shown in Fig. 4 in case of the elliptic hole. Along the hole boundary FI, 10 elements are generated, and the small boundary 17~ is identified with each element. Hence, 10 sub-functionals are generated for each cba and ~s. It is mentioned that whether adjoint or direct method is used, one should always evaluate u~,s, because the various terms in the boundary integrals of (13) or (28) are expressed in terms of ui and Pi as well as ui, s by the stress-strain relationship on the boundary 13,17. Since the linear elements are used in this study, ui, s becomes piecewise constant, which might degrade the accuracy by an order. To alleviate this problem, an equivalent ui, s having a linear variation is calculated by interpolation. The sensitivity results are given in Tables 1 and 2 for G

0

0

o

/ ElementNumber ! C~)"~- 5 67 D

0~

~

II

(29)

For the sensitivity study, two functionals are considered, both of which are imposed over a small part l~c of the hole. One is the average of the norm of displacements, and the other is the averaged Von-Mises stress, as follows:

r ~° Fig. 4. Boundary element mode/ f o r problem with a/b = 4

-I elliptic hole

Table 1. Sensitivity coeJJcients of displacement and stress functional in circular hole problem (a) Displacement norm functional Error against exact value (%)

where Ilull =u,~. ~s=Dui, ss~, and D = 2 ~ / ( 1 - ~). Since the traction is zero on F~, the Von-Mises stress as in (30) becomes simply the tangential stress which depends only on U~.s, i.e., the tangential derivative of ui. In the adjoint method, the adjoint condition for ~d is given by the traction with value ui/ll ull over I~e. On the other hand, the adjoint traction for cI,s becomes p*=Hni+

[ 5 ( x - - x e ) - - 5 ( x - - x i ) } s i on l"e

(31)

where x i and x e denote starting and ending points of

Element

Exact value

I 2 3 4 5 6 7 8 9 10

3.476E 3.379E 3.187E 2.905E 2.538E 2.096E 1.587E 1.033E 4.901E 1.168E

-

02 02 02 02 02 02 02 02 03 03

Direct method

Adjoint method

Finite diff.

1.85 3.40 3.36 3.57 3.29 2.16 0.48 0.59 4.58 71.70

3.48 1.14 2.94 3.84 2.97 0.24 5.71 12.48 18.11 13.69

2.27 1.12 0.52 0.40 1.04 0.94 9.90 1.62 54.63 91.54

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No, 1 43

A unified approach f o r a d j o i n t a n d direct m e t h o d : J o o H o C h o i a n d B y u n g M a n K w a k Table 1. (Continued) (b) Von-Mises stress functional Error against exact value (070) Element

Exact value

Direct method

Adjoint method

Finite diff.

1 2 3 4 5 6 7 8 9 10

- 4.836E - 02 - 3.109E - 01 - 7.083E - 01 5.345E - 01 1.099E + 00 7.877E - 01 1.325E - 01 - 7.001E - 01 - 1.464E + 00 - 1.919E + 00

712.8 39.32 2.81 8.21 9.59 13.69 55.51 2.03 4.23 5.35

4583.9 94.44 21.17 65.32 43.23 67.36 296.9 18.21 10.03 38.37

674.7 31.25 12.69 11.35 3.25 6.49 77.27 18.50 11.34 2.22

Table 2. Sensitivity coe.~cients of displacement and stress functional in elliptic hole problem (al b = 4) (a) Displacement norm functional Error against exact value (°7o) Element

Exact value

Direct method

Adjoint diff.

Finite diff.

1 2 3 4 5 6 7 8 9 10

3.474E 3.371E 3.167E 2.865E 2.473E 1.998E 1.456E 8.814E 3.592E 7.474E -

0.60 1.16 2.09 2.66 3.30 3.68 0.06 2.40 0.22 129.4

0.48 1.63 1.16 0.91 0.37 1.34 7.53 28.17 21.46 576.6

5.93 4.25 3.23 5.22 7.77 7.82 23.62 86.44 212.4 25.87

02 02 02 02 02 02 02 03 03 04

c o n t a i n a n y e r r o r s o u r c e . T h e y a r e e x a c t , b u t t h e discretization and round-off errors are inevitable. What a r e c o m p a r e d in t h e t a b l e s a r e j u s t t h e s e e r r o r s .

