An adaptive perfectly matched layer finite element method for acoustic-elastic interaction in periodic structures

TL;DR

This work develops an adaptive PML–FEM framework for time-harmonic acoustic–elastic scattering in unbounded periodic structures. It derives a truncated PML formulation with transparent boundary conditions for both the acoustic and elastic fields, proving exponential convergence of the PML error as the parameters $(\\delta_j, \\sigma_j)$ grow. A residual-based a posteriori error estimator that accounts for both finite element discretization and PML truncation is established, enabling an adaptive PML–FEM algorithm. Numerical experiments on geometries with corners and at high frequency demonstrate the method’s accuracy, robustness, and efficiency, highlighting its applicability to diffraction grating problems and related periodic scattering challenges.

Abstract

This paper considers the scattering of a time-harmonic acoustic plane wave by an elastic body with an unbounded periodic surface. The original problem can be confined to the analysis of the fields in one periodic cell. With the help of the perfectly matched layer (PML) technique, we can truncate the unbounded physical domain into a bounded computational domain. By respectively constructing the equivalent transparent boundary conditions of acoustic and elastic waves simultaneously, the well-posedness and exponential convergence of the solution to the associated truncated PML problem are established. The finite element method is applied to solve the PML problem of acoustic-elastic interaction. To address the singularity caused by the non-smooth surface of the elastic body, we establish a residual-type a posteriori error estimate and develop an adaptive PML finite element algorithm. Several numerical examples are presented to demonstrate the effectiveness of the proposed adaptive algorithm.

Figures (12)

• Figure 1: Problem geometry of acoustic-elastic interaction in periodic structures.
• Figure 2: Problem geometry of acoustic-elastic interaction in periodic structures with PML layers.
• Figure 3: Example 1. Real parts of the exact solutions (a)-(c) and numerical solutions (d)-(f) with $\kappa=1$.
• Figure 4: Example 1. Log-log curves of the error estimates versus $\mathrm{DOF}$ with $\delta = 1,2,3$: (a) A priori error; (b) A posteriori error.
• Figure 5: Example 2. Imaginary parts of the numerical solutions with $\kappa=1$ and $\delta = 3$.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2: See Chen03Jiang17-3
• Lemma 3
• proof
• Theorem 1
• proof
• Lemma 4
• proof
• Lemma 5
• proof