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Predicting Lightning Initiation using Deep Learning

Lightning occurrence presents safety challenges to people and property. The main challenge with lightning safety is that the majority of guidance is reactive. In other words, lightning has to have already occurred nearby before a person will respond and take shelter. Further, most injuries or fatalities occur as the storm approaches, or as it's moving away, when rainfall may not be present at the time of the flash. Thus, this project develops a physically-based deep learning model to produce lightning probabilities out to 15 minutes. The deep learning model combines a Convolutional Neural Network (CNN) with a Long Short-Term Memory (LSTM) network to capture both the spatial and temporal evolution of storms to predict the probability that lightning initiation will occur in the next 15 minutes. The model combines radar reflectivity, correlation coefficient and differential reflectivity to inferred storm hydrometer type and precipitation phase, which aids in the identification of electrification processes. The model is trained with data from the Geostationary Lightning Mapper (GLM), which is a near infrared sensor onboard the GOES-R series of satellites that measures optical brightness from lightning. This presentation will provide an overview of the project.

Andrew T White↗
DOE OSTI2026

Enabling probabilistic learning on manifolds through double diffusion maps

Here, we present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Double diffusion maps↗
DOE OSTI2023

A deep learning upscaling framework: Reactive transport and mineral precipitation in fracture-matrix systems

Pore-scale modeling has limited applicability at large scales due to its high computational cost. One common approach to upscale pore-scale models is the use of effective medium theories, which homogenize small-scale features in a porous structure and characterize the medium by macroscale properties (e.g., permeability) and equations (e.g., Darcy’s law). However, there are classes of physical processes for which effective medium approximations may become inaccurate, e.g., mineral precipitation and clogging during reactive transport. We have developed a deep learning upscaling framework, in which pore-scale modeling is directly employed in macroscale systems, without relying on effective medium approximations. The upscaling framework is first developed for general multiscale systems and then applied to modeling reactive transport with mineral precipitation in the altered layer in fracture-matrix structures. Solute transport from the fractures to the matrix is modeled as a wall boundary condition for the fractures, which, in turn, is predicted by recurrent neural networks using the concentration histories at the fracture-matrix boundary. Specifically, we consider a meter-scale fracture network embedded in sandstones, where the smallest feature is at the micron scale. Here the proposed framework allows us to span five orders of magnitude in length scales by capturing mineral precipitation in the altered layer of the rock matrix at the pore scale across the entire meter-scale fracture network.

42 ENGINEERING↗
DOE OSTI2021

An Analytic Benchmark for Neutron Boltzmann Transport with Downscattering—Part I: Flux and Eigenvalue Solutions

Computing in the energy dimension is one of the greatest challenges confronting present-day deterministic neutron transport solvers. Accurately resolving the neutron flux as neutrons downscatter across resonances in the nuclear cross sections currently requires considerable computing power and suffers from approximation errors. Flux uncertainty resulting from the uncertainty of the resonance structure is the single-largest cause of reactivity uncertainty. Any additional reference solution for the critical neutron downscattering problem with resonance phenomena would be a boon to verification and validation of neutronics codes. This paper establishes a benchmark to verify the accuracy of neutron transport criticality solvers along the energy dimension. For the first time, the analytic solution of the flux amplitude is derived in the particular case of an infinite homogeneous medium with isotropic scattering in the center of mass and an arbitrary number of no-threshold, neutral particle reaction resonances (e.g., radiative capture, fission, and resonance scattering). Furthermore, original analytic expressions are established to quantify the discrepancy between the ψ k (E) and ψ α (E) flux amplitudes, respective solutions of the multiplication factor k, or the exponential time-evolution frequency α eigenproblems. The physical study of these relations led to analysis of their first-order relative difference near the criticality condition α=0. Finally, numerical solutions are provided to a benchmark problem constituted of the first resonance of 239 Pu, the 6.67-eV resonance of 238 U, and a scattering isotope with a flat cross section, allowing for the computational verification of the energy resolution of current neutron transport criticality codes. Through these novel results, this analytic benchmark can serve as a reference to verify the energy resolution and sensitivity analysis of neutron transport criticality calculations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Modeling Volatility-Based Aerosol Phase State Predictions in the Amazon Rainforest

