Mesh Quality

The quality of the mesh is crucial to ensure simulation results align with expectations, ensuring good visualization, and in certain conditions, a low-quality mesh can cause simulations to diverge or even error in the first iteration.

Visually, we can assess the mesh quality based on its proportionality. However, this assessment is limited to the ability to judge proportionality and becomes significantly complex in evaluating detailed 3D domains. Hence, this chapter will discuss several indicators of mesh quality. Here are some commonly used mesh parameters:

3.3.1. Skewness

Skewness is used to indicate how tilted a mesh is. The more acute the angles of an element, the better the data transfer from one element to another. Hence, when the shape of an element becomes skewed, it requires significant corrections during the computation process, which reduces calculation quality and slows down computation.

Figure 3.8. Mesh Quality

Mathematically, skewness is defined as follows:

(3.2)

Figure 3.9. Mesh skewness calculation

With the theta is the equiangular face/cell (60 deg for tetra, and 90 deg for quads or hexa).

Please note that some software might have different definition of skewness, but you can understand the physical meaning from the equation (3.2) above.

Here are some general rules of thumb commonly used to assess mesh quality based on skewness (again, different software might have difference values):

Table 3.1. Mesh skewness Rule of thumb

Skewness	Cell quality
1	Degenerate
0.9 < 1	Bad
0.75 – 0.9	Poor
0.5 – 0.75	Fair
0.25 – 0.5	Good
>0 – 0.25	Excellent
0	Equilateral

3.3.2. Aspect ratio

Aspect ratio is the comparison between the longest length of an edge and the shortest length of an edge, where a larger aspect ratio results in a slender mesh.

The ideal value for the aspect ratio is 1, signifying that the length, width, and height are exactly the same. Higher aspect ratios reduce mesh quality.

Figure 3.10. AR = 1 (left), and high AR (right)

3.3.3. Orthogonality

Orthogonality defines the orientation between one element and another, where a more parallel orientation of vectors from the center to the center of an element indicates good mesh quality as it facilitates the flow of data transfer from one element to another.

Figure 3.11. Vectors definition for orthogonality calculation

From Figure 3.11 above, orthogonality can be calculated as follows:

(3.3)

Which indicates the dot product (cosine multiplication) between the surface normal vector A and the normal orientation vector from the center of element f.

Then, the orientation between the surface normal vector and the center-to-center orientation vector of the element, c is also calculated, given as follows:

(3.4)

The smaller the angle difference between A and , or c and f, the closer their dot product will be to 1. Thus, equations (3.3) and (3.4) will approach a value of zero, indicating good mesh quality.

Meanwhile, the worst skewness displayed on the mesh quality monitor is the highest value calculated from equations (3.3) and (3.4).