A computational framework for the large scale simulation of the dynamics of highly flexible filaments in a viscous flow

Abstract

In science and engineering, the behavior of filaments immersed in viscous flows is of significant interest for a wide range of biological applications. In order to simulate the dynamics of the filaments, in this thesis we develop, validate, and test a physics-based computational framework by coupling finite elements and boundary integral methods. Then we introduce a data-driven approach to complement the high-fidelity numerical methods with an inexpensive surrogate model based on neural networks.

First, we present the methods to model the dynamics of slender fibers immersed in a viscous fluid within a physics-based computational framework. Motivated by the very large deformations that the fibers experience especially in the case of extreme aspect ratios, we implement a finite element framework based on the exact Kirchhoff-Love beam formulation. This method constitutes a significant improvement with respect to other proposed approaches which simulate the fibers only in the regime of high rigidity, thus ignoring the complex non-linear deformations that very slender filaments experience. The long-range hydrodynamic interactions of the filaments are modeled through the slender body theory for Stokes flows, where the disturbance motion of the incompressible viscous flow due to the presence of the slender bodies can be approximated by a distribution of Stokeslets. We use GPU resources to solve the dense system generated by the fluid model and push the scale to a cloud of 𝑂(500) flexible filaments. Additionally, to address the numerical instabilities observed as filaments come close to each other, we incorporate a model to enforce rigid contact by adding a discrete repulsion force to the filaments when penetration is detected. We validate the integrated computational model against experimental data on the sedimentation of slender filaments in a viscous flow, a first amongst comparable work. Finally, we use the model to explore the very low stiffness regime and get new physical insights on the equilibrium velocities and stability of sedimenting filaments.

Then, we establish the framework to complement physics-based methods with data-driven machine learning approaches, and test our model on a proof of concept problem. More specifically, we employ a neural network functional to predict the macroscopic stiffness of porous structures from their geometries. We train the weights of the model with synthetic data associating the effective Young’s moduli of porous structures to the size of the pores present in the material. To mitigate the high cost of generating data with PDE solvers, we investigate recent advances in active learning to lower the number of training points required to reach a given level of accuracy. We then discuss the trade-off between predictive error and number of training points, and show that active learning can improve the performance of the model by more than an order of magnitude. Finally, in addition to the benefit of low predictive cost of the surrogate model, we take advantage of the functional nature of the neural network to solve the inverse problem consisting in finding the best geometry yielding a target macroscopic stiffness.