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A Solution to the Fundamental Linear Fractional Order Differential Equation

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

Hartley, Tom T.

Determination of a natural basis function set on thick, structured slabs using Prony's method

The calculation of currents on a thick, structured slab such as a thick slab of honeycomb is discussed. Unfortunately, for certain applications, the slab can be on the order of several wavelengths, so that the straightforward application of the method of moments using subdomain basis functions is too expensive. The authors discuss how to apply Prony's method to the currents calculated for a thin, structured slab to obtain a natural set of basis functions to represent the currents in the interior of a thick slab. Prony's method represents the currents as a series of complex exponential functions. The thick slab problem is then solved by the method of moments using subdomain basis functions near the slab interfaces and one or two of the complex exponentials as basis functions within the slab.

Jorgenson, Roy E.

Multioutput Convolutional Neural Network for Improved Parameter Extraction in Time-Resolved Electrostatic Force Microscopy Data

Time-resolved scanning probe microscopy methods, like time-resolved electrostatic force microscopy (trEFM), enable imaging of dynamic processes ranging from ion motion in batteries to electronic dynamics in microstructured thin film semiconductors for solar cells. Reconstructing the underlying physical dynamics from these techniques can be challenging due to the interplay of cantilever physics with the actual transient kinetics of interest in the resulting signal. Previously, quantitative trEFM used empirical calibration of the cantilever or feed-forward neural networks trained on simulated data to extract the physical dynamics of interest. Both these approaches are limited by interpreting the underlying signal as a single exponential function, which serves as an approximation but does not adequately reflect many realistic systems. Here, we present a multi-branched, multi-output convolutional neural network (CNN) that uses the trEFM signal in addition to the physical cantilever parameters as input. The trained CNN accurately extracts parameters describing both single-exponential and bi-exponential underlying functions, and more accurately reconstructs real experimental data in the presence of noise. This article demonstrates an application of physics-informed machine learning to complex signal processing tasks, enabling more efficient and accurate analysis of trEFM.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

Ocean feature recognition using genetic algorithms with fuzzy fitness functions (GA/F3)

A model for genetic algorithms with semantic nets is derived for which the relationships between concepts is depicted as a semantic net. An organism represents the manner in which objects in a scene are attached to concepts in the net. Predicates between object pairs are continuous valued truth functions in the form of an inverse exponential function (e sub beta lxl). 1:n relationships are combined via the fuzzy OR (Max (...)). Finally, predicates between pairs of concepts are resolved by taking the average of the combined predicate values of the objects attached to the concept at the tail of the arc representing the predicate in the semantic net. The method is illustrated by applying it to the identification of oceanic features in the North Atlantic.

Ankenbrandt, C. A.

An approximate solution for interlaminar stresses in laminated composites: Applied mechanics program

An approximate solution for interlaminar stresses in finite width, laminated composites subjected to uniform extensional, and bending loads is presented. The solution is based upon the principle of minimum complementary energy and an assumed, statically admissible stress state, derived by considering local material mismatch effects and global equilibrium requirements. The stresses in each layer are approximated by polynomial functions of the thickness coordinate, multiplied by combinations of exponential functions of the in-plane coordinate, expressed in terms of fourteen unknown decay parameters. Imposing the stationary condition of the laminate complementary energy with respect to the unknown variables yields a system of fourteen non-linear algebraic equations for the parameters. Newton's method is implemented to solve this system. Once the parameters are known, the stresses can be easily determined at any point in the laminate. Results are presented for through-thickness and interlaminar stress distributions for angle-ply, cross-ply (symmetric and unsymmetric laminates), and quasi-isotropic laminates subjected to uniform extension and bending. It is shown that the solution compares well with existing finite element solutions and represents an improved approximate solution for interlaminar stresses, primarily at interfaces where global equilibrium is satisfied by the in-plane stresses, but large local mismatch in properties requires the presence of interlaminar stresses.

Rose, Cheryl A.

A new algorithm for the integration of exponential and logarithmic functions

An algorithm for symbolic integration of functions built up from the rational functions by repeatedly applying either the exponential or logarithm functions is discussed. This algorithm does not require polynomial factorization nor partial fraction decomposition and requires solutions of linear systems with only a small number of unknowns. It is proven that if this algorithm is applied to rational functions over the integers, a computing time bound for the algorithm can be obtained which is a polynomial in a bound on the integer length of the coefficients, and in the degrees of the numerator and denominator of the rational function involved.

Rothstein, M.

