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NASA NTRS1985

Model-size reduction technique for the analysis of symmetric anisotropic structures

A two-step computational procedure is presented for reducing the size of the analysis model for an anisotropic symmetric structure to that of the corresponding orthotropic structure. The key elements of the procedure are: (1) decomposition of the stiffness matrix into the sum of an orthotropic and nonorthotropic (anisotropic) parts; and (2) successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique. The global approximation vectors are selected to be the solution corresponding to zero nonorthotropic matrix and its various-order derivatives with respect to an anisotropic tracing parameter (identifying the nonorthotropic material coefficients). The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure. The effectiveness of the proposed technique is demonstrated by means of numerical examples and its potential for solving other quasi-symmetric problems is discussed.

Noor, A. K.↗

NASA NTRS1988

Attitude determination and parameter estimation using vector observations

Procedures for attitude determination based on Wahba's loss function are generalized to include the estimation of parameters other than the attitude, such as sensor biases. Optimization with respect to the attitude requires either the singular value decomposition of a 3x3 matrix or finding the maximum eigenvalue and corresponding eigenvector of a 4x4 symmetric matrix, but does not require an a priori estimate of the attitude. Optimization with respect to the other parameters employs an iterative approach, which does require an a priori estimate of these parameters. Conventional state estimation methods require a priori estimates of both the parameters and the attitude, while the algorithms presented in this paper always compute the exact optimal attitude for given values of the parameters. The proposed method is shown to give the correct solution of an example problem. An expression for the covariance of the attitude and parameter estimates is derived.

Markley, F. Landis↗

NASA NTRS1989

Exploiting symmetries in the modeling and analysis of tires

A computational procedure is presented for reducing the size of the analysis models of tires having unsymmetric material, geometry and/or loading. The two key elements of the procedure when applied to anisotropic tires are: (1) decomposition of the stiffness matrix into the sum of an orthotropic and nonorthotropic parts; and (2) successive application of the finite-element method and the classical Rayleigh-Ritz technique. The finite-element method is first used to generate few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique. The proposed technique has high potential for handling practical tire problems with anisotropic materials, unsymmetric imperfections and asymmetric loading. It is also particularly useful for use with three-dimensional finite-element models of tires.

Noor, Ahmed K.↗

NASA NTRS1989

A block-corrected subdomain solution procedure for recirculating flow calculations

This paper describes a robust and efficient subdomain solution procedure for two-dimensional recirculating flows. The solution domain is divided into a number of overlapping subdomains, and a direct fully coupled solution is obtained for each subdomain using a sparse matrix form of LU decomposition. An effective parabolic block correction procedure, which calculates global corrections to the tentative solution by a marching technique similar to that used for boundary layer flows, is used to accelerate the convergence of the basic procedure. The use of effective block correction is found to be essential for the success of the subdomain approach on strongly recirculating flows. In a number of laminar two-dimensional flows, the new block-corrected method performed extremely well, rivaling the best direct methods in execution time, while requiring substantially less computer storage. The new method proved to be from two to ten times faster than conventional iterative methods, while requiring only a moderate increase in storage.

Braaten, M. E.↗

NASA NTRS1992

Isotopic and molecular analyses of hydrocarbons and monocarboxylic acids of the Murchison meteorite

The monocarboxylic acids and hydrocarbons of the Murchison meteorite (CM2) were isolated for isotropic analysis. The nonvolatile hydrocarbons were analyzed as crude methanol and benzene-methanol extracts and also after separation by silica gel chromatography into predominantly aliphatic, aromatic, and polar hydrocarbon fractions. The volatile hydrocarbons were obtained after progressive decomposition of the meteorite matrix by freeze-thaw, hot water, and acid treatment. Molecular analyses of the aromatic hydrocarbons showed them to comprise a complex suite of compounds in which pyrene, fluoranthene, phenanthrene, and acenaphthene were the most abundant components, a result similar to earlier analyses. The polar hydrocarbons also comprise a very complex mixture in which aromatic ketones, nitrogen, and sulfur heterocycles were identified. The monocarboxylic acids, aliphatic, aromatic, and polar hydrocarbons, and the indigenous volatile hydrocarbons were found to be D-rich. The deuterium enrichment observed in these compounds is suggestive. In two separate analyses, the delta-D values of the nonvolatile hydrocarbons were observed to increase in the following order: aliphatic-aromatic-polar. This finding is consistent with an early solar system or parent body conversion of aromatic to aliphatic compounds as well as the suggestion of pyrolytic formation of aromatic from aliphatic compounds.

Krishnamurthy, R. V.↗

NASA NTRS2000

Automated Design of Quantum Circuits

In order to design a quantum circuit that performs a desired quantum computation, it is necessary to find a decomposition of the unitary matrix that represents that computation in terms of a sequence of quantum gate operations. To date, such designs have either been found by hand or by exhaustive enumeration of all possible circuit topologies. In this paper we propose an automated approach to quantum circuit design using search heuristics based on principles abstracted from evolutionary genetics, i.e. using a genetic programming algorithm adapted specially for this problem. We demonstrate the method on the task of discovering quantum circuit designs for quantum teleportation. We show that to find a given known circuit design (one which was hand-crafted by a human), the method considers roughly an order of magnitude fewer designs than naive enumeration. In addition, the method finds novel circuit designs superior to those previously known.

