Efficient implementation of elastohydrodynamics via integral operators
Physical Review Fluids
... nsibility constraint, as well as the often nontrivial boundary conditions on the governing high-order PDEs. Formulating the problem for the evolving shape of a filament via an integral equation in the tangent angle has recently been found to greatly alleviate this numerical stiffness. The contribution of the present manuscript is to enable the simulation of nonlocal interactions of multiple filaments in a computationally efficient manner using the method of regularized stokeslets within this framework. The proposed method is benchmarked against a nonlocal bead and link model, and recent code utilizing a local drag velocity law. Systems of multiple filaments (1) in a background fluid flow, (2) under a constant body force, and (3) undergoing active self-motility are modeled efficiently. Buckling instabilities are analyzed by examining the evolving filament curvature, as well as by coarse graining the body frame tangent angles using a Chebyshev approximation for various choices of the relevant nondimensional parameters. From these experiments, insight is gained into how filamentfilament interactions can promote buckling, and further reveal the complex fluid dynamics resulting from arrays of these interacting fibers. By examining active moment-driven filaments, we investigate the speed of worm-and spermlike swimmers for different governing parameters. The MATLAB® implementation is made available as an open-source library, enabling flexible extension for alternate discretizations and different surrounding flows. A. L. HALL-MCNAIR et al. approaches in category (b), other ways of enforcing this condition are used. For example, the bead model of Jayaraman et al.  prescribes large spring constants between each bead, contributing to the numerical stiffness of the system. Equivalently, the gears model of Delmotte et al.  imposes a nonholonomic constraint to ensure nonpenetrability between adjacent beads, but as a result requires large numbers of points to represent a single filament. A recent promising development via Moreau et al. , referred to as coarse graining, is based on reformulating the problem via an integral equation with the filament tangent angle as the dependent variable. The method, initially developed using a local hydrodynamic drag law, provides an efficient framework for simulating noninteracting filament dynamics. This approach builds upon the early studies of Brokaw [5, 6] and Hines and Blum  and contrasts with Cartesian formulations [8, 9] . The contribution of the present manuscript is to enable efficient and accurate simulation of multiple, nonlocally interacting, passive and active filaments in ambient flows by incorporating recent developments in the regularized stokeslet method [10, 11] with the integral formulation in terms of the tangent angle of Moreau et al.  . The potential applications for a fast and accurate filament modeling framework are numerous. There has long been interest in understanding the mechanics and regulation of sperm flagellar movement, in particular problems relating to understanding the mechanical structure and motor regulation [5,    , investigating the response of the flagellar beat to its rheological environment    , understanding the dynamics of sperm due to surrounding solid walls [18, 19] , and studying the effect of viscosity on sperm swimming  . For a detailed review surrounding the importance of the sperm flagellum see Gallagher et al. . Furthermore, such a method could be used to investigate phenomena associated with epithelial cilia-driven flows such as cilia waveform modulation by length , the effects of flow induced by cilia on embryonic development  , studying the physical limits of flow sensing  , and investigating the mechanical structure of the axoneme in cilia  . Another avenue of active-filament research to which the proposed framework could be applied is magnetic swimmers  . These models have wider relevance in the field of synthetic biology, with particular application to microscopic bacteriophage-based fiber sensors    and flexible filament microbots  . The proposed framework could be used to further investigate the dynamics of bundles of filaments  and additionally has applications in the multiscale studies of complex polymeric fluids and flagellar movement through them [31, 32] . We will extend the framework introduced by Moreau et al. , augmenting and reformatting their formulation with the method of regularized stokeslets of Cortez et al. [33, 34] . These methods have proven to be accurate and effective in modeling the hydrodynamics in various multiplefiber scenarios [35, 36] . The method of regularized stokeslets enables the modeling of nonlocal hydrodynamics within and between filaments, and between filaments and surrounding structures. The method is implemented in a numerically efficient manner, retaining the computational economy and low hardware requirements inherited from the Moreau et al. formulation. The structure of this paper is as follows: in Sec. II the elastohydrodynamic integral formulation (EIF) for a single filament is proposed. In Sec. III alterations to the EIF for various single-and multifilament scenarios are presented. Verification and benchmarking of the method is given in Sec. IV. Simulation results of the problems formulated in Sec. III are then presented in Sec. V, followed by discussion of the results and of further possible applications in Sec. VI. The MATLAB® code for the methods described within this report are provided in the associated GitLab repository .