Numerical shape and topology optimization of regions supporting the boundary conditions of a physical problem

This article deals with a particular class of shape and topology optimization problems: the optimized design is a region $G$ of the boundary $\partial \Omega$ of a given domain $\Omega$, which supports a particular type of boundary conditions in the considered physical problem. In our analyses, we develop adapted versions of the notions of shape and topological derivatives, which are classically tailored to functions of a ``bulk'' domain. This leads to two complementary notions of derivatives for a quantity of interest $J(G)$ depending on a region $G \subset \partial \Omega$: on the one hand, we elaborate on the boundary variation method of Hadamard for evaluating the sensitivity of $J(G)$ with respect to ``small'' perturbations of the boundary of $G$ within $\partial \Omega$. On the other hand, we use techniques from asymptotic analysis to appraise the sensitivity of $J(G)$ with respect to the addition of a new connected component to the region $G$, shaped as a ``small'' surface disk. The calculation of both types of derivatives raises original difficulties, which are carefully detailed in a simple mathematical setting based on the conductivity equation. We notably propose formal arguments to calculate our derivatives with a minimum amount of technicality, and show how they can be generalized to handle more intricate problems, arising for instance in the contexts of acoustics and structural mechanics, respectively governed by the Helmholtz and linear elasticity equations. In numerical applications, our derivatives are incorporated into a recent algorithmic framework for tracking arbitrarily dramatic motions of a region $G$ within a fixed ambient surface, which combines the level set method with remeshing techniques to offer a clear, body-fitted discretization of the evolving region. Finally, various 3d numerical examples are presented to illustrate the salient features of our analysis.