Patran

Patran-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

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 (triangle)

The length of a side is divided by the height of the triangle from that side to its opposite node, then multiplied by ½ of the square root of 3. In a perfect equilateral triangle, this formula produces a value of 1. The process is performed for each of the three sides, and the largest value of the three is reported as the aspect ratio.

Aspect Ratio (quads)

If the element is not flat, it is 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 1 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.

HyperMesh CFD 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 (triangle)

Patran evaluates triangular skew by constructing a line from one of the triangle’s nodes to the midpoint of its opposite side, and another line connecting the midpoints of the remaining two sides.

An angle between these created lines is compared to 90 degrees to find its deviation from square. This process is then repeated for each of the remaining two nodes, and the largest of the three computed angle deviations is reported as the element’s skew.

Skew (Quad)

The skew test begins by bisecting the four element edges. This creates an origin at the vector average of the four corners, with the x-axis extending from the origin to the bisector on edge 2. Next, finding the cross-product of the x-axis and the vector that stretches from the origin to the midpoint of edge 3 defines the z-axis. With the x and z axes defined, their cross-product defines the y-axis.

Finally, subtracting the angle α (located between the y axis and the line bisecting edges 1 and 3) from 90 degrees reveals the element skew.

Taper

Patran calculates taper by first averaging the corner nodes to find the element center, and creating lines between this center and the corner nodes to split the element into four triangles.

t a p e r = \frac{4 α_{s m a l l e s t}}{α 1 + α 2 + α 3 + α 4}

The taper calculation is simply the smallest triangle’s area divided by the average of all the triangle areas—or, put another way, the taper is quadruple the area of the smallest triangle, divided by the sum of the areas of all four triangles:

t a p e r = \frac{4 α_{s m a l l e s t}}{α 1 + α 2 + α 3 + α 4}

Note: For the sake of display compatibility, HyperMesh CFD reports an equivalent value for Taper. Taper is determined as above, but is then subtracted from 1 to produce a number between zero and one. Thus, as the element taper decreases, the reported value approaches zero (a perfect square). Triangles are assigned a value of zero to prevent them from showing up as failed quads.

Warpage

The warpage test bisects the element edges, creating a point at the vector average of the element corners. This point serves as the base node for a plane, with the plane’s x-axis extending from the base node to the bisector on edge 2 of the element. The plane normal (z-axis) is in the direction of the cross-product of this x-axis and the vector from the origin to the bisector of edge 3. Each corner of the quad is then the same distance, h, from the plane. Next, Patran measures the length of each half-edge, and calculates the arcsine of the ratio of h to the shortest half-edge length (L):

Θ = \sin^{- 1} \frac{h}{L}

3D Element Only Checks

Vol. Aspect Ratio (Tetrahedron): Patran finds the aspect ratio of Tetra elements by finding the ratio between a vertex height and ½ the area of the opposing face. This process is repeated for each vertex, and the largest ratio found.
Figure 6. Vol. Aspect Ratio for Tetrahedrons

Next, Patran multiplies the largest ratio found by 0.805927, the corresponding ratio of an equilateral tetrahedron. The result is reported as the element’s aspect ratio, with a value of 1 representing a perfect equilateral tetrahedron.
Vol. Aspect Ratio (pyramid): Ratio of the element’s longest edge length to its shortest edge length.
Vol. Aspect Ratio (wedge): This test begins by averaging the triangular faces of the element to create a triangular mid-surface. Next, it finds the aspect ratio of the mid-surface, as for a tria element. Then it compares the average height (h1) of the wedge element to the mid-surface’s maximum edge length (h2).
Figure 7. Vol. Aspect Ratio for Wedges

If the wedge height h1 exceeds the edge length h2, the wedge’s aspect ratio equals the mid-surface aspect ratio multiplied by h2, then divided by the average distance between the triangular faces (h3).; If the wedge height h1 is less than the edge length h2, the wedge aspect ratio equals either the mid-surface aspect ratio, or the maximum edge length h2 divided by the average distance between the triangular faces (h3), whichever is greater.
$A s p e c t R a t i o = \frac{h_{4}}{h_{3}} \frac{\sqrt{3} h_{2}}{2 h_{1}}$
Vol. Aspect Ratio (hexahedron): Each face of the hex element is treated as a warped quadrilateral, and its center point found. The volume aspect ratio is simply the ratio of the largest distance h between the center points of any two opposing faces, to the smallest such distance.
Figure 8. Vol. Aspect Ratio for Hexahedrons

$A s p e c t R a t i o = \frac{\max (h_{1}, h_{2}, h_{3})}{\min (h_{1}, h_{2}, h_{3})}$