Scaled Jacobian

The Scaled Jacobian mesh quality criteria is a scalar measure of the deviation from the perfect element in the geometrical sense. This measure normalizes the range of reported values between [0,1] for a normal element, the value of 1 is considered a perfect element and 0 a element with a collapsed side. Negative values are also accepted for invalid elements.

The Scaled Jacobian is implemented for all volumetric elements (except for polyhedrons, returning 0).

For tetrahedrons and hexahedrons, the formulas are defined in The Verdict Library Reference Manual [1].

For pyramid, the minimum scaled jacobian of the four tetrahedrons formed in the four vertices of the pyramid base is reported.

For pentahedron, a decomposition into tetrahedron is also done.

For hexahedron prisms, the minimum scaled jacobian between two pentahedrons and one hexahedron is reported.

  • Geometrically the Scaled Jacobian of a tetrahedron can be understood by the follow figure:

    _images/scaled_jacobian_tetra.png

    The reported Scaled Jacobian will be 1 if \(\alpha=45^{\circ}\)

  • For hexadrons this measure return 1 for the perfect non rotated elements:

    _images/scaled_jacobian_hexa.png

    The reported Scaled Jacobian will be 1 if \(\alpha=0^{\circ}\) for all the vertices

To visualize the Scaled Jacobian quality criterion in your mesh:

  1. Display your mesh in the viewer.

  2. Choose Controls > Volume Controls > Scaled Jacobian or click “Scaled Jacobian” button img of the toolbar.

    Your mesh will be displayed in the viewer with its elements colored according to the applied mesh quality control criterion:

    _images/scaled_jacobian_mesh_hexa.png

See Also a sample TUI Script of a Scaled Jacobian filter.