MechE-CSE PhD Thesis Defense: Evan Massaro

Abstract:

Direct Simulation Monte Carlo (DSMC) remains the gold standard for rarefied gas dynamic simulations. However, flows featuring large density variations pose challenges for this method since they lead to highly non-uniform spatial distributions of computational particles. In such cases, meeting the minimum uncertainty and resolution requirements in the dilute regions requires a prohibitively large total number of computational particles across the system. In this thesis, we develop a suite of methods that address limitations associated with particle control in DSMC as well as related particle simulation methods for rarefied gas flows.

To this end, we introduce Weighted Particle Number Control (WPNC), a method for solving the Boltzmann equation that uses particles of different weights, as a means of controlling the number of particles and the resulting statistical uncertainties in different parts of the computational domain. Within WPNC, treatment of unequally weighted particles follows the established simulation approach known as Stochastic Weighted Particle Method (SWPM). Unfortunately, SWPM collision rules generate additional particles, which has previously limited its applicability, especially in collision-dominated flows. In this thesis we show that the tendency to create additional particles can be exploited to increase the number of particles in dilute regions while any instability in dense regions can be suppressed by a particle resampling technique such as stratified sampling, leading to a reliable and efficient particle control method. Overall, the proposed WPNC  scheme requires only small modifications to DSMC, and exactly conserves mass, momentum, and energy. In validation tests of systems with density ratios of order 10^2-10^4, we observe 10^1-10^2 times less variance in the dilute solution compared to a DSMC solution with the same number of particles, while introducing a moderate additional computational cost.

We also show that, instead of using particle resampling approaches, SWPM can be stabilized by pairing it with the Time Relaxed Monte Carlo (TRMC) method, a high-order integration method for the Boltzmann collision operator. The resulting method, referred to as Weighted Particle (WP)-TRMC, requires modest modifications to DSMC, is stable, conservative, and computationally more efficient over a wide range of length and time scales. We investigate the performance of the proposed WP-TRMC method relative to DSMC in several multiscale benchmark problems with large density ratios, including a propagating density discontinuity. In systems with density ratios of 10^1-10^4, we observe a 10^1-10^4 times reduction in the computational effort needed to reach a desired statistical uncertainty in the dilute solution.

Finally, we show how the TRMC methodology can be used to stabilize other weighted particle variants of DSMC, such as the Variance Reduced (VR)-DSMC method, leading to significant computational savings for flows featuring small deviations from equilibrium. In low signal flows with a wide range of rarefactions and with density ratios of 10^{-2}-10^{-1}, we observe a 10^2-10^4 times reduction in the computational effort needed to reach a desired statistical uncertainty in the overall solution. These diverse findings for both large and small signal flows help extend the range of problems for which direct Monte Carlo methods are practical and efficient.

Thesis Committee Members:

• Nicolas Hadjiconstantinou, Professor of Mechanical Engineering, MIT (Thesis supervisor)
• Wim van Rees, Associate Professor of Mechanical Engineering, MIT
• Jaime Peraire, Professor in Aeronautics and Astronautics, MIT