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Nonlinear stability of Halley comethosheath with transverse plasma motion

Weakly nonlinear Magneto Hydrodynamic (MHD) stability of the Halley cometosheath determined by the balance between the outward ion-neutral drag force and the inward Lorentz force is investigated including the transverse plasma motion as observed in the flanks with the help of the method of multiple scales. The eigenvalues and the eigenfunctions are obtained for the linear problem and the time evolution of the amplitude is obtained using the solvability condition for the solution of the second order problem. The diamagnetic cavity boundary and the adjacent layer of about 100 km thickness is found unstable for the travelling waves of certain wave numbers. Halley ionopause has been observed to have strong ripples with a wavelength of several hundred kilometers. It is found that nonlinear effects have stabilizing effect.

Srivastava, Krishna M.↗

Measuring Thermal Conductivity and Moisture Absorption of Cryo-Insulation Materials

NASA is seeking to develop thermal insulation material systems suitable for withstanding both extremely high temperatures encountered during atmospheric re-entry heating and aero- braking maneuvers, as well as extremely low temperatures existing in liquid fuel storage tanks. Currently, materials used for the high temperature insulation or Thermal Protection System (TPS) are different from the low temperature, or cryogenic insulation. Dual purpose materials are necessary to the development of reusable launch vehicles (RLV). The present Space Shuttle (or Space Transportation System, STS) employs TPS materials on the orbiter and cryo-insulation materials on the large fuel tank slung under the orbiter. The expensive fuel tank is jettisoned just before orbit is achieved and it burns up while re-entering over the Indian Ocean. A truly completely reusable launch vehicle must store aR cryogenic fuel internally. The fuel tanks will be located close to the outer surface. In fact the outer skin of the craft will probably also serve as the fuel tank enclosure, as in jet airliners. During a normal launch the combined TPS/cryo-insulation system will serve only as a low temperature insulator, since aerodynamic heating is relatively minimal during ascent to orbit. During re-entry, the combined TPS/cryo-insulation system will serve only as a high temperature insulator, since all the cryogenic fuel will have been expended in orbit. However, in the event of an.aborted launch or a forced/emergency early re-entry, the tanks will still contain fuel, and the TPS/cryo-insulation will have to serve as both low and high temperature insulation. Also, on long duration missions, such as to Mars, very effective cryo-insulation materials are needed to reduce bod off of liquid propellants, thereby reducing necessary tankage volume, weight, and cost. The conventional approach to obtaining both low and high temperature insulation, such as is employed for the X-33 and X-34 spacecraft, is to use separate TPS and cryo-insulation materials, which are connected by means of adhesives or stand-offs (spacers). Three concepts are being considered: (1) the TPS is bonded directly to the cryo-insulation which, in turn, is bonded to the exterior of the tank, (2) stand-offs are used to make a gap between the TPS and the cryo-insulation, which is bonded externally to the tank, (3) TPS is applied directly or with stand-offs to the exterior so the tank, and cryo-insulation is applied directly to the interior of the tank. Many potential problems are inherent in these approaches. For example, mismatch between coefficients of thermal expansion of the TPS and cryo-insulation, as well as aerodynamic loads, could lead to failure of the bond. Internal cryo-insulation must be prevent from entering the sump of the fuel turbo-pump. The mechanical integrity of the stand-off structure (if used) must withstand multiple missions. During ground hold (i.e., prior to launch) moisture condensation must be minimized in the gap between the cryo-insulation and the TPS. The longer term solution requires the development of a single material to act as cryo- insulation during ground hold and as TPS during re-entry. Such a material minimizes complexity and weight while improving reliability and reducing cost.

Lambert, Michael A.↗

Analysis of Complexity Evolution Management and Human Performance Issues in Commercial Aircraft Automation Systems

