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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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Final Technical Report for the SciDAC Center for the Study of Plasma Microturbulence

This final technical report for the SciDAC Center for the Study of Plasma Microturbulence documents contributions from MIT to the project over the period of the award (February 15, 2011 to February 14, 2020). This project focused on the development of neoclassical and gyrokinetic simulation capabilities relevant for the description of particle, momentum and energy transport in tokamak fusion plasmas. Extensive validation against experimental results from Alcator C-Mod and DIII-D experiments was performed. Overall areas of emphasis included gyrokinetic simulations, experiments, and observation of density gradient driven Trapped Electron Mode (TEM) turbulence relevant to burning plasmas; theory and gyrokinetic simulations of the parametric variation of the nonlinear upshift of the TEM critical density gradient; identification of a new mechanism for the isotope effect due to density gradient driven TEM turbulence; new synthetic diagnostics based on full-wave simulations which enabled unprecedented agreement between gyrokinetic simulations and measured fluctuation spectra with and without strong electron heating; development of new continuum pedestal neoclassical and turbulence codes; formulation and implementation of the first gyrokinetic exact Fokker-Planck collision operator in gyrokinetic turbulence simulations; development of new spectral approaches in velocity space for highly efficient treatments of collisions; new model collision operators implemented in the discontinuous Galerkin gyrokinetic edge turbulence code GKEYLL; and a new model and spectral code to study the interaction of ion- and electron-scale turbulence, which will serve as an efficient test-bed for multi-rate and multi-scale methods. These fundamental advances contribute to the development of a validated, predictive capability for turbulent and collisional particle, momentum and energy transport in both the core and edge of magnetically confined fusion plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Nonintrusive projection-based reduced order modeling using stable learned differential operators

Nonintrusive projection-based reduced order models (ROMs) are essential for dynamics prediction in multi-query applications where underlying governing equations are known but the access to the source of the underlying full order model (FOM) is unavailable; that is, FOM is a glass-box. This article proposes a learn-then-project approach for nonintrusive model reduction. In the first step of this approach, high-dimensional stable sparse learned differential operators (S-LDOs) are determined using the generated data. In the second step, the ordinary differential equations, comprising these S-LDOs, are used with suitable dimensionality reduction and low-dimensional subspace projection methods to provide equations for the evolution of reduced states. This approach allows easy integration into the existing intrusive ROM framework to enable nonintrusive model reduction while allowing the use of Petrov–Galerkin projections. The applicability of the proposed approach is demonstrated for Galerkin and LSPG projection-based ROMs through four numerical experiments: 1-D scalar advection, 1-D Burgers, 2-D scalar advection and 1-D scalar advection–diffusion–reaction equations. In conclusion, the results indicate that the proposed nonintrusive ROM strategy provides accurate and stable dynamics prediction.

42 ENGINEERING↗
DOE OSTI2020

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

Here, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ ≥ μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

42 ENGINEERING↗
DOE OSTI2021

Solution Irregularity Remediation for Spatial Discretization Error Estimation for S N Transport Solutions

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. Here, a metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A Novel Partitioned Approach for Reduced Order Model—Finite Element Model (ROM-FEM) and ROM-ROM Coupling

Partitioned methods allow one to build a simulation capability for coupled problems by reusing existing single-component codes. In so doing, partitioned methods can shorten code development and validation times for multiphysics and multiscale applications. In this work, we consider a scenario in which one or more of the “codes” being coupled are projection-based reduced order models (ROMs), introduced to lower the computational cost associated with a particular component. We simulate this scenario by considering a model interface problem that is discretized independently on two non-overlapping subdomains. Here we then formulate a partitioned scheme for this problem that allows the coupling between a ROM “code” for one of the subdomains with a finite element model (FEM) or ROM “code” for the other subdomain. The ROM “codes” are constructed by performing proper orthogonal decomposition (POD) on a snapshot ensemble to obtain a low-dimensional reduced order basis, followed by a Galerkin projection onto this basis. The ROM and/or FEM “codes” on each subdomain are then coupled using a Lagrange multiplier representing the interface flux. To partition the resulting monolithic problem, we first eliminate the flux through a dual Schur complement. Application of an explicit time integration scheme to the transformed monolithic problem decouples the subdomain equations, allowing their independent solution for the next time step. We show numerical results that demonstrate the proposed method’s efficacy in achieving both ROM-FEM and ROM-ROM coupling.

