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A discrete Perfectly Matched Layer for peridynamic scalar waves in two-dimensional viscous media

Yu Du, Yonglin Li, Jiwei Zhang

TL;DR

The paper tackles absorbing boundary treatment for two-dimensional peridynamic scalar waves, where nonlocality complicates traditional boundary conditions. It develops a discrete perfectly matched layer (PML) by first discretizing the PD model with a quadrature-based AC finite difference scheme, then performing discrete analytic continuation via discrete complex analysis to derive PML equations that absorb discrete plane waves without reflection. The resulting discrete PML uses damping profiles and auxiliary variables, achieving exponential attenuation and improved stability and efficiency over exact absorbing boundary conditions. This framework offers a practical, scalable boundary treatment for nonlocal wave problems and provides a path toward generalizations to other nonlocal operators and higher dimensions.

Abstract

In this paper, we propose a discrete perfectly matched layer (PML) for the peridynamic scalar wave-type problems in viscous media. Constructing PMLs for nonlocal models is often challenging, mainly due to the fact that nonlocal operators are usually associated with various kernels. We first convert the continua model to a spatial semi-discretized version by adopting quadrature-based finite difference scheme, and then derive the PML equations from the semi-discretized equations using discrete analytic continuation. The harmonic exponential fundamental solutions (plane wave modes) of the semi-discretized equations are absorbed by the PML layer without reflection and are exponentially damped. The excellent efficiency and stability of discrete PML are demonstrated in numerical tests by comparison with exact absorbing boundary conditions.

A discrete Perfectly Matched Layer for peridynamic scalar waves in two-dimensional viscous media

TL;DR

The paper tackles absorbing boundary treatment for two-dimensional peridynamic scalar waves, where nonlocality complicates traditional boundary conditions. It develops a discrete perfectly matched layer (PML) by first discretizing the PD model with a quadrature-based AC finite difference scheme, then performing discrete analytic continuation via discrete complex analysis to derive PML equations that absorb discrete plane waves without reflection. The resulting discrete PML uses damping profiles and auxiliary variables, achieving exponential attenuation and improved stability and efficiency over exact absorbing boundary conditions. This framework offers a practical, scalable boundary treatment for nonlocal wave problems and provides a path toward generalizations to other nonlocal operators and higher dimensions.

Abstract

In this paper, we propose a discrete perfectly matched layer (PML) for the peridynamic scalar wave-type problems in viscous media. Constructing PMLs for nonlocal models is often challenging, mainly due to the fact that nonlocal operators are usually associated with various kernels. We first convert the continua model to a spatial semi-discretized version by adopting quadrature-based finite difference scheme, and then derive the PML equations from the semi-discretized equations using discrete analytic continuation. The harmonic exponential fundamental solutions (plane wave modes) of the semi-discretized equations are absorbed by the PML layer without reflection and are exponentially damped. The excellent efficiency and stability of discrete PML are demonstrated in numerical tests by comparison with exact absorbing boundary conditions.

Paper Structure

This paper contains 9 sections, 6 theorems, 71 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Wave equation and finite difference scheme
  3. Discrete PML
  4. Discrete complex differentiation
  5. Discrete PML equations
  6. Derivation of discrete PML equations
  7. Implement
  8. Numerial tests
  9. Conclusion

Key Result

Proposition 3.1

Assume that $w_{\mathbf{i},\mathbf{j}}\in\mathbb{C}^{\mathbb{Z}^2\times\mathbb{Z}^2}$ is discrete-holomorphic on $(\Lambda, z^\alpha)$ for each fixed $\alpha\in\{1,2\}$, then for $\beta\in\{1,2\}$ and $\beta\neq\alpha$, it holds $D_{\alpha,\beta} w_{\mathbf{i},\mathbf{j}} = D_{\beta,\alpha} w_{\math which also indicates that $D_{\beta} w_{\mathbf{i},\mathbf{j}}$ is discrete-holomorphic on $(\Lambd

Figures (7)

  • Figure 1: (Example 1.) The decay rate $\mu(\boldsymbol{\kappa}^h,\sigma_0)$ as a function of $\sigma_0$ and $\kappa_1^h$ for fixed $\kappa_2^h$. The mesh size is $h=1/16$ and the horizon of the kernel is $\delta=1/4$.
  • Figure 2: (Example 1.) Left: The decay rate $\mu(\boldsymbol{\kappa}^h,\sigma_0)$ as a function of $\sigma_0$ for different $\boldsymbol{\kappa}^h$. Right: The reflection error $\max|U-U_{\mathrm{ref}}|$ over $\Omega$ with mesh size $h=1/16$. $U$ and $U_{\mathrm{ref}}$ are solutions to the problem with horizon $\delta=1/4$.
  • Figure 3: (Example 1.) Waves in the PML (shown as grids) are damped exponentially. The solution in the near field at different time instants. Top, from left to right: reference solutions at $t=1,1.5,2$. Bottom, from left to right: PML solutions at $t=1,1.5,2$.
  • Figure 4: (Example 1.) The errors in $l^2$-norm and max-norm between the PML solution and the reference solution at $t =1.5$. The dash line is a reference line with slope $-2$.
  • Figure 5: (Example 1.) Left: Time evolution of the solutions at $\mathbf{x}=(0.5,0.5)$. Right: The energy evolution of the discrete PML and reference solutions. Numerical solutions are solved under the mesh size $h=1/16$.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 3.1: Discrete holomorphicity
  • Definition 3.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 3 more