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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 OSTI2024

Learning Only on Boundaries: A Physics-Informed Neural Operator for Solving Parametric Partial Differential Equations in Complex Geometries

Recently, deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parameterized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from $O(N^d)$ to $O(N^{d-1}$), where $d$ is the domain’s dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Finally, our numerical experiments show the effectiveness of parameterized complex geometries and unbounded problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Seasonal and Geographical Variations in Fundamental Weather Patterns during Extreme Precipitation as Identified from Omega Equation Forcing

Abstract We investigate the large-scale weather patterns during extreme precipitation (PEx) events over the conterminous United States (CONUS) by applying a version of the quasigeostrophic (QG) omega equation. This work aims to develop a climatology of the weather patterns most related to PEx events during current climate. Extreme events are examined for each of seven regions defined by consistent annual cycles of precipitation and spanning the CONUS. For the CONUS we train several self-organizing maps (SOM) on a pressure–time series of vertical velocity from each of the advective forcing terms in the QG omega equation for each extreme event. The unsupervised learning of the SOM allows us to identify the most descriptive set of nine patterns in vertical velocity associated with precipitation extremes. This method finds multiple frontal- and cyclone-driven patterns while grouping primarily convective events into one pattern. Frontal events include a synoptic pattern consistent with West Coast atmospheric river events as well as pattern groups linked to developing and to mature (“occluded”) frontal cyclones. The primary patterns found during PEx events vary seasonally and geographically. Frontal cyclone patterns are most common during PEx events during summer in the part of the Great Plains and during winter for the Northeast, Southeast, Pacific Northwest, and Southwest. Convection is the most common pattern during summer in all regions. Except in the Southeast, the annual cycles of monthly number of PEx events and average precipitation match well, partially validating our choice of regions to aggregate PEx events.

Meteorology & Atmospheric Sciences↗
DOE OSTI2021

Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion (Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. Here, we find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel.

58 GEOSCIENCES↗
DOE OSTI2020

Equations of state, phase relations, and oxygen fugacity of the Ru-RuO 2 buffer at high pressures and temperatures

Experimental studies and measurements of inclusions in diamonds show that ferric iron components are increasingly stabilized with depth in the mantle. To determine the thermodynamic stability of such components, their concentration needs to be measured at known oxygen fugacities. The metal-oxide pair Ru and RuO 2 are ideal as an internal oxygen fugacity buffer in high-pressure experiments. Both phases remain solid to high temperatures and react minimally with silicates, only exchanging oxygen. To calculate oxygen fugacities at high pressure and temperature, however, requires information on the phase relations and equation of state properties of the solid phases. We have made in situ synchrotron X-ray diffraction measurements in a multi-anvil press on mixtures of Ru and RuO 2 to 19.4 GPa and 1473 K with which we have determined phase relations of the RuO 2 phases and derived thermal equations of state (EoS) parameters for both Ru and RuO 2 . Rutile-structured RuO 2 was found to undergo two phase transformations, first at ~7 GPa to an orthorhombic structure and then above 12 GPa to a cubic structure. The phase boundary of the cubic phase was constrained for the first time at high pressure and temperature. We have derived a continuous Gibbs free energy expression for the tetragonal and orthorhombic phases of RuO 2 by fitting the second-order phase transition boundary and P-V-T data for both phases, using a model based on Landau theory. The transition between the orthorhombic and cubic phases was then used along with EoS terms derived for both phases to determine a Gibbs free energy expression for the cubic phase. We have used these data to calculate the oxygen fugacity of the Ru + O 2 = RuO 2 equilibrium, which we have parameterized as a single polynomial across the stability fields of all three phases of RuO 2 . The expression is log 10 f O 2 (Ru – RuO 2 ) = (7.782 – 0.00996P + 0.001932P 2 – 3.76 × 10 –5 P 3 ) + (–13 763 + 592P – 3.955P 2 )/T + (–1.05 × 10 6 – 4622P)/T 2 , which should be valid from room pressure up to 25 GPa and 773–2500 K, with an estimated uncertainty of 0.2 log units. Our calculated fO 2 is shown to be up to 1 log unit lower than estimates that use previous expressions or ignore EoS terms.

