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NASA NTRS1995

The Schwinger Variational Method

Variational methods have proven invaluable in theoretical physics and chemistry, both for bound state problems and for the study of collision phenomena. The application of the Schwinger variational (SV) method to e-molecule collisions and molecular photoionization has been reviewed previously. The present chapter discusses the implementation of the SV method as applied to e-molecule collisions. Since this is not a review of cross section data, cross sections are presented only to server as illustrative examples. In the SV method, the correct boundary condition is automatically incorporated through the use of Green's function. Thus SV calculations can employ basis functions with arbitrary boundary conditions. The iterative Schwinger method has been used extensively to study molecular photoionization. For e-molecule collisions, it is used at the static exchange level to study elastic scattering and coupled with the distorted wave approximation to study electronically inelastic scattering.

Huo, Winifred M.↗
NASA NTRS1995

The Schwinger Variational Method

Variational methods have proven invaluable in theoretical physics and chemistry, both for bound state problems and for the study of collision phenomena. For collisional problems they can be grouped into two types: those based on the Schroedinger equation and those based on the Lippmann-Schwinger equation. The application of the Schwinger variational (SV) method to e-molecule collisions and photoionization has been reviewed previously. The present chapter discusses the implementation of the SV method as applied to e-molecule collisions.

Huo, Winifred M.↗
NASA NTRS1986

Assimilation of scatterometer winds into surface pressure fields using a variational method

A variational formulation was used to assimilate Seasat-A scatterometer (SASS) surface wind measurements near and during a severe storm in the North Atlantic into conventional National Meteorological Center sea level pressure fields. An estimate of the relative vorticity at every point on a grid was calculated using each of these two data sets. A solution to a modified geostrophic stream function is found subject to the constraints that (1) the relative vorticities calculated from the data agree as closely as possible with the relative vorticities from the variational solution, and that (2) the average kinetic energy is a minimum. Results are obtained which support the idea that averaged satellite data can be treated as synoptic data. Direct substitution rather than a time-weighted insertion made from SASS winds generally resulted in more accurate pressure analyses. In addition, this relatively simple model provides surface pressure fields which agree extremely well with surface truth and the results of other investigators who required additional sources of input data into more complex models. It will be possible to obtain improved wind field maps from future scatterometer pressure fields in mid-latitudes.

Harlan, J., Jr.↗
NASA NTRS1996

Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Ibrahim, A. H.↗
NASA NTRS1997

Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Ibrahim, A. H.↗
NASA NTRS1960

Application of Variational Methods to the Thermal Entrance Region of Ducts

A variational method is presented for solving eigenvalue problems which arise in connection with the analysis of convective heat transfer in the thermal entrance region of ducts. Consideration is given, to both situations where the temperature profile depends upon one cross-sectional coordinate (e.g. circular tube) or upon two cross-sectional coordinates (e.g. rectangular duct). The variational method is illustrated and verified by application to laminar heat transfer in a circular tube and a parallel-plate channel, and good agreement with existing numerical solutions is attained. Then, application is made to laminar heat transfer in a square duct as a check, an alternate computation for the square duct is made using a method indicated by Misaps and Pohihausen. The variational method can, in principle, also be applied to problems in turbulent heat transfer.

Sparrow, E. M.↗
NASA NTRS1959

APPLICATION OF VARIATIONAL METHODS TO RADIATION HEAT-TRANSFER CALCULATIONS

A variational method is presented for solving a class of integral equations which arise in radiation heat-transfer problems. First, to demonstrate the formulation of radiation problems in terms of integral equations, consideration is given to a system consisting of two nonblack, finite, parallel plates. After a general description of the variational method, its use is illustrated by application to the parallel-plate system. Comparisons are made which show very good agreement with exact solutions.

E. M. Sparrow↗