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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 487 records · Page 27
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Landmark-Warped Emulators for Models with Misaligned Functional Response

Many computer models output functional data, and in some cases, these functional data have similar, but misaligned, shape characteristics. In this paper, we introduce a general approach for building emulators for computer models that output misaligned functional data when key values in the functional response (landmarks) can be easily identified. This approach has two main parts: modeling the aligned (using the landmarks) functional data, and modeling the functions that map the misaligned data to the aligned space (warping functions). As the warping functions are required to be monotonic, we give special attention to modeling monotonic functional response data. We discuss how our approach can be easily applied for a variety of typical emulators, such as Gaussian processes, Bayesian multivariate adaptive regression splines, and Bayesian additive regression trees, and how sensitivity analysis can be performed. We demonstrate our approach by building emulators for two applications: (1) a high-energy-density physics computer model used to simulate inertial confinement fusion ignition experiments, where model outputs are highly misaligned, and (2) a multiphysics continuum hydrocode used to simulate high-velocity impact experiments, where model outputs are only slightly misaligned. In case (1) traditional methods cannot be applied, while in (2) they can be applied, but the proposed method performs significantly better.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2022

Validation of a High-Fidelity Supersonic Parachute Inflation Dynamics Model and Best Practice

The parachute inflation dynamics (PID) of a Disk-Gap-Band (DGB) parachute system travelling at supersonic velocities in the high-altitude Earth atmosphere is simulated using a high-fidelity, multiphysics, massively-parallel computational framework. The sensitivity of the simulation results with respect to critical modelling and simulation parameters, including the fluid mechanical governing equations (Euler vs Navier-Stokes with RANS turbulence closure vs Navier-Stokes with LES turbulence closer), the interaction between the flow and the suspension lines (accounted for vs not accounted for), the dynamics of the payload (restrained vs free), and the material model of the heterogeneous fabric canopy (Saint-Venant Kirchoff vs mechanics-informed multiscale neural network), is extensively investigated. The parameter investigation is enabled by advancements to the state-of-the-art in the modelling and simulation of the supersonic parachute inflation process, which are presented as part of the computational framework. The predictive quality of the computational framework in the context of supersonic PID is validated by comparison of the simulation results with the experimental results from NASA’s Advanced Supersonic Parachute Research (ASPIRE) project. Through the validation, best practices for the modelling and simulation of supersonic PID are developed and proposed.

Rabinovitch, Jason↗
DOE OSTI2023

Verification methods for drift–diffusion reaction models for plasma simulations

Abstract Compared to other computational physics areas such as codes for general computational fluid dynamics, the documentation of verification methods for plasma fluid codes remains under developed. Current analytical solutions for plasma are often highly limited in terms of testing highly coupled physics, due to the harsh assumptions needed to derive even simple plasma equations. This work highlights these limitations, suggesting the method of manufactured solutions (MMSs) as a potential option for future verification efforts. To demonstrate the flexibility of MMS in verifying these highly coupled systems, the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework was utilized. Thanks to the MOOSE framework’s robustness and modularity, as well as to its physics module capabilities and ecosystem applications (i.e. Zapdos and the chemical reaction network) developed for plasma physics modeling and simulation, this report lays the groundwork for a structured method of conducting plasma fluid code verification.

DeChant, Corey (ORCID:0000000293779732)↗
DOE OSTI2024

Sensitivity Analysis of Irradiated Fueled Experiments using the MOOSE Framework [Slides]

