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NASA NTRS1990

Mathematical theory of a relaxed design problem in structural optimization

Various attempts have been made to construct a rigorous mathematical theory of optimization for size, shape, and topology (i.e. layout) of an elastic structure. If these are represented by a finite number of parametric functions, as Armand described, it is possible to construct an existence theory of the optimum design using compactness argument in a finite dimensional design space or a closed admissible set of a finite dimensional design space. However, if the admissible design set is a subset of non-reflexive Banach space such as L(sup infinity)(Omega), construction of the existence theory of the optimum design becomes suddenly difficult and requires to extend (i.e. generalize) the design problem to much more wider class of design that is compatible to mechanics of structures in the sense of variational principle. Starting from the study by Cheng and Olhoff, Lurie, Cherkaev, and Fedorov introduced a new concept of convergence of design variables in a generalized sense and construct the 'G-Closure' theory of an extended (relaxed) optimum design problem. A similar attempt, but independent in large extent, can also be found in Kohn and Strang in which the shape and topology optimization problem is relaxed to allow to use of perforated composites rather than restricting it to usual solid structures. An identical idea is also stated in Murat and Tartar using the notion of the homogenization theory. That is, introducing possibility of micro-scale perforation together with the theory of homogenization, the optimum design problem is relaxed to construct its mathematical theory. It is also noted that this type of relaxed design problem is perfectly matched to the variational principle in structural mechanics.

Kikuchi, Noboru↗

NASA NTRS1995

An Accurate ab initio Quartic Force Field and Vibrational Frequencies for CH4 and Isotopomers

A very accurate ab initio quartic force field for CH4 and its isotopomers is presented. The quartic force field was determined with the singles and doubles coupled-cluster procedure that includes a quasiperturbative estimate of the effects of connected triple excitations, CCSD(T), using the correlation consistent polarized valence triple zeta, cc-pVTZ, basis set. Improved quadratic force constants were evaluated with the correlation consistent polarized valence quadruple zeta, cc-pVQZ, basis set. Fundamental vibrational frequencies are determined using second-order perturbation theory anharmonic analyses. All fundamentals of CH4 and isotopomers for which accurate experimental values exist and for which there is not a large Fermi resonance, are predicted to within +/- 6 cm(exp -1). It is thus concluded that our predictions for the harmonic frequencies and the anharmonic constants are the most accurate estimates available. It is also shown that using cubic and quartic force constants determined with the correlation consistent polarized double zeta, cc-pVDZ, basis set in conjunction with the cc-pVQZ quadratic force constants and equilibrium geometry leads to accurate predictions for the fundamental vibrational frequencies of methane, suggesting that this approach may be a viable alternative for larger molecules. Using CCSD(T), core correlation is found to reduce the CH4 r(e), by 0.0015 A. Our best estimate for r, is 1.0862 +/- 0.0005 A.

Lee, Timothy J.↗

NASA NTRS2010

Remembrance of Things Future: Prospective Memory in Laboratory, Workplace, and Everyday Settings

In this review, oriented to the human factors community, I will summarize and provide a perspective on recent research and theory on prospective memory. This will not be an exhaustive review of literature, which is already available in two excellent recent books that provide a wealth of detail on the current state of experimental research (Kliegel, McCaniel, & Einstein, 2008; McDaniel & Einstein, 2007; also see Brandimonte, Einstein, & McDaniel, 1996, for a still relevant overview of the field as it was emerging). Rather, I will explore the limits of existing experimental paradigms and theory, Vvilich, in my opinion, fail to capture some critical aspects of performance outside the laboratory. I will also review the relatively few studies in workplace and everyday settings and will discuss several studies that attempt to bridge between the bulk of experimental studies and these few naturalistic studies. Finally, I will describe countermeasures that can reduce vulnerability to forgetting to perform intended tasks, and I will propose a research agenda that would extend existing experimental and theoretical approaches and would support human factors practitioners by generating information on a wide range of issues relevant to prospective memory performance in natural settings.

Dismukes, R. Key↗

NASA NTRS1988

Parameter estimation in nonlinear distributed systems - Approximation theory and convergence results

An abstract approximation framework and convergence theory is described for Galerkin approximations applied to inverse problems involving nonlinear distributed parameter systems. Parameter estimation problems are considered and formulated as the minimization of a least-squares-like performance index over a compact admissible parameter set subject to state constraints given by an inhomogeneous nonlinear distributed system. The theory applies to systems whose dynamics can be described by either time-independent or nonstationary strongly maximal monotonic operators defined on a reflexive Banach space which is densely and continuously embedded in a Hilbert space. It is demonstrated that if readily verifiable conditions on the system's dependence on the unknown parameters are satisfied, and the usual Galerkin approximation assumption holds, then solutions to the approximating problems exist and approximate a solution to the original infinite-dimensional identification problem.

