Volume 29, 2023
|Number of page(s)||57|
|Published online||19 January 2023|
Phase-field methods for spectral shape and topology optimization
Fakulät für Mathematik, Universität Regensburg,
2 Mathematisches Institut, Universität Koblenz, 56070 Koblenz, Germany
3 Hausdorff Center for Mathematics, University of Bonn, 53115 Bonn, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 21 December 2022
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.
Mathematics Subject Classification: 35P05 / 35P15 / 35R35 / 49M05 / 49M41 / 49K20 / 49J20 / 49J40 / 49Q10 / 49R05
Key words: Eigenvalue optimization / shape optimization / topology optimization / PDE constrained optimization / phase-field approach / first order condition / sharp interface limit / Γ-limit / finite element approximation
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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