# 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

- 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.
- Chordal Deviation
- Largest distance between the centers of element edges and the associated surface.
- 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.
- 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".

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.
- 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.
- 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.

### 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.

- 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.

## 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.

- Aspect Ratio
- ratio1 = V1/H1
- 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.
- 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

- 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.
- 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.
- 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.

## 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.

- 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.
- 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.

### 3D Element Only Checks

- Volume Skew
- Only applicable to tetrahedral elements; all others are assigned values of zero.

## 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.

- 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).
- Aspect Ratio (quad)
- 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.
- 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.
- 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.

- 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.

### 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
- 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.
- 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:$$WC=2H/(D1+D2)$$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.