The last few years have seen an explosion of new numerical methods for filament hydrodynamics. Aside from their ubiquity in biology, physics, and engineering, filaments present unique challenges from an applied-mathematical point of view. Their slenderness, inextensibility, semiflexibility, and meso-scale nature all require numerical methods that can handle multiple lengthscales in the presence of constraints. Accounting for Brownian motion while keeping the dynamics in detailed balance and on the constraint is difficult, as is including a background solvent, which couples the dynamics of multiple filaments together in a suspension. In this paper, we present a simulation platform for deterministic and Brownian inextensible filament dynamics which includes nonlocal fluid dynamics and steric repulsion. We first review previous work, in which we formulated the equations and spatial discretization for deterministic and Brownian inextensible filament dynamics. We then present novel methods for nonlocal fluid dynamics and steric replusion. In the former case, we define the mobility on a single filament using line integrals of Rotne-Prager-Yamakawa regularized singularities, and numerically preserve the symmetric positive definite property by using a thicker regularization width for the nonlocal integrals than for the self term. For steric repulsion, we introduce a soft local repulsive potential defined as a double-integral over two filaments, then present a scheme to identify and evaluate the nonzero components of the integrand. We demonstrate that Langevin dynamics sample from the equilibrium distribution of free filament shapes, and that the modeling error in using the thicker regularization is small. We conclude with two examples, sedimenting filaments and cross-linked fiber networks, in which nonlocal hydrodynamics does and does not generate long-range flow fields, respectively.

翻译：近年来，针对细丝流体动力学的数值方法呈现爆发式增长。细丝不仅在生物学、物理学和工程学中无处不在，从应用数学的角度来看也提出了独特的挑战。其细长性、不可伸长性、半柔性及介观尺度特性，均要求数值方法能够在约束条件下处理多尺度问题。在保持动力学详细平衡并满足约束条件的同时考虑布朗运动是困难的，包含背景溶剂同样具有挑战性——溶剂会使悬浮液中多根细丝的动力学相互耦合。本文提出了一种用于确定性与布朗不可伸长细丝动力学的模拟平台，该平台包含非局部流体动力学与空间排斥效应。我们首先回顾了先前工作，其中我们建立了确定性与布朗不可伸长细丝动力学的控制方程及空间离散格式。随后，我们提出了处理非局部流体动力学与空间排斥的新方法。对于前者，我们通过Rotne-Prager-Yamakawa正则化奇点的线积分定义单根细丝上的迁移率，并通过在非局部积分中使用比自作用项更宽的正则化宽度，在数值上保持对称正定性。对于空间排斥，我们引入一种定义为双细丝上二重积分的软局部排斥势，并提出一种识别与计算被积函数非零分量的方案。我们证明朗之万动力学能够对自由细丝形状的平衡分布进行采样，且使用更宽正则化宽度带来的建模误差很小。最后通过两个示例——沉降细丝与交联纤维网络——分别展示了非局部流体动力学如何产生或不产生长程流场。