Surface and volumetric parametrisation using harmonic functions in non-convex domains
Date
2013-07-29
Authors
Klein, Richard
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Abstract
Many of the problems in mathematics have very elegant solutions. As complex, real–world geometries
come into play, however, this elegance is often lost. This is particularly the case with meshes of physical,
real–world problems. Domain mapping helps to move problems from some geometrically complex
domain to a regular, easy to use domain. Shape transformation, specifically, allows one to do this in 2D
domains where mesh construction can be difficult. Numerical methods usually work over some mesh on
the target domain. The structure and detail of these meshes affect the overall computation and accuracy
immensely. Unfortunately, building a good mesh is not always a straight forward task. Finite Element
Analysis, for example, typically requires 4–10 times the number of tetrahedral elements to achieve the
same accuracy as the corresponding hexahedral mesh. Constructing this hexahedral mesh, however, is a
difficult task; so in practice many people use tetrahedral meshes instead. By mapping the geometrically
complex domain to a regular domain, one can easily construct elegant meshes that bear useful properties.
Once a domain has been mapped to a regular domain, the mesh can be constructed and calculations can
be performed in the new domain. Later, results from these calculations can be transferred back to the
original domain. Using harmonic functions, source domains can be parametrised to spaces with many
different desired properties. This allows one to perform calculations that would be otherwise expensive
or inaccurate.
This research implements and extends the methods developed in Voruganti et al. [2006 2008] for
domain mapping using harmonic functions. The method was extended to handle cases where there are
voids in the source domain, allowing the user to map domains that are not topologically equivalent
to the equivalent dimension hypersphere. This is accomplished through the use of various boundary
conditions as the void is mapped to the target domains which allow the user to reshape and shrink the
void in the target domain. The voids can now be reduced to arcs, radial lines and even shrunk to single
points. The algorithms were implemented in two and three dimensions and ultimately parallelised to
run on the Centre for High Performance Computing clusters. The parallel code also allows for arbitrary
dimension genus-0 source domains. Finally, applications, such as remeshing and robot path planning
were investigated and illustrated.
Description
A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in
fulfillment of the requirements for the degree of Master of Science.
Johannesburg, 2013