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NASA NTRS2016

Application of Design of Experiments and Surrogate Modeling within the NASA Advanced Concepts Office, Earth-to-Orbit Design Process

Decisions made during early conceptual design have a large impact upon the expected life-cycle cost (LCC) of a new program. It is widely accepted that up to 80% of such cost is committed during these early design phases. Therefore, to help minimize LCC, decisions made during conceptual design must be based upon as much information as possible. To aid in the decision making for new launch vehicle programs, the Advanced Concepts Office (ACO) at NASA Marshall Space Flight Center (MSFC) provides rapid turnaround pre-phase A and phase A concept definition studies. The ACO team utilizes a proven set of tools to provide customers with a full vehicle mass breakdown to tertiary subsystems, preliminary structural sizing based upon worst-case flight loads, and trajectory optimization to quantify integrated vehicle performance for a given mission. Although the team provides rapid turnaround for single vehicle concepts, the scope of the trade space can be limited due to analyst availability and the manpower requirements for manual execution of the analysis tools. In order to enable exploration of a broader design space, the ACO team has implemented an advanced design methods (ADM) based approach. This approach applies the concepts of design of experiments (DOE) and surrogate modeling to more exhaustively explore the trade space and provide the customer with additional design information to inform decision making. This paper will first discuss the automation of the ACO tool set, which represents a majority of the development effort. In order to fit a surrogate model within tolerable error bounds a number of DOE cases are needed. This number will scale with the number of variable parameters desired and the complexity of the system's response to those variables. For all but the smallest design spaces, the number of cases required cannot be produced within an acceptable timeframe using a manual process. Therefore, automation of the tools was a key enabler for the successful application of an ADM approach to an ACO design study. Following the overview of the tool set automation, an example problem will be given to illustrate the implementation of the ADM approach. The example problem will first cover the inclusion of ground rules and assumptions (GR&A) for a study. The GR&A are very important to the study as they determine the constraints within which a trade study can be conducted. These trades must ultimately reconcile with the customer's desired output and any anticipated "what if" questions. The example problem will then illustrate the setup and execution of a DOE through the automated ACO tools. This process is accomplished more efficiently in this work by splitting the tools into two separate environments. The first environment encompasses the structural optimization and mass estimation tools, while the second is focused on trajectory optimization. Surrogate models are fit to the outputs of each environment and are "integrated" via connection of the surrogate equations. Throughout this process, checks are implemented to compare the output of the surrogates to the output of manually run cases to ensure that the error of the final surrogates is at an acceptable level. The conclusion of the example problem demonstrates the utility of the ADM based approach. Using surrogate models gives the ACO team the ability to visualize vehicle sensitivities to various design parameters and identify regions of interest within the design space. The ADM approach can thus be used to inform concept down selection and isolate promising vehicle configurations to be explored in more detail through the manual design process. In addition it provides the customer with an almost instantaneous turnaround on any ''what if" questions that may arise within the bounds of the surrogate model. This approach ultimately expands the ability of the ACO team to provide its customer with broad and rapid turnaround trade studies for launch vehicle conceptual design. The ability to identify a selection of designs which can meet the customer requirements will help ensure lower LCC of launch vehicle designs originating from ACO.

Zwack, Mathew R.↗

NASA NTRS1978

An algorithm for optimal structural design with frequency constraints

The paper presents a finite element method for minimum weight design of structures with lower-bound constraints on the natural frequencies, and upper and lower bounds on the design variables. The design algorithm is essentially an iterative solution of the Kuhn-Tucker optimality criterion. The three most important features of the algorithm are: (1) a small number of design iterations are needed to reach optimal or near-optimal design, (2) structural elements with a wide variety of size-stiffness may be used, the only significant restriction being the exclusion of curved beam and shell elements, and (3) the algorithm will work for multiple as well as single frequency constraints. The design procedure is illustrated with three simple problems.