Autoflight systems in the current generation of aircraft have been implicated in several recent incidents and accidents. A contributory aspect to these incidents may be the manner in which aircraft transition between differing behaviours or 'modes.' The current state of aircraft automation was investigated and the incremental development of the autoflight system was tracked through a set of aircraft to gain insight into how these systems developed. This process appears to have resulted in a system without a consistent global representation. In order to evaluate and examine autoflight systems, a 'Hybrid Automation Representation' (HAR) was developed. This representation was used to examine several specific problems known to exist in aircraft systems. Cyclomatic complexity is an analysis tool from computer science which counts the number of linearly independent paths through a program graph. This approach was extended to examine autoflight mode transitions modelled with the HAR. A survey was conducted of pilots to identify those autoflight mode transitions which airline pilots find difficult. The transitions identified in this survey were analyzed using cyclomatic complexity to gain insight into the apparent complexity of the autoflight system from the perspective of the pilot. Mode transitions which had been identified as complex by pilots were found to have a high cyclomatic complexity. Further examination was made into a set of specific problems identified in aircraft: the lack of a consistent representation of automation, concern regarding appropriate feedback from the automation, and the implications of physical limitations on the autoflight systems. Mode transitions involved in changing to and leveling at a new altitude were identified across multiple aircraft by numerous pilots. Where possible, evaluation and verification of the behaviour of these autoflight mode transitions was investigated via aircraft-specific high fidelity simulators. Three solution approaches to concerns regarding autoflight systems, and mode transitions in particular, are presented in this thesis. The first is to use training to modify pilot behaviours, or procedures to work around known problems. The second approach is to mitigate problems by enhancing feedback. The third approach is to modify the process by which automation is designed. The Operator Directed Process forces the consideration and creation of an automation model early in the design process for use as the basis of the software specification and training.

Vakil, Sanjay S.↗

On Multifunctional Collaborative Methods in Engineering Science

Multifunctional methodologies and analysis procedures are formulated for interfacing diverse subdomain idealizations including multi-fidelity modeling methods and multi-discipline analysis methods. These methods, based on the method of weighted residuals, ensure accurate compatibility of primary and secondary variables across the subdomain interfaces. Methods are developed using diverse mathematical modeling (i.e., finite difference and finite element methods) and multi-fidelity modeling among the subdomains. Several benchmark scalar-field and vector-field problems in engineering science are presented with extensions to multidisciplinary problems. Results for all problems presented are in overall good agreement with the exact analytical solution or the reference numerical solution. Based on the results, the integrated modeling approach using the finite element method for multi-fidelity discretization among the subdomains is identified as most robust. The multiple method approach is advantageous when interfacing diverse disciplines in which each of the method's strengths are utilized.

Ransom, Jonathan B.↗

Airborne-Managed Spacing in Multiple Arrival Streams

A significant bottleneck in the current air traffic system occurs at the runway. Expanding airports and adding new runways will help solve this problem; however, this comes at a significant cost, financially, politically and environmentally. A complementary solution is to safely increase the capacity of current runways. This can be achieved by precise spacing at the runway threshold with a resulting reduction in the spacing buffer required under today s operations. At the NASA Langley Research Center, the Advanced Air Transportation Technologies (AATT) Project is investigating airborne technologies and procedures that will assist the pilot in achieving precise spacing behind another aircraft. This new spacing clearance instructs the pilot to follow speed cues from a new on-board guidance system called Airborne Merging and Spacing for Terminal Arrivals (AMSTAR). AMSTAR receives Automatic Dependent Surveillance-Broadcast (ADS-B) reports from the leading aircraft and calculates the appropriate speed for the ownership to fly in order to achieve the desired spacing interval, time or distance-based, at the runway threshold. Since the goal is overall system capacity, the speed guidance algorithm is designed to provide system benefit over individual efficiency. This paper discusses the concept of operations and design of AMSTAR to support airborne precision spacing. Results from the previous stage of development, focused only on in-trail spacing, are discussed along with the evolution of the concept to include merging of converging streams of traffic. This paper also examines how this operation might support future wake vortex-based separation and other advances in terminal area operations. Finally, the research plan for the merging capabilities, to be performed during the summer and fall of 2004 is presented.

