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Stability Analysis of Monolithic Globally Divergence-Free ALE-HDG Methods for Fluid-Structure Interaction

Shuaijun Liu, Xiaoping Xie

Abstract

In this paper, we propose two monolithic fully discrete finite element methods for fluid-structure interaction (FSI) based on a novel Piola-type Arbitrary Lagrangian-Eulerian (ALE) mapping. For the temporal discretization, we apply the backward Euler method to both the non-conservative and conservative formulations. For the spatial discretization, we adopt arbitrary order hybridizable discontinuous Galerkin (HDG) methods for the incompressible Navier-Stokes and linear elasticity equations, and a continuous Galerkin (CG) method for the fluid mesh movement. We derive stability results for both the temporal semi-discretization and the fully discretization, and show that the velocity approximations of the fully discrete schemes are globally divergence-free. Several numerical experiments are performed to verify the performance of the proposed methods.

Stability Analysis of Monolithic Globally Divergence-Free ALE-HDG Methods for Fluid-Structure Interaction

Abstract

In this paper, we propose two monolithic fully discrete finite element methods for fluid-structure interaction (FSI) based on a novel Piola-type Arbitrary Lagrangian-Eulerian (ALE) mapping. For the temporal discretization, we apply the backward Euler method to both the non-conservative and conservative formulations. For the spatial discretization, we adopt arbitrary order hybridizable discontinuous Galerkin (HDG) methods for the incompressible Navier-Stokes and linear elasticity equations, and a continuous Galerkin (CG) method for the fluid mesh movement. We derive stability results for both the temporal semi-discretization and the fully discretization, and show that the velocity approximations of the fully discrete schemes are globally divergence-free. Several numerical experiments are performed to verify the performance of the proposed methods.

Paper Structure

This paper contains 17 sections, 11 theorems, 156 equations, 10 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Model problem and weak formulation
  3. Notation
  4. Model description
  5. Fluid equations
  6. Structure equations
  7. Coupling conditions
  8. The ALE mapping
  9. Weak formulations and stability
  10. Temporal semi-discretization
  11. The semi-discrete Piola-type ALE mapping
  12. Temporally semi-discrete schemes
  13. Energy stability analysis
  14. Full discretization
  15. Domain triangulation and fully discrete ALE mapping
  16. ...and 2 more sections

Key Result

Lemma 2.1

For any $\bm v \in \bm V _{\rm div}^{f}(t)$ and it holds that

Figures (10)

  • Figure 1: Schematic of the FSI domain: reference configuration at $t=0$ (left) and configuration at $t>0$ (right).
  • Figure 2: Unstructured triangular meshes generated by NETGEN with $\texttt{maxh}=1/20$ (left, Example 5.1) and $\texttt{maxh}=1/32$ (right, Example 5.2).
  • Figure 3: Energy $E^{n}_h$ of Scheme-C and Scheme-D with $k=2$ and $h=1/20$.
  • Figure 4: Geometric configuration of the benchmark problem.
  • Figure 5: Mesh with 1992 elements.
  • ...and 5 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.1
  • ...and 22 more