Kiusalaas, J.↗

NASA NTRS1981

Relative performance of algorithms for autonomous satellite orbit determination

Limited word size in contemporary microprocessors causes numerical problems in autonomous satellite navigation applications. Numerical error introduced in navigation computations performed on small wordlength machines can cause divergence of sequential estimation algorithms. To insure filter reliability, square root algorithms have been adopted in many applications. The optimal navigation algorithm requires a careful match of the estimation algorithm, dynamic model, and numerical integrator. In this investigation, the relationship of several square root filters and numerical integration methods is evaluated to determine their relative performance for satellite navigation applications. The numerical simulations are conducted using the Phase I GPS constellation to determine the orbit of a LANDSAT-D type satellite. The primary comparison is based on computation time and relative estimation accuracy.

Tapley, B. D.↗

NASA NTRS2010

Non-Targeted Effects and the Dose Response for Heavy Ion Tumorigenesis

BACKGROUND: There is no human epidemiology data available to estimate the heavy ion cancer risks experienced by astronauts in space. Studies of tumor induction in mice are a necessary step to estimate risks to astronauts. Previous experimental data can be better utilized to model dose response for heavy ion tumorigenesis and plan future low dose studies. DOSE RESPONSE MODELS: The Harderian Gland data of Alpen et al.[1-3] was re-analyzed [4] using non-linear least square regression. The data set measured the induction of Harderian gland tumors in mice by high-energy protons, helium, neon, iron, niobium and lanthanum with LET s ranging from 0.4 to 950 keV/micron. We were able to strengthen the individual ion models by combining data for all ions into a model that relates both radiation dose and LET for the ion to tumor prevalence. We compared models based on Targeted Effects (TE) to one motivated by Non-targeted Effects (NTE) that included a bystander term that increased tumor induction at low doses non-linearly. When comparing fitted models to the experimental data, we considered the adjusted R2, the Akaike Information Criteria (AIC), and the Bayesian Information Criteria (BIC) to test for Goodness of fit.In the adjusted R2test, the model with the highest R2values provides a better fit to the available data. In the AIC and BIC tests, the model with the smaller values of the summary value provides the better fit. The non-linear NTE models fit the combined data better than the TE models that are linear at low doses. We evaluated the differences in the relative biological effectiveness (RBE) and found the NTE model provides a higher RBE at low dose compared to the TE model. POWER ANALYSIS: The final NTE model estimates were used to simulate example data to consider the design of new experiments to detect NTE at low dose for validation. Power and sample sizes were calculated for a variety of radiation qualities including some not considered in the Harderian Gland data set and with different background tumor incidences. We considered different experimental designs with varying number of doses and varying low doses dependant on the LET of the radiation. The optimal design to detect a NTE for an individual ion had 4 doses equally spaced below a maximal dose where bending due to cell sterilization was < 2%. For example at 100 keV/micron we would irradiate at 0.03 Gy, 0.065 Gy, 0.13 Gy, and 0.26 Gy and require 850 mice including a control dose for a sensitivity to detect NTE with 80% power. Sample sizes could be improved by combining ions similar to the methods used with the Harderian Gland data.

Chappelli, Lori J.↗

NASA NTRS2010

Simultaneous Retrieval of Aerosol and Surface Optical Properties from Combined Airborne- and Ground-Based Direct and Diffuse Radiometric Measurements

This paper presents a new method for simultaneously retrieving aerosol and surface reflectance properties from combined airborne and ground-based direct and diffuse radiometric measurements. The method is based on the standard Aerosol Robotic Network (AERONET) method for retrieving aerosol size distribution, complex index of refraction, and single scattering albedo, but modified to retrieve aerosol properties in two layers, below and above the aircraft, and parameters on surface optical properties from combined datasets (Cloud Absorption Radiometer (CAR) and AERONET data). A key advantage of this method is the inversion of all available spectral and angular data at the same time, while accounting for the influence of noise in the inversion procedure using statistical optimization. The wide spectral (0.34-2.30 m) and angular range (180 ) of the CAR instrument, combined with observations from an AERONET sunphotometer, provide sufficient measurement constraints for characterizing aerosol and surface properties with minimal assumptions. The robustness of the method was tested on observations made during four different field campaigns: (a) the Southern African Regional Science Initiative 2000 over Mongu, Zambia, (b) the Intercontinental Transport Experiment-Phase B over Mexico City, Mexico (c) Cloud and Land Surface Interaction Campaign over the Atmospheric Radiation Measurement (ARM) Central Facility, Oklahoma, USA, and (d) the Arctic Research of the Composition of the Troposphere from Aircraft and Satellites (ARCTAS) over Elson Lagoon in Barrow, Alaska, USA. The four areas are dominated by different surface characteristics and aerosol types, and therefore provide good test cases for the new inversion method.

