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At least 91 records · Page 5

Updates on the DebriSat Hypervelocity Experiment and Characterization of Fragments in Support of Environmental Models

To develop, maintain, and update orbital debris environmental and break-up models, such as the NASA Orbital Debris Engineering Model (ORDEM) and the NASA Standard Satellite Breakup Model (SSBM), the NASA Orbital Debris Program Office (ODPO) relies on the most recent validated datasets from various measurement sources, both laboratory and environmental. One key project that will provide insight for break-up events using modern-day spacecraft materials and construction techniques is the DebriSat laboratory hypervelocity-impact test. Based on the mass of the target, the projectile, and the impact velocity, the expected number of fragments greater than 2 mm, generated using the NASA SSBM, was estimated to be close to 85,000. To date, the DebriSat fragment database continues to grow, with over 200,000 fragments collected that will help inform updates to the SSBM. Additionally, the growing fragment ensemble will support key parameters for the next release of NASA’s environmental models, employing fragment shapes, densities, and size distributions. To further support these environmental models, specifically the size parameter used in ground-based optical measurements, optical characterization on a subset of DebriSat fragments is being conducted in ODPO’s Optical Measurement Center (OMC). Broadband bidirectional reflectance distribution (BRDF) measurements will provide insight into the optical-based NASA Size Estimation Model (OSEM). The OSEM equates an object’s brightness to size (e.g., diameter of a disk or sphere) given several assumed parameters, including a defined phase function, albedo, and range. To address the first defined parameter, the ODPO has been using ray-tracing software to simulate light conditions in the OMC and to generate phase functions (i.e., specular, Lambertian, and experimentally-derived) of known shapes, materials, and sizes. This simulated data, in addition to the experimentally derived measurements collected in the OMC, will aid in determining whether a new phase function would be suitable for an updated OSEM. The OSEM also assumes a single-value albedo, thus preimpact spectral measurements on a subset of DebriSat materials were acquired for baseline material characterization and to provide insight into spacecraft material taxonomies. This DebriSat spectral data, along with spectral measurements from other known spacecraft material samples, will allow for better analyses of albedo variations and the effect on size calculations of defined laboratory targets, thus further supporting OSEM updates. This paper provides an overview of DebriSat, the status of the project, updates on the parameter distributions, an overview of the NASA SSBM, and ongoing fragment characterization efforts within the OMC.

Heather Cowardin↗

Modeling of Multicomponent Trace Contaminant Adsorption on Activated Carbon

Modeling of multicomponent trace contaminant adsorption on activated carbon for spacecraft life support applications presents challenges due to the very low contaminant concentrations in cabin air coupled with very strong adsorption of heavier components such as cyclic polydimethylsiloxanes. Both gas-phase and solid-phase micropore mass transfer resistances can be important, and competitive interactions between adsorbed components can result in behavior such as roll-up where lighter components displaced by heavier components have a higher concentration at the outlet than at the inlet. This paper describes the further development of a dynamic trace contaminant adsorption model based on Ideal Adsorbed Solution Theory (IAST). Alternative 1-dimensional and 2-dimensional versions of the model are described with different mass transfer resistance and driving force assumptions. Optimization of solution methods for speed and stability is also described. Mass balance equations are scaled to provide sensitivity over the wide range of component concentrations in cabin air. Model parameters are estimated from isotherm and breakthrough test data on single adsorbate and binary adsorbate systems. Multicomponent results are compared against predictions of a heritage model used extensively by NASA for trace contaminant control system (TCCS) carbon bed sizing. Finally, efforts to implement the model in a second software platform for wider access are described.

