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Planewave density interpolation methods for 3D Helmholtz boundary integral equations

Carlos Pérez-Arancibia, Catalin Turc, Luiz Faria

TL;DR

This work introduces planewave density interpolation (PWDI) to regularize all challenging surface integrals in 3D Helmholtz boundary integral equations, including weakly and hypersingular kernels and nearly singular configurations. By embedding a density interpolant $\Phi(\boldsymbol r,p)$ that solves the Helmholtz equation and satisfies Taylor-like interpolation conditions, PWDI produces kernel-regularized expressions whose integrands are bounded across target locations, enabling standard quadrature within both Nyström and boundary-element methods. The authors develop two PWDI constructions: a closed-form, low-order solution for $M=1$ and a purely algebraic high-order method for general $M$, with planning for compatibility with fast methods. Numerical experiments on simple and composite geometries (including touching/intersecting obstacles) demonstrate accurate far- and near-field results and robust convergence, illustrating PWDI’s potential to simplify BIE discretizations and improve practicality in complex scattering problems.

Abstract

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nyström and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nyström and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique.

Planewave density interpolation methods for 3D Helmholtz boundary integral equations

TL;DR

This work introduces planewave density interpolation (PWDI) to regularize all challenging surface integrals in 3D Helmholtz boundary integral equations, including weakly and hypersingular kernels and nearly singular configurations. By embedding a density interpolant that solves the Helmholtz equation and satisfies Taylor-like interpolation conditions, PWDI produces kernel-regularized expressions whose integrands are bounded across target locations, enabling standard quadrature within both Nyström and boundary-element methods. The authors develop two PWDI constructions: a closed-form, low-order solution for and a purely algebraic high-order method for general , with planning for compatibility with fast methods. Numerical experiments on simple and composite geometries (including touching/intersecting obstacles) demonstrate accurate far- and near-field results and robust convergence, illustrating PWDI’s potential to simplify BIE discretizations and improve practicality in complex scattering problems.

Abstract

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nyström and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nyström and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique.

Paper Structure

This paper contains 20 sections, 3 theorems, 87 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kernel regularization via density interpolation
  4. Taylor series on smooth surfaces
  5. Kernel-regularized boundary integral operators and layer potentials
  6. Multiple-scattering approach to scattering by composite surfaces
  7. Interpolating functions
  8. Closed-form planewave expansion functions in the case $M=1$
  9. Higher-order planewave expansion functions
  10. Numerical evaluation of integral operators and layer potentials
  11. Chebyshev-based Nyström method
  12. Galerkin boundary element method
  13. Numerical examples
  14. Validation of the density interpolation procedures
  15. Nyström and Boundary Element methods
  16. ...and 5 more sections

Key Result

Lemma 4.1

Let $\Phi:\mathbb{R}^3\times\Gamma\to\mathbb{C}$ be given by eq:interpolants, where $\Phi^{(j)}_\alpha:\mathbb{R}^3\times\Gamma\to\mathbb{C}$, $|\alpha|\leq M$, $j=1,2$, are the linear combinations of planewaves defined in eq:LCPW. Then, sufficient conditions for $\Phi$ to satisfy eq:cond_SL at $p\i for all sub-indices $\beta\in\mathbb{Z}^2_+$ such that $|\beta|\leq M$.

Figures (9)

  • Figure 1: Notation used for the the mesh triangles in the outer ($T$) and inner ($K$) integrals in \ref{['eq:discrete_integral']} and in the Algorithm \ref{['alg:BEM_BW']}.
  • Figure 2: (a) (resp. (d)): Plot of real and imaginary parts of $\rho$ defined in \ref{['eq:error_dens']} where the planewave interpolant $\Phi$ was constructed using the analytic (resp. numerical) procedure described in Section \ref{['Meq1']} (resp. \ref{['sec:higher_order']}). The interpolation point $p^*:=\bold{x}(\xi_1^*,\xi_2^*)=(-0.616, 0.310,0.599)$ is marked by a black dot. (b) and (c) (resp. (e) and (f)): Plots of the cross section of the partial derivatives $\partial^{\alpha}\rho$ for all $|\alpha|= 1$ (resp. $|\alpha|=3$) in the parameter space. Note that all the first (resp. third) order derivatives vanish exactly at the interpolation points $(\xi_1^*,\xi_2^*)=(-0.339, 0.790)$.
  • Figure 3: Grids utilized in the evaluation of the far- and near-field errors.
  • Figure 4: Far- and near-field errors in the solution of the Dirichlet problem \ref{['eq:Dirichlet']} produced by the Nyström method discretization of the BW integral equation \ref{['BW']} using the two proposed PWDI techniques for different interpolation orders $M$ and grid sizes $N$ (each quadrilateral surface patch is discretized using $N\times N$ quadrature points).
  • Figure 5: Far-field errors in the solution of the Dirichlet problem \ref{['eq:Dirichlet']} corresponding to the scattering of a plane-wave off three different surfaces using the Chebyshev-based Nyström method of Section \ref{['sec:Nystrom']} applied to the BW integral equation \ref{['BW']} with $k=\eta=1$. Both analytical (with $M=1$) and numerical (with $M=2$) PWDI procedures were used in this example.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Remark 5.1
  • Lemma A.1
  • ...and 2 more