A time-dependent adjoint approach for obtaining sensitivity derivatives for shape optimizations of two-dimensional acoustic metamaterials and phononic crystals is presented. The acoustic wave propagation problem is solved in the time-domain using a Streamline Upwind/Petrov Galerkin formulation. Surface parameterization is accomplished using control grids, which are based on a Laplacian-type equation. The gradient-based design procedure is suitable for large numbers of design variables, and results are shown on achieving effective material properties with a unit cell, and the broadband noise reduction with periodic arrays of stainless-steel cylinders.
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