KAWARADA, HIDEO

MAUNG, SAW WIN

Kyoto University Research Information Repository

TitleNumerical Study of a Shape Design Problem in Elasticity byuse of Auxiliary Domain MethodAuthor(s)MAUNG, SAW WIN; KAWARADA, HIDEOCitation数理解析研究所講究録 (1991), 744: 85-99Issue Date1991-02URL http://hdl.handle.net/2433/102173RightType Departmental Bulletin PaperTextversionpublisherKyoto University85Numerical Study of a Shape Design Problem in Elasticityby use of Auxiliary Domain MethodSAW WIN MAUNG*HIDEO KAWARADA**Abstract. In this paper, we discuss numerical computation obtaining solution ofa two-dimensional shape design problem in linear elasticity. Though this problemhas been solved by using finite element method in [3], we present an alternativeapproach in order to use flnite difference method. Auxiliary domain method isapplied so that computation, especially at the boundary, can be more easier. Oneof the reasons why we use flnite difference method is to avoid resetting of meshin numerical computing while we change the domain which we consider. In thatcase, there are some ways with fixed mesh to solve the problem, for example,grid generation method. But these methods have some disadvantages such aslosing linearity of the state equation and, obviously, become more complicated.Our technique provides to study numerical analysis of the problem as simple aspossible.1. Contact problem with frictionAt first we will present steady state problem. Let $a,$ $b,$ $c_{o}$ and $c$ , where$a<b$ and $0<c_{o}<c$ , be given. We consider two-dimensional elastic bodyrepresented by(1.1) $\Omega=\Omega(\alpha)=\{(x_{1}, x_{2})\in R^{2}| a<x_{1}<b, \alpha(x_{1})<x_{2}<c\}$as shown in the figure 1.1.$(\prime 1.2)$ $\Gamma_{c}=\Gamma_{c}(\alpha)=\{(x_{1}, x_{2})\in R^{2}| a<x_{1}<b, x_{2}=\alpha(x_{1})\}$contacts with the rigid body which is lower-half of the plane.Here $\alpha\in U_{ad}$ and the set $U_{ad}$ of admissible controls is defined by(1.3) $U_{ad}=\{\alpha\in C^{1,1}([a, b])|$ $0\leq\alpha(\dot{x}_{1})\leq c_{o}$ ,$|\alpha’(x_{1})|\leq c_{1}$ for $x_{1}\in[a, b]$ ,$|\alpha’’(x_{1})|\leq c_{2}$ a.e. in $(a, b)$ ,measure of $\Omega(\alpha)=c_{3}$ $\}$ .’Graduate School of Science and Technology, University of Chiba”Department of Industrial Mathematics, Faculty of Engineering, University of Chiba744 1991 85-9986figure 1.1The elastic body is subjected to forces and so deformation occurs. Wedenote the displacement vector by $u=(u_{1}(x_{1}, x_{2}),$ $u_{2}(x_{1}, x_{2}))$ . We assume thatstrain tensor(1.4) $e_{ij}(u)= \frac{1}{2}(\frac{\partial u:}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}})$ $i,j=1,2$and stress tensor $\mathcal{T}:j(u),$ $i,$ $j=1,2$ , satisfy the linearized Hooke’s law(1.5) $\tau_{ij}(u)=\lambda\delta_{ij}(e_{11}(u)+e_{22}(u))+2\mu e_{ij}(u)$ , $i,j=1,2$ .