Computational Geometry, Parallel Computation

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Short-term goals

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I. Distributed-memory Parallel Computation of the Voronoi Diagrams

 

I am now working on distributed parallel computation of the Voronoi diagrams. Using OpenMPI [1], I am investigating efficient communication strategies among different processes, that are robust for different particle distribution cases.

 

For example, if the particles are randomly and homogeneously distributed in the domain, and the domain is divided into subdomains defined on different processes, we expect minimum communications to be needed only within narrow banded regions in between nearby processes. This is the most ideal case, as inter-process communications are expensive.

 

However, there may be cases when particles are inhomogeneously distributed. For instance, particles cluster in some small regions but are sparsely distributed in other regions. In this case, when computing the Voronoi cell of a particle, it may require checking over a large region for nearby particles, and therefore, can result in significantly more inter-process communications. Efficient inter-process communication strategies are highly important for these cases.

 

II. 3D Multi-threaded Geometry Meshing Using the Delaunay Tetrahedralization

I also plan to develop a 3D version of the Delaunay meshing software. I plan to implement DistMesh for its time saving advantage. CVD meshing does not control mesh quality in 3D as well as in 2D, in particular, slivers can exist in the 3D mesh. Therefore, I plan to instead implement a variational tetrahedral meshing algorithm, ODT meshing, proposed by Alliez et al. [2], that minimizes an energy EODT. Alliez et al. showed that the ODT meshing leads to the elimination of slivers in the interior of the geometry domain. A few slivers may exist for geometry boundary vertices, but they can be fixed easily using vertex jittering method. I plan to develop a hybrid method of DistMesh and ODT meshing, to combine the advantages of both and generate an overall high-quality 3D mesh in a short time.

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Long-term goals

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I. A comprehensive software package integrating geometry meshing and FEM simulations, using parallel computing

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My long term goal is to publish a comprehensive software package that has multiple functionalities and can handle the full steps of FEM engineering simulations through an optimized pipeline: mesh generation, mesh editing and deformation, and FEM simulation. Furthermore, I would like to expand the mesh generation to allow for constrained Delaunay triangulation/tetrahedralization, forcing user specified points and edges to be part of the triangle mesh. This would make the software to be highly robust for general use in different research and applications.
 

II. Using ML in mesh generation

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Another goal is to explore incorporating machine learning (ML) techniques in the development of geometry meshing tools. Currently, I am implementing classical, iterative algorithms to obtain the desired particle positions for the meshes. However, I want to investigate using ML techniques to obtain the particle positions. For example, given a geometry shape, and a geometric adaptivity parameter, we can use ML to initialize point positions that give an overall high quality and adaptive mesh in a very short amount of time. Afterwards, we can use a few iterations of classical meshing algorithms to refine the mesh