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：在粘性（斯托克斯）流体中纤维的相互作用在工业和生物过程中起着关键作用，例如沉降、流变学、输运、细胞分裂和运动。数值模拟通常依赖于细长体理论（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大。然而，完全解析纤维表面的收敛边界积分方程（BIE）方法迄今因高纵横比三维几何中层面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，其成本与纵横比无关，使得此类BIE方法变得实用。该方法结合了中心线面板（每个面板具有少量环向傅里叶模式）、环向格林函数、广义切比雪夫求积、HPC并行实现以及FMM加速。我们还提出了新的细长体BIE公式，可在离散化后得到良态线性系统。我们针对具有（可能变化的）圆形横截面的细长刚性闭合纤维，在间距小至细长半径1/20的情况下，测试了拉普拉斯和斯托克斯狄利克雷问题以及斯托克斯迁移率问题，报告显示通常收敛至至少10位数字。我们利用此方法量化了数值SBT在接近接触的刚性纤维中的失效。我们还应用该方法对含有512个环（未知量高达165万）的沉降过程进行时间步进计算，精度约为7位数字。