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All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

A note on windowing for the waveform relaxation

The technique of windowing has been often used in the implementation of the waveform relaxations for solving ODE's or time dependent PDE's. Its efficiency depends upon problem stiffness and operator splitting. Using model problems, the estimates for window length and convergence rate are derived. The electiveness of windowing is then investigated for non-stiff and stiff cases respectively. lt concludes that for the former, windowing is highly recommended when a large discrepancy exists between the convergence rate on a time interval and the ones on its subintervals. For the latter, windowing does not provide any computational advantage if machine features are disregarded. The discussion is supported by experimental results.

Zhang, Hong↗
NASA NTRS1993

Optimal convolution SOR acceleration of waveform relaxation with application to semiconductor device simulation

In this paper we describe a novel generalized SOR (successive overrelaxation) algorithm for accelerating the convergence of the dynamic iteration method known as waveform relaxation. A new convolution SOR algorithm is presented, along with a theorem for determining the optimal convolution SOR parameter. Both analytic and experimental results are given to demonstrate that the convergence of the convolution SOR algorithm is substantially faster than that of the more obvious frequency-independent waveform SOR algorithm. Finally, to demonstrate the general applicability of this new method, it is used to solve the differential-algebraic system generated by spatial discretization of the time-dependent semiconductor device equations.

Reichelt, Mark↗
NASA NTRS2001

Relationship between early diastolic intraventricular pressure gradients, an index of elastic recoil, and improvements in systolic and diastolic function

BACKGROUND: Early diastolic intraventricular pressure gradients (IVPGs) have been proposed to relate to left ventricular (LV) elastic recoil and early ventricular "suction." Animal studies have demonstrated relationships between IVPGs and systolic and diastolic indices during acute ischemia. However, data on the effects of improvements in LV function in humans and the relationship to IVPGs are lacking. METHODS AND RESULTS: Eight patients undergoing CABG and/or infarct exclusion surgery had a triple-sensor high-fidelity catheter placed across the mitral valve intraoperatively for simultaneous recording of left atrial (LA), basal LV, and apical LV pressures. Hemodynamic data obtained before bypass were compared with those with similar LA pressures and heart rates obtained after bypass. From each LV waveform, the time constant of LV relaxation (tau), +dP/dt(max), and -dP/dt(max) were determined. Transesophageal echocardiography was used to determined end-diastolic (EDV) and end-systolic (ESV) volumes and ejection fractions (EF). At similar LA pressures and heart rates, IVPG increased after bypass (before bypass 1.64+/-0.79 mm Hg; after bypass 2.67+/-1.25 mm Hg; P<0.01). Significant improvements were observed in ESV, as well as in apical and basal +dP/dt(max), -dP/dt(max), and tau (each P<0.05). Overall, IVPGs correlated inversely with both ESV (IVPG=-0.027[ESV]+3.46, r=-0.64) and EDV (IVPG=-0.027[EDV]+4.30, r=-0.70). Improvements in IVPGs correlated with improvements in apical tau (Deltatau =5.93[DeltaIVPG]+4.76, r=0.91) and basal tau (Deltatau =2.41[DeltaIVPG]+5.13, r=-0.67). Relative changes in IVPGs correlated with changes in ESV (DeltaESV=-0.97[%DeltaIVPG]+23.34, r=-0.79), EDV (DeltaEDV=-1.16[%DeltaIVPG]+34.92, r=-0.84), and EF (DeltaEF=0.38[%DeltaIVPG]-8.39, r=0.85). CONCLUSIONS: Improvements in LV function also increase IVPGs. These changes in IVPGs, suggestive of increases in LV suction and elastic recoil, correlate directly with improvements in LV relaxation and ESV.

Non-NASA Center↗
NASA NTRS1992

Model experiment to study the effect of turbulence on risetime and waveform of N waves

Typical measured sonic boom rise times are two to five times longer than rise times calculated using molecular relaxation theory. The difference may be due to atmospheric turbulence. A model experiment was set up to study the influence of turbulence on waveform and rise time of spark-produced N waves. The N waves propagate through turbulence generated by a plane jet. The model turbulence is scaled down from atmospheric turbulence by approximately the same factor as the model N wave is scaled down from the sonic boom. Our experiments show that passage through the turbulence produces a wide variety of changes in the N waveform. Spiked and rounded N waves are observed, and average rise time is increased by a factor of about 2. A tentative observation based on data obtained so far is that rise time is always increased, never decreased, by turbulence.