Barmore, Bryan↗

Extended-Range Passive RFID and Sensor Tags

Extended-range passive radio-frequency identification (RFID) tags and related sensor tags are undergoing development. A tag of this type incorporates a retroreflective antenna array, so that it reflects significantly more signal power back toward an interrogating radio transceiver than does a comparable passive RFID tag of prior design, which does not incorporate a retroreflective antenna array. Therefore, for a given amount of power radiated by the transmitter in the interrogating transceiver, a tag of this type can be interrogated at a distance greater than that of the comparable passive RFID or sensor tag of prior design. The retroreflective antenna array is, more specifically, a Van Atta array, named after its inventor and first published in a patent issued in 1959. In its simplest form, a Van Atta array comprises two antenna elements connected by a transmission line so that the signal received by each antenna element is reradiated by the other antenna element (see Figure 1). The phase relationships among the received and reradiated signals are such as to produce constructive interference of the reradiated signals; that is, to concentrate the reradiated signal power in a direction back toward the source. Hence, an RFID tag equipped with a Van Atta antenna array automatically tracks the interrogating transceiver. The effective gain of a Van Atta array is the same as that of a traditional phased antenna array having the same number of antenna elements. Additional pairs of antenna elements connected by equal-length transmission lines can be incorporated into a Van Atta array to increase its directionality. Like some RFID tags here-to-fore commercially available, an RFID or sensor tag of the present developmental type includes one-port surface-acoustic-wave (SAW) devices. In simplified terms, the mode of operation of a basic one-port SAW device as used heretofore in an RFID device is the following: An interrogating radio signal is converted, at an input end, from an electrical signal to an acoustic wave that propagates along a surface and encounters multiple reflectors suitably positioned along the surface. Upon returning to the input end, the reflected acoustic wave is re-converted to an electrical signal, which, in turn, is reradiated from an antenna. The distances between the reflectors in the SAW device and the corresponding times between reflections encode the identifying or sensory information onto the reradiated signal. The fundamental problem in the present development is how to combine a Van Atta antenna array (which is inherently a multiple-port device) and one or more one-port SAW device(s) into a single, compact, passive unit that can function as a retroreflective RFID tag. The solution is to use one or more hybrid, half-power 90 couplers. A basic unit of this type, shown in Figure 2, includes a half-power 90 hybrid coupler; two identical SAW devices (SAW1 and SAW2) connected to ports 3 and 4 of the coupler, respectively; and antenna elements connected to ports 1 and 2 of the coupler. Necessarily omitting details for the sake of brevity, it must suffice to report that the phase relationships among the coupler inputs and outputs are such as to couple the incident signal from the antenna elements to the SAW devices and couple the reflected signals from the SAW devices back to the antenna elements in the phase relationships required for a Van Atta array. Hence, the reradiated signal is automatically directed back toward the interrogating transceiver and contains identifying and/or sensory information encoded in time intervals between reflections.

Fink, Patrick W.↗

A Novel Approach to Solve Forward/Inverse Problems in Remote Sensing Applications

Inversion of electromagnetic (EM) signals reflected from or transmitted through a medium, or emitted by it due to internal sources can be used to investigate the optical and physical properties of a variety of scattering/absorbing/emitting materials. Such media encompass planetary atmospheres and surfaces (including water/snow/ice), and plant canopies. In many situations the signals emerging from such media can be described by a linear transport equation which in the case of EM radiation is the radiative transfer equation (RTE). Solutions of the RTE can be used as a forward model to solve the inverse problem to determine the medium state parameters giving rise to the emergent (reflected/transmitted/emitted) EM signals. A novel method is developed to determine layer-by-layer contributions to the emergent signals from such stratified, multilayered media based on the solution of the pertinent RTE. As a specific example of how this approach may be applied, the radiation reflected from a multilayered atmosphere is used to solve the problem relevant for EM probing by a space-based lidar system. The solutions agree with those obtained using the standard lidar approach for situations in which single scattering prevails, but this novel approach also yields reliable results for optically thick, multiple scattering aerosol and cloud layers that cannot be provided by the traditional lidar approach.

forward modeling↗

A Numerical Simulation of Scattering from One-Dimensional Inhomogeneous Dielectric Random Surfaces

In this paper, an efficient numerical solution for the scattering problem of inhomogeneous dielectric rough surfaces is presented. The inhomogeneous dielectric random surface represents a bare soil surface and is considered to be comprised of a large number of randomly positioned dielectric humps of different sizes, shapes, and dielectric constants above an impedance surface. Clods with nonuniform moisture content and rocks are modeled by inhomogeneous dielectric humps and the underlying smooth wet soil surface is modeled by an impedance surface. In this technique, an efficient numerical solution for the constituent dielectric humps over an impedance surface is obtained using Green's function derived by the exact image theory in conjunction with the method of moments. The scattered field from a sample of the rough surface is obtained by summing the scattered fields from all the individual humps of the surface coherently ignoring the effect of multiple scattering between the humps. The statistical behavior of the scattering coefficient sigma(sup 0) is obtained from the calculation of scattered fields of many different realizations of the surface. Numerical results are presented for several different roughnesses and dielectric constants of the random surfaces. The numerical technique is verified by comparing the numerical solution with the solution based on the small perturbation method and the physical optics model for homogeneous rough surfaces. This technique can be used to study the behavior of scattering coefficient and phase difference statistics of rough soil surfaces for which no analytical solution exists.

