Determinant (3x3x3 stencil)

This is for hexahedral elements. This option is the same as the 2x2x2 stencil, but edge midpoints of blocks are added to the Jacobian computation.

The Jacobian determinants for hexahedras will be calculated at r,s,t = -1,0,1 of the natural coordinate system of the element (27 node positions). Next it calculates the maximum absolute determinant of the 27 determinants (3x3x3). If this is at position i with absolute determinant value max0, then for each of the 27 positions (j) (except i) the absolute distance of determinant j to determinant i will be calculated. The final result will then be 1 minus the maximum of the absolute distances divided by max0, so that the range of this quality criterion value will be between -1 and 1. The Jacobian determinant is the determinant of the Jacobian operator which connects the derivatives of the natural coordinates (r,s,t) with the derivatives of the local coordinates (x,y,z). J = ((dx/dr dy/dr dz/dr) (dx/ds dy/ds dz/ds) (dx/dt dy/dt dz/dt)).


Note:  A good book to understand the determinant calculation is: Finite Element Procedures, by K.J. Bathe, Prentice Hall, New Jersey 07632, 1996.