Anisotropic Helmholtz decomposition for controlled fluid simulation

Highlights：

• Anisotropic Helmholtz decomposition is unique up to the addition of a harmonic vector function

• Given an arbitrary vector field, it is possible to obtain a vector field that is divergence free with respect to the standard basis, by solving a T-Poisson equation, where T is a field composed of symmetric positive definite tensors. This process is called anisotropic projection

• The anisotropic projection is useful for find ing divergence free vector fields geometrically constrained by a tensor field. It can be applied for regularization of incompressible fluid transport.

• Numerical solution is challenging because the T-Poisson linear system coefficient matrices are not symmet ric.

• A Navier Stokes formulation whose all terms are constrained by a tensor field is provided. It s solution results in fluid flow ing along the tensor field geometric features.

• We provide a tensor field optimization method to reduce zero divergence error in numerical Helmholtz decomposition.

摘要

•Anisotropic Helmholtz decomposition is unique up to the addition of a harmonic vector function•Given an arbitrary vector field, it is possible to obtain a vector field that is divergence free with respect to the standard basis, by solving a T-Poisson equation, where T is a field composed of symmetric positive definite tensors. This process is called anisotropic projection•The anisotropic projection is useful for find ing divergence free vector fields geometrically constrained by a tensor field. It can be applied for regularization of incompressible fluid transport.•Numerical solution is challenging because the T-Poisson linear system coefficient matrices are not symmet ric.•A Navier Stokes formulation whose all terms are constrained by a tensor field is provided. It s solution results in fluid flow ing along the tensor field geometric features.•We provide a tensor field optimization method to reduce zero divergence error in numerical Helmholtz decomposition.