Higher-order time domain boundary elements for elastodynamics -- graded meshes and hp versions

TL;DR

This paper develops higher-order time domain boundary element methods with graded meshes and hp techniques for elastodynamics problems involving singularities at corners and edges, achieving quasi-optimal convergence.

Contribution

It introduces new hp and graded boundary element methods tailored for elastodynamics in polyhedral domains, with detailed singularity analysis and proven quasi-optimal estimates.

Findings

Numerical examples confirm theoretical convergence rates.

Methods effectively handle singularities at corners and edges.

Results demonstrate quasi-optimal approximation of solutions.

Abstract

The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasi-optimal estimates for piecewise polynomial approximations of the Dirichlet trace of the solution and the traction. The results are applied to hp and graded versions of the time domain boundary element method for the weakly singular and the hypersingular integral equations. Numerical examples confirm the theoretical results for the Dirichlet and Neumann problems for screens and for polygonal domains in 2d. They exhibit the expected quasi-optimal convergence rates and the singular behavior of the solutions.