Numerical Simulations of Fully Eulerian Fluid-Structure Contact Interaction using a Ghost-Penalty Cut Finite Element Approach

TL;DR

This work develops a fully Eulerian, monolithic fluid–structure interaction framework with contact using a cut finite element method stabilized by ghost penalties on a fixed mesh. Time integration employs backward Euler with implicit extensions of evolving domains, and spatial coupling is enforced via a unified Nitsche approach, while a semi-smooth Newton method handles non-smooth contact terms. The authors demonstrate the method on a challenging elastic ball benchmark, showing that accurate resolution of the interface and pressure field is crucial for realistic bounce dynamics, and that adaptive time stepping and ghost penalties contribute to stability and efficiency. The contributions advance unfitted, Eulerian FSI–contact modeling and provide a foundation for further convergence analysis and solver enhancements in complex multiphysics simulations.

Abstract

In this work, we develop a cut-based unfitted finite element formulation for solving nonlinear, nonstationary fluid-structure interaction with contact in Eulerian coordinates. In the Eulerian description fluid flow modeled by the incompressible Navier-Stokes equations remains in Eulerian coordinates, while elastic solids are transformed from Lagrangian coordinates into the Eulerian system. A monolithic description is adopted. For the spatial discretization, we employ an unfitted finite element method with ghost penalties based on inf-sup stable finite elements. To handle contact, we use a relaxation of the contact condition in combination with a unified Nitsche approach that takes care implicitly of the switch between fluid-structure interaction and contact conditions. The temporal discretization is based on a backward Euler scheme with implicit extensions of solutions at the previous time step. The nonlinear system is solved with a semi-smooth Newton's method with line search. Our formulation, discretization and implementation are substantiated with an elastic falling ball that comes into contact with the bottom boundary, constituting a challenging state-of-the-art benchmark.

Figures (8)

• Figure 1: Euclidean norm of the fluid velocity $\|v_f\|_2$ on cut cells using $w_{\textup{max}}=1$ (left) and $w_{\textup{max}}=3$ (right). The values are taken from the "modified flow around a cylinder" computation in FrKnStWeWi24_ENUMATH.
• Figure 2: Visualization of the sets $\mathcal{F}_G^s$ and $\mathcal{F}_{h,n}^{s,\textup{ext}}$ used in the definitions of the ghost penalty functions \ref{['gp']} and \ref{['gp_ext']} for a circular solid domain.
• Figure 3: Relaxation of the contact condition with a planar wall: We impose the no-penetration condition already at a distance $\epsilon>0$ from the lower wall.
• Figure 4: Initial configuration for the bouncing elastic ball. All units are given in meters.
• Figure 5: Minimal distance between the interface and the bottom boundary against time using the uniform mesh.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3