Modeling of Response of Thermoplastic Composites Under High Energy Impact

This work investigates the response of thermoplastic composites under impact loading. Panels fabricated with the material system under consideration – carbon fiber reinforced low melt semi-crystalline resin TC1225 LMPAEK T700G (T700/LMPAEK) – were subject to impact with an Aluminum projectile, at low and high velocities. At lower velocities, the projectile rebounded from the panel after impact and caused damage e.g. delaminations. Finite element models were developed to model this test using LS-DYNA®, with material model MAT299, a continuum damage model based on the LaRC04 criteria, to capture the dynamic response and damage modes in the T700/LMPAEK panels. Interlaminar delaminations were modeled using the cohesive contact formulation available within LS-DYNA. Previous work by the authors showed that delaminations were consistently underpredicted in size, relative to the test. It was surmised that this was due to effects of strain rate on intralaminar and interlaminar response of the T700/LMPAEK material system. Therefore, the current work is focused on incorporating strain rate effects in modeling the interlaminar response. Rate dependence was incorporated using ‘one-way tie-break contact’ in LS-DYNA, wherein, the Mode-II fracture toughness of the LMPAEK resin, G IIc , was assumed to be an exponential function of the strain rate. In particular, G IIc , was modeled to exponentially decrease at higher strain rates. Models with this form of rate dependence favorably correlated with experimental data regarding delamination extent and rebound velocity, relative to prior models without strain rate effects.

Composite Materials

Rain cell size statistics derived from radar observations at Wallops Island, Virginia

An investigation of two-dimensional rain cell size statistics has analyzed regression relations relating radar-determined rain rates to disdrometer data. This has yielded least-squares fits of radar reflectivity factors, and the application of a contouring program has generated 22,000 contours in which each isopleth belongs to predefined rain-rate intervals. An abundance of total and cell contours were observed belonging to all rain-rate categories. Both the computed number distributions and the conditional cumulative distributions as a function of contour diameter were found to be represented with good approximation by given exponential functions.

Goldhirsh, J.

Quantifying Leak Rates and Vacuum Impact on Granular Soil Column Pneumatic Seal

Introduction: All In-Situ Resource Utilization (ISRU) regolith reactor systems that aim to extract gases or vapors must maintain a seal along the flow path of the regolith from the inlet hopper to where extracted gases or vapors are generated. Using a column of regolith at the inlet has been proposed as an alternative to systems using series of mechanical valves to reduce mass and complexity. Preliminary investigations into the use of regolith itself as a barrier to escaping fluids have been conducted, which this research seeks to expand upon. Different vertical regolith simulant plug diameters and heights were tested under vacuum with dry nitrogen to observe leak rates and approximate steady state pressure gradient equilibria. The effect of absolute ambient pressure was also studied by repeating tests at incremental pressures ranging from 10e-5 Torr up to 1 atm to corroborate ambient bench tests from previous work. The impact of regolith dynamics was addressed to emulate continuous-feed regolith reaction processes by measuring leak rates through continuously draining simulant columns. OB-1 lunar highlands regolith simulant was used. Experimental Setup: Regolith simulant columns were constructed using a motorized butterfly valve separating upper acrylic tubes of various diameters from a larger, lower aluminum pipe plenum (See Figure 1). The butterfly valve was cracked open such that gas could freely pass through whereas regolith would bridge and stay suspended as a cohesive plug above in the acrylic portion. These assemblies were loaded into a vacuum chamber, and a compressed dry nitrogen line was routed into the lower plenum. Pressure transducers measured the absolute pressure of the lower plenum, which was compared to absolute chamber pressure to monitor the pressure gradient across the regolith simulant. A precision mass flow controller enabled fine control of plenum pressure. The plenum pressure was quasi-statically increased to set points informed by previous work [1] before shutting off the gas supply to allow the now pressurized gas volume to dissipate through the simulant into the chamber in what was referred to as a ‘leak test.’ In a separate set of tests, the pressure was allowed to quasi-statically build until visual disturbances in the soil were observed via remote camera feeds. Initial leak rate tests were repeated on the disturbed soil to observe changes in leak rates between initial as-poured and post-bubbling soil conditions. This test schedule was applied to simulant columns of three different diameters (approximately 2, 4, and 6 inches) as well as two different heights (around 8 and 16 inches) and was repeated at least three times for each configuration. Plenums were also pressurized while the butterfly valves were opened slightly further to facilitate slow, controlled, constant flowrate draining of the simulant to observe the impact that continuous-feed systems might have on leak rate through the regolith column. Finally, the regolith columns were reset and refilled and static leak rate tests were repeated approximately every decade in ambient pressure from 10e-5 Torr up to ambient conditions to observe the impact of transitioning out of the molecular flow regime on regolith column seal performance. Analysis: Trends with respect to each factor examined were constructed and reported alongside commentary on the most likely responsible physical phenomena. Confidence intervals were applied to the predictive findings of the data by quantifying instrumentation error and accounting for probabilistic impacts of curve fitting the generic exponential function onto pressure decay data. The leak rate is assumed directly proportional to the exponential term coefficient, while the steady-state static pressure gradient is assumed to be the asymptotic limit of each curve fit. Finally, a generic model developed by Ogino et al. [2] and later tuned for ambient testing was re-tuned using vacuum data to estimate leak rates on the lunar surface. Results: Pressure decay rates decreased up to 27-fold under vacuum when compared to ambient pressure, as shown in Figure 2. Simulant column diameter was found to exponentially correlate to both regolith column leak rate and steady-state equilibrium pressure delta. Height was found to be linearly related to leak rate –doubling the regolith column height consistently yielded around a 40% reduction in leak rate. Dynamic regolith draining was found to slightly increase leak rate, although to a lower degree than expected. Conclusion: Because of the transition into molecular gas flow through regolith, vacuum testing is required to properly estimate performance. Regolith columns should be designed as narrow as possible to avoid bridging and minimize leak rate, and as tall as is feasible to both further reduce leak rate and to help prevent bubbling, which has been shown to degrade the quality of the regolith seal. Disrupted soil was found to exhibit higher leak rates than pristine, as-poured soil, indicating the need to consider the impact of recovery from depressurization events. Regolith columns appear equally feasible for both dosed and continuous-feed reactors.

