# How Element Quality is Calculated

The quality of elements in a mesh can be gauged in many ways, and the methods used often depend not only on the element type, but also on the individual solver used.

When possible, the most common or standard methods are used, but there is no truly standardized set of element quality checks. When a solver does not support a specific check within Engineering Solutions, Engineering Solutions uses its own method to perform the check.

## HyperMesh

When possible, Engineering Solutions checks strive to maintain compatibility with popular solvers.

### 2D and 3D Element Checks

The following checks apply to both types of elements, but when applied to 3D elements they are generally applied to each face of the element. The value of the worst face is reported as the 3D element’s overall quality value.
Aspect Ratio
Ratio of the longest edge of an element to either its shortest edge or the shortest distance from a corner node to the opposing edge ("minimal normalized height"). Engineering Solutions uses the same method used for the Length (min) check.
For 3D elements, each face of the element is treated as a 2D element and its aspect ratio determined. The largest aspect ratio among these faces is returned as the 3D element’s aspect ratio.
Aspect ratios should rarely exceed 5:1
Chordal Deviation
Largest distance between the centers of element edges and the associated surface.
Second order elements return the same chordal deviation as first order, when the corner nodes are used due to the expensive nature of the calculations.
Interior Angles
Maximum and minimum interior angles are evaluated independently for triangles and quadrilaterals.
Jacobian
Deviation of an element from its ideal or "perfect" shape, such as a triangle’s deviation from equilateral.
The Jacobian value ranges from 0.0 to 1.0, where 1.0 represents a perfectly shaped element. The determinant of the Jacobian relates the local stretching of the parametric space which is required to fit it onto the global coordinate space.
Engineering Solutions evaluates the determinant of the Jacobian matrix at each of the element’s integration points (also called Gauss points) or at the element’s corner nodes, and reports the ratio between the smallest and the largest. In the case of Jacobian evaluation at the Gauss points, values of 0.7 and above are generally acceptable. You can select which method of evaluation to use (Gauss point or corner node) from the Check Element settings.
Length (min)
Minimum element lengths are calculated using one of two methods.
• The shortest edge of the element. This method is used for non-tetrahedral 3D elements.
• The shortest distance from a corner node to its opposing edge (or face, in the case of tetra elements); referred to as "minimal normalized height".
You can choose which method to use in the Check Element settings.
Note: This setting affects the calculation of the Aspect Ratio check.
Minimum Length / Size
Minimum element size is calculated using:
Shortest edge
Length of the shortest edge of each element is used.
Minimal normalized height
Is a more accurate, but more complex height.
For triangular elements, for each corner node (i), Engineering Solutions calculates the closest (perpendicular) distance to the ray including the opposite leg of the triangle, h(i). MNH = min(hi) * 2/sqrt(3.0). The scaling factor 2/sqrt(3.0) ensures that for equilateral triangles, the MNH is the length of the minimum side.
For quadrilateral elements, for each corner node, Engineering Solutions calculates the closest (perpendicular) distances to the rays containing the legs of the quadrilateral that do not include this node. The figure above depicts these lengths as red lines. Minimal normalized height is taken to be the minimum of all eight lines and the four edge lengths, thus, the minimum of 12 possible lengths.
Minimal height
The same as minimal normalized height, but without a scaling factor.
Skew
Skew of triangular elements is calculated by finding the minimum angle between the vector from each node to the opposing mid-side, and the vector between the two adjacent mid-sides at each node of the element. The minimum angle found is subtracted from ninety degrees and reported as the element’s skew.
Note: Skew for quads is part of the HyperMesh-Alt quality check.
Taper
Taper ratio for the quadrilateral element is defined by first finding the area of the triangle formed at each corner grid point. These areas are then compared to one half of the area of the quadrilateral.
Engineering Solutions then finds the smallest ratio of each of these triangular areas to ½ the quad element’s total area (in the diagram above, "a" is smallest). The resulting value is subtracted from 1, and the result reported as the element taper. This means that as the taper approaches 0, the shape approaches a rectangle.
$taper=1-{\left(\frac{{A}_{tri}}{0.5×{A}_{quad}}\right)}_{\mathrm{min}}$
Triangles are assigned a value of 0, in order to prevent Engineering Solutions from mistaking them for highly-tapered quadrilaterals and reporting them as "failed".
Warpage
Amount by which an element, or in the case of solid elements, an element face, deviates from being planar. Since three points define a plane, this check only applies to quads. The quad is divided into two trias along its diagonal, and the angle between the trias’ normals is measured.
Warpage of up to five degrees is generally acceptable.

