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NASA NTRS1990

Simultaneous expansion and orthogonalization of measured modes for structure identification

Tests of large structures on-orbit will be performed with measurements at a relatively few structure points. Values for the unmeasured degrees of freedom (dofs) can be estimated based on measured dofs and analytical model dynamic information. These 'expanded' mode shapes are useful for optimal-update identification and damage location as well as test/analysis correlation. A new method of expansion for test mode shape vectors is developed from the orthogonal Procrustes problem from computational linear algebra. A subspace defined by the set of measured dofs is compared to a subspace defined by mode shapes from an analytical model of the structure. The method simultaneously expands and orthogonalizes the mode shape vectors. Two demonstration problems are used to compare the new method to current expansion techniques. One demonstration uses test data from a laboratory scale-model truss structure. Performance of the new method is comparable or superior to that of the previous expansion methods which require separate orthogonalization.

Smith, Suzanne Weaver↗

NASA NTRS1990

Minimal parameter solution of the orthogonal matrix differential equation

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Bar-Itzhack, Itzhack Y.↗

NASA NTRS1989

Recursive, in-place algorithm for the hexagonal orthogonal oriented quadrature image pyramid

Pyramid image transforms have proven useful in image coding and pattern recognition. The hexagonal orthogonal oriented quadrature image pyramid (HOP), transforms an image into a set of orthogonal, oriented, odd and even bandpass subimages. It operates on a hexagonal input lattice and employs seven kernels, each of which occupies a neighborhood consisting of a point and a hexagon of six nearest neighbors. The kernels consist of one lowpass and six bandpass kernels that are orthogonal, self-similar, and localized in space, spatial frequency, orientation, and phase. The kernels are first applied to the image samples to create the first level of the pyramid, then to the lowpass coefficients to create the next level. The resulting pyramid is a compact, efficient image code. Here, a recursive, in-place algorithm for computation of the HOP transform is described. The transform may be regarded as a depth-first traversal of a tree structure. It is shown that the algorithm requires a number of operations that is on the order of the number of pixels.

Watson, Andrew B.↗

NASA NTRS1992

Orthogonal canonical forms for second-order systems

It is shown that a linear second-order system with arbitrary damping cannot be reduced to Hessenberg-triangular form by means of orthogonal transformations. However, it is also shown that such an orthogonal reduction is always possible for the modal damping commonly assumed for models of flexible structures. It is shown that modally damped models can be orthogonally reduced to a new triangular second-order Schur form.

Williams, Trevor↗

NASA NTRS1993

Nonlinear aerodynamic modeling using multivariate orthogonal functions

A technique was developed for global modeling of nonlinear aerodynamic coefficients using multivariate orthogonal functions based on the data. Each orthogonal function retained in the model was decomposed into an expansion of ordinary polynomials in the independent variables, so that the final model could be interpreted as selectively retained terms from a multivariable power series expansion. A predicted squared-error metric was used to determine the orthogonal functions to be retained in the model; analytical derivatives were easily computed. The approach was demonstrated on the Z-body axis aerodynamic force coefficient (Cz) wind tunnel data for an F-18 research vehicle which came from a tabular wind tunnel and covered the entire subsonic flight envelope. For a realistic case, the analytical model predicted experimental values of Cz very well. The modeling technique is shown to be capable of generating a compact, global analytical representation of nonlinear aerodynamics. The polynomial model has good predictive capability, global validity, and analytical differentiability.

