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

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.

💡 Research Summary

This paper addresses the time‑harmonic scattering of an acoustic plane wave by an elastic body whose surface is periodic and unbounded. The physical domain consists of a compressible inviscid fluid above the interface and an isotropic linear elastic solid below it. The governing equations are the Helmholtz equation for the acoustic pressure in the fluid and the Navier equation for the elastic displacement in the solid, coupled through continuity of normal velocity and traction on the interface. By exploiting the periodicity, the problem is reduced to a single periodic cell. Transparent boundary conditions (Dirichlet‑to‑Neumann operators) are derived for both the acoustic and elastic fields on artificial truncation boundaries placed above and below the cell.

A perfectly matched layer (PML) is introduced on both sides of the cell. Complex coordinate stretching with a damping profile s(·) yields modified differential operators with anisotropic coefficients. The authors prove that the truncated PML problem is well‑posed and that the error between the original and PML solutions decays exponentially with respect to the PML thickness and absorption strength, simultaneously for acoustic and elastic waves.

The variational formulation is discretized by a conforming finite element method using triangular elements. To handle the singularities caused by non‑smooth (re‑entrant) grating profiles, a residual‑type a posteriori error estimator is constructed. This estimator contains three contributions: element interior residuals, edge jump residuals, and a term accounting for the PML truncation error. The estimator is shown to be both reliable and efficient.

Based on the estimator, an adaptive algorithm is designed. The algorithm follows a standard marking‑refinement loop (e.g., Dörfler marking) and refines the mesh preferentially near geometric singularities, while the PML parameters are chosen so that the truncation error is negligible.

Numerical experiments are presented for three configurations: (i) a smooth periodic surface, (ii) a surface with a square re‑entrant corner, and (iii) a more complex multi‑periodic structure. In all cases, the adaptive PML‑FEM achieves the prescribed error tolerance with far fewer degrees of freedom than a uniform mesh approach. For the re‑entrant corner case, the adaptive refinement concentrates elements around the corner, leading to global L² errors below 10⁻⁶. The PML thickness and absorption parameters are shown to control the truncation error effectively, confirming the theoretical exponential convergence.

The main contributions of the work are: (1) a rigorous simultaneous transparent boundary condition for coupled acoustic‑elastic waves in periodic media, (2) a proof of exponential convergence of the PML truncation for the coupled system, (3) a residual‑based a posteriori error estimator that blends FEM discretization and PML truncation errors, and (4) a fully adaptive PML‑FEM that remains robust in the presence of geometric singularities. These results provide a powerful computational tool for applications such as underwater acoustics, ultrasonic nondestructive evaluation, and the design of diffraction gratings, where accurate and efficient simulation of coupled wave phenomena in periodic structures is essential.