Efficient Convergent Boundary Integral Methods for Slender Bodies

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 Nyström 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.

arXiv comment37 pages, 17 figures

References in corpus (18)

• Mixing and transport by ciliary carpets: a numerical study
• A fast platform for simulating flexible fiber suspensions applied to cell mechanics
• The boundary integral formulation of Stokes flows includes slender-body theory
• A general formulation of Bead Models applied to flexible fibers and active filaments at low Reynolds number
• Swirling Instability of the Microtubule Cytoskeleton
• A scalable computational platform for particulate Stokes suspensions
• A fast integral equation method for solid particles in viscous flow using quadrature by expansion
• An integral equation formulation for rigid bodies in Stokes flow in three dimensions
• A high-order Nystrom discretization scheme for boundary integral equations defined on rotationally symmetric surfaces
• An integral-based spectral method for inextensible slender fibers in Stokes flow
• A local target specific quadrature by expansion method for evaluation of layer potentials in 3D
• Boundary integral equation analysis for suspension of spheres in Stokes flow
• Integral equation methods for elastance and mobility problems in two dimensions
• Taylor States in Stellarators: A Fast High-order Boundary Integral Solver
• Regularized Single and Double Layer Integrals in 3D Stokes Flow
• A robust solver for elliptic PDEs in 3D complex geometries
• Kernel Aggregated Fast Multipole Method: Efficient summation of Laplace and Stokes kernel functions
• Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries