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NASA NTRS1971

Avalanche-diode oscillator circuit with tuning at multiple frequencies

Detailed theoretical analysis of three different modes or types of high efficiency oscillation in a PIN diode are presented. For the TRAPATT mode in a PIN diode, it is shown that a traveling avalanche zone is not necessary to generate a dense trapped plasma. An economical computer program for TRAPATT oscillations in a PIN diode is described. Typical results of diode power, dc-to-RF conversion efficiency, and required circuit impedances are presented for several different current waveforms. A semianalytical solution for a second type of high efficiency mode in a PIN diode is derived assuming a rectangular current waveform. A quasi-static approximation is employed to derive a semianalytical solution for the voltage across a PIN diode in a third mode, where avalanching occurs during a major portion of a half cycle. Calculations for this mode indicate that the power increases proportionally to the magnitude of the drive current with a small decrease in efficiency relative to the ordinary TRAPATT mode. An analytical solution is also given for a PIN diode, where it is assumed that the ionization coefficient is a step function. It is shown that the step-ionization approximation permits one to draw possible patterns of avalanche region in the depletion layer as a function of time. A rule governing admissible patterns is derived and an example solution given for one admissible pattern. Preliminary experimental results on the high-efficiency oscillations are presented and discussed. Two different experimental circuits, which used channel-dropping filters to provide independent harmonic tuning, are described. Simpler circuits used to produce high-efficiency oscillations are discussed. Results of experiments using inexpensive Fairchild FD300 diodes are given.

Parker, D.↗

NASA NTRS1977

Use of finite element techniques in the determination of the acoustic properties of turbofan inlets

This paper describes the application of the finite-element method in combination with Galerkin's method in the determination of the acoustic properties of turbofan inlets containing high-subsonic-Mach-number flows. An approximate solution to the steady inviscid flow field is obtained using an integral method for calculating the potential flow field in the inlet with a correction to account for compressibility effects. The accuracy of the finite-element technique in predicting the acoustic properties of annular ducts has been checked by comparison with available analytical solutions for the problems of planeand spinning-wave propagation through a hard-walled annular duct with a constant mean flow. Results are presented comparing low-frequency plane-wave propagation through a hard-walled turbofan inlet containing a one-dimensional flow with the same inlet containing a fully two-dimensional axisymmetric mean flow. It is shown that when one-dimensional mean flow is assumed to exist in the duct, the plane wave propagates with relatively little distortion. However, propagation of a plane wave through the fully two-dimensional flow field in the inlet produces severe distortion due to the excitation of higher-order modes.

Sigman, R. K.↗

NASA NTRS1992

The application of the principles of invariance to the radiative transfer equation in plant canopies

Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite acquired information. The general one-dimensional two-angle transport problem for a finite copy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar's X- and Y-functions by invoking the principles of invariance. A critical step in the formulation involves the decomposition of the integral of the scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are derived from the generalized solution. Various symmetries obeyed by the scattering operator and reciprocity relations are formally proved.

Ganapol, B. D.↗

NASA NTRS1995

Effects of Delaminations on the Damped Dynamic Characteristics of Composite Laminates: Mechanics and Experiments

Analytical and experimental work is presented on the damped free-vibration of delaminated laminates and beams. A laminate theory is developed where the unknown kinematic perturbations induced by a delamination crack are treated as additional degrees of freedom. The generalized stiffness, inertia and damping matrices of the laminate are formulated. An analytical solution is developed for the prediction of natural frequencies, modes and modal damping in composite beams with delamination cracks. Evaluations of the mechanics on various cantilever beams with a central delamination are performed. Experimental results for the modal frequencies and damping of composite beams with a single delamination are also presented and correlations between analytical predictions and measured data are shown. The effects of delamination vary based on crack size, laminate configuration, and mode order. The implications of the mechanics in developing delamination detection techniques are also discussed.