Gatebe, C. K.↗

NASA NTRS2016

Application of Design of Experiments and Surrogate Modeling within the NASA Advanced Concepts Office, Earth-to-Orbit Design Process

Zwack, Mathew R.↗

NASA NTRS2015

Compositional and Microstructural Evolution of Olivine During Pulsed Laser Irradiation: Insights Based on a FIB/Field-Emission TEM Study

Introduction: The use of pulsed laser irradiation to simulate the short duration, high-energy conditions characteristic of micrometeorite impacts is now an established approach in experimental space weathering studies. The laser generates both melt and vapor deposits that contain nanophase metallic Fe (npFe(sup 0)) grains with size distributions and optical properties similar to those in natural impact-generated melt and vapor deposits. There remains uncertainty, however, about how well lasers simulate the mechanical work and internal (thermal) energy partitioning that occurs in actual impacts. We are currently engaged in making a direct comparison between the products of laser irradiation and experimental/natural hypervelocity impacts. An initial step reported here is to use analytical TEM is to attain a better understanding of how the microstructure and composition of laser deposits evolve over multiple cycles of pulsed laser irradiation. Experimental Methods: We irradiated pressed-powder pellets of San Carlos olivine (Fo(sub 90)) with up to 99 rastered pulses of a GAM ArF excimer laser. The irradiated surface of the sample were characterized by SEM imaging and areas were selected for FIB cross sectioning for TEM study using an FEI Quanta dual-beam electron/focused ion beam instrument. FIB sections were characterized using a JEOL2500SE analytical field-emission scanning transmission electron microscope (FE-STEM) optimized for quantitative element mapping at less than 10 nm spatial resolutions. Results: In the SEM the 99 pulse pressed pellet sample shows a complex, inhomogeneous, distribution of laser-generated material, largely concentrated in narrow gaps and larger depressions between grains. Local concentrations of npFe0 spherules 0.1 to 1 micrometers in size are visible within these deposits in SEM back-scatter images. Fig. 1 shows bright-field STEM images of a FIB cross-section of a one of these deposits that continuously covers the top and sloping side of an olivine grain. The deposit has 3 microstructurally distinct sub-layers composed of silicate glass with varying modal fractions and size distributions of npFe( sup 0) spherules, along with nanocrystalline silicate material. A relatively thin (50-300 nm) topmost surface layer has a high-concentration of npFe0 spherules 5-20 nm in size. Element mapping shows the layer to be enriched in Fe by a factor of 2.5 relative to the olivine substrate, with Mg and Si depleted by 20% and 10% respectively. This is compositionally complementary to the underlying, middle layer of the deposit that is depleted in Fe, enriched in Mg and has a much lower npFe0 concentration. A third layer of nanocrystalline olivine occurs at the substrate interface. Discussion: The FE-STEM results suggest the topmost layer is a vapor deposit, underlain by a thicker microstructurally complex melt-generated layer. The compositional relations suggest the melt layer was partially vaporized, preferentially losing more volatile elements (e.g., Fe). The vaporized material re-condensed to form the thin, npFe(sup 0)-rich surface deposit during or immediately after the scan cycle. Nanocrystalline olivine that grew within the melt layer as it formed and cooled is similar in volume and microstructure to what we have observed in the impact melt lining of a micrometeorite impact crater in olivine. This suggest the time-temperature relations attained in the laser sample may not be too different from a micrometeorite impact. Our TEM observations, however, do not show evidence for the same level of mechanical dam-age (e.g., fracturing) seen around the natural micrometeorite crater.

Christoffersen, R.↗

NASA NTRS1984

Graphite/epoxy composite stiffened panel fabrication development

This report describes the manufacturing development procedures used to fabricate a series of carbon/epoxy panels with integrally molded stiffeners. Panel size was started at 6 inches by 18 inches and one stiffener and increased to 30 inches by 60 inches and six integral stiffeners. Stiffener concepts were optimized for minimum weight (or mass) to carry stress levels from 1500 lbs/inch to 25,000 lbs/inch compression load. Designs were created and manufactured with a stiffener configuration of integrally molded hat, J, I, sine wave I, solid blade, and honeycomb blade shapes. Successful and unsuccessful detail methods of tooling, lay-up methods, and bagging methods are documented. Recommendations are made for the best state-of-the-art manufacturing technique developed for type of stiffener construction.

