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DOE OSTI2026

Antarctic Meltwater Accelerates Southern Ocean Evolution Under Projected Atmospheric Warming

Increasing basal meltwater from Antarctic ice shelves may impact the Southern Ocean properties that feed back on the rate of melting. We investigate this feedback in a high‐emissions scenario using an Earth‐system model with interactive ice‐shelf basal melting, an improvement on previous studies that did not have the capability to evolve melt rates and the ocean state self‐consistently. We find that when interactive melt increases, it primarily accelerates the evolution of a spatial pattern of continental shelf warming and cooling that is initiated by freshening and sea‐ice formation decline due to projected atmospheric warming. The competition between enhanced warming at depth from reduced ventilation and enhanced continental shelf cooling from reduced dense water export leads to net ~35% reduction in ice‐shelf meltwater input into the Southern Ocean over the 21st century. Omitting this feedback introduces a bias in the timing of projected ocean‐melt‐driven ice loss from Antarctica.

58 GEOSCIENCES↗

NASA NTRS1998

Flight Testing the Linear Aerospike SR-71 Experiment (LASRE)

The design of the next generation of space access vehicles has led to a unique flight test that blends the space and flight research worlds. The new space vehicle designs, such as the X-33 vehicle and Reusable Launch Vehicle (RLV), are powered by linear aerospike rocket engines. Conceived of in the 1960's, these aerospike engines have yet to be flown, and many questions remain regarding aerospike engine performance and efficiency in flight. To provide some of these data before flying on the X-33 vehicle and the RLV, a spacecraft rocket engine has been flight-tested atop the NASA SR-71 aircraft as the Linear Aerospike SR-71 Experiment (LASRE). A 20 percent-scale, semispan model of the X-33 vehicle, the aerospike engine, and all the required fuel and oxidizer tanks and propellant feed systems have been mounted atop the SR-71 airplane for this experiment. A major technical objective of the LASRE flight test is to obtain installed-engine performance flight data for comparison to wind-tunnel results and for the development of computational fluid dynamics-based design methodologies. The ultimate goal of firing the aerospike rocket engine in flight is still forthcoming. An extensive design and development phase of the experiment hardware has been completed, including approximately 40 ground tests. Five flights of the LASRE and firing the rocket engine using inert liquid nitrogen and helium in place of liquid oxygen and hydrogen have been successfully completed.

Stephen Corda↗

DOE OSTI2025

A process-based evaluation of biases in extratropical stratosphere–troposphere coupling in subseasonal forecast systems

Abstract. Two-way coupling between the stratosphere and troposphere is recognized as an important source of subseasonal-to-seasonal (S2S) predictability and can open windows of opportunity for improved forecasts. Model biases can, however, lead to a poor representation of such coupling processes; drifts in a model's circulation related to model biases, resolution, and parameterizations have the potential to feed back on the circulation and affect stratosphere–troposphere coupling. We introduce a set of diagnostics using readily available data that can be used to reveal these biases and then apply these diagnostics to 22 S2S forecast systems. In the Northern Hemisphere, nearly all S2S forecast systems underestimate the strength of the observed upward coupling from the troposphere to the stratosphere, downward coupling within the stratosphere, and the persistence of lower-stratospheric temperature anomalies. While downward coupling from the lower stratosphere to the near surface is well represented in the multi-model ensemble mean, there is substantial intermodel spread likely related to how well each model represents tropospheric stationary waves. In the Southern Hemisphere, the stratospheric vortex is oversensitive to upward-propagating wave flux in the forecast systems. Forecast systems generally overestimate the strength of downward coupling from the lower stratosphere to the troposphere, even as most underestimate the radiative persistence in the lower stratosphere. In both hemispheres, models with higher lids and a better representation of tropospheric quasi-stationary waves generally perform better at simulating these coupling processes.

