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A Fully Parallelizable Loosely Coupled Scheme for Fluid-Poroelastic Structure Interaction Problems

Shihan Guo, Yizhong Sun, Yifan Wang, Xiaohe Yue, Haibiao Zheng

Abstract

We investigate the fluid-poroelastic structure interaction problem in a moving domain, governed by Navier-Stokes-Biot (NSBiot) system. First, we propose a fully parallelizable, loosely coupled scheme to solve the coupled system. At each time step, the solution from the previous time step is used to approximate the coupling conditions at the interface, allowing the original coupled problem to be fully decoupled into seperate fluid and structure subproblems, which are solved in parallel. Since our approach utilizes a loosely coupled scheme, no sub-iterations are required at each time step. Next, we conduct the energy estimates of this splitting method for the linearized problem (Stokes-Biot system), which demonstrates that the scheme is unconditionally stable without any restriction of the time step size from the physical parameters. Furthermore, we illustrate the first-order accuracy in time through two benchmark problems. Finally, to demonstrate that the proposed method maintains its excellent stability properties also for the nonlinear NSBiot system, we present numerical results for both $2D$ and $3D$ NSBiot problems related to real-world physical applications.

A Fully Parallelizable Loosely Coupled Scheme for Fluid-Poroelastic Structure Interaction Problems

Abstract

We investigate the fluid-poroelastic structure interaction problem in a moving domain, governed by Navier-Stokes-Biot (NSBiot) system. First, we propose a fully parallelizable, loosely coupled scheme to solve the coupled system. At each time step, the solution from the previous time step is used to approximate the coupling conditions at the interface, allowing the original coupled problem to be fully decoupled into seperate fluid and structure subproblems, which are solved in parallel. Since our approach utilizes a loosely coupled scheme, no sub-iterations are required at each time step. Next, we conduct the energy estimates of this splitting method for the linearized problem (Stokes-Biot system), which demonstrates that the scheme is unconditionally stable without any restriction of the time step size from the physical parameters. Furthermore, we illustrate the first-order accuracy in time through two benchmark problems. Finally, to demonstrate that the proposed method maintains its excellent stability properties also for the nonlinear NSBiot system, we present numerical results for both and NSBiot problems related to real-world physical applications.
Paper Structure (16 sections, 2 theorems, 47 equations, 9 figures, 2 tables, 2 algorithms)

This paper contains 16 sections, 2 theorems, 47 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem description
  3. Computational domains and mappings
  4. Fluid equation
  5. Structure equation
  6. Coupling condition
  7. Weak Formation
  8. The loosely coupled splitting scheme
  9. Robin-type boundary conditions
  10. Proposed parallel decoupling algorithm
  11. Stability analysis of the linear system
  12. Numerical examples
  13. Benchmark problem
  14. 2D fluid flow through a channel with poroelastic obstacles
  15. 3D simulation of the blood flow through a microfluidic chip with cylindrical poroelastic obstacles
  16. ...and 1 more sections

Key Result

Theorem 3.1

Assume that the solution of the NSBiot system in moving domain is unique. Let $\left(\boldsymbol{u}, p,\hat{\boldsymbol{\eta}},\hat{\boldsymbol{\xi}},\hat{\phi}\right)$ be the solution of the coupled NSBiot system (coupled), and let $\left(\boldsymbol{u}_r, p_r,\hat{\boldsymbol{\eta}}_r,\hat{\boldsy

Figures (9)

  • Figure 1: A sketch of the fluid-poroelastic structure interaction domain.
  • Figure 2: Convergence rate in time of the proposed decoupled algorithm for Case 1 (left) and Case 2 (right) at final time $T=1.0$.
  • Figure 3: An illustration of the $2D$ computation domain, where the region in red $\Omega_f$ stands for the fluid region, and regions in blue $\Omega_p$ denotes the poroelastic region. The size of the computational domain is $1.0\ cm \times 12.0\ cm$.
  • Figure 4: Superimposed streamlines and velocity magnitudes for fluid flow in the channel and Darcy flows through the poroelastic obstacles at $T=0.5$ s (top), $T=2.5$ s (middle) and $T=5.0$ s (bottom). The velocity is in the unit of $cm/s$.
  • Figure 5: Fluid pressure and Darcy pressure at final time $T=5.0$ s. The pressure is in units of $dynes/cm^2$.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Theorem 3.1: Equivalency
  • proof
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Remark 5.1