Williams, Colin P.↗

NASA NTRS2004

Determination of the Three-Dimensional Morphology of ALH84001 and Biogenic MV-1 Magnetite: Comparison of Results from Electron Tomography and Classical Transmission Electron Microscopy

Dated at approximately 3.9 billion years of age, carbonate disks, found within fractures of the host rock of Martian meteorite ALH84001, have been interpreted as secondary minerals that formed at low temperature in an aqueous medium. Heterogeneously distributed within these disks are magnetite nanocrystals that are of Martian origin. Approximately one quarter of these magnetites have morphological and chemical similarities to magnetite particles produced by magnetotactic bacteria strain MV-1, which are ubiquitous in aquatic habitats on Earth. Moreover, these types of magnetite particles are not known or expected to be produced by abiotic means either through geological processes or synthetically in the laboratory. The remaining three quarters of the ALH84001 magnetites are likely products of multiple processes including, but not limited to, precipitation from a hydrothermal fluid, thermal decomposition of the carbonate matrix in which they are embedded, and extracellular formation by dissimilatory Fe-reducing bacteria. We have proposed that the origins of magnetites in ALH84001 can be best explained as the products of multiple processes, one of which is biological. Recently the three-dimensional (3-D) external morphology of the purported biogenic fraction of the ALH84001 magnetites has been the subject of considerable debate. We report here the 3-D geometry of biogenic magnetite crystals extracted from MV-1 and of those extracted from ALH84001 carbonate disks using a combination of high resolution classical and tomographic transmission electron microscopy (TEM). We focus on answering the following questions: (1) which technique provides adequate information to deduce the 3-D external crystal morphology?; and, (2) what is the precise 3-D geometry of the ALH84001 and MV-1 magnetites?

Thomas-Keprta, Kathie L.↗

NASA NTRS1998

Automated Design of Quantum Circuits

evolutionary programming↗

DOE OSTI2024

Detecting damage in composites using volume decomposition analysis of tomographic data

Detection of damage in a single tow ceramic matrix composite specimen has been achieved using orthogonal decomposition of volumetric tomographic datasets collected at four tensile loads. This decomposition approach has been applied at two different length scales: (i) individual fibres and (ii) bulk volumes containing fibres and matrix material. Volumes were first decomposed to feature vectors, orders of magnitude smaller than the original volume they describe, and then comparisons between datasets at different load levels were made in feature vector space. The results show quantitative measurements of damage location, damage morphology and the relative growth of this damage with increased load when compared with a dataset with less or no damage. No prior knowledge of the dataset or training of algorithms is required for damage to be detected, it is only necessary that at least two datasets are available for comparison, e.g. from in situ or repeated scanning measurements. Results are generated on significantly shorter timescales when compared with previous automated approaches to tomography data processing. This approach has the potential to be applied to damage detection in a range of materials through comparisons of volumetric datasets from a range of measurement or computational techniques.

Middleton, Ceri A.↗

NASA NTRS2001

Dynamic modeling of gene expression data

We describe the time evolution of gene expression levels by using a time translational matrix to predict future expression levels of genes based on their expression levels at some initial time. We deduce the time translational matrix for previously published DNA microarray gene expression data sets by modeling them within a linear framework by using the characteristic modes obtained by singular value decomposition. The resulting time translation matrix provides a measure of the relationships among the modes and governs their time evolution. We show that a truncated matrix linking just a few modes is a good approximation of the full time translation matrix. This finding suggests that the number of essential connections among the genes is small.

NASA Discipline Plant Biology↗

NASA NTRS2009

Thermal Decomposition of an Impure (Roxbury) Siderite: Relevance to the Presence of Chemically Pure Magnetite Crystals in ALH84001 Carbonate Disks

The question of the origin of nanophase magnetite in Martian meteorite ALH84001 has been widely debated for nearly a decade. Golden et al. have reported producing nearly chemically pure magnetite from thermal decomposition of chemically impure siderite [(Fe, Mg, Mn)CO3]. This claim is significant for three reasons: first, it has been argued that chemically pure magnetite present in the carbonate disks in Martian meteorite ALH84001 could have formed by the thermal decomposition of the impure carbonate matrix in which they are embedded; second, the chemical purity of magnetite has been previously used to identify biogenic magnetite; and, third, previous studies of thermal decomposition of impure (Mg,Ca,Mn)-siderites, which have been investigated under a wide variety of conditions by numerous researchers, invariably yields a mixed metal oxide phase as the product and not chemically pure magnetite. The explanation for this observation is that these siderites all possess the same crystallographic structure (Calcite; R3c) so solid solutions between these carbonates are readily formed and can be viewed on an atomic scale as two chemically different but structurally similar lattices.