Modeling and simulation (M&S) methods are able to predict uncertainties in experimental parameters (e.g., power and fission density) during irradiation. A shortfall exists in predicting how sensitive some of the parameters will behave during the experimental process. Sensitivity and Uncertainty Quantification (SUQ) is critical in support of qualification and licensing reactor fuels. The application of a method to quantify the uncertainty in these experiments is critical to the prediction of their performance. In this work, we propose the use of a polynomial chaos expansion (PCE) method to quantify the sensitive parameters in these simulations and, in an extension, their experimental surrogates. We propose to perform M&S using PCE uncertainty quantification on a previously irradiated fueled experiment in order to provide a validation case for Griffin and expand its use as a verification and validation (V&V) tool for experiments with a neutronics component. Griffin is an advanced, deterministic neutronics analysis code built using the MOOSE (multiphysics object-oriented simulation environment) framework which can provide state-of-the-art neutronic analysis on M&S of experiments. We will use the stochastic tools module (STM) in MOOSE to provide PCE uncertainty quantification on the proposed experimental setup. Idaho National Laboratory (INL) does not yet have an in-house developed code with V&V approval for experiments performed on-site; this work would provide a necessary addition of support for experiments performed at INL. The Nuclear Regulatory Commission (NRC) has explicitly requested uncertainties in calculated values such as fuel power and burnup, and the development of this capability would benefit the relationship between INL and the NRC.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2026

Modeling Flow in a Microreactor Core: From High-Fidelity CFD to Subchannel Analysis

The accurate prediction of pressure drop in tightly packed, low–Reynolds number (Re) bare rod bundles is essential for the thermal-hydraulic design of the Microreactor Applications Research Validation and EvaLuation (MARVEL) reactor and other microreactor concepts. However, existing friction factor correlations, particularly the upgraded Cheng-Todreas (UCTD) correlation, have limited validation for the small pitch-to-diameter ratios (P/Ds) and transitional flow conditions characteristic of these systems. In this work, we perform high-fidelity large-eddy simulations (LESs) of both an infinite bare rod bundle and a finite bare 37-pin scalloped rod bundle across the range 1000≤ 𝑅𝑒 ≤5500. The simulations reveal strong gap vortex–driven transitional behavior and indicate that the UCTD may underpredict the friction factor by up to 28% at 𝑃/𝐷 = 1.05. Using the LES-calculated pressure drops, we formulate a new friction factor correlation that follows the Cheng-Todreas functional form but is calibrated for low-Re and tightly packed geometries representative of a MARVEL-like reactor. The correlation is implemented in the MOOSE (Multiphysics Object-Oriented Simulation Environment) subchannel module and compared against both the LES and UCTD predictions. Across all subchannel types, the proposed model reduces the streamwise velocity differences from as high as 44% (UCTD) to below 9%, and decreases the pressure gradient differences from 13% to 25% (UCTD) to 0.7% to 7% relative to the LES results. These results suggest that the new correlation has the potential to improve the pressure drop and flow field predictions for such geometries, highlighting the importance of high-fidelity simulations in supporting microreactor thermal-hydraulic model development and motivating future pressure drop experiments for compact rod bundles to further validate these findings.

21 - SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLAN↗
DOE OSTI2023

Enabling scientific machine learning in MOOSE using Libtorch

A neural-network-based machine learning interface has been developed for the Multiphysics Object-Oriented Simulation Environment (MOOSE). The interface relies on Libtorch, the C++ front-end of PyTorch, and enables an online interaction between modern machine learning algorithms and all the existing simulation, modeling, and analysis processes available in MOOSE. New capabilities in MOOSE include the native generation and training of artificial neural networks together with options to load pretrained neural networks in TorchScript format. Furthermore, the MOOSE stochastic tools module (MOOSE-STM) has been enhanced with neural network-based surrogate and reduced-order model generation options for efficient stochastic analyses. Lastly, a reinforcement learning capability has been added to MOOSE-STM for the interactive control and optimization of complex multiphysics problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