Palmer, R. J.↗

NASA NTRS1990

Numerical derivative techniques for trajectory optimization

The adoption of robust numerical optimization techniques in trajectory simulation programs has resulted in powerful design and analysis tools. These trajectory simulation/optimization programs are widely used, and a representative list includes the GTS system, the POST program, and newer collocation methods such as OTIS and FONPAC. All of these programs rely on optimization algorithms which require objective function and constraint gradient data during the iteration process. However, most trajectory optimization problems lack simple analytical expressions for these derivatives. In the general case a function evaluation involves integrating aerodynamic, propulsive, and gravity forces over multiple trajectory phases with complex control models. With the newer collocation methods, the integration is replaced by defect constraints and cubic approximations for the state. While analytic gradient expressions can sometimes be derived for trajectory optimization problems, the derivation is cumbersome, time consuming, and prone to mistakes. Fortunately, an alternate method exists for the gradient evaluation, namely finite difference approximations. In this paper some finite difference gradient techniques developed for use with the GTS system are presented. These techniques include methods for computing first and second partial derivatives of single and multiple sets of functions. A key feature of these methods is an error control mechanism which automatically adjusts the perturbation size to obtain accurate derivative values.

Hallman, Wayne P.↗

NASA NTRS2006

Status of Solar Sail Propulsion: Moving Toward an Interstellar Probe

NASA's In-Space Propulsion Technology Program has developed the first-generation of solar sail propulsion systems sufficient to accomplish inner solar system science and exploration missions. These first-generation solar sails, when operational, will range in size from 40 meters to well over 100 meters in diameter and have an areal density of less than 13 grams-per-square meter. A rigorous, multiyear technology development effort culminated last year in the testing of two different 20-meter solar sail systems under thermal vacuum conditions. This effort provided a number of significant insights into the optimal design and expected performance of solar sails as well as an understanding of the methods and costs of building and using them. In a separate effort, solar sail orbital analysis tools for mission design were developed and tested. Laboratory simulations of the effects of long-term space radiation exposure were also conducted on two candidate solar sail materials. Detailed radiation and charging environments were defined for mission trajectories outside the protection of the earth's magnetosphere, in the solar wind environment. These were used in other analytical tools to prove the adequacy of sail design features for accommodating the harsh space environment. Preceding, and in conjunction with these technology efforts, NASA sponsored several mission application studies for solar sails, including one that would use an evolved sail capability to support humanity's first mission into nearby interstellar space. The proposed mission is called the Interstellar Probe. The Interstellar Probe might be accomplished in several ways. A 200-meter sail, with an areal density approaching 1 gram-per-square meter, could accelerate a robotic probe to the very edge of the solar system in just under 20 years from launch. A sail using the technology just demonstrated could make the same mission, but take significantly longer. Conventional chemical propulsion systems would require even longer flight times. Spinner sails of the type being explored by the Japanese may also be a good option, but the level of maturity in that technology is not clear. While the technology to support a 200-meter, ultralightweight sail mission is not yet in hand, the recent NASA investments in solar sail technology are an essential first step toward making it a reality. This paper will describe the status of solar sail propulsion within NASA, near-term solar sail mission applications, and the plan to advance the technology to the point where the Interstellar Probe mission can be flown.

