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Unfitted Lattice Green's Function Method for Exterior Scattering in Complex Geometry

Siyuan Wang, Qing Xia

TL;DR

This work develops a finite-difference, unfitted boundary-algebraic framework for 2D exterior Helmholtz scattering from arbitrary geometries by leveraging a lattice Green's function (LGF) on an infinite lattice. Boundary conditions are imposed via local Lagrange interpolation on cut cells, and the solution is expressed through discrete single- and double-layer potentials, avoiding singular kernels inherent to classical boundary integrals. The method preserves dimension reduction and lack of artificial boundaries while enabling efficient reconstruction with FFTs, demonstrated through circular, triangular, and multi-body scattering tests that show second-order convergence and robustness across TM/TE polarizations. Its capacity to handle curved, non-aligned boundaries within a finite-difference setting, together with moderate conditioning and scalable memory usage, suggests strong potential for high-frequency extensions and 3D generalization.

Abstract

This paper develops a finite-difference analogue of the boundary integral/element method for the numerical solution of two-dimensional exterior scattering from scatterers of arbitrary shapes. The discrete fundamental solution, known as the lattice Green's function (LGF), for the Helmholtz equation on an infinite lattice is derived and employed to construct boundary algebraic equations through the discrete potentials framework. Unlike the continuous fundamental solution used in boundary integral methods, the LGF introduces no singularity, which simplifies numerical implementation. Boundary conditions are incorporated through local Lagrange interpolation on unfitted cut cells. The resulting method retains key advantages of boundary integral approaches-including dimension reduction and the absence of artificial boundary conditions--while enabling finite differences for complex geometries. Numerical results demonstrate the accuracy and robustness of the method for various scatterers, including circular, triangular, and multiple-body configurations.

Unfitted Lattice Green's Function Method for Exterior Scattering in Complex Geometry

TL;DR

This work develops a finite-difference, unfitted boundary-algebraic framework for 2D exterior Helmholtz scattering from arbitrary geometries by leveraging a lattice Green's function (LGF) on an infinite lattice. Boundary conditions are imposed via local Lagrange interpolation on cut cells, and the solution is expressed through discrete single- and double-layer potentials, avoiding singular kernels inherent to classical boundary integrals. The method preserves dimension reduction and lack of artificial boundaries while enabling efficient reconstruction with FFTs, demonstrated through circular, triangular, and multi-body scattering tests that show second-order convergence and robustness across TM/TE polarizations. Its capacity to handle curved, non-aligned boundaries within a finite-difference setting, together with moderate conditioning and scalable memory usage, suggests strong potential for high-frequency extensions and 3D generalization.

Abstract

This paper develops a finite-difference analogue of the boundary integral/element method for the numerical solution of two-dimensional exterior scattering from scatterers of arbitrary shapes. The discrete fundamental solution, known as the lattice Green's function (LGF), for the Helmholtz equation on an infinite lattice is derived and employed to construct boundary algebraic equations through the discrete potentials framework. Unlike the continuous fundamental solution used in boundary integral methods, the LGF introduces no singularity, which simplifies numerical implementation. Boundary conditions are incorporated through local Lagrange interpolation on unfitted cut cells. The resulting method retains key advantages of boundary integral approaches-including dimension reduction and the absence of artificial boundary conditions--while enabling finite differences for complex geometries. Numerical results demonstrate the accuracy and robustness of the method for various scatterers, including circular, triangular, and multiple-body configurations.

Paper Structure

This paper contains 20 sections, 46 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Lattice Green's Function
  3. Numerical Scheme
  4. Point sets
  5. Indirect formulations
  6. Single-layer formulation
  7. Double-layer formulation
  8. Boundary closure
  9. Reconstruction
  10. Algorithm
  11. Computational Cost
  12. Conditioning and Stability
  13. Memory Usage
  14. Numerical Simulations
  15. Planewave scattering outside a circle
  16. ...and 5 more sections

Figures (8)

  • Figure 1: Example of set $\gamma$ ($\gamma_{-}$: triangles; $\gamma_{+}$: squares)
  • Figure 2: An example of a cut cell ($\gamma_{-}$: triangles; $\gamma_{+}$: squares; $\zeta$: diamonds)
  • Figure 3: Exact scattering with a circle ($r=0.5,k=10$, single-layer formulation)
  • Figure 4: Convergence ($r=0.5,k=10$); SL: single layer; DL: double layer
  • Figure 5: TM scattering with a triangle
  • ...and 3 more figures