McKay, D.S.↗

DOE OSTI2020

Asymptotic behavior of meson transition form factors

One of the open issues in evaluations of the contribution from hadronic light-by-light scattering to the anomalous magnetic moment of the muon (g - 2)µ concerns the role of heavier scalar, axial-vector, and tensormeson intermediate states. The coupling of axial vectors to virtual photons is suppressed for small virtualities by the Landau–Yang theorem, but otherwise there are few rigorous constraints on the corresponding form factors. In this paper, we first derive the Lorentz decomposition of the two-photon matrix elements into scalar functions following the general recipe by Bardeen, Tung, and Tarrach. Based on this decomposition, we then calculate the asymptotic behavior of the meson transition form factors from a light-cone expansion in analogy to the asymptotic limits for the pseudoscalar transition form factor derived by Brodsky and Lepage. Finally, we compare our results to existing data as well as previous models employed in the literature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRSDate not supplied

Dynamic Mode Decomposition of Unsteady Pressure-Sensitive Paint Measurements for the NASA Unitary Plan Wind Tunnel Tests

This paper describes the Dynamic Mode Decomposition (DMD) of the pressures on the scale model of the Space Launch System (SLS) Block 1 cargo vehicle with the Unsteady Pressure-Sensitive Paint (uPSP) measurements, which were collected in the Ascent Transient Aerodynamics Tests with the Unitary Plan Wind Tunnel 11-by-11-foot Transonic Wind Tunnel in September 2019 at NASA Ames Research Center. The work described in this paper is a part of NASA’s development of a new state-of-the-art uPSP capability in production wind tunnels. The conventional DMD algorithm is based on the Singular Value Decomposition (SVD) of the data matrix. For the matrix of the uPSP measurements of the SLS ATAT, the number of rows is equal to the number of nodes in the grid of the scale model, and the number of columns is equal to the number of frames in the videos taken with 4 Phantom high-speed cameras. In this paper, it is verified that, for the time series with zero mean value, the DMD is equivalent to the decomposition with the Discrete Fourier Transform (DFT). Considering the uPSP is mainly used in the assessment of the unsteady, aerodynamic phenomena, the DMD of the uPSP measurements can be implemented in two steps: (1) subtract the mean value from the uPSP measurement on each of the grid nodes; (2) apply the Fast Fourier Transform (FFT) on the resulting zero-mean time series. The DMD of the uPSP measurements with FFT has two advantages: (1) the computational complexity of FFT is O(N*logN), where N is the length of the time series; (2) compared to the SVD-based DMD algorithm, the DMD with FFT can be easily implemented in parallel processing. A sample matrix of uPSP measurements, at the size of 341 grid nodes and 128 frames, is generated. Figures 1 and 2 show the eigenvalues and the ratios of the eigenvectors, respectively, of the sample matrix, without and with the mean value removed on each of the grid nodes, computed with the SVD-based DMD and the FFT. The figures demonstrate the equivalence of the SVD-based DMD and the decomposition with DFT/FFT for the time series with zero mean value. The results of DMD of the uPSP measurements of the SLS ATAT in September 2019 are presented in the paper. The DMD modes at different frequencies are shown, the aerodynamic phenomena (e.g. shockwave and vortex shedding) are demonstrated and the correlation of the DMD modes with the test configuration parameter (e.g., the Mach Number) is discussed. Figure 3 shows a software tool to visualize the DMD modes. The code to implement the algorithm described in this paper was written in C, with libraries of FFTW for FFT and MPI/OpenMP for parallel processing, and executed on the NASA Pleiades supercomputer. Funding for this research was provided by the NASA Aerosciences Evaluation and Test Capabilities Project.

Pressure-Sensitive Paint↗

DOE OSTI2021

Scattering amplitudes for monopoles: pairwise little group and pairwise helicity

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding “pairwise helicity” is identified with the quantized “cross product” of charges, e 1 g 2 - e 2 g 1 , for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}$ 2 ). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization X = W$\land$H has a unique W, a unique H (for a fixed W), and when both W and H are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMFk, and show on numerical examples that BMFk not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1987

Nonlinear beam kinematics by decomposition of the rotation tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

Danielson, D. A.↗

NASA NTRS1987

Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion extension, shear deformation, and warping of an initially curved and twisted beam.

Danielson, D. A.↗

NASA NTRS1988

Statistical analysis of effective singular values in matrix rank determination

A major problem in using SVD (singular-value decomposition) as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end, conference regions are derived for the perturbed singular values of matrices with noisy observation data. The analysis is based on the theories of perturbations of singular values and statistical significance test. Threshold bounds for perturbation due to finite-precision and i.i.d. random models are evaluated. In random models, the threshold bounds depend on the dimension of the matrix, the noisy variance, and predefined statistical level of significance. Results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered. Various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given.

Konstantinides, Konstantinos↗