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NASA NTRS1987

Nonlinear programming extensions to rational function approximations of unsteady aerodynamics

This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.

Tiffany, Sherwood H.↗
NASA NTRS1983

Microscopic Lagrangian description of warm plasmas. IV - Macroscopic approximation

The averaged-Lagrangian method is applied to linear wave propagation and nonlinear three-wave interaction in a warm magnetoplasma, in the macroscopic approximation. The microscopic Lagrangian treated by Kim and Crawford (1977) and by Galloway and Crawford (1977) is first expanded to third order in perturbation. Velocity integration is then carried out, before applying Hamilton's principle to obtain a general description of wave propagation and coupling. The results are specialized to the case of interaction between two electron plasma waves and an Alfven wave. The method is shown to be more powerful than the alternative possibility of working from the beginning with a macroscopic Lagrangian density.

Kim, H.↗
NASA NTRS1973

The constraint method: A new finite element technique

An approch to the finite element method which utilizes families of conforming finite elements based on complete polynomials is presented. Finite element approximations based on this method converge with respect to progressively reduced element sizes as well as with respect to progressively increasing orders of approximation. Numerical results of static and dynamic applications of plates are presented to demonstrate the efficiency of the method. Comparisons are made with plate elements in NASTRAN and the high-precision plate element developed by Cowper and his co-workers. Some considerations are given to implementation of the constraint method into general purpose computer programs such as NASTRAN.

Tsai, C.↗
NASA NTRS1988

Efficient approximation concepts using second order information

The application of second derivative information for solving structural optimization problems is considered. In the present method, rather than building approximate nonlinear forms for the objective function and constraints, only linear approximations are used. A separable quadratic approximation of the Lagrangian function is included in the subproblem statement. The method has been successfully used for simple problems that can be solved in closed form, in addition to the sizing optimization of trusses, and it is shown to converge faster than the convex linearization method or the method of moving asymptotes.

Fleury, Claude↗
NASA NTRS1980

Inversion and approximation of Laplace transforms

A method of inverting Laplace transforms by using a set of orthonormal functions is reported. As a byproduct of the inversion, approximation of complicated Laplace transforms by a transform with a series of simple poles along the left half plane real axis is shown. The inversion and approximation process is simple enough to be put on a programmable hand calculator.

Lear, W. M.↗
NASA NTRS2002

Finite Element A Posteriori Error Estimation for Heat Conduction

This research investigates residual-based a posteriori error estimates for finite element approximations of heat conduction in single-layer and multi-layered materials. The finite element approximation, based upon hierarchical modelling combined with p-version finite elements, is described with specific application to a two-dimensional, steady state, heat-conduction problem. Element error indicators are determined by solving an element equation for the error with the element residual as a source, and a global error estimate in the energy norm is computed by collecting the element contributions. Numerical results of the performance of the error estimate are presented by comparisons to the actual error. Two methods are discussed and compared for approximating the element boundary flux. The equilibrated flux method provides more accurate results for estimating the error than the average flux method. The error estimation is applied to multi-layered materials with a modification to the equilibrated flux method to approximate the discontinuous flux along a boundary at the material interfaces. A directional error indicator is developed which distinguishes between the hierarchical modeling error and the finite element error. Numerical results are presented for single-layered materials which show that the directional indicators accurately determine which contribution to the total error dominates.

Lang, Christapher G.↗
NASA NTRS1932

Approximate calculation of multispar cantilever and semicantilever wings with parallel ribs under direct and indirect loading

A method is presented for approximate static calculation, which is based on the customary assumption of rigid ribs, while taking into account the systematic errors in the calculation results due to this arbitrary assumption. The procedure is given in greater detail for semicantilever and cantilever wings with polygonal spar plan form and for wings under direct loading only. The last example illustrates the advantages of the use of influence lines for such wing structures and their practical interpretation.

Sanger, Eugen↗
NASA NTRS1981

Resizing structures for minimum weight

Approximation concepts and dual-method algorithms are combined in method of minimum-weight design for structures. Approximation Concepts Code for Efficient Structural Synthesis (ACCESS3) program is powerful research tool in which mathematical programming and optimality criteria are coalesced in efficient structural weight-minimization method.

Fleury, C.↗
NASA NTRS1971

A note on laminar shear flow over impulsively started bodies.

