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Numerical Analysis of a Cut Finite Element Approach for Fully Eulerian Fluid-Structure Interaction with Fixed Interface

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

Abstract

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.

Numerical Analysis of a Cut Finite Element Approach for Fully Eulerian Fluid-Structure Interaction with Fixed Interface

Abstract

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.

Paper Structure

This paper contains 18 sections, 10 theorems, 115 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and model
  3. Spatial and temporal discretization
  4. Spatial discretization
  5. Temporal discretization
  6. Stability analysis
  7. Extension of stability to the computational domains
  8. Stability estimate in the energy norm
  9. Stability of pressure
  10. A priori error estimates
  11. Projection and Interpolation operators
  12. Galerkin Orthogonality
  13. Consistency and interpolation errors
  14. Final a priori error estimate
  15. Numerical example: Lid-driven cavity
  16. ...and 3 more sections

Key Result

Lemma 4.1

Let $K \in \mathcal{T}^h$ and $v \in H^1(K)$. Then there exist $C_T, C_{\textup{TI}} > 0$ such that For a finite element function $v_h \in {\cal V}_i^{h,r}$, $i \in \{f,s\}$, it holds with a constant $C_{\textup{inv}}$

Figures (4)

  • Figure 1: Example of a domain partitioned into subdomains $\Omega_f$ and $\Omega_s$ (left) and a visualization of the corresponding computational domains with the sets of ghost penalty faces for the fluid ($\mathcal{F}_G^f$, middle) and the solid domain ($\mathcal{F}_G^s$, right).
  • Figure 2: Example of a cell $K$ with face $F = \mkern2mu \overline{\mkern-2mu K\mkern-2mu}\mkern2mu \cap \mkern2mu \overline{\mkern-2mu K\mkern-2mu}\mkern2mu'$ transformed by $\xi_K$ from the reference unit square $\hat{K}$ with vertical face $\hat{F}$, and a visualization of the (normal) vectors $\bm{n}$, $\hat{\bm{n}}$, $(\hat{\nabla} \xi_K) \hat{\bm{n}}$, $(\hat{\nabla} \xi_K)^{-1} \bm{n}$ and the projection $\pi_{\hat{F}}$ of a point $\hat{x} \in \hat{K}$ used in the proof of \ref{['lem_stab_gen']}.
  • Figure 3: Visualization of the set $\mathcal{T}_\Gamma^h$ and an example of a partition into patches $P_1,\dots,P_M$ (left), a single patch $P_j$ with the interface part $\Gamma_j^i$ and a suitable vertex $x_j$ (here: the midpoint of the left cell) (middle) and a visualization of the projection $\pi_h v$ on a patch $P_j$ for a scalar function $v$ (right) used in \ref{['lem.proj']}.
  • Figure 4: Configuration of lid-driven cavity test case with flow profile $v^{\textup{in}}$ at top boundary.

Theorems & Definitions (25)

  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.3
  • proof : Proof (\ref{['lem_stab_gen']})
  • Theorem 4.4: Stability estimate for the fully discretized bilinear form $\mathcal{A}^{kh}$
  • proof
  • Remark 4.5
  • Lemma 4.6
  • proof
  • ...and 15 more