Saravanos, D. A.↗

NASA NTRS1988

Charged-particle transport in one dimension

A numerical solution to high-energy charged-particle transport is found by evaluation of the integral equations obtained by inverting the Boltzmann differential operator. An algorithm has been written for either continuous or discrete spectra at the boundary, which allows efficient and accurate evaluation of this rather complicated problem. Present results are compared with analytic solutions based on the perturbation theory.

NASA Discipline Radiation Health↗

DOE OSTI2021

Optimal scheduling for profit maximization of energy storage merchants considering market impact based on dynamic programming

This paper analyzes how electricity merchants' market impact affects merchants' profit. Energy storage has long been studied for its role in maximizing profit, and merchant decisions are assumed to have no impact on market prices. However, the trading decisions of large-scale energy storage merchants (e.g., pumped storage hydro) will affect the market prices. This paper employs dynamic programming theory to investigate merchants' optimal economic dispatch considering the market impact and physical characteristics of storage systems. Our findings show that the State-of-Charge (SOC) based analytical solution significantly facilitates energy storage merchants' decision-making. The SOC range is segmented into three regions by two optimal SOC reference points, which depend on the available energy in storage, given prices, and market impact. By comparing the current storage SOC with the reference points, the merchant can get the corresponding optimal actions. We analytically show that if the merchant neglects the market impact on the power market, she will exaggerate her expected profit when the price-taker and price-maker merchants have the same generating and pumping upper limits offered to Independent System Operators (ISOs). Furthermore, the profit-maximizing merchant must, therefore, assay to balance the trade-off correctly between the intensity of market impact and the dispatched power. Our findings are verified by numerical simulation, and results demonstrate the ramifications for electricity merchants in energy arbitrage decisions.

25 ENERGY STORAGE↗

NASA NTRS1997

Verification of RadCAD: Specular Capabilities

As part of the RadCAD's development process, it is necessary to compare RadCAD's results with other radiation tools and exact solutions when and where possible. Form factor algorithms have been previously verified with exact solutions. This paper will consider RadCAD's specular capabilities. First, radiation exchange factors will be compared against exact solutions and results from TRASYS for various geometries. Critical dimensions and optical properties are changed for each geometry. Second, a specular adjunct plate system will be used to verify absorbed heat fluxes. This particular geometric problem has had some attention in the literature. Previous authors have used this problem to validate software results with exact analytical solution. This paper will compare absorbed heat rates against the exact solution and other published results from other thermal radiation tools. The agreement between RadCAD and the exact solutions is good. The maximum error for both specular and diffuse exchange factors for both geometries and all optical properties was 3%. The absorbed fluxes differed by a maximum of 4% for the adjunct plate problem.

Lucas, Sam↗

NASA NTRS1994

Azimuthally-dependent Finite Element Solution to the Cylindrical Resonator

The cylindrical cavity resonator loaded with an anisotropic dielectric is analyzed as a two-dimensional problem using a finite element approach that assumes sinusoidal dependence in azimuth. This methodology allows the first finite element treatment of the technically important case of a resonator containing a sapphire element with a cylindrically aligned c axis. Second order trial functions together with quadrilateral elements are adopted in the calculations. The method was validated through comparisons with the analytical solutions for the hollow metal cavity and a coaxial cavity, as well as through measurements on a shielded sapphire resonator.

