A proposal concerning terminology in reduced density matrix theory
Proposed terminology for reduced density matrix theory
Harriman, J. E.↗
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Semiclassical S-matrix theory of vibrationally inelastic collisions between two diatomic molecules
We derive a semiclassical S matrix for vibrationally inelastic collisions between two diatomic molecules, assuming a collinear geometry. Our theory incorporates a quantum mechanical superposition principle with classical dynamics and, as such, is an extension of the atom-diatomic molecule theory of Miller. The several approximations to the S matrix differ in the complexity with which the interference between various classical trajectories is treated. We report numerical calculations for H2-D2 and D2-D2 collisions based on two different interaction potentials. The cruder approximations yield transition probabilities which agree with exact quantum mechanical results to within a factor of 2. More sophisticated approximations to the S matrix yield excellent quantitative agreement with the quantum calculations.
Cohen, S. C.↗
A T-matrix theory of galactic heavy-ion fragmentation
The theory of galactic heavy ion fragmentation is furthered by incorporating a T matrix approach into the description of the three step process of abrasion, ablation, and final state interations. The connection between this T matrix and the interaction potential is derived. For resonant states, the substitution of complex energies for real energies in the transition rate is formerly justified for up to third order processes. The previously developed abrasion-ablation fragmentation theory is rederived from first principles and is shown to result from time ordering, classical probability, and zero width resonance approximations. Improvements in the accuracy of the total fragmentation cross sections require an alternative to the latter two approximations. A Lorentz invariant differential abrasion-ablation cross section is derived which explicitly includes the previously derived abrasion total cross sections. It is demonstrated that spectral and angular distributions can be obtained from the general Lorentz invariant form.
Norbury, J. W.↗
Matrix Theory for Data Association in PVS
Consider a collection of data generated by sensors from a set of aircraft. Data association is the process of connecting each sensor measurement with its corresponding aircraft. Furthermore once the data association has taken place, the state of the aircraft can be approximated using a Kalman filter. This talk aims to explore formal specification and verification of data association in the Prototype Verification System (PVS). Formal specification and verification of data association includes development of Kalman filters, Mahalanobis distance, and other topics of matrix analysis in PVS.
Linear Algebra↗
Theorems on symmetrics and flux conservation in radiative transfer using the matrix operator theory
The matrix operator approach to radiative transfer is shown to be a very powerful technique in establishing symmetry relations for multiple scattering in inhomogeneous atmospheres. Symmetries are derived for the reflection and transmission operators using only the symmetry of the phase function. These results will mean large savings in computer time and storage for performing calculations for realistic planetary atmospheres using this method. The results have also been extended to establish a condition on the reflection matrix of a boundary in order to preserve reciprocity. Finally energy conservation is rigorously proven for conservative scattering in inhomogeneous atmospheres.
Kattawar, G. W.↗
Theorems on symmetries and flux conservation in radiative transfer using the matrix operator theory.
The matrix operator approach to radiative transfer is shown to be a very powerful technique in establishing symmetry relations for multiple scattering in inhomogeneous atmospheres. Symmetries are derived for the reflection and transmission operators using only the symmetry of the phase function. These results will mean large savings in computer time and storage for performing calculations for realistic planetary atmospheres using this method. The results have also been extended to establish a condition on the reflection matrix of a boundary in order to preserve reciprocity. Finally energy conservation is rigorously proven for conservative scattering in inhomogeneous atmospheres.
Kattawar, G. W.↗
Matrix operator theory of radiative transfer. II - Scattering from maritime haze.
Matrix operator theory is used to calculate the reflected and transmitted radiance of photons that have interacted with plane-parallel maritime haze layers. The results are presented for three solar zenith angles, three values of the surface albedo, and a range of optical thicknesses from very thin to very thick. The diffuse flux at the lower boundary and the cloud albedo are tabulated. The forward peak and other features in the single-scattered phase function cause the radiance in many cases to be very different from that for Rayleigh scattering. In particular, the variation of the radiance with both the zenith or nadir angle and the azimuthal angle is more marked and the relative limb darkening under very thick layers is greater for haze M than for Rayleigh scattering. The downward diffuse flux at the lower boundary for A = 0 is always greater and the cloud albedo is always less for haze M than for Rayleigh layers.
