An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

TL;DR

This paper introduces an adaptive finite element method combined with a Dirichlet-to-Neumann operator truncation for efficiently solving elastic wave scattering problems by bi-periodic structures, with proven exponential error decay.

Contribution

It develops an a posteriori error estimate-based adaptive finite element DtN method that accounts for both discretization and truncation errors in elastic wave scattering.

Findings

The method achieves exponential decay of truncation error.

Numerical experiments confirm the effectiveness of the adaptive approach.

The approach accurately models elastic wave scattering by bi-periodic structures.

Abstract

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an open domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.