A fast direct solver for two-dimensional transmission problems of elastic waves
This provides a versatile and shape-independent solver for engineers and physicists analyzing elastic wave scattering, though it is incremental as it builds on existing boundary element methods with specific optimizations.
The paper tackles the problem of efficiently solving two-dimensional elastodynamic transmission problems for elastic wave scattering by inclusions, developing a fast direct boundary element method that achieves linear computational complexity at fixed low frequencies and is approximately 20% faster with the Burton–Miller formulation compared to PMCHWT.
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.