Phase-Field Methods for Spectral Shape and Topology Optimization
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.
Cite
@article{arxiv.2107.03159,
title = {Phase-Field Methods for Spectral Shape and Topology Optimization},
author = {Harald Garcke and Paul Hüttl and Christian Kahle and Patrik Knopf and Tim Laux},
journal= {arXiv preprint arXiv:2107.03159},
year = {2023}
}