Sarabandi, Kamal↗

Numerical Study of Turbulence Model Predictions for the MD 30P/30N and NHLP-2D Three-Element Highlift Configurations

This report details calculations for the McDonnell-Douglas 30P/30N and the NHLP-2D three-element highlift configurations. Calculations were performed with the Reynolds averaged Navier-Stokes code ISAAC to study the effects of various numerical issues on high lift predictions. These issues include the effect of numerical accuracy on the advection terms of the turbulence equations, Navier-Stokes versus the thin-layer Navier-Stokes approximation, an alternative formulation of the production term, and the performance of several turbulence models. The effect of the transition location on the NHLP-2D flow solution was investigated. Two empirical transition models were used to estimate the transition location.

Turbulence modeling↗

Feeding Ten Billion People Is Possible Within Four Terrestrial Planetary Boundaries

Global agriculture puts heavy pressure on planetary boundaries, posing the challenge to achieve future food security without compromising Earth system resilience. On the basis of process-detailed, spatially explicit representation of four interlinked planetary boundaries (biosphere integrity, land-system change, freshwater use, nitrogen flows) and agricultural systems in an internally consistent model framework, we here show that almost half of current global food production depends on planetary boundary transgressions. Hotspot regions, mainly in Asia, even face simultaneous transgression of multiple underlying local boundaries. If these boundaries were strictly respected, the present food system could provide a balanced diet (2,355 kcal per capita per day) for 3.4 billion people only. However, as we also demonstrate, transformation towards more sustainable production and consumption patterns could support 10.2 billion people within the planetary boundaries analysed. Key prerequisites are spatially redistributed cropland, improved water–nutrient management, food waste reduction and dietary changes. Adoption of the Sustainable Development Goals by all nations in 2015 is the first ever commitment to a world development path that safeguards the stability of the Earth system as a prerequisite for meeting universal human standards1. The longstanding challenge of achieving food security through sustainable agriculture is particularly acute in this context as world agriculture is a leading cause for the current transgressions of multiple planetary boundaries (PBs) globally and regionally2–5. The PB framework is a comprehensive scientific attempt to synoptically define our planet’s biogeophysical limits to anthropogenic interference. It suggests bounds to nine interacting processes that together delineate a Holocene-like Earth system state. The Holocene is chosen as the reference state as it is the only period known to provide a safe operating space for a world population of several billion people, and according to a precautionary principle, the PBs are set in sufficient distance from processes that may critically undermine Earth system resilience and global sustainability. A challenging question, thus, is whether human development goals such as food security can be met while maintaining multiple PBs along with their subglobal manifestations. Further PB transgressions could jeopardize the chances of providing sufficient food for a world population projected to be wealthier and reach >9 billion by 2050. This conundrum portrays a tradeoff between Earth’s biophysical carrying capacity and humankind’s rising food demand, calling in response for radical rethinking of food production and consumption patterns6–9. Yield gap closures, avoidance of excessive input use, shifts towards less resource-demanding diets, food waste reductions and efficient international trade are crucial options for sustainably increasing the food supply10–15. For example, enhancing water-use efficiency on irrigated and rain-fed farms can triple or quadruple crop yields in low-performing systems, suggesting possible global gains of >20% (ref. 16). Even higher gains appear feasible through globally optimized configurations of the land-use pattern17, and cutting food losses by half could generate food for another billion people18. Thus, collective large-scale implementation of such options could sustain food for a further growing world population19. Yet achieving this within a safe operating space as defined by PBs requires not only a halt to but actually a reversal of existing PB transgressions. Previous studies suggest that such a reconciliation might be possible, but these were based on aggregate representations of PBs (not accounting for the spatial patterns of limits, transgressions and interactions) or considered only one boundary in isolation17,20–23. Here, we systematically quantify to what extent current food production depends on local to global transgressions of the PBs for biosphere integrity, land-system change, freshwater use and nitrogen (N) flows, along with the potential of a range of solutions to avoid these transgressions and still increase food supply (Table 1). To this end, we configured an internally consistent process-based model of the terrestrial biosphere including agriculture (LPJmL) with multiple spatially distributed PBs and their interactions. LPJmL is among the longest-established and best-evaluated biosphere models, showing robust performance regarding simulation of, for example, carbon, water and crop yield dynamics (Supplementary Figs. 1 and 2 and Supplementary Table 1; see ref. 24 for a comprehensive benchmarking and Supplementary Methods for more detail on model evaluations). In principle following established definitions4, we refine the computation of some PBs with respect to their regional patterns and interactions (Methods), providing globally gridded precautionary limits to human interference with the Earth system at a level of great detail. In particular, we account for the evidence that many PBs need to be represented spatially explicitly4 to cover their