$\lambda\geq 0$ and $\mu>0$ are Lame’s coefficients and constants. On the boundary of$\Omega$ , we define(1.6) $T:= \tau_{:}(u).=\sum_{i=1}^{2}\tau_{ij}(u)n_{j}$ $i=1,2$ ,where $n=(n_{1}, n_{2})$ is outward unit normal vector.ProblemLet force $f=(f_{1}, f_{2})$ in $\Omega$ , stress $p=(p_{1}, p_{2})$ on $\Gamma_{p}$ and friction$g=constant$ on $\Gamma_{c}$ be given. Then for some given $\alpha\in U_{ad}$ , the problem isto find $u=u(\alpha)$ satisfying$(P)\{\begin{array}{l}-\sum_{j=l}^{2}\frac{\partial\tau_{j}(u)}{\partial x_{j}}=f.\cdot u_{i}=0T_{i}(u)=p.\cdot u_{2}+\alpha\geq 0,T_{2}(u)\geq 0|T_{1}(u)|\leq g,(g-|T_{l}(u)|)u_{1}=0\end{array}$,$i=1_{1},2,in_{0}i^{i_{=_{2}}}(u^{=_{u_{1}^{1}’}}+_{T_{1}^{2}(u)\leq^{o^{O}n\Gamma^{\Gamma_{p^{0}}}}}^{2_{\alpha)T_{2}(u)^{n_{=0}^{\Omega}}}}.$$on\Gamma_{c}on\Gamma_{c}$287Differentiation which will be used throughout is taken in the sense of distribution.In $(P)$ , the condition $u_{2}+\alpha\geq 0$ means that the elastic body can notpenetrate into rigid body. On $\Gamma_{c}$ , free boundaries will take place betweentouching parts where $u_{2}+\alpha=0,$ $T_{2}(u)>0$ and separating parts where$u_{2}+\alpha>0,$ $T_{2}(u)=0$ . There may be another free boundaries between slidingparts where $|T_{1}(u)|=g,$ $u_{1}\neq 0$ and static parts where $|T_{1}(u)|<g,$ $u_{1}=0$ .See the figure 1.2 below. The condition $u_{1}T_{1}(u)\leq 0$ states that, on slidingparts, $u_{1}$ and $T_{1}(u)$ are opposite.figure 1.22. Variational formulationTo express the problem $(P)$ in weak form, we introduce the space(2.1) $V=V(\alpha)=\{v\in(H^{1}(\Omega(\alpha)))^{2}|$ $v_{i}=0$ , $i=1,2$ , on $\Gamma_{0}\}$ ,nonempty closed convex set(2.2) $K=K(\alpha)=$ { $v\in V$ I $v_{2}(x_{1},$ $\alpha(x_{1}))+\alpha(x_{1})\geq 0$ a.e. in $(a,$ $b)$ },continuous bilinear form(2.3) $a(u, v)= \sum_{i,j=1}^{2}\int_{\Omega}\tau_{ij}(u)e_{ij}(v)dx$ , $u,$ $v\in V$ ,$= \int_{\Omega}$ $\{ \lambda div(u)div(v)+2\mu\sum_{i.j=1}^{2}e_{ij}(u)e_{ij}(v)\}dx$ ,continuous linear functional(2.4) $\{l,$ $v\rangle$ $= \sum_{i,j=1}^{2}\int_{\Omega}f_{i}v_{i}dx+\sum_{i,j=1}^{2}\int_{\Gamma_{p}}p;v;d\gamma$ , $v\in V$and continuous convex functional388(2.5) $j_{c}(v)= \int_{\Gamma_{c}}g|v_{1}|d\gamma$ , $v\in V$ .It can be shown that $(P)$ is equivalent to variational inequality(VI) $\{_{a(u,v-u)+j_{c}(v)-j_{c}(u)}u=u(\alpha)\in Kst\geq(l,$$v-u\rangle$ , for all $v\in K$ .For details, see [4]. Now the following classical result holds.Theorem 2.1Under the assumption of algebraic ellipticity condition for $\lambda$ and $\mu$ , thesolution of (VI) exists uniquely.Since $a(., .)