Shape optimization

If the cost function is potential energy, Corollaries 2.13 , 2.18 can be used directly. We now consider the case where the cost function is mean compliance , that is \begin{equation} F_c(u^{\varphi _t})=\int _{\Omega (t)}fu^{\varphi _t}dx+\int _{\Gamma _N(t)}gu^{\varphi _t} ds \tag{3.1} \end{equation} and the problem (2.1) is linear ((2.28) ), then we have \begin{equation} F_c(u^{\varphi _t})=-2\mathcal{E}(u^{\varphi _t};f,g,\Omega (t)) \tag{3.2} \end{equation} Hence the shape sensitivity of mean compliance expressed by the GJ-integral is \begin{equation} \left .\frac{d}{dt}F_c(u^{\varphi _t})\right |_{t=0} =2\left \{R_{\Omega }(u,\mu _{\varphi })+\int _{\partial \Omega }fu(\mu _{\varphi }\cdot n)ds\right \} \tag{3.3} \end{equation} The shape derivative $u'^{\varphi }$ is defined by $$ u'^{\varphi }(x)=\dot{u}^{\varphi }(x)-\nabla u(x)\cdot \mu _{\varphi }(x)$$ For $u\in W^{1,p}(\Omega ), v\in W^{1,q}(\mathbb{R}^d), \frac{1}{p}+\frac{1}{q}=1,p\ge 1, q\ge 1$, the following holds $$ \left .\frac{d}{dt}\int _{\Omega (t)}u^{\varphi _t}v\, dx\right |_{t=0} =\int _{\Omega }u'^{\varphi }v\,dx+\int _{\partial \Omega }uv(\mu _{\varphi }\cdot n)ds$$ If the variational problem (2.1) is linear (2.28) and Corollary 2.13 holds, then we have the following.

Theorem 3.1,GJ-Hadamard formula

[O-K00] Assume the problem (2.1) is linear. For a given function $\Phi \in H^1(\mathbb{R}^d;\mathbb{R}^m)$, let $u_{\Phi }^{\varphi _t}$ be the solution of the problem $$ \mathcal{E}(u_{\Phi }^{\varphi _t};\Phi ,0,\Omega (t)) =\min _{v\in V^{\varphi _t}(\Omega )}\mathcal{E}(v;\Phi ,0,\Omega (t))$$ and consider $\delta R_{\omega }(u,\mu )[w]=\lim _{\epsilon →0}\epsilon ^{-1}\left \{R_{\omega }(u+\epsilon w,\mu )-R_{\omega }(u,\mu )\right \}$. Then we have \begin{equation} \left .\frac{d}{dt}\int _{\Omega }\Phi u^{\varphi _t}\, dx\right |_{t=0} =\delta R_{\Omega }(u,\mu _{\varphi })[u_{\Phi }]+\int _{\partial \Omega }fu_{\Phi }(\mu _{\varphi }\cdot n)\,ds \tag{3.4} \end{equation}

Notes

[Corollary 2.81, Sok92], The mapping $t \mapsto u^{\varphi _t}\circ \varphi _t$ is weakly differentiable in $L^2(\Omega )$) and the weak material derivative 3.4 under Poisson with Dirichlet boundary condition. (see also [Corollary 3.1, N-S13]).

By defining $\delta P_{\omega }(u,\mu )[w]$ like $\delta R_{\omega }(u,\mu )[w]$, we get the following.

Corollary 3.2

If $u$ and $w$ is regular inside $\omega $, then $\delta P_{\omega }(u,\mu )[w]+\delta R_{\omega }(u,\mu )[w]=0$ for all $\mu \in W^{1,\infty }(\mathbb{R}^d;\mathbb{R}^d)$.

Consider the Poisson equation with Dirichlet boundary condition and its Green's function $\Delta _xG^{\varphi _t}(x,y)=-\delta (x-y)$($\delta $ is delta function); then we have $u^{\varphi _t}(z)=\int _{\Omega }G^{\varphi _t}(z,x)f(x)dy, u_{\Phi }^{\varphi _t}(z)=\int _{\Omega }G^{\varphi _t}(z,y)\Phi (y)dy$. From (2.46) , Theorem 3.1 and Corollary 3.2 we obtain Hadamard variational formula [Had68] \begin{equation} G'^{\varphi }(x,y)= \int _{\partial \Omega }\frac{\partial G}{\partial n_z}(x,z) \frac{\partial G}{\partial n_z}(y,z)(\mu (z)\cdot n(z))ds_z. \tag{3.5} \end{equation} Thus Theorem 3.1 is a generalization of Hadamard variational formula. Here we consider the const function by $\widehat{g}(z) \in C^2(\mathbb{R}^m)$ \begin{equation} F_g(u^{\varphi _t})=\int _{\Omega (t)}\widehat{g}(u^{\varphi _t})\, dx, \tag{3.6} \end{equation} When $\widehat{g}(z)=(z-u_{d})^2$, the cost function $F_g$ is least mean-square error. From Theorem 3.1 , $u'^{\varphi }$ belongs to $L^2(\Omega ;\mathbb{R}^m)$, we have \begin{equation} \left .\frac{d}{dt}F_g(u^{\varphi _t})\right |_{t=0} =\int _{\Omega }\nabla _z\widehat{g}(u)u'^{\varphi }dx+\int _{\partial \Omega } \widehat{g}(u)(\mu _{\varphi }\cdot n)\, ds. \tag{3.7} \end{equation} Let $u_g$ be the solution of the problem \begin{equation} \mathcal{E}(u_g;\nabla _z \widehat{g}(u),0,\Omega ) =\min _{v\in V(\Omega )} \mathcal{E}(v;\nabla _z \widehat{g}(u),0,\Omega ) \tag{3.8} \end{equation} Then from Theorem 3.1 , the first term in the right-hand side of (3.7) is written by $\delta R(u,\mu _{\varphi })[u_g]$ and we arrive at \begin{eqnarray} \left .\frac{d}{dt}F_g(u^{\varphi _t})\right |_{t=0} &=&\delta R_{\Omega }(u,\mu _{\varphi })[u_g]\notag \\ &&+\int _{\partial \Omega } (fu_g+\widehat{g}(u)-\nabla _z\widehat{g}(u)u)(\mu _{\varphi }\cdot n)ds \tag{3.9} \end{eqnarray} The method of erasing $u'^{\varphi }$ with $u_g$ is called the ``adjoint variable method'' and it is often used in shape optimization.

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