The Monolithic Heat Pipe Microreactor Reference Plant Model

This work introduces a reference plant model for a generic monolithic heat-pipe-cooled microreactor. The model will serve as a springboard to develop future evaluation models in the licensing process of similar microreactor designs at the U.S. Nuclear Regulatory Commission. This model has been developed with the Comprehensive Reactor Analysis Bundle and its specifications are based on open literature publications for the eVinci TM design. BlueCRAB is the U.S. Nu- clear Regulatory Commission non-light-water reactor analysis system based on the Multiphysics Object-Oriented Simulation Environment framework, which can couple the Griffin, BISON, and Sockeye applications to resolve the various physics that are essential for the safety analysis of this type of reactor system. The core specifications includes tristructural isotropic fuel, graphite monolith, graphite reflectors, and drums composed of graphite and B 4 C. No moderator or burnable poison pins are used in the design. The fuel enrichment is reduced to control excess reactivity in the core. This core design is not optimized and only serves for testing purposes, since the primary objective of this work is to exercise the multiphysics coupling for this type of reactor system. A three dimensional (3D) core heterogeneous Griffin discrete ordinates (SN) transport model allows the precise calculation of the flux distribution and pin powers. Griffin transfers the power density distribution and obtains a temperature distribution to and from BISON. The BISON model com- putes the 3D core temperature distribution and is coupled to 876 Sockeye subapplications running a heat pipe model. This 3D conduction model is coupled to the various heat pipes via heat flux boundary conditions. The model includes a small gap between the heat pipe and the monolith. Convective heat transfer boundaries with either ambient temperature or condenser temperature as heat sinks are imposed at the model boundaries. The 2D Sockeye heat pipe model uses a vapor- only methodology, which provides the needed resolution for transient calculations and allows the determination of various heat pipe limits. This approach is superior to the superconductor model traditionally used in steady-state calculations. BlueCRAB computes steady-state power and temperature distributions that serve as the initial condition for a loss-of-heat-sink transient simulation. The steady-state results show significant peaking due to the position of the control drum, but this is a characteristic of the particular design used, which is not optimized at this stage. The transient results show the reactor power slowly stabilizing towards a 3% power level after the partial loss of secondary heat removal. Several recriticalities are observed due to cooling through the secondary system but the reactor is self-stabilizing and behaves as expected.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2024

Challenging Common Assumptions of Thick-Wall Chamber Dynamics in Inertial Fusion Systems using MOOSE

As an increasing number of companies look toward commercial Inertial Fusion Energy (IFE) designs, there is a pressing need to understand the physics of thick-wall chamber gas dynamics. The thick liquid wall approach implements a renewable wall to mitigate the fusion target emissions, thereby reducing the radiation damage rate and significantly extending the lifetime of chamber structures, leading to increased plant availability and reduced waste streams in comparison to dry wall chamber designs. It is necessary, however, to assess the critical performance and safety aspects of these systems. For example, it is crucial to predict (1) where the mass ablated from the liquid walls will vent, which determines the placement of condensing surfaces; (2) debris propagation up the beam lines, which provides essential information for design and protection requirements; (3) peak pressures and impulse on chamber walls, which affect chamber structural design; and (4) momentum transfer to the liquid jets, which constrains the shape and positioning of the jets. In turn, the chamber design and its liquid walls affect shielding requirements, material activation, and tritium fuel cycle. Currently available simulation tools, however, are unable to accurately capture key thick-wall chamber dynamics. Significant assumptions are often made to simplify the system and reduce computational cost and modeling capability needs, but the impact of these assumptions on simulation predictions has not been evaluated. For example, no three-dimensional simulations can be found in the open literature to evaluate gas venting and momentum transfer to the jets with simulations using two-dimensional domains to represent complex three-dimensional geometries. Moreover, limited studies have been dedicated to jet breakup due to both turbulence and neutron heating, and no studies have been found that evaluate how jet breakup can impact shock-jet interaction. Furthermore, effects of radiative heat transfer have rarely been included for the hydrodynamic phase of shock propagation, and integration of proper equations of state in shock dynamics codes has been mostly exploratory. In this study, we use the flexible, high-fidelity Multiphysics Object-Oriented Simulation Environment (MOOSE) to model these complex phenomena and inform design and safety studies. Capabilities to model thick-wall chamber gas dynamics are being developed, and the impact of the assumptions listed above (i.e., two-dimensional vs three-dimensional, absence of jet breakout, no radiative heat transfer, and ideal gas behavior) are being quantified.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Drive-pressure optimization in ramp-wave compression experiments through differential evolution

