Linearly scalable fast direct solver based on proxy surface method for two-dimensional elastic wave scattering by cavity

TL;DR

The paper addresses efficient simulation of 2D elastic wave scattering by cavities, where standard boundary-element methods yield dense, large systems. It introduces a proxy-surface-based fast direct solver, a variant of the Martinsson-Rokhlin approach, formulated with a Galerkin Burton-Miller-type boundary integral equation to avoid fictitious eigenfrequencies and enable shared low-rank representations of off-diagonal blocks. The solver operates in two stages (upward compression and downward reconstruction) and leverages proxy surfaces to obtain shared coefficient matrices, achieving near-$O(N)$ complexity in the low-frequency regime and strong parallel scalability, including efficient handling of multiple right-hand sides. The results demonstrate accurate solutions, no fictitious frequency spikes, and substantial speedups, highlighting the method’s practical impact for elastodynamic analysis and its potential extension to three dimensions.

Abstract

This paper proposes an $O(N)$ fast direct solver for two-dimensional elastic wave scattering problems. The proxy surface method is extended to elastodynamics to obtain shared coefficients for low-rank approximations from discretized integral operators. The proposed method is a variant of the Martinsson-Rokhlin-type fast direct solver. Our variant avoids the explicit computation of the inverse of the coefficient matrix, thereby reducing the required number of matrix-matrix multiplications. Numerical experiments demonstrate that the proposed solver has a complexity of $O(N)$ in the low-frequency range and has a highly parallel computation efficiency with a strong scaling efficiency of 70\%. Furthermore, multiple right-hand sides can be solved efficiently; specifically, when solving problems with 180 right-hand side vectors, the processing time per vector from the second vector onward was approximately 28,900 times faster than that for the first vector. This is a key advantage of fast direct methods.

Figures (9)

• Figure 1: Diagram of elastic wave scattering by cavity.
• Figure 2: Binary tree decomposition with level 3 as deepest level. The root of the binary tree is level $0$ and has cell index $0$. A parent cell whose index is $i$ has two child cells whose indices are $2i+1$ and $2i+2$.
• Figure 3: Proxy surface setting. $\Gamma_i$ is a subset of $\Gamma$ with respect to the $i$-th cell at level $\ell$. The proxy surface is a local virtual boundary that encloses $\Gamma_{i}$. $\Gamma_{i}^{\prime}$ is a union of the proxy surface and subsets of $\Gamma$ enclosed by the local virtual boundary. $\Gamma_{+}$ is a subset of $\Gamma$ that is outside the proxy surface. By using the local virtual boundary instead of $\Gamma_{+}$ to calculate the interaction with $\Gamma_i$, we can make the fast direct solver have a complexity of $O(N)$.
• Figure 4: Scatter used in numerical examples. It was generated using $(x_1, x_2) = \qty((r + a \cos(b \theta)) \cos(\theta)/(1 + a), ((r + a \cos(b \theta)) \sin(\theta))/(1 + a))$ with $r = 1$, $a = 0.3$, and $b = 3$. It is supposed that the representative length of this mesh is 1.
• Figure 5: $N$ versus elapsed time for proposed fast direct solver (FDS) and conventional BEM (Conv). FDS and Conv were parallelized using OpenMP. The plots show results for 1 and 112 CPU cores. The plot of FDS (1 core) indicates that the proposed FDS has a complexity of $O(N)$. FDS (112 cores) is about 9.5 times faster than Conv (112 cores) at $N = 10240$. FDS (112 cores) is about 78 times faster than FDS (1 cores) at $N = 102400$. FDS (1 core) is faster than Conv (1 core) at $N \geq 800$. FDS (112 cores) is faster than Conv (112 cores) at $N \geq 3200$.