Organic aerosol (OA) is a complex matrix of various constituents—fresh (primary organic aerosols—POA) and aged via oxidation (secondary organic aerosols—SOA), generated from biogenic, anthropogenic, and biomass burning sources. The viscosity of OA can be critical in influencing new particle formation, reactive uptake processes that impact evaporation-growth kinetics, and the lifetime of particles in the atmosphere. We report this work utilizes a well-defined relationship between volatility and viscosity for pure compounds, which we incorporated within the Weather Research and Forecasting Model coupled to chemistry (WRF-Chem) to simulate the phase state and viscosity of bulk OA during the dry-to-wet transition season (September–October) in the Amazon rainforest during 2014. Our simulations indicate spatial and temporal heterogeneity in aerosol phase state often not captured by global-scale models. We show the strong role of water associated with organic aerosol ($w_s$) as the dominant factor that can be used to quantitatively estimate OA viscosity. Analysis of WRF-Chem simulations across the entire atmospheric column indicates a strong inverse log-linear relationship between $w_s$ and OA viscosity with a correlation coefficient approaching 1, in the background and biomass burning-influenced conditions. At high altitudes where relative humidity (RH) and temperatures are low, our simulations indicate that OA exists in a semisolid-/solid-like phase state, consistent with previous studies. OA hygroscopicity is strongly correlated (ca. –0.8) with OA viscosity at RH ca. 30–50%, but this RH range is found mostly at low OA concentrations and the middle troposphere (ca. 6–10 km altitudes) in our simulated domain. OA hygroscopicity is uncorrelated with viscosity at higher-RH (near surface) and lower-RH (upper troposphere) regimes. At the urban site near surface, where day–night differences in RH are significant, RH is found to drive the phase state. At the background forested site near surface, where day–night RH differences are small, biomass burning-influenced OA is semisolid and a significant OA associated with background conditions is liquid-like. Simulations indicate a long tail of OA viscosity frequency distributions extending in the semisolid/solid regimes over background biogenic-influenced conditions due to the role of low-volatility OA components such as monoterpene oxidation products.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Investigating Gas Dynamics and Reactions in Supersonic Jets

UF6 is highly reactive and readily hydrolyzes in contact with moisture to form uranyl fluoride (UO2 F2 ) and hydrofluoric acid (HF), however there is disagreement between experimental and computation investigations into the intermediates of the reaction. • As the hydrolysis reaction occurs at very fast time scales, it is difficult to investigate the intermediates and reaction kinetics experimentally. • Previous experiments at SRNL have had some success at halting and studying the UF6 hydrolysis reaction using cryogenic techniques, however isotopic substitution studies limit possible identity of intermediate compounds. • Primary goal of this project is to capture and study reaction intermediates produced during the gas phase hydrolysis of UF6 via supersonic expansion.

