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Decoupling methods for fluid-structure interaction with local time-stepping

Hemanta Kunwar, Hyesuk Lee

TL;DR

The paper develops two global-in-time domain decomposition schemes for fluid–structure interaction, formulating the coupling as a space–time interface problem to allow independent, potentially nonmatching time discretizations in the fluid and solid subdomains. One approach uses a time-dependent Steklov–Poincaré operator with a Lagrange multiplier to enforce interface conditions, while the other employs Robin transmission conditions leading to a Schwarz waveform relaxation (SWR) algorithm with provable convergence under suitable parameters. A nonconforming time discretization framework is introduced, enabling different time grids in each subdomain and data transfer via $L^2$ projections, with a backward Euler semi-discretization proving convergence when $\alpha_f=\alpha_s$. Numerical experiments on manufactured and hemodynamics-inspired tests demonstrate the methods’ accuracy and efficiency, highlighting the benefits of local time stepping for multiphysics problems and stability in the presence of added-mass effects.

Abstract

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and the Robin method, for solving a fluid-structure interaction system. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent phenomena. We present numerical tests for both non-physical and physical problems, with various mesh sizes and time step sizes to demonstrate the accuracy and efficiency of the proposed methods.

Decoupling methods for fluid-structure interaction with local time-stepping

TL;DR

The paper develops two global-in-time domain decomposition schemes for fluid–structure interaction, formulating the coupling as a space–time interface problem to allow independent, potentially nonmatching time discretizations in the fluid and solid subdomains. One approach uses a time-dependent Steklov–Poincaré operator with a Lagrange multiplier to enforce interface conditions, while the other employs Robin transmission conditions leading to a Schwarz waveform relaxation (SWR) algorithm with provable convergence under suitable parameters. A nonconforming time discretization framework is introduced, enabling different time grids in each subdomain and data transfer via projections, with a backward Euler semi-discretization proving convergence when . Numerical experiments on manufactured and hemodynamics-inspired tests demonstrate the methods’ accuracy and efficiency, highlighting the benefits of local time stepping for multiphysics problems and stability in the presence of added-mass effects.

Abstract

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and the Robin method, for solving a fluid-structure interaction system. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent phenomena. We present numerical tests for both non-physical and physical problems, with various mesh sizes and time step sizes to demonstrate the accuracy and efficiency of the proposed methods.
Paper Structure (11 sections, 3 theorems, 82 equations, 9 figures, 7 tables)

This paper contains 11 sections, 3 theorems, 82 equations, 9 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Model equations
  3. Global-in-time DD schemes
  4. Time-dependent Steklov-Poincaré operator
  5. Robin transmission conditions and the space-time interface problem
  6. Schwarz waveform relaxation (SWR) algorithm
  7. Nonconforming time discretization and SWR algorithm
  8. Numerical Examples
  9. Test 1 : With a Known Analytical Solution
  10. Test 2 : Hemodynamic Experiment
  11. Conclusion

Key Result

Lemma 2.1

Let ${\mathbf w} \in {\mathbf V}$ and suppose $\nabla \cdot {\mathbf w} \in C^0(\Omega_f)$. Then, there exists $C_*>0$ such that

Figures (9)

  • Figure 2.1: Two-dimensional domain formed by FSI system
  • Figure 4.2: The fluid and structure domains
  • Figure 4.3: Nonconforming time grids
  • Figure 4.4: Projection of nonconforming time grids in two dimensions
  • Figure 5.5: Errors at $T=0.2$ by the Steklov-Poincaré method using Taylor-Hood and P2 elements with $h=1/32$.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3: Regularity of Normal stress
  • Theorem 3.4
  • proof
  • Remark 4.1
  • Theorem 4.2
  • proof