A new finite element formulation unifying fluid-structure and fluid-fluid interaction problems

Abstract

When the accurate simulation of two materials that interact through their common and deformable interface is of interest, the efficient treatment of the interface determines the success or failure of a numerical method. In this work, we propose a new, robust and easy-to-code finite element formulation for such interaction problems. The remedy of the interface constraints, namely the continuity of velocities and stresses, is accomplished using a single-node approach and the same continuous basis functions for the velocities in both materials. Given that only Newtonian fluids will be examined, we do not have to introduce basis functions for the stress components. The XFEM method, which enriches locally the continuous basis function of a variable that presents a discontinuity, is employed to tackle the discontinuous behavior of the pressure across the interface. The incorporation of Petrov-Galerkin stabilization schemes enhances further our formulation and allows the usage of equal order interpolants for velocities and pressure. We solve the coupled system of equations in a monolithic manner to alleviate the convergence problems of the segregated approach. The novel aspect of our method is that its ingredients do not differentiate based on the constituent materials of the problem, and it can be used interchangeably for either a fluid-structure or a fluid-fluid interaction problem. The accuracy of the new finite element formulation is assessed by comparing its numerical results to those of the literature in three problems: i) the flow through a partially collapsible channel, ii) the induced motion of a flexible elastic plate, iii) the filament stretching of a Newtonian thread surrounded by another immiscible viscous fluid. In all cases, we are in agreement with the results of the literature. Furthermore, we conduct a challenging, 3D simulation for a setup that resembles the motion of a three-leaflet stented aortic heart valve.