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

TL;DR

This paper introduces an unfitted finite element method using cut finite elements and cubic splines for PDE-constrained shape optimization via shape gradient flow, achieving optimal convergence rates.

Contribution

It presents a novel unfitted finite element approach with cubic spline boundary evolution and ghost penalization, improving upon previous evolving finite element methods.

Findings

Proved optimal convergence rates for the proposed method.

Validated convergence rates through numerical experiments.

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.