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At least 523 records · Page 29

Physics-based model for electro-chemical process

Considering the kinetics of electrochemical reactions and mass transfer at the surface and near-surface of the electrode, a physics-based separation model for separating actinides from fission products in an electro-refiner is developed. The model, taking into account the physical, chemical and electrochemical processes at the electrode surface, can be applied to study electrorefining kinetics. One of the methods used for validation has been to apply the developed model to the computation of the cyclic voltammetry process of PuCl{sub 3} and UCl{sub 3} at a solid electrode in molten KCl-LiCl. The computed results appear to be similar to experimental measures. The separation model can be applied to predict materials flows under normal and abnormal operation conditions. Parametric studies can be conducted based on the model to identify the most important factors that affect the electrorefining processes.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Model-Free Voltage Control of Active Distribution System with PVs Using Surrogate Model-Based Deep Reinforcement Learning

Accurate knowledge of the distribution system topology and parameters is required to achieve good voltage control performance, but this is difficult to obtain in practice. This paper proposes a physical-model-free voltage control method based on a surrogate-model-enabled deep reinforcement learning approach. Specifically, a surrogate model is trained in a supervised manner using the recorded limited number of historical data to learn the relationship between the power injections and voltage fluctuations of each node. Then, the deep reinforcement learning algorithm is applied to learn an optimal control strategy from the experiences obtained by continuous interactions with the surrogate model. The proposed method can achieve physical-model-free control of unbalanced distribution network and inform real-time decisions to deal with fast voltage fluctuations caused by the rapid variation of PV generation. Simulation results on an unbalance IEEE 123-bus system show that the proposed method can achieve similar performance as that of perfect physical-model-based approaches while being advantageous over other traditional methods.

active distribution network↗

Assessment of Current MACCS Capabilities for Modeling Atmospheric Physical and Chemical Transformations

The physical and chemical transformation during atmospheric transport of radionuclides released into the environment has the possibility of impacting consequence modeling results. Accordingly, this report identifies physical and chemical transformations that may occur following release of chemically reactive radioactive species, how those transformations may affect modeling of consequences of release to the atmosphere and identifies current capabilities – in both MACCS and other state-of-practice atmospheric transport and dispersions models– to model those transformations. It was found that the inclusion of physical and chemical transformations is currently very limited in current state-of-practice codes for atmospheric dispersion of radionuclides. State-of-practice atmospheric dispersion codes appear to be typically limited to simulating either physical-chemical transformations or radioactive transformations, but not both. A state-of-practice atmospheric dispersion code capable of performing parallel physical, chemical, and radioactive transformation was not identified. A few atmospheric dispersion codes capable of modeling physical and chemical transport of specific species such as tritium or uranium hexafluoride were identified. Consequently, there is currently no information available that clearly suggests updates to the MACCS code are needed to bring it up to state-of-practice. However, investigations concluded that the MACCS computational framework can currently accommodate multiple physical/chemical forms in one simulation. Additionally, with some major assumptions, the computational framework in MACCS can accommodate parallel physical-chemical and radioactive transformations.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

Flow field Reconstruction for Inhomogeneous Turbulence using Data and Physics Driven Models

A methodology combining Large Eddy Simulation (LES) trained data and a physics driven wave packet model to obtain a reduced order reconstruction for broadband, three-dimensional, temporally stationary but spatially inhomogeneous, incompressible turbulence. Wake turbulence generated by an axisymmetric dragging disk with a turbulent co-flow serves as the benchmark test case. We begin by studying the proper-orthogonal decomposition of the turbulent fluctuations taken from a high-resolution LES to first identify whether the fields demonstrate a low-rank character. It is argued that the presence of the turbulent co-flow results in a largely broadband character lacking any tonal properties. This is especially true for Strouhal numbers greater than 1 and only a small fraction of energy is contained in the leading order Kelvin-Helmholtz modes. As such reconstructions and reduced order modeling purely relying on data from LES does not appear to be a lucrative solution - contrary to problems with strongly tonal character. To supplement the missing energy from a low order truncated mode expansion, we utilize a physics based super-resolution (enrichment) algorithm that relies on spatio-temporally localized Gabor wave packets whose time evolution is described using a set of ordinary differential equations. The reconstructed flow has single- and two-point correlations that are consistent with the reference high resolution simulation data.

