A Tutorial on Bayesian Analysis of Linear Shock Compression Data

Abstract

Gas gun and other shock compression experiments often produce shock wave velocity measurements that are linearly associated with particle velocity. Traditionally, this empirical relationship is quantified with a single Hugoniot curve that is estimated using least squares regression. However, for downstream modeling and simulation tasks, it is often more useful to have multiple Hugoniot curves in the pressure-volume plane that are consistent with the data. We employ Bayesian uncertainty quantification methods as a framework for propagating measurement uncertainty through to model parameters and predictions. Specifically, this tutorial shows how to sample multiple Hugoniot curves in the pressure-volume plane that are consistent with the shock wave-particle velocity measurements in a two-step Bayesian approach. First, we obtain an analytical expression for the posterior distribution of the linear model parameters using Bayesian linear regression. Second, we propagate samples from the posterior distribution through the Rankine-Hugoniot equations to yield Hugoniot curves in the pressure-volume plane. The procedure is demonstrated with publicly available data on argon, copper, and nickel, and compared against bootstrapping and linear regression. The Bayesian procedure is shown to be interpretable, computationally inexpensive, and less sensitive than an alternative bootstrapping approach to the removal of the point in the copper dataset that has the largest particle velocity. As a tutorial on Bayesian methodology for the shock compression community, we provide several derivations and explanations that make this paper self-contained, and made all code and data available at https://github.com/llnl/BALSCD.

Figures (14)

• Figure 1: Illustration of a shock compression experiment where the particle velocity is $U_\mathrm{p}$ and the shock wave velocity is $U_\mathrm{s}$. The centerline represents the shock front and the brown and blue regions are the material behind and in front of the shock front, respectively. The black circles represent the material behind the shock front and the arrows indicate the velocities.
• Figure 2: Shock Hugoniot data for three materials from marsh1980. The lower right hand corner indicates the number of points, $n$, in each dataset, along with the least squares estimates of $C_0$ and $S$ and the $R^2$ statistic. Note that the axes are not drawn to scale.
• Figure 3: The marginal posterior distribution of $C_0$ and a histogram of samples drawn from this distribution. The analytic distribution is useful because it fully characterizes the uncertainty in the Hugoniot model parameters, whereas the samples are useful for Monte Carlo and uncertainty propagation.
• Figure 4: Posterior distributions of $C_0$ and $S$. The red ellipses are 95% credible regions, meaning they contain 95% of the posterior probability.
• Figure 5: Credible intervals for the Hugoniot curve in the pressure-volume plane, obtained from the posterior distribution of $C_0$ and $S$.