de Castro, Amy↗
DOE OSTI2025

Asymptotic-preserving dynamical low-rank method for the stiff nonlinear Boltzmann equation

In kinetic theory, numerically solving the full Boltzmann equation is extremely expensive. This is because the Boltzmann collision operator involves a high-dimensional, nonlinear integral that must be evaluated at each spatial grid point and every time step. The challenge becomes even more pronounced in the fluid (strong collisionality) regime, where the collision operator exhibits strong stiffness, causing explicit time integrators to impose severe stability restrictions. In this paper, we propose addressing this problem through a dynamical low-rank (DLR) approximation. The resulting algorithm requires evaluating the Boltzmann collision operator only r 2 times, where r, the rank of the approximation, is much smaller than the number of spatial grid points. We propose a novel DLR integrator, called the XL integrator, which reduces the number of steps compared to the available alternatives (such as the projector splitting or basis update & Galerkin (BUG) integrator). For a class of problems including the Boltzmann collision operator which enjoys a separation property between physical and velocity space, we further propose a specialized version of the XL integrator, called the sXL integrator. This version requires solving only one differential equation to update the low-rank factors. Furthermore, the proposed low-rank schemes are asymptotic-preserving, meaning they can capture the asymptotic fluid limit in the case of strong collisionality. Our numerical experiments demonstrate the efficiency and accuracy of the proposed methods across a wide range of regimes, from non-stiff (kinetic) to stiff (fluid).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

Abstract Fractured porous media challenge modeling approaches due to high computational costs and excessive mesh refinement imposed by the extreme scale variability of fractures and the heterogeneity of the surrounding porous rock. To overcome such difficulties, we utilize the hierarchical finite element method ( Hi ‐FEM) that has been developed previously to simulate the electrical potential distribution in complex geologic environments. The method employs the hierarchical basis functions in classical finite element analysis to enable representation of material properties on each dimensional component of a given 3D unstructured finite element, thereby inherently allowing for interactions at the boundary between fracture and a host rock. In this study, we extend its application to transient fluid flow and heat conduction in the Laplace domain. Time‐domain flow solutions are obtained by numerical inverse Laplace transform. We evaluate the accuracy of the method using different flow models and demonstrate its robustness for large‐scale, rock mass models featuring complex fracture networks. Moreover, for the computation of nodal Darcian velocity fields in fractured porous media where the fractures are represented as 2D features, a new approach that employs the Yeh's Galerkin model for both volume and facet elements is proposed. Results show that Hi ‐FEM can produce accurate flow solutions for fractured porous media without any need of coupling or transfer mechanism while still being computationally economical and numerically robust, even for large‐scale simulations.

Beskardes, G. D.↗
DOE OSTI2021

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multigrid strategies. For both approaches, we start from an integral formulation of the continuous time dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multi-grid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies are compared for varying orders of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences in execution times

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Interface Problem Formulation Improvements with Application to Nuclear Fuel Performance Analysis

The U.S. Department of Energy’s Nuclear Energy Advanced Modeling and Simulation Program aims to develop predictive capabilities by applying computational methods to the analysis and design of advanced reactor and fuel cycle systems. This program has been providing engineering scale support for the development of BISON, a high-fidelity and high-resolution fuel performance tool. This report documents new developments and robustness improvements in mechanical and thermal (gap heat transfer) contact formulations. The improvements range from the migration of industrial level (“assessment”) nuclear fuel model setups to the usage of mortar formulations, the addition of frictional contact to one-dimensional layered representations of fuel and cladding components, and the addition of the Petrov-Galerkin approach to dual mortar, which improves performance on curved, relatively coarse meshes. In addition, the Lagrange-multiplier enforcement of mechanical mortar contact constraints has been extended to two additional types of enforcement: penalty and augmented Lagrange-Uzawa. We show that the latter approach yields the same interface results as dual mortar in the Multiphysics Object-Oriented Simulation Environment, with the advantage of not worsening the condition number of the system matrix—thereby enabling the use of some general implementations of iterative preconditioners, at the expense of additional system evaluations (i.e., augmentations).