SLS↗

Advanced in turbulence physics and modeling by direct numerical simulations

The advent of direct numerical simulations of turbulence has opened avenues for research on turbulence physics and turbulence modeling. Direct numerical simulation provides values for anything that the scientist or modeler would like to know about the flow. An overview of some recent advances in the physical understanding of turbulence and in turbulence modeling obtained through such simulations is presented.

Reynolds, W. C.↗

Physical Compact Model for Three-Terminal SONOS Synaptic Circuit Element

A well-posed physics-based compact model for a three-terminal silicon–oxide–nitride–oxide–silicon (SONOS) synaptic circuit element is presented for use by neuromorphic circuit/system engineers. Based on technology computer aided design (TCAD) simulations of a SONOS device, the model contains a nonvolatile memristor with the state variable QM representing the memristor charge under the gate of the three-terminal element. By incorporating the exponential dependence of the memristance on QM and the applied bias V for the gate, the compact model agrees quantitatively with the results from TCAD simulations as well as experimental measurements for the drain current. The compact model is implemented through VerilogA in the circuit simulation package Cadence Spectre and reproduces the experimental training behavior for the source–drain conductance of a SONOS device after applying writing pulses ranging from –12 V to +11 V, with an accuracy higher than 90%.

42 ENGINEERING↗

Transverse Emittance Reduction with Tapered Foil

The idea of reducing transverse emittance with tapered energy-loss foil is proposed by J.M. Peterson in 1980s and recently by B. Carlsten. In this paper, we present the physical model of tapered energy-loss foil and analyze the emittance reduction using the concept of eigen emittance. The study shows that, to reduce transverse emittance, one should collimate at least 4% of particles which has either much low energy or large transverse divergence. The multiple coulomb scattering is not trivial, leading to a limited emittance reduction ratio. Small transverse emittances are of essential importance for the accelerator facilities generating free electron lasers, especially in hard X-ray region. The idea of reducing transverse emittance with tapered energy-loss foil is recently proposed by B. Carlsten [1], and can be traced back to J.M. Peterson's work in 1980s [2]. Peterson illustrated that a transverse energy gradient can be produced with a tapered energy-loss foil which in turn leads to transverse emittance reduction, and also analyzed the emittance growth from the associated multiple coulomb scattering. However, what Peterson proposed was rather a conceptual than a practical design. In this paper, we build a more complete physical model of the tapered foil based on Ref. [2], including the analysis of the transverse emittance reduction using the concept of eigen emittance and confirming the results by various numerical simulations. The eigen emittance equals to the projected emittance when there is no cross correlation in beam's second order moments matrix [3]. To calculate the eigen emittances, it requires only to know the beam distribution at the foil exit. Thus, the analysis of emittance reduction and the optics design of the subsequent beam line section can be separated. In addition, we can combine the effects of multiple coulomb scattering and transverse energy gradient together in the beam matrix and analyze their net effect. We find that,when applied to an electron linac or electron beam line, the energy spread increase and angular growth due to multiple scattering are not trivial; as a result, the transverse emittance can only be reduced with a limited ratio, e.g. down to about 65% the original value. The contents of this paper are arranged as follows. In Sec. II, we build the physical model of the tapered foil, derive the transverse eigen emittance and discuss the emittance reduction criteria. In Sec. III, we implement numerical simulations to verify the physical model; and in Sec. IV, we present numerical experiments and subsequent beam line to remove the transverse energy gradient to demonstrate the applicability of such method. Conclusions are given in the last section.

