Smoothed corners and scattered waves

Charles L. Epstein, Michael O'Neil

Research output: Contribution to journalArticle

Abstract

We introduce an arbitrary order, computationally efficient method to smooth corners on curves in the plane, as well as edges and vertices on surfaces in ℝ3. The method is local, only modifying the original surface in a neighborhood of the geometric singularity, and preserves desirable features like convexity and symmetry. The smoothness of the final surface is an explicit parameter in the method, and the band-limit of the smoothed surface is proportional to its smoothness. Several numerical examples are provided in the context of acoustic scattering. In particular, we compare scattered fields from smoothed geometries in two dimensions with those from polygonal domains. We observe that significant reductions in computational cost can be obtained if merely approximate solutions are desired in the near- or far-field. Provided that the smoothing is subwavelength, the error of the scattered field is proportional to the size of the geometry that is modified.

Original languageEnglish (US)
Pages (from-to)A2665-A2698
JournalSIAM Journal on Scientific Computing
Volume38
Issue number5
DOIs
StatePublished - 2016

Keywords

  • Corners
  • Helmholtz
  • Lipschitz domain
  • Potential theory
  • Quadrature
  • Rounding
  • Scattering
  • Smoothing

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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