Johnson, Les↗

NASA NTRSDate not supplied

Microwave Structure Construction Capability Year One Accomplishments

The Microwave Structure Construction Capability (MSCC) element, part of the Moon to Mars Planetary Autonomous Construction Project (MMPACT) was initiated in 2020. MSCC is responsible for creating horizontal and vertical infrastructure on the moon using microwave energy. Microwave energy was selected since it is the only method to volumetrically heat the regolith. All other sintering/melting methods rely on thermal conduction through the very low conductivity surface, resulting in an inefficient process. Advances were achieved in materials characterization and understanding, microwave sintering in vacuum, and microwave design and analyses. Two dielectric property testing systems have been developed at Radiance Technologies and JPL. These will examine dielectric properties at cryogenic temperatures and over a broad frequency range. Permittivity and permeability testing at -60 ̊C in vacuum from 0.05 to 3 GHz has been generated at JPL. Additional modifications will be made to go to -190 ̊C (LN2). Radiance Technologies created a test system to measure dielectric properties at greater than 10 GHz and initiated work on developing a vacuum capable, portable test system to measure dielectric properties of Apollo regolith and simulants from 100 MHz to 18 GHz. These tests are to identify optimal heating frequencies and protocols. During microwave sintering at about 1100 ̊C, volatiles were creating difficulty in achieving a reasonably dense specimen. Due to processing in vacuum and the nature of the lunar regolith, some volatiles and porosity are expected. However, the Earth produced simulants have non-lunar materials in them that create volatiles that aren’t representative of lunar regolith. Therefore, a five month effort was conducted to establish a heat treat method to remove these non-lunar materials. Tests were conducted using TGA mass spectrometry, heating in vacuum and conducting mass spectrometry, dielectric and DTA, Raman, BET, particle size analysis, morphological analysis, carbon and sulfur chemical content determination and microscopy. The process has been scaled-up to 6 kg batch size and undergoing evaluation. A 36 kg batch size is the target for JSC-1A and other limited availability simulants. These calcining protocols will be standard for NASA and beyond. MSCC has also created scalable processes for fabricating synthetic lunar materials. Processes to fabricate Anorthite (plagioclase CaAl2Si2O8), Diopside (pyroxene CaMgSi₂O₆), and Enstatite (pyroxene Mg2Si2O6) have been generated. These materials will enable generation of microwave sintering models to bound various composition ranges anticipated on the Moon, therefore mitigating the need for a precise simulant with respect to location on the Moon. Successful microwave sintering in air using a horn applicator was demonstrated. All previous microwave vacuum sintering in the literature was at small scale and in a contained enclosure thus taking advantage of reflections. This is the first to use a lunar like microwave applicator to sinter ceramic in a bed as it would be done on the Moon. Small scale and inert sintering were conducted to assist in developing protocols with quicker turnaround times than larger scale testing. Testing has anchored thermal analysis predicting heat flow in vacuum during microwave sintering. Thermal conductivity testing was also initiated. Microwave coupling to the regolith has been modeled by multiple organizations and with different software packages. At least six horn designs and applicator configurations for both magnetron and solid state sources are being examined. Optimal simulant container designs for microwaves have also been generated. The power and electronics design for the solid state microwave system has been initiated. Concept designs for a lander based microwave sintering have been evaluated.

microwave↗

NASA NTRS1996

Optimal Trajectories for the Helicopter in One-Engine-Inoperative Terminal-Area Operations

This paper presents a summary of a series of recent analytical studies conducted to investigate One-Engine-Inoperative (OEI) optimal control strategies and the associated optimal trajectories for a twin engine helicopter in Category-A terminal-area operations. These studies also examine the associated heliport size requirements and the maximum gross weight capability of the helicopter. Using an eight states, two controls, augmented point-mass model representative of the study helicopter, Continued TakeOff (CTO), Rejected TakeOff (RTO), Balked Landing (BL), and Continued Landing (CL) are investigated for both Vertical-TakeOff-and-Landing (VTOL) and Short-TakeOff-and-Landing (STOL) terminal-area operations. The formulation of the nonlinear optimal control problems with considerations for realistic constraints, solution methods for the two-point boundary-value problem, a new real-time generation method for the optimal OEI trajectories, and the main results of this series of trajectory optimization studies are presented. In particular, a new balanced- weight concept for determining the takeoff decision point for VTOL Category-A operations is proposed, extending the balanced-field length concept used for STOL operations.