43 PARTICLE ACCELERATORS↗

Physics-Based Model to Represent Membrane-Electrode Assemblies of Solid-Oxide Fuel Cells Based on Gadolinium-Doped Ceria

This paper reports a physics-based model that predicts membrane-electrode assembly (MEA) performance of solid-oxide fuel cells (SOFCs) with Ce 0.9 Gd 0.1 O 2− δ (GDC10) electrolyte membranes. The paper derives self-consistent thermodynamic and transport properties for GDC1o mobile charged defects (oxide vacancies and reduced-ceria small polarons) by fitting published measurements of oxygen non-stoichiometry and conductivity over ranges of temperature and O 2 partial pressures. The button-cell model is applied to evaluate how mixed ionic-electronic conductivity influences the performance of an SOFC MEA with a GDC10 electrolyte sandwiched between a porous, composite Ni-GDC10 anode and a porous, composite cathode of Sm 0.5 Sr 0.5 CoO 3− δ (i.e., SSC) and GDC10. SSC properties are also derived by fitting published conductivity and oxygen non-stoichiometry measurements. Mixed conductivity of GDC10 and competing charge transfer reactions at both electrodes reduce open circuit voltages due to leakage current and buildup of defect concentrations at electrode-electrolyte interfaces. To fit polarization data, the button-cell model includes heterogeneous reaction rates for defect incorporation on the GDC10 surface along with Butler–Volmer expressions derived for competing charge transfer reaction rates from rigorous analyses assuming rate-limiting, elementary charge transfer reactions for each electrode. The calibrated MEA model can support rigorous SOFC modeling with GDC10 electrolytes over the range of conditions within a fully operating cell.

Electrochemistry↗

A multi-physics constitutive model to predict hydrolytic aging in quasi-static behaviour of thin cross-linked polymers

The effect of hydrolytic aging on mechanical quasi-static responses of rubber-like materials, in particular, the idealized Mullins effect and permanent set have been modeled. The effect of hydrolytic damage on the mechanical integrity of the polymer matrix is modeled as the direct competition of two micro-structural phenomena (i) chain scission and (ii) reduction of cross-links. Both phenomena and their correlation were modeled and thus, the strain energy of the polymer matrix is written with respect to three independent mechanisms; i) the shrinking original matrix that has not been attacked by water, ii) conversion of the first network to a new network due to the reduction of the cross-links, and iii) energy loss from network degradation due to water attacks to polymer active agents. The proposed model satisfies the Clausius-Duhem inequality and is thus physically feasible. The model is validated with respect to sets of our experimental data and other sets available in the literature. The proposed model is based on the assumption of homogeneous diffusion and mainly relevant for thin samples. In view of its accuracy, interpret-ability, and deep insight it provides into the nature of damage accumulation, the model is a good choice for further implementation in FE applications.

42 ENGINEERING↗

Estimating Vegetation Parameters from Interferometric and Polarimetric Radar Using Physical Scattering Models

Radar data from vegetated land surfaces depend on many structural and compositional parameters describing the terrain. Because early, noninterferometric radar systems usually constituted an insufficient observation set from which to estimate parameters of the terrain, statistical regression techniques were used which incorporated some level of apriori knowledge or field measurements. With the advent of radar interferometry and polarimetric interferometry, potentially at multiple baselines, the observation set is now approaching that required to quantitatively estimate the parameters describing a vegetated land surface. Quantitative estimation entails formulating a physical scattering model relating the radar observations to the vegetation and surface parameters on which they depend. This paper describes the physics of candidate scattering models, and shows how the models determine the estimable parameter set. It also indicates the measurement accuracy of parameters such as vegetation height, height-to-base-of-live-crown, and surface topography with multibaseline polarimetric interferometry.