Kevin E Lange↗

In-Situ Resource Utilization Modeling of a Lunar Water Processing System

A key element of achieving a sustained surface presence, such as defined in NASA’s Artemis plan, is In-Situ Resource Utilization (ISRU). ISRU is the practice of using local resources to provide mission consumables that reduce system launch mass requirements, and regenerate resources (chiefly, water and oxygen) for propulsion and life support supporting both Lunar and Martian missions. ISRU systems require multiple complex processes, such as excavation, chemical reactors, and electrolysis subsystems that must operate in harmony to optimize the overall system process from beginning to end. The Mission Analysis and Integration Tool (MAIT) was previously developed with MATLAB in FY22 to connect individual subsystem models into a customized, flexible framework for the purpose of technology downselect, optimization, and end-to-end process planning. Beginning in FY24, MAIT was updated and became the capital program in the Systems Engineering and Integration (SE&I) ISRU Modeling and Analysis (SIMA) project. Prior work was leveraged and evolved using MATLAB/Simulink due to its ability to communicate with a vast number of other programming languages and makes up the backbone of data flow between inputs and outputs to the subsystem models. MAIT initially evaluated a suite of ISRU-related technologies, including the water processing Lunar Auger Dryer for ISRU (LADI) system with integrated upstream excavation and downstream electrolysis subsystems. With individual models consolidated, the MAIT tool generated over 5,000 cases during its first round of parametric sweeps on the water processing architecture at multiple production targets; the system analysis produced valuable insight into the optimal LADI geometry that minimized energy demands, estimated effects to cold trap size and radiator requirements, and calculated the power dynamics of the electrolysis unit and liquid oxygen storage volume. Additional efforts are being made to demonstrate the ability to scale ISRU technologies supporting the Space Technology Mission Directorate’s (STMD) commercialization strategy and increase the MAIT software capability. Work is ongoing to handle a wide array of ISRU system models beyond the Lunar environment, e.g. production of propellant for a Martian lander.

Avery Carlson↗

In-Situ Resource Utilization Modeling of a Lunar Water Processing System

A key element of achieving a sustained surface presence, such as defined in NASA’s Artemis plan, is In-Situ Resource Utilization (ISRU). ISRU is the practice of using local resources to provide mission consumables that reduce system launch mass requirements, and regenerate resources (chiefly, water and oxygen) for propulsion and life support supporting both Lunar and Martian missions. ISRU systems require multiple complex processes, such as excavation, chemical reactors, and electrolysis subsystems that must operate in harmony to optimize the overall system process from beginning to end. The Mission Analysis and Integration Tool (MAIT) was previously developed with MATLAB in FY22 to connect individual subsystem models into a customized, flexible framework for the purpose of technology downselect, optimization, and end-to-end process planning. Beginning in FY24, MAIT was updated and became the capital program in the Systems Engineering and Integration (SE&I) ISRU Modeling and Analysis (SIMA) project. Prior work was leveraged and evolved using MATLAB/Simulink due to its ability to communicate with a vast number of other programming languages and makes up the backbone of data flow between inputs and outputs to the subsystem models. MAIT initially evaluated a suite of ISRU-related technologies, including the water processing Lunar Auger Dryer for ISRU (LADI) system with integrated upstream excavation and downstream electrolysis subsystems. With individual models consolidated, the MAIT tool generated over 5,000 cases during its first round of parametric sweeps on the water processing architecture at multiple production targets; the system analysis produced valuable insight into the optimal LADI geometry that minimized energy demands, estimated effects to cold trap size and radiator requirements, and calculated the power dynamics of the electrolysis unit and liquid oxygen storage volume. Additional efforts are being made to demonstrate the ability to scale ISRU technologies supporting the Space Technology Mission Directorate’s (STMD) commercialization strategy and increase the MAIT software capability. Work is ongoing to handle a wide array of ISRU system models beyond the Lunar environment, e.g. production of propellant for a Martian lander.

Avery Carlson↗

Robust Vision-based Multi-spacecraft Guidance Navigation and Control using CNN-based Pose Estimation

In this paper, we present an end-to-end simulation framework for tracking an uncooperative Target spacecraft in Low Earth Orbit using a CubeSat-class Ego spacecraft outfitted with a camera. Currently, capturing high-fidelity realistic images in space for this scenario is difficult and exorbitantly expensive. Therefore, we developed a framework to simulate the spacecraft orbits in Basilisk software and generate high-fidelity realistic images of spacecraft in Unreal Engine, including the effects from Sun, Earth, Moon and stars. The Ego spacecraft uses cameras to capture images of the uncooperative Target and estimates its position and attitude using a CNN based 6DOF pose estimation pipeline, eliminating need for large SWAP-C(Size, Weight, Power and Cost) sensors like LIDAR or reliance on inter-spacecraft communication, This CNN, which is motivated by ESA’s Pose Estimation challenge of 2019, is trained using simulated data from our end-to-end simulation framework. We compare the performance of two distinct CNNbased algorithms for pose estimation along a nominal trajectory. In presence of non-Gaussian modeling uncertainties, the statedependent estimation error is characterized with a quadratic upper-bound. The quadratically-bounded error can be used by a robust controller to maneuver