Waldron, Abigail M.↗
DOE OSTI2022

Mass Transport Limitations and Kinetic Consequences of Corn Stover Deacetylation

Alkaline pretreatment of herbaceous feedstocks such as corn stover prior to mechanical refining and enzymatic saccharification improves downstream sugar yields by removing acetyl moieties from hemicellulose. However, the relationship between transport phenomena and deacetylation kinetics is virtually unknown for such feedstocks and this pretreatment process. Here, we report the development of an experimentally validated reaction–diffusion model for the deacetylation of corn stover. A tissue-specific transport model is used to estimate transport-independent kinetic rate constants for the reactive extraction of acetate, hemicellulose and lignin from corn stover under representative alkaline conditions (5–7 g L -1 NaOH, 10 wt% solids loadings) and at low to mild temperatures (4–70°C) selected to attenuate individual component extraction rates under differential kinetic regimes. The underlying transport model is based on microstructural characteristics of corn stover derived from statistically meaningful geometric particle and pore measurements. These physical descriptors are incorporated into distinct particle models of the three major anatomical fractions (cobs, husks and stalks) alongside an unsorted, aggregate corn stover particle, capturing average Feret lengths of 917–1239 μm and length-to-width aspect ratios of 1.8–2.9 for this highly heterogeneous feedstock. Individual reaction–diffusion models and their resulting particle model ensembles are used to validate and predict anatomically-specific and bulk feedstock performance under kinetic-controlled vs. diffusion-controlled regimes. In general, deacetylation kinetics and mass transfer processes are predicted to compete on similar time and length scales, emphasizing the significance of intraparticle transport phenomena. Critically, we predict that typical corn stover particles as small as ~2.3 mm in length are entirely diffusion-limited for acetate extraction, with experimental effectiveness factors calculated to be 0.50 for such processes. Debilitatingly low effectiveness factors of 0.021–0.054 are uncovered for cobs—implying that intraparticle mass transfer resistances may impair observable kinetic measurements of this anatomical fraction by up to 98%. These first-reported quantitative maps of reaction vs. diffusion control link fundamental insights into corn stover anatomy, biopolymer composition, practical size reduction thresholds and their kinetic consequences. These results offer a guidepost for industrial deacetylation reactor design, scale-up and feedstock selection, further establishing deacetylation as a viable biorefinery pretreatment for the conversion of lignocellulosics into value-added fuels and chemicals.

09 BIOMASS FUELS↗
DOE OSTI2023

Multiscale Reactive Model for 1,3,5-Triamino-2,4,6-trinitrobenzene Inferred by Reactive MD Simulations and Unsupervised Learning

When high-energy-density materials are subjected to thermal or mechanical insults at extreme conditions (shock loading), a coupled response between the thermo-mechanical and chemical behaviors is systematically induced. Herein we develop a reaction model for the fast chemistry of 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) at the mesoscopic scale, where the chemical behavior is determined by underlying microscopic reactive simulations. The slow carbon cluster formation is not discussed in the present work. All-atom reactive molecular dynamics (MD) simulations are performed with the ReaxFF potential, and a reduced-order chemical kinetics model for TATB is fitted to isothermal and adiabatic simulations of single crystal chemical decomposition. Unsupervised machine learning techniques based on non-negative matrix factorization are applied to MD trajectories to model the decomposition kinetics of TATB in terms of a four-component model. The associated heats of reaction are fit to the temperature evolution from adiabatic decomposition trajectories. Using a chemical species analysis, we show that non-negative matrix factorization captures the main chemical decomposition steps of TATB and provides an accurate estimation of their evolution with temperature. The final analytical formulation, coupled to a diffusion term, is incorporated into a continuum formalism, and simulation results are compared one-to-one against MD simulations of 1D reaction propagation along different crystallographic directions and with different initial temperatures. A good agreement is found for both the temporal and spatial evolution of the temperature field.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Flow-Driven Stretch Fluctuations Govern the Nonlinear Viscoelasticity of Elongating Associative Polymer Networks

We use nonequilibrium molecular dynamics simulations to verify recent tube-model predictions that associative polymer networks exhibit broad stretch fluctuations during elongational flow. Simulations further show that these fluctuating dynamics give rise to the rate-dependent extensional viscosity 𝜂 𝐸 measured in filament stretching experiments on H-bonding networks. Simulations model bivalent associative networks with a reactive bead-spring model for varying association strength and extensional strain rate. We observe that stretch fluctuations are driven by a new form of chain tumbling, where chains continually collapse and elongate as their associations break and reform within the advecting network. This produces a broad, nearly uniform distribution of chain stretch over a wide range of strain rates, manifesting as a rate-independent plateau in the extensional stress. Our results show that the nonlinear viscoelasticity of associative networks is dominated by large fluctuations in molecular response, which cannot be captured by current mean-field models.