Banks, H. T.↗

NASA NTRS1985

A new theory for multistep discretizations of stiff ordinary differential equations: Stability with large step sizes

A large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one is considered. A small parameter epsilon characterizes the stiffness of these systems. A system of o.d.e.s. in this set is approximated by a general class of multistep discretizations which includes both one-leg and linear multistep methods. Sufficient conditions are determined under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of o.d.e.s., when the step size resolves the slow time scale, but not the fast one. This property is called stability with large step sizes. The theory presented lets one compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff o.d.e.s. In particular, it is shown that one-leg methods have better stability properties with large step sizes than their linear multistep counter parts. The theory also allows one to relate the concept of D-stability to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes.

Majda, G.↗

NASA NTRS1985

Statistical theory of cubic Langmuir turbulence

The cubic direct interaction approximation is applied to a truncated (in Fourier space) version of the cubically nonlinear Schroedinger equation model of Langmuir physics. The results are compared (in the three-mode case) to those for an ensemble of numerical solutions of the dynamical equations with 10,000 different sets of Gaussianly distributed initial conditions. In the undriven, undamped case, the statistical theory (but not the ensemble) evolves to a state of thermal equilibrium. In the driven, damped case, the statistical theory appears to evolve to a state close to that corresponding to one of the limit cycles of the dynamical equations.

Sun, G.-Z.↗

NASA NTRS2010

An Analytical Explanation for the X-43A Flush Air Data Sensing System Pressure Mismatch Between Flight and Theory

Following the successful Mach 7 flight test of the X-43A, unexpectedly low pressures were measured by the aft set of the onboard Flush Air Data Sensing System s pressure ports. These in-flight aft port readings were significantly lower below Mach 3.5 than was predicted by theory. The same lower readings were also seen in the Mach 10 flight of the X-43A and in wind-tunnel data. The pre-flight predictions were developed based on 2-dimensional wedge flow, which fails to predict some of the significant 3-dimensional flow features in this geometry at lower Mach numbers. Using Volterra s solution to the wave equation as a starting point, a three-dimensional finite wedge approximation to flow over the X-43A forebody is presented. The surface pressures from this approximation compare favorably with the measured wind tunnel and flight data at speeds of Mach 2.5 and 3.

Ellsworth, Joel C.↗

NASA NTRS2001

Impact of Functionally Graded Cylinders: Theory

This final report summarizes the work funded under the Grant NAG3-2411 during the 04/05/2000-04/04/2001 period. The objective of this one-year project was to generalize the theoretical framework of the two-dimensional higher-order theory for the analysis of cylindrical functionally graded materials/structural components employed in advanced aircraft engines developed under past NASA Glenn funding. The completed generalization significantly broadens the theory's range of applicability through the incorporation of dynamic impact loading capability into its framework. Thus, it makes possible the assessment of the effect of damage due to fuel impurities, or the presence of submicron-level debris, on the life of functionally graded structural components. Applications involving advanced turbine blades and structural components for the reusable-launch vehicle (RLV) currently under development will benefit from the completed work. The theory's predictive capability is demonstrated through a numerical simulation of a one-dimensional wave propagation set up by an impulse load in a layered half-plane. Full benefit of the completed generalization of the higher-order theory described in this report will be realized upon the development of a related computer code.

Aboudi, Jacob↗

NASA NTRS1971

Algebraic theory of dimensional analysis.

An algorithm has been developed to calculate a complete optimized set of dimensionless products of the variables associated with a physical phenomenon from any complete set of dimensionless products. The algorithm is nonexhaustive and is readily adaptable for a digital computer. Illustrative examples are also given.

Chen, W.-K.↗

NASA NTRS1991

Fuzzy set methods for object recognition in space applications

During the reporting period, the development of the theory and application of methodologies for decision making under uncertainty was addressed. Two subreports are included; the first on properties of general hybrid operators, while the second considers some new research on generalized threshold logic units. In the first part, the properties of the additive gamma-model, where the intersection part is first considered to be the product of the input values and the union part is obtained by an extension of De Morgan's law to fuzzy sets, is explored. Then the Yager's class of union and intersection is used in the additive gamma-model. The inputs are weighted to some power that represents their importance and thus their contribution to the compensation process. In the second part, the extension of binary logic synthesis methods to multiple valued logic synthesis methods to enable the synthesis of decision networks when the input/output variables are not binary is discussed.

Keller, James M.↗

NASA NTRS1997

Positive Anharmonicities: The Oxonide Anion as an Example

An accurate ab initio quartic force field for the ozonide anion has been determined at the singles and doubles coupled-cluster level of theory that includes a perturbational estimate of the effects of connected triple excitations, denoted CCSD(T), using the augmented valence triple-zeta correlation consistent one-particle basis set. Convergence of the harmonic frequencies with respect to the one-particle basis set has been demonstrated by computing quadratic force fields at the CCSD(T) level using augmented valence double-zeta and quadruple-zeta basis sets. Fundamental vibrational frequencies have been determined via second-order ro-vibrational perturbation theory and also using exact variational methods. Agreement is very good, and they both show that the antisymmetric O-O stretch, v 3, possesses a positive anharmonic correction (that is, the fundamental frequency is predicted to be higher in energy than the harmonic frequency). Comparison of the O_3 and O3 quartic force fields shows that the positive anharmonic correction is the result of a particularly large F3333 symmetry internal coordinate force constant. The reasonableness of this force constant has been tested by computing both F33 and F3333 at the CCSD, CCSD(T), and CASPT2 levels of theory. A discussion of known positive anharmonicities for stretching modes is presented.