Rahmani, Amir↗

Validation methods for flight crucial systems

Research to develop techniques that can aid in determining the reliability and performance of digital electronic fault-tolerant systems, that have probability of catastrophic system failure on the order of 10 to the -9th at 10 hours, is reviewed. The computer-aided reliability estimation program (CARE III) provides general-purpose reliability analysis and a design tool for fault-tolerant systems; large reduction of state size; and a fault-handling model based on probabilistic description of detection, isolation, and recovery mechanisms. The application of design proof techniques as part of the design and development of the software implemented fault-tolerance computer is mentioned. Emulation techniques and experimental procedures are verified using specimens of fault-tolerant computers and the capabilities of the validation research laboratory, AIRLAB.

Holt, H. M.↗

Control Software for Piezo Stepping Actuators

A control system has been developed for the Space Interferometer Mission (SIM) piezo stepping actuator. Piezo stepping actuators are novel because they offer extreme dynamic range (centimeter stroke with nanometer resolution) with power, thermal, mass, and volume advantages over existing motorized actuation technology. These advantages come with the added benefit of greatly reduced complexity in the support electronics. The piezo stepping actuator consists of three fully redundant sets of piezoelectric transducers (PZTs), two sets of brake PZTs, and one set of extension PZTs. These PZTs are used to grasp and move a runner attached to the optic to be moved. By proper cycling of the two brake and extension PZTs, both forward and backward moves of the runner can be achieved. Each brake can be configured for either a power-on or power-off state. For SIM, the brakes and gate of the mechanism are configured in such a manner that, at the end of the step, the actuator is in a parked or power-off state. The control software uses asynchronous sampling of an optical encoder to monitor the position of the runner. These samples are timed to coincide with the end of the previous move, which may consist of a variable number of steps. This sampling technique linearizes the device by avoiding input saturation of the actuator and makes latencies of the plant vanish. The software also estimates, in real time, the scale factor of the device and a disturbance caused by cycling of the brakes. These estimates are used to actively cancel the brake disturbance. The control system also includes feedback and feedforward elements that regulate the position of the runner to a given reference position. Convergence time for smalland medium-sized reference positions (less than 200 microns) to within 10 nanometers can be achieved in under 10 seconds. Convergence times for large moves (greater than 1 millimeter) are limited by the step rate.

Shields, Joel F.↗

Cloud Thickness from Diffusion of Lidar Pulses in Clouds

Measurements of the distribution of reflected light from a laser beam incident on an aqueous suspension of particles or "cloud" with known thickness and particle size distribution are reported. The distribution is referred to as the "cloud radiative Green's function", G. In the diffusion domain, G is sensitive to cloud thickness, allowing that important quantity to be retrieved. The goal of the laboratory simulation is to provide preliminary estimates of sensitivity of G to cloud thickness,for use in the optimal design of an offbeam Lidar instrument for remote sensing of cloud thickness (THOR, Thickness from Offbeam Returns). These clouds of polystyrene microspheres suspended in water are analogous to real clouds of water droplets suspended in air. The microsphere size distribution is roughly lognormal, from 0.5 microns to 25 microns, similar to real clouds. Density of suspended spheres is adjusted so mean-free-path of visible photons is about 10 cm, approximately 1000 times smaller than in real clouds. The light source is a ND:YAG laser at 530 nm. Detectors are flux and photon-counting Photomultiplier Tube (PMTS), with a glass probe for precise positioning. A Labview 5 VI controls positioning, and data acquisition, via an NI Motion Control board connected to a stepper motor driving an Edmund linear slider, and a 16-channel 16-bit NI-DAQ board. The stepper motor is accurate to 10 microns, and step size is selectable from the VI software. Far from the incident beam, the rate of exponential increase as the direction of the incident beam is approached scales as expected from diffusion theory, linearly with the cloud thickness, and inversely as the square root of the reduced optical thickness, and is independent of particle size. Near the beam the signal begins to increase faster than exponential, due to single and low-order scattering near the backward direction, and here the distribution depends on particle size. Results are being used to verify 3D Monte Carlo radiative transfer simulations, used to estimate signal-to-noise ratios for remotely sensed off beam returns, for both homogeneous and inhomogeneous clouds. Signal-to-noise estimates show that unfiltered observations are straight forward at night, while narrow band pass filters are being studied for day.