Kattawar, G. W.↗
Postbuckling analysis using a general purpose code
A new capability for solving postbuckling problems in shell structures is described. The matrix theory to adapt Newton's method to nonlinear finite element shell analysis is outlined first. The matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular. The matrix theory suggests a change of variables as part of the usual iterative procedure in Newton's method. The change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures. Numerical results from a new option that has been programmed in an existing general purpose code are presented. The analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples.
Thurston, G. A.↗
Matrix operator theory of radiative transfer. I - Rayleigh scattering.
An entirely rigorous method for the solution of the equations for radiative transfer based on the matrix operator theory is reviewed. The advantages of the present method are: (1) all orders of the reflection and transmission matrices are calculated at once; (2) layers of any thickness may be combined, so that a realistic model of the atmosphere can be developed from any arbitrary number of layers, each with different properties and thicknesses; (3) calculations can readily be made for large optical depths and with highly anisotropic phase functions; (4) results are obtained for any desired value of the surface albedo including the value unity and for a large number of polar and azimuthal angles; (5) all fundamental equations can be interpreted immediately in terms of the physical interactions appropriate to the problem; and (6) both upward and downward radiance can be calculated at interior points from relatively simple expressions.
Plass, G. N.↗
Direct calculation of second-order density matrix. 1 - Theory of the Green's function technique
Second order density matrix calculations using Green function
Simons, J.↗
Mechanical behavior of fiber reinforced SiC/RBSN ceramic matrix composites: Theory and experiment
The mechanical behavior of continuous fiber reinforced SiC/RBSN (Reaction Bonded Silicon Nitride) composites with various fiber contents is evaluated. Both catastrophic and noncatastrophic failures are observed in tensile specimens. Damage and failure mechanisms are identified via in-situ monitoring using NDE (nondestructive evaluation) techniques through the loading history. Effects of fiber/matrix interface debonding (splitting) parallel to fibers are discussed. Statistical failure behavior of fibers is also observed, especially when the interface is weak. Micromechanical models incorporating residual stresses to calculate the critical matrix cracking strength, ultimate strength, and work of pull-out are reviewed and used to predict composite response. For selected test problems, experimental measurements are compared to analytical predictions.
Chulya, Abhisak↗
Mechanical behavior of fiber reinforced SiC/RBSN ceramic matrix composites - Theory and experiment
The mechanical behavior of continuous fiber reinforced SiC/RBSN (Reaction Bonded Silicon Nitride) composites with various fiber contents is evaluated. Both catastrophic and noncatastrophic failures are observed in tensile specimens. Damage and failure mechanisms are identified via in-situ monitoring using NDE (nondestructive evaluation) techniques throughout the loading history. Effects of fiber/matrix interface debonding (splitting) parallel to fibers are discussed. Statistical failure behavior of fibers is also observed, especially when the interface is weak. Micromechanical models incorproating residual stresses to calculate the critical matrix cracking strength, ultimate strength, and work of pull-out are reviewed and used to predict composite response. For selected test problems, experimental measurements are compared to analytical predictions.
Chulya, Abhisak↗
North Alabama Lightning Mapping Array (LMA): VHF Source Retrieval Algorithm and Error Analyses
Two approaches are used to characterize how accurately the North Alabama Lightning Mapping Array (LMA) is able to locate lightning VHF sources in space and in time. The first method uses a Monte Carlo computer simulation to estimate source retrieval errors. The simulation applies a VHF source retrieval algorithm that was recently developed at the NASA Marshall Space Flight Center (MSFC) and that is similar, but not identical to, the standard New Mexico Tech retrieval algorithm. The second method uses a purely theoretical technique (i.e., chi-squared Curvature Matrix Theory) to estimate retrieval errors. Both methods assume that the LMA system has an overall rms timing error of 50 ns, but all other possible errors (e.g., multiple sources per retrieval attempt) are neglected. The detailed spatial distributions of retrieval errors are provided. Given that the two methods are completely independent of one another, it is shown that they provide remarkably similar results. However, for many source locations, the Curvature Matrix Theory produces larger altitude error estimates than the (more realistic) Monte Carlo simulation.
