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.

翻译：本文提出了一种用于模拟悬浮在Stokes流体中可变形且可扩展囊泡的新型数值方案。该方案的核心特征在于采用基于单位分解法的曲面表示，相较于基于球谐函数的表示方法，可实现渐近加速的计算。我们采用边界积分方程公式来表征和离散化流体动力学相互作用，其中边界积分具有弱奇异性。基于正则化Stokes核建立了求积方案，并利用基于单位分解的有限差分法计算界面力。对于含N个点的曲面离散化，本数值方案具有四阶精度和O(N)渐近复杂度，优于采用乘积法则求积的球谐函数谱方法（O(N²logN)复杂度）。我们通过GPU加速实现高分辨率复杂形状模拟，研究了抵抗剪切与拉伸的囊泡在剪切流和泊肃叶流中的动力学行为，验证了方案的收敛性，并与现有最先进方法进行了对比。