42 ENGINEERING↗
DOE OSTI2020

Time-stepping DPG formulations for the heat equation

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for computations and software implementations. The sole examination of time-stepping DPG formulations was performed by Führer, Heuer, and Gupta, who applied Rothe’s method to an ultraweak formulation of the heat equation to develop an implicit time-stepping scheme; their work emphasized theoretical results, including error estimates in time and space. Here, we follow Führer, Heuer, and Gupta in examining the heat equation; our focus is on numerical experiments, examining the stability and accuracy of several formulations, including primal as well as ultraweak, and explicit as well as implicit and Crank–Nicolson time-stepping schemes. We are additionally interested in communication-avoiding algorithms, and we therefore include a highly experimental formulation that places all the trace terms on the right-hand side of the equation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Model fusion with physics-guided machine learning: Projection-based reduced-order modeling

The unprecedented amount of data generated from experiments, field observations, and large-scale numerical simulations at a wide range of spatiotemporal scales has enabled the rapid advancement of data-driven and especially deep learning models in the field of fluid mechanics. Although these methods are proven successful for many applications, there is a grand challenge of improving their generalizability. This is particularly essential when data-driven models are employed within outer-loop applications like optimization. In this work, we put forth a physics-guided machine learning (PGML) framework that leverages the interpretable physics-based model with a deep learning model. Leveraging a concatenated neural network design from multi-modal data sources, the PGML framework is capable of enhancing the generalizability of data-driven models and effectively protects against or inform about the inaccurate predictions resulting from extrapolation. We apply the PGML framework as a novel model fusion approach combining the physics-based Galerkin projection model and long- to short-term memory (LSTM) network for parametric model order reduction of fluid flows. We demonstrate the improved generalizability of the PGML framework against a purely data-driven approach through the injection of physics features into intermediate LSTM layers. Our quantitative analysis shows that the overall model uncertainty can be reduced through the PGML approach, especially for test data coming from a distribution different than the training data. Moreover, we demonstrate that our approach can be used as an inverse diagnostic tool providing a confidence score associated with models and observations. The proposed framework also allows for multi-fidelity computing by making use of low-fidelity models in the online deployment of quantified data-driven models.

42 ENGINEERING↗
DOE OSTI2022

Fluid-Kinetic Coupling: Advanced Discretizations for Simulations on Emerging Heterogeneous Architectures (LDRD FY20-0643)

