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A Stable Loosely-Coupled Dirichlet-Neumann Scheme for Fluid-Structure Interaction with Large Added Mass

Francesca Renzi, Christian Vergara

TL;DR

A new strongly-coupled (SC) partitioning strategy is presented for the solution of the FSI problem, from which a stable LC scheme is derived based on Dirichlet and Neumann interface conditions, demonstrating the effectiveness and applicability of the proposed schemes.

Abstract

Solving fluid-structure interaction (FSI) problems when the densities are similar (large added mass), such as in hemodynamics, is challenging since the stability and convergence of the adopted numerical scheme could be compromised. In particular, while loosely coupled (LC) partitioned approaches are appealing due to their computational efficiency, the stability issues arising in high added mass regimes limit their applicability. In this work, we present a new strongly-coupled (SC) partitioning strategy for the solution of the FSI problem, from which we derive a stable LC scheme based on Dirichlet and Neumann interface conditions. We analyse the convergence of the new SC scheme on a benchmark problem, demonstrating enhanced behaviour over the standard DN method for specific ranges of a parameter $α$, without additional relaxation. Building on this, we introduce a new LC scheme by performing a single iteration per time step. Stability analysis on a benchmark problem proves that the proposed LC scheme is conditionally stable in large added mass regimes, under a constraint on a parameter $α$. Numerical experiments in large added mass settings confirm the theoretical results, demonstrating the effectiveness and applicability of the proposed schemes.

A Stable Loosely-Coupled Dirichlet-Neumann Scheme for Fluid-Structure Interaction with Large Added Mass

TL;DR

A new strongly-coupled (SC) partitioning strategy is presented for the solution of the FSI problem, from which a stable LC scheme is derived based on Dirichlet and Neumann interface conditions, demonstrating the effectiveness and applicability of the proposed schemes.

Abstract

Solving fluid-structure interaction (FSI) problems when the densities are similar (large added mass), such as in hemodynamics, is challenging since the stability and convergence of the adopted numerical scheme could be compromised. In particular, while loosely coupled (LC) partitioned approaches are appealing due to their computational efficiency, the stability issues arising in high added mass regimes limit their applicability. In this work, we present a new strongly-coupled (SC) partitioning strategy for the solution of the FSI problem, from which we derive a stable LC scheme based on Dirichlet and Neumann interface conditions. We analyse the convergence of the new SC scheme on a benchmark problem, demonstrating enhanced behaviour over the standard DN method for specific ranges of a parameter , without additional relaxation. Building on this, we introduce a new LC scheme by performing a single iteration per time step. Stability analysis on a benchmark problem proves that the proposed LC scheme is conditionally stable in large added mass regimes, under a constraint on a parameter . Numerical experiments in large added mass settings confirm the theoretical results, demonstrating the effectiveness and applicability of the proposed schemes.

Paper Structure

This paper contains 16 sections, 3 theorems, 30 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem settings
  3. Model equations
  4. Time discretization and algebraic setting
  5. A new strongly coupled partitioned scheme: SC-DN-$\alpha$ method
  6. Definition of the scheme
  7. Convergence of the SC-DN-$\alpha$ algorithm
  8. Numerical results
  9. Test I:
  10. Test II:
  11. Test III:
  12. A new loosely coupled partitioned scheme: LC-DN-$\alpha$ algorithm
  13. Stability of the LC-DN-$\alpha$ algorithm
  14. Numerical results
  15. Test IV:
  16. ...and 1 more sections

Key Result

Proposition 1

The SC-DN-$\alpha$eq:fluid-step-modelSC-eq:structure-step_modelSC algorithm converges to the solution of eq:toy_discrete if and only if:

Figures (5)

  • Figure 1: Fluid and structure domains.
  • Figure 2: Fluid and structure domains for the benchmark problem.
  • Figure 3: SC-DN-$\alpha$ scheme. Mean pressure (top), obtained as average on the radial section of the fluid domain, and fluid domain radius (bottom) at three instants. Test I.
  • Figure 4: Fluid mean pressure over the cross-section at $z = 3\,cm$: different values of $\alpha$ for $\rho_s/\rho_f = 0.9$ (left); different values of $\rho_f$ for $\rho_s = 1$ and $\alpha=0.11$ (right). $\Delta t=10^{-3}s$. Test IV.
  • Figure 5: Comparison between the SC-DN-$\alpha$ and the LC-DN-$\alpha$ fluid mean pressure over the cross-section at $z = 3\,cm$, for different values of $\rho_s/\rho_f$ and $\Delta t$. Test V.

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • Remark 3
  • Remark 4
  • Proposition 3
  • Remark 5