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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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Conforming window functions for meshfree methods

Window functions provide a base for the construction of approximation functions in many meshfree methods. They control the smoothness and extent of the approximation functions and are commonly defined using Euclidean distances which helps eliminate the need for a meshed discretization, simplifying model development for some classes of problems. However, for problems with complicated geometries such as nonconvex or multi-body domains, poor solution accuracy and convergence can occur unless the extents of the window functions, and thus approximation functions, are carefully controlled, often a time consuming or intractable task. Here in this paper, we present a method to provide more control in window function design, allowing efficient and systematic handling of complex geometries. “Conforming” window functions are constructed using Bernstein–Bézier splines defined on local triangulations with constraints imposed to control smoothness. Graph distances are used in conjunction with Euclidean metrics to provide adequate information for shaping the window functions. The conforming window functions are demonstrated using the Reproducing Kernel Particle Method showing improved accuracy and convergence rates for problems with challenging geometries. Conforming window functions are also demonstrated as a means to simplify the imposition of essential boundary conditions.

, rapid design-to-analysis↗
DOE OSTI2024

Physically interpretable approximations of many-body spectral functions

The rational function approximation provides a natural and interpretable representation of response functions such as the many-body spectral functions. We apply the vector fitting (VFIT) algorithm to fit a variety of spectral functions calculated from the Holstein model of electron-phonon interactions. We show that the resulting rational functions are highly efficient in their fitting of sharp features in the spectral functions, and could provide a means to infer physically relevant information from a spectral data set. The position of the peaks in the approximated spectral function are determined by the location of poles in the complex plane. Additionally, we developed a variant of VFIT that incorporates regularization to improve the quality of fits. With this procedure, we demonstrate it is possible to achieve accurate spectral function fits that vary smoothly as a function of physical conditions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1973

Swirling flows in streamtubes of variable cross section.

The behavior of vortex flow at high swirls is investigated. The problem was first formulated in the method of weighted residuals, using Bessel functions as approximating functions. Convergent solutions require a large number of steps in the axial direction and do not seem to offer an advantage over traditional methods. As the results do not differ substantially from those of the zeroth approximation (the solution for 'cylindrical' flow), this approximation is then explored and discussed in more detail. It is used to derive a relationship predicting a critical radius for which stagnation on the axis (vortex breakdown) can be expected for a given swirl. Flow patterns with open and closed vortex bubbles are explored. At high swirls, even minute variations in outer streamtube radius have profound effects on the flow. Vortex breakdown bubbles in divergences, tubes, and free flows can be explained.

Bossel, H. H.↗
DOE OSTI2022

High-throughput predictions of metal–organic framework electronic properties: theoretical challenges, graph neural networks, and data exploration

Abstract With the goal of accelerating the design and discovery of metal–organic frameworks (MOFs) for electronic, optoelectronic, and energy storage applications, we present a dataset of predicted electronic structure properties for thousands of MOFs carried out using multiple density functional approximations. Compared to more accurate hybrid functionals, we find that the widely used PBE generalized gradient approximation (GGA) functional severely underpredicts MOF band gaps in a largely systematic manner for semi-conductors and insulators without magnetic character. However, an even larger and less predictable disparity in the band gap prediction is present for MOFs with open-shell 3 d transition metal cations. With regards to partial atomic charges, we find that different density functional approximations predict similar charges overall, although hybrid functionals tend to shift electron density away from the metal centers and onto the ligand environments compared to the GGA point of reference. Much more significant differences in partial atomic charges are observed when comparing different charge partitioning schemes. We conclude by using the dataset of computed MOF properties to train machine-learning models that can rapidly predict MOF band gaps for all four density functional approximations considered in this work, paving the way for future high-throughput screening studies. To encourage exploration and reuse of the theoretical calculations presented in this work, the curated data is made publicly available via an interactive and user-friendly web application on the Materials Project.

36 MATERIALS SCIENCE↗
NASA NTRS1999

Structural Reliability Analysis and Optimization: Use of Approximations

This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different approximations including the higher-order reliability methods (HORM) for representing the failure surface. This report is divided into several parts to emphasize different segments of the structural reliability analysis and design. Broadly, it consists of mathematical foundations, methods and applications. Chapter I discusses the fundamental definitions of the probability theory, which are mostly available in standard text books. Probability density function descriptions relevant to this work are addressed. In Chapter 2, the concept and utility of function approximation are discussed for a general application in engineering analysis. Various forms of function representations and the latest developments in nonlinear adaptive approximations are presented with comparison studies. Research work accomplished in reliability analysis is presented in Chapter 3. First, the definition of safety index and most probable point of failure are introduced. Efficient ways of computing the safety index with a fewer number of iterations is emphasized. In chapter 4, the probability of failure prediction is presented using first-order, second-order and higher-order methods. System reliability methods are discussed in chapter 5. Chapter 6 presents optimization techniques for the modification and redistribution of structural sizes for improving the structural reliability. The report also contains several appendices on probability parameters.