Garfinkel, Chaim I. (ORCID:000000017258666X)↗

NASA NTRS2010

Development of a Compact Eleven Feed Cryostat for the Patriot 12-m Antenna System

The Eleven antenna has constant beam width, constant phase center location, and low spillover over a decade bandwidth. Therefore, it can feed a reflector for high aperture efficiency (also called feed efficiency). It is equally important that the feed efficiency and its subefficiencies not be degraded significantly by installing the feed in a cryostat. The MIT Haystack Observatory, with guidance from Onsala Space Observatory and Chalmers University, has been working to integrate the Eleven antenna into a compact cryostat suitable for the Patriot 12-m antenna. Since the analysis of the feed efficiencies in this presentation is purely computational, we first demonstrate the validity of the computed results by comparing them to measurements. Subsequently, we analyze the dependence of the cryostat size on the feed efficiencies, and, lastly, the Patriot 12-m subreflector is incorporated into the computational model to assess the overall broadband efficiency of the antenna system.

Beaudoin, Christopher↗

DOE OSTI2026

Automatic Volume Balancing for Online Refueling in Molten Salt Reactor Simulations with SCALE

This work introduces recently implemented capabilities in the SCALE code system’s TRITON reactor physics sequence that improve molten salt reactor (MSR) modeling: (1) a volume balancing option, which automatically balances the volumes of the fed material with a corresponding material removal to maintain fixed mixture volumes in the neutron transport model and accurate densities, and (2) continuous feed from mixtures, which enables users to define material feed streams directly from salt mixture definitions rather than individual nuclides. The new capabilities were demonstrated via SCALE/TRITON simulations of two representative MSR concepts. Simulations of the 180 MWth molten chloride fast reactor—a system with a fixed fuel salt volume in the reactor core—applied continuous refueling with fresh fuel salt. The results confirm approximately constant reactivity when the new volume balancing capability is used. Additionally, excellent agreement with a manual feed-and-drain method for volume balancing further verified the new implementation. Simulations of the 400 MWth EIRENE reactor, an integral MSR in which the salt volume may grow over time within the reactor core, demonstrated the capability to represent growing salt volume within the TRITON depletion calculation. Compared with reference solutions, the results show consistent trends in reactivity and isotopic evolution, confirming that these new user-friendly capabilities provide accurate, physically consistent approaches for modeling MSRs in SCALE.

Elzohery, Rabab [ORNL] (ORCID:0000000160043633)↗

NASA NTRS2012

Customizing NASA's Earth Science Research Products for addressing MENA Water Challenges

As projected by IPCC 2007 report, by the end of this century the Middle East North Mrica (MENA) region is projected to experience an increase of 3 C to 5 C rise in mean temperatures and a 20% decline in precipitation. This poses a serious problem for this geographic zone especially when majority of the hydrological consumption is for the agriculture sector and the remaining amount is for domestic consumption. In late 2011, the World Bank, USAID and NASA have joined hands to establishing integrated, modem, up to date NASA developed capabilities for various countries in the MENA region for addressing water resource issues and adapting to climate change impacts for improved decision making for societal benefits. The main focus of this undertaking is to address the most pressing societal issues which can be modeled and solved by utilizing NASA Earth Science remote sensing data products and hydrological models. The remote sensing data from space is one of the best ways to study such complex issues and further feed into the decision support systems. NASA's fleet of Earth Observing satellites offer a great vantage point from space to look at the globe and provide vital signs necessary to maintain healthy and sustainable ecosystem. NASA has over fifteen satellites and thirty instruments operating on these space borne platforms and generating over 2000 different science products on a daily basis. Some of these products are soil moisture, global precipitation, aerosols, cloud cover, normalized difference vegetation index, land cover/use, ocean altimetry, ocean salinity, sea surface winds, sea surface temperature, ozone and atmospheric gasses, ice and snow measurements, and many more. All of the data products, models and research results are distributed via the Internet freely through out the world. This project will utilize several NASA models such as global Land Data Assimilation System (LDAS) to generate hydrological states and fluxes in near real time. These LDAS products will then be further compared with other NASA satellite observations (MODIS, VIIRS, TRMM, etc.) and other discrete models to compare and optimize evapotranspiration, soil moisture and crop irrigation, drought assessment and water balance. The floods being a critical disaster in many of the MENA countries, NASA's global flood mapping and modeling framework (CREST) will be customized for country specific needs and delivered to the remote sensing organizations for their future use. Training is an important component under this activity and adequate level of training will be offered to build basic capacity to work with NASA provided data products, models for their future use. This paper provides a comprehensive introduction to NASA's Earth Science mission for understanding the behavior of our home Planet, projecting its health for future generations and applying research results solving societal issues.