Treuhaft, Robert N.↗

MPEX AI Digital Twins

All magnetically confined plasma fusion power plant concepts (Tokamak, Spherical Tokamak, Stellarator, Mirror, ...) must exhaust the heat and plasma from the core confinement region to the material walls. The primary channel for this exhaust is through a plasma divertor which directs plasma along open magnetic field lines to a material target. The Material Plasma Exposure eXperiment (MPEX) illustrated in Figure 1, is a high-power, steady-state linear plasma device designed to produce the plasma material interaction (PMI) conditions of the divertor of future magnetic confinement fusion power plants: energy flux 20MW/m 2 , ion fluence 1031/m 2 , pulse duration 106 sec. These goals of plasma exposure in MPEX are well beyond those achieved in magnetic fusion experimental devices. Successfully achieving these high power steady state conditions for long pulses requires operational control of the heating and particle sources and the plasma flux to the walls and target. The MPEX AI Hot Spot Controller, proposed in this project, will help achieve the operational milestones of MPEX. The MPEX device will begin commissioning at the end of FY26. A smaller proto-MPEX was operated for 14,666 plasma discharges and will resume operation in September of 2025 as proto-MPEX-lite, with reduced capability, to test a new window for the Helicon plasma source. The proto-MPEX data has undergone surrogate modeling with machine learning methods (R. Archibald, 2022 IEEE International Conference on Big Data). This proto-MPEX data will be used to begin development of the AI digital twins described in this white paper. The scientific mission of MPEX is to qualify materials of different composition for use in the high energy and plasma flux conditions of a fusion power plant. The materials exposed in MPEX will in some cases be exposed to high neutron fluxes at other ORNL facilities to measure the changes to their PMI properties. The targets exposed in MPEX will be transported under vacuum to a Surface Analysis Station (SAS). The SAS will be equipped with the following diagnostics: Focused Ion Beam (FIB) for trench milling, 100-400 angstrom resolution scanning electron microscope (SEM), surface mapping x-ray spectrometer, high resolution camera, and a future upgrade to a laser induced breakdown spectroscopy quadruple mass spectrometer (LIBS-QMS). The MPEX experiments will generate diverse pre- and post-exposure measurement data of detailed material properties down to the crystal grain level in 3D for post-exposure assessment of PMI damage (e.g. cracking, melting, erosion and redeposition of the material). Physics models for the PMI, and how the material composition and manufacturing impact its performance under high energy plasma exposure, need to be validated with MPEX data to guide the selection of new candidate materials. Our vision for the MPEX AI Digital Twins project is to supply experimental and physics model simulation data to train Artificial Intelligence (AI) models for data processing, analysis, operational control, PMI and materials simulation to maximize the scientific output of the MPEX device. Ultimately, an AI digital twin of MPEX material assessment metrics for tested and synthetic material types with simulated PMI will be trained by the AI Modeling Teams on the experimental and physics simulation data submitted to the American Science Cloud by this project. A purely empirical search for the best material is inefficient given the finite number of samples that can be tested on MPEX. In order to expand the material properties database for training the MPEX Material Assessment AI Digital Twin, and to gain physics understanding of the PMI processes, physics models of the material properties and PMI processes are required. The physics simulations provide detailed simulation data, like impact angles for plasma ions, sputtering yields, transport of the ionized sputtered target material in the plasma, and redeposition locations. This simulation data expands the measurement data for deeper physics understanding. The experimental data is essential to validate the PMI and material structure simulation models. The validated models can then be used to generate new simulation data of MPEX material assessments for synthetic material compositions that have not been exposed in MPEX. These predictive simulations, plus the whole experimental dataset, will be used to train the MPEX Material Assessment AI Digital Twin allowing a rapid generative AI search for new materials with reduced PMI damage by interpolating the domain of the training set. These new optimum materials can be simulated with the physics codes and/or tested in MPEX. The ability of AI neural networks to interpolate multi-dimensional parameter spaces and generate virtual data is exploited for a more efficient search for optimum materials. The advent of the Transformational AI Models Consortium (TAIMC) is an opportunity to engage with state of the art private and public AI developers to achieve the goals of the AI digital twins and AI accelerated physics models proposed in this project. Our partners at ORNL from the Advance Scientific Computing Research (ASCR) organization will collaborate in accelerating the integrated plasma material interaction simulation framework. This simulation framework will provide a platform for generating simulation data across a range of physical fidelities, including hybrid methods that produce multi-fidelity results. This data will be leveraged for AI model development, both for generation of surrogates and the automation of simulation campaigns. A part of the research below will include collaborative efforts with the TAIMC to (i) adapt data storage approaches to ensure AI-readiness, (ii) provide a protypical exemplar to inform and exercise constructed workflows, and (iii) generate and share data, using the TAIMC unified AI data standard, for foundational models that will be trained from multiple sources across the DOE complex. We will also collaborate with the TAIMC, as well as the planned AI modeling teams, to develop approaches for reducing the cost of data generation. These include tailored multi-fidelity approaches as well as fine-tuning strategies to augment general, large-scale foundational models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

