Sharp interface limit for a phase field model in structural optimization (Q2813311)

The authors analyze a structural optimization problem for a mixture of two elastic materials in a fixed container \(\Omega \subset \mathbb{R}^{d}\), \( d=2,3\). The bounded and Lipschitz domain \(\Omega \) may be decomposed as \( \Omega _{1}\cup \Omega _{2}\cup \Gamma \) and the interface \(\Gamma \) may be decomposed as \(\Gamma =\Gamma _{D}\cup \Gamma _{g}\) with \(\Gamma _{D}\cap \Gamma _{g}=\emptyset \) and \(\mathcal{H}^{d-1}(\Gamma _{D})>0\). The elastic materials satisfy the equilibrium equations \(-\nabla \cdot (D_{2}W_{i}(x, \mathcal{E}(u))=f\) in \(\Omega _{i}\) and the boundary conditions \( D_{2}W_{i}(x,\mathcal{E}(u))\cdot n=g\) on \(\Gamma _{g}\cap \partial \Omega _{i}\), \(u=u_{D}\) on \(\Gamma _{D}\cap \partial \Omega _{i}\). Here \(D_{2}W_{i}\) means the derivative with respect to the second component, \(g\in L^{2}(\Gamma _{g})\), \(f\in L^{2}(\Omega )\), \(\mathcal{E}(u)=\frac{1}{2} (\nabla u+\nabla u^{T})\) and \(W_{i}(x,\mathcal{E})=\frac{1}{2}(\mathcal{E}- \overline{\mathcal{E}}_{i}):\mathcal{C}_{i}(\mathcal{E}-\overline{\mathcal{E} }_{i})\), where \(\mathcal{C}_{i}\) is the elasticity tensor which satisfies the usual properties and \(\overline{\mathcal{E}}_{i}\) is the value of the strain when the \(i\)th material is unstressed. The design variable is a measurable function \(\varphi :\Omega \rightarrow \mathbb{R}\) such that \(\{x\in \Omega \mid \varphi (x)=1\}=\Omega _{1}\) (resp. \(\{x\in \Omega \mid \varphi (x)=-1\}=\Omega _{2}\)). Introducing a variational formulation of the problem and using classical arguments, the authors prove the existence of a unique weak solution to this problem, leading to a solution operator \( S:\{\varphi \in L^{1}(\Omega )\mid \left| \varphi \right| \leq 1\) a.e. in \(\Omega \}\rightarrow H_{D}^{1}(\Omega )\). The shape problem consists to minimize \(\mathbb{H}(u)=\int_{\Omega }h_{\Omega }(x,u)dx+\int_{\Gamma _{g}}h_{\Gamma }(s,u)ds\), for Caratheodory functions \( h_{\Omega }\) and \(h_{\Gamma }\) which satisfy further hypotheses. The authors add two different regularizing terms. In the first case, they add the term \( \gamma c_{0}P_{\Omega }(\{\varphi =1\})\) where \(P_{\Omega }\) denotes the perimeter operator. In the second case, they add the term \(\gamma \int_{\Omega }\frac{\varepsilon }{2}\left| \nabla \varphi \right| ^{2}+\frac{1}{\varepsilon }\psi (\varphi )dx\). In both cases, they prove an existence and uniqueness result and they build the necessary optimality conditions. In the second case, they prove a \(\Gamma \)-convergence result taking the limit when \(\varepsilon \) goes to 0. The paper is completed with the presentation of numerical simulations obtained using the projected gradient method.