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NASA NTRS2007

Lightning Charge Retrievals: Dimensional Reduction, LDAR Constraints, and a First Comparison with LIS Satellite Data

A "dimensional reduction" ("DR") method is introduced for analyzing lightning field changes ((Delta)Es) whereby the number of unknowns in a discrete two-charge model is reduced from the standard eight (x, y, z, Q, x', y', z', Q') to just four (x, y, z, Q). The four unknowns (x, y, z, Q) are found by performing a numerical minimization of a chi-square function. At each step of the minimization, an overdetermined fixed matrix (OFM) method is used to immediately retrieve the best "residual source" (x', y', z', Q'), given the values of (x, y, z, Q). In this way, all eight parameters Ix. y, z, Q, x', y', z', Q') are found, yet a numerical search of only four parameters (x, y, z, Q) is required. The DR method has been used to analyze lightning-caused (Delta)Es derived from multiple ground-based electric field measurements at the NASA Kennedy Space Center (KSC) and U.S. Air Force Eastern Range (ER). The accuracy of the DR method has been assessed by comparing retrievals with data provided by the lightning detection and ranging (LDAR) system at the KSC-ER. and from least squares error estimation theory, and the method is shown to be a useful "stand alone" charge retrieval tool. Since more than one charge distribution describes a finite set of (Delta)Es (i.e., solutions are nonunique), and since there can be appreciable differences in the physical characteristics of these solutions, not all DR solutions are physically acceptable. Hence. an alternative and more accurate method of analysis is introduced that uses LDAR data to constrain the geometry of the charge solutions. thereby removing physically unacceptable retrievals. The charge solutions derived from this method are shown to compare well with independent satellite- and ground-based observations of lightning m several Florida storms.

Koshak, W. J.↗
NASA NTRS2007

Lightning Charge Retrievals: Dimensional Reduction, LDAR Constraints, and a First Comparison with LIS Satellite Data

A "dimensional reduction" (DR) method is introduced for analyzing lightning field changes (DELTAEs) whereby the number of unknowns in a discrete two-charge model is reduced from the standard eight (x, y, z, Q, x', y', z', Q') to just four (x, y, z, Q). The four unknowns (x, y, z, Q) are found by performing a numerical minimization of a chi-square function. At each step of the minimization, an Overdetermined Fixed Matrix (OFM) method is used to immediately retrieve the best "residual source" (x', y', z', Q'), given the values of (x, y, z, Q). In this way, all 8 parameters (x, y, z, Q, x', y', z', Q') are found, yet a numerical search of only 4 parameters (x, y, z, Q) is required. The DR method has been used to analyze lightning-caused DeltaEs derived from multiple ground-based electric field measurements at the NASA Kennedy Space Center (KSC) and USAF Eastern Range (ER). The accuracy of the DR method has been assessed by comparing retrievals with data provided by the Lightning Detection And Ranging (LDAR) system at the KSC-ER, and from least squares error estimation theory, and the method is shown to be a useful "stand-alone" charge retrieval tool. Since more than one charge distribution describes a finite set of DELTAEs (i.e., solutions are non-unique), and since there can exist appreciable differences in the physical characteristics of these solutions, not all DR solutions are physically acceptable. Hence, an alternative and more accurate method of analysis is introduced that uses LDAR data to constrain the geometry of the charge solutions, thereby removing physically unacceptable retrievals. The charge solutions derived from this method are shown to compare well with independent satellite- and ground-based observations of lightning in several Florida storms.

Koshak, W. J.↗
NASA NTRS1989

Electrochemical Studies Of Aluminum Coated With Primer

Technical paper describes experiments using ac-impedance method of analyzing corrosion of metal surfaces and breakdown of protective coatings. Alternating-current-impedance scans provide more detailed information, indicating changes in corrosion rates, progressive deterioration of coatings, changes in metal surfaces, and reaction mechanisms. Changes in conditions of coatings detected by ac method before dc methods show increase in metal-corrosion current. Measurements of impedance as function of frequency provides sufficient data to enable resistances and capacitances to be determined by performing least-squares fit. Knowledge of dependence of these values on time leads to more detailed understanding of corrosion process.

