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

Peijun Li, Xiaokai Yuan

TL;DR

This work develops a PML framework for scattering of biharmonic waves in a periodic array of cavities within a Kirchhoff–Love plate, governed by $\Delta^2 u-\kappa^4 u=0$ with quasi-periodic and clamped boundary conditions. It proves the well-posedness of the PML-augmented variational problem via Garding's inequality and Fredholm theory and shows exponential convergence of the PML solution using an auxiliary-problem approach, controlled by a layer-efficiency factor $\Theta$. Three mixed finite element decompositions ($(q,p)$, $(u,q)$, and decoupled $(p,q)$) with interior penalty are proposed to discretize the PML problem, each enabling stable, accurate absorption of outgoing waves in the PML layers. Numerical experiments with plane incidence validate exponential PML convergence, demonstrate effective damping of the bending moment near cavity surfaces, and indicate roughly first-order convergence with mesh refinement, supporting the practicality of the approach for periodic biharmonic-wave scattering and suggesting its applicability to broader complex scattering problems.

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.

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

TL;DR

This work develops a PML framework for scattering of biharmonic waves in a periodic array of cavities within a Kirchhoff–Love plate, governed by with quasi-periodic and clamped boundary conditions. It proves the well-posedness of the PML-augmented variational problem via Garding's inequality and Fredholm theory and shows exponential convergence of the PML solution using an auxiliary-problem approach, controlled by a layer-efficiency factor . Three mixed finite element decompositions (, , and decoupled ) with interior penalty are proposed to discretize the PML problem, each enabling stable, accurate absorption of outgoing waves in the PML layers. Numerical experiments with plane incidence validate exponential PML convergence, demonstrate effective damping of the bending moment near cavity surfaces, and indicate roughly first-order convergence with mesh refinement, supporting the practicality of the approach for periodic biharmonic-wave scattering and suggesting its applicability to broader complex scattering problems.

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.
Paper Structure (18 sections, 8 theorems, 182 equations, 10 figures)

This paper contains 18 sections, 8 theorems, 182 equations, 10 figures.

Table of Contents

  1. introduction
  2. Problem formulation
  3. The reduced problem
  4. The PML problem
  5. The PML formulation
  6. The variational problem
  7. The well-posedness
  8. Convergence analysis
  9. Numerical methods
  10. $(q, p)$-decomposition
  11. $(u, q)$-decomposition
  12. Decoupled $(p, q)$-decomposition
  13. Numerical experiments
  14. Example 1
  15. Example 2
  16. ...and 3 more sections

Key Result

Theorem 3.1

There exists a unique weak solution $u\in H^2_{{\rm qp}, \Gamma_c}(\Omega)$ to the variational problem variationalTBC, with the exception of a discrete set of wavenumbers $\kappa$.

Figures (10)

  • Figure 1: Schematic of the problem geometry.
  • Figure 2: The schematic of the PML problem.
  • Figure 3: The solution of $(q, p)$-decomposition for a circle-shaped cavity.
  • Figure 4: The solution of $(u, q)$-decomposition for a circle-shaped cavity.
  • Figure 5: The solution of decoupled $(q, p)$-decomposition for a circle-shaped cavity.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Theorem 3.1
  • Theorem 4.1
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • proof
  • Lemma 6.1
  • proof
  • ...and 4 more