Phase-field methods for spectral shape and topology optimization

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