Danford, Merlin D.↗
NASA NTRS1993

A general rough-surface inversion algorithm: Theory and application to SAR data

Rough-surface inversion has significant applications in interpretation of SAR data obtained over bare soil surfaces and agricultural lands. Due to the sparsity of data and the large pixel size in SAR applications, it is not feasible to carry out inversions based on numerical scattering models. The alternative is to use parameter estimation techniques based on approximate analytical or empirical models. Hence, there are two issues to be addressed, namely, what model to choose and what estimation algorithm to apply. Here, a small perturbation model (SPM) is used to express the backscattering coefficients of the rough surface in terms of three surface parameters. The algorithm used to estimate these parameters is based on a nonlinear least-squares criterion. The least-squares optimization methods are widely used in estimation theory, but the distinguishing factor for SAR applications is incorporating the stochastic nature of both the unknown parameters and the data into formulation, which will be discussed in detail. The algorithm is tested with synthetic data, and several Newton-type least-squares minimization methods are discussed to compare their convergence characteristics. Finally, the algorithm is applied to multifrequency polarimetric SAR data obtained over some bare soil and agricultural fields. Results will be shown and compared to ground-truth measurements obtained from these areas. The strength of this general approach to inversion of SAR data is that it can be easily modified for use with any scattering model without changing any of the inversion steps. Note also that, for the same reason it is not limited to inversion of rough surfaces, and can be applied to any parameterized scattering process.

Moghaddam, M.↗
NASA NTRS2008

JPEG 2000 Encoding with Perceptual Distortion Control

An alternative approach has been devised for encoding image data in compliance with JPEG 2000, the most recent still-image data-compression standard of the Joint Photographic Experts Group. Heretofore, JPEG 2000 encoding has been implemented by several related schemes classified as rate-based distortion-minimization encoding. In each of these schemes, the end user specifies a desired bit rate and the encoding algorithm strives to attain that rate while minimizing a mean squared error (MSE). While rate-based distortion minimization is appropriate for transmitting data over a limited-bandwidth channel, it is not the best approach for applications in which the perceptual quality of reconstructed images is a major consideration. A better approach for such applications is the present alternative one, denoted perceptual distortion control, in which the encoding algorithm strives to compress data to the lowest bit rate that yields at least a specified level of perceptual image quality. Some additional background information on JPEG 2000 is prerequisite to a meaningful summary of JPEG encoding with perceptual distortion control. The JPEG 2000 encoding process includes two subprocesses known as tier-1 and tier-2 coding. In order to minimize the MSE for the desired bit rate, a rate-distortion- optimization subprocess is introduced between the tier-1 and tier-2 subprocesses. In tier-1 coding, each coding block is independently bit-plane coded from the most-significant-bit (MSB) plane to the least-significant-bit (LSB) plane, using three coding passes (except for the MSB plane, which is coded using only one "clean up" coding pass). For M bit planes, this subprocess involves a total number of (3M - 2) coding passes. An embedded bit stream is then generated for each coding block. Information on the reduction in distortion and the increase in the bit rate associated with each coding pass is collected. This information is then used in a rate-control procedure to determine the contribution of each coding block to the output compressed bit stream.

Watson, Andrew B.↗
NASA NTRS1989

Methods for local gravity field approximation

The most widely known modern method for estimating gravity field values from observed data is least-squares collocation. Its advantages are that it can make estimates at arbitrary locations based on irregularly spaced observations, and that it makes use of statistical information about errors in the input data while providing corresponding information about the quality of the output estimates. Disadvantages of collocation include the necessity of inverting square matrices of dimension equal to the number of data values and the need to assume covariance models for the gravity field and the data errors. Fourier methods are an important alternative to collocation; having the advantage of greater computational efficiency, but requiring data estimates to be on a regular grid and not using or providing statistical accuracy information. The GEOFAST algorithm is an implementation of collocation that achieves high computational efficiency by transforming the estimation equations into the frequency domain where an accurate approximation may be made to reduce the workload. The forward and inverse Fast Fourier Transforms (FFTs) are utilized. The accuracy and computational efficiency of the GEOFAST algorithm is demonstrated using two sets of synthetic gravity data: marine gravity for an ocean trench region including wavelengths longer than 200 km; and local land gravity containing wavelengths as short as 5 km. These results are discussed along with issues such as the advantages of first removing reference field models before carrying out the estimation algorithm.