$ is symmetric, we have equivalent minimization problem$(M)$ $u=u(\alpha)\in Ks.t$ . $J(u)= \min_{v\in K}J(v)$ ,where(2.6) $J(v)= \frac{1}{2}a(v, v)-\langle l, v\rangle+j_{c}(v)$which is, in physical meaning, total potential energy.3. Setting of the problemFor $\alpha\in U_{ad}$ , we define cost functional by(3.1) $E(\alpha)=J(u(\alpha))$ , where $u=u(\alpha)$ is the solution of (VI).Then shape design problem is to find$(SD)$ $\alpha^{*}\in U_{ad}$ s.t. $E( \alpha^{*})=\min_{\alpha\in U_{ad}}E(\alpha)$ .We can explain that it is to look for an optimal domain on which displacementfield $u$ causes minimum of total potential energy. Existence of such domain isproved in the following.Theorem 3.1 (J. Haslinger and P. Neittaanmaki) [3]There exists at least one solution of the problem $(SD)$4894. Regulari.zation and penalization‘l’o approxiinate $nondif\Gamma erentiable$ functional $j_{c}$ by a family of convex anddifferentiable functionals, we can regularize it as(4.1) $j_{c\epsilon_{r}}=j_{ce_{r}}(v)= \int_{\Gamma_{c}}g\sqrt{v_{1}^{2}+\epsilon_{f}^{2}}d\gamma$ , $(\epsilon_{f}>0)$ .And to replace the fact that $u\in K$ , a penalization operator $\beta$ : $v\beta(v)$can be defined by(4.2) $( \beta(v))w)=-\frac{1}{\epsilon_{p}}\int_{\Gamma_{c}}(v_{2}+\alpha)^{-}w_{2}d\gamma$ , $w\in V$ , $(\epsilon_{p}>0)$Denote $\epsilon=$ $(\epsilon_{f} , \epsilon_{p})$ Now we consider the following regularized andpenalized variational equation$(VE)$ $\{_{a(u_{e},v)+}^{u_{e}=u_{e}(\alpha)}(j_{ce}(u_{e}^{s.t})+\beta(u_{\epsilon}), v)\in_{/}V_{r}=\langle l, v\rangle$, for all $v\in V$ .Note that the operator $v j_{c\epsilon_{r}}’(v)$ and $\beta$ are monotone. Furthermorekernel of $\beta=K$ and $\beta$ is lipschitz continuous. By using these facts, we getthe following convergence result. $[1][2]$Theorem 4.1There exists unique solution $u_{\epsilon}$ of $(VE)$ and $u$. $arrow u$ : solution of (VI)strongly in V as $\epsilon=(\epsilon_{f}, \epsilon_{p})arrow(0,0)$As before, $(VE)$ is the same as the problem$(M_{e})$ $u_{\epsilon}=u_{\epsilon}(\alpha)\in V$ s.t. $J_{\epsilon}(u_{e})= \min$ $J_{\epsilon}(v)$ ,$v\in V$where(4.3) $J_{\epsilon}(v)= \frac{1}{2}a(v, v)-\{l,$ $v$ ) $+j_{c\epsilon_{r}}(v)+ \frac{1}{2\epsilon_{p}}\Psi(v)$and(4.4) $\Psi(v)=\int_{\Gamma_{c}}[(v_{2}+\alpha)^{-}]^{2}d\gamma$ .590The problem becomes to find $u_{e}=(u_{\epsilon 1}, u_{\epsilon 2})$ satisfying$(P_{\epsilon})$ $\{\begin{array}{l}-\sum_{j=1}^{2}\frac{\partial\tau_{|j}(u_{e})}{\partial x_{j}}=f_{i},i=1,2’u_{e}.\cdot=0,i=1,2’T_{j}(u_{\epsilon})=p_{i}’ i=1,2’T_{2}(u_{e})=\frac{1}{\epsilon_{p}}(u_{e2}+\alpha)^{-}T_{1}(u_{\epsilon})=-g\frac{u_{e1}}{\sqrt{u_{\epsilon l}^{2}+\epsilon_{r}^{2}}}\end{array}$ $o_{on^{\Gamma_{\Gamma^{c_{\epsilon}}}^{\Gamma_{p}}}}^{o_{o^{n}n^{\Omega_{\Gamma^{0}}}}}in_{n}$.One can notice that the boundary conditions on $\Gamma_{c}$ become more simplerbecause of regularization and penalization. It will also be helpful in calculatingthe derivative of cost functional.5. Auxiliary domain methodWe construct an auxiliary domain $\tilde{\Omega}$ as shown in the following figure 5.1and denote $\hat{\Omega}=(a, b)\cross(c\sim, c)$Let