Optimized Algorithm for Collision Probability Calculations in Cubic Geometry

An optimized algorithm for implementing a recently developed method of computing collision probabilities (CPs) in three dimensions is reported in this work for the case of a homogeneous cube. Use is made of the geometrical regularity of the domain to rewrite, in a very compact way, the approximate formulas for calculating CPs in general three-dimensional geometry that were derived in a previous work by the author. The ensuing gain in computation time is found to be substantial: While the computation time associated with the general formulas increases as K², where K is the number of elements used in the calculation, that of the specific formulas increases only linearly with K. Accurate numerical results are given for several test cases, and an extension of the algorithm for computing the self-collision probability for a hexahedron is reported at the end of the work.