A fast direct solver for two-dimensional transmission problems of elastic waves

Abstract

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.

Figures (20)

• Figure 1: Diagram of transmission problems for an elastic wave
• Figure 2: Overview of the proxy method. This figure corresponds to the case of $\{ \varphi^r \}$. The circles with dashed lines are proxy boundaries. The boundary $\Gamma$ is partitioned to cells, which correspond to the sum of each support of the basis functions denoted as $\bigcup_{r=1}^{n} \mathop{\mathrm{supp}}\nolimits(\varphi^{J_{i}^{\varphi}(r)})$ for $i = 1, 2, \ldots, p$. Cells are merged to form a parent-level cell by computing and factorizing the interactions between a cell and its corresponding proxy boundary, and the interactions between a cell and its adjacent cells
• Figure 3: Correspondence between index set ${J_i^{\prime}}^{\varphi}$ and boundaries. The circle represented by the dashed line indicates a proxy boundary. The index ${J_i^{\prime}}^{\varphi}$ corresponds to the union of indices on the proxy boundary enclosing the cell associated with $J_i^{\varphi}$, and indices on the enclosed part of the adjacent cells of $J_i^{\varphi}$
• Figure 4: Geometry for the unit circle
• Figure 5: Geometry for the square

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5