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Modeling and numerical simulation of fully Eulerian fluid-structure interaction using cut finite elements

Stefan Frei, Tobias Knoke, Marc C. Steinbach, Anne-Kathrin Wenske, Thomas Wick

TL;DR

This work develops a monolithic, fully Eulerian FSI formulation on a fixed mesh using cut finite elements (CutFEM) with Nitsche-type interface conditions and ghost penalty stabilization. A novel weight function balances ghost penalties across poorly cut cells, enabling robust simulations with backward Euler time stepping and Taylor-Hood spatial discretization. The approach is demonstrated on a fixed-interface benchmark, showing convergence and good agreement with corresponding ALE results, thereby enabling stable simulations of large deformations and potential topology changes without mesh regeneration. Overall, the method provides a robust, unfitted framework for fully Eulerian FSI that can handle challenging interface motions and complex solid behavior without remeshing.

Abstract

We present a monolithic finite element formulation for (nonlinear) fluid-structure interaction in Eulerian coordinates. For the discretization we employ an unfitted finite element method based on inf-sup stable finite elements. So-called ghost penalty terms are used to guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The resulting system is solved in a monolithic fashion using Newton's method. Our developments are tested on a numerical example with fixed interface.

Modeling and numerical simulation of fully Eulerian fluid-structure interaction using cut finite elements

TL;DR

This work develops a monolithic, fully Eulerian FSI formulation on a fixed mesh using cut finite elements (CutFEM) with Nitsche-type interface conditions and ghost penalty stabilization. A novel weight function balances ghost penalties across poorly cut cells, enabling robust simulations with backward Euler time stepping and Taylor-Hood spatial discretization. The approach is demonstrated on a fixed-interface benchmark, showing convergence and good agreement with corresponding ALE results, thereby enabling stable simulations of large deformations and potential topology changes without mesh regeneration. Overall, the method provides a robust, unfitted framework for fully Eulerian FSI that can handle challenging interface motions and complex solid behavior without remeshing.

Abstract

We present a monolithic finite element formulation for (nonlinear) fluid-structure interaction in Eulerian coordinates. For the discretization we employ an unfitted finite element method based on inf-sup stable finite elements. So-called ghost penalty terms are used to guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The resulting system is solved in a monolithic fashion using Newton's method. Our developments are tested on a numerical example with fixed interface.
Paper Structure (8 sections, 11 equations, 3 figures, 2 tables)

This paper contains 8 sections, 11 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fluid-structure interaction system
  3. Strong form
  4. Weak formulation
  5. Discretization and ghost penalties
  6. Nonlinear solution
  7. Numerical test: modified "flow around a cylinder benchmark"
  8. Acknowledgement

Figures (3)

  • Figure 1: Configuration of laminar flow around an elastic shell with center at (0.2, 0.2), inner radius of 0.01 and outer radius of 0.05.
  • Figure 2: From top to bottom: Fluid speed $\lVert v_f\rVert_2$, pressure and displacement using the fully Eulerian approach at time $T = 25$ with $w_{\text{max}} = 3$. As the results using the ALE approach are visually very similar to the fully Eulerian approach, we refrain from including their plots.
  • Figure 3: Profiles of fluid speed $\lVert v_f\rVert_2$ (left) and pressure (right) at time $T = 25$ with $w_{\text{max}}=3$ along three vertical lines: at $x = 0.15$ in front of the solid (top), at $x = 0.25$ behind the solid (middle), and at $x = 2.2$ at the outflow boundary (bottom); ALE = $\color{blue}\times$, Euler = $\circ$.