Sailor, R. V.↗
NASA NTRS2011

High-Temperature Thermometer Using Cr-Doped GdAlO3 Broadband Luminescence

A new concept has been developed for a high-temperature luminescence-based optical thermometer that both shows the desired temperature sensitivity in the upper temperature range of present state-of-the-art luminescence thermometers (above 1,300 C), while maintaining substantial stronger luminescence signal intensity that will allow these optical thermometers to operate in the presence of the high thermal background radiation typical of industrial applications. This objective is attained by using a Cr-doped GdAlO3 (Cr:GdAlO3) sensor with an orthorhombic perovskite structure, resulting in broadband luminescence that remains strong at high temperature due to the favorable electron energy level spacing of Cr:GdAlO3. The Cr:GdAlO3 temperature (and pressure) sensor can be incorporated into, or applied onto, a component s surface when a non-contact surface temperature measurement is desired, or alternatively, the temperature sensor can be attached to the end of a fiber-optic probe that can then be positioned at the location where the temperature measurement is desired. In the case of the fiber-optic probe, both the pulsed excitation and the luminescence emission travel through the fiber-optic light guide. In either case, a pulsed light source provides excitation of the luminescence, and the broadband luminescence emission is collected. Real-time temperature measurements are obtain ed using a least-squares fitting algorithm that determines the luminescence decay time, which has a known temperature dependence established by calibration. Due to the broad absorption and emission bands for Cr:GdAlO3, there is considerable flexibility in the choice of excitation wavelength and emission wavelength detection bands. The strategic choice of the GdAlO3 host is based on its high crystal field, phase stability, and distorted symmetry at the Cr3+ occupation sites. The use of the broadband emission for temperature sensing at high temperatures is a key feature of the invention and is novel since broadband luminescence emission normally shows severe thermal quenching. The tightly bound AlO6 octahedra in GdAlO3 results in a larger energy barrier to nonradiative decays than in other materials and therefore makes using broadband emission for temperature sensing possible at high temperatures. This approach results in a substantial increase in temperature capability. For example, the most commonly used Cr-doped crystal used for luminescence-based temperature measurements, ruby, has only been demonstrated up to 600 C, whereas the Cr:GdAlO3 optical thermometer under development has already been shown to exhibit useful luminescence up to 1,300 C. Because GdAlO3 is non-reactive and is stable in harsh, high-temperature environments, sensors composed of Cr:GdAlO3 will be very well suited for remote high-temperature measurements in engine or industrial environments where its intense high-temperature luminescence will stand out above significant thermal radiation background levels.

Eldridge, Jeffrey↗
NASA NTRS1978

Are supernovae radio sources - A search for radio emission from young supernova remnants

A search has been conducted for radio emission at 11 and 3.7 cm from 46 recent supernovae having accurately determined positions and ages of a few months to 79 years. None of these supernovae was detected at a flux density greater than 5-10 mJy. These negative results cannot be explained by internal absorption and are thus due to intrinsically weak synchrotron emission in young supernova remnants. There are two possibilities: either (1) relativistic particles are accelerated not by the supernova outburst but by processes occurring much later (at least about 75 years) in the remnant or (2) the magnetic field in the young remnants evolves very slowly, in proportion to the inverse square root of time, so that its value in the remnants observed was no more than about 0.002 gauss. The constraints the observations place on these possibilities and on the energy in cosmic rays in young remnants are discussed. Gamma-ray observations at times of no more than about 1 year following an outburst will allow one to discriminate between the two alternative explanations of the radio results.

Brown, R. L.↗
NASA NTRS2017

Geometrically Flexible and Efficient Flow Analysis of High Speed Vehicles Via Domain Decomposition, Part 1: Unstructured-Grid Solver for High Speed Flows