Plasma physics simulations are vital for a host of Sandia mission concerns, for fundamental science, and for clean energy in the form of fusion power. Sandia's most mature plasma physics simulation capabilities come in the form of particle-in-cell (PIC) models and magnetohydrodynamics (MHD) models. MHD models for a plasma work well in denser plasma regimes when there is enough material that the plasma approximates a fluid. PIC models, on the other hand, work well in lower-density regimes, in which there is not too much to simulate; error in PIC scales as the square root of the number of particles, making high-accuracy simulations expensive. Real-world applications, however, almost always involve a transition region between the high-density regimes where MHD is appropriate, and the low-density regimes for PIC. In such a transition region, a direct discretization of Vlasov is appropriate. Such discretizations come with their own computational costs, however; the phase-space mesh for Vlasov can involve up to six dimensions (seven if time is included), and to apply appropriate homogeneous boundary conditions in velocity space requires meshing a substantial padding region to ensure that the distribution remains sufficiently close to zero at the velocity boundaries. Moreover, for collisional plasmas, the right-hand side of the Vlasov equation is a collision operator, which is non-local in velocity space, and which may dominate the cost of the Vlasov solver. The present LDRD project endeavors to develop modern, foundational tools for the development of continuum-kinetic Vlasov solvers, using the discontinuous Petrov-Galerkin (DPG) methodology, for discretization of Vlasov, and machine-learning (ML) models to enable efficient evaluation of collision operators. DPG affords several key advantages. First, it has a built-in, robust error indicator, allowing us to adapt the mesh in a very natural way, enabling a coarse velocity-space mesh near the homogeneous boundaries, and a fine mesh where the solution has fine features. Second, it is an inherently high-order, high-intensity method, requiring extra local computations to determine so-called optimal test functions, which makes it particularly suited to modern hardware in which floating-point throughput is increasing at a faster rate than memory bandwidth. Finally, DPG is a residual-minimizing method, which enables high-accuracy computation: in typical cases, the method delivers something very close to the $L^2$ projection of the exact solution. Meanwhile, the ML-based collision model we adopt affords a cost structure that scales as the square root of a standard direct evaluation. Moreover, we design our model to conserve mass, momentum, and energy by construction, and our approach to training is highly flexible, in that it can incorporate not only synthetic data from direct-simulation Monte Carlo (DSMC) codes, but also experimental data. We have developed two DPG formulations for Vlasov-Poisson: a time-marching, backward-Euler discretization and a space-time discretization. We have conducted a number of numerical experiments to verify the approach in a 1D1V setting. In this report, we detail these formulations and experiments. We also summarize some new theoretical results developed as part of this project (published as papers previously): some new analysis of DPG for the convection-reaction problem (of which the Vlasov equation is an instance), a new exponential integrator for DPG, and some numerical exploration of various DPG-based time-marching approaches to the heat equation. As part of this work, we have contributed extensively to the Camellia open-source library; we also describe the new capabilities and their usage. We have also developed a well-documented methodology for single-species collision operators, which we applied to argon and demonstrated with numerical experiments. We summarize those results here, as well as describing at a high level a design extending the methodology to multi-species operators. We have released a new open-source library, MLC, under a BSD license; we include a summary of its capabilities as well.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Forward sensitivity analysis and mode dependent control for closure modeling of Galerkin systems

Model reduction by projection-based approaches is often associated with losing some of the important features that contribute towards the dynamics of the retained scales. As a result, a mismatch occurs between the predicted trajectories of the original system and the truncated one. We put forth a framework to apply a continuous time control signal in the latent space of the reduced order model (ROM) to account for the effect of truncation. We set the control input using parameterized models by following energy transfer principles. Our methodology relies on observing the system behavior in the physical space and using the projection operator to restrict the feedback signal into the latent space. Then, we leverage the forward sensitivity method (FSM) to derive relationships between the feedback and the desired mode-dependent control. We test the performance of the proposed approach using two test cases, corresponding to viscous Burgers and vortex merger problems at high Reynolds number. Results show that the ROM trajectory with the applied FSM control closely matches its target values in both the data-dense and data-sparse regimes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

Herein, we present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional sub-spaces whose basis functions are realizations of a sequence of neural networks. Here, the finite-dimensional subspaces are then used to define a standard Galerkin approximation of the variational equation. This approach enjoys a number of advantages, including: the sequential nature of the algorithm offers a systematic approach to enhancing the accuracy of a given approximation; the sequential enhancements provide a useful indicator for the error that can be used as a criterion for terminating the sequential updates; the basic approach is largely oblivious to the nature of the partial differential equation under consideration; and, some basic theoretical results are presented regarding the convergence (or otherwise) of the method which are used to formulate basic guidelines for applying the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Extended Galerkin Neural Network Approximation of Singular Variational Problems with Error Control

We present extended Galerkin neural networks, a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs, and (2) an “extended” feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Furthermore, numerical results are presented for several problems, including steady Stokes flow around reentrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.

a posteriori error estimate↗