cylindrical resonator two-dimensional finite eleme↗

DOE OSTI2021

Solving the sample size problem for resource selection functions

Abstract Sample size sufficiency is a critical consideration for estimating resource selection functions (RSFs) from GPS‐based animal telemetry. Cited thresholds for sufficiency include a number of captured animals and as many relocations per animal N as possible. These thresholds render many RSF‐based studies misleading if large sample sizes were truly insufficient, or unpublishable if small sample sizes were sufficient but failed to meet reviewer expectations. We provide the first comprehensive solution for RSF sample size by deriving closed‐form mathematical expressions for the number of animals M and the number of relocations per animal N required for model outputs to a given degree of precision. The sample sizes needed depend on just 3 biologically meaningful quantities: habitat selection strength, variation in individual selection and a novel measure of landscape complexity, which we define rigorously. The mathematical expressions are calculable for any environmental dataset at any spatial scale and are applicable to any study involving resource selection (including sessile organisms). We validate our analytical solutions using globally relevant empirical data including 5,678,623 GPS locations from 511 animals from 10 species (omnivores, carnivores and herbivores living in boreal, temperate and tropical forests, montane woodlands, swamps and Arctic tundra). Our analytic expressions show that the required M and N must decline with increasing selection strength and increasing landscape complexity, and this decline is insensitive to the definition of availability used in the analysis. Our results demonstrate that the most biologically relevant effects on the utilization distribution (i.e. those landscape conditions with the greatest absolute magnitude of resource selection) can often be estimated with much fewer than animals. We identify several critical steps in implementing these equations, including (a) a priori selection of expected model coefficients and (b) regular sampling of background (pseudoabsence) data within a given definition of availability. We discuss possible methods to identify a priori expectations for habitat selection coefficients, effects of scale on RSF estimation and caveats for rare species applications. We argue that these equations should be a mandatory component for all future RSF studies.

Street, Garrett M.↗

DOE OSTI2024

Studying the Random Number Generators in MCNP6 using an Analytic Benchmark

An analytic solution to a previously studied toy problem is derived and used as a code verification benchmark. Using various Random Number Generators (RNGs) in MCNP6, including the newest SFC64 RNG available in MCNP6.3.1, and their various properties (e.g., RNG stride), we show how these RNGs perform and how to correct or workaround potential issues with respect to the analytic benchmark problem.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1990

On the numerical solution of the dynamically loaded hydrodynamic lubrication of the point contact problem

The transient analysis of hydrodynamic lubrication of a point-contact is presented. A body-fitted coordinate system is introduced to transform the physical domain to a rectangular computational domain, enabling the use of the Newton-Raphson method for determining pressures and locating the cavitation boundary, where the Reynolds boundary condition is specified. In order to obtain the transient solution, an explicit Euler method is used to effect a time march. The transient dynamic load is a sinusoidal function of time with frequency, fractional loading, and mean load as parameters. Results include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency. The results are compared with the analytic solution to the transient step bearing problem with the same dynamic loading function. The similarities of the results suggest an approximate model of the point contact minimum film thickness solution.

Lim, Sang G.↗

NASA NTRS1992

Analytic description of the electron temperature behavior in the upper ionosphere and plasmasphere

Approximate analytic solutions to the well-known and commonly used time-dependent electron energy balance equation for the upper ionosphere and plasmasphere have been obtained and are discussed. The various potential heating sources for the terrestrial plasmasphere are summarized and the corresponding electron temperatures and related characteristic heating and cooling times are calculated. A comparison between the analytic expressions for the temperature variations and relevant measurements shows excellent agreement.

Khazanov, G. V.↗

DOE OSTI2021

An extended numerical manifold method for unsaturated soil-water interaction analysis at micro-scale

To investigate unsaturated soil-water interaction at micro-scale, this work extends the numerical manifold method (NMM) by incorporating a soil-water coupling model considering specific capillary water distribution and capillary force calculation. The soil skeleton is constructed by a soil skeleton generation algorithm with random polygons. To more realistically capture the interaction between soil grains and capillary water, a capillary mechanics-based geometric algorithm is proposed to iteratively calculate the capillary water distribution. The capillary forces corresponding to the capillary water distribution are calculated based on the Young-Laplace equation. The proposed capillary water solving framework is first verified by reproducing the soil-water characteristic curve and the capillary water distribution of an ideal contact-disk model against analytical solutions. To further validate the ability of the capillary water solving framework to predict hydraulic behavior of the real soil, a laboratory test on the Toyoura sand is reproduced numerically. Then an ideal direct shear test is performed to further validate the two-way soil-water coupling procedure, in which a comparison between the numerical and analytical results regarding the shear strength and matric suction is presented. Finally, microscopic hydraulic and compression tests are conducted on two soil specimens with the same porosity and mean grain diameter but different uniformity coefficients. The results elucidate that the extended method is a potential tool to explore unsaturated soil behaviors at micro-scale.

