Open Access

ARTICLE

A Spectrally Accurate Quadrature for 3-D Boundary Integrals

Gregory Baker¹, Huaijian Zhang¹

1 Department of Mathematics, The Ohio State University, 231 W. 18th Ave, Columbus, OH43210, baker@math.ohio-state.edu

Computer Modeling in Engineering & Sciences 2011, 80(3&4), 219-232. https://doi.org/10.3970/cmes.2011.080.219

Abstract

Boundary integral methods have proved very useful in the simulation of free surface motion, in part, because only information at the surface is necessary to track its motion. However, the velocity of the surface must be calculated quite accurately, and the error must be reasonably smooth, otherwise the surface buckles as numerical inaccuracies grow, leading to a failure in the simulation. For two-dimensional motion, the surface is just a curve and the boundary integrals are simple poles that may be removed, allowing spectrally accurate numerical integration. For three-dimensional motion, the singularity in the integrand, although weak, presents a greater challenge to the design of spectrally accurate quadrature. One way forward is to take advantage of a polar coordinate representation around the singularity point. Of course the typical grids used in boundary integral methods don't lend themselves to transformation to local polar coordinates, but the range of integration can be split into two regions, one near the singularity where polar coordinates can be used with suitable interpolation and an outer region where standard methods apply. We provide details and results of some tests that confirm spectral accuracy in the method.

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