Gerten, Dieter↗

Discrete ordinate solution of the radiative transfer equation in the 'polarization normal wave representation'

The transfer equations for normal waves in finite, inhomogeneous and plane-parallel magnetoactive media are solved using the discrete ordinate method. The physical process of absorption, emission, and multiple scattering are accounted for, and the medium may be forced both at the top and bottom boundary by anisotropic radiation as well as by internal anisotropic sources. The computational procedure is numerically stable for arbitrarily large optical depths, and the computer time is independent of optical thickness.

Kylling, A.↗

Evolvable Mars Campaign Long Duration Habitation Strategies: Architectural Approaches to Enable Human Exploration Missions

The Evolvable Mars Campaign (EMC) is the current NASA Mars mission planning effort which seeks to establish sustainable, realistic strategies to enable crewed Mars missions in the mid-2030s timeframe. The primary outcome of the Evolvable Mars Campaign is not to produce "The Plan" for sending humans to Mars, but instead its intent is to inform the Human Exploration and Operations Mission Directorate near-term key decisions and investment priorities to prepare for those types of missions. The FY'15 EMC effort focused upon analysis of integrated mission architectures to identify technically appealing transportation strategies, logistics build-up strategies, and vehicle designs for reaching and exploring Mars moons and Mars surface. As part of the development of this campaign, long duration habitats are required which are capable of supporting crew with limited resupply and crew abort during the Mars transit, Mars moons, and Mars surface segments of EMC missions. In particular, the EMC design team sought to design a single, affordable habitation system whose manufactured units could be outfitted uniquely for each of these missions and reused for multiple crewed missions. This habitat system must provide all of the functionality to safely support 4 crew for long durations while meeting mass and volume constraints for each of the mission segments set by the chosen transportation architecture and propulsion technologies. This paper describes several proposed long-duration habitation strategies to enable the Evolvable Mars Campaign through improvements in mass, cost, and reusability, and presents results of analysis to compare the options and identify promising solutions. The concepts investigated include several monolithic concepts: monolithic clean sheet designs, and concepts which leverage the co-manifested payload capability of NASA's Space Launch System (SLS) to deliver habitable elements within the Universal Payload Adaptor between the SLS upper stage and the Orion/Service module on the top of the vehicle. Multiple modular habitat options for Mars surface and in-space missions are also considered with various functionality and volume splits between modules to find the best balance of reducing the single largest mass which must be delivered to a destination and reducing the number of separate elements which must be launched. Analysis results presented for each of these concepts in this paper include mass/volume/power sizing using parametric sizing tools, identification of unique operational constraints, and limited comments on the additional impacts of reusability/dormancy on system design. Finally, recommendations will be made for promising solutions which will be carried forward for consideration in the Evolvable Mars Campaign work.

Simon, Matthew A.↗

The ATS-F/Nimbus-F tracking and orbit determination experiment

The experiment described was conducted to demonstrate a procedure for tracking a near-earth satellite via a geostationary satellite without the aid of multiple ground station tracking. Another objective of the experiment was connected with the utilization of the broad tracking coverage provided by the geostationary satellite to obtain an improved geopotential solution. Questions of overall experiment implementation are discussed along with details regarding ground equipment, the ATS-F transponder, and the Nimbus-F transponder. Aspects of measurement evaluation are also examined, taking into account basic measurements, measurement interpretation, and approaches for orbit computation.