Habib, Shahid↗

NASA NTRS2005

Small Body GN&C Research Report: A Robust Model Predictive Control Algorithm with Guaranteed Resolvability

A robustly stabilizing MPC (model predictive control) algorithm for uncertain nonlinear systems is developed that guarantees the resolvability of the associated finite-horizon optimal control problem in a receding-horizon implementation. The control consists of two components; (i) feedforward, and (ii) feedback part. Feed-forward control is obtained by online solution of a finite-horizon optimal control problem for the nominal system dynamics. The feedback control policy is designed off-line based on a bound on the uncertainty in the system model. The entire controller is shown to be robustly stabilizing with a region of attraction composed of initial states for which the finite-horizon optimal control problem is feasible. The controller design for this algorithm is demonstrated on a class of systems with uncertain nonlinear terms that have norm-bounded derivatives, and derivatives in polytopes. An illustrative numerical example is also provided.

model predictive control algorithm (MPC)↗

NASA NTRS2023

Multiphysics Modeling of Electrode Dissolution under Hydrodynamic Conditions in Low Conductivity Water

Efficient plug-flow electrochemical reactors are characterized by their ability to effectively convert species with significant ionic strength and/or with the assistance of a well-behaved electrolyte. Because of the classic electroconductivity requirement, electrochemical reactors are not particularly attractive options for applications with low-conductivity conditions, especially for low-pressure feeds. However, these reactors can still find application in dedicated processes such as ultra-clean water conditioning and biological-coupled separation systems. Consequently, a multi-physics model was developed using COMSOL Multiphysics to characterize different reactor-design conditions based on the Tertiary Current Distribution which accounts for the effects of expected variations in electrolyte composition and ionic strength on the electrochemical process, as well as solution resistance and electrode kinetics. The modelling framework presented here employs single-phase laminar fluid flow, the Nernst-Planck equation, the water-based electroneutrality condition, and concentration-dependent overpotentials. This investigation particularly explores the electrolysis process of “anodic dissolution,” in which elemental species dissociate from the solid matrix of the electrode and enter the liquid phase of the electrolyte as soluble ions. The reactor configuration of interest has a rectangular parallel-electrode design and is operated galvanostatically. A series of parametric studies are carried out to inspect the response of the system when hydrodynamic, kinetic, and geometric conditions are changed. The parametric sweep resulted in variable effluent concentrations and system voltage, which reveal underlaying optimization patterns for electrode dissolution in low conductivity water.

Multiphysics↗

NASA NTRS1979

Total aircraft flight-control system - Balanced open- and closed-loop control with dynamic trim maps

The availability of the airborne digital computer has made possible a Total Aircraft Flight Control System (TAFCOS) that uses virtually the complete nonlinear propulsive and aerodynamic data for the aircraft to construct dynamic trim maps that represent an inversion of the aircraft model. The trim maps, in series with the aircraft, provide essentially a linear feed-forward path. Basically, open-loop trajectory control is employed with only a small perturbation feedback signal required to compensate for inaccuracy in the aircraft model and for external disturbances. Simulation results for application to an automatic carrier-landing system are presented. Flight-test results for a STOL aircraft operating automatically over a major portion of its flight regime are presented. The concept promises a more rapid and straightforward design from aerodynamic principles, particularly for highly nonlinear configurations, and requires substantially less digital computer capacity than conventional automatic flight-control system designs.

Smith, G. A.↗

NASA NTRS1996

Biomechanically Induced and Controller Coupled Oscillations Experienced on the F-16XL Aircraft During Rolling Maneuvers

During rapid rolling maneuvers, the F-16 XL aircraft exhibits a 2.5 Hz lightly damped roll oscillation, perceived and described as 'roll ratcheting.' This phenomenon is common with fly-by-wire control systems, particularly when primary control is derived through a pedestal-mounted side-arm controller. Analytical studies have been conducted to model the nature of the integrated control characteristics. The analytical results complement the flight observations. A three-degree-of-freedom linearized set of aerodynamic matrices was assembled to simulate the aircraft plant. The lateral-directional control system was modeled as a linear system. A combination of two second-order transfer functions was derived to couple the lateral acceleration feed through effect of the operator's arm and controller to the roll stick force input. From the combined systems, open-loop frequency responses and a time history were derived, describing and predicting an analogous in-flight situation. This report describes the primary control, aircraft angular rate, and position time responses of the F-16 XL-2 aircraft during subsonic and high-dynamic-pressure rolling maneuvers. The analytical description of the pilot's arm and controller can be applied to other aircraft or simulations to assess roll ratcheting susceptibility.