Cahalan, Robert F.↗

Analyzing Responses of Chemical Sensor Arrays

NASA is developing a third-generation electronic nose (ENose) capable of continuous monitoring of the International Space Station s cabin atmosphere for specific, harmful airborne contaminants. Previous generations of the ENose have been described in prior NASA Tech Briefs issues. Sensor selection is critical in both (prefabrication) sensor material selection and (post-fabrication) data analysis of the ENose, which detects several analytes that are difficult to detect, or that are at very low concentration ranges. Existing sensor selection approaches usually include limited statistical measures, where selectivity is more important but reliability and sensitivity are not of concern. When reliability and sensitivity can be major limiting factors in detecting target compounds reliably, the existing approach is not able to provide meaningful selection that will actually improve data analysis results. The approach and software reported here consider more statistical measures (factors) than existing approaches for a similar purpose. The result is a more balanced and robust sensor selection from a less than ideal sensor array. The software offers quick, flexible, optimal sensor selection and weighting for a variety of purposes without a time-consuming, iterative search by performing sensor calibrations to a known linear or nonlinear model, evaluating the individual sensor s statistics, scoring the individual sensor s overall performance, finding the best sensor array size to maximize class separation, finding optimal weights for the remaining sensor array, estimating limits of detection for the target compounds, evaluating fingerprint distance between group pairs, and finding the best event-detecting sensors.

Zhou, Hanying↗

Avoiding Stair-Step Artifacts in Image Registration for GOES-R Navigation and Registration Assessment

In developing software for independent verification and validation (IVV) of the Image Navigation and Registration (INR) capability for the Geostationary Operational Environmental Satellite R Series (GOES-R) Advanced Baseline Imager (ABI), we have encountered an image registration artifact which limits the accuracy of image offset estimation at the subpixel scale using image correlation. Where the two images to be registered have the same pixel size, subpixel image registration preferentially selects registration values where the image pixel boundaries are close to lined up. Because of the shape of a curve plotting input displacement to estimated offset, we call this a stair-step artifact. When one image is at a higher resolution than the other, the stair-step artifact is minimized by correlating at the higher resolution. For validating ABI image navigation, GOES-R images are correlated with Landsat-based ground truth maps. To create the ground truth map, the Landsat image is first transformed to the perspective seen from the GOES-R satellite, and then is scaled to an appropriate pixel size. Minimizing processing time motivates choosing the map pixels to be the same size as the GOES-R pixels. At this pixel size image processing of the shift estimate is efficient, but the stair-step artifact is present. If the map pixel is very small, stair-step is not a problem, but image correlation is computation-intensive. This paper describes simulation-based selection of the scale for truth maps for registering GOES-R ABI images.

stair-step artifact↗

Avoiding Stair-Step Artifacts in Image Registration for GOES-R Navigation and Registration Assessment

In developing software for independent verification and validation (IVV) of the Image Navigation and Registration (INR) capability for the Geostationary Operational Environmental Satellite R Series (GOES-R) Advanced Baseline Imager (ABI), we have encountered an image registration artifact which limits the accuracy of image offset estimation at the subpixel scale using image correlation. Where the two images to be registered have the same pixel size, subpixel image registration preferentially selects registration values where the image pixel boundaries are close to lined up. Because of the shape of a curve plotting input displacement to estimated offset, we call this a stair-step artifact. When one image is at a higher resolution than the other, the stair-step artifact is minimized by correlating at the higher resolution. For validating ABI image navigation, GOES-R images are correlated with Landsat-based ground truth maps. To create the ground truth map, the Landsat image is first transformed to the perspective seen from the GOES-R satellite, and then is scaled to an appropriate pixel size. Minimizing processing time motivates choosing the map pixels to be the same size as the GOES-R pixels. At this pixel size image processing of the shift estimate is efficient, but the stair-step artifact is present. If the map pixel is very small, stair-step is not a problem, but image correlation is computation-intensive. This paper describes simulation-based selection of the scale for truth maps for registering GOES-R ABI images.

image registration↗

Study of software application of airborne laser doppler system for severe storms measurement

Significant considerations are described for performing a Severe Storms Measurement program in real time. Particular emphasis is placed on the sizing and timing requirements for a minicomputer-based system. Analyses of several factors which could impact the effectiveness of the system are presented. The analyses encompass the problems of data acquisition, data storage, data registration, correlation, and flow field computation, and error induced by aircraft motion, moment estimation, and pulse integration.