biomimetic & bio-inspired materials↗
DOE OSTI2023

Modeling simultaneous particle shrinkage, dissolution and breakage using the modified moving grid technique

Simultaneous shrinkage, dissolution and breakage are important particle size reduction phenomena that characterize processes like the reactive degradation of solid chemicals. The dynamics of the particle size distribution (PSD) for such processes are non-trivial to model due to the number expending processes brought about by the eventual dissolution of particles. To this end, Population Balance Model (PBM) resolved through the sectional techniques is the natural approach. Here, we introduce a modified Moving Grid technique (m-MGT) to accurately resolve the particle size reduction phenomena. Our technique mimics the perpetual particle shrinkage through a continuously left-moving size grid and incorporates a strategic grid removal routine to capture the disappearance of particles. Coupled with the Fixed Pivot (FP) discretization for breakage, our m-MGT not only preserves the moment-related properties, but also benchmarked very well against the analytical number densities and exhibited a minimum of first-order convergence in all assessed case studies.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Control-oriented core-SOL-divertor model to address integrated burn and divertor control challenges in ITER

The real-time regulation of a burning plasma’s temperature and density, or burn control, will be necessary to produce high fusion power in future tokamaks like ITER. This is made more challenging due to the plasma’s nonlinear characteristics and the interdependence between the core-plasma and edge-plasma regions. For example, a raising plasma temperature leads to increasing reactivity and therefore to more alpha-particle heating, which further increases temperature. Furthermore, a raise of the fusion power increases the heat flow through the scrape-off-layer (SOL), which can compromise the integrity of the divertor without proper safeguards. For control design, a model-based approach is attractive because it can directly incorporate the nonlinear, coupled, burning-plasma dynamics into the design. To facilitate this design approach, a control-oriented core-SOL-divertor (CSD) model is presented in this work. In this CSD model, a core-plasma model captures the nonlinear dynamics of the core’s density and temperature, and a SOL-divertor model defines the plasma conditions at the separatrix and divertor including the heat load on the target plates. The core-plasma and SOL-divertor models are coupled through the exchange of various variables. In particular, the SOL-divertor model yields the separatrix temperature and the influx of recycled particles into the core-plasma. Further, these variables influence the power and particle balances captured by the core-plasma model. In return, the core-plasma model determines the intensity of the heat and particles fluxes across the separatrix, and this outflow strongly impacts the SOL-divertor model. Therefore, the power and density of the core-plasma, which can be readily modulated through external heating systems and pellet injection, can be viewed as control knobs for the SOL-divertor region in addition to the gas puffing. In simulations of the CSD model, it is demonstrated how external actuation can be utilized to meet burn control and divertor control objectives simultaneously.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Development and Validation of Two-Phase Flow Models in MOOSE and Application to Molten Salt Reactors

Two-phase flow in Molten Salt Reactors (MSRs) is important as it impacts reactivity evolution, reactor transient response, and the removal of species dissolved in the molten salt through gas phase transfer. Therefore, accurately predicting the gas distribution and the associated liquid-gas interface area in MSRs is essential for their design and operation. Recently, we integrated a new two-phase model into Idaho National Laboratory (INL)’s Multiphysics Object-Oriented Simulation Environment (MOOSE): a multi-D generalization of a mixture drift-flux model. It provides greater computational efficiency, which is typically preferred for modeling reactor transients. However, the mixture model’s accuracy in capturing void distribution and interfacial area in MSRs still needs to be assessed. This article begins with a description of the mathematical framework for the two-phase model implemented in MOOSE. It then presents validation of these models against relevant experimental data. Finally, the model is applied to the Molten Salt Reactor Experiment case study, analyzing various operational conditions such as different rates of fission product volatilization and diverse cover gas entrainment scenarios at the reactor pump. The article concludes by assessing the suitability of the mixture drift-flux model for capturing the two-phase flow dynamics critical to MSR operations

