Boundary integral formulations for transient linear thermoelasticity with combined-type boundary conditions

TL;DR

This paper develops boundary integral formulations for transient thermoelastic problems with mixed boundary conditions, proving coercivity and solution existence using Laplace domain techniques, supporting advanced numerical methods.

Contribution

It introduces coercive boundary integral operators for thermoelastic equations with mixed boundary conditions and establishes existence and uniqueness of solutions in the time domain.

Findings

Proves coercivity of combined boundary integral operators.

Establishes existence and uniqueness of solutions.

Provides mathematical foundation for boundary element and convolution quadrature methods.

Abstract

We study boundary integral formulations for an interior/exterior initial boundary value problem arising from the thermo-elasto-dynamic equations in a homogeneous and isotropic domain. The time dependence is handled, based on Lubich's approach, through a passage to the Laplace domain. We focus on the cases where one of the unknown fields satisfies a Dirichlet boundary condition, while the other one is subject to conditions of Neumann type. In the Laplace domain, combined single- and double-layer potential boundary integral operators are introduced and proven to be coercive. Based on the Laplace domain estimates, it is possible to prove the existence and uniqueness of solutions in the time domain. This analysis complements previous results that may serve as the mathematical foundation for discretization schemes based on the combined use of the boundary element method and convolution quadrature.