first light 18th International Conference on the Physics of Non-Ideal Plasmas, Oxford, 16th – 19th September 2024 EoS uncertainty quantification applied to First Light Fusion amplifier design J. R. Allison, D. Adekanye, V. Beltrán, T. Edwards, J. Pecover, D. Chapman, N. Joiner, N. Hawker, R. Bordas [email protected] FLF Endor amplifier • A projectile-driven layered design that amplifies a planar-to-planar shock from an impacting projectile • A relatively inexpensive platform for accessing high pressures Diagram components: Projectile (Tungsten, PMMA), Endor (PMMA, Tantalum). Image: Skidmore et al. (PRB to be submitted) [Redacted due to copyright] 1-layer amplifier variant results from Sandia STAR 2SLGG • STAR Two-stage Light-Gas Gun (Image credit: Sandia National Laboratories) • Shock velocity in Quartz block (Skidmore et al., PRB to be submitted) [Redacted due to copyright] Uncertainty quantification methodology • We adopt a non-intrusive Monte-Carlo approach to uncertainty quantification, whereby a discrete sample is drawn from probability distributions for each uncertain feature • We have an in-house sampling framework that optimizes for space-filling and orthogonal Latin Hypercube designs using simulated annealing (Joseph & Hung 2008) • This input sample is used to drive an ensemble of 2D hydro simulations to predict the distribution of amplifier responses Workflow steps: 1. Identify uncertain features of the model 2. Construct statistical models for each uncertain feature 3. Draw a discrete sample from the statistical models 4. Run an ensemble of 2D hydrodynamic simulations 5. Carry out sensitivity analysis on Amplifier responses Amplifier uncertainties • The principal uncertainties are projectile velocity variance (aleatoric) and Equation-of-State model error (epistemic) • We use a PERT distribution to model projectile velocity uncertainty, with {min, mode, max} = {6, 6.5, 7} km s⁻¹ • Our EoS tables are generated using the QEOS global model (More et al. 1988) with the FEOS package (Faik et al. 2018) • We model EoS uncertainty as random deviations from our tables, but these must satisfy thermodynamic consistency between the state variables Plots shown: PERT distribution, Pressure (log₁₀(P/Pa)), Energy (log₁₀(u / J kg⁻¹)) EoS UQ using Gaussian Processes • We use Gaussian Processes as a function of (ρ, T) to generate the random sample of EoS deviations • GPs define distributions of functions of the input space, which can be further conditioned (i.e. constrained) by observed data • A GP is uniquely defined by its mean and covariance functions: f(x) ~ GP(f̄(x), k_f(x, x')) (where f̄(x) is the Mean function, k_f(x, x') is the Covariance function, and f(x) is the Random function drawn from GP) • Hyperparameters describing the behaviour of these functions can be optimised with respect to the observed data using a likelihood function (i.e. "training the GP") EoS UQ using Gaussian Processes • We model pressure deviations from FEOS as a single GP: ΔP(x) ~ δ_P(x) P̄(x) GP(0, k̂_f(x, x')) (where δ_P(x) is the Relative uncertainty in FEOS-modelled pressure) • Then numerically solve for deviations in internal energy using Maxwell relations: Δu(x) = ∫ (1/ρ) (ΔP - ∂_ln T(ΔP)) d(ln ρ) + Δu_T • And likewise for entropy: Δs(x) = ∫ (1/T) ∂_ln T(Δu) d(ln T) + Δs_ρ Δs_ρ = -∫ ((1/(ρT)) ∂_ln T(ΔP) + ∂_ln T(Δs)) d(ln ρ) + Δs_0 EoS UQ model for Amplifier • Pressure uncertainties on Hugoniot states are modelled by comparing with experimental LASL data (Marsh et al. 1980) Plots shown for Principal Shock Hugoniot: - PMMA FEOS original vs PMMA LASL data (Pressure [Pa] vs Density [kg m⁻³]) - Tantalum FEOS original vs Tantalum LASL data (Pressure [Pa] vs Density [kg m⁻³]) EoS UQ model for Amplifier • Pressure uncertainties on Hugoniot states are modelled by comparing with experimental LASL data (Marsh et al. 1980) • Uncertainties for off-Hugoniot states are then modelled using a transition to a conservative global value of 100% uncertainty Uncertainty maps for pressure EoS in PMMA and Tantalum (log₁₀(σ_P / P) over log₁₀(ρ / kg m⁻³) and log₁₀(T / K)) EoS UQ model for Amplifier • Pressure uncertainties on Hugoniot states are modelled by comparing with experimental LASL data (Marsh et al. 1980) • Uncertainties for off-Hugoniot states are then modelled using a