Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Generalized functions”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,027 records · Page 57
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Environmental remediation with functional aerogels and xerogels

Several different types of aerogel and/or xerogel scaffolds have been demonstrated as effective sorbents for the capture and immobilization of radionuclides in gaseous form [e.g., iodine gas or I2(g), Xe] as well as ionic form (e.g., Ce4+, Cs+, I–, IO3-, Rb+, Sr2+, 99Tc7+, and U6+). These scaffolds have unique properties, which include high specific surface areas, high pore volumes, varieties of pore sizes, and functionalities that provide methods for binding radionuclides through physisorption, chemisorption, or a combination thereof. This combination of properties and functionalities make these types of materials ideal scaffolds for use as sorbents for capturing radionuclides. The primary base materials that will be discussed in this chapter include Ag0-functionalized silica aerogels, Ag+-impregnated aluminosilicate aerogels, Ag0-functionalized aluminosilicate aerogels, metal-impregnated (non-Ag) aluminosilicate aerogels and xerogels, sulfide-based aerogels, and carbon-based aerogel composites. For the capture of I2(g), the materials reported herein show some of the highest iodine loadings ever reported for inorganic sorbents. For the capture of ionic species, these materials also show promise to be some of the next generations of materials for active radionuclide remediation. This progress report will describe how these materials are fabricated, the general properties of these materials, as well as an overview of how they have been used for different applications in environmental remediation of radionuclides.

aerogel, xerogel, iodine, radionuclide remediation↗
DOE OSTI2021

Gauge invariance and Ward identities in nonlinear response theory

We present a formal analysis of nonlinear response functions in terms of correlation functions in real- and imaginary-time domains. In particular, we show that causal nonlinear response functions, expressed in terms of nested commutators in real time, can be obtained from the analytic continuation of time-ordered response functions, which are more easily amenable to diagrammatic calculation. This generalizes the well-known result of linear response theory. We then use gauge invariance arguments to derive exact relations between second-order response functions in density and current channels. These identities, which are non-perturbative in the strength of inter-particle interactions, allow us to establish exact connections between nonlinear optics calculations done in different electromagnetic gauges.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Empirical Study of the Effect of Thermal Loading on the Heating Efficiency of Variable-Speed Air Source Heat Pumps

Heating buildings with air source heat pumps (ASHPs) has the potential to save energy compared to utilizing conventional heat sources. Accurate understanding of the efficiency of ASHPs is important to maximize the energy savings. While it is well understood that, in general, ASHP efficiency decreases with decreasing outdoor temperature, it is not well understood how the ASHP efficiency changes with different levels of thermal loading, even though it is an important consideration for sizing and controlling ASHPs. The goal of this study was to create an empirical model of the ASHP efficiency as a function of two independent variables–outside temperature and level of thermal loading. Four ductless mini-split ASHPs were evaluated in a cold chamber where the temperature (representing the outdoor temperature) was varied over a wide range. For each temperature, the ASHP performance data were collected at several levels of thermal loading. The data for all four ASHPs were combined and approximated with an analytical function that can be used as a general model for the ASHP steady-state efficiency as a function of the outside temperature and level of thermal loading. To the knowledge of the authors, no such empirical model that is solely based on third-party test data has been published before. While limitations exist, the model can be used to help guide future selection and operation of ASHPs.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
NASA NTRS1981

Variation Principles and Applications in the Study of Cell Structure and Aging

In this report we have attempted to show that "some reality lies concealed in biological variation". This "reality" has its principles, laws, mechanisms, and rules, only a few of which we have sketched. A related idea we pursued was that important information may be lost in the process of ignoring frequency distributions of physiological variables (as is customary in experimental physiology and gerontology). We suggested that it may be advantageous to expand one's "statistical field of vision" beyond simple averages +/- standard deviations. Indeed, frequency distribution analysis may make visible some hidden information not evident from a simple qualitative analysis, particularly when the effect of some external factor or condition (e.g., aging, dietary chemicals) is being investigated. This was clearly illustrated by the application of distribution analysis in the study of variation in mouse liver cellular and fine structure, and may be true of fine structural studies in general. In living systems, structure and function interact in a dynamic way; they are "inseparable," unlike in technological systems or machines. Changes in fine structure therefore reflect changes in function. If such changes do not exceed a certain physiologic range, a quantitative analysis of structure will provide valuable information on quantitative changes in function that may not be possible or easy to measure directly. Because there is a large inherent variation in fine structure of cells in a given organ of an individual and among individuals, changes in fine structure can be analyzed only by studying frequency distribution curves of various structural characteristics (dimensions). Simple averages +/- S.D. do not in general reveal all information on the effect of a certain factor, because often this effect is not uniform; on the contrary, this will be apparent from distribution analysis because the form of the curves will be affected. We have also attempted to show in this chapter that similar general statistical principles and mechanisms may be operative in biological and technological systems. Despite the common belief that most biological and technological characteristics of interest have a symmetric bell-shaped (normal or Gaussian) distribution, we have shown that more often than not, distributions tend to be asymmetric and often resemble a so-called log-normal distribution. We saw that at least three general mechanisms may be operative, i.e., nonadditivity of influencing factors, competition among individuals for a common resource, and existence of an "optimum" value for a studied characteristic; more such mechanisms could exist.