Lee, Timothy J.↗

NASA NTRS1991

The sensitivity of fracture location distribution in brittle materials

The Weibull weak-link theory allows for the computation of distribution functions for both the fracture location and the applied far-field stress. Several authors have suggested using the fracture location information from tests to infer Weibull parameters, and others have used the predictive capabilities of the theory to calculate average fracture locations for brittle bodies. By a simple set of example calculations, it is shown that the fracture location distribution function is distinctly more sensitive to perturbations in the stress state than the fracture stress distribution function is. In general, the average fracture location is more subject to stress perturbations than the average fracture stress. The results indicate that care must be exercised in applying fracture location theory.

Wetherhold, Robert C.↗

NASA NTRS1969

Geomorphology of a sand ridge.

Sand ridge origin and dynamical setting, discussing morphology-tidal current system equilibrium indicated by theory and field measurements

Smith, J. D.↗

NASA NTRS1973

A kernel function method for computing steady and oscillatory supersonic aerodynamics with interference.

The method presented uses a collocation technique with the nonplanar kernel function to solve supersonic lifting surface problems with and without interference. A set of pressure functions are developed based on conical flow theory solutions which account for discontinuities in the supersonic pressure distributions. These functions permit faster solution convergence than is possible with conventional supersonic pressure functions. An improper integral of a 3/2 power singularity along the Mach hyperbola of the nonplanar supersonic kernel function is described and treated. The method is compared with other theories and experiment for a variety of cases.

Cunningham, A. M., Jr.↗

NASA NTRS1975

A formalism for cosmic ray propagation studies

The continuity equation for cosmic ray propagation is used to derive a set of linear equations interrelating the fluxes of multiply charged nuclei as observed at any particular part of the galaxy. The derivation leads to model independent definitions for cosmic ray storage time, mean density of target nuclei and effective mass traversed. The set of equations form a common framework for comparisons of theories and observations. As an illustration, it is shown that there exists a large class of propagation models which give the same result as the exponential path length model. The formalism is shown to accommodate dynamic as well as equilibrium models of production and propagation.

Golden, R. L.↗

NASA NTRS1976

A quasi-linear kinetic equation for cosmic rays in the interplanetary medium

A kinetic equation for interplanetary cosmic rays is set up with the aid of weak-plasma-turbulence theory for an idealized radially symmetric model of the interplanetary magnetic field. As a starting point, this treatment invokes the Vlasov equation instead of the traditional Fokker-Planck equation. Quasi-linear theory is applied to obtain a momentum diffusion equation for the heliocentric frame of reference which describes the interaction of cosmic rays with convecting magnetic irregularities in the solar-wind plasma. Under restricted conditions, the well-known equation of solar modulation can be obtained from this kinetic equation.

Luhmann, J. G.↗

NASA NTRS1983

An upper bound on the radius of a highly electrically conducting lunar core

Parker's (1980) nonlinear inverse theory for the electromagnetic sounding problem is converted to a form suitable for analysis of lunar day-side transfer function data by: (1) transforming the solution in plane geometry to that in spherical geometry; and (2) transforming the theoretical lunar transfer function in the dipole limit to an apparent resistivity function. The theory is applied to the revised lunar transfer function data set of Hood et al. (1982), which extends in frequency from 10 to the -5th to 10 to the -3rd Hz. On the assumption that an iron-rich lunar core, whether molten or solid, can be represented by a perfect conductor at the minimum sampled frequency, an upper bound of 435 km on the maximum radius of such a core is calculated. This bound is somewhat larger than values of 360-375 km previously estimated from the same data set via forward model calculations because the prior work did not consider all possible mantle conductivity functions.

Hobbs, B. A.↗

NASA NTRS1986

Microwave spectrum and structure of bicyclo /1.1.1/ pentanone and comparison with several theoretical structures

The microwave spectra of five isotopic species of bicyclo (1.1.1) pentanone have been investigated. The rotational constants along with various centrifugal distortion constants for each species have been determined. From the rotational constants, a complete r(s) structure has been determined for the heavy atoms. Analysis of Stark effect measurements has shown the dipole moment to be along the a principal inertial axis with a magnitude of 3.164 (5) D. These results are compared with those obtained by four current theoretical methods: molecular mechanics (MM2), MNDO, and Hartree-Fock ab initio theory with STO-3G and 3-21G basis sets.

Mcrae, Glenn A.↗