The natural Helmholtz-Hodge decomposition (nHHD) is a new variant of the well-known vector field decompositions called the Helmholtz-Hodge decomposition (HHD). The HHD describes a vector field as the sum of an incompressible, an irrotational, and a vector field. It is a fundamental tool used in a wide variety of applications such as climate modeling, combustion, graphics, astrophysics, geophysics, etc.
The Helmholtz-Hodge decomposition decomposes a flow into an incompressible, an irrotational, and a harmonic vector field.
However, for bounded domains, the HHD is not uniquely defined; traditionally, boundary conditions are imposed to obtain a unique solution. Unfortunately, the traditional boundary conditions may not be compatible with a given data, which can lead to sometimes drastic artifacts and distortions in all three components, hence producing unphysical results. Especially for applications in visualization and analysis, this may cause problems, since the boundary conditions used during the simulation/experiment may not be known, or the simulation may have used an open boundry; in both these cases, imposing (traditional) boundary conditions on the decompsostion gives incorrect results.
We developed the natural HHD, which computes the three vector field components using a completely data-driven approach. the nHHD obtains uniqueness without assuming boundary conditions a priori. As a result, it enables a reliable and artifact-free analysis for flows with open boundaries or unknown boundary conditions.
|The decomposed flow using standard boundary conditions (left) shows unphysical behavior as it is forced to be aligned with the boundary. The natural HHD (right), on the other hand, captures the underlying phenomena correctly as the flow is decomposed "naturally", i.e., without boundary conditions.|
The technical details on the nHHD can be found in the following paper.
The Natural Helmholtz-Hodge Decomposition for Open-Boundary Flow Analysis.
Harsh Bhatia, Valerio Pascucci, and Peer-Timo Bremer.
IEEE Transactions on Visualization and Computer Graphics (TVCG), vol. 20, no. 11, pp. 1566–1578, Nov. 2014, doi:10.1109/TVCG.2014.2312012.
A basic serial implementation of the nHHD for rectilinear grids can be found at https://github.com/bhatiaharsh/naturalHHD.