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Boundary Integral Formulations for Flexural Wave Scattering in Thin Plates

Peter Nekrasov, Zhaosen Su, Travis Askham, Jeremy G. Hoskins

TL;DR

The paper addresses exterior flexural-wave scattering in thin plates under clamped, supported, and free boundary conditions by developing second-kind boundary integral equations. The authors extend Fredholm-layer-potential representations via carefully chosen kernels for the clamped and supported cases, and introduce a novel Hilbert-transform-based representation for the free plate to achieve a SKIE. They provide rigorous jump analyses, establish compactness results, and implement a high-order Nyström discretization with GGQ to achieve accurate solutions on complex geometries, including large-scale multi-object scattering. The approach enables efficient, high-accuracy simulations of flexural waves in applications ranging from geophysics to engineering, with potential extensions to interior problems, weaker geometries, and static regimes.

Abstract

In this paper, we develop second kind integral formulations for flexural wave scattering problems involving the clamped, supported, and free plate boundary conditions. While the clamped plate problem can be solved with layer potentials developed for the biharmonic equation, the free plate problem is more difficult due to the order and complexity of the boundary conditions. In this work, we describe a representation for the free plate problem that uses the Hilbert transform to cancel singularities of certain layer potentials, ultimately leading to a Fredholm integral equation of the second kind. Additionally, for the supported plate problem, we improve on an existing representation to obtain a second kind integral equation formulation. With these representations it is possible to solve flexural wave scattering problems with high-order-accurate methods, examine the far field patterns of scattering objects, and solve large problems involving multiple scatterers.

Boundary Integral Formulations for Flexural Wave Scattering in Thin Plates

TL;DR

The paper addresses exterior flexural-wave scattering in thin plates under clamped, supported, and free boundary conditions by developing second-kind boundary integral equations. The authors extend Fredholm-layer-potential representations via carefully chosen kernels for the clamped and supported cases, and introduce a novel Hilbert-transform-based representation for the free plate to achieve a SKIE. They provide rigorous jump analyses, establish compactness results, and implement a high-order Nyström discretization with GGQ to achieve accurate solutions on complex geometries, including large-scale multi-object scattering. The approach enables efficient, high-accuracy simulations of flexural waves in applications ranging from geophysics to engineering, with potential extensions to interior problems, weaker geometries, and static regimes.

Abstract

In this paper, we develop second kind integral formulations for flexural wave scattering problems involving the clamped, supported, and free plate boundary conditions. While the clamped plate problem can be solved with layer potentials developed for the biharmonic equation, the free plate problem is more difficult due to the order and complexity of the boundary conditions. In this work, we describe a representation for the free plate problem that uses the Hilbert transform to cancel singularities of certain layer potentials, ultimately leading to a Fredholm integral equation of the second kind. Additionally, for the supported plate problem, we improve on an existing representation to obtain a second kind integral equation formulation. With these representations it is possible to solve flexural wave scattering problems with high-order-accurate methods, examine the far field patterns of scattering objects, and solve large problems involving multiple scatterers.
Paper Structure (22 sections, 3 theorems, 147 equations, 7 figures)

This paper contains 22 sections, 3 theorems, 147 equations, 7 figures.

Table of Contents

  1. Background
  2. Mathematical framework
  3. The clamped plate problem
  4. The supported plate problem
  5. The free plate problem
  6. Numerical implementation
  7. Details of the discretization
  8. Catastrophic cancellation
  9. Results
  10. Discussion
  11. Code availability
  12. Acknowledgements
  13. Heuristic strategy for deriving the integral kernels
  14. Asymptotics of the Green's function
  15. Formulae for the derivatives of the biharmonic Green's function
  16. ...and 7 more sections

Key Result

Theorem 1

Let ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ be the layer potentials corresponding to the kernels $K_1$ and $K_2$, respectively. Let ${\mathcal{K}}_{11},{\mathcal{K}}_{12},{\mathcal{K}}_{21},$ and ${\mathcal{K}}_{22}$ be the boundary operators corresponding to the kernels $K_{11},K_{12},K_{21}$, and for ${\mathbf{x}}_0 \in \partial\Omega$, where ${\mathbf{x}}_0^+$ corresponds to the limit from the

Figures (7)

  • Figure 1: Convergence of the three boundary integral equations as a function of the number of discretization points $N$ ($k = 8, \nu = 1/3$). The solutions converge at sixteenth order (left). The equations were solved on the boundary of a droplet (right), with the green dot representing the location of the point source and the black dots representing the locations where the error was measured. The discretization plotted on the right was the finest one tested ($N_p=16, n_{\textrm{GL}}=16$). The black box represents the region near the boundary that was chosen for closer investigation (see Figure \ref{['near_boundary_figure']}).
  • Figure 2: The accuracy of evaluation for points near the boundary was checked for the same geometry and discretization as Figure \ref{['convergence_figure']}. The relative error for the free plate problem is plotted in some region near the boundary (left). The error was also calculated at a set of points that get exponentially close to the boundary, indicated by black dots, and plotted for the clamped, free, and supported plate problems (right).
  • Figure 3: Wave scattering off a droplet with free plate boundary conditions for three different wavenumbers. The field is incident from the left and the magnitude of the total field $|u|$ is plotted.
  • Figure 4: Wave scattering in free plates with three different Poisson's ratios: $\nu = 0.3$ (ice), $\nu = 0$ (cork), and $\nu = -1$ (auxetic). The field is incident from the left and the wavenumber is $k = 12$. The quantity that is plotted is the real part of the total field. The condition numbers for these three cases were 1.37E+04 ($\nu = 0.3$), 1.90E+04 ($\nu = 0$), and 2.46E+04 ($\nu = -1$). For concave domains, Poisson's ratio has a large effect on wave reflection and resonant behavior.
  • Figure 5: Comparison of the scattered field for various boundary conditions ($k = 3, \nu = 1/3$), measured in the far field. The geometry is a starfish (left) with arrows indicating the direction of the incident field and the angle $\theta$. Different boundary conditions lead to differences in both magnitude (middle) and phase (right) of the scattered field .
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • proof : Proof of Theorem \ref{['thm:clamped']}.
  • Theorem 2
  • proof
  • proof : Proof of Theorem \ref{['thm:supported']}.
  • Theorem 3
  • proof
  • proof : Proof of Theorem \ref{['thm:free']}.