Regularized boundary integral equation methods for open-arc scattering problems in thermoelasticity

TL;DR

This paper introduces new boundary integral equation methods for thermoelastic scattering problems involving open-arcs, effectively reducing computational complexity and improving accuracy through regularization and spectral quadrature.

Contribution

The paper develops novel regularized boundary integral equation formulations for open-arc thermoelastic scattering, explicitly handling edge singularities and reducing iteration counts.

Findings

Reduced iteration numbers in solving discretized systems

High accuracy demonstrated with spectral quadrature

Effective handling of edge singularities in open-arc problems

Abstract

This paper devotes to developing novel boundary integral equation (BIE) solvers for the problem of thermoelastic scattering by open-arcs with four different boundary conditions in two dimensions. The proposed methodology is inspired by the Calder\'on formulas, whose eigenvalues are shown to accumulate at particular points depending only on Lam\'e parameters, satisfied by the thermoelastic boundary integral operators (BIOs) on both closed- and open-surfaces. Regularized BIEs in terms of weighted BIOs on open-arc that explicitly exhibits the edge singularity behavior, depending on the types of boundary conditions, of the unknown potentials are constructed to effectively reduce the required iteration number to solve the corresponding discretized linear systems. We implement the new formulations utilizing regularizations of singular integrals, which reduces the strongly- and hyper-singular integrals into weakly-singular integrals. Combined with spectrally accurate quadrature rules, numerical examples are presented to illustrate the accuracy and efficiency of the proposed solvers.