Morelli, Eugene A.↗

NASA NTRS1993

Nested Krylov methods and preserving the orthogonality

Recently the GMRESR inner-outer iteraction scheme for the solution of linear systems of equations was proposed by Van der Vorst and Vuik. Similar methods have been proposed by Axelsson and Vassilevski and Saad (FGMRES). The outer iteration is GCR, which minimizes the residual over a given set of direction vectors. The inner iteration is GMRES, which at each step computes a new direction vector by approximately solving the residual equation. However, the optimality of the approximation over the space of outer search directions is ignored in the inner GMRES iteration. This leads to suboptimal corrections to the solution in the outer iteration, as components of the outer iteration directions may reenter in the inner iteration process. Therefore we propose to preserve the orthogonality relations of GCR in the inner GMRES iteration. This gives optimal corrections; however, it involves working with a singular, non-symmetric operator. We will discuss some important properties, and we will show by experiments that, in terms of matrix vector products, this modification (almost) always leads to better convergence. However, because we do more orthogonalizations, it does not always give an improved performance in CPU-time. Furthermore, we will discuss efficient implementations as well as the truncation possibilities of the outer GCR process. The experimental results indicate that for such methods it is advantageous to preserve the orthogonality in the inner iteration. Of course we can also use iteration schemes other than GMRES as the inner method; methods with short recurrences like GICGSTAB are of interest.

Desturler, Eric↗

NASA NTRS2007

Zernike-like Orthogonal Basis Functions for Wavefront Characterization over Sampled, Irregular Apertures

For optical systems with circular apertures, wavefronts are often analyzed using Zernike polynomials, and individual Zernike functions are associated with familiar optical aberrations. For systems with noncircular apertures, or in practical situations in which wavefronts are measured at a limited number of points in the aperture, the Zernike polynomials are no longer an orthogonal basis for the measured data. Although there are an endless number of ways to create a basis for such measured data, a "Zernike-like" basis is useful to connect with our experience with the usual optical aberrations. In this paper, the steps required to identify a Zernike-like basis for wavefronts over sampled, irregular apertures are presented, based on the Gram-Schmidt orthogonalization technique. The benefits of analyzing optical wavefronts using an orthogonal basis specific to an optical system's aperture shape and wavefront sampling, instead of using the traditional Zernike polynomials, are detailed in two examples, from image-based wavefront sensing on a segmented-aperture telescope (the James Webb Space Telescope Testbed Telescope at Ball Aerospace) and from interferometer characterization for surface metrology of a hexagonal mirror segment.

Aronstein, David L.↗

NASA NTRS1999

Phase array calibration orthogonal phase sequence

Methods and systems for calibrating an array antenna are described. The array antenna has a plurality of antenna elements each having a signal with a phase and an amplitude forming an array antenna signal. For calibration, the phase of each element signal is sequentially switched one at a time through four orthogonal phase states. At each orthogonal phase state, the power of the array antenna signal is measured. A phase and an amplitude error for each of the element signals is determined based on the power of the array antenna signal at each of the four orthogonal phase states. The phase and amplitude of each of the element signals is then adjusted by the corresponding phase and amplitude errors.

Sorace, Ronald E.↗

DOE OSTI2026

Offline Maximizing Minimally Invasive Proper Orthogonal Decomposition for Reduced-Order Modeling of S n Radiation Transport

Deterministic solutions to the Sn radiation transport equation can be computationally expensive to calculate. Reduced-order modeling enables efficient approximation of the full-order model (FOM) solution. We propose a novel method for constructing reduced-order models (ROMs) of the S n radiation transport equation, offline maximizing minimally invasive (OMMI) proper orthogonal decomposition (POD). POD uses the method of snapshots to create a reduced-order basis for constructing an ROM. Minimally invasive POD leverages the sweep infrastructure existing in deterministic transport codes to create a POD-based ROM, even when infeasible by traditional methods. Offline maximizing minimally invasive proper orthogonal decomposition (OMMI-POD) extends minimally invasive POD by performing sweeps offline, therefore maximizing the potential speedup. OMMI-POD does so by creating a library of reduced systems from a training set. This library of reduced systems is then interpolated to provide a rapid approximate solution of the S n radiation transport equation. The model is evaluated on a set of test problems, achieving a low error with a 466 times speedup over the FOM. Also presented is a study of the effect of sampling method on the performance of OMMI-POD, specifically comparing naive uniform sampling to the more accurate and computationally expensive greedy sampling.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Feature learning and generalization in deep networks with orthogonal weights