Alley, P. L.↗

Image Guider Subsystem Analysis for the GHAPS Project

The Gondola for High-Altitude Planetary Science (GHAPS) project is a balloon-borne astronomical observatory designed operate in the UV, Visible, and near-mid IR spectral region. The GHAPS Optical Telescope Assembly (OTA) is designed around a one meter aperture narrow field-of-view telescope with near diffraction limited performance. GHAPS will utilize Wallops Arc-Second Pointing System (WASP) for pointing the OTA with an accuracy of 1 arc second or better. WASP relies heavily on a self-contained star tracker assembly to determine the OTA line of sight. Preliminary structural analysis indicated that potential misalignments could be present between the OTA line of sight and the star tracker FOV center during the expected flight conditions that could compromise GHAPS pointing accuracy. An analysis is presented that estimates the worst case long-term (or bias) pointing misalignments for the GHAPS opto-mechanical configuration, examines the need for additional hardware to correct pointing errors, and determines the best hardware and software implementation to do so. The analysis includes the development of a photometric model to predict the limiting magnitude of a GHAPS Facility Guide System (FGS) for a variety of image sensor technologies from visible to mid IR. FGS system performance is estimated using star magnitude and spectral data from actual star catalogs for more realistic predictions. Based on expected performance and other factors, such as size, weight, power, and cost, a final recommendation is made for the GHAPS configuration.

Telescope Guider↗

Multidisciplinary Optimization Branch Experience Using iSIGHT Software

The Multidisciplinary Optimization (MDO) Branch at NASA Langley Research Center is investigating frameworks for supporting multidisciplinary analysis and optimization research. An optimization framework call improve the design process while reducing time and costs. A framework provides software and system services to integrate computational tasks and allows the researcher to concentrate more on the application and less on the programming details. A framework also provides a common working environment and a full range of optimization tools, and so increases the productivity of multidisciplinary research teams. Finally, a framework enables staff members to develop applications for use by disciplinary experts in other organizations. Since the release of version 4.0, the MDO Branch has gained experience with the iSIGHT framework developed by Engineous Software, Inc. This paper describes experiences with four aerospace applications: (1) reusable launch vehicle sizing, (2) aerospike nozzle design, (3) low-noise rotorcraft trajectories, and (4) acoustic liner design. All applications have been successfully tested using the iSIGHT framework, except for the aerospike nozzle problem, which is in progress. Brief overviews of each problem are provided. The problem descriptions include the number and type of disciplinary codes, as well as all estimate of the multidisciplinary analysis execution time. In addition, the optimization methods, objective functions, design variables, and design constraints are described for each problem. Discussions on the experience gained and lessons learned are provided for each problem. These discussions include the advantages and disadvantages of using the iSIGHT framework for each case as well as the ease of use of various advanced features. Potential areas of improvement are identified.

Padula, S. L.↗

Structural Weight Estimation for Launch Vehicles

This paper describes some of the work in progress to develop automated structural weight estimation procedures within the Vehicle Analysis Branch (VAB) of the NASA Langley Research Center. One task of the VAB is to perform system studies at the conceptual and early preliminary design stages on launch vehicles and in-space transportation systems. Some examples of these studies for Earth to Orbit (ETO) systems are the Future Space Transportation System [1], Orbit On Demand Vehicle [2], Venture Star [3], and the Personnel Rescue Vehicle[4]. Structural weight calculation for launch vehicle studies can exist on several levels of fidelity. Typically historically based weight equations are used in a vehicle sizing program. Many of the studies in the vehicle analysis branch have been enhanced in terms of structural weight fraction prediction by utilizing some level of off-line structural analysis to incorporate material property, load intensity, and configuration effects which may not be captured by the historical weight equations. Modification of Mass Estimating Relationships (MER's) to assess design and technology impacts on vehicle performance are necessary to prioritize design and technology development decisions. Modern CAD/CAE software, ever increasing computational power and platform independent computer programming languages such as JAVA provide new means to create greater depth of analysis tools which can be included into the conceptual design phase of launch vehicle development. Commercial framework computing environments provide easy to program techniques which coordinate and implement the flow of data in a distributed heterogeneous computing environment. It is the intent of this paper to present a process in development at NASA LaRC for enhanced structural weight estimation using this state of the art computational power.