Smith, John W.↗

NASA NTRS2004

Observing the Global Water Cycle from Space

This paper presents an approach to measuring all major components of the water cycle from space. Key elements of the global water cycle are discussed in terms of the storage of water-in the ocean, air, cloud and precipitation, in soil, ground water, snow and ice, and in lakes and rivers, and in terms of the global fluxes of water between these reservoirs. Approaches to measuring or otherwise evaluating the global water cycle are presented, and the limitations on known accuracy for many components of the water cycle are discussed, as are the characteristic spatial and temporal scales of the different water cycle components. Using these observational requirements for a global water cycle observing system, an approach to measuring the global water cycle from space is developed. The capabilities of various active and passive microwave instruments are discussed, as is the potential of supporting measurements from other sources. Examples of space observational systems, including TRMM/GPM precipitation measurement, cloud radars, soil moisture, sea surface salinity, temperature and humidity profiling, other measurement approaches and assimilation of the microwave and other data into interpretative computer models are discussed to develop the observational possibilities. The selection of orbits is then addressed, for orbit selection and antenna size/beamwidth considerations determine the sampling characteristics for satellite measurement systems. These considerations dictate a particular set of measurement possibilities, which are then matched to the observational sampling requirements based on the science. The results define a network of satellite instrumentation systems, many in low Earth orbit, a few in geostationary orbit, and all tied together through a sampling network that feeds the observations into a data-assimilative computer model.

Hildebrand, P. H.↗

DOE OSTI2025

Investigating Kinetic Mechanisms of Soot Formation in Plasma Pyrolysis of Methane via Active Learning (Final Technical Report)

Plasma pyrolysis of methane is an effective route for zero-carbon hydrogen production. Yet, soot generated from pyrolysis of hydrocarbons is detrimental to the climate and human health. There is ample experimental and theoretical evidence that suggests polycyclic aromatic hydrocarbons (PAHs) are the molecular precursors to soot particles. The reaction pathways of PAH formation are intricately dependent on a multitude of process parameters, whose kinetic mechanisms are not well-understood in plasma pyrolysis. This project aims to leverage advances in the kinetic modeling of soot formation in combustion, as well as in surrogate modeling and active learning, to systematically investigate the effects of process parameter on the kinetics of PAH formation in plasma pyrolysis of methane. To this end, we propose to use the PAH formation kinetics model developed by the PPPL/PU group based on the well-established ABF and HACA mechanisms, coupled with low-temperature plasma models. We will develop an active learning (AL) framework based on Bayesian optimization to systematically and data-efficiently explore the complex and multivariable parameter space of plasma pyrolysis in order to quantify the effects of plasma and feed parameters on the ABF and HACA kinetic pathways. AL is the branch of machine learning concerned with systematically querying samples from a system (experimental or computational) to train a data-driven model that maps design parameters to a performance criterion. We will use the data generated via AL to perform global sensitivity analysis, combined with uncertainty quantification, to elucidate the impact of different reaction pathways on minimizing formation of soot precursors. This study will result in an improved understanding of kinetics of PAH formation in plasma pyrolysis and can pave the way for more advanced mechanistic studies (e.g., soot nucleation mechanisms). Additionally, the findings will be useful for establishing practical strategies for increasing the pyrolysis efficiency and producing high-grade carbon for synthesis of nanomaterials.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1998

Incorporating Parallel Computing into the Goddard Earth Observing System Data Assimilation System (GEOS DAS)

Atmospheric data assimilation is a method of combining actual observations with model forecasts to produce a more accurate description of the earth system than the observations or forecast alone can provide. The output of data assimilation, sometimes called the analysis, are regular, gridded datasets of observed and unobserved variables. Analysis plays a key role in numerical weather prediction and is becoming increasingly important for climate research. These applications, and the need for timely validation of scientific enhancements to the data assimilation system pose computational demands that are best met by distributed parallel software. The mission of the NASA Data Assimilation Office (DAO) is to provide datasets for climate research and to support NASA satellite and aircraft missions. The system used to create these datasets is the Goddard Earth Observing System Data Assimilation System (GEOS DAS). The core components of the the GEOS DAS are: the GEOS General Circulation Model (GCM), the Physical-space Statistical Analysis System (PSAS), the Observer, the on-line Quality Control (QC) system, the Coupler (which feeds analysis increments back to the GCM), and an I/O package for processing the large amounts of data the system produces (which will be described in another presentation in this session). The discussion will center on the following issues: the computational complexity for the whole GEOS DAS, assessment of the performance of the individual elements of GEOS DAS, and parallelization strategy for some of the components of the system.

