We have been investigating applications of a topology optimisation method with the level set method. In this study, to further enhance the applicability of the method, we investigate a topology optimisation method for threedimensional scalar wave scattering problems which can be defined in an unbounded domain. To this end, the fast multipole boundary element method (FMBEM), which can deal with the unbounded domain accurately and efficiently, is implemented in the proposed optimisation method. A detail of the algorithm of the topology optimisation with the level set method and the FMBEM is presented. Also, a rigorous derivation of the topological derivative, which characterises the sensitivity of the objective function when an infinitely small spherical object appears, using spherical functions is presented. After validating the topological derivatives with approximated ones, we show the efficiency of the proposed optimisation method with a numerical benchmark. Through these numerical experiments, we conclude that the proposed topological optimisation with the level set method and the FMBEM can be applied to scattering problems in acoustics.
This paper presents a structural optimisation method in three-dimensional acoustic-elastic coupled problems. The proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points. In the process of the optimisation, configuration of the elastic materials is expressed with a level set function, and the distribution of the level set function is iteratively updated with the help of the topological derivative. The topological derivative is associated with state and adjoint variables which are the solutions of the acoustic-elastic coupled problems. In this paper, the acoustic-elastic coupled problems are solved by a BEM-FEM coupled solver, in which the fast multipole method (FMM) and a multi-frontal solver for sparse matrices are efficiently combined. Along with the detailed formulations for the topological derivative and the BEM-FEM coupled solver, we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves, from which we confirm the effectiveness of the present method.
SUMMARYPreconditioning methods based on Calderon's formulae for the periodic fast multipole method for elastodynamics in 3D are investigated. Three different types of formulations are proposed. The first type is a preconditioning just by appropriately ordering the coefficient matrix without multiplying preconditioners. The other two types utilise preconditioners constructed using matrices needed in the main fast multipole method algorithms. We make several numerical experiments with proposed preconditioners to confirm the efficiency of these proposed methods. We also conclude that the preconditioning of the first type is faster with respect to the computational time than other preconditioning methods discussed in this article. Copyright
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