Helmholtz decomposition-based SPH

Abstract

Background: SPH method has been widely used in the simulation of water scenes. As a numerical method of partial differential equations, SPH can easily deal with the distorted and complex boundary. In addition, the implementation of SPH is relatively simple, and the results are stable and not easy to diverge. However, SPH method also has its own limitations. In order to further improve the performance of SPH method and expand its application scope, a series of key and difficult problems restricting the development of SPH need to be improved. Methods: In this paper, we introduce the idea of Helmholtz decomposition into the framework of smoothed particle hydrodynamics (SPH) and propose a novel velocity projection scheme for three-dimensional water simulation. First, we apply Helmholtz decomposition to a three-dimensional velocity field and decompose it into three orthogonal subspaces. Then, our method combines the idea of spatial derivatives in SPH to obtain a discrete Poisson velocity equation. Finally, the conjugate gradient (CG) is utilized to efficiently solve the Poisson equation. Results: The experimental results show that the proposed scheme is suitable for various situations and has higher efficiency than the current SPH projection scheme. Conclusion: Compared with the previous projection scheme, our solution does not need to modify the particle velocity indirectly by pressure projection, but directly by velocity field projection. The new scheme can be well integrated into the existing SPH framework, and can be applied to the interaction of water with static and dynamic obstacles, even for viscous fluid.

Keywords

• Water simulation
• SPH
• Helmholtz decomposition
• Conjugate Gradient