Optimization of the shape of regions supporting boundary conditions

Abstract

This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a ‘physical’ partial differential equation. At first, we analyze a model situation involving the solution $$u_\varOmega $$ to a Laplace equation in a domain $$\varOmega $$ ; the boundary $$\partial \varOmega $$ is divided into three parts $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ , supporting respectively homogeneous Dirichlet, homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative $$J^\prime (\varOmega )(\theta )$$ of a general objective function $$J(\varOmega )$$ of the domain is calculated in the framework of Hadamard’s method when the considered deformations $$\theta $$ are allowed to modify the geometry of $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ (i.e. $$\theta $$ does not vanish on the boundary of these regions). The structure of this shape derivative turns out to depend very much on the regularity of $$u_\varOmega $$ near the boundaries of the regions $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ . For this reason, in particular, $$J^\prime (\varOmega )$$ is difficult to calculate and to evaluate numerically when the transition $$\overline{\varGamma _D} \cap {\overline{\varGamma }}$$ between homogeneous Dirichlet and homogeneous Neumann boundary conditions is subject to optimization. To overcome this difficulty, an approximation method is proposed, in which the considered ‘exact’ Laplace equation with mixed boundary conditions is replaced with a ‘smoothed’ version, featuring Robin boundary conditions on the whole boundary $$\partial \varOmega $$ with coefficients depending on a small parameter $$\varepsilon $$ . We prove the consistency of this approach in our model context: the approximate objective function $$J_\varepsilon (\varOmega )$$ and its shape derivative converge to their exact counterparts as $$\varepsilon $$ vanishes. Although it is rigorously justified only in a model problem, this approximation methodology may be adapted to many more complex situations, for example in three space dimensions, or in the context of the linearized elasticity system. Various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process.