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Batis August 29, 2010 20:22

Determinant 3x3x3

could You explain me the definition of the Determinant 3x3x3 quality parameter? Beceause, the one from ICEM documentation isn't clear for me.


camoesas November 16, 2010 07:47

Hey Batis,

The Determinant3x3x3 is supposed to be higher than 0.2; if you have negative values your volumes are supposed to be negative.

The Determinant(3x3x3) is defined as the Determinant of the Jacobian Matrix...

But I donīt know what this means in detail. I would also be gratefull if somebody could clarify it.

- How can I get from my cell to a matrix?
- What does it mean for my cell if I have low values (between 0 and 0.2)? Because just from seeing the cells with low Det. I canīt notice anything bad.



PSYMN December 5, 2010 20:33

Hey all, I will put the official Help for determinant below, but basically, 2X2X2 is for a linear hexa element (2 end points on each edge). The 3X3X3 is for quadratic Hexas (3 points on each edge). You will usually get the same answer for both checks because you have linear elements. The 2X2X2 is faster, so just use that.

Greater than 0.2 is great. But most solvers will handle greater than 0.05 or 0.01.

Determinant (2x2x2 stencil)
The Determinant, more properly defined as the relative determinant, is the ratio of the smallest determinant of the Jacobian matrix divided by the largest determinant of the Jacobian matrix. In this option, the determinant at each corner of the hexahedron is found. The default range is 0–1 with a Determinant value of 1 indicating a perfectly regular mesh element and 0 indicating an element degenerate in one or more edges. Negative values indicate inverted elements.

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 hexahedrals 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)).

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

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