### 3D Element Only Checks

Minimum Length / Size
Two methods are used to calculate the minimum element size.
Shortest edge
Length of the shortest edge of each element is used.
Minimal normalized height
More accurate, but more complex.
Engineering Solutions calculates the closest (perpendicular) distances to the planes formed by the opposite faces for each corner node.
The resulting minimum length/size is the minimum of all such measured distances.
Tetra Collapse
The height of the tetra element is measured from each of the four nodes to its opposite face, and then divided by the square root of the face’s area. The minimum of the four resulting values (one per node) is then normalized by dividing it by 1.24. As the tetra collapses, the value approaches 0.0, while a perfect tetra has a value of 1.0. Non-tetrahedral elements are given values of 1 so that Engineering Solutions will not mistake them for bad tetra elements.
Vol. Aspect Ratio
Tetrahedral elements are evaluated by finding the longest edge length and dividing it by the shortest height (measured from a node to its opposing face). Other 3D elements, such as hex elements, are evaluated based on the ratio of their longest edge to their shortest edge.
Volume Skew
Only applicable to tetrahedral elements; all others are assigned values of zero. Volume Skew is defined as 1-shape factor, so a skew of 0 is perfect and a skew of 1 is the worst possible value.
The shape factor for a tetrahedral element is determined by dividing the element’s volume by the volume of an ideal (equilateral) tetrahedron of the same circumradius. In the case of tetrahedral elements, the circumradius is the radius of a sphere passing through the four vertices of the tetrahedron.

## HyperMesh-Alt

Engineering Solutions includes some alternate methods of calculating certain element types, which only apply to quads or rectangular faces of solids, and only include alternate checks for Aspect Ratio, Skew, Taper and Warpage.

Note: Because these methods apply only to certain quality checks, in order to use them you must choose the set individually option in the Check Element settings.
Aspect Ratio
ratio1 = V1/H1
ratio2 = V2/H2
Skew value is larger of ratio1 or ratio2.
Skew
First, Engineering Solutions constructs lines connecting the midpoints of each edge of the quad, dotted in the picture below. Next, Engineering Solutions constructs a third line, green in the picture below, perpendicular to one of the initial lines, then finds the angle between this third line and the remaining initial line – with which is it most likely not perpendicular, unless the quad is a perfect rectangle.
α is the skew (angle) value.
Taper
First, the quad’s nodes are projected to plane defined by the orthonormal vectors U-V found as follows:
• Z = X × Y
• V = Z × X
• U = X
In Engineering Solutions, Taper angle is defined as: $\theta =\mathrm{max}\left(\frac{{\theta }_{1}}{2};\frac{{\theta }_{2}}{2}\right)$.
The optimal value is 0°, and a generally acceptable limit is. <= 30°. The The ultimate limit, which the Taper angle cannot exceed is 45°.
Warpage
Only applies to quads or rectangular faces of solids. Warpage = 100 * h / max { Li }, where h is the minimum distance between the diagonals.

## OptiStruct

For the most part, OptiStruct uses the same checks as HyperMesh. However, OptiStruct uses its own method of calculating Aspect Ratio, and it does not support 3D element checks.

