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We investigate the problem of finding the optimal shape and topology of asystem of acoustic lenses in a dissipative medium. The sound propagation isgoverned by a general semilinear strongly damped wave equation. We introduce aphase-field formulation of this problem through diffuse interfaces between thelenses and the surrounding fluid. The resulting formulation is shown to bewell-posed and we prove that the corresponding optimization problem has aminimizer. By analyzing properties of the reduced objective functional andwell-posedness of the adjoint problem, we rigorously derive first-orderoptimality conditions for this problem. Additionally, we consider the$\Gamma$-limit of the reduced objective functional and in this way establish arelation between the diffuse interface problem and a perimeter-regularizedsharp interface shape optimization problem.

A phase-field approach to shape and topology optimization of acoustic waves in dissipative media