5. D I S C U S S I O N S

The present work provides a unified approach for the a d j o i n t a n d d i r e c t m e t h o d in t h e s h a p e D S A u s i n g t h e b o u n d a r y i n t e g r a l f o r m u l a t i o n s . A l t h o u g h t h e y are d e v e l o p e d in t h e B I E f o r m u l a t i o n , t h e y c a n be a p p l i e d to o t h e r b o u n d a r y f o r m u l a t i o n s s u c h as t h e T r e f f t z f o r m u l a t i o n , since t h e b o u n d a r y i n t e g r a l i d e n t i t y , w h i c h holds for any functions satisfying the governing equat i o n , is u t i l i z e d t h r o u g h o u t t h e p r o c e d u r e . T h e effect o f d i s c o n t i n u i t y o n t h e b o u n d a r y is also i n c l u d e d in t h e d e r i v a t i o n , a n d is i l l u s t r a t e d b y a n e x a m p l e w i t h a p a t c h o f u n i f o r m p r e s s u r e o n a s e m i - i n f i n i t e plate. While the numerical accuracy of the direct d i f f e r e n t i a t i o n m e t h o d seems b e t t e r t h a n t h e o t h e r s , t h e c h o i c e b e t w e e n d i r e c t a n d a d j o i n t m e t h o d c a n be m a d e f r o m t h e p o i n t o f efficiency a n d m a y b e f r o m p e r s o n a l p r e f e r e n c e . It is n o t e d , h o w e v e r , t h a t in t h e a d j o i n t m e t h o d , a c o n c e n t r a t e d f o r c e o r m o m e n t s h o u l d be t r e a t e d in t h e b o u n d a r y e l e m e n t a n a l y s i s . B o t h m e t h o d s r e q u i r e t h e t a n g e n t i a l d e r i v a t i v e s o f uj o n t h e b o u n d a r y . While they are determined by a numerical d i f f e r e n t i a t i o n in this p a p e r , a n o t h e r B I E f o r m u l a t i o n m a y be e m p l o y e d t o d e t e r m i n e t h e t a n g e n t i a l d e r i v a t i v e s o f ui directly. T h e so c a l l e d d e r i v a t i v e B E M in t h e P o t e n t i a l p r o b l e m b y the a u t h o r s 2~ suggests o n e w a y t o g o in t h e f u t u r e .

REFERENCES

Table 2. (Continued) (b) Von-Mises stress functional

1 Error against exact value (070) 2

Element

Exact value

1 2 3 4 5 6 7 8 9 10

-

6.217E - 03 4.465E - 02 1.275E - 01 2.682E - 01 4.907E - 01 6.541E - 01 1.314E + 00 1.759E + 00 6.620E - 01 - 5.289E + 00

Direct method

Adjoint method

Finite diff.

1017.4 8.35 16.06 4.75 0.41 48.56 9.66 10.76 80.18 2.43

5830.8 169.7 36.50 3.40 11.27 72.47 24.28 49.62 68.63 6.39

1394,7 84.73 27.56 8.05 2.72 52.14 10.62 31.66 119.0 4.97

3 4 5

6 7

t h e c i r c u l a r a n d elliptic h o l e p r o b l e m , r e s p e c t i v e l y . I n t h e s e t a b l e s , t h e results b y finite d i f f e r e n c e s a r e also compared where the boundary is p e r t u r b e d by tSb = 0 . 0 0 l b . C o m p a r i n g t h e results o f t h e t w o f u n c t i o n a l s , t h e sensitivities o f t h e stress a r e less a c c u r a t e t h a n t h o s e o f t h e d i s p l a c e m e n t , since t h e y i n v o l v e ui,~ in t h e f u n c t i o n a l as s h o w n in (30). F r o m t h e o v e r a l l c o m parisons, the direct method shows better accuracies than t h e a d j o i n t m e t h o d o r t h e finite d i f f e r e n c e s as e x p e c t e d . A l l t h e t a b l e s s h o w p o o r a c c u r a c i e s at e l e m e n t s w i t h s m a l l s e n s i t i v i t y c o e f f i c i e n t s o r w h e r e t h e sign is changed. This does not mean that the formulas derived

