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All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

A new formulation for anisotropic radiative transfer problems. I - Solution with a variational technique

The equations of radiative transfer in anisotropically scattering media are reformulated as linear operator equations in a single independent variable. The resulting equations are suitable for solution by a variety of standard mathematical techniques. The operators appearing in the resulting equations are in general nonsymmetric; however, it is shown that every bounded linear operator equation can be embedded in a symmetric linear operator equation and a variational solution can be obtained in a straightforward way. For purposes of demonstration, a Rayleigh-Ritz variational method is applied to three problems involving simple phase functions. It is to be noted that the variational technique demonstrated is of general applicability and permits simple solutions for a wide range of otherwise difficult mathematical problems in physics.

Cheyney, H., III↗
NASA NTRS1979

Variational description of the positive column with two-stem ionization

The ionization balance in diffusion dominated discharges which depends on both one and two step ionization processes is considered. The Spenke diffusion equation (D sq delta n + neutrino n + sq kn =0) describing such conditions is solved by the Rayleigh-Ritz variational method. Simple analytic approximations to the density profile, and the similarity relation between neutrino,k,D and the discharge dimensions, are derived for planar and cylindrical geometry, and compared with exact computations for certain limiting cases.

Crawford, F. W.↗
NASA NTRS1980

Variational description of the positive column with two-step ionization

The ionization balance in diffusion-dominated discharges may depend on both one- and two-step ionization processes. The Spenke diffusion equation describing such conditions is solved in this paper by the Rayleigh-Ritz variational method. Simple analytic approximations to the density profile, and the similarity relation between the equation parameter, and the discharge dimensions, are derived for planar and cylindrical geometry and compared with exact computations for certain limiting cases.

Crawford, F. W.↗
NASA NTRS1985

Mixed models and reduction method for dynamic analysis of anisotropic shells

A time-domain computational procedure is presented for predicting the dynamic response of laminated anisotropic shells. The two key elements of the procedure are: (1) use of mixed finite element models having independent interpolation (shape) functions for stress resultants and generalized displacements for the spatial discretization of the shell, with the stress resultants allowed to be discontinuous at interelement boundaries; and (2) use of a dynamic reduction method, with the global approximation vectors consisting of the static solution and an orthogonal set of Lanczos vectors. The dynamic reduction is accomplished by means of successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate the global approximation vectors. Then the Rayleigh-Ritz technique is used to generate a reduced system of ordinary differential equations in the amplitudes of these modes. The temporal integration of the reduced differential equations is performed by using an explicit half-station central difference scheme (Leap-frog method). The effectiveness of the proposed procedure is demonstrated by means of a numerical example and its advantages over reduction methods used with the displacement formulation are discussed.

Noor, A. K.↗
NASA NTRS1985

Model-size reduction technique for the analysis of symmetric anisotropic structures

A two-step computational procedure is presented for reducing the size of the analysis model for an anisotropic symmetric structure to that of the corresponding orthotropic structure. The key elements of the procedure are: (1) decomposition of the stiffness matrix into the sum of an orthotropic and nonorthotropic (anisotropic) parts; and (2) successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique. The global approximation vectors are selected to be the solution corresponding to zero nonorthotropic matrix and its various-order derivatives with respect to an anisotropic tracing parameter (identifying the nonorthotropic material coefficients). The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure. The effectiveness of the proposed technique is demonstrated by means of numerical examples and its potential for solving other quasi-symmetric problems is discussed.

Noor, A. K.↗
NASA NTRS1951

Analytical determination of coupled bending-torsion vibrations of cantilever beams by means of station functions

A method based on the concept of station functions is presented for calculating the modes and the frequencies of nonuniform cantilever beams vibrating in torsion, bending, and coupled bending-torsion motion. The method combines some of the advantages of the Rayleigh-Ritz and Stodola methods, in that a continuous loading function for the beam is used, with the advantages of the influence-coefficient method, in that the continuous loading function is obtained in terms of the displacements of a finite number of stations along the beam.

Mendelson, Alexander↗
NASA NTRS1982

Development of a reduced basis technique for transient thermal analysis

A technique to reduce the degrees of freedom in static and dynamic problems, the reduced basis method, is described. The method combines the classical Rayleigh-Ritz approximation with contemporary finite element methods to retain modeling versatility as the degrees of freedom are reduced. Applications to a nonlinear dynamic response problem are discussed efforts to apply the method to nonlinear transient thermal response problems are summarized. The selection of basis vectors for reducing the system of equations is addressed.

