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Convergence of the PML method for thermoelastic wave scattering problems

Qianyuan Yin, Changkun Wei, Bo Zhang

TL;DR

This paper investigates 3D time-harmonic thermoelastic obstacle scattering governed by Biot's system with a Dirichlet boundary and Kupradze radiation. It develops a uniaxial Cartesian PML based on complex coordinate stretching to truncate the unbounded domain and proves well-posedness of the truncated PML problem for all frequencies except a discrete set using analytic Fredholm theory. The main result is the exponential convergence of the PML method with respect to the PML thickness $d$ and absorption parameter $\alpha_0$, established via precise decay of the PML extension and an error bound between the original and PML Dirichlet-to-Neumann maps. The analysis relies on coercivity/ellipticity under explicit parameter constraints and a careful treatment of the PML layer. The work advances numerical analysis for thermoelastic scattering and provides a framework for future exploration of alternative PML geometries and boundary conditions.

Abstract

This paper is concerned with the thermoelastic obstacle scattering problem in three dimensions. A uniaxial perfectly matched layer (PML) method is firstly introduced to truncate the unbounded scattering problem, leading to a truncated PML problem in a bounded domain. Under certain constraints on model parameters, the well-posedness for the truncated PML problem is then proved except possibly for a discrete set of frequencies, based on the analytic Fredholm theory. Moreover, the exponential convergence of the uniaxial PML method is established in terms of the thickness and absorbing parameters of PML layer. The proof is based on the PML extension technique and the exponential decay properties of the modified fundamental solution. As far as we know, this is the first convergence result of the PML method for the time-harmonic thermoelastic scattering problem.

Convergence of the PML method for thermoelastic wave scattering problems

TL;DR

This paper investigates 3D time-harmonic thermoelastic obstacle scattering governed by Biot's system with a Dirichlet boundary and Kupradze radiation. It develops a uniaxial Cartesian PML based on complex coordinate stretching to truncate the unbounded domain and proves well-posedness of the truncated PML problem for all frequencies except a discrete set using analytic Fredholm theory. The main result is the exponential convergence of the PML method with respect to the PML thickness and absorption parameter , established via precise decay of the PML extension and an error bound between the original and PML Dirichlet-to-Neumann maps. The analysis relies on coercivity/ellipticity under explicit parameter constraints and a careful treatment of the PML layer. The work advances numerical analysis for thermoelastic scattering and provides a framework for future exploration of alternative PML geometries and boundary conditions.

Abstract

This paper is concerned with the thermoelastic obstacle scattering problem in three dimensions. A uniaxial perfectly matched layer (PML) method is firstly introduced to truncate the unbounded scattering problem, leading to a truncated PML problem in a bounded domain. Under certain constraints on model parameters, the well-posedness for the truncated PML problem is then proved except possibly for a discrete set of frequencies, based on the analytic Fredholm theory. Moreover, the exponential convergence of the uniaxial PML method is established in terms of the thickness and absorbing parameters of PML layer. The proof is based on the PML extension technique and the exponential decay properties of the modified fundamental solution. As far as we know, this is the first convergence result of the PML method for the time-harmonic thermoelastic scattering problem.
Paper Structure (6 sections, 13 theorems, 130 equations)

This paper contains 6 sections, 13 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Thermoelastic scattering problem
  3. The PML method for thermoelastic wave
  4. The PML equation in truncated domain and layer
  5. Convergence analysis of the PML method
  6. Conclusion

Key Result

Lemma 2.1

The solution $U$ of homogeneous thermoelastic system in ${\mathbb R}^3\backslash\overline{B}_1$ can be written in the form where $\boldsymbol{u}^1,\boldsymbol{u}^2,\boldsymbol{u}^3$ and $p^1,p^2$ satisfy Here, the wave number $\lambda_1,\lambda_2$ are the roots of the characteristic relation equations chara1, and $\lambda_3=\sqrt{\frac{\rho}{\mu}}\omega$ is the the wave number of shear waves.

Theorems & Definitions (24)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3: Constant Convention
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • Lemma 4.2: Ellipticity lemma
  • ...and 14 more