The Challenge of Grounding Planning in Simulation with an Interactive Model Development Environment

A principal obstacle to fielding automated planning systems is the difficulty of modeling. Physical systems are modeled conventionally based on specification documents and the modeler's understanding of the system. Thus, the model is developed in a way that is disconnected from the system's actual behavior and is vulnerable to manual error. Another obstacle to fielding planners is testing and validation. For a space mission, generated plans must be validated often by translating them into command sequences that are run in a simulation testbed. Testing in this way is complex and onerous because of the large number of possible plans and states of the spacecraft. Though, if used as a source of domain knowledge, the simulator can ease validation. This paper poses a challenge: to ground planning models in the system physics represented by simulation. A proposed, interactive model development environment illustrates the integration of planning and simulation to meet the challenge. This integration reveals research paths for automated model construction and validation.

radio occultation↗

Physics-Informed Neural Network Solution of Point Kinetics Equations for a Nuclear Reactor Digital Twin

A digital twin (DT) for nuclear reactor monitoring can be implemented using either a differential equations-based physics model or a data-driven machine learning model. The challenge of a physics-model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, whereas the challenge of a data-driven DT consists of extensive training requirements and a potential lack of predictive ability. We investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. We develop a PINN model to solve the point kinetic equations (PKEs), which are time-dependent, stiff, nonlinear, ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring spatial dependence of the neutron flux. The PINN model solution of PKEs is developed to monitor the start-up transient of Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. For the test cases considered, the extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

A Review of Physical and Numerical Approaches for the Study of Gas Stirring in Ladle Metallurgy

This article presents a review of the research into gas stirring in ladle metallurgy carried out over the past few decades. Herein, the physical modeling experiments are divided into four major areas: (1) mixing and homogenization in the ladle; (2) gas bubble formation, transformation, and interactions in the plume zone; (3) inclusion behavior at the steel–slag interface and in the molten steel; and (4) open eye formation. Several industrial trials have also been carried out to optimize gas stirring and open eye formation. Approaches for selecting criteria for scaling to guarantee flow similarity between industrial trials and physical modeling experiments are discussed. To describe the bubble behavior and two-phase plume structure, four main mathematical models have been used in different research fields: (1) the quasi-single-phase model, (2) the volume of fluid (VOF) model, (3) the Eulerian multiphase (E–E) model, and (4) the Eulerian–Lagrangian (E–L) model. In recent years, the E–E model has been used to predict gas stirring conditions in the ladle, and specific models in commercial packages, as well as research codes, have been developed gradually to describe the complex physical and chemical phenomena. Furthermore, the coupling of turbulence models with multiphase models is also discussed. For physical modeling, some general empirical rules have not been analyzed sufficiently. Based on a comparison with the available experimental results, it is found that the mathematical models focusing on the mass transfer phenomenon and inclusion behaviors at the steel-slag interface, vacuum degassing at the gas–liquid interface, dissolution rate of the solid alloy at the liquid–solid interface, and the combination of fluid dynamics and thermodynamics need to be improved further. To describe industrial conditions using mathematical methods and improve numerical modeling, the results of physical modeling experiments and industrial trials must offer satisfactory validations for the improvement of numerical modeling.

36 MATERIALS SCIENCE↗

The Heliotail: Theory and Modeling

Physical processes are discussed related to the heliotail which is formed when the solar wind interacts with the local interstellar medium. Although astrotails are commonly observed, the heliotail observations are only indirect. As a consequence, the direct comparison of the observed astrophysical objects and the Sun is impossible. This requires proper theoretical understanding of the heliotail formation and evolution, and numerical simulations in sufficiently large computational boxes. In this paper, we review some previous results related to the heliotail flow and show new simulations which demonstrate that the solar wind collimation inside the Parker spiral field lines diverted by the heliopause toward the heliotail is unrealistic. On the contrary, solar cycle effects ensure that the solar wind density reaches its largest values near the solar equatorial plane. We also argue that a realistic heliotail should be very long to account for the observed anisotropy of 1-10 TeV cosmic rays.

