Yasuhiro Matsumoto

Yasuhiro Matsumoto, Taizo Maruyama

This paper describes a fast direct boundary element method for elastodynamic transmission problems in two dimensions, which can be used for analyzing elastic wave scattering by an inclusion. We develop an efficient solver based on a discretization method that is broadly applicable regardless of the inclusion shape. From the smoothness of the solutions of the Navier--Cauchy equation, it is reasonable that the displacement is approximated by the piecewise linear bases and the traction is approximated by the piecewise constant bases. However, in this mixed bases strategy, Calderón preconditioning, that is, an analytical preconditioning with excellent performance, cannot be applied. To circumvent this issue, we developed a fast direct solver formulated using both Burton--Miller and Poggio--Miller--Chang--Harrington--Wu--Tsai (PMCHWT) boundary integral equations. Our method uses a technique based on the proxy method for low-rank approximation of the coefficient matrix's off-diagonal blocks. To handle transmission problems, the proposed fast direct solver uses separate binary tree partitions for nodes and elements. Numerical examples demonstrate that our solver achieves linear computational complexity at fixed low frequencies and can efficiently handle problems with multiple right-hand sides. Notably, the solver based on the Burton--Miller formulation is approximately 20\% faster than the one using the PMCHWT formulation. Our new method provides a versatile, fast solver, whose performance is relatively independent of the shape of inclusions and computational parameters, such as density, for elastodynamic transmission problems.

Yasuhiro Matsumoto

This paper discusses a fast direct solver using boundary integral equations for Helmholtz transmission problems involving multiple inclusions in two dimensions. Efficiently addressing scattering problems in the presence of numerous inclusions remains a key challenge for various practical applications. For problems involving a large number of scatterers, the number of iterations in Krylov subspace methods is known to increase significantly. This occurs even when using second-kind boundary integral equations, which are typically recognized for their rapid convergence. We consider a fast direct solver as an alternative, an approach that has been less commonly explored for transmission problems with disjoint multiple inclusions. The low-rank approximation based on the proxy method achieve speedup by calculating interactions between disjoint scatterers without the terms derived from the internal integral representation. Notably, this advantage applies to the Poggio--Miller--Chang--Harrington--Wu--Tsai (PMCHWT) formulation but breaks down in the Burton--Miller case. Numerical examples demonstrate that the proposed solver can compress the system of linear algebraic equations to a size of $O(ωD)$, where $ω$ is the frequency of the incident wave and $D$ is the diameter of the (smallest) bounding box enclosing the multiple inclusions. The total computational cost scales as $O(N^{1.5})$ $(= O(\sqrt{N}^3))$ at most for a fixed $ω$ when the inclusions are arranged on a grid. Moreover, the PMCHWT formulation, that omits the interior term in the proxy method, is approximately six times faster than the Burton--Miller formulation when treating each inclusion as a cell. Furthermore, in the same setting, the former can compress the size of the system of linear algebraic equations by half compared to the latter.