42 - ENGINEERING↗
DOE OSTI2026

Oxide nucleation via threshold-driven volume instability in Ni oxidation

Oxide nucleation dictates the onset of metal oxidation, yet the atomistic pathways bridging initial oxygen adsorption to bulk phase transformation remain elusive. Here, we utilize in-situ atomic-resolution imaging to directly capture the nucleation of nickel oxide, revealing a threshold-driven, cooperative transformation of Ni into NiO. Our observations identify a distinct incubation period during which oxygen progressively accumulates in subsurface layers. Once a critical concentration and penetration depth are reached, a collective lattice reconfiguration is triggered, abruptly converting multiple Ni layers into NiO. Atomistic simulations corroborate this mechanism, identifying a cooperative lattice instability induced by subsurface oxygen saturation. These results establish subsurface oxygen incorporation as the missing mechanistic link between surface adsorption and three-dimensional oxide formation, providing an atomistic framework to understand and control reactive phase transformations in materials synthesis, catalysis, and degradation.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Closures for multicomponent reacting flows based on dispersion analysis

This work presents algebraic closure models associated with advective transport and nonlinear reactions in a Reynolds-averaged Navier-Stokes context for a system of species subject to binary reactions and transport by advection and diffusion. Expanding upon analysis originally developed for non-reactive transport in the context of Taylor dispersion of scalars, this work extends the modified gradient diffusion model explicated by Peters [N. Peters, Turbulent Combustion, Cambridge Monographs on Mechanics (Cambridge University Press, Cambridge, 2000)] and based on work by Corrsin [S. Corrsin, The reactant concentration spectrum in turbulent mixing with a first-order reaction, J. Fluid Mech. 11, 407 (1961)] beyond single-component transport phenomena and involving nonlinear reactions. The presented model forms, from this weakly nonlinear extension of the original dispersion theory, lead to an analytic expression for the eddy diffusivity matrix that explicitly captures the influence of the reaction kinetics on the closure operators. Furthermore, we demonstrate that the derived model form directly translates between flow topologies through a priori and a posteriori testing of a binary species system subject to homogeneous isotropic turbulence. Using two- and three-dimensional direct numerical simulations involving laminar and turbulent flows, it is shown that this framework improves prediction of mean quantities compared to previous results. Lastly, the presented model form, collapses to the earlier gradient diffusion and its modified version derived by Corrsin in the limits of nonreactive species and linear reactions, respectively.

42 ENGINEERING↗
DOE OSTI2021

New reactions of diazene and related species for modelling combustion of amine fuels

Potential energy surfaces for reactions involving N 2 H 2 isomers of diazene (diimide) have been explored using density functional theory, with energies based on coupled-cluster theory. A focus is on processes that create or consume these species, and isomerisation between the E (trans) and Z (cis) forms of HNNH. Furthermore, these include isomerisation and dissociation pathways for HNNH, addition of H atoms to form N 2 H 3 , abstraction by H atoms yielding short-lived NNH, and abstraction reactions of H with N 2 H 3 . Transition state and capture theories are applied for high-pressure-limiting behaviour, while low-pressure and falloff regions are characterised via the methods of Troe and coworkers. Rate constants and thermochemistry are provided to improve models of diamine chemistry, relevant to the combustion of NH 3 especially at high concentrations, high pressures or under reducing conditions. Results indicate that amine radical recombination mainly yields the E HNNH isomer, while H-abstraction from N 2 H 3 results in E HNNH and H 2 NN. However, at elevated temperature E → Z isomerisation becomes competitive, and Z HNNH, being more reactive, acts to enhance the diazene consumption rate.

