Trajectory-based and State-to-State DSMC


As initially proposed by Koura (Physics of Fluids 9 (11), 1997, pp. 3543-3549), it is possible to imbed trajectory calculations within the DSMC method, thereby effectively replacing stochastic DSMC collision models. Similar to Molecular Dynamics (MD), the only simulation input is a potential energy surface (PES) that dictates the interaction forces between atoms. With modern computer power, such simulations are becoming feasible for flow fields of interest. Since pure Molecular Dynamics simulations are also possible, we are in a position to directly compare trajectory-based DSMC solutions to pure MD solutions where the same PES is the only model input to both techniques. If implemented correctly, trajectory-based DSMC solutions should agree precisely with pure MD solutions (for dilute gases) down to the level of the velocity and internal energy distribution functions. Alternatively, trajectory calculations may be performed individually and integrated to obtain state-to-state cross-sections. If full state-to-state cross-section databases can be converged, then state-to-state DSMC calculations would be as accurate as trajectory-based DSMC calculations but much more computationally efficient. 

Results and Publications

Trajectory-based DSMC Simulations:

    Norman, P., Valentini, P., and Schwartzentruber, T.E., “GPU-Accelerated Classical Trajectory

    Calculation Direct Simulation Monte Carlo Applied to Shock Waves”, J. Comp. Phys., 247 (2013),

    pp. 153-167.

We have implemented classical trajectories into our 3D parallel DSMC code (the MGDS code). The figure (top right) shows a 3D trajectory-based DSMC solution to hypersonic argon flow over a planetary probe geometry using a standard Lennard-Jones interatomic potential. The simulation contains approximately 20 million simulation particles, and therefore computes a very large number of trajectories over the course of the simulation.

The figure (bottom right) shows a 2D trajectory-based DSMC solution to hypersonic nitrogen flow over a cylinder. This simulation uses the Ling-Rigby + Harmonic Oscillator potential to model forces between N atoms  in the simulation and thus involves rotating and vibrating molecules. The inset figure shows the translational, rotational, and vibrational temperatures post-processed  from the simulation along the stagnation line.

Thus, full flow fields can now be computed using only a potential energy surface (PES) as the model input. Research is now focused on what can be done with much more sophisticated PES developed from computational chemistry. In the above article we outline algorithms to incorporate trajectories into a standard DSMC code using the No-Time-Counter (NTC) algorithm, and we also outline algorithms for threading on Graphical Processing Units (GPUs) and multi-core CPUs. Since the majority of the simulation time is spent integrating a large number trajectories, this type of numerical method scales very well on large computing clusters. In the above paper, we also demonstrate exact agreement between pure MD and trajectory-based DSMC solutions for argon and nitrogen shock waves.

State-to-State DSMC Simulations:

   Zhang, C. and Schwartzentruber, T.E, “Consistent Implementation of State-to-State Collision    

   Models for Direct Simulation Monte Carlo”, AIAA Paper 2014-0866, Jan. 2014, presented at the

   52nd Aerospace Sciences Meeting, AIAA SciTech, National Harbor, Maryland.

We have recently developed the capability to use  general state-to-state cross-sections within DSMC simulations. The above conference paper completely details how to implement state-resolved DSMC models. Two aspects that are stressed in  detail are (1) how to ensure a state-resolved model satisfies detailed balance and equipartition of energy at equilibrium, and (2) how state-resolved models are linked to transport properties. It turns out that both aspects, (1) and (2), are related.

In the above paper, a state-resolved DSMC implementation of the Forced Harmonic Oscillator (FHO) model is presented. The figure (image top right) depicts vibrational temperature histories for a number of isothermal relaxation calculations obtained from DSMC-FHO state-resolved simulations. Here, the gas is seen to reach thermal equilibrium and obey equipartition of energy at large times. By fitting the vibrational temperature histories with the Landau-Teller equation, the vibrational relaxation time constant is determined and compared with the Millikan-White experimental correlation (image bottom right). It is noted that the results are obtained considering only vibration-translation (V-T) transitions in the FHO model and thus exact agreement is not expected.

Complete details are contained in the above paper.