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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 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Double-power-law Feature of Energetic Particles Accelerated at Coronal Shocks

Recent observations have shown that in many large solar energetic particle (SEP) events the event-integrated differential spectra resemble double power laws. We perform numerical modeling of particle acceleration at coronal shocks propagating through a streamer-like magnetic field by solving the Parker transport equation, including protons and heavier ions. We find that for all ion species the energy spectra integrated over the simulation domain can be described by a double power law, and the break energy depends on the ion charge-to-mass ratio as E B ~ (Q/A) α , with α varying from 0.16 to 1.2 by considering different turbulence spectral indices. We suggest that the double-power-law distribution may emerge as a result of the superposition of energetic particles from different source regions where the acceleration rates differ significantly due to particle diffusion. The diffusion and mixing of energetic particles could also provide an explanation for the increase of Fe/O at high energies as observed in some SEP events. Although further mixing processes may occur, our simulations indicate that either a power-law break or rollover can occur near the Sun and predict that the spectral forms vary significantly along the shock front, which may be examined by upcoming near-Sun SEP measurements from the Parker Solar Probe and Solar Orbiter.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

A multiresolution adaptive wavelet method for nonlinear partial differential equations

We report the multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to solve partial differential equations (PDEs) with features evolving on a wide range of spatial and temporal scales. To meet these challenges, we present a multiresolution wavelet algorithm to solve PDEs with significant data compression and explicit error control. We discretize in space by projecting fields and spatial derivative operators onto wavelet basis functions. We provide error estimates for the wavelet representation of fields and their derivatives. Then, our estimates are used to construct a sparse multiresolution discretization which guarantees the prescribed accuracy. Additionally, we embed a predictor-corrector procedure within the temporal integration to dynamically adapt the computational grid and maintain the accuracy of the solution of the PDE as it evolves. We present examples to highlight the accuracy and adaptivity of our approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A direct-adjoint approach for material point model calibration with application to plasticity

Here, this paper proposes a new approach for the calibration of material parameters in local elastoplastic constitutive models. The calibration is posed as a constrained optimization problem, where the constitutive model evolution equations for a single material point serve as constraints. The objective function quantifies the mismatch between the stress predicted by the model and corresponding experimental measurements. To improve calibration efficiency, a novel direct-adjoint approach is presented to compute the Hessian of the objective function, which enables the use of second-order optimization algorithms. Automatic differentiation is used for gradient and Hessian computations. Two numerical examples are employed to validate the Hessian matrices and to demonstrate that the Newton–Raphson algorithm consistently outperforms gradient-based algorithms such as L-BFGS-B.

36 MATERIALS SCIENCE↗
DOE OSTI2023

A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integro-differential operator consists of both local convection-diffusion and nonlocal diffusion operators. Here, our numerical scheme is based on the direct approximation of the nonlinear Feynman–Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman–Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows

Herein we demonstrate that the reformulation of renormalization group (RG) flow equations as nonlinear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional Z 2 -symmetric model that the dissipative character of generic nonlinear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semigroup property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dissipative character of RG flows, its irreversibility and the entropy production during the RG flow may be linked to the existence of a so-called C– / A-function. In total, this introduces an asymmetry in the so-called RG time—in complete analogy to the thermodynamic arrow of time—and allows for an interpretation of infrared actions as equilibrium solutions of dissipative RG flows equations. The impossibility of resolving microphysics from macrophysics is evident in this framework. Furthermore, we directly link the irreversibility and the entropy production in RG flows to an explicit numerical entropy production, which is manifest in diffusive and non-linear partial differential equations (PDEs) and a standard mathematical tool for the analysis of PDEs. Using exactly solvable zero-dimensional Z 2 -symmetric models, we explicitly compute the (numerical) entropy production related to the total variation nonincreasing property of the PDE during RG flows toward the infrared limit. Finally, we discuss generalizations of our findings and relations to the C– / A-theorem as well as how our work may help to construct truncations of RG flow equations in the future, including numerically stable schemes for solving the corresponding PDEs.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Quandary

Quandary numerically simulates and optimizes the time-evolution of open quantum systems. The underlying dynamics are modelled by Lindblad's master equation, a linear ordinary differential equation (ODE) describing quantum systems interacting with the environment. Quandary solves this ODE numerically by applying a time-stepping integration scheme, and utilizes a gradient-based optimization approach to determine optimal control pulses that drive the quantum system to a desired target state. Two optimization objectives are considered: (a) Unitary gate optimization that finds controls to realize a unitary gate transformation, and (b) optimal reset that aims to drive the quantum system to the ground states. Gradient-based optimization schemes utilizing Petsc's Tao optimization package are applied to generate control pulses that minimize the respective measure. To evaluate the gradient of the objective function, the discrete adjoint method is used while leveraging techniques from Algorithmic Differentiation to produce exact and consistent gradients. To mitigate excessive execution run times, the software can be build together with the XBraid software library which provides a parallelization strategy to distribute the time-evolution of the underlying dynamics onto multiple processor.