Ramp-wave dynamic-compression experiments are used to examine quasi-isentropic loading paths in materials. The gradual and continuous increase in pressure created by ramp waves make these types of experiments ideal for studying nonequilibrium material behavior, such as solidification kinetics. In ramp-wave compression experiments, the input drive pressure to the experimental setup may be exerted through one of a number of different mechanisms (e.g., magnetic fields, gas-gun-driven impactors, or high-energy lasers) and is generally required for simulating such experiments. Yet, regardless of the specific mechanism, this drive pressure cannot be measured directly (measurements are generally taken at a location near the back of the experimental setup through a transparent window), leading to an inverse problem where one must determine the drive pressure at the front of the experimental setup (i.e., the input) that corresponds to the particle velocity (the output) measured near the back of the experimental setup. Furthermore, we solve this inverse problem using a heuristic optimization algorithm, known as differential evolution, coupled with a multiphysics, hydrodynamics code that simulates the compression of the experimental setup. By running many rounds of forward simulations of the experimental setup, our optimization process iteratively searches for a drive pressure that is optimized to closely reproduce the experimentally measured particle velocity near the back of the experimental setup. While our optimization methodology requires a significant number of hydrodynamics simulations to be conducted, many of these can be performed in parallel, which greatly reduces the time cost of our methodology. One novel aspect of our method for determining the drive pressure is that it does not require physical modeling of the drive mechanism and can thus be broadly applied to many types of ramp-compression experiments, regardless of the drive mechanism.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

FY26 Progress on Demonstration of a Multiphysics Steady State Capability for Modeling Core Radial Expansion in SFRs

Under the U.S. Department of Energy Office of Nuclear Energy Advanced Modeling and Simulation (NEAMS) Program, an integrated multiphysics approach is being developed to model the core bowing phenomena important to liquid metal-cooled fast reactors. Core bowing is an important passive safety mechanism in liquid metal-cooled fast reactors and involves multiphysics effects including radiation transport, fluid flow, heat transfer, and mechanical response to temperature and flux gradients. This report summarizes recent progress on developing a multiphysics, MOOSE-based workflow to predict core bowing and associated reactivity feedback. Significant new capabilities in the reactor physics code Griffin - sodium backfill and pin power reconstruction for deformed geometries - were applied in this effort. This year’s work included verification, code comparisons, sensitivity studies, and coupled demonstrations that advance the state of MOOSE-based core bowing workflow. Griffin’s sodium backfill capability was verified by demonstrating that its automated treatment of geometry expansion and material-density updates reproduces manual calculations exactly, confirming solid mass conservation and proper coolant backfilling in expanded geometries. Reconstructed pin powers were compared for Griffin’s ductheterogeneous and ring-heterogeneous treatments in single-, seven-, and nineteen-assembly cases, with best agreement observed in lower-leakage configurations and the duct-heterogeneous approach offering substantially lower computational cost. Thermal-hydraulic sensitivity sensitivities showed that MOOSE SCM, SAM, and CFD are expected to produce similar deformation predictions despite variances in their temperature predictions, and that explicit treatment of inter-assembly flow becomes increasingly important as gap flow rate increases. Finally, coupled demonstrations on small multi-assembly configurations using Griffin, MOOSE Solid Mechanics, MOOSE SCM, and Heat Conduction produced physically consistent reactivity feedback from thermal expansion and bowing. The coupled demonstrations simulated grid plate expansion as well as resultant core bowing at full power conditions. Simplifications were made in current workflow, namely the assumption of instantaneous full power conditions following hot zero power, and pre-expanding the Griffin geometry axially due to lack of an axial fuel pin expansion model and temperature feedback to Griffin.

