Finite Element Mesh Quality Metrics

The quality metrics for a mesh in finite element analysis include angle skewness, aspect ratio, middle point deviation, shape, and warpage. These cell quality metrics can be used to identify poor quality cells that could produce inaccurate results or slow convergence. Once a finite element model is selected, the associated field functions can be computed on hexahedral, tetrahedral, wedge, and pyramid element types with or without mid-side nodes, and on 2D elements such as quadrilaterals and triangles.

These qualities are not defined on arbitrary cells; an arbitrary 3D mesh cell is not always suitable for FE computation—the quality metrics cannot be computed, so, in such cases, the default value is set to –∞.

All the quantities express the deviation of the finite element cell to an ideal element. The ideal element is defined as an element of given type with unit edge length.

Each metric has a specific formula for computing cell quality $q$ .

Angle Skewness

Angle skewness is a measure of the equiangular skew and is defined as:

$$q=1-\max (\frac{\theta -{\theta }_c}{180-{\theta }_c},\frac{{\theta }_c-\theta }{{\theta }_c})$$

(4840)

where $\theta$ is the angle over a face vertex in the element and ${\theta }_c$ is the angle of the corresponding vertex in a face of an ideal element (60° for triangular face and 90° for quad face). The minimum of $q$ is taken over all corner vertices of the element.

$q$ is in the interval [0, 1], where 0 is a bad cell and 1 is an ideal cell. This metric is not able to recognize flat hex elements.

Aspect Ratio

The quality $q$ is computed as the maximum edge length $h_{\text{max}}$ over the minimum edge length $h_{\text{min}}$ :

$$q=\frac{h_{\text{max}}}{h_{\text{min}}}$$

(4841)

$q$ is in the inverval [1,∞], where $q=1$ is best quality and $q\gg 1$ is bad quality. For good elements, $q<10$ .

Middle-Point Deviation

This metric is suitable for elements with mid-side vertices. Every element edge (corner vertices ${\mathbf{x}}_i$ and ${\mathbf{x}}_j$ , mid-side vertex ${\mathbf{x}}_m$ ) is inspected by looking at the placement of the mid-side vertex and the deviation from a straight line between the corner vertices.

$$q=\text{max}\left[\frac{d}{l_e}\right]=\text{max}\left[\frac{\Vert {\mathbf{x}}_m-\frac{1}{2}({\mathbf{x}}_i+{\mathbf{x}}_j)\Vert }{l_e}\right]$$

(4842)

where

• $d$ is the distance from the midpoint between ${\mathbf{x}}_i$ and ${\mathbf{x}}_j$ —that is, $\frac{1}{2}({\mathbf{x}}_i+{\mathbf{x}}_j)$ —to ${\mathbf{x}}_m$ .
• $l_e$ is the average edge length.

$q$ is in the interval [0,∞], where $q=0$ is an ideal element. For good elements, $q<0.25$ .

Shape

This metric is based on the Jacobi determinant (Jacobian) of the transformation matrix $\mathbf{T}$ that transforms the coordinates from an ideal element to the physical element. The Jacobian is computed on all element vertices, including mid-side vertices. The shape function quality $q$ is taken as the minimum of the values at all element vertices, computed as:

$$q=D\frac{\text{det}{\left(\mathbf{T}\right)}^{\frac{2}{D}}}{{\left|\mathbf{T}\right|}^2}$$

(4843)

where:

• $q$ is in the interval [0,1] $\cup$ {–1}, where $q=1$ is an ideal element. For elements with a negative Jacobian, $q=-1$ .
• $D$ = 3 for three-dimensional cases, or 2 for two-dimensional cases.
• $\left|\mathbf{T}\right|$ is the Frobenius matrix norm.

Warpage

This metric is the maximum distance $h$ of the face vertices from the face plane, divided by the average edge length $l_e$ .

$$q=\frac{\text{max}h}{l_e}=\text{max}\frac{\mathbf{n}\cdot ({\mathbf{x}}_j-{\mathbf{x}}_c)}{l_e}$$

(4844)

where:

• $\mathbf{n}$ is the normal vector.
• ${\mathbf{x}}_j$ is the face vertex.
• ${\mathbf{x}}_c$ is any point from the face plane.

For volumetric elements, it is the maximum normalized distance of all faces from the face plane. For a triangular face with corner vertices ${\mathbf{x}}_1$ , ${\mathbf{x}}_2$ , and ${\mathbf{x}}_3$ , with face center ${\mathbf{x}}_c$ , the normal vector is computed as:

$$\mathbf{n}=\frac{({\mathbf{x}}_2-{\mathbf{x}}_1)\times ({\mathbf{x}}_3-{\mathbf{x}}_1)}{\Vert ({\mathbf{x}}_2-{\mathbf{x}}_1)\times ({\mathbf{x}}_3-{\mathbf{x}}_1)\Vert }\text{, }{\mathbf{x}}_c=\frac{{\mathbf{x}}_1+{\mathbf{x}}_2+{\mathbf{x}}_3}{3}$$

For a quadrilateral face with corner vertices ${\mathbf{x}}_1$ , ${\mathbf{x}}_2$ , ${\mathbf{x}}_3$ , and ${\mathbf{x}}_4$ , the normal vector is computed as the normal to the diagonals $\mathbf{n}={\mathbf{t}}_1\times {\mathbf{t}}_2$ , where:

$$\begin{array}{l}{{\mathbf{d}}_1={\mathbf{x}}_3-{\mathbf{x}}_1,\mathbf{d}}_2={\mathbf{x}}_2-{\mathbf{x}}_4\\ \begin{array}{c}{\mathbf{t}}_1=\frac{{\mathbf{d}}_1+{\mathbf{d}}_2}{\Vert {\mathbf{d}}_1+{\mathbf{d}}_2\Vert },{\mathbf{t}}_2=\frac{{\mathbf{d}}_1-{\mathbf{d}}_2}{\Vert {\mathbf{d}}_1-{\mathbf{d}}_2\Vert }\end{array}\end{array}$$

and

$${\mathbf{x}}_c=\frac{{\mathbf{x}}_1+{\mathbf{x}}_2+{\mathbf{x}}_3+{\mathbf{x}}_4}{4}$$

$q$ is in the interval [0,∞], where $q=0$ is an ideal element. For good elements, $q\le 0.5$ .