Armstrong, Katherine↗
DOE OSTI2020

H2OI95: A Stand-Alone Fortran Code for Evaluating the IAPWS-95 Equation-of-State Model for Water (Rev. 1)

H2OI95 is a stand-alone Fortran code for evaluating the IAPWS-95 equation-of-state model for the thermodynamic properties of water. It also evaluates the corresponding thermochemical properties of water consistent with the CODATA recommendations. The IAPWS-95 model is based on a model equation for the dimensionless Helmholtz energy for which the primary variables are the inverse reduced temperature (τ = T cr /T ) and reduced density (δ = ρ/ρ cr ). Here T is the absolute temperature (K), ρ is density (kg/m³), and the subscript “ cr ” refers to the critical point of water (647.096 K and 22.064 MPa pressure in this model, for which ρ cr is 322 kg/m³). The code solves four basic types of problems, distinguished by the specified inputs: 1. Temperature (K) and density (ρ kg/m³) or reduced density (δ) 2. Temperature (K) and pressure (MPa). 3. Temperature (K) on the saturation (liquid-vapor equilibrium) curve 4. Pressure (MPa) on the saturation curve. Each type of problem is run using a corresponding input (text) file. All but the first type of problem require iteration. For example, to solve for desired temperature and pressure, the reduced density must be adjusted to give the desired pressure. Iteration is accomplished using the Newton-Raphson method, though the secant method is also used in solving the fourth type of problem. For the last three types of problems, H2OI95 has been used to conduct numerical studies of convergence and the problem of multiple numerical solutions, some of which are not physically valid. Obtaining valid results depends mainly on appropriate choice of starting values for the density. The default values used in H2OI95 appear to consistently lead to generally desired results. With modification (not addressed here), H2OI95 can be used to support SUPCRT92 and similar codes that compute chemical thermodynamic properties of species and reactions over a wide range of temperature and pressure (273.16-1273K and 0-1000 MPa). Version 1.1 of this software calculates an extended set of water properties, including some derivable solely from IAPWS-95, such as the isobaric thermal expansivity and the isothermal compressibility, and still other properties based on supplementary models sanctioned by IAPWS, including the dynamic viscosity, thermal conductivity, surface tension, and static dielectric constant, and various other properties (notably Debye-Hückel parameters) derivable from the dielectric constant and its partial derivatives with respect to temperature and pressure. A coding error in Version 1.0 was identified while testing the implementation of the viscosity model. This error is described in Appendix C. It mainly affects calculation of “higher order” properties such as compressibility and heat capacity near the critical point.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Deuteron-Deuteron Elastic and Three and Four-Body Breakup Scattering Using the Faddeev-Yakubovskii Equations

The deuteron-deuteron elastic and three and four-body breakup scattering cross section have been calculated using the Faddeev-Yakubovskii (FY) chain-of-partition momentum-space equations. In this calculation the initial two-cluster potential is split into separable and non-separable components, and the effective potential is reduced to elastic two-body and three and four-body breakup open-channels and a closed-channel many-body contribution. The closed-channel contribution is determined by minimizing a variational bound. The Coulomb interaction was included by expanding the initial and final Coulomb states in a Coulomb-Sturmian basis. The three sets of chain-of-partition integral equations were solved for the elastic and three and four-body breakup scattering amplitudes. The calculations were performed for the S = 2 spin/L = 0 angular-momentum state. The elastic and double-breakup calculations were performed for energies up to E = 5.48 MeV, while the single-breakup calculations were performed for energies up to E = 4.17 MeV. In the case of elastic scattering, the calculated scattering length of 5 a dd = 7.8 ± 0.3 fm is in good agreement with a FY cluster reduction calculation. The calculated phase shift is smaller than that predicted by the resonating group model and this difference is believed to be due to the differences in the potential and calculation methods. The breakup cross sections were calculated as a function of initial deuteron momentum and fragmented-deuteron momentum. Here, the d+d → d+n+p cross sections were compared with neutron yield measurements and, while the measurements also included the L > 0 components, the general features were consistent. Estimates of the calculational uncertainties/bias are provided.

42 ENGINEERING↗
DOE OSTI2021

Verification of Combined VR Techniques, Derivation of Future Time Equation, and Integration of LLNL Pulsed Sphere V&V Suite [Slides]

To determine whether the combination of forced-collision and DXTRAN variance-reduction (VR) techniques is unbiased, and to gain insight into the operation of these techniques, proof that first-moment estimates from Monte Carlo simulations employing both techniques are unbiased is developed. A general background on the forced-collision and DXTRAN VR techniques and their combination is given. Proof of an unbiased simulation is outlined by showing the equivalence of the history score moment equations of simulations with these techniques in use. A report with detailed proof of this equivalence is available upon request. The derivation of the future time equation using a similar approach, as well as a summary of the addition of the LLNL Pulsed Sphere experiments to the MCNP verification and validation suite, is also briefly discussed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Prompt Gamma Analysis of ARIES Materials and Updates to the 2015 Calibration Equations

