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.

翻译：细长纤维在粘性（斯托克斯）流体中的相互作用对工业与生物过程（如沉降、流变学、输运、细胞分裂及运动）至关重要。数值模拟通常依赖细长体理论（SBT），这是一种渐近非收敛近似，当纤维相互靠近时其误差会急剧增加。然而，完全解析纤维表面的收敛边界积分方程（BIE）方法此前因高纵横比三维几何中层势求积的高昂成本而难以实用。我们提出了一种高阶Nyström求积方案，其计算成本与纵横比无关，使得此类BIE方法变得可行。该方法结合了中心线面板（每个面板包含少量极向傅里叶模式）、环形格林函数、广义切比雪夫求积、高性能计算并行实现以及FMM加速。我们还提出了细长体的新型BIE公式，经离散化后能生成良态线性系统。我们针对具有（可能变化的）圆形截面的细长刚性闭合纤维，测试了拉普拉斯与斯托克斯狄利克雷问题及斯托克斯迁移率问题，纤维间距可小至细长半径的1/20，报告显示收敛精度通常达到至少10位有效数字。以此量化了近接触刚性纤维的数值SBT的失效情况。我们将该方法应用于512个环的沉降时间步进模拟，未知量多达165万，计算精度约为7位有效数字。