A topology optimisation for three-dimensional acoustics with the level set method and the fast multipole boundary element method

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.

[1]  M. Wang,et al.  Piecewise constant level set method for structural topology optimization , 2009 .

[2]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[3]  Marc Bonnet,et al.  FM-BEM and Topological Derivative Applied to Acoustic Inverse Scattering , 2007 .

[4]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[5]  R. Feijóo,et al.  Topological sensitivity analysis , 2003 .

[6]  Martin P. Bendsøe,et al.  Optimization of Structural Topology, Shape, And Material , 1995 .

[7]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[8]  Takayuki Yamada,et al.  A Level Set-Based Topology Optimization Method for Maximizing Thermal Diffusivity in Problems Including Design-Dependent Effects , 2011 .

[9]  Ole Sigmund,et al.  Towards all-dielectric, polarization-independent optical cloaks , 2012 .

[10]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[11]  N. Nishimura.  Fast multipole accelerated boundary integral equation methods , 2002 .

[12]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[13]  R. Duraiswami,et al.  Fast Multipole Methods for the Helmholtz Equation in Three Dimensions , 2005 .

[14]  Toshiro Matsumoto,et al.  A Level Set-Based Topology Optimization Method Using the Boundary Element Method in Three Dimension , 2012 .

[15]  V. Rokhlin.  Rapid Solution of Integral Equations of Scattering Theory , 1990 .

[16]  Kazuhisa Abe,et al.  A BE-based shape optimization method enhanced by topological derivative for sound scattering problems , 2010 .

[17]  Takayuki Yamada,et al.  A topology optimization method based on the level set method incorporating a fictitious interface energy , 2010 .

[18]  N. Olhoff,et al.  Topological design of vibrating structures with respect to optimum sound pressure characteristics in a surrounding acoustic medium , 2010 .

[19]  N. Nishimura,et al.  Calderon's preconditioning for periodic fast multipole method for elastodynamics in 3D , 2012 .