Efficient Convergent Boundary Integral Methods for Slender Bodies

TL;DR

This paper introduces a high-order, efficient boundary integral method for simulating slender fibers in viscous fluids, achieving high accuracy and convergence even at close proximities, surpassing traditional slender body theory limitations.

Contribution

It develops a convergent boundary integral equation approach with aspect-ratio independent cost, enabling accurate simulations of slender fibers at close contact, which was previously computationally impractical.

Findings

Achieved at least 10-digit convergence for slender fiber problems.

Quantified the breakdown of slender body theory at close separations.

Successfully simulated sedimentation of 512 loops with high accuracy.

Abstract

The interaction of fibers in a viscous (Stokes) fluid plays a crucial role in industrial and biological processes, such as sedimentation, rheology, transport, cell division, and locomotion. Numerical simulations generally rely on slender body theory (SBT), an asymptotic, nonconvergent approximation whose error blows up as fibers approach each other. Yet convergent boundary integral equation (BIE) methods which completely resolve the fiber surface have so far been impractical due to the prohibitive cost of layer-potential quadratures in such high aspect-ratio 3D geometries. We present a high-order Nystr\"om quadrature scheme with aspect-ratio independent cost, making such BIEs practical. It combines centerline panels (each with a small number of poloidal Fourier modes), toroidal Green's functions, generalized Chebyshev quadratures, HPC parallel implementation, and FMM acceleration. We also present new BIE formulations for slender bodies that lead to well conditioned linear systems upon discretization. We test Laplace and Stokes Dirichlet problems, and Stokes mobility problems, for slender rigid closed fibers with (possibly varying) circular cross-section, at separations down to $1/20$ of the slender radius, reporting convergence typically to at least 10 digits. We use this to quantify the breakdown of numerical SBT for close-to-touching rigid fibers. We also apply the methods to time-step the sedimentation of 512 loops with up to $1.65$ million unknowns at around 7 digits of accuracy.