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At least 775 records · Page 43

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DOE OSTI2021

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1972

Comparison of satellite theories

The accuracy of five mathematical models in computing a nominal orbit for the Vanguard 2 satellite by using a position velocity vector is considered. Either numerical integration or analytical theories are used in all models as well as the same force model that corresponds to a potential with the zonal harmonics to order four. The amounts of spread in the values of the total energy and the z-component of the angular momentum for a set of times are considered as measures of accuracy.

Hertz, H. G.↗

NASA NTRS1976

A limiting analysis for edge effects in angle-ply laminates

A zeroth order solution for edge effects in angle ply composite laminates using perturbation techniques and a limiting free body approach was developed. The general method of solution for laminates is developed and then applied to the special case of a graphite/epoxy laminate. Interlaminar stress distributions are obtained as a function of the laminate thickness to width ratio h/b and compared to existing numerical results. The solution predicts stable, continuous stress distributions, determines finite maximum tensile interlaminar normal stress for two laminates, and provides mathematical evidence for singular interlaminar shear stresses.

Hsu, P. W.↗

NASA NTRS1978

A simple element for multilayer beams in NASTRAN thermal stress analysis

In the application of NASTRAN, structural members are usually represented by bar elements with multipoint constraint cards to enforce the interface conditions. While this is a very powerful method in principle, it was found that in practice the process for specification of constraints became tedious and error prone, unless the geometry was simple and the number of grid points low. An alternative approach was found within the framework of the NASTRAN program. This approach made use of the idea that a thermal distortion in a multilayer beam may be similar to a homogeneous beam with a thermal gradient across the cross section. The exact mathematical derivation for the equivalent beam, and all the necessary formulae for the equivalent parameters in NASTRAN analysis are presented. Some numerical examples illustrate the simplicity and ease of this approach for finite element analysis.

Chen, W. T.↗

NASA NTRS1981

Analysis of coherent satellite communication systems in the presence of interference and noise

The general analysis presented can be used to evaluate the performance of M-ary phase shift keying (MPSK) used over a nonlinear satellite channel in the presence of interference (CW tone and ISI) and channel noise. The numerical evaluation is based on the classical moment technique for approximating the probability distribution from moments of a random variable. The mathematical formulation of error probabilities of MPSK signals transmitted over nonlinear satellite repeaters is presented to include the effect of CW tone interference as well as the intersymbol interference. The approach is general and can be extended to include the effect of the synchronization.

Huang, T.-C.↗

NASA NTRS1984

Thermal shock in a circumferentially cracked hollow cylinder with cladding

An theoretical analysis is presented which demonstrates the effect of cladding on the thermal resistance of a circumferentially cracked hollow cylinder. The cladding is assumed to be bonded to the inner wall of the hollow cylinder. The axisymmetric circumferential crack may be either embedded in the cylinder wall or may be an edge crack which passes through the clad and opens into the inner wall of the hollow cylinder. The problem is formulated mathematically and a solution is found which is in the form of a single integral equation. The integral equation is solved numerically and yields estimates of transient temperature distributions, thermal stresses in the uncracked cylinder, and stress intensity factors as a function of time for various cladding thickness to cylinder wall thickness ratios. It is shown that yielding of the clad under certain conditions can result in a reduction in the magnitude of the stress intensity factor for the crack tip in the elastic base material.

Nied, H. F.↗

NASA NTRS1989

Mechanical response of thick laminated beams and plates subject to out-of-plane loading

The use of simplified elasticity solutions to determine the mechanical response of thick laminated beams and plates subject to out-of-plane loading is demonstrated. Excellent results were obtained which compare favorably with theoretical, numerical and experimental analyses from other sources. The most important characteristic of the solution methodology presented is that it combines great mathematical precision with simplicity. This symbiosis has been needed for design with advanced composite materials.

Hiel, C. C.↗

NASA NTRS1988

Mathematical and statistical analysis

The goal of the mathematical and statistical analysis component of RICIS is to research, develop, and evaluate mathematical and statistical techniques for aerospace technology applications. Specific research areas of interest include modeling, simulation, experiment design, reliability assessment, and numerical analysis.

Houston, A. Glen↗

NASA NTRS2001

Geomagnetic Cutoff Rigidity Computer Program: Theory, Software Description and Example

The access of charged particles to the earth from space through the geomagnetic field has been of interest since the discovery of the cosmic radiation. The early cosmic ray measurements found that cosmic ray intensity was ordered by the magnetic latitude and the concept of cutoff rigidity was developed. The pioneering work of Stoermer resulted in the theory of particle motion in the geomagnetic field, but the fundamental mathematical equations developed have 'no solution in closed form'. This difficulty has forced researchers to use the 'brute force' technique of numerical integration of individual trajectories to ascertain the behavior of trajectory families or groups. This requires that many of the trajectories must be traced in order to determine what energy (or rigidity) a charged particle must have to penetrate the magnetic field and arrive at a specified position. It turned out the cutoff rigidity was not a simple quantity but had many unanticipated complexities that required many hundreds if not thousands of individual trajectory calculations to solve. The accurate calculation of particle trajectories in the earth's magnetic field is a fundamental problem that limited the efficient utilization of cosmic ray measurements during the early years of cosmic ray research. As the power of computers has improved over the decades, the numerical integration procedure has grown more tractable, and magnetic field models of increasing accuracy and complexity have been utilized. This report is documentation of a general FORTRAN computer program to trace the trajectory of a charged particle of a specified rigidity from a specified position and direction through a model of the geomagnetic field.

