Phase-field methods for spectral shape and topology optimization

¹ Fakulät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany
² Mathematisches Institut, Universität Koblenz, 56070 Koblenz, Germany
³ Hausdorff Center for Mathematics, University of Bonn, 53115 Bonn, Germany

^* Corresponding author: paul.huettl@mathematik.uni-regensburg.de

Received: 7 July 2021
Accepted: 21 December 2022

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.