Lipkens, Bart↗
NASA NTRS1994

High-Speed Research: 1994 Sonic Boom Workshop: Atmospheric Propagation and Acceptability Studies

The workshop proceedings include papers on atmospheric propagation and acceptability studies. Papers discussing atmospheric effects on the sonic boom waveform addressed several issues. It has long been assumed that the effects of molecular relaxation are adequately accounted for by assuming that a steady state balance between absorption and nonlinear wave steepening exists. It was shown that the unsteadiness induced by the nonuniform atmosphere precludes attaining this steady state. Further, it was shown that the random atmosphere acts as a filter, effectively filtering out high frequency components of the distorted waveform. Several different propagation models were compared, and an analysis of the sonic boom at the edge of the primary carpet established that the levels there are bounded. Finally, a discussion of the levels of the sonic boom below the sea surface was presented.

Mccurdy, David A.↗
NASA NTRS1992

A numerical model for sonic boom propagation through an inhomogeneous, windy atmosphere

The ZEPHYRUS computer model calculates sonic boom distortion during propagation through the atmosphere. The model includes the effects of nonlinear distortion, attenuation, dispersion, and wind. Trial runs with the model indicate that, in general, stable shocks have not formed when the sonic boom reaches the ground. Also, the rise time of the lead shock may strongly depend on the overall waveform shape, and may be significantly increased by purely dispersive effects when matching occurs between the rise time and the characteristic oxygen molecular relaxation time.

Robinson, Leick D.↗
NASA NTRS1992

Analysis of sonic boom data to quantify distortions of shock profiles

Researchers at Penn State have been examining some sonic boom waveforms recorded during overflights by the Air Force which have become available to NASA and its contractors. The quality of the digitized data and the supporting meteorological data was such that one could test the applicability of molecular relaxation theories. In the late sixties, it had been supposed that the finite rise times in the absence of turbulence had neglected the vibrational relaxation of nitrogen molecules. Bass et al. have demonstrated that molecular relaxation definitely gives the correct order of magnitude of the observed rise times. However, the Air Force data in conjunction with the recent steady-state shock profile model theory of Kang and Pierce give the first opportunity to make a detailed quantitative assessment of the molecular relaxation hypothesis. Currently an investigation is ongoing to establish a method of quantifying the distortion of a sonic boom wave from a classic N-wave shape using the Air Force data taken at Edwards AFB in 1987. Using the premise that energy will be conserved approximately for a sonic boom wave both before and after the boom passes through the Earth's turbulent boundary layer, a classic undistorted waveform is constructed from the distorted signature received at the ground. A correlation between the mean-squared deviation of the distorted and undistorted waveforms and the distance the boom travels through the turbulence is sought.

Gionfriddo, Thomas A.↗
NASA NTRS1983

Influence of modulation frequency in rubidium cell frequency standards

The error signal which is used to control the frequency of the quartz crystal oscillator of a passive rubidium cell frequency standard is considered. The value of the slope of this signal, for an interrogation frequency close to the atomic transition frequency is calculated and measured for various phase (or frequency) modulation waveforms, and for several values of the modulation frequency. A theoretical analysis is made using a model which applies to a system in which the optical pumping rate, the relaxation rates and the RF field are homogeneous. Results are given for sine-wave phase modulation, square-wave frequency modulation and square-wave phase modulation. The influence of the modulation frequency on the slope of the error signal is specified. It is shown that the modulation frequency can be chosen as large as twice the non-saturated full-width at half-maximum without a drastic loss of the sensitivity to an offset of the interrogation frequency from center line, provided that the power saturation factor and the amplitude of modulation are properly adjusted.

Audoin, C.↗
NASA NTRS1994

Progress in modeling atmospheric propagation of sonic booms

The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by humidity and turbulence. A realistic atmosphere is invariably somewhat turbulent, and may be characterized by an ambient fluid velocity v and sound speed c that vary from point to point. The absolute humidity will also vary from point to point, although possibly not as irregularly. What is ideally desired is a relatively simple scheme for predicting the probable spreads in key sonic boom signature parameters. Such parameters could be peak amplitudes, rise times, or gross quantities obtainable by signal processing that correlate well with annoyance or damage potential. The practical desire for the prediction scheme is that it require a relatively small amount of knowledge, possibly of a statistical nature, concerning the atmosphere along, the propagation path from the aircraft to the ground. The impact of such a scheme, if developed, implemented, and verified, would be that it would give the persons who make planning decisions a tool for assessing the magnitude of environmental problems that might result from any given overflight or sequence of overflights. The technical approach that has been followed by the author and some of his colleagues is to formulate a hierarchy of simple approximate models based on fundamental physical principles and then to test these models against existing data. For propagation of sonic booms and of other types of acoustic pulses in nonturbulent model atmospheres, there exists a basic overall theoretical model that has evolved as an outgrowth of geometrical acoustics. This theoretical model depicts the sound as propagating within ray tubes in a manner analogous to sound in a waveguide of slowly varying cross-section. Propagation along the ray tube is quasi-one-dimensional, and a wave equation for unidirectional wave propagation is used. A nonlinear term is added to this equation to account for nonlinear steepening, and the formulation has been carried through to allow for spatially varying sound speed, ambient density, and ambient wind velocities. The model intrinsically neglects diffraction, so it cannot take into account what has previously been mentioned in the literature as possibly important mechanisms for turbulence-related distortion. The model as originally developed could predict an idealized N-waveform which often agrees with data in terms of peak amplitude and overall positive phase duration. It is possible, moreover, to develop simple methods based on the physics of relaxation processes for incorporating molecular relaxation into the quasi-one-dimensional model of nonlinear propagation along ray tubes.

Pierce, Allan D.↗
NASA NTRS1992

Relaxation and turbulence effects on sonic boom signatures

The rudimentary theory of sonic booms predicts that the pressure signatures received at the ground begin with an abrupt shock, such that the overpressure is nearly abrupt. This discontinuity actually has some structure, and a finite time is required for the waveform to reach its peak value. This portion of the waveform is here termed the rise phase, and it is with this portion that this presentation is primarily concerned. Any time characterizing the duration of the rise phase is loosely called the 'rise time.' Various definitions are used in the literature for this rise time. In the present discussion the rise time can be taken as the time for the waveform to rise from 10 percent of its peak value to 90 percent of its peak value. The available data on sonic booms that appears in the open literature suggests that typical values of shock over-pressure lie in the range of 30 Pa to 200 Pa, typical values of shock duration lie in the range of 150 ms to 250 ms, and typical values of the rise time lie in the range of 1 ms to 5 ms. The understanding of the rise phase of sonic booms is important because the perceived loudness of a shock depends primarily on the structure of the rise phase. A longer rise time typically implies a less loud shock. A primary question is just what physical mechanisms are most important for the determination of the detailed structure of the rise phase.

Pierce, Allan D.↗
NASA NTRS1995

The making of an Alfvenic fluctuation: The resolution of a second-order analysis

Ulysses observations of the high speed polar streams show that they are largely occupied by very large amplitude Alfvenic fluctuations accompanied by many rotational discontinuities. These fluctuations have a nearly constant magnetic intensity or amplitude, and the magnetic field direction per wave cycle sweeps only through a limited arc, much as a car wiperblade would do. Barnes and Hollweg (JGR, 79, 2302, 1974) suggested that this unusual waveform could arise from an obliquely propagating and linearly polarized Alfven wave of finite amplitude. From a second-order analysis, they showed that the existence of a particular solution with a constant amplitude but could not resolve the outcome of the homogeneous solution which consisted of fast waves. They suggested that Landau damping of these fast waves may be needed to get the observed waveform. We present a 1 1/2 D hybrid simulation which is fully nonlinear and correctly describes the ion kinetics for an initially monochromatic and linearly polarized Alfven wave propagating obliquely to the background magnetic field. The wave has a large amplitude and a wavelength so long that it can be considered dispersionless for simulation times. At early times, the second harmonic in density and in magnetic field transverse to the initial wave magnetic field are generated and have more power than other harmonics. Steepening is observed with a weak fast shock emerging, but no rotational discontinuity is left behind, and instead a constant amplitude and an arc-shaped waveform is made. The compressional component which develops after the shocks have dissipated is to zeroth order better described as a pure acoustic wave than as a fast wave. This might be explained by the relaxing of the Alfven wave to a state where its ponderomotive force vanishes so that the compressional component can travel almost independently of it.

Vasquez, Bernard J.↗
NASA NTRS1992

Wave equations and computational models for sonic boom propagation through a turbulent atmosphere

The improved simulation of sonic boom propagation through the real atmosphere requires greater understanding of how the transient acoustic pulses popularly termed sonic booms are affected by atmospheric turbulence. A nonlinear partial differential equation that can be used to simulate the effects of smaller-scale atmospheric turbulence on sonic boom waveforms is described. The equation is first order in the time derivative and involves an extension of geometrical acoustics to include diffraction phenomena. Various terms in the equation are explained in physical terms. Such terms include those representing convection at the wave speed, diffraction, molecular relaxation, classical dissipation, and nonlinear steepening. The atmospheric turbulence enters through an effective sound speed, which varies with all three spatial coordinates, and which is the sum of the local sound speed and the component of the turbulent flow velocity projected along a central ray that connects the aircraft trajectory with the listener.

Pierce, Allan D.↗
NASA NTRS2005

Measured Effects of Turbulence on the Loudness and Waveforms of Conventional and Shaped Minimized Sonic Booms

Turbulence has two distinctive effects on sonic booms: there is distortion in the form of random perturbations that appear behind the shock waves, and shock rise times are increased randomly. A first scattering theory by S.C. Crow in the late 1960s quantified the random distortions, and Crow's theory was shown to agree with available flight test data. A variety of theories for the shock thickness have been presented, all supporting the role of turbulence in increasing rise time above that of a basic molecular-relaxation structure. The net effect of these phenomena on the loudness of shaped minimized booms is of significant interest. Initial analysis suggests that there would be no change to average loudness, but this had not been experimentally investigated. The January 2004 flight test of the Shaped Sonic Boom Demonstrator (SSBD), together with a reference unmodified F-5E, included a 12500- foot linear ground sensor array with 28 digitally recorded sensor sites. This data set provides an opportunity to re-test Crow's theory for the post-shock perturbations, and to examine the net effect of turbulence on the loudness of shaped sonic booms.

Plotkin, Kenneth J.↗
NASA NTRS1981

On a theory of temporal fluctuations in the electrostatic potential structures associated with auroral arcs

A possible mechanism is presented for the generation of large-amplitude temporal fluctuations in the structure of the electron energization region associated with auroral arcs. The mechanism is based on the observation that the auroral arc system resembles a laboratory circuit consisting of the series connection of battery, resistance and a forward biased diode containing collisionless plasma in which large-amplitude relaxation oscillations are sometimes observed to be superimposed on the steady-state current. It is shown that in both the laboratory and auroral systems, in which a localized auroral arc dynamo, the ionosphere and the electron energization region are involved, the oscillations are controlled by the times for ions and electrons to traverse the acceleration region, which also characterize the low- and high-frequency structure of the fluctuating waveform. It is demonstrated that a plausible one-dimensional double-layer model of the auroral arc acceleration region exhibits the dynamic negative resistance necessary for the generation of oscillations by the present mechanism. Finally, consideration is given to two kinds of auroral phenomena which might be associated with the mechanism: the 10-Hz quasi-periodic flickering aurora and 10-Hz modulations in the intensity of electrostatic hydrogen cyclotron waves.

Silevitch, M. B.↗
NASA NTRSDate not supplied

Discretization Error Estimation and Control for Farfield Acoustic Signatures

We investigate the utility of adjoint-based error estimates for sonic boom farfield simulations governed by solutions of the augmented Burgers’ equation. Solution of this nonlinear system uses operator splitting with a second-order finite volume discretization in space and second-order Runge-Kutta time marching, while the absorption and molecular relaxation are solved using second-order central differencing. The discretization error in selected ground sonic boom cost functionals is estimated using the method of adjoint-weighted residuals. Key elements of the implementation process are emphasized with details provided on the practical aspects as appliedto the sonic boom farfield propagation. We establish the accuracy of the adjoint solutions usingcomplex step and finite difference approaches, and examine the accuracy of the error estimates using analytical N-wave solutions. We then apply it to a pressure waveform corresponding to the X-59 research aircraft. The investigations demonstrate that the method of adjoint-weighted residuals accurately predicts the level of discretization error present in sonic boom farfield simulations while offering insight into which features of the near field signal are the primary drivers of ground noise metrics. The numerical results indicate that at sampling frequencies as low as50kHz, discretization error in the propagation is under 0.01 dB[A] for realistically complex examples.

CST↗
NASA NTRS2016

Vibration Penalty Estimates for Indoor Annoyance Caused by Sonic Boom

Commercial supersonic flight is currently forbidden over land because sonic booms have historically caused unacceptable annoyance levels in overflown communities. NASA is providing data and expertise to noise regulators as they consider relaxing the ban for future quiet supersonic aircraft. One key objective is a predictive model for indoor annoyance based on factors such as noise and indoor vibration levels. The current study quantified the increment in indoor sonic boom annoyance when sonic booms can be felt directly through structural vibrations in addition to being heard. A shaker mounted below each chair in the sonic boom simulator emulated vibrations transmitting through the structure to that chair. The vibration amplitudes were determined from numeric models of a large range of residential structures excited by the same sonic boom waveforms used in the experiment. The analysis yielded vibration penalties, which are the increments in sound level needed to increase annoyance as much as the vibration does. For sonic booms at acoustic levels from 75 to 84 dB Perceived Level, vibration signals with lower amplitudes (+1 sigma) yielded penalties from 0 to 5 dB, and vibration signals with higher amplitudes (+3 sigma) yielded penalties from 6 to 10 dB.

Rathsam, Jonathan↗
NASA NTRS2018

Simulation and Analysis of Opportunistic MSPA for Multiple Cubesat Deployments

We describe a software approach for simultaneous demodulation and decoding of multiple frequency- multiplexed spacecraft across multiple ground stations. The approach involves the use of a single ground antenna with wide enough aperture to receive multiple angularly adjacent spacecraft simultaneously. This technique uses a wide-band RF signal digitizer coupled with easily and cheaply duplicated software-receiver modules to independently process the frequency-channelized downlink signals from the spacecraft. This approach relaxes the need for realizable modems at each antenna, for example in the Deep-SpaceNetwork (DSN), and can also function as a delayed data retrieval method for cubesat missions with routine science data return. Thus, we hope that this solution can enable more efficient utilization of DSN assets. We concentrate on a simulation similar to the proposed Exploration Mission 1 (EM-1) mission from a geometric perspective. To that end, the simulated scenario involves ten secondary cubesats deployed and tracked for a span of four days (the modeled motion over the first four days of the mission does not include any Trajectory Correction Maneuvers (TCMs) that may be necessary as the cubesats approach the moon). Transmissions from the cubesats may be received through a combination of DSN ground sites based on visibility. The cubesats are assumed to utilize typical cubesat transmit powers and a waveform similar to that of the Iris radio. One cubesat is chosen as the “target” and is considered tracked by all ground-stations throughout the simulation (i.e., it is consistently at the center of the main lobe of each ground station antenna when the ground station is in view of the cubesat in terms of elevation angle). The simulation effort involves synthesizing a wideband signal that includes the ten cubesats across a large bandwidth due to each cubesat having its own center-frequency in X-band, near 8.4 GHz. Each signal is characterized by its own Doppler-frequency shift and free-space path-loss computed through the underlying geometry of the simulation as well as the cubesat transmit power. Finally, each cubesat signal experiences a unique antenna gain at each ground station due to the underlying antenna pattern and spacecraft-ground station geometry. We establish two interesting findings: First, the link-budget for the EM-1-like scenario is almost completely limited by the angle between the center of the antenna main beam and the cubesat, which means that the signal-to-noise ratio is wholly adequate for demodulation and decoding at the simulated bitrates unless the cubesat exits the main beam. Secondly, and fortunately due to the use of a software radio architecture, we show that it is possible to successfully receive signals from most cubesats for the entire 4-day simulation. Due to the signal-to-noise ratio being sufficient for demodulation even after the side-lobe’s 17dB reduction in SNR, the software radio can still demodulate such cubesats as long as the radio is capable of re-establishing carrier lock as the cubesats leave the main beam and enter the side-lobe. Considering that the target application utilizes offline processing, a loss of lock can be detected and lock can be re-established by iterating over the data. Extensive simulations demonstrate these results.

Abraham, Douglas S.↗