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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS1962

On the Long-Period Effects in the Motion of an Artificial Satellite Caused by the Ellipticity of the Equator of the Earth

If a satellite has a mean daily motion nearly commensurable with the angular velocity of rotation of the earth, long-period perturbations will influence its motion. The eccentricity of such a commensurable orbit is not necessarily small, as the orbit of Explorer VI shows. This circumstance caused the author to develop a semianalytical method of treating these perturbations, which avoids the development of the disturbing function into powers of the eccentricity and consequently includes the cases of elongated orbits and orbits with moderate eccentricities.

MOTION↗
NASA NTRS1979

On some theoretical and practical aspects of multigrid methods

A description and explanation of a simple multigrid algorithm for solving finite element systems is given. Numerical results for an implementation are reported for a number of elliptic equations, including cases with singular coefficients and indefinite equations. The method shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.

Nicolaides, R. A.↗
NASA NTRS1980

Application of the multigrid method to grid generation

The multigrid method (MGM), used to numerically solve the pair of nonlinear elliptic equations commonly used to generate two dimensional boundary-fitted coordinate systems is discussed. Two different geometries are considered: one involving a coordinate system fitted about a circle and the other selected for an impinging jet flow problem. Two different relaxation schemes are tried: one is successive point overrelaxation and the other is a four-color scheme vectorizeable to take advantage of a parallel processor computer for greater computational speed. Results using MGM are compared with those using SOR (doing successive overrelaxations with the corresponding relaxation scheme on the fine grid only). It is found that MGM becomes significantly more effective than SOR as more accuracy is demanded and as more corrective grids, or more grid points, are used. For the accuracy required, it is found that MGM is two to three times faster than SOR in computing time. With the four-color relaxation scheme as applied to the impinging jet problem, the advantage of MGM over SOR is not as great. This may be due to the effect of a poor initial guess on MGM for this problem.

Ohring, S.↗
NASA NTRS1980

Three-layer interactive method for computing supersonic laminar separated flows

An interactive model for numerical computation of complicated two-dimensional flowfields including regions of reversed flow is proposed. The present approach is one of dividing the flowfield into three regions, in each of which a simplified mathematical model is applied: (1) outer, supersonic flow for which the full potential equation (hyperbolic) is used; (2) viscous, laminar layer in which the compressible boundary-layer model (parabolic) is used; and (3) recirculating flow modeled by the incompressible Navier-Stokes equations (elliptic). For matching of the numerical solutions in the three layers, two interaction models are developed: one for pressure interaction, the other for interaction between the shear layer and the recirculating flow. The uniform solution for the whole flowfield is then obtained by iteration of the local solutions under the constraints imposed by matching. The three-layer interactive model is used for solution of the flowfield past an asymmetric cavity. The method is shown to be capable of dealing with backflow without encountering problems at separation, characteristic to the boundary-layer approach.

Brandeis, J.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal estimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed.

Rodriguez, G.↗
NASA NTRS1983

Quasiconformal mappings and grid generation

A finite difference scheme is developed for constructing quasiconformal mappings for arbitrary simply and doubly connected regions. Computational grids are generated to reduce elliptic equations to canonical form. Examples of conformal mappings on surfaces are also included.

Mastin, C. W.↗
NASA NTRS1984

Transformation of two and three-dimensional regions by elliptic systems

The research during this period continued to expand the class of numerical algorithms that can be accurately and efficiently implemented on overlapping grids. Whereas previous calculations have been used to solve elliptic equations and to find the steady-state solution of parabolic equations, the present work is aimed towards developing time-accurate solution techniques for parabolic and hyperbolic equations. The primary difficulty here is in the correct treatment of the interior boundary nodes that must be updated at each iteration. The implementation of explicit methods is straightforward. However, the common practice of lagging these values when using an implicit methods leads to inconsistencies in the difference equation. One way to avoid this problem is to alternately calculate with an implicit and an explicit method on each subgrid. With this procedure, the explicit method generates boundary values at the next time level which are then used by the implicit step. It can be shown that when a backward implicit method is combined with a forward explicit method, the composite method is second order accurate and unconditionally stable for linear problems. A second area in which progress can be reported is in the distribution of grid points on curves and surfaces.

Mastin, E. W.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal esimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed. While this paper focuses on theoretical developments, applications arising in the area of large structure static shape determination are contained in a closely related paper (Rodriguez and Scheid, 1982).

Rodriguez, G.↗
NASA NTRS1985

Multigrid solutions to quasi-elliptic schemes

Quasi-elliptic schemes arise from central differencing or finite element discretization of elliptic systems with odd order derivatives on non-staggered grids. They are somewhat unstable and less accurate then corresponding staggered-grid schemes. When usual multigrid solvers are applied to them, the asymptotic algebraic convergence is necessarily slow. Nevertheless, it is shown by mode analyses and numerical experiments that the usual FMG algorithm is very efficient in solving quasi-elliptic equations to the level of truncation errors. Also, a new type of multigrid algorithm is presented, mode analyzed and tested, for which even the asymptotic algebraic convergence is fast. The essence of that algorithm is applicable to other kinds of problems, including highly indefinite ones.

Brandt, A.↗
NASA NTRS1985

Multigrid solutions to quasi-elliptic schemes

Quasi-elliptic schemes arise from central differencing or finite element discretization of elliptic systems with odd order derivatives on non-staggered grids. They are somewhat unstable and less accurate than corresponding staggered-grid schemes. When usual multigrid solvers are applied to them, the asymptotic algebraic convergence is necessarily slow. Nevertheless, it is shown by mode analyses and numerical experiments that the usual FMG algorithm is very efficient in solving quasi-elliptic equations to the level of truncation errors. Also, a new type of multigrid algorithm is presented, mode analyzed and tested, for which even the asymptotic algebraic convergence is fast. The essence of that algorithm is applicable to other kinds of problems, including highly indefinite ones.