Petersson, NilsA.↗
DOE OSTI2024

Sparsified time-dependent Fourier neural operators for fusion simulations

This paper presents a sparsified Fourier neural operator for coupled time-dependent partial differential equations (ST-FNO) as an efficient machine learning surrogate for fluid and particle-based fusion codes such as NIMROD (Non-Ideal Magnetohydrodynamics with Rotation - Open Discussion) and GTC (Gyrokinetic Toroidal Code). ST-FNO leverages the structures in the governing equations and utilizes neural operators to represent Green's function-like numerical operators in the corresponding numerical solvers. Once trained, ST-FNO can rapidly and accurately predict dynamics in fusion devices compared with first-principle numerical algorithms. In general, ST-FNO represents an efficient and accurate machine learning surrogate for numerical simulators for multi-variable nonlinear time-dependent partial differential equations, with the proposed architectures and loss functions. The efficacy of ST-FNO has been demonstrated using quiescent H-mode simulation data from NIMROD and kink-mode simulation data from GTC. The ST-FNO H-mode results show orders of magnitude reduction in memory and central processing unit usage in comparison with the numerical solvers in NIMROD when computing fields over a selected poloidal plane. The ST-FNO kink-mode results achieve a factor of 2 reduction in the number of parameters compared to baseline FNO models without accuracy loss.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Multiscale simulations for multi-continuum Richards equations

In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. Furthermore, to tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled and coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Transported PDF Modeling of Compressible Turbulent Reactive Flows by using the Eulerian Monte Carlo Fields Method

Although the transported probability density function (PDF) method has been developed for decades, its application has been mainly focused on the low-Mach number flow problems. This work extends the transported PDF method to compressible flow problems. The Eulerian Monte Carlo fields (EMCF) solution method is employed to solve the transported PDF equation for compressible flow problems. A pseudo stagnation enthalpy is introduced and its stochastic partial differential equation is derived to ensure total energy conservation numerically. A new mixing model called interaction by partial exchange with mean (IPEM) is introduced to expand the available choices of mixing models for the EMCF method. The consistency of the EMCF method is examined for solving the transported PDF equation. Numerical implementation details are discussed, such as the density coupling between the compressible flow solver and the EMCF solver, discretization schemes for the mixing terms and the stochastic terms. The implemented compressible flow solver coupled with the EMCF solver is verified and validated in a series of test cases with increasing level of complexity, ranging from a statistically one-dimensional turbulent mixing layer to a self-excited resonance model rocket combustor. It is observed that in general with the increase of compressibility, there is an increase in the sensitivity of the modeling results to the different models and algorithms. This makes it necessary to develop a thorough understanding of the model sensitivity in order to develop a robust and accurate simulation solver for highly compressible turbulent reactive flows. The thermo-acoustic instability inside the model rocket combustor case is captured reasonably, which demonstrates the overall capability of the developed compressible turbulent combustion solver based on the transported PDF method.

42 ENGINEERING↗
DOE OSTI2023

Evolution of coupled weakly driven waves in a dissipative plasma

The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. In conclusion, the obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

On a simplified solution of climate-carbon dynamics in idealized flat10MIP simulations

Abstract. Idealized experiments with coupled climate-carbon Earth system models (ESMs) provide a basis for understanding the response of the carbon cycle to external forcing and for quantifying climate-carbon feedbacks. Here, we analyze globally-averaged results from idealized esm-flat10 experiments and show that most models exhibit a quasi-linear relationship between cumulative carbon uptake on land and in the ocean during a period of constant fossil fuel emissions of 10 Pg C yr−1. We hypothesize that this relationship does not depend on emission pathways. Further, as a simplification, we quantify the relationship between cumulative ocean carbon uptake and changes in ocean heat content using a linear approximation. In this way, changes in oceanic heat content and atmospheric CO2 concentration become interdependent variables, reducing the coupled temperature-CO2 system to just one differential equation. The equation can be solved analytically or numerically for the atmospheric CO2 concentration as a function of fossil fuel emissions. This approach leads to a simplified description of global carbon and climate dynamics, which could be used for applications beyond existing analytical frameworks.