Prompt gamma (PG) analysis is a nondestructive, nuclear, elemental analysis technique that uses charged particle reactions to interrogate a sample, and elements present in the sample matrix are identified through the characteristic gamma-rays emitted from the product nuclei in alpha-p and alpha-n nuclear reactions. This technique has been applied to plutonium oxide packaged in over 4,000 individual 3013 containers, and the concentrations of certain light elements were determined from the integrated peak areas based on a calibration that was published previously. This report provides the results for a new population of 3013 containers packaged with oxide materials produced by the conversion of metal by Advanced Recovery and Integrated Extraction System (ARIES) project using the direct metal oxidation (DMO) process and muffle furnaces. New PG and analytical chemistry data collected since 2015 were added to the existing calibration data sets to refine the calibration parameters. Calibration equations were also developed for determining beryllium and fluorine at low concentrations in high-purity ARIES product oxides. The new fluorine calibration provides an order of magnitude greater sensitivity and results in an additional 1,408 containers in the original population identified as having fluorine as an impurity. Additionally, equations for calculating the lower limits of detection (LLDs) as a function of the actual counting time (live time) were obtained using WLS regression technique for samples in the calibration data set. This resulted in changes to the LLDs published previously.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions. Today, tools such as transfer matrix and finite elements enable the calculation of the rotor dynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems. In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered. The purpose of this paper is to show examples of how the linear vs. nonlinear formulations differ. This paper will also show how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.

42 ENGINEERING↗
DOE OSTI2022

A 6th Order Mehrstellen Finite Volume Discretization of Poisson's Equation in Three Dimensions

We discuss the derivation of a new, sixth-order finite volume scheme for Poisson’s equation on 3D Cartesian equispaced grids. The scheme is based on a discretization of the Laplace operator with a compact (Mehrstellen) 27-point stencil. To achieve sixth order convergence the right hand side of the equation is replaced with a discrete operator that involves the discrete Laplace and Biharmonic operators and the sum of discrete fourth-order cross derivatives applied to the charge function. Numerical tests demonstrate the superiority of the proposed method compared to the well known schemes associated with the 7-point and 19-point discretizations of the Laplacian.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Comprehensive uncertainty quantification (UQ) for full engineering models by solving probability density function (PDF) equation

This report details a new method for propagating parameter uncertainty (forward uncertainty quantification) in partial differential equations (PDE) based computational mechanics applications. The method provides full-field quantities of interest by solving for the joint probability density function (PDF) equations which are implied by the PDEs with uncertain parameters. Full-field uncertainty quantification enables the design of complex systems where quantities of interest, such as failure points, are not known apriori. The method, motivated by the well-known probability density function (PDF) propagation method of turbulence modeling, uses an ensemble of solutions to provide the joint PDF of desired quantities at every point in the domain. A small subset of the ensemble is computed exactly, and the remainder of the samples are computed with approximation of the driving (dynamics) term of the PDEs based on those exact solutions. Although the proposed method has commonalities with traditional interpolatory stochastic collocation methods applied directly to quantities of interest, it is distinct and exploits the parameter dependence and smoothness of the dynamics term of the governing PDEs. The efficacy of the method is demonstrated by applying it to two target problems: solid mechanics explicit dynamics with uncertain material model parameters, and reacting hypersonic fluid mechanics with uncertain chemical kinetic rate parameters. A minimally invasive implementation of the method for representative codes SPARC (reacting hypersonics) and NimbleSM (finite- element solid mechanics) and associated software details are described. For solid mechanics demonstration problems the method shows order of magnitudes improvement in accuracy over traditional stochastic collocation. For the reacting hypersonics problem, the method is implemented as a streamline integration and results show very good accuracy for the approximate sample solutions of re-entry flow past the Apollo capsule geometry at Mach 30.

42 ENGINEERING↗
DOE OSTI2020

Finite Volume Discretization of the Euler Equations in Pronghorn

Modeling flow and heat transfer in high temperature gas reactors (HTGR) requires the ability to model a wide range of flow speeds from slow (natural convection), to intermediate (forced-flow conditions), to supersonic regimes (depressurization) for a wide range of geometries including the pebble bed, upper and lower plenum, and risers. In previous work, Pronghorn has effectively modeled low-to-medium speed flows in scenarios such as the one described in the two-dimensional PBMR-400 benchmark, using its finite-element-based streamline-upwind Petrov-Galerkin (SUPG) stabilized implementation of the Euler equations. However, limitations of this method become apparent when dealing with more complicated geometries (e.g. imposing slip boundary conditions at nodes belonging to two different boundaries) and when gas speeds are fast enough for shocks and supersonic flow to occur. For these problems, the finite-element-based solver lacks robustness and is plagued by slow iterative convergence or even divergence. In order to address these challenges, the Pronghorn code at INL has been updated with new, modified versions of its original equations. The new Pronghorn models are built on the finite volume method with a Harten-Lax-van Leer-Contact (HLLC) Riemann solver based numerical flux method, which (1) allows imposing slip boundary conditions much more robustly and (2) performs well for a wide range of flow speeds. The finite-volume-based flow solver will form the basis for a robust coarse-mesh thermal-hydraulics capability in Pronghorn.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Machine-learning based model reduction for partial differential equations