Fully-connected deep neural networks with weights initialized from independent Gaussian distributions can be tuned to criticality, which prevents the exponential growth or decay of signals propagating through the network. However, such networks still exhibit fluctuations that grow linearly with the depth of the network, which may impair the training of networks with width comparable to depth. We show analytically that rectangular networks with tanh activations and weights initialized from the ensemble of orthogonal matrices have corresponding preactivation fluctuations which are independent of depth, to leading order in inverse width. Moreover, we demonstrate numerically that, at initialization, all correlators involving the neural tangent kernel (NTK) and its descendants at leading order in inverse width—which govern the evolution of observables during training—saturate at a depth of ~20, rather than growing without bound as in the case of Gaussian initializations. We speculate that this structure preserves finite-width feature learning while reducing overall noise, thus improving both generalization and training speed in deep networks with depth comparable to width. We provide some experimental justification by relating empirical measurements of the NTK to the superior performance of deep non-linear orthogonal networks trained under full-batch gradient descent on the MNIST and CIFAR-10 classification tasks.

97 MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

Spectral Proper Orthogonal Decomposition of uPSP Measurements in Recent NASA Ames Wind Tunnel Test

This paper discusses Spectral Proper Orthogonal Decomposition (SPOD) of the Unsteady Pressure-Sensitive Paint (uPSP) measurements in recent NASA Ames wind tunnel test. The uPSP measurements were collected using Innovative Scientific Solutions, Inc. (ISSI) porous, fast-response pressure-sensitive paint, 40 ISSI four-inch air-cooled Light-Emitting Diodes, and 8 Phantom v2512 high-speed cameras at 10,000 frames per second in the uPSP Launch Vehicle Demonstration Test (LVDT) of the Space Launch System (SLS) vehicle in the 11-by 11-foot transonic test section of the Unitary Plan Wind Tunnel at NASA Ames Research Center in April 2024. SPOD is derived from a space-time proper orthogonal decomposition problem for statistically stationary flows. SPOD modes are determined in the frequency domain. Each SPOD mode oscillates at a single frequency. SPOD can be viewed as an extension of the Discrete Fourier Transform composition and the Dynamic Mode Decomposition. In this paper, the outputs of SPOD of the uPSP measurements in the uPSP LVDT are presented and the effectiveness of SPOD in the identification, diagnosis and analysis of the aerodynamic and aeroacoustic phenomena is demonstrated. The unsteady and dynamic property of the pressure field on the surface of the SLS Block 1B crew vehicle is presented with the visualization of the SPOD modes of the uPSP measurements in the tests of a Mach sweep run of the uPSP LVDT. The SPOD outputs were generated with the execution in parallel of a code in Python, with the library of Message Passing Interface for parallel processing, on the NASA Pleiades supercomputer. 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. Funding was provided by the NASA Aerosciences Evaluation and Test Capabilities Portfolio Office.

Aeroacoustics↗

DOE OSTI2025

The overlapping fragment approach for non-orthogonal configuration interaction with fragments

The non-orthogonal configuration interaction with fragments (NOCI-F) approach is extended opening the possibility to study intramolecular processes and materials with covalent or ionic lattices. So far, NOCI-F has been applied to study intermolecular energy and electron transfer employing ensembles of fragments that do not have atoms or bonds in common. The here presented approach divides the target system into two overlapping fragments that share one or more atoms and/or one or more bonds. After the construction of a collection of (multiconfigurational) fragment wave functions in a state specific optimization procedure, the fragment wave functions are combined to form many-electron basis functions for the non-orthogonal configuration interaction of the whole system. The orbitals in the overlapping fragment are defined by a corresponding orbital transformation of the fragment orbitals through a singular value decomposition. The overlapping fragments approach is first illustrated for a model system and then used to highlight some possible applications of NOCI with overlapping fragments. In conclusion, the results of excited state diffusion in transition metal oxide, intramolecular singlet fission and magnetic interactions in organic biradicals and ionic compounds are discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1971

Orthogonal transform feasibility study

The application of various orthogonal transformations to communication was investigated, with particular emphasis placed on speech and visual signal processing. The fundamentals of the one- and two-dimensional orthogonal transforms and their application to speech and visual signals are treated in detail.

