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.

Figures (18)

• Figure 1: Left: A slender body is described by a centerline $\gamma$ parameterized as ${\bm{x}}_c({s})$, and cross-sectional radius ${\varepsilon}({s})$. The vectors $d{\bm{x}}_c/ds$, ${\bm{e}}_1({s})$ and ${\bm{e}}_2({s})$ are orthogonal, with ${\bm{e}}_1({s})$ setting the ${\theta}$ angular origin at each ${s}$. Right: The slender body surface $\Gamma$ ($=\partial\Omega_b$ for the $b^\text{th}$ body) is partitioned into surface elements $\Gamma_k$, $k=1,\dots,K$, each discretized by the tensor product of $N_{{s}}^{(k)}$ Gauss--Legendre "panel" nodes in ${s}$ with $N_{{\theta}}^{(k)}$ equispaced nodes in ${\theta}$.
• Figure 2: The slender body surface is discretized into elements $\Gamma_k$. For a given quadrature accuracy tolerance $\epsilon_{\textrm{quad}}$, there is a region $\mathcal{N}_{\Gamma_k}$ around each element such that for target points outside this region, standard quadrature rules (tensor product of Gauss--Legendre and periodic trapezoidal) can be used. For target points inside $\mathcal{N}_{\Gamma_k}$ we use special quadrature rules. We approximate the region $\mathcal{N}_{\Gamma_k}$ by a set of overlapping spheres centered at the surface discretization nodes. This simplifies the task of identifying the target points within $\mathcal{N}_{\Gamma_k}$.
• Figure 3: A circular source loop $\gamma_{circ}$ (in red) of unit radius, parameterized by ${\theta}$ with points on the loop given by ${\bm{y}}_{circ}({\theta})$. An annular sheet (in blue) with radius between $\alpha$ and $2\alpha$, and a target point ${\bm{x}}_k$ on this annulus. We use the methods described in \ref{['sss:modal-greens-fn']} and \ref{['s:cheb-quad']} to build a generalized Chebyshev quadrature rule in ${\theta}$ that can integrate the potential at any target point on the annulus.
• Figure 4: A slender body element $\Gamma_k$ with a target point ${\bm{x}}$ close to it. To evaluate the potential at ${\bm{x}}$, for the outer integral we use a panel based Gauss--Legendre quadrature rule in ${s}$. Panels are refined adaptively near the target ${\bm{x}}$, giving a total of $m_a$ nodes.
• Figure 5: Illustration of our special quadrature rule in ${s}$ for an on-surface target point ${\bm{x}}={\bm{y}}({s}_0,{\theta}_0)$. The ${s}$-integral is singular with the asymptotic forms shown. Top: This integral could be approximated using dyadically refined Gauss--Legendre panel quadrature, plus a special singular quadrature rule for the panels on either side of ${\bm{x}}$. Bottom: For efficiency we replace such a panel quadrature rule by a single generalized Chebyshev quadrature rule with far fewer ($m_0$) nodes.

Theorems & Definitions (8)

• Remark 1
• Proposition 2
• Theorem 3
• Remark 4
• Remark 5: Collision handling
• proof : Proof of \ref{['p:null']}
• proof : Proof of \ref{['t:BIE']}
• Remark 6