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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 757 records · Page 42
All sourcesNASA NTRSDOE OSTI
NASA NTRS2007

Application of the Finite Element Method in Atomic and Molecular Physics

The finite element method (FEM) is a numerical algorithm for solving second order differential equations. It has been successfully used to solve many problems in atomic and molecular physics, including bound state and scattering calculations. To illustrate the diversity of the method, we present here details of two applications. First, we calculate the non-adiabatic dipole polarizability of Hi by directly solving the first and second order equations of perturbation theory with FEM. In the second application, we calculate the scattering amplitude for e-H scattering (without partial wave analysis) by reducing the Schrodinger equation to set of integro-differential equations, which are then solved with FEM.

Shertzer, Janine↗
DOE OSTI2023

Leverage Score Sampling for Parametric PDEs (Final Technical Report)

This final technical report summarizes the accomplishments of work performed under DOE Office of Science Award DE-SC0022266, which is titled “Leverage Score Sampling for Parametric PDEs”. The goal of the project was to extend methods from Randomized Numerical Linear Algebra (RandNLA) to tackle central computational challenges in model order reduction and uncertainty quantification (UQ) for parametric partial differential equations (PDEs). In particular, we sought to use importance sampling methods originally developed for RandNLA to develop sample efficient active learning algorithms for approximating high-dimensional scalar functions, e.g. by polynomials, Gaussian process models, and simple neural networks. Such methods can be immediately applied to developing surrogate models or to approximating quantity of interest (QoI) surfaces. In the context of PDEs, each sample used for learning equates to the solution of the differential equation for a particular set of parameters, so sample efficiency translates to improved computational efficiency for a variety of downstream tasks.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1992

Comparison of level-1 and level-2 buckling and postbuckling solutions

The accuracy of the solutions obtained with the Level-1 computational modules of DISDECO (Delft Interactive Shell Design Code) are investigated by comparing them with solutions obtained with ANILISA, a Level-2 computational module. Whereas the Level-1 solutions are based on the use of trigonometric functions to represent the out-of-plane displacement W and involve Galerkin type approximations to reduce the governing partial differential equations to simple algebraic expressions, the Level-2 solutions employ a truncated Fourier expansion in the circumferential direction and solve the resulting ordinary differential equations forming a nonlinear eigenvalue problem by a Stodola like technique. This involves the numerical integration of the set of ordinary differential equations, which in turn makes it possible to satisfy the specified boundary conditions exactly. It is shown that in many applications the Level-1 predictions concerning the critical buckling load of the perfect structure and its imperfection sensitivity are quite accurate and reliable. For those cases where this is not true, it is shown how the pre and postprocessing facilities of DISDECO can be used efficiently to arrive at a reliable prediction. This involves always the use of the Level-2 computational module ANILISA with its rigorous prebuckling solution and its feature which makes it possible to satisfy the specified boundary conditions exactly.

Arbocz, J.↗
NASA NTRS2006

Model Predictive Optimal Control of a Time-Delay Distributed-Parameter Systems

This paper presents an optimal control method for a class of distributed-parameter systems governed by first order, quasilinear hyperbolic partial differential equations that arise in many physical systems. Such systems are characterized by time delays since information is transported from one state to another by wave propagation. A general closed-loop hyperbolic transport model is controlled by a boundary control embedded in a periodic boundary condition. The boundary control is subject to a nonlinear differential equation constraint that models actuator dynamics of the system. The hyperbolic equation is thus coupled with the ordinary differential equation via the boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to implement a model predictive control design for a wind tunnel to eliminate a transport delay effect that causes a poor Mach number regulation.