Aspect Ratio
Ratio between the minimum and maximum side lengths.
3D elements are evaluated by treating each face of the element as a 2D element, finding the aspect ratio of each face, and then returning the most extreme aspect ratio found.
Chordal Deviation
Chordal deviation of an element is calculated as the largest distance between the centers of element edges and the associated surface. 2nd order elements return the same chordal deviation as 1st order, when the corner nodes are used due to the expensive nature of the calculations.
Interior Angles
Maximum and minimum values are evaluated independently for triangles and quadrilaterals.
Jacobian
Deviation of an element from its ideal or "perfect" shape, such as a triangle’s deviation from equilateral. The Jacobian value ranges from 0.0 to 1.0, where 1.0 represents a perfectly shaped element. The determinant of the Jacobian relates the local stretching of the parametric space which is required to fit it onto the global coordinate space.
Engineering Solutions evaluates the determinant of the Jacobian matrix at each of the element’s integration points, also called Gauss points, or at the element’s corner nodes, and reports the ratio between the smallest and the largest. In the case of Jacobian evaluation at the Gauss points, values of 0.7 and above are generally acceptable. You can select which method of evaluation to use, Gauss point or corner node, from the Check Element settings.
Length (min)
Minimum element lengths are calculated using one of two methods:
• The shortest edge of the element. This method is used for non-tetrahedral 3D elements.
• The shortest distance from a corner node to its opposing edge (or face, in the case of tetra elements); referred to as "minimal normalized height".
Skew
Skew of triangular elements is calculated by finding the minimum angle between the vector from each node to the opposing mid-side, and the vector between the two adjacent mid-sides at each node of the element. The minimum angle found is subtracted from ninety degrees and reported as its skew.
Warpage
Amount by which an element, or in the case of solid elements, an element face, deviates from being planar. Since three points define a plane, this check only applies to quads. The quad is divided into two trias along its diagonal, and the angle between the trias’ normals is measured.
Warpage of up to five degrees is generally acceptable.

## Abaqus

Abaqus-specific checks used to calculate element quality for 2D and 3D elements.

### 2D and 3D Element Checks

These checks apply to both types of elements, but when applied to 3D elements they are generally applied to each face of the element. The value of the worst face is reported as the 3D element’s overall quality value.

Additional element checks not listed here are not part of the solver’s normal set of checks, and therefore use HyperMesh check methods.
Aspect Ratio
Ratio of the longest edge of an element to its shortest edge.
When applied to 3D elements, the same method is used (longest edge divided by shortest edge) rather than evaluating each face individually and taking the worst face result.
Interior Angles
Maximum and minimum values are evaluated independently for triangles and quadrilaterals.
Jacobian
Deviation of an element from its ideal or "perfect" shape, such as a triangle’s deviation from equilateral. The Jacobian value ranges from 0.0 to 1.0, where 1.0 represents a perfectly shaped element. The determinant of the Jacobian relates the local stretching of the parametric space which is required to fit it onto the global coordinate space.
Engineering Solutions evaluates the determinant of the Jacobian matrix at each of the element’s integration points, also called Gauss points, or at the element’s corner nodes, and reports the ratio between the smallest and the largest. In the case of Jacobian evaluation at the Gauss points, values of 0.7 and above are generally acceptable. You can select which method of evaluation to use (Gauss point or corner node) from the Check Element settings.
Length (min)
Minimum element lengths are calculated using one of two methods:
• The shortest edge of the element. This method is used for non-tetrahedral 3D elements.
• The shortest distance from a corner node to its opposing edge (or face, in the case of tetra elements); referred to as "minimal normalized height".

Skew (tria only)
Defined by shape factor. Abaqus determines triangular element shape factor by dividing the element’s area by the area of an ideally shaped element. The ideally shaped element is defined as an equilateral triangle with the same circumradius—the radius of a circle that passes through the three vertices of the triangle—as the element.
This shape factor converts to skew by subtracting it from 1. Thus, a perfect equilateral tria element has a skew of 0 and the worst tria has a value of 1.0.
Quadrilaterals are simply assigned a value of 0.