Shore, C. P.↗
NASA NTRS1982

Application of the reduced basis method to nonlinear transient thermal analysis

An effort to apply the reduced basis method to nonlinear transient thermal analysis is described. The method combines the classical Rayleigh-Ritz and modal superposition techniques with contemporary finite element methods to retain modeling versatility as the degrees of freedom in a problem are reduced. The essence of the method is to use a few thermal modes from eigenvalue analyses as basis vectors to represent the temperature response for a given thermal problem similar to the use of vibration modes to represent displacements in a dynamic response problem. Approximate temperature distributions were obtained using the reduced basis method for a small section of the Shuttle Orbiter lower wing undergoing reentry heating. Good agreement was obtained between the reduced basis method solutions and full system solutions with reductions in the degrees of freedom of up to a factor of four. The good agreement indicates the reduced basis method has the potential for significant reduction in computing effort for thermal analysis; however, considerable work remains to determine techniques for selecting the type and number of basis vectors needed for approximate solutions to more complex transient thermal problems.

Shore, C. P.↗
NASA NTRS1985

Reduction method for thermal analysis of complex aerospace structures

A reduction method which combines classical Rayleigh-Ritz modal superposition techniques with contemporary finite-element methods is applied to transient nonlinear thermal analysis of aerospace structures. The essence of the method is the use of a few thermal modes from eigenvalue analyses as basis vectors to represent the temperature response in the structure. The method is used to obtain approximate temperature histories for a portion of the Shuttle orbiter wing subject to reentry heating and for a large space antenna reflector subject to heating associated with a low Earth orbit. The reduction method has excellent potential for significant size reduction for radiation-dominated problems such as the antenna reflector. However, for conduction-dominated problems such as the Shuttle wing, especially those with complex spatial and temporal variations in the applied heating, additional work appears necessary to find alternate sources of basis vectors which will permit significant problem size reductions.

Shore, C. P.↗
NASA NTRS1974

Compressible seal flow analysis using the finite element method with Galerkin solution technique

High pressure gas sealing involves not only balancing the viscous force with the pressure gradient force but also accounting for fluid inertia--especially for choked flow. The conventional finite element method which uses a Rayleigh-Ritz solution technique is not convenient for nonlinear problems. For these problems, a finite element method with a Galerkin solution technique (FEMGST) was formulated. One example, a three-dimensional axisymmetric flow formulation has nonlinearities due to compressibility, area expansion, and convective inertia. Solutions agree with classical results in the limiting cases. The development of the choked flow velocity profile is shown.

Zuk, J.↗
NASA NTRS1986

Model-size reduction for the buckling and vibration analyses of anisotropic panels

A computational procedure is presented for reducing the size of the model used in the buckling and vibration analyses of symmetric anisotropic panels to that of the corresponding orthotropic model. The key elements of the procedure are the application of an operator splitting technique through the decomposition of the material stiffness matrix of the panel into the sum of orthotropic and nonorthotropic (anisotropic) parts and the use of a reduction method through successive application of the finite element method and the classical Rayleigh-Ritz technique. The effectiveness of the procedure is demonstrated by numerical examples.

Noor, A. K.↗
NASA NTRS1982

Systematic generation of nonlinear discretized dynamic equilibrium equations of spinning cantilevers

General nonlinear discretized governing equations of motion of spinning elastic solids and structures are adjusted for the case of a spinning cantilever with initial geometric imperfections. Consideration is given to second degree nonlinearities in the strain-displacement and velocity-displacement relationships. Parameters of the discretization are developed to include the type and number of the coordinate functions used in the admissible trial solution in order to unify the discretization approaches associated with stationarity principles. The coordinate functions comprise both sets of continuous and piecewise continuous functions employed in the Rayleigh-Ritz and the finite element methods, respectively. Coefficient matrices are provided which contain the energy density expressions and which are adaptable to computer programming.

El-Essawi, M.↗
NASA NTRS1974

Transverse vibrations of non-uniform beams

The free vibrations of elastic beams with nonuniform characteristics are investigated theoretically by a new method. The new method is seen to combine the advantages of a finite element approach and of a Rayleigh-Ritz analysis. Comparison with the known analytical results for uniform beams shows good convergence of the method for natural frequencies and modes. For internal shear forces and bending moments, the rate of convergence is less rapid. Results from experiments conducted with a cantilevered helicopter blade with strong nonuniformities and also from alternative theoretical methods, indicate that the theory adequately predicts natural frequencies and mode shapes. General guidelines for efficient use of the method are presented.

Klein, L.↗
NASA NTRS1973

A theoretical analysis of the free vibrations of ring- and/or stringer-stiffened elliptical cylinders with arbitrary end conditions. Volume 1: Analytical derivation and applications

The derivation and application of a Rayleigh-Ritz modal vibration analysis are presented for ring and/or stringer stiffened noncircular cylindrical shells with arbitrary end conditions. Comparisons with previous results from experimental and analytical studies showed this method of analysis to be accurate for a variety of end conditions. Results indicate a greater effect of rings on natural frequencies than of stringers.

Boyd, D. E.↗