58 GEOSCIENCES↗

NASA NTRS1990

One dimensional heavy ion beam transport: Energy independent model

Attempts are made to model the transport problem for heavy ion beams in various targets, employing the current level of understanding of the physics of high-charge and energy (HZE) particle interaction with matter are made. An energy independent transport model, with the most simplified assumptions and proper parameters is presented. The first and essential assumption in this case (energy independent transport) is the high energy characterization of the incident beam. The energy independent equation is solved and application is made to high energy neon (NE-20) and iron (FE-56) beams in water. The numerical solutions is given and compared to a numerical solution to determine the accuracy of the model. The lower limit energy for neon and iron to be high energy beams is calculated due to Barkas and Burger theory by LBLFRG computer program. The calculated values in the density range of interest (50 g/sq cm) of water are: 833.43 MeV/nuc for neon and 1597.68 MeV/nuc for iron. The analytical solutions of the energy independent transport equation gives the flux of different collision terms. The fluxes of individual collision terms are given and the total fluxes are shown in graphs relative to different thicknesses of water. The values for fluxes are calculated by the ANASTP computer code.

Farhat, Hamidullah↗

NASA NTRS1994

Completed Beltrami-Michell Formulation for Analyzing Radially Symmetrical Bodies

A force method formulation, the completed Beltrami-Michell formulation (CBMF), has been developed for analyzing boundary value problems in elastic continua. The CBMF is obtained by augmenting the classical Beltrami-Michell formulation with novel boundary compatibility conditions. It can analyze general elastic continua with stress, displacement, or mixed boundary conditions. The CBMF alleviates the limitations of the classical formulation, which can solve stress boundary value problems only. In this report, the CBMF is specialized for plates and shells. All equations of the CBMF, including the boundary compatibility conditions, are derived from the variational formulation of the integrated force method (IFM). These equations are defined only in terms of stresses. Their solution for kinematically stable elastic continua provides stress fields without any reference to displacements. In addition, a stress function formulation for plates and shells is developed by augmenting the classical Airy's formulation with boundary compatibility conditions expressed in terms of the stress function. The versatility of the CBMF and the augmented stress function formulation is demonstrated through analytical solutions of several mixed boundary value problems. The example problems include a composite circular plate and a composite circular cylindrical shell under the simultaneous actions of mechanical and thermal loads.

Kaljevic, Igor↗

NASA NTRS1996

Computational Analysis of Static and Dynamic Behaviour of Magnetic Suspensions and Magnetic Bearings

