Discrete element modelling (DEM) according to Cundall and Strack [
] of a large set of particles with long-ranging forces like gravitation or electrostatics has the disadvantage of a computation time dependence on the number of particles of O(N
). This can be overcome by the usage of a hierarchical tree code [
] which groups particles which are far away to virtual pseudoparticles. This reduces the number of force calculations so that the computation time dependence can be reduced to O(N log N). The criterion which determines a possible grouping of particles is the theta parameter which is a measure for the reciprocal distance of the particle of interest to the neighboring particles.
The paper here shows that the tree algorithm can be also used for an efficient collision checking routine since particles that the algorithm determines to be far enough from the particle of interest can certainly be excluded for a collision checking. It is shown, however, that for a particle set with a uniform radius distribution an upper limit for the theta parameter exists. When this upper limit is exceeded collisions will be suppressed or artificial collisions will occur so that the simulation result is severely falsified. This limit can be overcome by extending the theta parameter to include the radius of the particle of interest. This allows the theta parameter to be increased significantly over the previously found limit, thus reducing the computation time by a further 30 % without introducing much additional error. The prerequisite for such a high theta parameter is that the simulated particle set is rather densely packed (packing density > 10%) so that the total behaviour is dominated by collisions not by the long-ranging forces.
The algorithm is applied to the simulation of electrostatically charged toner particles used for the electrophotographic print process. To provide a test case for the simulation a simple transfer experiment is chosen: A roller covered thickly with charged toner is positioned next to a second roller with a thin air gap between them. An external voltage is applied to the rollers causing the particles to jump to the second roller. The results of this experiment can be easily measured and the comparison with the simulation shows an error below 5%.