us define again the space(5.1) $\hat{V}=\{v\in(H^{1}(\hat{\Omega}))^{2}|$ $v;=0,$ $i=1,2$ , on $\Gamma_{0}\cup\tilde{\Gamma}_{0}\}$Let $\epsilon_{d}>0$ Our aim is to suppose the minimization problem, in placeof $(M_{e})$ ,$(\overline{M}_{\epsilon})$ $\wedge u_{e}=\wedge u_{e}(\alpha)\in\hat{V}$ s.t. $J_{\epsilon}(u_{e}) \wedge\wedge=\min$ $J_{e}(v)\wedge$ ,$v\in\hat{V}$691where(5.2) $J_{e}(v)= \frac{1}{2}a(v, v)+\frac{1}{2}\epsilon_{d}\sim a(v, v)+\frac{1}{2}\int_{\Omega}\sim|v|^{2}dx$$- \langle l, v\rangle+j_{ce_{r}}(v)+\frac{1}{2\epsilon_{p}}\Psi(v)$ .$\sim a(., .)$ is as in (2.3), but integration is based on the domain $\tilde{\Omega}$ Here$\epsilon=$$(\epsilon_{r} , \epsilon_{p}, \epsilon_{d} )$ . We can explain $(\overline{M}_{e})$ asAfter all shape design problem to be considered is$(\overline{SD}_{e})$$\alpha_{\epsilon}^{*}\in U_{ad}$ $s.t$ . $\hat{E}_{e}(\alpha_{e}^{*})=$ $\min$ $\hat{E}_{\epsilon}(\alpha)$ ,$\alpha\in U_{ad}$and the cost functional to be optimized is(5.3) $E_{\epsilon}(\alpha)=\hat{J_{\epsilon}}(u_{e}\wedge(\alpha))$ , where $u_{e}\wedge=u_{e}\wedge(\alpha)$ is the solution of $(\overline{M_{e}})$ .6. Derivative of cost functionalWe need to calculate Frechet derivative of $\hat{E}_{e}(\alpha)$ to be used in optimizer.Let us state two lemmas which are useful.Lemma 6.1Let $C(\alpha)$ be smooth function which is defined in $\Omega(\alpha)$ Then$\frac{d}{d\alpha}[\int_{\Omega(\alpha)}C(\alpha)dx]\delta\alpha=\int_{\Omega(\alpha)}\frac{\partial C(\alpha)}{\partial\alpha}\delta\alpha dx$$+ \int_{\partial\Omega(\alpha)}C(\alpha)n.\nu\delta\alpha d\gamma$ .792Lemma 6.2Under the same conditions as in Lemma 6.1, we have$\frac{d}{d\alpha}[\int_{\partial\Omega(\alpha)}C(\alpha)d\gamma]\delta\alpha=\int_{\partial\Omega(\alpha)}\frac{\partial C(\alpha)}{\partial\alpha}\delta\alpha d\gamma$$+ \int_{\partial\Omega(\alpha)}\{\nabla(C(\alpha)). n+HC(\alpha)\}n.\nu\delta\alpha d\gamma$ .Here $\nabla$ is gradient operator with respect to $x,$ $H$ is curvature of the curve$\partial\Omega(\alpha)$ and $\nu$ is to be taken as $(0,1)$ or $(0, -1)$ .Let us.denote $w=(w_{1}, w_{2})=(\neq^{\partial u_{\alpha}}\wedge$ $,$ $\neq^{\partial u_{\alpha}}\wedge)$ With the help ofLemmas, we derive to obtain the followings.(6.1) $\frac{d}{d\alpha}[\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^{2}\tau_{ij}(u_{\epsilon}\wedge)e_{ij}(u_{e}\wedge)dx]\delta\alpha$$=- \int_{\Omega}\sum_{i,j=1}^{2}\frac{\partial\tau_{ij}(u_{e}\wedge)}{\partial x_{j}}w_{i}dx+\int_{\Gamma_{c}}\sum_{i=1}^{2}\tau_{:}(u_{e}\wedge)w_{i}d\gamma$$+ \int_{\Gamma_{p}}\sum_{i=1}^{2}\tau_{:}(u_{\epsilon}\wedge)w_{i}d\gamma$$+ \int_{\Gamma_{c}}\frac{1}{2}\sum_{i,j=1}^{2}\tau_{ij}(u_{e}\wedge)e_{ij}(u_{e}\wedge)n.