Grandhi, Ramana V.↗
DOE OSTI2022

Double and Charge-Transfer Excitations in Time-Dependent Density Functional Theory

Time-dependent density functional theory has emerged as a method of choice for calculations of spectra and response properties in physics, chemistry, and biology, with its system-size scaling enabling computations on systems much larger than otherwise possible. While increasingly complex and interesting systems have been successfully tackled with relatively simple functional approximations, there has also been increasing awareness that these functionals tend to fail for certain classes of approximations. Here I review the fundamental challenges the approximate functionals have in describing double excitations and charge-transfer excitations, which are two of the most common impediments for the theory to be applied in a black-box way. At the same time, I describe the progress made in recent decades in developing functional approximations that give useful predictions for these excitations.

Chemistry↗
DOE OSTI2017

Compressed Continuous Computation v. 12/20/2016

A library for performing numerical computation with low-rank functions. The (C3) library enables performing continuous linear and multilinear algebra with multidimensional functions. Common tasks include taking "matrix" decompositions of vector- or matrix-valued functions, approximating multidimensional functions in low-rank format, adding or multiplying functions together, integrating multidimensional functions.

Gorodetsky, Alex↗
NASA NTRS1997

An X-Ray Analysis Database of Photoionization Cross Sections Including Variable Ionization

Results of research efforts in the following areas are discussed: review of the major theoretical and experimental data of subshell photoionization cross sections and ionization edges of atomic ions to assess the accuracy of the data, and to compile the most reliable of these data in our own database; detailed atomic physics calculations to complement the database for all ions of 17 cosmically abundant elements; reconciling the data from various sources and our own calculations; and fitting cross sections with functional approximations and incorporating these functions into a compact computer code.Also, efforts included adapting an ionization equilibrium code, tabulating results, and incorporating them into the overall program and testing the code (both ionization equilibrium and opacity codes) with existing observational data. The background and scientific applications of this work are discussed. Atomic physics cross section models and calculations are described. Calculation results are compared with available experimental data and other theoretical data. The functional approximations used for fitting cross sections are outlined and applications of the database are discussed.

Wang, Ping↗
NASA NTRS1976

The Relation of Finite Element and Finite Difference Methods

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

Vinokur, M.↗
NASA NTRS1980

Polynomial approximation of functions in Sobolev spaces

Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

Dupont, T.↗
NASA NTRS1993

LVQ and backpropagation neural networks applied to NASA SSME data

Feedfoward neural networks with backpropagation learning have been used as function approximators for modeling the space shuttle main engine (SSME) sensor signals. The modeling of these sensor signals is aimed at the development of a sensor fault detection system that can be used during ground test firings. The generalization capability of a neural network based function approximator depends on the training vectors which in this application may be derived from a number of SSME ground test-firings. This yields a large number of training vectors. Large training sets can cause the time required to train the network to be very large. Also, the network may not be able to generalize for large training sets. To reduce the size of the training sets, the SSME test-firing data is reduced using the learning vector quantization (LVQ) based technique. Different compression ratios were used to obtain compressed data in training the neural network model. The performance of the neural model trained using reduced sets of training patterns is presented and compared with the performance of the model trained using complete data. The LVQ can also be used as a function approximator. The performance of the LVQ as a function approximator using reduced training sets is presented and compared with the performance of the backpropagation network.

Doniere, Timothy F.↗
NASA NTRS1992

A finite element formulation for scattering from electrically large 2-dimensional structures

A finite element formulation is given using the scattered field approach with a fictitious material absorber to truncate the mesh. The formulation includes the use of arbitrary approximation functions so that more accurate results can be achieved without any modification to the software. Additionally, non-polynomial approximation functions can be used, including complex approximation functions. The banded system that results is solved with an efficient sparse/banded iterative scheme and as a consequence, large structures can be analyzed. Results are given for simple cases to verify the formulation and also for large, complex geometries.

Ross, Daniel C.↗
DOE OSTI2024

Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots

Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.

Science & Technology - Other Topics↗
DOE OSTI2017

Approximation of functions and their conjugates in variable Lebesgue spaces

One-sided Steklov means are used to introduce moduli of continuity of natural order in variable L{sub 2π}{sup p(⋅)}-spaces. A direct theorem of Jackson- Stechkin type and an inverse theorem of Salem-Stechkin type are given. Similar results are obtained for the conjugate functions. Bibliography: 24 titles. (paper)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work

Abstract Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary . In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

Johnson, Charles A.↗
NASA NTRS1972

Spline smoothing of histograms by linear programming

An algorithm for an approximating function to the frequency distribution is obtained from a sample of size n. To obtain the approximating function a histogram is made from the data. Next, Euclidean space approximations to the graph of the histogram using central B-splines as basis elements are obtained by linear programming. The approximating function has area one and is nonnegative.

Bennett, J. O.↗