Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids

Abstract: When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentric pressures. Second, meshes must align with potentially complex moving surfaces and boundaries, necessitating continuous remeshing. Unfortunately, sacrificing mesh quality in favour of speed yields inaccurate velocities and simulation artifacts. We describe how to eliminate the boundary-matching constraint by adapting recent embedded boundary techniques to tetrahedra, so that neither air nor solid boundaries need to align with mesh geometry. This enables the use of high quality, arbitrarily graded, non-conforming Delaunay meshes, which are simpler and faster to generate. Temporal coherence can also be exploited by reusing meshes over adjacent timesteps to further reduce meshing costs. Lastly, our free surface boundary condition eliminates the spurious currents that previous methods exhibited for slow or static scenarios. We provide several examples demonstrating that our efficient tetrahedral embedded boundary method can substantially increase the flexibility and accuracy of adaptive Eulerian fluid simulation.

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Authors:
Christopher Batty - University of British Columbia
Stefan Xenos - Exocortex Technologies, Inc.
Ben Houston - Exocortex Technologies, Inc.

Citation: C. Batty, S. Xenos and B. Houston. Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids. In Proceedings of Eurographics 2010.

Bibtex:

@inproceedings{tetrahedra2010,
 author = {Christopher Batty and Stefan Xenos and Ben Houston},
 title = {Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids},
 booktitle = {Proceedings of Eurographics},
 year = {2010},
}

Funding:
Exocortex Technologies, Inc.
This research is being commercialized by Exocortex as part of their MaelstromFX software. For inquiries and further information, please contact info@exocortex.com .

Related Projects:
A Fast Variational Framework for Accurate Solid-Fluid Coupling
The embedded boundary method used in the tetrahedra method above grew out of our prior work on incorporating embedded solid boundaries (and rigid bodies) into Cartesian grid simulations.

Matching Fluid Simulation Elements to Surface Geometry and Topology
This follow-up paper extends the above method to Voronoi meshes with a smarter pressure sampling scheme, coupled to a mesh-based surface tracker to capture thin sheets, and an improved surface tension model.

Hindsights:
At locations where the adaptive BCC mesh changes resolution, circumcentres (pressure samples) of adjacent tetrahedra will often be coincident. This can cause issues when computing pressure gradients (ie. divide by zero). A simple hack to circumvent this is to divide by an epsilon instead, as done by Sin et al. in their Voronoi-based approach. However, a better solution in our case is to collapse the two offending tetrahedra into a single polyhedral cell.