Brovkin, Victor↗
DOE OSTI2023

Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A multi-dimensional parametric study of variability in multi-phase flow dynamics during geologic CO 2 sequestration accelerated with machine learning

Successful geologic CO 2 storage projects depend on numerical simulations to predict reservoir performance during site selection, injection verification, and post-injection monitoring phases of the project. These numerical simulations solve non-linear sets of coupled partial differential equations, while accounting for multi-phase fluid dynamics on the basis of constitutive equations that are embedded into the solution scheme. As a consequence, individual simulations often require tens to hundreds of hours to complete on high-performance computing clusters. Moreover, laboratory experiments reveal that parametric functions for capillary pressure and relative permeability exhibit substantial variability, even within the same rock type. This combination of computational expense and wide-ranging parametric variability means that there remains substantial uncertainty in the behavior of multi-phase CO 2 -water systems, particularly in the context of feedbacks between relative permeability and capillary pressure. To bridge this knowledge gap, here we develop a novel workflow that utilizes physics-based numerical simulation to train an artificial neural network (ANN) emulator for interrogating the multivariate parameter space that governs both capillary pressure and relative permeability. With this approach, the ANN is trained to emulate both fluid pressure distribution and CO 2 saturation, which are then interrogated quantitatively to generate parametric response surface mappings with high-fidelity resolution. Results from this study initially show that capillary entry pressure is the dominant control on both CO 2 plume geometry and fluid pressure propagation when considering the combined effects of capillary pressure and relative permeability, particularly when phase interference is low and residual CO 2 saturation is high. Moreover, the ANN emulator provides tremendous computational speed-up by computing 2691 individual simulations in several minutes; whereas, the same simulation ensemble would have required ~3 years of simulation time using only physics-based simulation methods (25,000 times speed up).

58 GEOSCIENCES↗
DOE OSTI2024

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimension. In this paper, we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control. We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians. Additionally, we present an efficient, grid-free numerical solver based on our representation formulas, which is shown to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality. Using existing optimization methods and the min-plus technique, we extend our numerical solvers to address more general classes of convex and nonconvex initial costs. We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit (CPU) and a field-programmable gate array (FPGA). In several cases, our FPGA implementation obtains over a 10 times speedup compared to the CPU, which demonstrates the promising performance boosts FPGAs can achieve. Furthermore, our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An extended trajectory-mechanics approach for calculating two-phase flow paths

A technique originating in quantum dynamics is used to derive a trajectory-based, semi-analytical solution for two-phase flow. The partial differential equation governing the evolution of the aqueous phase is equivalent to a family of ordinary differential equations defined along a path through the porous medium. The trajectories may be found by solving the differential equations directly or by post-processing the output of a numerical solution to the full set of governing equations. The trajectories, which differ from conventional streamlines, are found to bend downward in response to gravitational forces. The curvature is more pronounced as the dip of the porous layer containing the flow increases. Subtle changes in the relative permeability curve can lead to significant variations in the trajectories. The ordinary differential equation for the trajectory provides an expression for the travel time along the path. The expression produces a semi-analytical approximation to the model parameter sensitivities, the partial derivatives of the travel times with respect to changes in the permeability model. The semi-analytical trajectory-based sensitivities generally agree with those computed using a numerical reservoir simulator and a perturbation approach. The sensitivities are useful in tomographic imaging algorithms designed to estimate the spatial variation in permeability within a porous medium using multiphase observations.

58 GEOSCIENCES↗
DOE OSTI2022

Automatic Differentiation of C++ Codes on Emerging Manycore Architectures with Sacado

Automatic differentiation (AD) is a well-known technique for evaluating analytic derivatives of calculations implemented on a computer, with numerous software tools available for incorporating AD technology into complex applications. However, a growing challenge for AD is the efficient differentiation of parallel computations implemented on emerging manycore computing architectures such as multicore CPUs, GPUs, and accelerators as these devices become more pervasive. In this work, we explore forward mode, operator overloading-based differentiation of C++ codes on these architectures using the widely available Sacado AD software package. In particular, we leverage Kokkos, a C++ tool providing APIs for implementing parallel computations that is portable to a wide variety of emerging architectures. Here we describe the challenges that arise when differentiating code for these architectures using Kokkos, and two approaches for overcoming them that ensure optimal memory access patterns as well as expose additional dimensions of fine-grained parallelism in the derivative calculation. We describe the results of several computational experiments that demonstrate the performance of the approach on a few contemporary CPU and GPU architectures. We then conclude with applications of these techniques to the simulation of discretized systems of partial differential equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Learning viscoelasticity models from indirect data using deep neural networks

In this study, we propose a novel approach to model viscoelasticity materials, where rate-dependent and non-linear constitutive relationships are approximated with deep neural networks. We assume that inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input–output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multiple-physics systems. We show that limited displacement data holds sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation – modeling viscoelasticity properties from observed displacement data – as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and implicit function differentiation rules. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

97 MATHEMATICS AND COMPUTING↗