Anisotropic Helmholtz decomposition for controlled fluid simulation

The combination of fluids and tensors has been explored in recent works to constrain the fluid flow along specific directions for simulation applications. Such an approach points towards the necessity to model flows using anisotropic fluid dynamics. The formalism behind anisotropic Helmholtz decomposition plays a central role given the foundations to model directional constraints. In this paper, we apply the anisotropic Helmholtz theory to obtain a divergent free velocity field that respects the constraints of a symmetric positive tensor field. A major contribution is a well-defined anisotropic projection method suitable for anisotropic transport. An anisotropic projection is paramount for stable incompressible advection in anisotropic medium. Aiming to show how to benefit from the anisotropic projection, we customize the Navier-Stokes equations to use tensor information for locally modifying fluid momentum and material advection. Besides, we develop a stable numerical method to integrate the obtained system of partial differential equations and a methodology to optimize the tensor field to reduce numerical errors. Experiments show that tensor fields with different anisotropic features can provide distinct projected divergence-free vector fields. Our results also show that the proposed method forms a basis for fluid flow simulation following meaningful paths induced by the tensor field geometry and topology.