10 SYNTHETIC FUELS↗
DOE OSTI2020

Facilitated Transport Membranes with Tunable Amine-CO 2 Chemistry for CO 2 /H 2 Separation

CO 2 -selective, amine-containing facilitated transport membranes are of great interest for syngas purification since high-pressure H 2 can be retained upon CO 2 removal. A variety of amine-containing polymers have shown decent chemical and thermal stability at aggressive conditions, but their CO 2 /H 2 separation properties are largely limited by the hydrostatic compaction and severe carrier saturation associated with the high syngas pressure. Herein, we report a new approach to enhance the CO 2 permeability by manipulating the steric hindrance of the amine carriers. A series of α-aminoacids with different alkyl or hydroxyethyl substituents are deprotonated by 2-(1-piperazinyl)ethylamine, leading to nonvolatile amine carriers with different degrees of steric hindrance. For hosting the low MW amine carriers, a highly crosslinked poly(vinyl acetal) is synthesized as a water-swellable polymer network. In order to avoid membrane compaction, perforated graphene oxide mono-sheets are dispersed as reinforcement fillers. In the presence of moisture, a bulkier alkyl substituent to the amino site (increasing steric hindrance) destabilizes the carbamate adduct and thus drastically increases the chemisorption of CO 2 , while the incorporation of ethylene oxide groups provides additional physisorption of CO 2 . The enhanced CO 2 solubility significantly mitigates the carrier saturation behavior, and an unprecedented CO 2 /H 2 selectivity greater than 100 is demonstrated at 107 °C and 12.5 atm of CO 2 partial pressure. As the CO 2 partial pressure reduces to 0.4 atm, a less hindered amine yields a higher reactive diffusivity of CO 2 , resulting in a CO 2 permeance of 435 GPU with a selectivity greater than 500. These reaction-mediated polymeric membranes are well above the theoretical upper bound, and they are of great interest for designing a highly-selective membrane process for syngas purification.

08 HYDROGEN↗
DOE OSTI2023

Polyimide dynamically compressed to decomposition pressures: Two-wave structures captured by velocimetry and modeling

We performed a series of six plate impact experiments on polyimide and modeled them using new reactant and products equations of state combined with an Arrhenius rate model. The first experiment was diagnosed with embedded electromagnetic velocity gauges through which we directly observed attenuation of the lead shock to an approximately constant state over a propagation distance of roughly 4 mm. Simulated gauge profiles were in excellent qualitative agreement with experiment and suggested a sluggish chemical reaction that did not proceed to completion. The remaining five experiments were conducted in a transmission geometry and diagnosed velocimetrically at the sample/window interface. All five of these yielded profiles with a sharp shock followed by a more gradual approach to maximum interface velocity that was “rounded” to varying degree. These profiles proved difficult to interpret unambiguously due to the convolution of the reactive wave upon first shock with reflection of the lead wave and reshock or release by the window. Comparison with thermochemical calculations strongly suggests that the point of maximum interface velocity corresponds to the equilibrium reshock or release locus. We discuss the implications of this point for the practice of impedance matching based on the reflected Hugoniot of reactive materials such as polymers. The reactant and thermochemical products equations of state are developmental SESAME tables 97710 and 97720, respectively.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Stochastic Modeling in a Multimaterial Continuum Mixture Shock Physics Code

Stochastic modelling approaches are presented to capture random effects at multiple time and length scales. Random processes that occur at the microscale produce nondeterministic effects at the macroscale. Here we present three stochastic modeling approaches that describe random processes at microscopic length scales and map these processes to the macroscopic length scale. The first stochastic modeling approach is based upon a particle based numerical technique to solve a Stochastic Differential Equation (SDE) using an arbitrary diffusion process to capture random processes at the microstructural level. The second approach prescribes a Probability Density Function (PDF) for the drift and diffusion of the random variable derived using the forward and backward Kolmogorov equations. This method requires mean and drift evolution PDF transport equations. The third approach is the coupling of multiple random variables which are dependent on each other. The relationship of the PDFs and a coupling function, known as a copula, produces a Joint Probability Density Function (JPDF). These stochastic modeling approaches are implemented into a Multiple Component (MC) shock physics computational code and used to model statistical fracture and reactive flow applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