Jacobian Ratio
The Jacobian Ratio metric compares the shape of a given element to that of an ideal element. If an element has a poor Jacobian ratio, the element may not map well from element space to real space, thereby making computations based on the element shape less reliable. The ideal shape of a hexahedron has all flat sides and 90 degree angles.
There are two ways to calculate the Jacobian Ratio, either based on Corner Nodes (nodal points) or based on Gauss Points (integration points). The corner node calculation is more restrictive while the Gauss points calculation is less restrictive. In either case, the value is bounded by -1 (worst) and 1 (best). An element with Jacobian ratio <= 0 should be avoided.
An element's Jacobian ratio is computed by the following steps and using all available nodes for the element:
At each sampling location listed in the table below, the determinant of the Jacobian matrix is computed and called RJ. At a given point, RJ represents the magnitude of the mapping function between the element's natural coordinates and real space. In an ideally-shaped element, RJ is relatively constant over the element, and does not change sign.
Element Shape RJ Sampling Locations for corner nodes calculation RJ Sampling Locations for corner nodes calculation 8-node bricks Corner nodes Choose the optimal number of Gauss quadature points for integration 20-node bricks All nodes and centroid Eight Gauss quadrature points The Jacobian ratio of the element is the ratio of the minimum to the maximum sampled value of RJ.
The Jacobian ratio of an ideal hexahedral element is 1, indicating (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 worsens. If the node is moved significantly, the Jacobian ratio will become negative and the element is invalid.
The figure below illustrates the Jacobian ratio metric by color for various element shapes.