We develop a novel synergistic approach between model reduction and machine learning. The specific goal of this project is to aid in the construction of reduced order models for basis functions that are custom-made to represent the solution of partial differential equations. Partial differential equations (PDEs) are one of the main mathematical tools for describing physical phenomena. However, due to either efficiency or necessity, for many real-world problems, we are interested in constructing reduced order models (ROMs) which focus only on the explicit computation of subsets of the active spatio-temporal scales in the problem, while treating the interaction with the rest of the scales approximately. The task of accurate representation of such interactions (usually called memory terms) constitutes a vast area of research known as model reduction. PI Stinis has significant expertise in the construction of ROMs for complex systems. In addition, in recent work with the project key participant Qadeer, they have utilized machine learning to acquire custom-made basis functions (CBFs) to expand the solutions of PDEs. In the proposed work, we will merge the two concepts by constructing ROMs for subsets of the CBFs needed to represent the solution of a PDE. Specifically, we will use the Mori-Zwanzig model reduction formalism to construct ROMs for subsets of CBFs for nonlinear PDEs of various complexity, as well as investigate the usage of CBFs in the spectral vanishing viscosity method for problems that can form shocks in finite time. The outcome of the research is aimed to be proof-of-concept about a novel synergistic approach between model reduction and machine learning, thus advancing the field of scientific machine learning. Such a capability will benefit the efficient modeling of physical systems appearing in various areas of interest to the DOE.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Model-form Error Correction using Universal Differential Equations for an Agent-Based Model of Infectious Disease

This report demonstrates universal differential equations (UDEs) as an approach to bridge the gap between ordinary differential equations (ODE) models and agent-based models (ABMs). Using UDE models as surrogates for ABMs allows us to preserve the foundational ODE that represents global disease dynamics while coupling it with a neural network model to approximate functions for the local behaviors of the ABM.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

A Quantum Approach for Implementing Fixed-Point Arithmetic in Solving Ordinary Differential Equations

Differential equations (DEs) serve as fundamental tools in mathematical modeling across scientific disciplines, yet classical numerical solvers face limitations with large-scale or computationally intensive problems. This study explores a quantum-inspired approach to solving DEs, combining quantum- inspired techniques with classical methods. It focuses on fixed- point arithmetic on quantum circuits, utilizing basic quantum gates to manipulate DE solutions. We expand upon the techniques introduced by Zanger et al. [Quantum, 5, 502 (2021)] by offering a precise computation for a fixed-point signed multiplication scheme, while also presenting a quantum circuit capable of executing the fixed-point division algorithm. We demonstrate the feasibility of our approach through the simulation of a linear Ordinary Differential Equation (ODE), where initial conditions and parameters are encoded into quantum circuits using fixed- point representation. By executing sequences of quantum gates mimicking numerical integration steps, we obtain approximate solutions to the ODE with specified fixed-point precision.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Poisson Equation for a (General) Homogeneous d-Dimensional Ellipsoid with Applications to Beam Envelope Tracking

This note describes the solution of the free-space Poisson equation in the interior of a $d$-dimensional homogeneous ellipsoid, and the associated space charge fields. An explicit formula (\ref{Sformula}) is provided that relates the $d\times d$ matrix describing the space charge (quadratic) potential to the $d\times d$ covariance matrix of the ellipsoid. For the cases $d=2$ and $d=3$, this result is used to determine the linear map corresponding to a space charge kick, that may be used to push the beam $6\times 6$ covariance matrix during envelope tracking. The treatment of upright ellipsoids for $d=2$ and $d=3$ is well-represented in the literature. However, the approach taken here emphasizes a general ellipsoid with arbitrary correlations in any dimension. The Appendix provides a general solution of the free-space Poisson equation in dimension $d$ for a source distribution with ellipsoidal symmetry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

The Parabolic Wave Equation Generalized to a Spatially Varying Plasma in a Vertical Magnetic Field

This report walks through the process of deriving the wave equation for a vertical wave being propagated through a spatially varying plasma in a vertical magnetic field. Additionally, this report shows how to derive the index of refraction of the plasma corresponding to three different modes of the vertical magnetic field. Finally, this paper briefly explores the next steps planned to successfully simulate and test the equations in a pre-existing phase screen scintillation code.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