Robinson, G. S.↗

NASA NTRS1972

FORTRAN program for generating a two-dimensional orthogonal mesh between two arbitrary boundaries

A FORTRAN 4 program is presented which computes and plots coordinates for a two-dimensional orthogonal mesh in the region between the walls of a flow channel. The program is designed for a channel containing a body about which flow passes and which spans the channel from one wall to the other. However, the condition that the channel contain an immersed body can be easily removed from the program. Input to the program consists of spline points of the channel walls and the body geometry. Output includes printed and plotted coordinates of the generated orthogonal mesh and angles of the mesh with the horizontal plane.

Mcnally, W. D.↗

NASA NTRS1972

Determination of a third orthogonal velocity component using two rotationally displaced laser Doppler velocimeter systems.

Transformation equations are derived that may be used to determine a third orthogonal velocity component from measurements made at a common point by two rotationally displaced, two orthogonal component laser Doppler velocimeter (LDV) systems. These equations also may be used to relate velocity measurements made in a particular coordinate system to any other coordinate system. It is shown that the set of smallest angles that may be used to separate the two, two component, LDV systems is very sensitive to the relative magnitudes and directions of the individual velocity components. When the magnitudes of all velocity components are about the same, it is found that the minimum angle of separation is of the same order as the instrumental error of the LDV systems.

Farmer, W. M.↗

NASA NTRS1977

A generalized orthogonal coordinate system for describing families of axisymmetric and two-dimensional bodies

A generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and two-dimensional body shapes of interest to aerodynamicists. Such body shapes include spheres, ellipses, spherically capped cones, flat-faced cylinders with rounded corners, circular disks, and planetary probe vehicles. A set of transformation equations is also developed whereby a uniform velocity field approaching a body at any angle of attack can be resolved in the transformed coordinate system. The Navier-Stokes equations are written in terms of a generalized orthogonal coordinate system to show the resultant complexity of the governing equations.

Gnoffo, P. A.↗

NASA NTRS1979

Gravitational time delay in orthogonally polarized radiation passing by the sun

Two parallel investigations into the degree, if any, to which orthogonally polarized rays are deflected differently on passing through the gravitational field of the sun were previously conducted. The first involved very long and intermediate length baseline radio interferometry. The second was initially based on observations of radiation transmitted by the Pioneer 6 spacecraft, on passing behind the sun in 1968. This work was extended by using Helios-A and Helios-B spacecraft. It was calculated that the differential deflection between orthogonally polarized components is less than one part in 10 to the 7th power of the total gravitational deflection, or less than about 10 to the -7th power arc sec, in total.

Harwit, M.↗

NASA NTRS1980

Application of a numerically generated orthogonal coordinate system to the solution of inviscid axisymmetric supersonic flow over blunt bodies

A numerically generated orthogonal coordinate system (with the body surface and shock wave as opposite boundaries) was applied with a time asymptotic method to obtain steady flow solutions for axisymmetric inviscid flow over several blunt bodies including spheres, paraboloids, ellipsoids, hyperboloids, hemisphere cylinders, spherically blunted cones, and a body with a concavity in the stagnation region. Comparisons with experimental data and with the results of other computational methods are discussed. The numerically generated orthogonal coordinate system is described and applications of the method to complex body shapes, particularly those with concave regions, are discussed.

Hamilton, H. H., II↗