Larson, Jay W.↗

NASA NTRS1986

A xenon ion propulsion module for enhanced spacecraft capability

Design, development and preliminary testing of a two-engine xenon ion propulsion module for early flight evaluation is described. Extensive use is made of flight spare propellant system components and also engineering model 30-cm ion engines and assemblies originally developed for the Solar Electric Propulsion System program. Significant design features include a redundant propellant feed system that incorporates a novel gas pulse assembly for rapid and completely reliable engine startup, a central neutralizer subsystem with dual neutralizers for redundancy, and major ion engine performance improvements resulting in a nearly doubling of the 30-cm engine thrust in addition to operating on xenon rather than mercury propellant. At a module input power of 10.0 kw, maximum thrust and specific impulse are projected to be 0.4 N and 3,500 sec. respectively, for a total module efficiency of 67 percent. Total mass of the xenon ion module, including the propellant tank, is only 70.2 kg. The technical approaches taken towards developing and integrating the subsystems comprising this ion propulsion module are presented and discussed in detail.

Aston, G.↗

NASA NTRS2004

Simulation-Based Verification of Autonomous Controllers via Livingstone PathFinder

AI software is often used as a means for providing greater autonomy to automated systems, capable of coping with harsh and unpredictable environments. Due in part to the enormous space of possible situations that they aim to addrs, autonomous systems pose a serious challenge to traditional test-based verification approaches. Efficient verification approaches need to be perfected before these systems can reliably control critical applications. This publication describes Livingstone PathFinder (LPF), a verification tool for autonomous control software. LPF applies state space exploration algorithms to an instrumented testbed, consisting of the controller embedded in a simulated operating environment. Although LPF has focused on NASA s Livingstone model-based diagnosis system applications, the architecture is modular and adaptable to other systems. This article presents different facets of LPF and experimental results from applying the software to a Livingstone model of the main propulsion feed subsystem for a prototype space vehicle.

Lindsey, A. E.↗

NASA NTRS1977

A dual-loop model of the human controller in single-axis tracking tasks

A dual loop model of the human controller in single axis compensatory tracking tasks is introduced. This model possesses an inner-loop closure which involves feeding back that portion of the controlled element output rate which is due to control activity. The sensory inputs to the human controller are assumed to be system error and control force. The former is assumed to be sensed via visual, aural, or tactile displays while the latter is assumed to be sensed in kinesthetic fashion. A nonlinear form of the model is briefly discussed. This model is then linearized and parameterized. A set of general adaptive characteristics for the parameterized model is hypothesized. These characteristics describe the manner in which the parameters in the linearized model will vary with such things as display quality. It is demonstrated that the parameterized model can produce controller describing functions which closely approximate those measured in laboratory tracking tasks for a wide variety of controlled elements.

