Simulation of the fiber orientation through a finite element approach to solve the Fokker–Planck equation

Abstract

This work aims to introduce a groundbreaking approach by directly computing the Fokker–Planck equation, providing a mesoscopic scale orientation indicator based on the 2D-probability density function (PDF) of the fibers’ orientation state. Unlike conventional methods that rely on pre-averaged quantities and closure approximations, our method offers enhanced accuracy and information preservation. The model’s enhanced accuracy can be served as a foundational tool for future studies, enabling the development of comprehensive models describing the fluid-flow coupling problem with precision. Consequently, this advancement facilitates the simulation of real-case scenarios, such as the dynamic motion of fibers during the injection phase of molten thermoplastics within a mold cavity. The novelty of this work lies in its application of the Streamline-Upwind/Petrov–Galerkin (SUPG) finite element method, on both orientation and physical spaces. Our model shows the potential to improve the understanding and prediction of fiber behavior in industrial applications, offering valuable insights into process optimization and design. Implemented within a finite element framework, a comprehensive investigation is conducted into the effects of mesh refinement, time scheme, and time stepping on the computational modeling of the PDF evolution, aiming to strike an optimal balance between model precision and computational efficiency. The validation tests were conducted for the case of simple shear flow to examine the influence of the interaction coefficient $C_I$ and the fiber shape factor $\lambda$ on the resolution of the probability distribution function. The numerical results demonstrate the evolution of fiber orientation over time under Poiseuille flow conditions.