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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
[58,73,86,100,115,120,122,141,144]
in order to improve the shape of the elements and to eliminate (especially in
3D) poor quality elements. This mesh enhancement yields an 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 positions of nodes
without modifying the topology of the mesh, is the most commonly used method for
this purpose. It consists in solving the Laplacian equation for the
locations of the interior nodes with given positions of the boundary
nodes. The solution can be accomplished in a computationally inexpensive way
by an iterative process in which each internal node is repositioned
into the centre (appropriate metric should be used when element
stretch and orientation is desirable) of the polygon or polyhedron formed by the
surrounding nodes. This process is repeated until there is (almost) no
movement in the mesh (the mesh is in an equilibrium if a spring analogy is considered).
Although the application of the Laplacian smoothing typically improves the quality
of the mesh in average, the quality of the most distorted elements may be
reduced even further (elements with negative area or volume may arise).
This is typical for 3D tetrahedral meshes where
flat tetrahedrons (called slivers) bounded by well-shaped triangles may occur.
It is therefore essential to combine the smoothing with some
topology and/or optimization based technique. The topology based
approaches rest usually upon an edge and face swapping
[58,100].
The optimization based techniques, on the other hand, exploit either variational methods
[73] for the detection of the global optimum with
respect to a particular cost function, or line search methods
[161] to optimize the element in the local
neighbourhood of nodes subjected to the smoothing.
However, the penalty paid for the extreme effectiveness of optimization
based methods in elimination of the most severely distorted elements is their
computational expense.

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** Previous:** Generation of Quadrilateral and Hexahedral Meshes
*Daniel Rypl *

2005-12-07