Zhao, Yiyuan↗

NASA NTRS2000

Efficient Simulation of Wing Modal Response: Application of 2nd Order Shape Sensitivities and Neural Networks

At the preliminary design stage of a wing structure, an efficient simulation, one needing little computation but yielding adequately accurate results for various response quantities, is essential in the search of optimal design in a vast design space. In the present paper, methods of using sensitivities up to 2nd order, and direct application of neural networks are explored. The example problem is how to decide the natural frequencies of a wing given the shape variables of the structure. It is shown that when sensitivities cannot be obtained analytically, the finite difference approach is usually more reliable than a semi-analytical approach provided an appropriate step size is used. The use of second order sensitivities is proved of being able to yield much better results than the case where only the first order sensitivities are used. When neural networks are trained to relate the wing natural frequencies to the shape variables, a negligible computation effort is needed to accurately determine the natural frequencies of a new design.

Kapania, Rakesh K.↗

NASA NTRS1999

Power System Optimization Codes Modified

A major modification of and addition to existing Closed Brayton Cycle (CBC) space power system optimization codes was completed. These modifications relate to the global minimum mass search driver programs containing three nested iteration loops comprising iterations on cycle temperature ratio, and three separate pressure ratio iteration loops--one loop for maximizing thermodynamic efficiency, one for minimizing radiator area, and a final loop for minimizing overall power system mass. Using the method of steepest ascent, the code sweeps through the pressure ratio space repeatedly, each time with smaller iteration step sizes, so that the three optimum pressure ratios can be obtained to any desired accuracy for each of the objective functions referred to above (i.e., maximum thermodynamic efficiency, minimum radiator area, and minimum system mass). Two separate options for the power system heat source are available: 1. A nuclear fission reactor can be used. It is provided with a radiation shield 1. (composed of a lithium hydride (LiH) neutron shield and tungsten (W) gamma shield). Suboptions can be used to select the type of reactor (i.e., fast spectrum liquid metal cooled or epithermal high-temperature gas reactor (HTGR)). 2. A solar heat source can be used. This option includes a parabolic concentrator and heat receiver for raising the temperature of the recirculating working fluid. A useful feature of the code modifications is that key cycle parameters are displayed, including the overall system specific mass in kilograms per kilowatt and the system specific power in watts per kilogram, as the results for each temperature ratio are computed. As the minimum mass temperature ratio is encountered, a message is printed out. Several levels of detailed information on cycle state points, subsystem mass results, and radiator temperature profiles are stored for this temperature ratio condition and can be displayed or printed by users.

Juhasz, Albert J.↗

NASA NTRS2007

Solar Sail Propulsion Technology at NASA

NASA's In-Space Propulsion Technology Program developed the first generation of solar sail propulsion systems sufficient to accomplish inner solar system science and exploration missions. These first generation solar sails, when operational, will range in size from 40 meters to well over 100 meters in diameter and have an area density of less than 13 grams per square meter. A rigorous, multi-year technology development effort culminated in 2005 with the testing of two different 20-m solar sail systems under thermal vacuum conditions. This effort provided a number of significant insights into the optimal design and expected performance of solar sails as well as an understanding of the methods and costs of building and using them. In addition, solar sail orbital analysis tools for mission design were developed and tested. Laboratory simulations of the effects of long-term space radiation exposure were also conducted on two candidate solar sail materials. Detailed radiation and charging environments were defined for mission trajectories outside the protection of the earth's magnetosphere, in the solar wind environment. These were used in other analytical tools to prove the adequacy of sail design features for accommodating the harsh space environment. The presentation will describe the status of solar sail propulsion within NASA, near-term solar sail mission applications, and near-term plans for further development.

Johnson, Charles Les↗

NASA NTRS2000

Flexible Wing Model for Structural Sizing and Multidisciplinary Design Optimization of a Strut-Braced Wing

This paper describes a structural and aeroelastic model for wing sizing and weight calculation of a strut-braced wing. The wing weight is calculated using a newly developed structural weight analysis module considering the special nature of strut-braced wings. A specially developed aeroelastic model enables one to consider wing flexibility and spanload redistribution during in-flight maneuvers. The structural model uses a hexagonal wing-box featuring skin panels, stringers, and spar caps, whereas the aerodynamics part employs a linearized transonic vortex lattice method. Thus, the wing weight may be calculated from the rigid or flexible wing spanload. The calculations reveal the significant influence of the strut on the bending material weight of the wing. The use of a strut enables one to design a wing with thin airfoils without weight penalty. The strut also influences wing spanload and deformations. Weight savings are not only possible by calculation and iterative resizing of the wing structure according to the actual design loads. Moreover, as an advantage over the cantilever wing, employment of the strut twist moment for further load alleviation leads to increased savings in structural weight.