Economos, Angelos C.↗
DOE OSTI2021

Energy-Resolved Information Scrambling in Energy-Space Lattices

Here, weakly interacting Fermi gases simulate spin lattices in energy space, offering a rich platform for investigating information spreading and spin coherence in a large many-body quantum system. We show that the collective spin vector can be determined as a function of energy from the measured spin density, enabling general energy-space resolved protocols. We measure an out-of-time-order correlation function in this system and observe the energy dependence of the many-body coherence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Shortcomings of meta-GGA functionals when describing magnetism

Several recent studies have shown that SCAN, a functional belonging to the meta-generalized gradient approximation (MGGA) family, leads to significantly overestimated magnetic moments in itinerant ferromagnetic metals. However, this behavior is not inherent to the MGGA level of approximation since TPSS, for instance, does not lead to such severe overestimations. In order to provide a broader view of the accuracy of MGGAfunctionals for magnetism, we extend the assessment to more functionals but also to antiferromagnetic solids. The results show that to describe magnetism there is overall no real advantage in using a MGGA functional compared to GGAs. For both types of approximation, an improvement in ferromagnetic metals is necessarily accompanied by a deterioration (underestimation) in antiferromagnetic insulators, and vice versa. Furthermore, we also provide some analysis in order to understand in more detail the relation between the mathematical form of the functionals and the results.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Engineering local strain for single-atom nuclear acoustic resonance in silicon

Mechanical strain plays a key role in the physics and operation of nanoscale semiconductor systems, including quantum dots and single-dopant devices. In this report we describe the design of a nanoelectronic device, where a single nuclear spin is coherently controlled via nuclear acoustic resonance (NAR) through the local application of dynamical strain. The strain drives spin transitions by modulating the nuclear quadrupole interaction. We adopt an AlN piezoelectric actuator compatible with standard silicon metal–oxide–semiconductor processing and optimize the device layout to maximize the NAR drive. We predict NAR Rabi frequencies of order 200 Hz for a single 123Sb nucleus in a wide region of the device. Spin transitions driven directly by electric fields are suppressed in the center of the device, allowing the observation of pure NAR. Using electric field gradient-elastic tensors calculated by the density-functional theory, we extend our predictions to other high-spin group-V donors in silicon and to the isoelectronic 73Ge atom.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

On the minimum number of radiation field parameters to specify gas cooling and heating functions

Fast and accurate approximations of gas cooling and heating functions are needed for hydrodynamic galaxy simulations. We use machine learning to analyze atomic gas cooling and heating functions in the presence of a generalized incident local radiation field computed by Cloudy. We characterize the radiation field through binned radiation field intensities instead of the photoionization rates used in our previous work. We find a set of 6 energy bins whose intensities exhibit relatively low correlation. We use these bins as features to train machine learning models to predict Cloudy cooling and heating functions at fixed metallicity. We compare the relative SHapley Additive exPlanation (SHAP) value importance of the features. From the SHAP analysis, we identify a feature subset of 3 energy bins (0.5-1, 1-4, and 13-16Ry) with the largest importance and train additional models on this subset. We compare the mean squared errors and distribution of errors on both the entire training data table and a randomly selected 20% test set withheld from model training. The machine learning models trained with 3 and 6 bins, as well as 3 and 4 photoionization rates, have comparable accuracy everywhere, with errors ≳10 times smaller than for the interpolation table of Gnedin and Hollon (2012). We conclude that 3 energy bins (or 3 analogous photoionization rates: molecular hydrogen photodissociation, neutral hydrogen HI, and fully ionized carbon CVI) are sufficient to characterize the dependence of the gas cooling and heating functions on our assumed incident radiation field model.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2004

