The last few years have witnessed 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. For nonlocal hydrodynamics, 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. Using a temporal integrator developed in previous work, 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. In the latter case, we show that the effect of hydrodynamics can be accounted for through steric repulsion.

翻译：近年来，细丝流体力学数值方法呈现爆发式增长。细丝不仅在生物学、物理学和工程学中普遍存在，从应用数学视角来看也提出了独特挑战。其细长性、不可拉伸性、半柔性及介观尺度特性均要求数值方法能够处理约束条件下的多尺度问题。在保持动力学详细平衡与约束条件的同时考虑布朗运动十分困难，包含背景溶剂的情况亦然——后者会使悬浮液中多根细丝的动力学相互耦合。本文提出一种用于确定性与布朗不可拉伸细丝动力学的模拟平台，该平台包含非局部流体力学与空间排斥效应。对于非局部流体力学，我们通过Rotne-Prager-Yamakawa正则化奇点的线积分定义单根细丝的迁移率，并采用比自作用项更宽的正则化宽度计算非局部积分，从而在数值上保持对称正定性。对于空间排斥，我们引入定义为双细丝二重积分的软局部排斥势，并提出识别与计算被积函数非零分量的方案。基于先前工作开发的时间积分器，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，且采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——分别展示非局部流体力学是否产生长程流场的情形。在后一案例中，我们证明流体力学效应可通过空间排斥机制加以解释。