Gern, Frank H.↗

NASA NTRS1992

Multidisciplinary optimization of aeroservoelastic systems using reduced-size models

Efficient analytical and computational tools for simultaneous optimal design of the structural and control components of aeroservoelastic systems are presented. The optimization objective is to achieve aircraft performance requirements and sufficient flutter and control stability margins with a minimal weight penalty and without violating the design constraints. Analytical sensitivity derivatives facilitate an efficient optimization process which allows a relatively large number of design variables. Standard finite element and unsteady aerodynamic routines are used to construct a modal data base. Minimum State aerodynamic approximations and dynamic residualization methods are used to construct a high accuracy, low order aeroservoelastic model. Sensitivity derivatives of flutter dynamic pressure, control stability margins and control effectiveness with respect to structural and control design variables are presented. The performance requirements are utilized by equality constraints which affect the sensitivity derivatives. A gradient-based optimization algorithm is used to minimize an overall cost function. A realistic numerical example of a composite wing with four controls is used to demonstrate the modeling technique, the optimization process, and their accuracy and efficiency.

Karpel, Mordechay↗

NASA NTRS2007

Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets

Scattered data interpolation is a problem of interest in numerous areas such as electronic imaging, smooth surface modeling, and computational geometry. Our motivation arises from applications in geology and mining, which often involve large scattered data sets and a demand for high accuracy. The method of choice is ordinary kriging. This is because it is a best unbiased estimator. Unfortunately, this interpolant is computationally very expensive to compute exactly. For n scattered data points, computing the value of a single interpolant involves solving a dense linear system of size roughly n x n. This is infeasible for large n. In practice, kriging is solved approximately by local approaches that are based on considering only a relatively small'number of points that lie close to the query point. There are many problems with this local approach, however. The first is that determining the proper neighborhood size is tricky, and is usually solved by ad hoc methods such as selecting a fixed number of nearest neighbors or all the points lying within a fixed radius. Such fixed neighborhood sizes may not work well for all query points, depending on local density of the point distribution. Local methods also suffer from the problem that the resulting interpolant is not continuous. Meyer showed that while kriging produces smooth continues surfaces, it has zero order continuity along its borders. Thus, at interface boundaries where the neighborhood changes, the interpolant behaves discontinuously. Therefore, it is important to consider and solve the global system for each interpolant. However, solving such large dense systems for each query point is impractical. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. The problems arise from the fact that the covariance functions that are used in kriging have global support. Our implementations combine, utilize, and enhance a number of different approaches that have been introduced in literature for solving large linear systems for interpolation of scattered data points. For very large systems, exact methods such as Gaussian elimination are impractical since they require 0(n(exp 3)) time and 0(n(exp 2)) storage. As Billings et al. suggested, we use an iterative approach. In particular, we use the SYMMLQ method, for solving the large but sparse ordinary kriging systems that result from tapering. The main technical issue that need to be overcome in our algorithmic solution is that the points' covariance matrix for kriging should be symmetric positive definite. The goal of tapering is to obtain a sparse approximate representation of the covariance matrix while maintaining its positive definiteness. Furrer et al. used tapering to obtain a sparse linear system of the form Ax = b, where A is the tapered symmetric positive definite covariance matrix. Thus, Cholesky factorization could be used to solve their linear systems. They implemented an efficient sparse Cholesky decomposition method. They also showed if these tapers are used for a limited class of covariance models, the solution of the system converges to the solution of the original system. Matrix A in the ordinary kriging system, while symmetric, is not positive definite. Thus, their approach is not applicable to the ordinary kriging system. Therefore, we use tapering only to obtain a sparse linear system. Then, we use SYMMLQ to solve the ordinary kriging system. We show that solving large kriging systems becomes practical via tapering and iterative methods, and results in lower estimation errors compared to traditional local approaches, and significant memory savings compared to the original global system. We also developed a more efficient variant of the sparse SYMMLQ method for large ordinary kriging systems. This approach adaptively finds the correct local neighborhood for each query point in the interpolation process.

Memarsadeghi, Nargess↗