An Interface-Driven Adaptive Variational Procedure for Fully Eulerian Fluid-Structure Interaction via Phase-field Modeling

In this paper, we present a novel interface-driven adaptive variational procedure using a fully Eulerian description of fluid-structure interaction. The proposed fully-Eulerian procedure involves a fixed background unstructured mesh on which the fluid-structure interface is treated implicitly. We model the fluid-structure interaction by the phase-field finite element formulation relying on a partitioned staggered integration of the convective Allen-Cahn equation with the unified momentum equation for both solid and fluid dynamics. We employ the positivity preserving variational scheme for a bounded and stable solution of the Allen-Cahn phase-field equation. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize the residual based error indicators and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. The proposed nonlinear adaptive partitioned procedure restricts the coarsening step to the last non-linear iteration while simultaneously ensuring convergence properties of the coupled governing equations. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangle domain. We next systematically assess the performance of the adaptive procedure in terms of conservation properties for the increasing complexity of problems. Finally, we demonstrate our fully-Eulerian interface-driven adaptive FSI model to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.