**Mesh quality measures for 2d meshes**

There are
several methods to measure the quality of a mesh element (cell). The quality of
a cell is measured with respect to a reference element (cell) that is claimed
to have the best quality. The reference element is dependent on the application
where this mesh is going to be used. Therefore, there is no absolute definition
of mesh quality. Considering the applications of multi-physics simulation, the
mesh elements (cells) are required to be almost uniform. For a 2d case, the
triangular cell is required to be an equilateral triangle, the quad element is
required to be a square, and so on. The following are some techniques to
measure mesh quality:

**1.**

**Minimum angle criteria**

Quality is
measured based on a predefined optimum value of minimum angle.

**2.**

**Maximum angle criteria**

Quality is
measured based on a predefined optimum value of maximum angle.

**3.**

**Perimeter and area criteria**

The quality
is measured as the ration between the

__square of perimeter__divided by the area of the element. This method is suitable for any polygon element. An additional advantage is that this method can be used for mixed-element meshes.**4.**

**Inscribed and circumscribed circles criteria**

In fact,
this method can be used only with triangular elements. The quality is defined
as

__radius of the circumscribed circle__divided by__radius of the inscribed circle__.**5.**

**Perimeter and edge ratio criteria**

This method
also is used only for triangular elements. For any cell compute the perimeter
and divide it by number of edges of the cell to get

__edge length of the uniform polygon having the same perimeter__(reference edge length). Then get the ratios of edge lengths to the reference edge length. Finally, compute the arithmetic average of these ratios to get the quality of this cell.**6.**

**Edge ratio criteria**

Used for
triangular cells. Quality is the ratio of

__longest edge length__to__shortest edge length__