79 ASTRONOMY AND ASTROPHYSICS↗

The NASA-AMES Research Center Stratospheric Aerosol Model. 1. Physical Processes and Computational Analogs

A time-dependent one-dimensional model of the stratospheric sulfate aerosol layer is presented. In constructing the model, a wide range of basic physical and chemical processes are incorporated in order to avoid predetermining or biasing the model predictions. The simulation, which extends from the surface to an altitude of 58 km, includes the troposphere as a source of gases and condensation nuclei and as a sink for aerosol droplets. The size distribution of aerosol particles is resolved into 25 categories with particle radii increasing geometrically from 0.01 to 2.56 microns such that particle volume doubles between categories.

Turco, R. P.↗

Super Resolving Unrolled Neural Networks for Remote Sensing

In remote sensing systems, the capabilities of the system are constrained by the complex interactions between size, weight, and power (SWAP) of potential designs. In electro-optical (EO) systems, examples of these critical parameters include the system’s sensitivity and resolution. Those parameters can be increased by ever larger optical apertures and focal planes but at the cost of more SWAP. Multi-image super resolution (MISR) techniques allow resolution to be enhanced via computation rather than more sophisticated optical hardware. These algorithms combine multiple images together into a single, higher resolution image, trading temporal resolution and computation for spatial resolution. Fielded MISR techniques, such as Drizzle, can require several hundred images to create a single super resolved image, implying reduced temporal resolution, increased data acquisition load, and limiting mission applications. Iterative techniques, such as model-based image reconstruction and compressive sensing, have been shown to create super resolved images using fewer images than Drizzle. They do this by posing an optimization problem that balances accuracy between a highly accurate physical model and an image model. In the case of super resolution, the physical model is defined by the relation between low resolution input images and the desired high resolution output image. The image model encodes some assumptions about the super resolved image. These assumptions are meant to suppress reconstruction artifacts that arise due to deterministic physical model error, stochastic measurement noise, and potential undersampling. In practice, the performance of iterative methods are limited by imaging models compatible with optimization. Deep learning-based methods can effectively learn image models of arbitrary complexity, but lack the theoretical explainability and robustness of iterative techniques. Consensus equilibrium (CE) generalizes the iterative techniques beyond optimization, enabling blackbox algorithms such as traditional and neural image denoisers to be used as the image model. CE-based approaches retain much of the explainability and robustness of iterative techniques while allowing the expressiveness of machine learning image models to be used. Additionally, by unrolling iterations of CE with an embedded image denoiser, the image denoiser can be further trained and specialized to the specific application with potentially higher quality reconstructions. Under this project, we demonstrated the feasibility of training an unrolled neural network based upon CE. While we didn’t train one, we showed that the CE process is differentiable and its gradient can be tractably computed. We also explored the usage of a variants of CE akin to generative neural works. Most importantly, we applied the CE framework to a number of problems including non-blind deconvolution, upsampling, single-image super resolution, MISR, event-based sensing, and saturated deconvolution. Our MISR prototype creates high quality reconstructions with an order of magnitude fewer images than previous approaches and, critically, produces these reconstructions fast enough for practical usage.

47 OTHER INSTRUMENTATION↗

Π4U: A high performance computing framework for Bayesian uncertainty quantification of complex models

We present Π4U,{sup 1} an extensible framework, for non-intrusive Bayesian Uncertainty Quantification and Propagation (UQ+P) of complex and computationally demanding physical models, that can exploit massively parallel computer architectures. The framework incorporates Laplace asymptotic approximations as well as stochastic algorithms, along with distributed numerical differentiation and task-based parallelism for heterogeneous clusters. Sampling is based on the Transitional Markov Chain Monte Carlo (TMCMC) algorithm and its variants. The optimization tasks associated with the asymptotic approximations are treated via the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). A modified subset simulation method is used for posterior reliability measurements of rare events. The framework accommodates scheduling of multiple physical model evaluations based on an adaptive load balancing library and shows excellent scalability. In addition to the software framework, we also provide guidelines as to the applicability and efficiency of Bayesian tools when applied to computationally demanding physical models. Theoretical and computational developments are demonstrated with applications drawn from molecular dynamics, structural dynamics and granular flow.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