Boundary integral equation analysis for spheroidal suspensions

TL;DR

This paper introduces a spectrally accurate, fast boundary integral method for evaluating interactions in suspensions of spheroids, applicable to Laplace and Stokes problems, enabling efficient simulations of dense particle systems.

Contribution

The paper develops a novel spectral boundary integral method using spheroidal harmonics and fast multipole acceleration for suspensions of spheroids, improving accuracy and efficiency.

Findings

Achieves spectrally accurate evaluation of near-field interactions.

Efficiently handles dense suspensions of hundreds of spheroids.

Applicable to both Laplace and Stokes flow problems.

Abstract

In this work, we provide a fast, spectrally accurate method for the evaluation of boundary integral operators (BIOs) on a suspension of prolate and oblate spheroids. We first derive formulas for the standard layer potential operators for the Laplace equation applied to an expansion of the integral densities in the appropriate spheroidal harmonic basis. These then lead to analytical expressions in solid harmonics that allow spectrally accurate evaluation of near-field particle interactions. Finally, a standard quadrature scheme is used to evaluate smooth, far-field interactions; these are then accelerated using the fast multipole method. Through a number of numerical test cases, we verify the accuracy and efficiency of our BIO evaluation framework for dense, polydisperse suspensions of spheroids. Through the use of standard formulas linking Stokes and Laplace potentials, we show our scheme can be readily applied to problems involving particulate suspension flows. For both Laplace and Stokes, our method allows us to evaluate BIOs for suspensions up to hundreds of particles on a single processor.