  A gradient-based method for shape optimization problems constrained by the
acoustic wave equation is presented. The method makes use of high-order
accurate finite differences with summation-by-parts properties on multiblock
curvilinear grids to discretize in space. Representing the design domain
through a coordinate mapping from a reference domain, the design shape is
obtained by inverting for the discretized coordinate map. The adjoint state
framework is employed to efficiently compute the gradient of the loss
functional. Using the summation-by-parts properties of the finite difference
discretization, we prove stability and dual consistency for the semi-discrete
forward and adjoint problems. Numerical experiments verify the accuracy of the
finite difference scheme and demonstrate the capabilities of the shape
optimization method on two model problems with real-world relevance.