### 3D Element Only Checks

Volume Skew
Only applicable to tetrahedral elements; all others are assigned values of zero.
Volume Skew is defined as 1 minus the shape factor, so a skew of 0 is perfect and a skew of 1 is the worst possible value.
The shape factor for a tetrahedral element is determined by dividing the element’s volume by the volume of an ideal (equilateral) tetrahedron of the same circumradius. In the case of tetrahedral elements, the circumradius is the radius of a sphere passing through the four vertices of the tetrahedron.

## ANSYS

ANSYS-specific checks used to calculate element quality for 2D and 3D elements.

### 2D and 3D Element Checks

These checks apply to both types of elements, but when applied to 3D elements they are generally applied to each face of the element. The value of the worst face is reported as the 3D element’s overall quality value.

Additional element checks not listed here are not part of the solver’s normal set of checks, and therefore use HyperMesh check methods.
Aspect Ratio (tria)
For tria elements, a line is drawn from one node to the midpoint of the opposite edge. Next, another line is drawn between the midpoints of the remaining two sides. These lines are typically not perpendicular to each other or to any of the element edges, but provide four points (three midpoints plus the vertex).
Then, a rectangle is created for each of these two lines, such that one line perpendicularly meets the midpoints of two opposing edges of the rectangle, and the remaining edges of the rectangle pass through the end points of the remaining line. This results in two rectangles, one perpendicular to each of the two lines.
Third, this process is repeated for each of the remaining two nodes of the tria element, resulting in the construction of four additional rectangles (six in total).
Finally, each rectangle is examined to find the ratio of its longest side to its shortest side. Of these six values—one for each rectangle—the most extreme value is then divided by the square root of three to produce the tria aspect ratio.
The best aspect ratio (an equilateral tria) is 1. Higher numbers indicate greater deviation from equilateral.
If the element is not flat, it’s projected to a plane which is based on the average of the element’s corner normals. All subsequent calculations are based on this projected element rather than the original (curved) element.
Next, two lines are created which bisect opposite edges of the element. These lines are typically not perpendicular to each other or to any of the element edges, but they provide four midpoints.
Third, a rectangle is created for each line, such that the line perpendicularly bisects two opposing edges of the created rectangle, and the remaining two edges of the rectangle pass through the remaining line’s endpoints. This creates two rectangles—one perpendicular to each line.
Finally, the rectangles are compared to find the one with the greatest length ratio of longest side to shortest side. This value is reported as the quad’s aspect ratio. A value of one indicates a perfectly equilateral element, while higher numbers indicate increasingly greater deviation from equilateral.
Interior Angles
Maximum and minimum values are evaluated independently for triangles and quadrilaterals.
Jacobian
Deviation of an element from its ideal or "perfect" shape, such as a triangle’s deviation from equilateral. The Jacobian value ranges from 0.0 to 1.0, where 1.0 represents a perfectly shaped element. The determinant of the Jacobian relates the local stretching of the parametric space which is required to fit it onto the global coordinate space.
Engineering Solutions evaluates the determinant of the Jacobian matrix at each of the element’s integration points, also called Gauss points, or at the element’s corner nodes, and reports the ratio between the smallest and the largest. In the case of Jacobian evaluation at the Gauss points, values of 0.7 and above are generally acceptable. You can select which method of evaluation to use (Gauss point or corner node) from the Check Element settings.
Length (min)
Minimum element lengths are calculated using one of two methods:
• The shortest edge of the element. This method is used for non-tetrahedral 3D elements.
• The shortest distance from a corner node to its opposing edge (or face, in the case of tetra elements); referred to as "minimal normalized height".
Angle Deviation (Skew)
Only applicable to quadrilateral elements, and relies upon the angles between adjacent legs at each corner node (that is, the interior angles at each corner). Each angle is compared to a base of 90 degrees, and the one with the largest deviation from 90 is reported as the angle deviation. Triangular elements are given a value of zero.
Warping Factor
Only applicable to quadrilateral elements as well as the quadrilateral faces of 3D bricks, wedges, and pyramids.
Calculated by creating a normal from the vector product of the element’s two diagonals. Next, the element’s area is projected to a plane through the average normal. Finally, the difference in height is measured between each node of the original element and its corresponding node on the projection. For flat elements, this is always zero, but for warped elements one or more nodes will deviate from the plane. The greater the difference, the more warped the element is.
The warping factor is calculated as the edge height difference divided by the square root of the projected area.