44

AND CONCLUSIONS

Haftka, R. T. and Grandhi, R. V. 'Structural Shape Optimization - a Survey, Computer Methods in Applied Mechanics and Engineering, 1986, 57(1), 91-106 Haug, E. J., Choi, K. K. and Komkov, V. Design Sensitivity Analysis of Structural Systems, Academic Press, New York, 1986 Wang, S.-Y., Sun, Y. and GaUagher, R. H. Sensitivity Analysis in Shape Optimization of Continuum Structures, Computers and Structures, 1985, 20(5), 855-867 Prasad, G. and Emerson, J. F. A General Capability of Design Sensitivity for Finite Element Systems, AIAA/ASME/ ASCE/AHS 23rd SDM Conference, May 1982, 175-186 Dems, K. and Mroz, Z. Variational Approach by means of Adjoint Systems to Structural Optimization and Sensitivity Analysis-ll. Structural Shape Variation, International Journal for Solids and Structures, 1984 26(6), 527-552 Choi, K. K. and Haug, E. J. Shape Design Sensitivity Analysis of Elastic Structures, Journal of Structural Mechanics, 1983, 11(2), 231-269 Mota Soares, C. A., Leal, R. P. and Choi, K. K. Boundary Elements in Shape Optimal Design of Structural Components, in C. A. Mota Soares (ed.), Proc. of the NATO ASI on

Computer Aided Optimal Design: Structural and Mechanical Systems, Springer-Verlag, Berlin Heidelberg, 1987, 605-631 8 9 10

Wu, S. J. Applications of the Boundary Element Method for Structural Shape Optimization, Ph.D. Thesis, The University of Missouri, Columbia, MO, 1986 Kane, J. H. Optimization of Continuum Structures Using a Boundary Element Formulation, Ph.D. Thesis, The University of Connecticut, 1986 Meric, R. A. Boundary Elements in Shape Design Sensitivity Analysis of Thermoelastic Solids, in C. A. Mota Soares (ed.), Proc. of the NATO ASI on Computer Aided Optimal Design: Structural and Mechanical Systems, Springer-Verlag, Berlin Heidelberg, 1987, 643-652

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Choi, J. H. and Kwak, B. M. Shape Design Sensitivity Analysis of Elliptic Problems in Boundary Integral Equation Formulation, Mechanics of Structures and Machines, 1988, 16(2), 147-165 Kwak, B. M. and Choi, J. H. Shape Design Sensitivity Analysis using Boundary Integral Equation for Potential Problem, in C. A. Mota Soares (ed.), Proc. of the NATO ASI on Computer

16

17 18

Aided Optimal Design: Structural and Mechanical Systems, 13

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Springer-Verlag, Berlin Heidelberg, 1987, 633-642 Choi, J. H. and Kwak, B. M. Boundary Integral Equation Method for Shape Optimization of Elastic Structures, Int. Journal for Numerical Methods in Engineering, 1988, 26, 1579-1595 Barone, M. R. and Yang, R. J. Boundary Integral Equations for Recovery of Design Sensitivities in Shape Optimization, AIAA Journal, 1988, 26(5), 589-594 Choi, J. H. and Choi, K. K. Direct Differentiation Method for Shape Design Sensitivity Analysis Using Boundary Integral Formulation, submitted to Computers and Structures, 1988

19 20 21

Patterson, C. and Sheikh, M. A. Regular Boundary Integral Equations for Stress Analysis, in C. A. Brebbia (ed), Boundary Elements in Engineering, Proc. 3rd Int. Seminar, SpringerVerlag, Berlin Heidelberg, 1981, 85-104 Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C. Boundary Element Techniques, Springer-Verlag, New York, 1984 Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956 Brebbia, C. A. and Butterfield, R. Formal Equivalence of Direct and Indirect Boundary Element Methods, AppL Math. Modelling, 1978, 2, 132-134 Timoshenko, S. P. and Goodier, J. N. Theory of Elasticity, McGraw-Hill, Tokyo, 1970 Choi, J. H. and Kwak, B. M. A Boundary Integral Equation Formulation in Derivative Unknowns for Two Dimensional Potential Problems, Journal of Applied Mechanics, 1989, 111, 617-623

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