2019/12/20 by Andrew Gibbs, David G. Hewett, Gibbs, Andrew +5
Engineering · Physics and Astronomy · #65N38 #65R20 #78A45 #78M15 #78M35 #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1912.09916
openalex publication_date 2019/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM)\nfor high frequency scattering by two-dimensional screens and apertures, whose\ncomputational cost to achieve any prescribed accuracy remains bounded with\nincreasing frequency. Our method is a collocation implementation of the high\norder hp HNA approximation space of Hewett et al. IMA J. Numer. Anal. 35\n(2015), pp.1698- 1728, where a Galerkin implementation was studied.\n An advantage of the current collocation scheme is that the one-dimensional\nhighly oscillatory singular integrals appearing in the BEM matrix entries are\nsignificantly easier to evaluate than the two-dimensional integrals appearing\nin the Galerkin case, which leads to much faster computation times. Here we\ncompute the required integrals at frequency-independent cost using the\nnumerical method of steepest descent, which involves complex contour\ndeformation.\n The change from Galerkin to collocation is nontrivial because naive\ncollocation implementations based on square linear systems suffer from severe\nnumerical instabilities associated with the numerical redundancy of the HNA\nbasis, which produces highly ill-conditioned BEM matrices. In this paper we\nshow how these instabilities can be removed by oversampling, and solving the\nresulting overdetermined collocation system in a weighted least-squares sense\nusing a truncated singular value decomposition. On the basis of our numerical\nexperiments, the amount of oversampling required to stabilise the method is\nmodest (around 25% typically suffices) and independent of frequency. As an\napplication of our method we present numerical results for high frequency\nscattering by prefractal approximations to the middle-third Cantor set.\n