Constraint Penalization Method in the Lattice Boltzmann Method (LBM) for Fluid-Structure Interaction

TL;DR

This work introduces a constraint penalization method within the Lattice Boltzmann Method (LBM) to model fluid–structure interactions with rigid bodies. By extending the fictitious-domain concept, rigid-body motion is enforced through a penalization term applied directly to the fluid velocity, avoiding explicit interface forces and Lagrange multipliers, while preserving the locality and efficiency of the LBM. The velocity-based D2Q9 LBM is augmented with a penalization source, yielding an effective viscosity $\nu_{eff} = \frac{\nu}{1 - \alpha \tau}$ and requiring $\alpha < \frac{1}{\tau}$ for stability; this enables a monolithic coupling of fluid and solid regions. The method is validated across five benchmarks—lateral migration, single and elliptical particle settling, drafting–kissing–tumbling, and dense suspensions—demonstrating accurate rigid-body dynamics and hydrodynamic interactions with negligible computational overhead. Overall, the approach provides a robust, simpler alternative to Uzawa-based FSI for rigid solids in 2D, with strong potential for scalable, efficient simulations of particulate flows.

Abstract

A constraint penalization method is introduced within the Lattice Boltzmann (LBM) framework to model fluid-structure interactions involving rigid bodies. The proposed approach extends the fictitious domain concept by enforcing the rigid-body motion through a penalization term directly applied to the fluid velocity field, eliminating the need for explicit Lagrange multipliers or interface force computation. This formulation preserves the locality and simplicity of the LBM algorithm while ensuring an implicit coupling between the fluid and solid regions. Numerical experiments demonstrate that the method accurately reproduces rigid-body motion and hydrodynamic interactions with minimal additional computational cost. The method is applied to particle sedimentation, starting with a simple example and progressing to increasingly complex cases.

Figures (10)

• Figure 1: Discrete velocity set for the two-dimensional nine-velocity (D2Q9) lattice used in the present LBM implementation.
• Figure 2: Lateral migration of a neutrally buoyant circular particle in a plane Couette flow. The upper and lower walls move at velocities $\pm U_w/2$, generating a linear shear profile that drives the particle toward the channel mid-plane.
• Figure 3: (a) Couette flow configuration with streamlines, (b) Neutrally buoyant particle migration towards the channel centerline.
• Figure 4: Sedimentation of a circular particle under gravity in a vertical channel
• Figure 5: (a) Particle centroid trajectory, (b) Vorticity field around the settling particle.