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NASA NTRS1990

The vibrating ribbon problem revisited

A revised formal solution of the vibrating ribbon problem of hydrodynamic stability is presented. The initial formulation of Gaster (1965) is modified by application of the Briggs method and a careful treatment of the complex double Fourier transform inversions. Expressions are obtained in a natural way for the discrete spectrum as well as for the four branches of the continuous spectra. These correspond to discrete and branch-cut singularities in the complex wavenumber plane. The solutions from the continuous spectra decay both upsteam and downstream of the ribbon, with the decay in the upstream direction being much more rapid than that in the downstream direction. Comments and clarification of related prior work are made.

Ashpis, David E.↗

DOE OSTI2023

Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions

The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. So our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.

42 ENGINEERING↗

DOE OSTI2022

Slepians.jl

Slepians.jl is a package that solves the concentration problem of Slepian, Landau and Pollak, numerically. Loosely speaking, we find a function that is both finite in extent and has a Fourier transform which lies in a certain domain. In a single dimension, with discrete sampling, the discrete prolate spheroidal sequences solve this optimization problem.

HALEY, CHARLOTTE ANNA LISA↗

NASA NTRS1975

A study of sound generation in subsonic rotors, volume 2

Computer programs were developed for use in the analysis of sound generation by subsonic rotors. Program AIRFOIL computes the spectrum of radiated sound from a single airfoil immersed in a laminar flow field. Program ROTOR extends this to a rotating frame, and provides a model for sound generation in subsonic rotors. The program also computes tone sound generation due to steady state forces on the blades. Program TONE uses a moving source analysis to generate a time series for an array of forces moving in a circular path. The resultant time series are than Fourier transformed to render the results in spectral form. Program SDATA is a standard time series analysis package. It reads in two discrete time series and forms auto and cross covariances and normalizes these to form correlations. The program then transforms the covariances to yield auto and cross power spectra by means of a Fourier transformation.

Chalupnik, J. D.↗

DOE OSTI2022

Bell state analyzer for spectrally distinct photons

We demonstrate a Bell state analyzer that operates directly on frequency mismatch. Based on electro-optic modulators and Fourier-transform pulse shapers, our quantum frequency processor design implements interleaved Hadamard gates in discrete frequency modes. Experimental tests on entangled-photon inputs reveal fidelities of ∼ <#comment/> 98 % <#comment/> for discriminating between the | Ψ <#comment/> + ⟩ <#comment/> and | Ψ <#comment/> − <#comment/> ⟩ <#comment/> frequency-bin Bell states. Our approach resolves the tension between wavelength-multiplexed state transport and high-fidelity Bell state measurements, which typically require spectral indistinguishability.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1984

Spectral Information from Gapped Data. a Comparison of Technique

The fast Fourier transformations (FFT) is used to estimate power spectra of continuous signals evenly sampled on discrete domains. The problem of finding power spectra on unevenly sampled domains, in particular a regularly spaced domain with gaps is discussed. The analysis of the ACRIM solar bolometric intensity data, obtained with a 3/5 on and 2/5 off duty cycle of approximately 100 minutes, would benefit from the techniques. The comparative effectiveness of three different analysis techniques applied to synthetic data generated on gapped domain is reported.

Kuhn, J. R.↗

DOE OSTI2025

A Linear-Complexity Tensor Butterfly Algorithm for Compressing High-Dimensional Oscillatory Integral Operators

This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a d-dimensional (d > 1) integral operator discretized into a 2d-mode tensor with n2d entries, the overall CPU time and memory requirement scale as O(nd), in stark contrast to the O(nd log n) complexity of existing matrix algorithms such as matrix butterfly algorithms and fast Fourier transforms (FFTs), where n is the number of points per direction. When comparing with other tensor algorithms such as quantized tensor train (QTT), the proposed algorithm also shows superior CPU and memory performance for tensor contraction. Remarkably, the tensor butterfly algorithm can efficiently model high-frequency Green's function interactions between two unit cubes, each spanning 512 wavelengths per direction, which represents problems of scale over 512× larger than that existing butterfly algorithms can handle, with the same amount of computation resources. On the other hand, for a problem representing 64 wavelengths per direction, which is the largest size existing algebraic matrix algorithms can handle, our tensor butterfly algorithm exhibits 200x speedups and 30× memory reduction compared with existing ones. Moreover, the tensor butterfly algorithm also permits O(nd)-complexity FFTs and Radon transforms up to d = 6 dimensions.

Kielstra, P Michael↗

NASA NTRS1978

The determination of gravity anomalies from geoid heights using the inverse Stokes' formula, Fourier transforms, and least squares collocation

A numerical method for the determination of gravity anomalies from geoid heights is described using the inverse Stokes formula. This discrete form of the inverse Stokes formula applies a numerical integration over the azimuth and an integration over a cubic interpolatory spline function which approximates the step function obtained from the numerical integration. The main disadvantage of the procedure is the lack of a reliable error measure. The method was applied on geoid heights derived from GEOS-3 altimeter measurements in the calibration area of the GEOS-3 satellite.

