The thesis explores the generation of anisotropic and boundary conforming
Voronoi regions and Delaunay triangulations, high-order mesh
quality and the development of mesh enhancement techniques which
incorporate quality measures to preserve mesh validity longer.
In the first part an analogy with crystal growth is proposed to handle
mesh anisotropy and boundary conformity in Voronoi diagrams
and Delaunay mesh generation. A Voronoi partition of a domain corresponds
to the steady-state configuration of many crystals growing
from their seed points. Mesh anisotropy is incorporated and the shape
of the boundary of an isolated crystal is guided by re-interpreting a
user-defined Riemann metric in terms of the velocity of the crystal
boundary. A straightforward implementation of conformity to boundaries
is achieved by treating the boundary of the computational domain
as the boundary of a stationary crystal.
The second part attempts to answer the question: what is a good highorder
element? A review of a priori mesh quality measures suitable
for high-order elements is presented. A systematic analysis of the
quality measures for interior and boundary elements is then carried
out utilising a number of test cases that consist of a set of carefully
selected elements with various degrees of distortion. Their ability
to identify severe geometrical distortion is discussed. The effect of
boundary curvature on the performance of quality measures is also
investigated.
The last part proposes improvements to a conventional mesh deformation
method based on the equations of elasticity to maintain highorder
mesh validity and enhance mesh quality. This is accomplished
by incorporating additional terms, that can be interpreted as body
forces and thermal stresses in the elastic analogy. Different test cases
are designed to prolong mesh validity, and their performance is reported.
A proposal of how to formulate these terms to incorporate
anisotropy is also presented.