The ability to solve the equations governing the hypersonic turbulent flow of a real gas on unstructured grids using a spatially-elliptic, 2nd-order accurate, cell-centered, finite-volume method has been recently implemented in the VULCAN-CFD code. This paper describes the key numerical methods and techniques that were found to be required to robustly obtain accurate solutions to hypersonic flows on non-hex-dominant unstructured grids. The methods and techniques described include: an augmented stencil, weighted linear least squares, cell-average gradient method, a robust multidimensional cell-average gradient-limiter process that is consistent with the augmented stencil of the cell-average gradient method and a cell-face gradient method that contains a cell skewness sensitive damping term derived using hyperbolic diffusion based concepts. A data-parallel matrix-based symmetric Gauss-Seidel point-implicit scheme, used to solve the governing equations, is described and shown to be more robust and efficient than a matrix-free alternative. In addition, a y+ adaptive turbulent wall boundary condition methodology is presented. This boundary condition methodology is deigned to automatically switch between a solve-to-the-wall and a wall-matching-function boundary condition based on the local y+ of the 1st cell center off the wall. The aforementioned methods and techniques are then applied to a series of hypersonic and supersonic turbulent flat plate unit tests to examine the efficiency, robustness and convergence behavior of the implicit scheme and to determine the ability of the solve-to-the-wall and y+ adaptive turbulent wall boundary conditions to reproduce the turbulent law-of-the-wall. Finally, the thermally perfect, chemically frozen, Mach 7.8 turbulent flow of air through a scramjet flow-path is computed and compared with experimental data to demonstrate the robustness, accuracy and convergence behavior of the unstructured-grid solver for a realistic 3-D geometry on a non-hex-dominant grid.

White, Jeffery A.↗
NASA NTRS2002

Computational Thermochemistry of Jet Fuels and Rocket Propellants

The design of new high-energy density molecules as candidates for jet and rocket fuels is an important goal of modern chemical thermodynamics. The NASA Glenn Research Center is home to a database of thermodynamic data for over 2000 compounds related to this goal, in the form of least-squares fits of heat capacities, enthalpies, and entropies as functions of temperature over the range of 300 - 6000 K. The chemical equilibrium with applications (CEA) program written and maintained by researchers at NASA Glenn over the last fifty years, makes use of this database for modeling the performance of potential rocket propellants. During its long history, the NASA Glenn database has been developed based on experimental results and data published in the scientific literature such as the standard JANAF tables. The recent development of efficient computational techniques based on quantum chemical methods provides an alternative source of information for expansion of such databases. For example, it is now possible to model dissociation or combustion reactions of small molecules to high accuracy using techniques such as coupled cluster theory or density functional theory. Unfortunately, the current applicability of reliable computational models is limited to relatively small molecules containing only around a dozen (non-hydrogen) atoms. We propose to extend the applicability of coupled cluster theory- often referred to as the 'gold standard' of quantum chemical methods- to molecules containing 30-50 non-hydrogen atoms. The centerpiece of this work is the concept of local correlation, in which the description of the electron interactions- known as electron correlation effects- are reduced to only their most important localized components. Such an advance has the potential to greatly expand the current reach of computational thermochemistry and thus to have a significant impact on the theoretical study of jet and rocket propellants.

Crawford, T. Daniel↗
NASA NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗
NASA NTRS2022

Spatial Grid-Based Object Localization from A Single Passive Sensor: A Deep Learning-Integrated Approach

Ongoing efforts at NASA’s Langley Research Center have produced a single passive sensor system for detecting ground objects and pinpointing their real-world location to a desired level of precision. The Langley center serves as a test range for unmanned aerial systems (UAS) and real-time knowledge about the location of people on campus is needed to inform least-risk UAS flight operations. The proposed system provides this knowledge through a camera combined with a convolutional neural network and an algorithm that projects an imaginary grid of square cells from the ground plane onto the perspective view of the camera. The position of detected objects on the camera’s projected grid determines their location in the real-world. The imaginary grid is easily mapped to a universal coordinate system, such as longitude and latitude, to provide both relative and absolute positional information of the detected objects. This simple system is shown to be accurate and effective, with decisive advantages over alternative multi-sensor and active sensor approaches. Extensions to the system are described to allow adaptation to a variety of other use cases.

object localization↗
NASA NTRS1993

Numerical investigation of sound transmission through double wall cylinders with respect to active noise control