Hess, R. A.↗

NASA NTRSDate not supplied

An Updated Secondary Lunar Meteoroid Ejecta Model for Engineering Design

Introduction: The surface of the Moon is constantly being bombarded by a flux of meteoroids of various sizes. Impacts due to these meteoroids produce secondary ejecta material at much lower speeds but with a total mass larger than the original impactor. Details about the secondary ejecta are important for planning missions on the lunar surface. In this work, an updated ejecta model is presented called the Meteoroid Model of Secondary Ejecta (MeMoSeE), to re-place the Apollo-era ejecta model, NASA SP-8013 [1], in the SLS-SPEC 159 Design Specification for Natural Environments (DSNE) [2]. The model pro-duces secondary ejecta flux environments for a user-specified location on the lunar surface, and sorts the incoming secondary flux by angular direction and speed. Methods: MeMoSeE is separated into three parts: the inputs, the conversion step, and the integration of fluxes. Inputs to the model include the primary meteoroid fluxes and the primary near-Earth object (NEO) fluxes. Meteoroid fluxes, both asteroidal and cometary, are calculated using the Meteoroid Engineering Model (MEM3) [3] for different locations on the Moon. For each surface location, an ephemeris is generated using the JPL HORIZONS System [4] that feeds into MEM3. The NEO fluxes are approximated by the high-density population of MEM3 (i.e., only the directionality), where the speed distribution of the NEO fluxes is renormalized to match observations [5]. The regolith properties are used as defined in the DSNE [2]. The conversion step utilizes scaling laws given by Housen & Holsapple 2011 [6] to convert the primary impactor flux to the total mass of secondary ejecta. The ejecta distribution, at the point-of-impact (POI), is separated into a zenith angle and azimuthal angle distribution. The zenith angle distribution follows a beta distribution where the peak depends on the impact altitude angle and the impact azimuth [7]. We employ an ejecta azimuth distribution that is based on Rival & Mandeville 1999 [8] which focuses ejecta in the downstream direction for more oblique impacts. Finally, during the integration step, we sum secondary ejecta number fluxes at a particular region-of-interest (ROI) that originated from many POI locations over the entire surface of the Moon. We keep track of both altitude and azimuth angle bins as well as a range of speed bins, following the igloo gridding as done in MEM3 [9]. The ejecta particle size distribution and density is assumed to be the same as the lunar regolith [10, 2]. Results: The primary fluxes are computed for one Metonic cycle (19 years) for various locations over the lunar surface with a fixed orientation. Both the angular and speed distributions of the primary fluxes are dependent on the latitude and longitude. The over-all primary fluxes show a roughly 13% increase from the eastern limb to the western limb. In general, the speed distribution of the secondary fluxes span from a user-defined minimum speed to the escape speed of the Moon (2.38 km/s), roughly following a power-law relation [6]. Different parts of the speed distribution come from different primary impact locations on the Moon. The secondary ejecta is dominated by the slowest speeds, where these particles originate nearby the ROI. For speeds around 71% of the escape speed, the secondary ejecta originates from locations near the antipodal point. On the other hand, for secondary ejecta speeds that exceed roughly 90% the escape speed, the ejecta particles come from all over the lunar surface to the ROI. Comparing the secondary ejecta fluxes from MeMoSeE with NASA SP-8013 [1], there is a reduction by about 2-3 orders of magnitude for secondary ejecta particles greater than 1 µg. These estimates agree with recent findings from Bjorkman & Christiansen 2019 [11]. The secondary ejecta fluxes are also compared with the primary fluxes, where the ejecta fluxes are roughly an order of magnitude greater than the primary fluxes. References: [1] Cour-Palais, B. G., (1969) NASA SP-8013. [2] NASA SLS-SPEC-159 Rev. H (2020). [3] Moorhead, A. V., et al. (2019) JS&R, 1-17. [4] Giorgini, J. D., (2015) IAUGA, 29, 2256293. [5] Moorhead, A. V., (2020) Memo OSMA/MEO/Lunar-001. [6] Housen, K. R., and Holsapple, K. A., (2011) Icarus, 211(1), 856-875. [7] Gault, D. E., and Wedekind, J. A., (1978) L&PSCP, 9, 3843-3875. [8] Rival, M., and Mandeville, J., (1999) Space Debris, 1(1), 45-57. [9] Moorhead, A. V., (2019) MEM3 User Guide. [10] Carrier, W. D., (1973) The Moon, 6(3-4), 250-263. [11] Bjorkman, M. D. & Christiansen, E. L., (2019) ODC

impacts↗

NASA NTRS1975

One-third scale model of an offset fed scanning antenna for the shuttle imaging microwave system

An experimental study is reported, involving the design, fabrication, and measurement of a one-third scale model of the 4m diameter parabolic torus reflector with a Gregorian feed for the SIMS application. Antenna pattern measurements of the model at 60 and 30 GHz are presented to verify the predictions of a geometrical optics analysis upon which the antenna design was based. It is shown that the antenna measurements bear out the geometrical optics analysis in every respect and that the Gregorian subreflector feed is usable at all frequencies above at least 10 Ghz in the full size SIMS antenna. The fabrication of the model antenna is discussed, including a computerized technique for fabricating the transcendental subreflector shape with a numerically controlled machine.

Gustincic, J. H.↗