Nguyen, Nhan↗
NASA NTRS1982

Integrating matrix formulations for vibrations of rotating beams including the effects of concentrated masses

By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

Lakin, W. D.↗
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↗
NASA NTRS1995

Nozzle flow with vibrational nonequilibrium

This research concerns the modeling and numerical solutions of the coupled system of compressible Navier-Stokes equations in cylindrical coordinates under conditions of equilibrium and nonequilibrium thermodynamics. The problem considered was the modeling of a high temperature diatomic gas N2 flowing through a converging-diverging high expansion nozzle. The problem was modeled in two ways. The first model uses a single temperature with variable specific heats as functions of this temperature. For the second model we assume that the various degrees of freedom all have a Boltzmann distribution and that there is a continuous redistribution of energy among the various degrees of freedom as the gas passes through the nozzle. Each degree of freedom is assumed to have its own temperature and, consequently, each system state can be characterized by these temperatures. This suggests that formulation of a second model with a vibrational degree of freedom along with a rotational-translation degree of freedom, each degree of freedom having its own temperature. Initially the vibrational degree of freedom is excited by heating the gas to a high temperature. As the high temperature gas passes through the nozzle throat there is a sudden drop in temperature along with a relaxation time for the vibrational degree of freedom to achieve equilibrium with the rotational-translation degree of freedom. That is, we assume that the temperature change upon passing through the throat is so great that the changes in the vibrational degree of freedom occur at a much slower pace and consequently lags behind the rotational-translational energy changes. This lag results in a finite relaxation time. In this context the term nonequilibrium is used to denote the fact that the energy content of the various degrees of freedom are characterized by two temperatures. We neglect any chemical reactions which could also add nonequilibrium effects. We develop the energy equations for the nonequilibrium model from first principles. The resulting equations, which model the nozzle flow, can be expressed in various forms. In most forms the resulting equations are coupled systems of nonlinear partial differential equations subject to certain boundary conditions. To solve the resulting coupled system of nonlinear partial differential equations, several numerical techniques were investigated: (1) the explicit MacCormack method, (2) the explicit-implicit MacCormack method, (3) the method of operator splitting, (4) factorization schemes, and (5) the Steger-Warming scheme.

Heinbockel, J. H.↗
NASA NTRS1987

Parallel processing architecture for computing inverse differential kinematic equations of the PUMA arm

In advanced robot control problems, on-line computation of inverse Jacobian solution is frequently required. Parallel processing architecture is an effective way to reduce computation time. A parallel processing architecture is developed for the inverse Jacobian (inverse differential kinematic equation) of the PUMA arm. The proposed pipeline/parallel algorithm can be inplemented on an IC chip using systolic linear arrays. This implementation requires 27 processing cells and 25 time units. Computation time is thus significantly reduced.

Hsia, T. C.↗
DOE OSTI2023

How machine learning can extend electroanalytical measurements beyond analytical interpretation

Electroanalytical measurements are routinely used to estimate material properties exhibiting current and voltage signatures. Analysis of such measurements relies on analytical expressions of material properties to describe the experiments. The need for analytical expressions limits the experiments that can be used to measure properties as well as the properties that can be estimated from a given experiment. Such analytical relations are essentially solutions of the physics-based differential equations (with properties as coefficients) describing the material behavior under certain specific conditions. In recent years, a new machine learning-based approach has been gaining popularity wherein the differential equations are numerically solved to interpret the electroanalytical experiments in terms of corresponding material properties. Since the physics-based differential equations are solved, one can additionally estimate underlying fields, e.g., concentration profile, using such an approach. To exemplify the characteristics of such a machine learning assisted interpretation of electroanalytical measurements, we use data from the Hebb–Wagner test on a magnesium spinel intercalation host. In conclusion, as compared to the traditional analytical expression-based interpretation, the emerging approach decreases experimental efforts to characterize relevant material properties as well as provides field information that was previously inaccessible.

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

Parallel projection—An improved return mapping algorithm for finite element modeling of shape memory alloys

Here, we present a novel finite element analysis of inelastic structures containing Shape Memory Alloys (SMAs). Phenomenological constitutive models for SMAs lead to material nonlinearities, that require substantial computational effort to resolve. Finite element analysis methods, which rely on Gauss quadrature integration schemes, must solve two sets of coupled differential equations: one at the global level and the other at the local, i.e. Gauss point level. In contrast to the conventional return mapping algorithm, which solves these two sets of coupled differential equations separately using a nested Newton procedure, we propose a scheme to solve the local and global differential equations simultaneously. In the process we also derive closed-form expressions used to update the internal/constitutive state variables, and unify the popular closest-point and cutting plane methods with our formulas. Numerical testing indicates that our method allows for larger thermomechanical loading steps and provides increased computational efficiency, over the standard return mapping algorithm.