Market pressure for more fuel efficient air travel has led to increased use of turboprop and higher bypass turbofan engines. The low frequency components of propeller, jet and boundary layer noise are difficult to attenuate with conventional passive techniques. Weight and geometric restrictions for sound absorbing meterials limit the amount and type of treatment that may be applied. An active noise control (ANC) method is providing to be an attractive alternative. The approach taken in this paper uses a numerical finite/boundary element method (FEM/BEM) that may be easilty adapted to arbitrary geometries. A double walled cylinder is modeled using commercially available software. The outer shell is modeled as an aluminum cylinder, similar to that of aircraft skins. The inner shell is modeled as a composite material representative of a lightweight, stiff trim panel. Two different inner shell materials are used. The first is representative of current trim structure, the second a much stiffer composite. The primary source is generated by an exterior acoustic monopole. Control fields are generated using normal force inputs to the inner cylindrical shell. A linear least mean square (LMS) algorithm is used to determine amplitudes of control forces that minimize the interior acoustic field. Coupling of acoustic and structural modes and noise reductions are discussed for each of the inner shell materials.

Coats, T. J.↗
NASA NTRS1992

Parameterization of solar cells

The aggregation (sorting) of the individual solar cells into an array is commonly based on a single operating point on the current-voltage (I-V) characteristic curve. An alternative approach for cell performance prediction and cell screening is provided by modeling the cell using an equivalent electrical circuit, in which the parameters involved are related to the physical phenomena in the device. These analytical models may be represented by a double exponential I-V characteristic with seven parameters, by a double exponential model with five parameters, or by a single exponential equation with four or five parameters. In this article we address issues concerning methodologies for the determination of solar cell parameters based on measured data points of the I-V characteristic, and introduce a procedure for screening of solar cells for arrays. We show that common curve fitting techniques, e.g., least squares, may produce many combinations of parameter values while maintaining a good fit between the fitted and measured I-V characteristics of the cell. Therefore, techniques relying on curve fitting criteria alone cannot be directly used for cell parameterization. We propose a consistent procedure which takes into account the entire set of parameter values for a batch of cells. This procedure is based on a definition of a mean cell representing the batch, and takes into account the relative contribution of each parameter to the overall goodness of fit. The procedure is demonstrated on a batch of 50 silicon cells for Space Station Freedom.

Appelbaum, J.↗
NASA NTRS1992

An evaluation of corrosion protection by two epoxy primers on 2219-T87 and 7075-T73 aluminum

A comparison of the corrosion protection provided by two amine epoxy primers was made using salt fog, alternate immersion, and total immersion as exposure media. The study is the result of a request to use an unqualified low volatile organic carbon (VOC) primer (AKZO 463-6-78) in place of the current primer (AKZO 463-6-3) because environmental regulations have eliminated use of the current primer in many states. Primed, scribed samples of 2219-T87 and 7075-T73 aluminum were exposed to 5-percent NaCl salt fog and 3.5-percent NaCl alternate immersion for a period of 90 days. In addition, electrode samples immersed in 3.5-percent NaCl were tested using electrochemical impedance spectroscopy (EIS). The EG&G model 368 ac impedance measurement system was used to monitor changing properties of AKZO 463-6-78 and AKZO 463-6-3 primed 2219-T87 aluminum for a period of 30 days. The response of the corroding system of a frequency scan can be modeled in terms of an equivalent circuit consisting of resistors and capacitors in a specific arrangement. Each resistor/capacitor combination represents physical processes taking place within the electrolyte, at the electrolyte/primer surface, within the coating, and at the coating/substrate surface. Values for the resistors and capacitors are assigned following a nonlinear least squares fit of the data to the equivalent circuit. Changes in the values of equivalent circuit parameters during the 30-day exposure allow assessment of the time to and mechanism of coating breakdown.

Mendrek, M. J.↗
NASA NTRS2007

Tracking and Data Relay Satellite (TDRS) Orbit Estimation Using an Extended Kalman Filter

