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

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.