42 ENGINEERING↗
NASA NTRS1984

Dynamics analysis of electrodynamic satellite tethers. Equations of motion and numerical solution algorithms for the tether

The equations of motion are developed for a perfectly flexible, inelastic tether with a satellite at its extremity. The tether is attached to a space vehicle in orbit. The tether is allowed to possess electrical conductivity. A numerical solution algorithm to provide the motion of the tether and satellite system is presented. The resulting differential equations can be solved by various existing standard numerical integration computer programs. The resulting differential equations allow the introduction of approximations that can lead to analytical, approximate general solutions. The differential equations allow more dynamical insight of the motion.

Nacozy, P. E.↗
NASA NTRS1991

Discussion summary: Fictitious domain methods

Fictitious Domain methods are constructed in the following manner: Suppose a partial differential equation is to be solved on an open bounded set, Omega, in 2-D or 3-D. Let R be a rectangle domain containing the closure of Omega. The partial differential equation is first solved on R. Using the solution on R, the solution of the equation on Omega is then recovered by some procedure. The advantage of the fictitious domain method is that in many cases the solution of a partial differential equation on a rectangular region is easier to compute than on a nonrectangular region. Fictitious domain methods for solving elliptic PDEs on general regions are also very efficient when used on a parallel computer. The reason is that one can use the many domain decomposition methods that are available for solving the PDE on the fictitious rectangular region. The discussion on fictitious domain methods began with a talk by R. Glowinski in which he gave some examples of a variational approach to ficititious domain methods for solving the Helmholtz and Navier-Stokes equations.

Glowinski, Rowland↗
NASA NTRS2018

Atmospheric Chemistry Modeling Using a Regression Forest Model

Atmospheric chemistry is central to many environmental issues such as air pollution, climate change, and stratospheric ozone loss. Chemistry Transport Models (CTM) are a central tool for understanding these issues, whether for research or for forecasting. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O1D to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a CTM. We have investigated a machine learning approach to solving the differential equations instead of solving them numerically. From an annual simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry. This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, centennial scale climate simulations etc. We discuss our approches' speed and accuracy, and highlight some potential future directions for improving this approach.

Keller, Christoph A.↗
NASA NTRS2019

Atmospheric Chemistry Modeling and Air Quality Forecasting Using Machine Learning

Atmospheric chemistry models are a central tool to study the impact of chemical constituents on the environment, vegetation and human health. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O1D to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a chemistry model.We have investigated a machine learning approach to emulate the chemistry instead of solving the differential equations numerically. From a one-month simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry.This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, etc. We discuss speed and accuracy of our approach, and highlight some potential future directions for improving it.

Keller, Christoph A.↗
NASA NTRS2019

Atmospheric Chemistry Modeling Using Machine Learning

Atmospheric chemistry models are a central tool to study the impact of chemical constituents on the environment, vegetation and human health. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O1D to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a chemistry model. We have investigated a machine learning approach to emulate the chemistry instead of solving the differential equations numerically. From a one-month simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry. This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, etc. We discuss speed and accuracy of our approach, and highlight some potential future directions for improving it.

Keller, Christoph A.↗
NASA NTRS2019

Machine Learning Application to Atmospheric Chemistry Modeling

Atmospheric chemistry models are a central tool to study the impact of chemical constituents on the environment, vegetation and human health. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O (sup 1) D (Deuterium) to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a chemistry model. We have investigated a machine learning approach to emulate the chemistry instead of solving the differential equations numerically. From a one-month simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry. This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, etc. We discuss speed and accuracy of our approach, and highlight some potential future directions for improving it.

Keller, Christoph A.↗
NASA NTRS1976

Test particle propagation in magnetostatic turbulence. 2: The local approximation method

An approximation method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local approximation method. Simplification of the integrodifferential equation is achieved through approximation to the Laplace transform of its kernel. The approximation is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this approximation method. The n'th level of approximation is constructed formally, and the first five levels of approximation are calculated explicitly. It is shown that each level of approximation is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The order in time of these partial differential equations is found to increase as n does. At n = 0 the most local first order partial differential equation which governs the Markovian limit is regained.

Klimas, A. J.↗