Phase-Field Methods for Spectral Shape and Topology Optimization

TL;DR

This paper introduces a phase-field approach for spectral shape and topology optimization, enabling the adjustment of domain shapes to optimize Laplace eigenvalues with rigorous analysis and numerical validation.

Contribution

It develops a novel phase-field method for spectral shape optimization, deriving optimality conditions and analyzing the sharp interface limit.

Findings

Successful numerical simulations demonstrate the method's effectiveness.

Theoretical derivation of optimality conditions and sharp interface limit.

Extension of eigenvalue problems via phase-field dependent coefficients.

Abstract

We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem.