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All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Incremental triangulation by way of edge swapping and local optimization

This document is intended to serve as an installation, usage, and basic theory guide for the two dimensional triangulation software 'HARLEY' written for the Silicon Graphics IRIS workstation. This code consists of an incremental triangulation algorithm based on point insertion and local edge swapping. Using this basic strategy, several types of triangulations can be produced depending on user selected options. For example, local edge swapping criteria can be chosen which minimizes the maximum interior angle (a MinMax triangulation) or which maximizes the minimum interior angle (a MaxMin or Delaunay triangulation). It should be noted that the MinMax triangulation is generally only locally optical (not globally optimal) in this measure. The MaxMin triangulation, however, is both locally and globally optical. In addition, Steiner triangulations can be constructed by inserting new sites at triangle circumcenters followed by edge swapping based on the MaxMin criteria. Incremental insertion of sites also provides flexibility in choosing cell refinement criteria. A dynamic heap structure has been implemented in the code so that once a refinement measure is specified (i.e., maximum aspect ratio or some measure of a solution gradient for the solution adaptive grid generation) the cell with the largest value of this measure is continually removed from the top of the heap and refined. The heap refinement strategy allows the user to specify either the number of cells desired or refine the mesh until all cell refinement measures satisfy a user specified tolerance level. Since the dynamic heap structure is constantly updated, the algorithm always refines the particular cell in the mesh with the largest refinement criteria value. The code allows the user to: triangulate a cloud of prespecified points (sites), triangulate a set of prespecified interior points constrained by prespecified boundary curve(s), Steiner triangulate the interior/exterior of prespecified boundary curve(s), refine existing triangulations based on solution error measures, and partition meshes based on the Cuthill-McKee, spectral, and coordinate bisection strategies.

Wiltberger, N. Lyn↗
NASA NTRS1998

Phase Inversion: Inferring Solar Subphotospheric Flow and Other Asphericity from the Distortion of Acoustic Waves

A method is proposed for analyzing an almost monochromatic train of waves propagating in a single direction in an inhomogeneous medium that is not otherwise changing in time. An effective phase is defined in terms of the Hilbert transform of the wave function, which is related, via the JWKB approximation, to the spatial variation of the background state against which the wave is propagating. The contaminating effect of interference between the truly monochromatic components of the train is eliminated using its propagation properties. Measurement errors, provided they are uncorrelated, are manifest as rapidly varying noise; although that noise can dominate the raw phase-processed signal, it can largely be removed by low-pass filtering. The intended purpose of the analysis is to determine the distortion of solar oscillations induced by horizontal structural variation and material flow. It should be possible to apply the method directly to sectoral modes. The horizontal phase distortion provides a measure of longitudinally averaged properties of the Sun in the vicinity of the equator, averaged also in radius down to the depth to which the modes penetrate. By combining such averages from different modes, the two-dimensional variation can be inferred by standard inversion techniques. After taking due account of horizontal refraction, it should be possible to apply the technique also to locally sectoral modes that propagate obliquely to the equator and thereby build a network of lateral averages at each radius, from which the full three-dimensional structure of the Sun can, in principle, be determined as an inverse Radon transform.

Gough, Douglas↗
NASA NTRS2016

Spectral and Wavefront Error Performance of WFIRST-AFTA Bandpass Filter Coating Prototypes

The Cycle 5 design baseline for the Wide-Field Infrared Survey Telescope Astrophysics Focused Telescope Assets (WFIRST/AFTA) instrument includes a single wide-field channel (WFC) instrument for both imaging and slit-less spectroscopy. The only routinely moving part during scientific observations for this wide-field channel is the element wheel (EW) assembly. This filter-wheel assembly will have 8 positions that will be populated with 6 bandpass filters, a blank position, and a Grism that will consist of a three-element assembly to disperse the full field with an undeviated central wavelength for galaxy redshift surveys. All filter elements in the EW assembly will be made out of fused silica substrates (110 mm diameter) that will have the appropriate bandpass coatings according to the filter designations (Z087, Y106, J129, H158, F184, W149 and Grism). This paper presents and discusses the performance (including spectral transmission and reflected/transmitted wavefront error measurements) of a subset of bandpass filter coating prototypes that are based on the WFC instrument filter compliment. The bandpass coating prototypes that are tested in this effort correspond to the Z087, W149, and Grism filter elements. These filter coatings have been procured from three different vendors to assess the most challenging aspects in terms of the in-band throughput, out of band rejection (including the cut-on and cutoff slopes), and the impact the wavefront error distortions of these filter coatings will have on the imaging performance of the wide-field channel in the WFIRST/AFTA observatory.

cryogenic temperatures↗
NASA NTRS2016

Spectral and Wavefront Error Performance of WFIRST/AFTA Prototype Filters

The Cycle 5 design baseline for the Wide-Field Infrared Survey Telescope Astrophysics Focused Telescope Assets (WFIRSTAFTA) instrument includes a single wide-field channel (WFC) instrument for both imaging and slit-less spectroscopy. The only routinely moving part during scientific observations for this wide-field channel is the element wheel (EW) assembly. This filter-wheel assembly will have 8 positions that will be populated with 6 bandpass filters, a blank position, and a Grism that will consist of a three-element assembly to disperse the full field with an undeviated central wavelength for galaxy redshift surveys. All filter elements in the EW assembly will be made out of fused silica substrates (110 mm diameter) that will have the appropriate bandpass coatings according to the filter designations (Z087, Y106, J129, H158, F184, W149 and Grism). This paper presents and discusses the performance (including spectral transmission and reflectedtransmitted wavefront error measurements) of a subset of bandpass filter coating prototypes that are based on the WFC instrument filter compliment. The bandpass coating prototypes that are tested in this effort correspond to the Z087, W149, and Grism filter elements. These filter coatings have been procured from three different vendors to assess the most challenging aspects in terms of the in-band throughput, out of band rejection (including the cut-on and cutoff slopes), and the impact the wavefront error distortions of these filter coatings will have on the imaging performance of the de-field channel in the WFIRSTAFTA observatory.

cryogenic temperatures↗
DOE OSTI2020

Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains

In this work, we propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modeling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2013

Ground-Based Lidar for Atmospheric Boundary Layer Ozone Measurements

Ground-based lidars are suitable for long-term ozone monitoring as a complement to satellite and ozonesonde measurements. However, current ground-based lidars are unable to consistently measure ozone below 500 m above ground level (AGL) due to both engineering issues and high retrieval sensitivity to various measurement errors. In this paper, we present our instrument design, retrieval techniques, and preliminary results that focus on the high-temporal profiling of ozone within the atmospheric boundary layer (ABL) achieved by the addition of an inexpensive and compact mini-receiver to the previous system. For the first time, to the best of our knowledge, the lowest, consistently achievable observation height has been extended down to 125 m AGL for a ground-based ozone lidar system. Both the analysis and preliminary measurements demonstrate that this lidar measures ozone with a precision generally better than 10% at a temporal resolution of 10 min and a vertical resolution from 150 m at the bottom of the ABL to 550 m at the top. A measurement example from summertime shows that inhomogeneous ozone aloft was affected by both surface emissions and the evolution of ABL structures.

differential absorption lidar↗
DOE OSTI2023

Photometry, Centroid and Point-spread Function Measurements in the LSST Camera Focal Plane Using Artificial Stars

Abstract The Vera C. Rubin Observatory’s LSST Camera (LSSTCam) pixel response has been characterized using laboratory measurements with a grid of artificial stars. We quantify the contributions to photometry, centroid, point-spread function size, and shape measurement errors due to small anomalies in the LSSTCam CCDs. The main sources of those anomalies are quantum efficiency variations and pixel area variations induced by the amplifier segmentation boundaries and “tree-rings”—circular variations in silicon doping concentration. This laboratory study using artificial stars projected on the sensors shows overall small effects. The residual effects on point-spread function (PSF) size and shape are below 0.1%, meeting the ten-year LSST survey science requirements. However, the CCD mid-line presents distortions that can have a moderate impact on PSF measurements. This feature can be avoided by masking the affected regions. Effects of tree-rings are observed on centroids and PSFs of the artificial stars and the nature of the effect is confirmed by a study of the flat-field response. Nevertheless, further studies of the full-focal plane with stellar data should more completely probe variations and might reveal new features, e.g., wavelength-dependent effects. The results of this study can be used as a guide for the on-sky operation of LSSTCam.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1990

Minimizing distortion in truss structures via Tabu search

The shape control of large flexible space structures is of great interest to structural designers. A related problem is to seek ways to minimize the need for active controls by careful design and construction of the space structure. A tetrahedral truss structure that is used to support a precision segmented reflector or antenna surface is considered. The structure has a hexagonal platform and is characterized by the number of rings of members in the truss. For simplicity it is assumed that a flat truss geometry exists. Hence, all structural members and ball joints are required to have the same nominal length and diameter, respectively. Inaccuracies in the length of member or diameters of joints may produce unacceptable levels of surface distortion and internal forces. In the case of a truss structure supporting an antenna, surface distortions may cause unacceptable gain loss or pointing errors. The focus is solely on surface distortion, however, internal forces may be treated in a similar manner. To test the Tabu search code for DSQRMS the appropriate influence matrices are used for a flat, two-ring tetrahedral reflector truss generated by Green and Haftka (1989). In this example there are 102 members (NMEMB) and 31 ball joints (NJOINT) of the same nominal length, respectively. Hence, all the members may be interchanged and all the joints may be interchanged. In addition, 19 positions on the surface of the truss (NNODES) were used to measure error influences. After a variety of experiments a set of good parameters was choosen for Tabu search. The sample size at each iteration is 10*NMEMB and the short term memory size is 40. In addition four pruning rules were used to accelerate the search..

Kincaid, Rex K.↗
DOE OSTI2020

Network-Scale Ubiquitous Volume Estimation Using Tree-Based Ensemble Learning Methods

Currently ubiquitous volume data for roadway networks remains the key missing dimension in traffic operations. Most volume data are average annual daily traffic (AADT) measures derived from the Highway Performance Monitoring System (HPMS). Although methods to factor the AADT to hourly averages for typical day of week exist, actual volume data is limited to a sparse collection of locations in which volumes are continuously recorded. This paper/poster explores the use of state-of-art machine learning techniques to estimate accurate volume measures that span the highway network providing ubiquitous coverage in space, and point-in-time measures for a specific date and time. Three tree-based ensemble learning models, random forest (RF), gradient boost machine (GBM), and extreme gradient boost (XGBoost), were tested for volume estimation by learning from combined dataset of commercial probe data provided by TomTom, the FHWA's Travel Monitoring Analysis System (TMAS) data, and other infrastructure attributes such as number of lanes, speed limit, and weather. The methods were tested on major corridors and freeways in the metropolitan area of Denver. All three machine learning methods were able to provide hourly volume estimates 24 hours a day, 7 days a week, and 365 days a year with around 18% mean absolute error to true volume and about 5% of error with respect to roadway capacity. The low error measures allow the potential application by transportation agencies.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2020

Data-Consistent Inversion for Stochastic Input-to-Output Maps

Data-consistent inversion is a recently developed measure-theoretic framework for solving a stochastic inverse problem involving models of physical systems. The goal is to construct a probability measure on model inputs (i.e., parameters of interest) whose associated push-forward measure matches (i.e., is consistent with) a probability measure on the observable outputs of the model (i.e., quantities of interest). Previous implementations required the map from parameters of interest to quantities of interest to be deterministic. This work generalizes this framework for maps that are stochastic, i.e., contain uncertainties and variation not explainable by variations in uncertain parameters of interest. Generalizations of previous theorems of existence, uniqueness, and stability of the data-consistent solution are provided while new theoretical results address the stability of marginals on parameters of interest. A notable aspect of the algorithmic generalization is the ability to query the solution to generate independent identically distributed samples of the parameters of interest without requiring knowledge of the so-called stochastic parameters. This work therefore extends the applicability of the data-consistent inversion framework to a much wider class of problems. This includes those based on purely experimental and field data where only a subset of conditions are either controllable or can be documented between experiments while the underlying physics, measurement errors, and any additional covariates are either uncertain or not accounted for by the researcher. Finally, numerical examples demonstrate application of this approach to systems with stochastic sources of uncertainties embedded within the modeling of a system and a numerical diagnostic is summarized that is useful for determining if a key assumption is verified among competing choices of stochastic maps.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2007

Portable Microleak-Detection System

The figure schematically depicts a portable microleak-detection system that has been built especially for use in testing hydrogen tanks made of polymer-matrix composite materials. (As used here, microleak signifies a leak that is too small to be detectable by the simple soap-bubble technique.) The system can also be used to test for microleaks in tanks that are made of other materials and that contain gases other than hydrogen. Results of calibration tests have shown that measurement errors are less than 10 percent for leak rates ranging from 0.3 to 200 cm3/min. Like some other microleak-detection systems, this system includes a vacuum pump and associated plumbing for sampling the leaking gas, and a mass spectrometer for analyzing the molecular constituents of the gas. The system includes a flexible vacuum chamber that can be attached to the outer surface of a tank or other object of interest that is to be tested for leakage (hereafter denoted, simply, the test object). The gas used in a test can be the gas or vapor (e.g., hydrogen in the original application) to be contained by the test object. Alternatively, following common practice in leak testing, helium can be used as a test gas. In either case, the mass spectrometer can be used to verify that the gas measured by the system is the test gas rather than a different gas and, hence, that the leak is indeed from the test object.

Rivers, H. Kevin↗
NASA NTRS2019

Further Development Of Aperture: A Precise Extremely Large Reflective Telescope Using Re-Configurable Elements: NASA Innovative Advanced Concepts (NIAC) - Phase II

One of the pressing needs for space ultraviolet-visible astronomy is a design to allow larger mirrors than the James Webb Space Telescope primary. The diameter of the rocket fairing limits the mirror diameter such that all future missions calling for mirrors up to 16 meters in diameter or larger will require a mirror that is deployed post-launch. In response to the deployment requirement, we address the issues of this concept called "A Precise Extremely Large Reflective Telescope Using Reconfigurable Elements (APERTURE) with both hardware experiments and software simulations... We designed and built several fixtures with O-rings to hold a membrane. We established a coating process to make a membrane that was coated on one side with Cr and the other side with Cr-Terfenol-D-NiCo. The Terfenol-D (T-D hereafter) is the MSM (Magnetic Smart Memory) we use. We bought and established a procedure for measuring a deformation over time and purchased a Shack Hartmann system from Imagine Optic (https://www.imagine-optic.com). The first substrate we used was DuPont (TM) Kapton® polyimide film. Due to material creep, we found the stability over a 48-hour period with a Kapton substrate was not as good as desired (greater than 1 micron). We then switched to CP1 Polyimide. We found the CP1 much more stable to creep, being stable from about 3 hours to 48 hours to within a measurement error to below approximately 0.1 micron. We produced a 13 micron maximum deviation on a 50-millimeter-diameter piece of CP1 (25 microns thick). The T-D coating was about 2 microns, and the other layers, about 10 nanometers. The magnetic field at the base was about 0.1 teslas. We can make the T-D film at least 5 times thicker and the magnetic field at least 5 times stronger, and hence make deformations as much as 25 times larger. We have a formed a collaboration produced at the NIAC (NASA Innovative Advanced Concepts) mid-term review with Dr. Ron Shiri of Goddard Space Flight Center (GSFC) to explore making controlled deviations on lambda/14-lambda/20 scales which are required to bring a surface to the diffraction limit. We carried out only preliminary work on Si using a Coordinate Measuring Machine (CMM), which produced deviations on the 1 micron level. We are still working on a program to bring to GSFC a flat enough (radius of curvature greater than 10 microns) -coated a Si piece with Cr, T-D, NiCo. Then we plan to carry out tests with an interferometer. Further, we formed a new collaboration with Prof. Rajan Vaidyanathan of the University of Central Florida to replace the CP1 with a shape memory alloy (SMA). With his collaboration, we acquired new Federal funding outside of NASA to explore the use of SMAs (we use NiTi). Our preliminary results indicate that we can produce deformations greater than 1 micron on approximately 100 microns thick. Furthermore we have shown that the NiTi can deploy to better than 1 micron of its set original and then trained shape.

Re-Configurable Elements↗
DOE OSTI2023

Impact of Irradiation-Induced Filter Heating on Calibration of NIR-Longpass-Filtered Reference Solar Cells

Reference solar cells play a crucial role in determining the performance of photovoltaic (PV) devices. In the performance calibration chain of mainstream single-junction PV technologies, common reference cell types include Si, KG-filtered Si, and GaAs cells. For emerging multijunction (MJ) PV technologies, such as CdTe/Si and perovskite/Si, Si reference cells with colored glass near-infrared-longpass (LP) filters like Schott RG-715 and 850 glass have been proposed. They offer a better spectral response that matches the bottom junctions of the emerging MJs, which could lead to lower uncertainties in performance measurements. However, this article reveals a prominent decrease in short-circuit current (ISC) during National Renewable Energy Laboratory's primary calibration over the course of minutes when using this type of LP-filtered reference cell, which could result in unacceptable measurement errors. Unlike quartz or KG filter glasses, LP colored glass filters demonstrate temperature-sensitive cut-on wavelength. When incident irradiance reaches these LP-filtered reference cells, the increased temperature due to light absorption causes a shift of the cut-on to longer wavelength. As a result, the device I SC exhibits a continuous decrease (approximately a 2.5% drop with RG850 LP filter in a 5-min duration) even when the device temperature is controlled at 25 degrees C. To address the temperature gradient issue between colored glass LP filter and the reference cell, we propose the direct integration of thin-film LP semiconductor layer on reference cells. This type of LP-filter-integrated cell has shown minimal temperature-related I SC variation and can serve as a more reliable reference cell source for accurate performance measurements.

14 SOLAR ENERGY↗
DOE OSTI2019

Using Low Energy Atmospheric Neutrinos for Precision Measurement of the Mixing Parameters

Use of low energy atmospheric neutrinos is considered for precision measurement of neutrino mixing parameters. At around energy $E \simeq$ a few $\times$100 MeV and baseline $L$ of a few $\times$1000 km, CP phase effect is $\sim$10 times larger than that of the conventional LBL accelerator neutrino experiments. We report here a few progresses: (1) To analyze physics in the region, a new perturbative framework at around the solar-scale enhancement is developed. (2) To know the characteristic features of CP $\delta$ dependence of the atmospheric neutrinos at low energies, we plot the ratio of the $\nu_{e}$ and $\bar{\nu}_{e}$ fluxes $F(\delta)/F(\delta=0)$ for $\delta=\pm \frac{\pi}{2}$ and $\pi$ as a function of $E$ using the Honda {\it et al.} flux. Interestingly, it shows $\delta$ dependence of $\simeq$5-10\% level, with positive (negative) sign for $\nu_{e}$ ($\bar{\nu}_{e}$). (3) To reduce the flux systematic errors, measurement of muon energy distribution around $\sim$1 GeV at high altitude may be useful. Water Cherenkov or muon range detectors may be the options with advantage in the latter if it is magnetized in view of the $\delta$ dependence of the flux.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1990

Sea level static calibration of a compact multimission aircraft propulsion simulator with inlet flow distortion

Wind tunnel tests of propulsion-integrated aircraft models have identified inlet flow distortion as a major source of compressor airflow measurement error in turbine-powered propulsion simulators. Consequently, two Compact Multimission Aircraft Propulsion Simulator (CMAPS) units were statically tested at sea level ambient conditions to establish simulator operating performance characteristics and to calibrate the compressor airflow against an accurate bellmouth flowmeter in the presence of inlet flow distortions. The distortions were generated using various-shaped wire mesh screens placed upstream of the compressor. CMAPS operating maps and performance envelopes were obtained for inlet total pressure distortions (ratio of the difference between the maximum and minimum total pressures to the average total pressure) up to 35 percent, and were compared to baseline simulator operating characteristics for a uniform inlet. Deviations from CMAPS baseline performance were attributed to the coupled variation of both compressor inlet-flow distortion and Reynolds number index throughout the simulator operating envelope for each screen configuration. Four independent methods were used to determine CMAPS compressor airflow; direct compressor inlet and discharge measurements, an entering/exiting flow-balance relationships, and a correlation between the mixer pressure and the corrected compressor airflow. Of the four methods, the last yielded the least scatter in the compressor flow coefficient, approximately + or - 3 percent over the range of flow distortions.

Won, Mark J.↗
DOE OSTI2023

Optimal PMU design based on sampling model and sensitivity analysis

The precise measurements of the synchrophasor and frequency from phasor measurement units (PMUs) are widely used in power grid applications. With the improvement of the technique, applications always require stable dynamic performance and higher accuracy for the synchrophasor and frequency measurements, which is challenging for PMU development. To evaluate the contribution of PMU hardware to measurement accuracy, this paper proposes a general-purpose sampling model to analyze the measurement error. In the proposed sampling model, a strict mathematical derivation is derived, where its error is purely determined by the parameters of the PMU hardware. The sensitivity analysis is carried out by three methods, including mathematical analysis, computer simulation, and variance-based sensitivity analysis. Through the sensitivity analysis, this paper establishes the systematic formulation and the inclusion of synchrophasor, frequency, and ROCOF. Experimental results based on the real-world testbench involving distribution-level PMUs match the mathematical analysis conclusion, which verifies the correctness of the general-purpose sampling model. Furthermore, a strategy for the optimal PMU design is proposed, which could guide PMU design in the future.

42 ENGINEERING↗
NASA NTRS1999

A Spherical Earth Solution for TOA Lightning Location Retrieval

With the advent of high-speed digital computers, the method of chi square minimization is a highly practical means for analyzing a wide variety of (otherwise intractable) nonlinear inversion problems in applied mathematical physics. Little thought or effort is required to apply the chi square method to obtain quick and reasonable estimates of a solution, and the method offers a means to assess retrieval errors. Because the method is simple and practical it is sometimes hastily applied to problems that can be solved by formal analytic or quasi-analytic means. Presently, Global Atmospherics Inc. (GAI) finds the minimum of a chi square function to analyze time-of-arrival (TOA) and magnetic bearing data derived from the National Lightning Detection Network (NLDN); ellipsoidal Earth geometry is assumed. An analytic solution to this problem has not yet been attained, but the consideration and solving of less general problem statements might eventually lead to a final solution. In the present study, the problem of retrieving lightning ground-strike location on a spherical Earth surface using a network of 4 or more time-of-arrival (TOA) sensors is considered. It is shown that this problem has an analytic solution and therefore does not require the use of nonlinear estimation theory (such as the chi square method mentioned above). The mathematical robustness of the analytic solution is tested using computer-generated lightning sources and simulated TOA measurement errors. A quasi-analytic extension of the spherical Earth solution for an oblate spheroidal Earth geometry is considered in a related study. The incorporation of magnetic bearing information into these analytic solutions would lead to a general and elegant analytic retrieval scheme that would most likely replace the chi square estimation theory currently employed by Global Atmospherics Inc. (GAI).

Koshak, William J,↗
NASA NTRS2015

Optimization of Microphone Locations for Acoustic Liner Impedance Eduction

Two impedance eduction methods are explored for use with data acquired in the NASA Langley Grazing Flow Impedance Tube. The first is an indirect method based on the convected Helmholtz equation, and the second is a direct method based on the Kumaresan and Tufts algorithm. Synthesized no-flow data, with random jitter to represent measurement error, are used to evaluate a number of possible microphone locations. Statistical approaches are used to evaluate the suitability of each set of microphone locations. Given the computational resources required, small sample statistics are employed for the indirect method. Since the direct method is much less computationally intensive, a Monte Carlo approach is employed to gather its statistics. A comparison of results achieved with full and reduced sets of microphone locations is used to determine which sets of microphone locations are acceptable. For the indirect method, each array that includes microphones in all three regions (upstream and downstream hard wall sections, and liner test section) provides acceptable results, even when as few as eight microphones are employed. The best arrays employ microphones well away from the leading and trailing edges of the liner. The direct method is constrained to use microphones opposite the liner. Although a number of arrays are acceptable, the optimum set employs 14 microphones positioned well away from the leading and trailing edges of the liner. The selected sets of microphone locations are also evaluated with data measured for ceramic tubular and perforate-over-honeycomb liners at three flow conditions (Mach 0.0, 0.3, and 0.5). They compare favorably with results attained using all 53 microphone locations. Although different optimum microphone locations are selected for the two impedance eduction methods, there is significant overlap. Thus, the union of these two microphone arrays is preferred, as it supports usage of both methods. This array contains 3 microphones in the upstream hard wall section, 14 microphones opposite the liner, and 3 microphones in the downstream hard wall section.

Jones, M. G.↗