A monolithic finite element method for phase-field modeling of fully Eulerian fluid–structure interaction

In this paper, we introduce a fully-monolithic, implicit finite element method designed for investigating fluid–structure interaction problems within a fully Eulerian framework. Our approach employs a coupled Navier–Stokes Cahn–Hilliard phase-field model, recently developed by Mokbel et al. (2018). This model adeptly addresses significant challenges such as large solid deformations, topology changes, stable incorporation of surface tensions, and eliminates the need for remeshing methods. While the original model was primarily tested for axisymmetric problems, our work extends its application to encompass a range of two- and three-dimensional verification tests. Additionally, we advance the model to handle multi-solid–fluid interaction scenarios through the integration of a multi-body contact algorithm. Assuming both the solid and fluid to be incompressible, we describe them using Navier–Stokes equations. For the solid, a hyperelastic neo-Hookean material is assumed, and the elastic solid stress is computed based on the left Cauchy–Green deformation tensor, which is governed by an Oldroyd-B like equation. We employ a residual-based variational multiscale method for solving the full Navier–Stokes equations, a stabilized Galerkin finite element method using Streamline-Upwind/Petrov–Galerkin (SUPG) stabilization for solving the Oldroyd-B equation, and a mixed finite element splitting scheme for the Cahn–Hilliard equation. The system of partial differential equations is solved using an implicit, monolithic scheme based on the generalized- α time integration method. Our approach is verified through two-dimensional numerical examples, including the deformation of an elastic wall by flow, the deformation and motion of a solid disk in a lid-driven cavity flow, and the bouncing of an elastic ball, showcasing the method’s ability to handle solid-wall contact. Furthermore, we extend the application to multi-body contact problems and verify the model’s accuracy by solving three-dimensional benchmark problems, such as the motion of an elastic solid sphere in lid-driven cavity flow and the falling of an elastic sphere onto an elastic block.