Mesh Quality

The mesh quality can have a considerable impact on the computational analysis in terms of the quality of the solution and the time needed to obtain it. This aspect becomes especially important if poorly conditioned problems, non-linear, and/or transient analysis are considered. From this point of view, the evaluation of the quality of the mesh is very useful because it provides some indication of how suitable a particular discretization is for the analysis type under consideration.

There are several ways how to compute the quality of individual elements and how to quantify the overall quality of a mesh. In the presented work, three elementary criteria have been adopted. The first one evaluates the element quality with respect to the equilateral simplex (as the best possible element). For a triangular element, the quality is expressed as

(2.102) |

where represents the area of the triangle, , , and are the lengths of its sides and is a normalizing coefficient which justifies the quality of an equilateral triangle to . A similar formula is also used to evaluate the quality of a tetrahedral element

(2.103) |

where denotes the volume of the tetrahedron, , , , and are the areas of its faces and is again a normalizing coefficient ensuring that the quality of an equilateral tetrahedron is equal to . Alternative relations for the evaluation of the element quality are usually based on the ratio of areas (volumes) of circles (spheres) inscribed and circumscribed to a triangle (tetrahedron). The overall mesh quality is evaluated separately for triangular and tetrahedral elements in terms of the arithmetic mean by

(2.104) |

or as the harmonic mean by

(2.105) |

where stands for the number of elements (of a particular type). The advantage of the harmonic mean as the overall mesh quality indicator resides in the fact that the harmonic mean is highly sensitive to the occurrence of elements with an extremely poor quality. It is also convenient to provide the worst quality or the quality distribution. The second criterion examines the dihedral angles looking for the minimum and maximum extremes and calculating the average. Again, the distribution of the dihedral angles offers a valuable indication about the mesh quality. The optimal dihedral angles corresponding to the equilateral simplices are and , respectively. Note that the dihedral angles may be also used to evaluate the quality of individual elements using the relation

(2.106) |

where is the minimal and the maximal dihedral angle in a particular element. The last criterion is based on the mesh connectivity and evaluates the valency (number of connected edges) for nodes classified either to a region or to a surface, patch, or shell. While the optimal valency for a uniform triangular mesh is exactly , only the approximate value of is available for a uniform tetrahedral mesh. The more the valency of a particular node differs from the optimal value the more irregular the mesh is in the neighbourhood of that node.

*Daniel Rypl
2005-12-07*