Static modelling of magnetic bearings is often carried out using magnetic circuit theory. This theory cannot easily include nonlinear effects such as magnetic saturation or the fringing of flux in air-gaps. Modern computational tools are able to accurately model complex magnetic bearing geometries, provided some care is exercised. In magnetic suspension applications, the magnetic fields are highly three-dimensional and require computational tools for the solution of most problems of interest. The dynamics of a magnetic bearing or magnetic suspension system can be strongly affected by eddy currents. Eddy currents are present whenever a time-varying magnetic flux penetrates a conducting medium. The direction of flow of the eddy current is such as to reduce the rate-of-change of flux. Analytic solutions for eddy currents are available for some simplified geometries, but complex geometries must be solved by computation. It is only in recent years that such computations have been considered truly practical. At NASA Langley Research Center, state-of-the-art finite-element computer codes, 'OPERA', 'TOSCA' and 'ELEKTRA' have recently been installed and applied to the magnetostatic and eddy current problems. This paper reviews results of theoretical analyses which suggest general forms of mathematical models for eddy currents, together with computational results. A simplified circuit-based eddy current model proposed appears to predict the observed trends in the case of large eddy current circuits in conducting non-magnetic material. A much more difficult case is seen to be that of eddy currents in magnetic material, or in non-magnetic material at higher frequencies, due to the lower skin depths. Even here, the dissipative behavior has been shown to yield at least somewhat to linear modelling. Magnetostatic and eddy current computations have been carried out relating to the Annular Suspension and Pointing System, a prototype for a space payload pointing and vibration isolation system, where the magnetic actuator geometry resembles a conventional magnetic bearing. Magnetostatic computations provide estimates of flux density within airgaps and the iron core material, fringing at the pole faces and the net force generated. Eddy current computations provide coil inductance, power dissipation and the phase lag in the magnetic field, all as functions of excitation frequency. Here, the dynamics of the magnetic bearings, notably the rise time of forces with changing currents, are found to be very strongly affected by eddy currents, even at quite low frequencies. Results are also compared to experimental measurements of the performance of a large-gap magnetic suspension system, the Large Angle Magnetic Suspension Test Fixture (LAMSTF). Eddy current effects are again shown to significantly affect the dynamics of the system. Some consideration is given to the ease and accuracy of computation, specifically relating to OPERA/TOSCA/ELEKTRA.

Britcher, Colin P.↗

NASA NTRS1989

Compression behavior of delaminated composite plates

The response of delaminated composite plates to compressive in-plane loads was investigated. The delaminated region may be either circular or elliptical, and may be located between any two plies of the laminate. For elliptical delaminations, the axes of the ellipse may be arbitrarily oriented with respect to the applied loads. A model was developed that describes the stresses, strains, and deformation of the sublaminate created by the delamination. The mathematical model is based on a two dimensional nonlinear plate theory that includes the effects of transverse shear deformation. The model takes into account thermal and moisture induced strains, transverse pressures acting on the sublaminate, and contact between the sublaminate and plate. The solution technique used is the Ritz method. A computationally efficient computer implementation of the model was developed. The code can be used to predict the nonlinear-load-strain behavior of the sublaminate including the buckling load, postbuckling behavior, and the onset of delamination growth. The accuracy of the code was evaluated by comparing the model results to benchmark analytical solutions. A series of experiments was conducted on Fiberite T300/976 graphite/epoxy laminates bonded to an aluminum honeycomb core forming a sandwich panel. Either circles or ellipses made from Teflon film were embedded in the laminates, simulating the presence of a delamination. Each specimen was loaded in compression and the strain history of the sublaminate was recorded far into the postbuckling regime. The extent of delamination growth was evaluated by C-scan examination of each specimen. The experimental data were compared to code predictions. The code was found to describe the data with reasonable accuracy. A sensitivity study examined the relative importance of various material properties, the delamination dimensions, the contact model, the transverse pressure differential, the critical strain energy release rate, and the relative growth direction on the buckling load, the postbuckling behavior, and the growth load of the sublaminate.

Peck, Scott O.↗

NASA NTRS1992

Minimization of vibration in elastic beams with time-variant boundary conditions

This paper presents an innovative method for minimizing the vibration of structures with time-variant boundary conditions (supports). The elastic body is modeled in two ways: (1) the first model is a letter seven type beam with a movable mass not to exceed the lower tip; (2) the second model has an arm that is a hollow beam with an inside mass with adjustable position. The complete solutions to both problems are carried out where the body is undergoing large rotation. The quasi-static procedure is used for the time-variant boundary conditions. The method developed employs partial differential equations governing the motion of the beam, including the effects of rigid-body motion, time-variant boundary conditions, and calculus of variations. The analytical solution is developed using Laplace and Fourier transforms. Examples of elastic robotic arms are given to illustrate the effectiveness of the methods developed.

Amirouche, F. M. L.↗