Warping factor is computed and tested for some quadrilateral shell elements, and the quadrilateral faces of bricks, wedges, and pyramids. A high factor may indicate a condition the underlying element formulation cannot handle well, or may simply hint at a mesh generation flaw.
Warping Factor Calculation for Quadrilateral Shell Elements
A quadrilateral element's warping factor is computed from its corner node positions and other available data by the following steps:
Note: When computing the warping factor for a quadrilateral shell element, the Meshing application assumes 0 thickness for the shell.
An average element normal is computed as the vector (cross) product of the 2 diagonals (Figure 43: Shell Average Normal Calculation).
The projected area of the element is computed on a plane through the average normal (the dotted outline on Figure 44: Shell Element Projected onto a Plane).
The difference in height of the ends of an element edge is computed, parallel to the average normal. In Figure 44: Shell Element Projected onto a Plane, this distance is 2h. Because of the way the average normal is constructed, h is the same at all four corners. For a flat quadrilateral, the distance is zero.
The "area warping factor" (
) for the element is computed as the edge height
difference divided by the square root of the projected area.For all shells except those in the "membrane stiffness only" group, if the thickness is available, the "thickness warping factor" is computed as the edge height difference divided by the average element thickness. This could be substantially higher than the area warping factor computed in 4 (above).
The warping factor tested against warning and error limits (and reported in warning and error messages) is the larger of the area factor and, if available, the thickness factor.
The best possible quadrilateral warping factor, for a flat quadrilateral, is zero.
Figure 45: Quadrilateral Shell Having Warping Factor shows a "warped" element plotted on top of a flat one. Only the right-hand node of the upper element is moved. The element is a unit square, with a real constant thickness of 0.1.
When the upper element is warped by a factor of 0.01, it cannot be visibly distinguished from the underlying flat one.
When the upper element is warped by a factor of 0.04, it just begins to visibly separate from the flat one.
Warping of 0.1 is visible given the flat reference, but seems trivial; however, it is well beyond the error limit for a membrane shell. Warping of 1.0 is visually unappealing. This is the error limit for most shells.
Warping beyond 1.0 would appear to be obviously unacceptable. However, SHELL181 permits even this much distortion. Furthermore, the warping factor calculation seems to peak at about 7.0. Moving the node further off the original plane, even by much larger distances than shown here, does not further increase the warping factor for this geometry. Users are cautioned that manually increasing the error limit beyond its default of 5.0 for these elements could mean no real limit on element distortion.
Warping Factor Calculation for 3-D Solid Elements
The warping factor for a 3-D solid element face is computed as though the 4 nodes make up a quadrilateral shell element with no real constant thickness available, using the square root of the projected area of the face as described in 4 (above).
The warping factor for the element is the largest of the warping factors computed for the 6 quadrilateral faces of a brick, 3 quadrilateral faces of a wedge, or 1 quadrilateral face of a pyramid. Any brick element having all flat faces has a warping factor of zero (Figure 46: Warping Factor for Bricks).
Twisting the top face of a unit cube by 22.5° and 45° relative to the base produces warping factors of about 0.2 and 0.4, respectively.