Crystallographic texture approximation by quadratic programming

This paper considers the problem of approximating a given crystallite orientation distribution function (codf) by a set of texture components. Problems of this type arise for example if the codf has to be reconstructed from discrete orientations or if one looks for a physical interpretation of the codf. The same problem is encountered if crystallographic texture based constitutive models have to be specified. The equivalence of these tasks to a mixed integer quadratic programming problem (MIQP) - a standard but challenging problem in optimization theory - is shown. Special emphasis is given to the generation of a class of approximations with an increasing number of texture components. Furthermore, the constraints resulting from the non-negativity, the normalization, and the symmetry of the codf are analyzed. Finally, a set of approximations of three different experimental textures determined with this solution scheme is presented and discussed. Based on these hierarchical solutions, the engineer can decide in what detail the microstructure is considered.