Smart, D. F.↗

DOE OSTI2021

Practical CO2—WAG Field Operational Designs Using Hybrid Numerical-Machine-Learning Approaches

Machine-learning technologies have exhibited robust competences in solving many petroleum engineering problems. The accurate predictivity and fast computational speed enable a large volume of time-consuming engineering processes such as history-matching and field development optimization. The Southwest Regional Partnership on Carbon Sequestration (SWP) project desires rigorous history-matching and multi-objective optimization processes, which fits the superiorities of the machine-learning approaches. Although the machine-learning proxy models are trained and validated before imposing to solve practical problems, the error margin would essentially introduce uncertainties to the results. In this paper, a hybrid numerical machine-learning workflow solving various optimization problems is presented. By coupling the expert machine-learning proxies with a global optimizer, the workflow successfully solves the history-matching and CO2 water alternative gas (WAG) design problem with low computational overheads. The history-matching work considers the heterogeneities of multiphase relative characteristics, and the CO2-WAG injection design takes multiple techno-economic objective functions into accounts. This work trained an expert response surface, a support vector machine, and a multi-layer neural network as proxy models to effectively learn the high-dimensional nonlinear data structure. The proposed workflow suggests revisiting the high-fidelity numerical simulator for validation purposes. The experience gained from this work would provide valuable guiding insights to similar CO2 enhanced oil recovery (EOR) projects.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2021

Caravel: A C++ framework for the computation of multi-loop amplitudes with numerical unitarity

We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is composed of modules for the D-dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. Here, we provide programs that showcase Caravel's main functionalities and allow to compute selected one- and two-loop amplitudes.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

A Continuing Search for a Near-Perfect Numerical Flux Scheme

While enjoying demonstrated improvement in accuracy, efficiency, and robustness over existing schemes, the Advection Upstream Splitting Scheme (AUSM) was found to have some deficiencies in extreme cases. This recent progress towards improving the AUSM while retaining its advantageous features is described. The new scheme, termed AUSM+, features: unification of velocity and Mach number splitting; exact capture of a single stationary shock; and improvement in accuracy. A general construction of the AUSM+ scheme is layed out and then focus is on the analysis of the a scheme and its mathematical properties, heretofore unreported. Monotonicity and positivity are proved, and a CFL-like condition is given for first and second order schemes and for generalized curvilinear co-ordinates. Finally, results of numerical tests on many problems are given to confirm the capability and improvements on a variety of problems including those failed by prominent schemes.

Liou, Meng-Sing↗

NASA NTRS2010

DigitalHuman (DH): An Integrative Mathematical Model ofHuman Physiology

Mathematical models and simulation are important tools in discovering the key causal relationships governing physiological processes and improving medical intervention when physiological complexity is a central issue. We have developed a model of integrative human physiology called DigitalHuman (DH) consisting of -5000 variables modeling human physiology describing cardiovascular, renal, respiratory, endocrine, neural and metabolic physiology. Users can view time-dependent solutions and interactively introduce perturbations by altering numerical parameters to investigate new hypotheses. The variables, parameters and quantitative relationships as well as all other model details are described in XML text files. All aspects of the model, including the mathematical equations describing the physiological processes are written in XML open source, text-readable files. Model structure is based upon empirical data of physiological responses documented within the peer-reviewed literature. The model can be used to understand proposed physiological mechanisms and physiological interactions that may not be otherwise intUitively evident. Some of the current uses of this model include the analyses of renal control of blood pressure, the central role of the liver in creating and maintaining insulin resistance, and the mechanisms causing orthostatic hypotension in astronauts. Additionally the open source aspect of the modeling environment allows any investigator to add detailed descriptions of human physiology to test new concepts. The model accurately predicts both qualitative and more importantly quantitative changes in clinically and experimentally observed responses. DigitalHuman provides scientists a modeling environment to understand the complex interactions of integrative physiology. This research was supported by.NIH HL 51971, NSF EPSCoR, and NASA

Hester, Robert L.↗

NASA NTRS1972

Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures

A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.

Scott, R. K.↗

DOE OSTI2025

Hardware acceleration for HPS algorithms in two and three dimensions

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincaré–Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU; in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [1] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems.

Fast direct solvers↗

DOE OSTI2024

Calculation of machine precision second order derivatives using dual-complex numbers

It is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision 2nd order derivatives. To address this limitation, it is demonstrated in this paper that combined dual-complex numbers can be used to compute machine precision 1st and 2nd order derivatives. The dual-complex approach is simpler than utilizing multicomplex or hyper-dual numbers as existing dual libraries can be used as is or easily augmented to accept complex numbers, and the complexity of developing, integrating, and deploying multicomplex or hyper-dual libraries is avoided. The efficacy of this approach is demonstrated for both univariate and multivariate functions. Finally, source code examples using the Python, Julia, and Mathematica languages are provided as supplemental material.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Finally, our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.

Auto-encoder↗

DOE OSTI2022

Deep Learning-Assisted Real-Time Forward Modeling of Electromagnetic Logging in Complex Formations

Higher dimensional (i.e., 2-D and 3-D) modeling is indispensable to correctly evaluate the responses of electromagnetic logging tools in complex formation environments. However, limited by the high computational cost of rigorous modeling such as finite difference method and the finite element method, the real-time applications in the well logging industry primarily rely on the 1-D forward solver, which would result in erroneous formation evaluation for complex scenarios. As a result, aiming at realizing fast modeling for electromagnetic logging tools in complex formations, this paper proposes a general framework assisted by deep neural networks (DNNs). The framework consists of three modules: earth model classification, parameter extraction, and surrogate construction. Separate DNNs are trained and tested for different modules. The accuracy and efficiency of the DNN assisted fast modeling are validated by several experiments. Here, this study finds that the fast modeling assisted by DNNs is able to calculate the tool responses and reconstruct the subsurface formations in real-time.

97 MATHEMATICS AND COMPUTING↗