Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Adaptive Finite Element DtN Method for the Acoustic-Elastic Interaction Problem in Periodic Structures

Lei Lin, Junliang Lv

TL;DR

This work develops an adaptive finite element method with Dirichlet-to-Neumann (DtN) boundary conditions for the time-harmonic acoustic-elastic scattering problem in periodic structures. By formulating a coupled variational problem with quasi-periodic spaces and dual DtN operators for acoustic and elastic waves, the authors derive a duality-based a posteriori error estimate that splits the error into finite element and DtN truncation components, with the truncation error decaying exponentially in the truncation parameter $N$. An adaptive algorithm is proposed and implemented in MATLAB, using residual-based indicators and a controlled truncation error $oldsymbol{ ext{ε}}_N$ to drive mesh refinement and DtN parameters; numerical experiments on flat, cornered, and curved interfaces demonstrate quasi-optimal convergence and the method's robustness to $N$. The results establish adaptive DtN-FEM as an effective, PML-free approach for unbounded-domain wave problems in periodic media, with potential extension to electromagnetic-elastic interactions as future work.

Abstract

Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm.

An Adaptive Finite Element DtN Method for the Acoustic-Elastic Interaction Problem in Periodic Structures

TL;DR

This work develops an adaptive finite element method with Dirichlet-to-Neumann (DtN) boundary conditions for the time-harmonic acoustic-elastic scattering problem in periodic structures. By formulating a coupled variational problem with quasi-periodic spaces and dual DtN operators for acoustic and elastic waves, the authors derive a duality-based a posteriori error estimate that splits the error into finite element and DtN truncation components, with the truncation error decaying exponentially in the truncation parameter . An adaptive algorithm is proposed and implemented in MATLAB, using residual-based indicators and a controlled truncation error to drive mesh refinement and DtN parameters; numerical experiments on flat, cornered, and curved interfaces demonstrate quasi-optimal convergence and the method's robustness to . The results establish adaptive DtN-FEM as an effective, PML-free approach for unbounded-domain wave problems in periodic media, with potential extension to electromagnetic-elastic interactions as future work.

Abstract

Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm.

Paper Structure

This paper contains 8 sections, 11 theorems, 131 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Problem
  3. Finite Element Approximation
  4. The A Posteriori Error Analysis
  5. Implementation and Numerical Examples
  6. Adaptive Algorithm
  7. Numerical Examples
  8. Conclusion

Key Result

Lemma 1

Let $D \subset \mathbb{R}^2$ be a domain in which the divergence theorem holds. Then, for any vector fields $\bm{u} \in C^2(\overline{D})^2$ and $\bm{v} \in C^1(\overline{D})^2$, the first generalized Betti formula holds: where

Figures (9)

  • Figure 1: A two-dimensional schematic of the problem geometry for the acoustic-elastic interaction in periodic structures
  • Figure 2: The real parts of the exact solutions (a,b,c) and numerical solutions (d,e,f) with $\kappa = 1$ in Example 1
  • Figure 3: Quasi-optimality of the a priori and a posteriori errors in Example 1
  • Figure 4: The real parts of the numerical solutions with $\kappa =1$ in Example 2
  • Figure 5: Quasi-optimality of the a posteriori errors with different $\kappa$ (a); the adaptive mesh with 5,966 elements after 8 refinement iterations (b) in Example 2
  • ...and 4 more figures

Theorems & Definitions (18)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3: See Jiang17-3
  • Remark 1
  • Lemma 4: See Chen03
  • Lemma 5: See Wang15
  • Lemma 6: SeeBrenner08
  • Lemma 7
  • proof
  • ...and 8 more