Numerical simulation of an extensible capsule using regularized Stokes kernels and overset finite differences

TL;DR

The paper presents a fourth-order accurate numerical framework for simulating an extensible capsule in Stokes flow, leveraging a partition-of-unity surface representation with six overlapping patches to enable nonuniform surface discretization and efficient evaluation of boundary-integral operators. It combines a boundary-integral formulation with regularized Stokes kernels, overset finite differences for interfacial forces, and fourfold upsampling of singular quadrature, all accelerated on GPUs; a single-level FMM can further reduce complexity to linear scaling. Comprehensive convergence tests and physically realistic simulations in shear and Poiseuille flows validate accuracy against state-of-the-art spherical-harmonics methods and demonstrate robust long-time stability. The work provides a scalable, high-accuracy alternative for microfluidic capsule dynamics with potential extensions to multiple capsules and adaptive surface representations.

Abstract

In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that enables asymptotically faster computations compared to spherical-harmonics based representations. We use a boundary integral equation formulation to represent and discretize hydrodynamic interactions. The boundary integrals are weakly singular. We use the quadrature scheme based on the regularized Stokes kernels. We also use partition-of unity based finite differences that are required for the computational of interfacial forces. Given an N-point surface discretization, our numerical scheme has fourth-order accuracy and O(N) asymptotic complexity, which is an improvement over the O(N^2 log(N)) complexity of a spherical harmonics based spectral scheme that uses product-rule quadratures. We use GPU acceleration and demonstrate the ability of our code to simulate the complex shapes with high resolution. We study capsules that resist shear and tension and their dynamics in shear and Poiseuille flows. We demonstrate the convergence of the scheme and compare with the state of the art.

Figures (17)

• Figure 1: A representation of the problem setup. The grey filled region is the interior of the capsule with membrane $\gamma$. Exterior of the capsule is filled with a Newtonian fluid and the capsule is suspended freely in it. $\boldsymbol u_{\infty}$ is the imposed background fluid velocity.
• Figure 2: Here we summarize the notation for the atlas construction. The unit sphere $\mathbb{S}^{2}$ is on the left and the capsule $\gamma$ is on the right with the diffeomorphism $\phi:\mathbb{S}^{2} \longrightarrow \gamma$. We show two overlapping patches colored in red and blue. The first patch $\mathcal{P}_{1}^{0}$ is the red colored arc from point $a_{0}$ to $b_{0}$ on $\mathbb{S}^{2}$. Its corresponding patch $\mathcal{P}_{1}$ on the capsule surface $\gamma$ is shown in red as the arc from points $a_{1}$ to $b_{1}$. The second patch $\mathcal{P}_{2}^{0}$ is the blue colored arc from point $c_{0}$ to $d_{0}$ on $\mathbb{S}^{2}$. Its corresponding patch $\mathcal{P}_{2}$ on the capsule surface $\gamma$ is shown in blue as the arc from points $c_{1}$ to $d_{1}$. Their corresponding coordinate domains $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ are also shown in the red and blue color. The coordinate charts $\eta_{i}^{0}: \mathcal{U}_{1} \longrightarrow \mathcal{P}^{0}_{i}, \ i=1,2$, are shown as dashed lines in the respective colors. The diffeomorphism $\phi$ also gives the coordinate charts $\eta_{i}:\mathcal{U}_{1} \longrightarrow \mathcal{P}_{i},\ i=1,2$, for the patches on the capsule surface $\gamma$ shown as colored dashed lines from $\mathcal{U}_{i}$ to $\mathcal{P}_{i}$. The corresponding partition of unity functions $\psi^{0}_{i},i=1,2,$ with $\mathsf{supp}(\psi^{0}_{i}) \subset \mathcal{P}^{0}_{i}$ are drawn over coordinate domains $\mathcal{U}_{i}$ for visual clarity since $\mathcal{P}_{i}^{0}$ is diffeomorphic to $\mathcal{U}_{i}$.
• Figure 3: The six hemispherical patches forming an open cover of the unit sphere $\mathcal{S}^{2}$. Each one is represented by the black grid. a)--f) $\mathcal{P}_{i}^{0}$ for $i=1,2,\ldots,6$.
• Figure 4: Representation of discretization of a unit sphere using $m_{\mathrm{th}}$-order grids for (a) $m=8$, (b) $m=16$, (c) $m=32$.
• Figure 5: Representation of discretization of the ellipsoid $x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2}=1$ with $a=0.5,b=1,c=1$, using $m_{\mathrm{th}}$-order grids for (a) $m=8$, (b) $m=16$, (c) $m=32$.