ISRU

Exponential approximations in optimal design

One-point and two-point exponential functions have been developed and proved to be very effective approximations of structural response. The exponential has been compared to the linear, reciprocal and quadratic fit methods. Four test problems in structural analysis have been selected. The use of such approximations is attractive in structural optimization to reduce the numbers of exact analyses which involve computationally expensive finite element analysis.

Belegundu, A. D.

Dynamical influences on the moment of inertia tensor from lateral viscosity variations inferred from seismic tomographic models

We have investigated the influences of lateral variations of viscosity on the moment of inertia tensor from viscous flows due to the density anomalies in the mantle inferred from seismic tomographic models. The scaling relations between the density and the seismic anomalies is taken as either a constant or a function increasing with depth in accord with the recent high-pressure experimental studies. The viscosity is taken as an exponential function of the 3D density anomaly. In models with an isoviscous background, the effects on the perturbed moment of inertia tensor from the lateral viscosity variations are smaller than those due to variations in the radial viscosity profiles. In mantle models with a background viscosity increasing with depth, the influences of the lateral viscosity variations are significant. The most striking feature in the latter case is that the two off-diagonal elements delta I(sub xz) and delta I(sub yz) in the inertia tensor exhibit greatest sensitivity to lateral variations of the viscosity. While the other elements of the inertia change by only about a few tens of percent in the range of lateral viscosity contrast considered (less than 300), delta I(sub xz) and delta I(sub yz) can vary up to 40 times even with a change in sign, depending on the radial viscosity stratification and the location of the strongest lateral variations. The increase in the velocity-density scaling relation with depth can reduce the influences of the lateral viscosity variations, but it does not change the overall sensitive nature of delta I(sub xz) and delta I(sub yz). This study demonstrates clearly that the lateral viscosity variations, especially in the upper mantle, must be considered in the determination of long-term polar wander, since the variations in the delta I(sub xz) and delta I(sub yz) terms are directly responsible for exciting rotational movements.

Zhang, Shuxia

An accurate method for evaluating the kernel of the integral equation relating lift to downwash in unsteady potential flow

The method is capable of generating approximations of arbitrary accuracy. It is based on approximating the algebraic part of the nonelementary integrals in the kernel by exponential functions and then integrating termwise. The exponent spacing in the approximation is a geometric sequence. The coefficients and exponent multiplier of the exponential approximation are computed by least squares so the method is completely automated. Exponential approximates generated in this manner are two orders of magnitude more accurate than the exponential approximation that is currently most often used for this purpose. The method can be used to generate approximations to attain any desired trade-off between accuracy and computing cost.

Desmarais, R. N.

Multiple scattering theory of radiative transfer in inhomogeneous atmospheres.

In this paper we treat the multiple scattering theory of radiative transfer in plane-parallel inhomogeneous atmospheres. The treatment presented here may be adopted to model atmospheres characterized by an optical depth dependent coherent scattering phase function. For the purpose of illustration we consider the semi-infinite medium in which the absorption property of the atmosphere is characterized by an exponential function. The methodology employed here is the extension of the case treated previously by the author for homogeneous atmospheres.