Rummel, R.↗

NASA NTRS1982

Propagation of wide bandwidth signals through strongly turbulent ionized media

Analytic and numerical techniques are presented which directly address the problem of propagation of wide bandwidth signals through random ionized media. This work is applicable to the problems of satellite communication and space based radar observation through a disturbed ionospheric propagation channel that would result from a high altitude chemical release or nuclear detonation. An analytic solution is presented for the two-position, two-frequency mutual coherence function for spherical wave propagation in the strong scattering limit. This solution is used to derive simple expressions for the power impulse response function and to determine its relationship to the important parameters of decorrelation distance and coherence bandwidth which describe the disturbed propagation channel. Results for mean time delay and time delay jitter are presented and compared to direct simulation results and to other theoretical calculations. A numerical/analytical solution to the parabolic wave equation is presented in the form of a multiple phase-screen (MPS) propagation simulation. In this very general solution technique, the random medium is divided into a finite number of layers. The field fluctuation through each layer is obtained by replacing the layer by a centrally located thin phase-screen, whose statistical properties are determined from the statistics of the electron-density irregularities. The waveform then propagates from phase-screen to phase-screen via an exact solution to the Fresnel-Kirchnoff equation. For wide bandwidth waveforms, numerical solutions are obtained at a number of discrete frequencies centered about the carrier and then time-domain solutions are obtained by Fourier transform techniques. Detailed results are presented for a number of interesting cases including propagation of a 20 MHz bandwidth signal through a finite barium cloud at a carrier frequency of 100 MHz. One of the major uses of the MPS propagation simulation has been to provide realizations of the received signal after propagation through a disturbed channel. The MPS simulation obtains a general solution of the parabolic wave equation under both weak and strong scattering conditions. A second technique to directly obtain realizations of wide bandwidth waveforms is presented here. This technique is limited to the case of strong scattering but requires only a fraction of the computer resources needed for MPS signal generation. Detailed comparisons of the two signal generation techniques are presented.

Random Media↗

NASA NTRS2005

Schlieren Measurements of Buoyancy Effects on Flow Transition in Low-Density Gas Jets

The transition from laminar to turbulent flow in helium jets discharged into air was studied using Rainbow Schlieren Deflectometry technique. In particular, the effects of buoyancy on jet oscillations and flow transition length were considered. Experiments to simulate microgravity were conducted in the 2.2s drop tower at NASA Glenn Research Center. The jet Reynolds numbers varied from 800 to1200 and the jet Richardson numbers ranged between 0.01 and 0.004. Schlieren images revealed substantial variations in the flow structure during the drop. Fast Fourier Transform (FFT) analysis of the data obtained in Earth gravity experiments revealed the existence of a discrete oscillating frequency in the transition region, which matched the frequency in the upstream laminar regime. In microgravity, the transition occurred farther downstream indicating laminarization of the jet in the absence of buoyancy. The amplitude of jet oscillations was reduced by up to an order of magnitude in microgravity. Results suggest that jet oscillations were buoyancy induced and that the brief microgravity period may not be sufficient for the oscillations to completely subside.

Pasumarthi, Kasyap S.↗

NASA NTRS2021

Fast reconstruction of off-axis digital holograms

Hologram reconstruction via the angular spectrum method utilizes fast Fourier transforms which quickly but inefficiently shift between image and pupil planes. We demonstrate hologram reconstruction using discrete transforms over limited regions, decreasing computation time 10x.

Wallace, James K↗

NASA NTRS1990

The theoretical, discrete, and actual response of the Barnes objective analysis scheme for one- and two-dimensional fields

The response of the Barnes objective analysis scheme is studied as a function of wavenumber or wavelength. The first- and second-pass theoretical response functions for continuous two-dimensional fields are derived using Fourier transforms. The results are compared with Barnes' (1973) responses for one-dimensional waves. The continuous theoretical response for one- and two-dimensional waves is compared with the response for discrete applications using uniformly spaced observations for the case where interpolation points and observation points are coincident and for the case where interpolation points are midway between observation points. The actual response of an idealized discrete application of the Barnes scheme is examined, confirming the results of the analysis of the discrete theoretical response.