### 3D Element Only Checks

ANSYS does not use any exclusively 3D checks within Engineering Solutions, but Engineering Solutions does use its own when ANSYS is set as the solver. For details on 3D checks, refer to HyperMesh.

## Nastran

Nastran-specific checks used to calculate element quality for 2D and 3D elements.

Additional element checks not listed here are not part of the solver’s normal set of checks, and therefore use HyperMesh check methods.

### 2D and 3D Element Checks

Aspect Ratio
Ratio of the longest edge of an element to its shortest edge.
Interior Angles
Maximum and minimum values are evaluated independently for triangles and quadrilaterals.
Jacobian
Deviation of an element from its ideal or "perfect" shape, such as a triangle’s deviation from equilateral. The Jacobian value ranges from 0.0 to 1.0, where 1.0 represents a perfectly shaped element. The determinant of the Jacobian relates the local stretching of the parametric space which is required to fit it onto the global coordinate space.
Engineering Solutions evaluates the determinant of the Jacobian matrix at each of the element’s integration points, also called Gauss points, or at the element’s corner nodes, and reports the ratio between the smallest and the largest. In the case of Jacobian evaluation at the Gauss points, values of 0.7 and above are generally acceptable. You can select which method of evaluation to use (Gauss point or corner node) from the Check Element settings.
Skew
Engineering Solutions creates lines between the midpoints of opposite sides of the element, then measures the angles between these lines. The angle with the greatest deviation from the ideal value is used to determine skew.
Taper
Engineering Solutions finds the taper of quadrilateral elements by treating each node as the corner of a triangle, using one of the quad’s diagonals as the triangle’s third leg. The areas of each of these four "virtual" triangles are compared to one half of the total area of the quadrilateral element to produce a ratio; the largest of these ratios is then compared to the tolerance value. A value of 1.0 is a perfect quadrilateral, and higher numbers denote greater taper.
However, for the sake of consistency within Engineering Solutions, an equivalent taper is reported instead. This means that the smallest area ratio found (instead of the largest ratio) is subtracted from 1, so that 0 represents a perfect quadrilateral element instead of 1.0, and greater deviation from 0 indicates greater taper. Triangle elements are simply assigned a value of 0 to prevent Engineering Solutions from incorrectly identifying them as failed (highly-tapered) quads.
Warpage
First, Engineering Solutions constructs a plane based on the mean of the quad’s four points. This means that the corner points of a warped quad are alternately H units above and below the constructed plane. This value is then used along with the length of the element’s diagonals in the following equation:
$WC=2H/\left(D1+D2\right)$
Where WC is the Warping Coefficient, H is the "height" or distance of the nodes from the constructed plane, and D1 and D2 are the lengths of the diagonals. Thus, a perfect quad has a WC of zero.

### 3D Element Only Checks

Vol. Aspect Ratio
Engineering Solutions evaluates Tetrahedral elements by finding the longest edge length and dividing it by the shortest height, measured from a node to its opposing face. Other 3D elements, such as hex elements, are evaluated based on the ratio of their longest edge to their shortest edge.
Warpage
Engineering Solutions evaluates warpage on solid element faces by dividing the quad face into two trias along its diagonal, and measuring the cosine of the angle between the trias’ normals. This value will be 1.0 for a face where all nodes lie on the same plane.