An accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions

TL;DR

This paper introduces a new boundary integral equation method for dynamic poroelasticity in two dimensions, featuring regularized operators and spectral properties that improve accuracy and computational efficiency.

Contribution

It proposes a novel regularized boundary integral equation and new formulations for strongly-singular and hyper-singular operators in 2D dynamic poroelasticity.

Findings

Eigenvalues of the new integral equation are bounded away from zero and infinity.

The reformulated operators improve numerical stability and accuracy.

Numerical examples demonstrate the method's effectiveness.

Abstract

This paper is concerned with the boundary integral equation method for solving the exterior Neumann boundary value problem of dynamic poroelasticity in two dimensions. The main contribution of this work consists of two aspescts: the proposal of a novel regularized boundary integral equation, and the presentation of new regularized formulations of the strongly-singular and hyper-singular boundary integral operators. Firstly, turning to the spectral properties of the double-layer operator and the corresponding Calder\'{o}n relation of the poroelasticity, we propose the novel low-GMRES-iteration integral equation whose eigenvalues are bounded away from zero and infinity. Secondly, with the help of the G\"{u}nter derivatives, we reformulate the strongly-singular and hyper-singular integral operators into combinations of the weakly-singular operators and the tangential derivatives. The accuracy and efficiency of the proposed methodology are demonstrated through several numerical examples.