Wozniak, Nicholas↗
DOE OSTI2023

On Practical Aspects of Variational Consistency in Contact Dynamics

Usage of contact mechanics methodologies is a pervasive modeling requirement in dynamic simulations. While for some trivial problems, solutions taken from analytical geometry are available, use of a finite element framework is common to achieve formulation generality. This work explores two dynamic contact formulations: one based on the traditional node-to-segment (NTS) approach, and a variationally consistent segment-to-segment (STS) mortar formulation. The NTS formulation employed here enforces the constraints kinematically (i.e., the interpenetration is enforced to the solver tolerance), whereas the mortar approach uses Lagrange multipliers to enforce the contact constraints. Both approaches are implemented in the open-source finite element framework Multiphysics Object-Oriented Simulation Environment (MOOSE). The results highlight two relevant contact-interface-related dynamic phenomena in finite element simulations. First, stabilization of contact constraints is discussed, taking into account the evolution of the total energy in a benchmark problem. Second, the influence of finite element discretization on both of the aforementioned contact formulations is analyzed by exercising a large-deformation example with continuous relative sliding. Variationally consistent contact approaches such as the mortar formulation lead to improved energy preservation and avoid spurious excitation of the system's frequencies. This is especially relevant in settings where inertia and vibrations are of importance.

42 ENGINEERING↗
DOE OSTI2024

Comparison of excess free energy at an interface according to the applied interpolation scheme for elasticity: A phase-field method

Phase-field modeling is an effective simulation technique for modeling microstructure evolution of elastically anisotropic systems. To introduce the elastic energy contribution in a phase field model, an interpolation scheme is used to define the mechanical properties within the phases and across the continuous interface. Several existing interpolation schemes introduce a potential excess elastic energy at the interface, which undesirable effect on microstructure evolution needs to be evaluated. In this study, we focused on three interpolation schemes including Khachaturyan’ scheme (KHS), Voigt–Taylor’s scheme (VTS), and Steinbach–Apel’s scheme (SAS). Comparisons of these schemes’ performances were performed in three configuration types using the MOOSE (Multiphysics Object-Oriented Simulation Environment) framework: bi-crystal, isotropic particle-matrix and anisotropic particle-matrix. The contribution of excess elastic energy on the interface energy as a function of interface width and the computational time to steady-state were evaluated in these three configurations. SAS introduces the lowest excess elastic energy contribution and the VTS has the biggest contribution amongst the considered schemes. Moreover, when modeling precipitation in an anisotropic elastic material, the SAS approach seems to predict more physical convex shapes during growth, making it preferable to KHS and VTS. Finally, as currently implemented, SAS requires the largest computational time and KHS requires the smallest time to reach steady-state amongst the considered schemes.

36 MATERIALS SCIENCE↗
DOE OSTI2024

The MOOSE Thermal Hydraulics Module

The Multiphysics Object-Oriented Simulation Environment (MOOSE) is an open-source object-oriented finite element framework written in C++ (Lindsay et al., 2022). The Thermal Hydraulics Module (THM) is an optional MOOSE physics module that provides capabilities for studying thermal hydraulic systems. Its core capability lies in assembling a network of coupled components, for instance, pipes, junctions, and valves. THM provides several new systems to MOOSE to enable and facilitate thermal hydraulic simulations, most notably the Components system, which provides a higher-level syntax to MOOSE’s lower-level objects. This system is extensible by the user, but the current library primarily includes components based on a one-dimensional, single-phase, variable-area, compressible flow model, as well as heat conduction.

42 ENGINEERING↗
DOE OSTI2023

NEAMS Technical Area Support in MOOSE

The Multiphysics Object-Oriented Simulation Environment (MOOSE) framework is a foundational capability used by the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program to create over 15 different simulation tools for advanced nuclear reactors. Due to this ubiquity, improvements to the framework in support of modeling and simulation goals are critical to the program. These improvements can take many forms, including optimization, improved user experience, streamlined application programming interfaces (APIs), parallelism, and new capabilities. The work transcribed in this report was conducted in direct support of the simulation tools and has already been deployed. The capabilities outlined in this report include addition of Times and Positions systems, redesign of mechanical contact constraints to enable the augmented Lagrange algorithm, overhaul of the restart system, and incorporation of p-refinement in MOOSE.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

BISON Fuel Fragmentation Relocation and Dispersal (FFRD) Assessment Database for Eventual Use in Bayesian Calibration

