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.
Funder
Deutsche Forschungsgemeinschaft
Subject
Computational Mathematics,Control and Optimization,Control and Systems Engineering
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