Algorithms for Particle-Field Simulations with Collisions

Abstract

We develop an efficient algorithm for detecting collisions among a large number of particles moving in a velocity field, when the field itself is possibly coupled to the particle motions. We build on ideas from molecular dynamics simulations and, as a byproduct, give a literature survey of methods for hard sphere molecular dynamics. We analyze the complexity of the algorithm in detail and present several experimental results on performance which corroborate the analysis. An optimal algorithm for collision detection has cost scaling at least like the total number of collisions detected. We argue, both theoretically and experimentally, that with the appropriate parameter choice and when the number of collisions grows with the number of particles at least as fast as for billiards, the algorithm we recommend is optimal.

Simulation of Cell Signaling Communications Using Event-Driven Algorithms

“…Then it is clear that some changes need to be made on the previous algorithm to improve the performance and scalability [8]. The first one is to save collision times, since on two consecutive iterations the computed time on the first iteration will likely be the same on the second one.…”

“…We see that, despite the higher running time values obtained due to the simulation time used, the overall behavior of the algorithm is the same, indicating that the simulator is well behaved. Herir Sigurgeirsson, Andrew Stuart and Wing-Lok Wan in 2001 [8] report that for a 5000 particle system, their algorithm handles about 16,000 collisions per second on a Pentium III PC. For the same amount of particles our algorithm handles around 30,600 collisions per second using the cell list algorithm.…”

“…This oversimplified algorithm can be applied as long as the number of particles in the system is not very large. But this algorithm does not scale well with the total number of particles since the required number of calculations that have to be performed each iteration depends on the number of particle pairs as followsThen it is clear that some changes need to be made on the previous algorithm to improve the performance and scalability [8]. The first one is to save collision times, since on two consecutive iterations the computed time on the first iteration will likely be the same on the second one.…”

“…Finally, in Fig. 12 we run the algorithm using similar parameters to those used in [8], that is: n=50000 particles and ρ=15%. We see that, despite the higher running time values obtained due to the simulation time used, the overall behavior of the algorithm is the same, indicating that the simulator is well behaved.…”

A model for preferential concentration

“…There is some literature on the topic of two-point motion in the context of particle tracers. 27,28 In conclusion we show the following: ͑i͒ that a simple Gaussian random field model for the velocity field, coupled with Stokes' law for particle motion, provides remarkably good agreement with some of the experimental data concerning preferential concentration; ͑ii͒ the model allows for an elucidation of various scaling limits that either confirm experimental observation ͑for small or large Stokes number͒ or provide new insight into large time behavior, predicting preferential concentration in some cases, and giving stochastic differential equations governing the phenomenon; ͑iii͒ the model is fast to simulate, giving an order of magnitude speedup over DNS simulations; ͑iv͒ the effect of collisions can be studied numerically, using the fast algorithm in Sigurgeirsson et al, 29 and our results show that the effect of collisions is negligible here; to the extent that they are no- ticeable they create an interesting antidiffusive behavior, sharpening concentration lines in the particle distributions.…”