Prediction of the shear flow around bodies impulsively set into motion at a uniform velocity. Information is presented on the local wall shear stress, velocity distribution, steady flow times, and thermal response for wedge flows where local flow acceleration occurs. The essential features of the flow field are found to be describable by the approximate series expansion method of Goldstein and Rosenhead (1936). This method would appear to be useful in rapidly calculating the viscous drag on the forward face of various shaped bodies where local flow acceleration occurs.

Back, L. H.↗
NASA NTRS1972

An implicit finite-difference solution to the viscous shock layer, including the effects of radiation and strong blowing

An implicit finite-difference scheme is developed for the fully coupled solution of the viscous, radiating stagnation-streamline equations, including strong blowing. Solutions are presented for both air injection and injection of carbon-phenolic ablation products into air at conditions near the peak radiative heating point in an earth entry trajectory from interplanetary return missions. A detailed radiative-transport code that accounts for the important radiative exchange processes for gaseous mixtures in local thermodynamic and chemical equilibrium is utilized in the study. With minimum number of assumptions for the initially unknown parameters and profile distributions, convergent solutions to the full stagnation-line equations are rapidly obtained by a method of successive approximations. Damping of selected profiles is required to aid convergence of the solutions for massive blowing. It is shown that certain finite-difference approximations to the governing differential equations stabilize and improve the solutions. Detailed comparisons are made with the numerical results of previous investigations. Results of the present study indicate lower radiative heat fluxes at the wall for carbonphenolic ablation than previously predicted.

Garrett, L. B.↗
NASA NTRS1973

Detection and Estimation of an Optical Image by Photon-Counting Techniques

Statistical description of a photoelectric detector is given. The photosensitive surface of the detector is divided into many small areas, and the moment generating function of the photo-counting statistic is derived for large time-bandwidth product. The detection of a specified optical image in the presence of the background light by using the hypothesis test is discussed. The ideal detector based on the likelihood ratio from a set of numbers of photoelectrons ejected from many small areas of the photosensitive surface is studied and compared with the threshold detector and a simple detector which is based on the likelihood ratio by counting the total number of photoelectrons from a finite area of the surface. The intensity of the image is assumed to be Gaussian distributed spatially against the uniformly distributed background light. The numerical approximation by the method of steepest descent is used, and the calculations of the reliabilities for the detectors are carried out by a digital computer.

Wang, Lily Lee↗
NASA NTRS1974

The internal structure of the convective envelope of the sun

An axisymmetric hydrodynamic model of the sun consisting of a core at radiant equilibrium and a convective envelope is studied with consideration of gravitational forces, rotation, turbulence effects, and electromagnetic forces. The dimensionless parameters that control the equations of plasma motion within the sun and the boundary conditions are investigated; estimates are given for these parameters on the basis of experimental data and the existing theoretical models. Stability conditions are derived for convective equilibrium, developed cellular convection, and developed turbulent convection with consideration of rotation and turbulent Reynolds stresses. These conditions characterize the possible physical state of the matter in the convective envelope. Using estimates of the dimensionless parameters and models of some of them, a method of successive approximations is proposed for solution of the problem as a whole.

Yavorskaya, I. M.↗
NASA NTRS1975

The effect of thermal neutron field slagging caused by cylindrical BF3 counters in diffusion media

Computations are carried out in transport approximation (first collision method) for the attenuation of the field of thermal neutrons formed in counters of the CHM-8 and CHMO-5 type. The deflection of the thermal neutron field is also obtained near the counters and in the air (shade effect) and in various decelerating media (water, paraffin, plexiglas) for which the calculations are carried out on the basis of diffusion theory. To verify the calculations, the distribution of the density of the thermal neutrons at various distances from the counter in the water is measured.

Gorshkov, G. V.↗
NASA NTRS1975

An analysis method for two-dimensional transonic viscous flow

A method for the approximate calculation of transonic flow over airfoils, including shock waves and viscous effects, is described. Numerical solutions are obtained by use of a computer program which is discussed in the appendix. The importance of including the boundary layer in the analysis is clearly demonstrated, as well as the need to improve on existing procedures near the trailing edge. Comparisons between calculations and experimental data are presented for both conventional and supercritical airfoils, emphasis being on the surface pressure distribution, and good agreement is indicated.

Bavitz, P. C.↗