Schmid, P. E.↗

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

Geddes, K. O.↗

A three-dimensional application with the numerical grid generation code: EAGLE (utilizing an externally generated surface)

Program EAGLE (Eglin Arbitrary Geometry Implicit Euler) is a multiblock grid generation and steady-state flow solver system. This system combines a boundary conforming surface generation, a composite block structure grid generation scheme, and a multiblock implicit Euler flow solver algorithm. The three codes are intended to be used sequentially from the definition of the configuration under study to the flow solution about the configuration. EAGLE was specifically designed to aid in the analysis of both freestream and interference flow field configurations. These configurations can be comprised of single or multiple bodies ranging from simple axisymmetric airframes to complex aircraft shapes with external weapons. Each body can be arbitrarily shaped with or without multiple lifting surfaces. Program EAGLE is written to compile and execute efficiently on any CRAY machine with or without Solid State Disk (SSD) devices. Also, the code uses namelist inputs which are supported by all CRAY machines using the FORTRAN Compiler CF177. The use of namelist inputs makes it easier for the user to understand the inputs and to operate Program EAGLE. Recently, the Code was modified to operate on other computers, especially the Sun Spare4 Workstation. Several two-dimensional grid configurations were completely and successfully developed using EAGLE. Currently, EAGLE is being used for three-dimension grid applications.

Houston, Johnny L.↗

Automated design of gravity-assist trajectories to Mars and the outer planets

In this paper, a new approach to planetary mission design is described which automates the search for gravity-assist trajectories. This method finds all conic solutions given a range of launch dates, a range of launch energies and a set of target planets. The new design tool is applied to the problems of finding multiple encounter trajectories to the outer planets and Venus gravity-assist trajectories to Mars. The last four-planet grand tour opportunity (until the year 2153) is identified. It requires an earth launch in 1996 and encounters Jupiter, Uranus, Neptune, and Pluto. Venus gravity-assist trajectories to Mars for the 30 year period 1995-2024 are examined. It is shown that in many cases these trajectories require less launch energy to reach Mars than direct ballistic trajectories.

Longuski, James M.↗

Development of deterministic transport methods for low energy neutrons for shielding in space

Transport of low energy neutrons associated with the galactic cosmic ray cascade is analyzed in this dissertation. A benchmark quality analytical algorithm is demonstrated for use with BRYNTRN, a computer program written by the High Energy Physics Division of NASA Langley Research Center, which is used to design and analyze shielding against the radiation created by the cascade. BRYNTRN uses numerical methods to solve the integral transport equations for baryons with the straight-ahead approximation, and numerical and empirical methods to generate the interaction probabilities. The straight-ahead approximation is adequate for charged particles, but not for neutrons. As NASA Langley improves BRYNTRN to include low energy neutrons, a benchmark quality solution is needed for comparison. The neutron transport algorithm demonstrated in this dissertation uses the closed-form Green's function solution to the galactic cosmic ray cascade transport equations to generate a source of neutrons. A basis function expansion for finite heterogeneous and semi-infinite homogeneous slabs with multiple energy groups and isotropic scattering is used to generate neutron fluxes resulting from the cascade. This method, called the FN method, is used to solve the neutral particle linear Boltzmann transport equation. As a demonstration of the algorithm coded in the programs MGSLAB and MGSEMI, neutron and ion fluxes are shown for a beam of fluorine ions at 1000 MeV per nucleon incident on semi-infinite and finite aluminum slabs. Also, to demonstrate that the shielding effectiveness against the radiation from the galactic cosmic ray cascade is not directly proportional to shield thickness, a graph of transmitted total neutron scalar flux versus slab thickness is shown. A simple model based on the nuclear liquid drop assumption is used to generate cross sections for the galactic cosmic ray cascade. The ENDF/B V database is used to generate the total and scattering cross sections for neutrons in aluminum. As an external verification, the results from MGSLAB and MGSEMI were compared to ANISN/PC, a routinely used neutron transport code, showing excellent agreement. In an application to an aluminum shield, the FN method seems to generate reasonable results.

Ganapol, Barry↗

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Coutsias, Evangelos A.↗