Jacobian Ratio
Jacobian ratio is computed and tested for all elements except triangles and tetrahedra that (a) are linear (have no midside nodes) or (b) have perfectly centered midside nodes. A high ratio indicates that the mapping between element space and real space is becoming computationally unreliable.
Jacobian Ratio Calculation
An element's Jacobian ratio is computed by the following steps, using the full set of nodes for the element:
At each sampling location listed in the table below, the determinant of the Jacobian matrix is computed and called RJ.
RJ at a given point represents the magnitude of the mapping function between element natural coordinates and real space. In an ideally-shaped element, RJ is relatively constant over the element, and does not change sign.
Element Shape Element Shape Element Shape RJ Sampling Locations 10-node tetrahedra corner nodes 5-node or 13-node pyramids base corner nodes and near apex node (apex RJ factored so that a pyramid having all edges the same length will produce a Jacobian ratio of 1) 8-node quadrilaterals corner nodes and centroid 20-node bricks all nodes and centroid all other elements corner nodes - The Jacobian ratio of the element is the ratio of the maximum to the minimum sampled value of RJ. If the maximum and minimum have opposite signs, the Jacobian ratio is arbitrarily assigned to be -100 (and the element is clearly unacceptable).
- If the element is a midside-node tetrahedron, an additional RJ is computed for a fictitious straight-sided tetrahedron connected to the four corner nodes. If that RJ differs in sign from any nodal RJ (an extremely rare occurrence), the Jacobian ratio is arbitrarily assigned to be -100.
- If the element is a line element having a midside node, the Jacobian matrix is not square (because the mapping is from one natural coordinate to 2D or 3D space) and has no determinant. For this case, a vector calculation is used to compute a number which behaves like a Jacobian ratio. This calculation has the effect of limiting the arc spanned by a single element to about 106 degrees.
A triangle or tetrahedron has a Jacobian ratio of 1 if each midside node, if any, is positioned at the average of the corresponding corner node locations. This is true no matter how otherwise distorted the element may be. Hence, this calculation is skipped entirely for such elements. Moving a midside node away from the edge midpoint position will increase the Jacobian ratio. Eventually, even very slight further movement will break the element. Refer to the figure below. This is described as "breaking" the element because it suddenly changes from acceptable to unacceptable (that is, "broken").
Any rectangle or rectangular parallelepiped having no midside nodes, or having midside nodes at the midpoints of its edges, has a Jacobian ratio of 1. Moving midside nodes toward or away from each other can increase the Jacobian ratio. Eventually, even very slight further movement will break the element. Refer to the figure below.
A quadrilateral or brick has a Jacobian ratio of 1 if (a) its opposing faces are all parallel to each other, and (b) each midside node, if any, is positioned at the average of the corresponding corner node locations. As a corner node moves near the center, the Jacobian ratio climbs. Eventually, any further movement will break the element. Refer to the figure below.
