The thrust of our software is the representation of the orientation space using Rodrigues parameters. Rodrigues parameters are particularly efficient for handling crystal symmetries. Under the Rodigues parameterization, the orientation space is reduced to a polyhedral region after accounting for crystal symmetries. Points interior to polyhedron are the unique representatives of their class of symmetric orientations, and points on the boundary have equivalent orientations on the opposite face.
The polyhedral representation of orientation space then makes it easy to apply finite element methods. After meshing the orientation space parameterization, the ODF (Orientation Distribution Function) can be easily represented using piecewise polynomial interpolation.
Pole figures and inverse pole figures can be easily computed from the representation used above. Similarly, one can discretize the sphere using finite elements.
Our software offers alternatives to the traditional methods which use Euler angles and spherical harmonics. Please try them out.
Currently, there are two main pieces of software for distribution, the material point simulator and the ODF/PF function set. The ODF/PF function set is a collection of matlab functions which operate on ODF's (orientation distribution functions) and PF's (pole figures). It handles plotting of the ODF using Rodrigues parameters, plotting of pole figures and inverse pole figures, evaluation of pole figures inverse pole figures from ODF's, and it provides many tools for computing ODF's from pole figures. The material point simulator is a FORTRAN program which models viscoplastic polycrystal deformation for a discrete aggregate of orientations. The core of the material point simulator is computation of the stress/strain rate relationship for single crystals and for upper-bound and lower-bound polycrystal models. It has a flexible script-based interface, which allows evolution of orientation as well as yield surface generation. The data formats allow input and output in a number of orientation conventions, including Euler angles and angle/axis conventions. It also interfaces well with the ODF/PF function set for ease of visualization and post-processing. For more details, see:
For more information about OpenDX, see the OpenDX home page.
The development of the software and the research behind that development were also supported in large part by the ONR. Other sponsors include the NSF and PET.
People involved in the research and development are many. Some are Paul Dawson, Matt Miller, Kapil Mathur, Armand Beaudoin, Ashish Kumar, Gorti Sarma, Sergey Myagchilov, Dave Mika, Nathan Barton, Joel Bernier and Donald Boyce.
Thanks also to Chris Pelkie for his valuable assistance with openDX.
Also, see the useful web page: Encyclopaedia of Cubature Formulas, by Ronald Cools, from which several quadrature and cubature formulas for simplices were taken.