Pauley, Patricia M.↗

NASA NTRS1979

Local atomic and electronic structure of oxide/GaAs and SiO2/Si interfaces using high-resolution XPS

The chemical structures of thin SiO2 films, thin native oxides of GaAs (20-30 A), and the respective oxide-semiconductor interfaces, have been investigated using high-resolution X-ray photoelectron spectroscopy. Depth profiles of these structures have been obtained using argon ion bombardment and wet chemical etching techniques. The chemical destruction induced by the ion profiling method is shown by direct comparison of these methods for identical samples. Fourier transform data-reduction methods based on linear prediction with maximum entropy constraints are used to analyze the discrete structure in oxides and substrates. This discrete structure is interpreted by means of a structure-induced charge-transfer model.

Grunthaner, F. J.↗

NASA NTRS1992

A fast direct solver for a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of two-dimensional separable elliptic equations on the sphere is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wavenumber and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

Moorthi, Shrinivas↗

DOE OSTI2022

Bosonic field digitization for quantum computers

Quantum simulation of quantum field theory is a flagship application of quantum computers that promises to deliver capabilities beyond classical computing. The realization of quantum advantage will require methods that can accurately predict error scaling as a function of the resolution and parameters of the model and that can be implemented efficiently on quantum hardware. In this paper, we address the representation of lattice bosonic fields in a discretized field amplitude basis, develop methods to predict error scaling, and present efficient qubit implementation strategies. A low-energy subspace of the bosonic Hilbert space, defined by a boson occupation number cutoff, can be represented with exponentially good accuracy by a low-energy subspace of a finite-size Hilbert space. The finite representation construction and the associated errors are directly related to the accuracy of the Nyquist-Shannon sampling and the finite Fourier transforms of the boson number states in the field and the conjugate-field bases. We analyze the relation between the boson mass, the discretization parameters used for wave function sampling, and the finite representation size. Numerical simulations of small size Φ 4 problems demonstrate that the boson mass optimizing the sampling of the ground state wave function is a good approximation to the optimal boson mass yielding the minimum low-energy subspace size. However, we find that accurate sampling of general wave functions does not necessarily result in accurate representation. Finally, we develop methods for validating and adjusting the discretization parameters to achieve more accurate simulations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

Powers of magnetic graph matrix: Fourier spectrum, walk compression, and applications

Magnetic graphs, originally developed to model quantum systems under magnetic fields, have recently emerged as a powerful framework for analyzing complex directed networks. Existing research has primarily used the spectral properties of the magnetic graph matrix to study global and stationary network features. However, their capacity to model local, nonequilibrium behaviors, often described by matrix powers, remains largely unexplored. We present a combinatorial interpretation of the magnetic graph matrix powers through directed walk profiles—counts of graph walks indexed by the number of edge reversals. Crucially, we establish that walk profiles correspond to a Fourier transform of magnetic matrix powers. The connection allows exact reconstruction of walk profiles from magnetic matrix powers at multiple discrete potentials, and more importantly, an even smaller number of potentials often suffices for accurate approximate reconstruction in real networks. This shows the empirical compressibility of the information captured by the magnetic matrix. This fresh perspective suggests further applications; for example, we illustrate how powers of the magnetic matrix can identify frustrated directed cycles (e.g., feedforward loops) and can be effectively employed for link prediction by encoding local structural details in directed graphs.

complex networks↗

NASA NTRS1975

MTF and point-spread function for a large-area CCD imager

The MTF degradation due to lateral diffusion is calculated for a back illuminated CCD imager for typical device parameters. The discrete nature of the CCD and finite size of the photosensitive elements result in an additional MTF degradation. The Fourier transform approach is utilized to calculate the effective point spread function for these processes in the time domain. Experimental data are presented on the point spread function for a three phase, double level anodized aluminum 160 x 100 thinned and back illuminated CCD imager and compared with the theoretical results. A simple modification of the Crowell and Labuda model suggested by these results is presented.

Ando, K. J.↗

DOE OSTI2023

A fast Fourier transform-based solver for elastic micropolar composites

This work presents a spectral micromechanical formulation for obtaining the full-field and homogenized response of elastic micropolar composites. The algorithm relies on a coupled set of convolution integral equations for the micropolar strains, where periodic Green’s operators associated with a linear homogeneous reference medium are convolved with functions of the Cauchy and couple stress fields that encode the material’s heterogeneity, as well as any potential material nonlinearity. Such convolution integral equations take an algebraic form in the reciprocal Fourier space that can be solved iteratively. In this vein, the fast Fourier transform (FFT) algorithm is leveraged to accelerate the numerical solution, resulting in a mesh-free formulation in which the periodic unit cell representing the heterogeneous material can be discretized by a regular grid of pixels in two dimensions (or voxels in three dimensions). For verification, the numerical solutions obtained with the micropolar FFT solver are compared with analytical solutions for a matrix with a dilute circular inclusion subjected to plane strain loading. The developed computational framework is then used to study length-scale effects and effective (micropolar) moduli of composites with various topological configurations.

97 MATHEMATICS AND COMPUTING↗