Kanal, M.

The BMW analytic aerodynamic drag method for the Vinti satellite theory

In order to retain separability in the Vinti theory of earth satellite motion when a non conservative force such as air drag is considered, a set of variational equations for the orbital elements are introduced, and expressed as functions of the transverse, radial, and normal components of the nonconservative forces acting on the system. In particular, the atmospheric density profile is written as a fitted exponential function of the eccentric anomaly, which reproduces tabular values of static model atmospheric densities at all altitudes to within ninety-eight percent and simultaneously reduces the variational equations to indefinite integrals with closed form evaluations, whose limits are in terms of the eccentric anomaly. The values of the limits for any arbitrary time interval are obtained from the Vinti program. Results of the BMW (Bonavito, Mistretta, Watson) theory for the case of the intense air drag satellites San Marco-2 and Air Force Cannonball are given. These results indicate that the satellite ephemerides produced by the BMW theory in conjunction with the Vinti program are of very high accuracy. In addition, since the program is entirely analytic, several months of ephemerides can be obtained within a few seconds of computer time.

Watson, J. S.

Ultraviolet radiation effects on the infrared damage rate of a thermal control coating

The effects of ultraviolet radiation on the infrared reflectance of ZnO silicone white thermal coatings were investigated. Narrow band ultraviolet radiation for wavelengths in the 2200A to 3500A range by a monochromator and a high pressure, 150-W Eimac xenon lamp. The sample was irradiated while in a vacuum of at least 0.000001 torr, and infrared reflectance was measured in situ with a spectroreflectometer at 19,500A. Reflectance degradation was studied as a function of wavelength, time, intensity, and dose. Damage was wavelength dependent at constant exposure, but no maximum was evident above the shortest wavelength investigated here. The degradation rate at constant intensity was an exponential function of time and varies with intensity.

Bass, J. A.

Development of a winter wheat adjustable crop calendar model

The author has identified the following significant results. Least squares techniques were applied for parameter estimation of functions to predict winter wheat phenological stage with daily maximum temperature, minimum temperature, daylength, and precipitation as independent variables. After parameter estimation, tests were conducted using independent data. It may generally be concluded that exponential functions have little advantage over polynomials. Precipitation was not found to significantly affect the fits. The Robertson triquadratic form, in general use for spring wheat, yielded good results, but special techniques and care are required. In most instances, equations with nonlinear effects were found to yield erratic results when utilized with averaged daily environmental values as independent variables.

Baker, J. R.

The Oscillating Circular Airfoil on the Basis of Potential Theory

Proceeding from the thesis by W. Kinner the present report treats the problem of the circular airfoil in uniform airflow executing small oscillations, the amplitudes of which correspond to whole functions of the second degree in x and y. The pressure distribution is secured by means of Prandtl's acceleration potential. It results in a system of linear equations the coefficients of which can be calculated exactly with the aid of exponential functions and Hankel's functions. The equations necessary are derived in part I; the numerical calculation follows in part II.

Schade, T.

Multispectral Emission of the Sun during the First Whole Sun Month: Magnetohydrodynamic Simulations

We demonstrate that a three-dimensional magnetohydrodynamic (MHD) simulation of the corona can model its global plasma density and temperature structure with sufficient accuracy to reproduce many of the multispectral properties of the corona observed in extreme ultraviolet (EW) and X-ray emission. The key ingredient to this new type of global MHD model is the inclusion of energy transport processes (coronal heating, anisotropic thermal conduction, and radiative losses) in the energy equation. The calculation of these processes has previously been confined to one-dimensional loop models, idealized two-dimensional computations, and three-dimensional active region models. We refer to this as the thermodynamic MHD model, and we apply it to the time period of Carrington rotation 1913 (1996 August 22 to September 18). The form of the coronal heating term strongly affects the plasma density and temperature of the solutions. We perform our calculation for three different empirical heating models: (1) a heating function exponentially decreasing in radius; (2) the model of Schrijver et al.; and (3) a model reproducing the heating properties of the quiet Sun and active regions. We produce synthetic emission images from the density and temperature calculated with these three heating functions and quantitatively compare them with observations from E W Imaging Telescope on the Solar and Heliospheric Observatory and the soft X-ray telescope on Yohkoh. Although none of the heating models provide a perfect match, heating models 2 and 3 provide a reasonable match to the observations.

Lionello, Roberto