Convergence of the PML solution for elastic wave scattering by biperiodic structures

TL;DR

This paper analyzes elastic wave scattering by biperiodic structures using PML, establishing convergence and well-posedness, and extends previous results from 1D to 2D structures with numerical validation.

Contribution

It develops an exact transparent boundary condition and proves exponential convergence of the PML method for 3D elastic wave scattering by biperiodic structures, extending prior 1D results.

Findings

Proves well-posedness of the PML truncated problem

Establishes exponential convergence of the PML solution

Numerical experiments confirm the method's effectiveness

Abstract

This paper is concerned with the analysis of elastic wave scattering of a time-harmonic plane wave by a biperiodic rigid surface, where the wave propagation is governed by the three-dimensional Navier equation. An exact transparent boundary condition is developed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing a PML equivalent transparent boundary condition. The proofs rely on a careful study of the error between the two transparent boundary operators. The work significantly extend the results from the one-dimensional periodic structures to the two-dimensional biperiodic structures. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.