A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications

Oscar P. Bruno, Leonid Kunyansky

Research output: Contribution to journalArticle

154 Citations (Scopus)

Abstract

We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds.

Original languageEnglish (US)
Pages (from-to)80-110
Number of pages31
JournalJournal of Computational Physics
Volume169
Issue number1
DOIs
StatePublished - May 1 2001
Externally publishedYes

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Surface scattering
scattering
Fast Fourier transforms
Scattering
acoustic scattering
Boundary integral equations
evaluation
personal computers
multiplication
Personal computers
integral equations
unity
partitions
breakdown
Acoustics
sampling
Sampling
Data storage equipment
estimates
approximation

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

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abstract = "We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order {"}two-face{"} equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds.",
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