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##

Mesh Optimization

Most of the above described methods for the mesh generation are
accompanied by some sort of mesh optimization technique
[27,33,41,53,63]
in order to improve the shape of the elements, their connectivity and to eliminate
poor quality elements. This mesh enhancement yields the improved
accuracy and stability of the numerical solution and reduces the number of
elements required to capture the underlying physical phenomenon.
The Laplacian smoothing technique, changing the position of nodes
without modifying the topology of the mesh, is the most commonly used
method applied to 2D and 3D meshes. It consists in solving Laplacian equation for the
locations of the interior nodes with given positions of boundary
nodes. The solution can be accomplished in a computationally inexpensive way
by iterative process in which each interior node is repositioned
into the centre (appropriate metric should be used if an anisotropic mesh is
considered) of a polygon (in 2D) or a polyhedron (in 3D) formed by
connected elements. This process is repeated until there is (almost) no
movement in the mesh. The concept of the Laplacian smoothing can be also extended to
3D surfaces. In the case of smoothing the anisotropic mesh in the
parametric space, the relevant metric must be considered. In the real
space, however, regardless whether working with the polygon
projected to the tangent plane at the node being smoothed or with
the raw ``non-planar'' polygon, the smoothed node has to be projected
back to the surface to satisfy the surface constraint. Although the
application of the Laplacian smoothing typically improves the quality
of the mesh in average, the quality of the worst element may be
spoiled even more. It is therefore essential to combine the smoothing with some
topology and/or optimization based technique. The topology based
approaches for triangular meshes rest usually upon the edge swapping [27].
The optimization based techniques, on the other hand, exploit either variational methods
[33] for the detection of the global optimum with
respect to a particular cost function or line search methods
[74] to optimize the elements in the local
neighbourhood of the node subjected to the smoothing.
However, the penalty paid for the high effectiveness of optimization
based methods in the elimination of the most severely distorted elements is their
computational expense.

** Next:** Indirect Triangulation of 3D Surfaces
**Up:** Related Research
** Previous:** Discretization of Discrete 3D Surfaces
*Daniel Rypl *

2005-12-07