A three-dimensional shape optimization for transient acoustic scattering problems using the time-domain boundary element method

We develop a three-dimensional shape optimization (SO) framework for the wave equation with taking the unsteadiness into account. Resorting to the adjoint variable method, we derive the shape derivative (SD) with respect to a deformation (perturbation) of an arbitrary point on the target surface of acoustic scatterers. Successively, we represent the target surface with non-uniform rational B-spline patches and then discretize the SD in term of the associated control points (CPs), which are useful for manipulating a surface. To solve both the primary and adjoint problems, we apply the time-domain boundary element method (TDBEM) because it is the most appropriate when the analysis domain is the ambient air and thus infinitely large. The issues of the severe computational cost and instability of the TDBEM are resolved by exploiting the fast and stable TDBEM proposed by the present authors. Instead, since the TDBEM is mesh-based and employs the piecewise-constant element for space, we introduce some approximations in evaluating the discretized SD from the two solutions of TDBEM. By regarding the evaluation scheme as the computation of the gradient of the objective functional, given as the summation of the absolute value of the sound pressure over the predefined observation points, we can solve SO problems with a gradient-based non-linear optimization solver. To assess the developed SO system, we performed several numerical experiments from the perspective of verification and application with satisfactory results.