An unfitted finite element method for PDE-constrained shape optimization via shape gradient flow

Abstract

In this paper, we propose an unfitted finite element method to solve PDE-constrained shape optimization problems via shape gradient flow. The shape gradient flow system consists of the state equation, the adjoint equation, the velocity equation, as well as the flow map that generates the evolution of the boundary driven by the velocity field, which can be viewed as a limit system of the classical shape gradient descent algorithm. In \cite{GongLiRao} the authors proposed an evolving finite element method to solve the shape gradient flow system. Instead, in this paper, we propose an unfitted finite element method in which the evolution of the boundary is realized by cubic splines and the equations are solved by cut finite element methods with ghost penalization. Under reasonable assumptions, we are able to prove some optimal convergence rates that are further validated by numerical experiments.