Cerro, Jeff↗

Flight and Direct to Earth/Space Relay Communication System Architecture for GSFC CubeSat Missions

The CubeSat platform is finding increasing use in space science applications due to its low cost and comparative ease of launch. It is becoming a key scientific discovery tool in low Earth orbit (LEO) and beyond, including geosynchronous equatorial orbit (GEO), the Lagrange Points, Lunar missions, and more. The increasing complexity of these missions and their scientific goals must be supported by equal advancements in communications technology. Higher data rates and greater reliability are required every year. However, the reduced Size, Weight, and Power (SWaP) constraints of CubeSat platforms introduce unique challenges in the area of satellite communications. There is currently a lack of communication equipment tailored specifically to the CubeSat platform. This lack of standardized, tested equipment extends development time and reduces mission confidence. Furthermore, missions utilizing the CubeSat platform are often subject to more difficult design constraints. Antenna placement, size, and pointing are often subordinate to the requirements of the payload instruments and mission goals. Traditional link margin estimation techniques are insufficient in these cases, as they emphasize worst case scenarios. In reality the actual link parameters may vary widely even during a single pass. This presents new challenges in predicting communications performance and scheduling ground station contacts, but also new opportunities for improving efficiency. This paper presents the integration, testing, and validation process for a new software defined radio (SDR) designed for the CubeSat platform in conjunction with Vulcan Wireless, Inc. The SDR is planned for use on 5 upcoming CubeSat missions at NASAs Goddard Space Flight Center (GSFC) including a Geosynchronous Transfer Orbit (GTO) mission and it may also serve as a standard and well-tested option for future missions by enabling a standardized, rapid and low cost CubeSat communication system network integration process. Detailed simulations have been developed to estimate the communication performance of these missions, taking the unique antenna placements and attitude behavior of each satellite into account. These simulations allow a much more accurate analysis of the expected link margin, which varies considerably during each pass for the NASA Space Relay (SR) and Direct to Earth (DTE) network. The modelling procedures are outlined, and the results are used to predict communications performance of the missions.

Space Networks↗

Uncertainty Quantification of a Rotorcraft Conceptual Sizing Toolsuite

A computational framework to support the quantification of system uncertainties and sensitivities for rotorcraft applications is presented using the NASA Design and Analysis of Rotorcraft (NDARC) conceptual sizing tool. A 90 passenger conceptual tiltrotor configuration was used for case demonstration in the modeling of uncertainties in NDARCs emission module. A non-intrusive forward propagation uncertainty quantification approach was applied to ensemble simulations using a Monte Carlo methodology with stratified Latin hypercube sampling. An off-the-shelf software, DAKOTA, which supports trade studies and design space exploration, including optimization, surrogate modeling and uncertainty analysis was used to address the research goals. A toolsuite was further developed incorporating DAKOTA with automated design processes and methods using function wrappers to execute program routines including support for data post-processing. Uncertainties in rotorcraft emissions modeling using the Average Temperature Response metric for a set mission profile were studied. It was shown that for the current study, using the base-line best estimate modeling parameters for the Average Temperature Response metric, NDARC under-estimates the effects of emissions when compared with results from Monte Carlo simulations. A global sensitivity analysis was further undertaken to quantify the contribution of the various emission species on output sensitivity, hence uncertainty. The work demonstrates that the developed toolsuite is robust and will support the quantification of system uncertainties and sensitivities in future rotorcraft design efforts.

Rotorcraft↗

NASTRAN processing at Johnson Space Center

The processing behavior of NASTRAN, a general purpose system of computer software used for solving engineering analysis problems by means of the finite element displacement method, is discussed. The host system for the NASTRAN is the Univac 1110, and the primary function is performing static and dynamic structural analyses of the Space Shuttle. The processing implications of the NASTRAN structural analysis are addressed and the current NASTRAN situation on the Univac 1110 is examined. The current NASTRAN workload is described and the effect of workload growth due to increased structure size is estimated. A cost and performance analysis is presented.

Richards, F. M., Jr.↗