\nu\delta\alpha d\gamma$(6.2) $\frac{d}{d\alpha}[\frac{\epsilon_{d}}{2}\int_{\Omega}\sim\sum_{i,j=1}^{2}\tau_{1i}(u_{e}\wedge)e_{ij}(u_{\epsilon}\wedge)dx]\delta\alpha$$=- \epsilon_{d}\int_{\Omega}\sim\sum_{:,i=1}^{2}\frac{\partial\tau_{ij}(\wedge u_{e})}{\partial x_{j}}w_{1}dx+\epsilon_{d}\int_{\Gamma_{c}}\sum_{:=1}^{2}\tilde{\tau_{:}}(u_{e}\wedge)w_{i}d\gamma$$- \int_{\Gamma_{c}}\frac{\epsilon_{d}}{2}\sum_{i.j=1}^{2}\sim_{ij}\tau(u_{e}\wedge)e\sim:;(u_{e}\wedge)n.\nu\delta\alpha d\gamma$(6.3) $\frac{d}{d\alpha}[\frac{1}{2}\int_{\Omega}\sim\sum_{:=1}^{2}\wedge u_{\epsilon i}^{2}dx]\delta\alpha$$= \int_{\tilde{\Omega}}\sum_{i=1}^{2}u_{ei}\wedge w_{i}dx-\int_{\Gamma_{c}}\frac{1}{2}\sum_{1=1}^{2}u_{ei}^{2}\wedge n.\nu\delta\alpha d\gamma$.893(6.4) $\frac{d}{d\alpha}[-\int_{\Omega}\sum_{1=1}^{2}f_{i}u_{e}\wedge;dx]\delta\alpha$$=- \int_{\Omega}\sum_{i=1}^{2}f_{i}w_{i}dx-\int_{\Gamma_{e}}\sum_{:=1}^{2}f_{i}u_{\epsilon i}\wedge n.\nu\delta\alpha d\gamma$(6.5) $\frac{d}{d\alpha}[-\int_{\Gamma_{p}}\sum_{i=1}^{2}p_{i}\wedge u_{e1}\cdot d\gamma]\delta\alpha=-\int_{\Gamma_{p}}\sum_{i=1}^{2}pwd\gamma$(6.6) $\frac{d}{d\alpha}[\int_{\Gamma_{c}}g\sqrt{u_{e1}^{2}+\epsilon_{f}^{2}\wedge}d\gamma]\delta\alpha$$= \int_{\Gamma_{c}}g\frac{\wedge u_{e1}}{\sqrt{\wedge u_{e1}^{2}+\epsilon_{f}^{2}}}w_{1}d\gamma$$+ \int_{\Gamma_{c}}g\{\nabla(\sqrt{u_{e1}^{2}+\epsilon_{f}^{2}\wedge})$ . $n+H\sqrt{\wedge u_{\epsilon 1}^{2}+\epsilon_{f}^{2}}\}n.\nu\delta\alpha d\gamma$(6.7) $\frac{d}{d\alpha}[\frac{1}{2\epsilon_{p}}\int_{\Gamma_{c}}[(u_{e2}\wedge+\alpha)^{-}]^{2}d\gamma]\delta\alpha$$=- \frac{1}{\epsilon_{p}}\int_{\Gamma_{c}}(u_{\epsilon 2}\wedge+\alpha)^{-}w_{2}d\gamma$$+ \frac{1}{2\epsilon_{p}}\int_{\Gamma_{c}}\{\nabla$ $([ (u_{e2}\wedge+\alpha)^{-}]^{2})$ . $n$$+H[(u_{e2}\wedge+\alpha)^{-}]^{2}\}n.\nu\delta\alpha d\gamma$$(6.1)\sim(6.7)$ and equations in $(\hat{P}_{e})$ give the final result :(6.8) $\frac{d}{d\alpha}[\hat{E}_{e}(\alpha)]\delta\alpha$$= \int_{\Gamma_{c}}[\frac{1}{2}\sum_{i,j=1}^{2}\{\tau_{ij}(u_{\epsilon}\wedge)e_{ij}(u_{e}\wedge)-\epsilon_{d^{\mathcal{T}}}^{\sim_{ij}}1^{\wedge}e(u_{e}\wedge)\}$$- \sum_{:=1}^{2}\{\frac{1}{2}u_{ei}^{2}\wedge+f_{i}u_{ei}\wedge\}$$+ \nabla(g\sqrt{u_{e1}^{2}+\epsilon_{f}^{2}\wedge}+\frac{1}{2\epsilon_{p}}[(u_{e2}\wedge+\alpha)^{-}]^{2}).n$$+H \{g\sqrt{u_{e1}^{2}+\epsilon_{f}^{2}\wedge}+\frac{1}{2\epsilon_{p}}[(u_{e2}\wedge+\alpha)^{-}]^{2}\}]n.\nu\delta\alpha d\gamma$ .994Since $w$ is cancelled, adjoint equaion is not necessary to be considered. We willdenote the right side of (6.8) by $\int_{\Gamma_{c}}\phi\delta\alpha d\gamma$In $ac$tual computation, we want to work out with the admissible set(6.9) $W_{ad}=\{\alpha\in C^{1,1}([a,b])|$ $0\leq\dot{\alpha}(x_{1})\leq c_{o}$ ,$|\alpha’(x_{1})|\leq c_{1}$ for $x_{1}\in[a, b]$ ,$|\alpha’’(x_{1})|\leq c_{2}$ a.e. in $(a, b)$ $\}$from which restriction for measure of $\Omega(\alpha)$ is excluded. But it is unavoidable toretain that condition. Therefore we try to include it again in cost functional as(6.10) $\hat{F}_{e}(\alpha)=\hat{E}_{e}(\alpha)+k|c_{3}-m(\Omega)|^{2}$ ,where $k$ is a positive constant and $m(\Omega)$ is notation for measure of $(\Omega(\alpha))$ .At last it becomes to study numerically the shape design problem$(\overline{SD}_{e})$ $\alpha_{\epsilon}^{*}\in W_{ad}$ s.t. $\hat{F}_{e}(\alpha_{e}^{*})=$ $\min$ $\hat{F}_{e}(\alpha)$ .