Existing light-water reactor (LWR) fuel vendors have been interested in seeking increased discharge burnups of nuclear fuel rods for improved economics for quite a few years. With increased burnups come additional challenges that must be addressed. It has been experimentally observed that average burnups higher than the current regulatory limit of 62 MWd/kgU may undergo fuel fragmentation, relocation, and dispersal (FFRD) during a loss-of-coolant accident (LOCA). Industry must demonstrate approaches to mitigate FFRD. In an effort to support industry, the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program within the U.S. Department of Energy (DOE) has for several years invested in developing multiscale models and creating a validation/assessment database for these models to study the mechanisms driving FFRD. This report provides an update on changes made to the assessment database and new models added to BISON to support the study of fuel rod behavior during FFRD. An effort has been initiated this year to begin adding dedicated inputs to the publicly available Virtual Test Bed (VTB) repository for industry use. A section of this report details the efforts made in this area. NEAMS has recently developed new capabilities in the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework’s stochastic tools module for calibration using Bayesian inference. The goal in the future is to use these capabilities to calibrate and identify weaknesses in the existing BISON models for FFRD. The report concludes with a discussion on the models most likely to benefit the most from such calibration.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

NEAMS Workbench MOOSE Integration Update

The Nuclear Energy Advanced Modeling and Simulation (NEAMS) Workbench is a graphical user interface (GUI) that provides a common analysis environment for the accelerated use of the NEAMS toolkit. To improve design and analysis of current and future nuclear energy systems, the NEAMS Workbench provides an integrated development environment for model creation, review, execution, output review, and visualization for integrated tools. In addition to a GUI, the NEAMS Workbench provides the Workbench Analysis Sequence Processor, an open-source tool set that facilitates input-content assistance for supported domain-specific input languages and user-friendly syntaxes. These supported syntaxes enable accelerated development and deployment of enhanced user inputs and workflows. The Multiphysics Object-Oriented Simulation Environment (MOOSE) has been supported in the NEAMS Workbench since 2016 and receives continuous updates. Efforts to improve MOOSE-based tool integration in the NEAMS Workbench are ongoing. This document details the current and planned enhancements of the NEAMS Workbench and the MOOSE framework to improve integration and usability of MOOSE-based applications within the NEAMS Workbench.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

A Multicomponent Reactive Transport Model for Integrated Surface‐Subsurface Hydrology Problems

Abstract Despite the widespread use of integrated hydrology models in a variety of applications, consideration of multicomponent reactive transport is still not common. The implementation of these processes requires coupling transport at the surface‐subsurface interface and efficient solution of the non‐linear geochemical model that is consistent with the integrated hydrology solution. The Advanced Terrestrial Simulator provides a flexible multiphysics framework that facilitated this process. In this work, the integrated reactive transport process kernel (PK) was weakly coupled to the integrated hydrology PK. In turn, integrated transport and reactions were coupled using an operator splitting approach. This splitting enabled an explicit solution of the integrated transport problem, including a novel algorithm to calculate exchange fluxes across the surface‐subsurface interface and a point‐by‐point solution of the geochemical problem. Geochemical capabilities were added using well‐established external codes, but rather than using a custom interface to each, a generic interface was used that clearly specifies the variables and operations used by the chemistry PK. The implementation is demonstrated with two example simulations: transport of a tracer in a soil column as it saturates over time and water ponds on the surface and reactive transport in a hillslope driven by successive wet‐dry cycles that result in infiltration, runoff and exfiltration processes.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Useability and Optimization Improvements in MOOSE

The Multiphysics Object-Oriented Simulation Environment (MOOSE) framework is a foundational capability used by the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program to create over 15 different simulation tools for advanced nuclear reactors. Due to this broad use, improvements to the framework in support of modeling and simulation goals are critical to the program. These improvements can take many forms, including optimization, improved user experience, streamlined application programming interfaces (APIs), parallelism, and new capabilities. The work transcribed in this report was conducted in direct support of the simulation tools and has already been deployed or will be deployed in the coming months. The capabilities were implemented in the same order as they are covered in this report: increased support of face variables, arbitrary spatial and temporal evaluation of material properties, and the addition of a triangular meshing library in libMesh.

97 MATHEMATICS AND COMPUTING↗