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A SIMPLE-Based Preconditioned Solver for the Direct-Forcing Immersed Boundary Method

Rachel Yovel, Eran Treister, Yuri Feldman

TL;DR

The proposed approach introduces a novel and accessible framework for immersed boundary simulations requiring strong pressure-force coupling, based on a preconditioned SIMPLE algorithm and the spectral equivalence between the Schur complement and the discrete Laplacian is rigorously demonstrated.

Abstract

We present a robust and scalable solver for direct-forcing immersed boundary simulations, based on a preconditioned SIMPLE algorithm. The method applies block elimination to the pressure-force coupled system, and utilizes the discrete Laplacian operator as an efficient preconditioner for the resulting Schur complement. We rigorously demonstrate the spectral equivalence between the Schur complement and the discrete Laplacian, ensuring convergence behavior that is independent of grid resolution and physical parameters. This enables accurate, stable, and efficient two-way coupled fluid-structure interaction (FSI) simulations with moving boundaries and significant added-mass effects. These simulations are all executable on standard computing platforms. Extensive validation and verification - including simulations of oscillating, sedimenting, and buoyant spheres, as well as configurations involving multiple immersed bodies - confirm the solver's accuracy and efficiency across a broad range of FSI scenarios. The proposed approach introduces a novel and accessible framework for immersed boundary simulations requiring strong pressure-force coupling.

A SIMPLE-Based Preconditioned Solver for the Direct-Forcing Immersed Boundary Method

TL;DR

The proposed approach introduces a novel and accessible framework for immersed boundary simulations requiring strong pressure-force coupling, based on a preconditioned SIMPLE algorithm and the spectral equivalence between the Schur complement and the discrete Laplacian is rigorously demonstrated.

Abstract

We present a robust and scalable solver for direct-forcing immersed boundary simulations, based on a preconditioned SIMPLE algorithm. The method applies block elimination to the pressure-force coupled system, and utilizes the discrete Laplacian operator as an efficient preconditioner for the resulting Schur complement. We rigorously demonstrate the spectral equivalence between the Schur complement and the discrete Laplacian, ensuring convergence behavior that is independent of grid resolution and physical parameters. This enables accurate, stable, and efficient two-way coupled fluid-structure interaction (FSI) simulations with moving boundaries and significant added-mass effects. These simulations are all executable on standard computing platforms. Extensive validation and verification - including simulations of oscillating, sedimenting, and buoyant spheres, as well as configurations involving multiple immersed bodies - confirm the solver's accuracy and efficiency across a broad range of FSI scenarios. The proposed approach introduces a novel and accessible framework for immersed boundary simulations requiring strong pressure-force coupling.
Paper Structure (27 sections, 1 theorem, 38 equations, 12 figures, 6 tables)

This paper contains 27 sections, 1 theorem, 38 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Governing equations for the dynamics of a spherical particle in a Newtonian flow.
  4. Updating the rotation matrix.
  5. Interpolation and regularization operators.
  6. Method: time stepping and the pressure--force-density corrections system
  7. One-way coupled moving boundary
  8. Two-way coupled FSI
  9. Iterative Coupling Algorithm
  10. Step 1: Explicit prediction of particle position and orientation
  11. Step 2: Eulerian velocity prediction
  12. Step 3: Iterative correction procedure
  13. Method: preconditioning
  14. The Laplacian as a preconditioner
  15. Efficient matrix-free implementation
  16. ...and 12 more sections

Key Result

Theorem 5.1

For every generalized saddle-point matrix of the form where $G$ is an $n\times m$ matrix and $R$ an $n\times k$ matrixTypically, $n>m>k,$ but this assumption is not necessary for the Theorem's statement., the leading block $L$ is spectrally equivalent to the primal Schur complement $S_p$ from Eq. eq:SchurPrimal, in the sense

Figures (12)

  • Figure 1: An Eulerian discretization cell in 3D and two adjacent Lagrangian points on the surface of the immersed body.
  • Figure 2: An oscillating sphere of diameter $D$ and a box of dimensions $4D\times 4D \times 6D$
  • Figure 3: Time evolution of the drag coefficient $C_D=8f_z/\pi$ as a function of time for $A/D=1$, and maximal values of the drag coefficient, $\max(C_D)$, as a function of $A/D$. The dotted line shows the position of the sphere's center (right axis).
  • Figure 4: Schematic representation of porous spheres modeled by arrays of (a) 7 and (b) 14 solid sub-spheres packed without overlap within a bounding sphere of diameter $D$.
  • Figure 5: Time evolution of external forces applied to individual sub-spheres in the 7-sphere array and in the 14-sphere array during oscillatory motion. Each solid curve corresponds to the force applied to a sub-sphere identified by its serial number from Table \ref{['tab:spheresCenters']}.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Remark 4.1
  • Remark 4.2
  • Theorem 5.1
  • proof
  • Remark 5.1
  • Remark 5.2