$\alpha\in W_{ad}$It can easibly be seen that(6.11) $\frac{d}{d\alpha}[\hat{F}_{e}(\alpha)]\delta\alpha$$= \int_{\Gamma_{c}}\phi\delta\alpha d\gamma+2k(c_{3}-m(\Omega))\int_{\Gamma_{c}}\frac{1}{\sqrt{1+(\alpha’(x_{1}))^{2}}}\delta\alpha d\gamma$ .For certain $\alpha$ , we can write the right side of (6.11) like as $\int_{\Gamma_{c}}\Phi\delta\alpha d\gamma$7. optimizing algorithmNumerical computations are carried out as stated below.Step $0$ : Initial $\alpha\in W_{ad}$ is given.Step 1 : Solve $(\overline{M}_{\epsilon})$ .Step 2 : Compute $\Phi$ .Step 3 : Take $\delta\alpha=-\rho\Phi$ $(\rho>0)$Step 4 : Set $\alphaarrow\alpha+\delta\alpha$Go to Step 1.1095Relaxation metbod is used to solve $(\overline{M}_{e})$ To optimize $F_{e}$ , accordingto Pironneau’s inct,bod, wc take $\delta\alpha$ as $\delta\alpha=-\rho\Phi$ $(\rho>0)$ so tlurt$\frac{d}{d\alpha}[\hat{F}_{\epsilon}(\alpha)]\delta\alpha\leq 0$ . Suitable $\rho$ had to be selected in order to that $\alpha+\delta\alpha$ liesin the admissible set $W_{ad}$ .8. Numerical examplesWe take $f=(f_{1}, f_{2})=(0,0)$ for simplicity and use the data:$a=-1.0,$ $b=1.0,$ $c=1.5$ $,$ $\sim c=-0.5$ mesh size $h= \frac{1}{8}$ , $\frac{1}{16}$$c_{0}=0.5,$ $c_{1}=0.5,$ $c_{2}=4.0,$ $c_{3}=2.75$$\lambda=1.0,$ $\mu=1.0,$ $g=0.OO1$$\epsilon_{f},$$\epsilon_{d}=2h^{2},$ $h^{2},$ $\frac{2h^{2}}{2}$$k=8.0$$\epsilon_{p}=h,$ $\frac{h}{2}$ $\frac{2h}{5}$For $p=(p\iota,p_{2})$ , see the following figures.1. F.E. Browder, The fixed-poin $t$ theory of multi-valu $\epsilon d$ problems, Math. Annalen 1977(1968), 283-301.2. R. Glowinski, J.L. Lions and R. Tremolieres, ”Numerical Analysis of Variational Inequal-ities,” Studies in Mathematics and Its Applications, $vol8$ , North-Holland Publishing co.,1981.3. J. Haslinger and P. Neittanmaki, “Finite Element Approximation for Optimal ShapeDesign,” John Wiley and sons, 1988.4. I. Hlavacek, J. Haslinger, J. Necas and J. Lovisek, “Solution of Variational Inequalitiesin Mechanics,” Springer-Verlag, 1988.5. H. Kawarada, “Free BOundary problems,” Tokyo University, 1989. (in Japanese)6. O. Pironneau, “Optimal Shape Design for Elliptic System,” Springer-Verlag, 1983.1196979899I 5

Numerical Study of a Shape Design Problem in Elasticity by use of Auxiliary Domain Method

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Numerical Study of a Shape Design Problem in Elasticity by use of Auxiliary Domain Method

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