Parametric Studies of Square Solar Sails Using Finite Element Analysis

Parametric studies are performed on two generic square solar sail designs to identify parameters of interest. The studies are performed on systems-level models of full-scale solar sails, and include geometric nonlinearity and inertia relief, and use a Newton-Raphson scheme to apply sail pre-tensioning and solar pressure. Computational strategies and difficulties encountered during the analyses are also addressed. The purpose of this paper is not to compare the benefits of one sail design over the other. Instead, the results of the parametric studies may be used to identify general response trends, and areas of potential nonlinear structural interactions for future studies. The effects of sail size, sail membrane pre-stress, sail membrane thickness, and boom stiffness on the sail membrane and boom deformations, boom loads, and vibration frequencies are studied. Over the range of parameters studied, the maximum sail deflection and boom deformations are a nonlinear function of the sail properties. In general, the vibration frequencies and modes are closely spaced. For some vibration mode shapes, local deformation patterns that dominate the response are identified. These localized patterns are attributed to the presence of negative stresses in the sail membrane that are artifacts of the assumption of ignoring the effects of wrinkling in the modeling process, and are not believed to be physically meaningful. Over the range of parameters studied, several regions of potential nonlinear modal interaction are identified.

Sleight, David W.↗
DOE OSTI2024

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), https://doi.org/10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. Here, we examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 x 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), doi:10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. We examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 × 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2015

Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier- Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle

Miller, Steven A. E.↗
NASA NTRS2015

Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier-Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle.

Miller, Steven A. E.↗
NASA NTRS2014

Transient Thermoelectric Solution Employing Green's Functions

The study works to formulate convenient solutions to the problem of a thermoelectric couple operating under a time varying condition. Transient operation of a thermoelectric will become increasingly common as thermoelectric technology permits applications in an increasing number of uses. A number of terrestrial applications, in contrast to steady-state space applications, can subject devices to time varying conditions. For instance thermoelectrics can be exposed to transient conditions in the automotive industry depending on engine system dynamics along with factors like driving style. In an effort to generalize the thermoelectric solution a Greens function method is used, so that arbitrary time varying boundary and initial conditions may be applied to the system without reformulation. The solution demonstrates that in thermoelectric applications of a transient nature additional factors must be taken into account and optimized. For instance, the materials specific heat and density become critical parameters in addition to the thermal mass of a heat sink or the details of the thermal profile, such as oscillating frequency. The calculations can yield the optimum operating conditions to maximize power output andor efficiency for a given type of device.

thermoelectric power generation↗
DOE OSTI2024

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1978

Design development of the Apollo command and service module thrust vector attitude control systems

Development of the Apollo thrust vector control digital autopilot (TVC DAP) was summarized. This is the control system that provided pitch and yaw attitude control during velocity change maneuvers using the main rocket engine on the Apollo service module. A list of ten primary functional requirements for this control system are presented, each being subordinate to a more general requirement appearing earlier on the list. Development process functions were then identified and the essential information flow paths were explored. This provided some visibility into the particular NASA/contractor interface, as well as relationships between the many individual activities.

William H. Peters↗
NASA NTRS1941

Two-dimensional potential flow past a smooth wall with partly constant curvature

The speed of a two-dimensional flow potential flow past a smooth wall, which evinces a finite curvature jump at a certain point and approximates to two arcs in the surrounding area, has a vertical tangent of inflection in the critical point as a function of the arc length of the boundary curve. This report looks at a general theorem of the local character of the conformal function at the critical point as well as the case of the finite curvature jump.

Koppenfels, Werner Von↗
NASA NTRS2017

New Mathematical Functions for Vacuum System Analysis

A new bivariate function has been found that provides solutions of integrals having the form u (sup minus eta) e (sup u) du which arise when developing predictions for the behavior of pressure within a rigid volume under high vacuum conditions in the presence of venting as well as sources characterized by power law transient decay over the range [0,1] for eta and for u greater than or equal to 0. A few properties of the new function are explored in this work. For instance the eta equals 1/2 case reproduces the Dawson function. In addition, a slight variation of the solution technique reproduces the exponential integral for eta equals 1. The technique used to generate these functions leads to an approach for solving a more general class of nonlinear ordinary differential equations, with the potential for identifying other new functions that solve other integrals.

Dawson Function↗