Numerical solution to the PML problem of the biharmonic wave scattering in periodic structures

TL;DR

This paper develops and analyzes numerical methods for solving the biharmonic wave scattering problem in periodic structures with PML, demonstrating exponential convergence and effective wave absorption through finite element techniques.

Contribution

It introduces new finite element formulations and decomposition methods for the PML problem of biharmonic waves, with proven well-posedness and convergence.

Findings

Proven exponential convergence of the PML solution.

Effective wave absorption demonstrated in numerical experiments.

Suppression of oscillations in the bending moment near cavities.

Abstract

Consider the interaction of biharmonic waves with a periodic array of cavities, characterized by the Kirchhoff--Love model. This paper investigates the perfectly matched layer (PML) formulation and its numerical soution to the governing biharmonic wave equation. The study establishes the well-posedness of the associated variational problem employing the Fredholm alternative theorem. Based on the examination of an auxiliary problem in the PML layer, exponential convergence of the PML solution is attained. Moreover, it develops and compares three decomposition methods alongside their corresponding mixed finite element formulations, incorporating interior penalty techniques for solving the PML problem. Numerical experiments validate the effectiveness of the proposed methods in absorbing outgoing waves within the PML layers and suppressing oscillations in the bending moment of biharmonic waves near the cavity's surface.