Abstract
It is quite important, in industrial communities, to design devices in order to let them have required capabilities. In the design process, we sometimes use the topology optimisation, which is considered as the most flexible computer-aided design method. For the topology optimisation in wave problems which is defined in infinite domain, the boundary element method (BEM) is considered as the most suitable numerical method because of its capability of dealing with the infinite domain. Since the numerical complexity of the BEM is at best O(N^2), where N is the degree of freedom, the BEM is required to be combined with an acceleration technique. As one of the most accepted acceleration techniques, we usually use the fast multipole boundary element method (FMBEM) whose computational complexity is O(N log^α N) with small α. In the FMBEM, an algebraic equation Ax=b is solved with an iterative solver in which matrix-vector products are carried out with the fast multipole method (FMM). It is, however, not appropriate to use the iterative-solver-based FMBEM for the optimisation problems since, in the optimisation problems, it is required to solve two boundary value problems (BVP), called the forward and the adjoint problems. Hence, in this study, we utilise a direct-solver-based FMM to solve the BVPs, in which the coefficient matrix A is compressed with an algebraic method. We show that the proposed direct solver can effectively compute the topological derivative in the optimisation process and can solve a topology optimisation problem for 2D acoustics with the impedance boundary conditions.