Brandt, A.↗
NASA NTRS1987

A method of calculating the total flow from a given sea surface topography

Using a simple dynamical model of a wind-driven ocean circulation of the Stommel type, and an analytical basis developed to objectively analyze the sea surface height residuals from an altimeter and, in the process, to determine the total flow instead of just the near surface geostrophic component associated with the given sea surface topography. The method is based on first deriving the solution to the forced problem for a given wind stress required to develop a hypothetical true or perfect data field and to establishing the basis for the objective analysis. The stream function and the surface height field for the forced problem are developed in terms of certain characteristic functions with the same expansion coefficients for both fields. These characteristic functions are simply the solutions for a homogeneous elliptic equation for the stream function and the solutions of an inhomogeneous balance equation for the height field. For the objective analysis, using a sample of randomly selected height values from the true data field, the height field characteristic functions are used to fit the given topography in a least squares sense. The resulting expansion coefficients then permit the synthesis of the total flow field via the stream function characteristic modes and the solution is perfectly well behaved even along the equator. The method of solution is easily adaptable to realistic ocean basis by straight forward numerical methods. The analytical basis of the theory and the results for an ideal rectangular basin on a beta plane are described.

Rao, Desiraju B.↗
NASA NTRS1988

The use of multigrid techniques in the solution of the Elrod algorithm for a dynamically loaded journal bearing

A numerical solution to a theoretical model of vapor cavitation in a dynamically loaded journal bearing is developed, utilizing a multigrid iterative technique. The code is compared with a presently existing direct solution in terms of computational time and accuracy. The model is based on the Elrod algorithm, a control volume approach to the Reynolds equation which mimics the Jakobssen-Floberg and Olsson cavitation theory. Besides accounting for a moving cavitation boundary and conservation of mass at the boundary, it also conserves mass within the cavitated region via liquid striations. The mixed nature of the equations (elliptic in the full film zone and nonelliptic in the cavitated zone) coupled with the dynamic aspects of the problem create interesting difficulties for the present solution approach. Emphasis is placed on the methods found to eliminate solution instabilities. Excellent results are obtained for both accuracy and reduction of computational time.

Woods, Claudia M.↗
NASA NTRS1987

Unstructured multigrid methods

The use of the multigrid procedure with a sequence of unnested coarser grids is discussed. Validity of the procedure is assessed by considering the solution of a single linear elliptic equation. It is demonstrated how a scheme with the optimum order of operations can be constructed. Application to the solution of the Euler equations is considered.

Loehner, R.↗
NASA NTRS1988

Isothermal magnetostatic atmospheres. II - Similarity solutions with current proportional to the magnetic potential cubed

The paper presents a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry. The distributed current in the model J is directed along the x-axis, where x is the horizontal ignorable coordinate. The current J is taken to be proportional to the cube of the magnetostatic potential A and falls off exponentially with distance vertical to the base with an e-folding distance equal to the gravitational scale height. A range of similarity solution examples are displayed depending on the values of the similarity parameters. Each similarity parameter corresponds to a symmetry of the underlying nonlinear elliptic equation for A. The similarity parameters also determine the source currents for the potential field solution of the family. The solutions show the interplay between the gravitational force, the J & B force (B, magnetic field induction) and the gas pressure gradient.

Webb, G. M.↗
NASA NTRS1988

Body force circulation and the Antarctic Ozone minimum

A dynamical mechanism, which is relevant to the ozone-depletion problem and which involves a 'body force' circulation discussed by Eliassen (1951), is considered. The body force problem and its application to 'generic' upper-tropospheric wave-driving are discussed theoretically. Following a review of the interhemispheric differences in the winter and early-spring circulation, the elliptic equation governing the residual mean meridional streamfunction is derived, and its analytic-solution properties in negligible shear are discussed. Finally, a numerical solution is derived using the observed August winds under the assumption of radiative equilibrium, and the effect of diabatic heating is discussed.

Dunkerton, Timothy J.↗
NASA NTRS1988

Incompressible and anelastic flow simulations on numerically generated grids

In the numerical simulation of incompressible and anelastic flows, it is necessary to solve an elliptic equation at each time step. When the boundaries of such flows are nonrectangular, it may be advantageous to solve the equations on a new, numerically generated coordinate grid, in which the property of orthogonality has been preserved. Flow equations in general curvilinear coordinates maintaining the conservative form are given for both anelastic models using the momentum equations, and for incompressible models, using the vorticity equation. The general problem of grid-generation in two dimensions is presented, and a quasi-conformal transformation technique is discussed in detail. Some examples of grids generated by this technique are exhibited. Three examples of the flow of a stratified fluid over obstacles are presented, in which the grid-generation permits some new results to be obtained.

Sharman, R. D.↗
NASA NTRS1990

A multigrid solver for semi-implicit global shallow-water models

A multigrid solver is developed for the discretized two-dimensional elliptic equation on the sphere that arises from a semiimplicit time discretization of the global shallow-water equations. Different formulations of the semiimplicit scheme result in variable-coefficient Helmholtz-type equations for which no fast direct solvers are available. The efficiency of the multigrid solver is optimal, in the sense that the total operation count is proportional to the number of unknowns. Numerical experiments using initial data derived from actual 300-mb height and wind velocity fields indicate that the present model has very good accuracy and stability properties.

Barros, Saulo R. M.↗