Surface discretization considerations for the boundary-element method applied to three-dimensional ellipsoidal particles in Stokes flow

Abstract

The boundary-element method has often been used for simulating particle motion in Stokes flow, yet there is a scarcity of quantitative studies examining local errors induced by meshing highly elongated particles. In this paper, we study the eigenvalues and eigenfunctions of the double layer operator for an ellipsoid in an external linear or quadratic flow. We examine the local and global errors induced by changing the interpolation order of the geometry (flat or curved triangular elements) and the interpolation order of the double layer density (piecewise-constant or piecewise-linear over each element). Our results show that local errors can be quite large even when the global errors are small, prompting us to examine the distribution of local errors for each parameterization. Interestingly, we find that increasing the interpolation orders for the geometry and the double layer density does not always guarantee smaller errors. Depending on the nature of the meshing near high curvature regions, the number of high aspect ratio elements, and the flatness of the particle geometry, a piecewise-constant density can exhibit lower errors than piecewise-linear density, and there can be little benefit from using curved triangular elements. Overall, this study provides practical insights on how to appropriately discretize and parameterize three-dimensional boundary-element simulations for elongated particles with prolate-like and oblate-like geometries.