Fast VEM Fluid Simulation

2026-07-20

AI Summary

This work addresses the challenges of fluid simulation under complex boundaries, where body-fitted meshes often lead to ill-conditioned pressure projection systems and high computational costs. To overcome these issues, the authors propose FastVEM, a novel framework that co-designs several key components: a virtual element method (VEM) discretization, a convexity-preserving cut-cell mesh generation strategy, a VEM-based polynomial-space particle-in-grid advection scheme, and a Galerkin geometric multigrid solver with diffusion-free prolongation. Together, these elements form a boundary-aware nested grid hierarchy that robustly and efficiently enforces incompressibility and handles intricate boundary conditions on irregular body-fitted meshes. Compared to existing cut-cell fluid simulators, FastVEM achieves up to 100× acceleration in the pressure projection phase while supporting significantly more complex geometric boundaries.

Abstract

The intricate motion arising from fluid--boundary interactions is visually compelling, yet notoriously difficult and computationally expensive to simulate in the presence of complex boundaries. Accurately resolving boundary geometry requires body-fitted grids constructed via cut-cell methods, which often leads to poorly conditioned linear systems in the pressure projection stage and, consequently, prohibitive computational cost. We present FastVEM, an efficient boundary-conforming fluid simulation framework that enables high-fidelity flow--boundary interaction at substantially reduced cost. Computational efficiency is achieved through a coordinated, top-down design spanning numerical discretization, grid construction, and linear solvers. FastVEM adopts a Virtual Element Method (VEM) discretization to robustly enforce incompressibility and boundary conditions on irregular body-fitted grids, and employs a VEM polynomial-space Particle-in-Cell scheme for advection. Complementing this discretization, a convexity-preserving cut-cell strategy is introduced to construct simulation-friendly body-fitted grids. To accelerate pressure projection, we develop a Galerkin geometric multigrid solver featuring a diffusion-free prolongation operator that prevents coarse-level matrix densification, along with a nested, boundary-aware grid hierarchy that ensures well-posed placement of coarse-level degrees of freedom. Compared to prior cut-cell--based fluid simulators, FastVEM speeds up the computationally dominant pressure projection stage by up to 100x, while robustly handling even more challenging boundary geometries.

Problem

• fluid simulation
• complex boundaries
• cut-cell methods
• pressure projection
• computational cost

Innovation

• Virtual Element Method
• cut-cell
• geometric multigrid
• boundary-conforming simulation
• fluid simulation