Alternatives to the Tracking and Data Relay Satellite (TDRS) orbit estimation procedure were studied to develop a technique that both produces more reliable results and is more amenable to automation than the prior procedure. The Earth Observing System (EOS) Terra mission has TDRS ephemeris prediction 3(sigma) requirements of 75 meters in position and 5.5 millimeters per second in velocity over a 1.5-day prediction span. Meeting these requirements sometimes required reruns of the prior orbit determination (OD) process, with manual editing of tracking data to get an acceptable solution. After a study of the available alternatives, the Flight Dynamics Facility (FDF) began using the Real-Time Orbit Determination (RTOD(Registered TradeMark)) Kalman filter program for operational support of TDRSs in February 2007. This extended Kalman filter (EKF) is used for daily support, including within hours after most thrusting, to estimate the spacecraft position, velocity, and solar radiation coefficient of reflectivity (C(sub R)). The tracking data used are from the Bilateration Ranging Transponder System (BRTS), selected TDRS System (TDRSS) User satellite tracking data, and Telemetry, Tracking, and Command (TT&C) data. Degraded filter results right after maneuvers and some momentum unloads provided incentive for a hybrid OD technique. The results of combining EKF strengths with the Goddard Trajectory Determination System (GTDS) Differential Correction (DC) program batch-least-squares solutions, as recommended in a 2005 paper on the chain-bias technique, are also presented.

Ward, Douglas T.↗
NASA NTRS2017

Linear Subpixel Learning Algorithm for Land Cover Classification from WELD using High Performance Computing

In this work, we use a Fully Constrained Least Squares Subpixel Learning Algorithm to unmix global WELD (Web Enabled Landsat Data) to obtain fractions or abundances of substrate (S), vegetation (V) and dark objects (D) classes. Because of the sheer nature of data and compute needs, we leveraged the NASA Earth Exchange (NEX) high performance computing architecture to optimize and scale our algorithm for large-scale processing. Subsequently, the S-V-D abundance maps were characterized into 4 classes namely, forest, farmland, water and urban areas (with NPP-VIIRS-national polar orbiting partnership visible infrared imaging radiometer suite nighttime lights data) over California, USA using Random Forest classifier. Validation of these land cover maps with NLCD (National Land Cover Database) 2011 products and NAFD (North American Forest Dynamics) static forest cover maps showed that an overall classification accuracy of over 91 percent was achieved, which is a 6 percent improvement in unmixing based classification relative to per-pixel-based classification. As such, abundance maps continue to offer an useful alternative to high-spatial resolution data derived classification maps for forest inventory analysis, multi-class mapping for eco-climatic models and applications, fast multi-temporal trend analysis and for societal and policy-relevant applications needed at the watershed scale.

Subpixel↗
NASA NTRS1993

Space transportation vehicle design evaluation using saturated designs

An important objective in the preliminary design and evaluation of space transportation vehicles is to find the best values of design variables that optimize the performance characteristic (e.g. dry weight). For a given configuration, the vehicle performance can be determined by the use of complex sizing and performance evaluation computer programs. These complex computer programs utilize iterative algorithms and they are generally too expensive and/or difficult to use directly in multidisciplinary design optimization. An alternative is to use response surface methodology (RSM) and obtain quadratic polynomial approximations to the functional relationships between performance characteristics and design variables. In RSM, these approximation models are then used to determine optimum design parameter values and for rapid sensitivity studies. Constructing a second-order model requires that 'n' design parameters be studied at least at 3 levels (values) so that the coefficients in the model can be estimated. There, 3(n) factorial experiments (point designs or observations) may be necessary. For small values of 'n' such as two or three, this design works well. However, when a large number of design parameters are under study, the number of design points required for a full-factorial design may become excessive. Fortunately, these quadratic polynomial approximations can be obtained by selecting an efficient design matrix using central composite designs (CCD) from design of experiments theory. Each unique point design from the CCD matrix is then conducted using computerized analysis tools (e.g. POST, CONSIZ, etc.). In the next step, least squares regression analysis is used to calculate the quadratic polynomial coefficients from the data. However, in some multidisciplinary applications involving a large number of design variables and several disciplines, the computerized performance synthesis programs may get too time consuming and expensive to run even with the use of efficient central composite designs. In such cases, it may be preferable to keep the number of design points to an absolute minimum and trade some model accuracy with cost. For this purpose, another class of experimental designs, called saturated D-optimal designs may be utilized for generating a matrix of vehicle designs. A design is called saturated when the number of design points is exactly equal to the number of terms in the model to be fitted. As a result, saturated designs require the absolute minimum number of design points ((n+1)(n+2)/s) to estimate the quadratic polynomial model coefficients. Saturated designs can be generated using the D-optimality criterion. A good saturated design should give rise to least squares estimates with minimum generalized variance.

Unal, Resit↗