transition to a conservative global value of 100% uncertainty • Finally, a sample of 500 EoS tables is drawn from the UQ model to drive an ensemble of hydrodynamic simulations Distributions of the mean deviation from FEOS for a sample of 500 EoS tables: - PMMA Ion: Pressure ⟨(P - P̄)/P̄⟩_{ρ,T} = -0.11_{-0.16}^{+0.14}, Internal Energy ⟨(u - ū)/ū⟩_{ρ,T} = 0.14_{-0.17}^{+0.20} - Tantalum Ion: Pressure ⟨(P - P̄)/P̄⟩_{ρ,T} = -0.03_{-0.06}^{+0.07}, Internal Energy ⟨(u - ū)/ū⟩_{ρ,T} = 0.05_{-0.09}^{+0.10} Results – Amplifier response to EoS and projectile uncertainties Scatter plot matrix and distributions showing correlations among: - Temporal pressure deviation [%] - Mean pressure [GPa] - Planarity - Flow component Results – Amplifier response to EoS and projectile uncertainties Comparison of 2D hydro simulations for: - min(P̄) - Baseline - max(P̄) Color bar: P [TPa] ranging from 0 to 1 Results – Amplifier response to EoS and projectile uncertainties Comparison of vorticity / flow fields for: - min(P̄) - Baseline - max(P̄) Color bar: ω = (∇ × v)·ĵ [s⁻¹] from -2.5e+08 to 2.5e+08 Results – Sensitivity analysis of impact of uncertainties Regression model: ŷ_i = ∑_j^n SRC_{i,j} x̂_j Heatmap of Standardized Regression Coefficients (SRC) showing increasing impact on amplifier: - Explained variance (R²_adj): Mean pressure (0.87), Flow component (0.75), Temporal pressure deviation (0.31), Planarity (0.12) - Features evaluated: Projectile velocity, PMMA amplifier fill electron/ion internal energy & pressure, Quartz block electron/ion internal energy & pressure, Tantalum amplifier anvil electron/ion internal energy & pressure, PMMA projectile sabot electron/ion internal energy & pressure, Tantalum amplifier foil electron/ion internal energy & pressure, Tantalum projectile tip electron/ion internal energy & pressure, Quartz block ion/electron pressure, PMMA projectile sabot electron/ion pressure, Tantalum amplifier foil electron pressure/internal energy, Tantalum projectile tip ion internal energy/pressure, Tantalum projectile tip electron pressure, Tantalum amplifier anvil ion internal energy. Comparison with experimental results from STAR 2SLGG • Recent experimental shots with the STAR two-stage light-gas gun at Sandia National Laboratories • Mean pressure inferred from PDV velocities of shock in Quartz block consistent with a more performant amplifier than predicted in silico • This offset from prediction is explained by our EoS uncertainty model • The distribution is biased towards higher PMMA internal energies and hence higher amplifier pressures (Experimental measurement with STAR two-stage light-gas gun: Skidmore et al., PRB to be submitted) Closing the loop - conditioning the EoS tables on experiment Bayesian formulation: Pr(y_EOS | D, M) ∝ Pr(D | y_EOS, M) Pr(y_EOS | M) - Pr(y_EOS | D, M): Updated EoS distribution (Posterior) - Pr(D | y_EOS, M): Likelihood function - Pr(y_EOS | M): Input EoS distribution (Prior) • Given additional experimental data we can update our EoS distribution (Bayesian inference) • Ideally this is done using a dedicated Monte-Carlo method for posterior sampling or neural density estimation • Here we simply demonstrate the approach by weighting the prior distribution by the likelihood function Histograms compare Prior vs Posterior probability density for PMMA Ion and Tantalum Ion pressure and internal energy deviations. Summary • Physical shortcomings in tabulated EoS data are expected to be a source of uncertainty in our projectile-driven amplifier design at First Light Fusion • We have developed a thermodynamically-consistent uncertainty-quantification model for our EoS using Gaussian Processes (GPs) • We have carried out a sensitivity study of our Endor amplifier design to both EoS and projectile velocity • The mean shock pressure is most sensitive to the projectile velocity and EoS of the low-impedance PMMA fill material • The shock direction is most sensitive to the EoS of the low-impedance PMMA fill material • The distribution of output pressures is biased by ~50 GPa above the baseline prediction, consistent with experimental results on the STAR two-stage light-gas gun first light Thank you for your attention Please get in touch [email protected] firstlightfusion.com | powering a world worth inheriting | FLFusion