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.

翻译：考虑双调和波与一组以Kirchhoff–Love模型为特征的周期排列腔体的相互作用。本文研究了完美匹配层(PML)公式及其对控制双调和波动方程的数值求解。该研究利用Fredholm择一性定理建立了相关变分问题的适定性。基于对PML层中辅助问题的考察，实现了PML解的指数收敛。此外，本文开发并比较了三种分解方法及其相应的混合有限元形式，引入了内部罚函数技术来解决PML问题。数值实验验证了所提方法在PML层内吸收出射波以及抑制腔体表面附近双调和波弯矩振荡的有效性。