Noncohesive granular materials in slowly rotated containers mix by discrete avalanches; such a process can be described mathematically as a mapping of avalanching wedges. A natural decomposition is thus proposed : a geometrical part consisting of a mapping wedge --> wedge, which captures large-scale aspects of the problem; a dynamical part confined to the avalanche itself, which captures details emanating from differences in size/density/morphology. Both viewpoints are developed and comparisons with experiments are used to verify the predictions of the models. In this article, we develop a model of granular mixing and show how to extend the model in order that it may : (1) handle complicated geometries, (2) be applicable for 3-D mixers, (3) rapidly test mixing enhancement strategies, and (4) incorporate differences in particle properties. In addition, an optimal fill level is determined for several 2-D mixing geometries, and a novel hybrid - geometrical/dynamical - computational technique is proposed. By merging the geometrical and dynamical viewpoints, this technique reduces the computational time of a typical molecular-dynamics-type simulation by a factor of 15. The ultimate goal is to provide fundamental understanding and tools for the rational design and optimization of granular mixing devices.