Koshak, W. J.↗
Electron excitation cross sections for the 2s^2 2p ^2p^o-2s2p^2 ^4p and 2s2p^2 ^2D transitions in O^3+
Comparison is made with results of new calculations in a 25-state R-matrix theory and with results in a previous eight-state R-matrix calculation. The presence of rich resonance structure is confirmed in both experiment and theory.
electron excitation electron temperature and densi↗
Flap-Lag-Torsion Stability in Forward Flight
An aeroelastic stability of three-degree flap-lag-torsion blade in forward flight is examined. Quasisteady aerodynamics with a dynamic inflow model is used. The nonlinear time dependent periodic blade response is calculated using an iterative procedure based on Floquet theory. The periodic perturbation equations are solved for stability using Floquet transition matrix theory as well as constant coefficient approximation in the fixed reference frame. Results are presented for both stiff-inplane and soft-inplane blade configurations. The effects of several parameters on blade stability are examined, including structural coupling, pitch-flap and pitch-lag coupling, torsion stiffness, steady inflow distribution, dynamic inflow, blade response solution and constant coefficient approximation.
Panda, B.↗
Flap-lag-torsion aeroelastic stability of a circulation control rotor in forward flight
The aeroelastic stability of a circulation control rotor blade undergoing three degrees of motion (flap, lag, and torsion) is investigated in forward flight. Quasi-steady strip theory is used to evaluate the aerodynamics forces; and the airfoil characteristics are from data tables. The propulsive and the auxiliary power trims are calculated from vehicle and rotor equilibrium equations through the numerical integration of element forces in azimuth as well as in radial directions. The nonlinear time dependent periodic blade response is calculated using an iterative procedure based on Floquet theory. The periodic perturbation equations are solved for stability using Floquet transition matrix theory. The effects of several parameters on blade stability are examined, including advance ratio, collective pitch, thrust level, shaft tilt, structural stiffnesses variation, and propulsive and auxiliary power trims.
Chopra, Inderjit↗
A viscoplastic constitutive theory for metal matrix composites at high temperature
A viscoplastic theory is presented for representing the high-temperature deformation behavior of metal matrix composites. The point of view taken is a continuum one where the composite is considered a material in its own right, with its own properties that can be determined for the composite as a whole. It is presumed that a single preferential (fiber) direction is identifiable at each material point (continuum element) admitting the idealization of local transverse isotropy. A key ingredient in this work is the specification of an experimental program for the complete determination of the material functions and parameters for characterizing a particular metal matrix composite. The parameters relating to the strength of anisotropy can be determined through tension/torsion tests on longitudinally and circumferentially reinforced thin-walled tubes. Fundamental aspects of the theory are explored through a geometric interpretation of some basic features analogous to those of the classical theory of plasticity.
Robinson, D. N.↗
Insensitive control technology development
THe investigation of two insensitive controller synthesis techniques was reported. The finite dimensional inverse approach produces a time varying insensitive controller and/or parameter identifier by constructing inverse functions derived from a finite number of input output pair relationships. The MD/IM concept relies on the information matrix theory that was developed in the estimation and identification field. The MD/IM synthesis technique is based on the hypothesis that minimizing the information matrix will reduce system identifiability and consequently system sensitivity to uncertain parameters. The controllers designed with both techniques were evaluated on a realistic C-5A aircraft flight control problem. Results indicate that the FDI controller is more suited to trajectory type problems because of its time varying nature. The MD/IM controller performed as well as the top-rated controllers of the initial effort and has direct application to aircraft flight control problems.
Harvey, C. A.↗