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NASA NTRS1972

Current methods of radio occultation data inversion

The methods of Abel integral transform and ray-tracing inversion have been applied to data received from radio occultation experiments as a means of obtaining refractive index profiles of the ionospheres and atmospheres of Mars and Venus. In the case of Mars, certain simplifications are introduced by the assumption of small refractive bending in the atmosphere. General inversion methods, independent of the thin atmosphere approximation, have been used to invert the data obtained from the radio occultation of Mariner 5 by Venus; similar methods will be used to analyze data obtained from Jupiter with Pioneers F and G, as well as from the other outer planets in the Outer Planet Grand Tour Missions.

Kliore, A. J.↗
NASA NTRS1972

A crack problem with four distinct harmonic functions

The problem of an elastic solid containing a semi-infinite plane crack subjected to concentrated shears parallel to the edge of the crack is considered. A closed form solution using four distinct harmonic functions (none of which can be taken arbitrarily) is found to satisfy the finite displacement and inverse square root stress singularity at the edge of the crack. Explicit expressions in terms of elementary functions are given for the distribution of stress and displacement in the solid. These are obtained by employing Fourier and Kontorovich-Lebedev integral transforms and certain singular solutions of Laplace equations in three dimensions. The variations of the intensity of the local stress field along the crack border are shown graphically. An example is presented, which is in contrast with the conclusion established in the literature that one of the four Papkovich-Neuber functions in three-dimensional elasticity may be arbitrarily set to zero.

Sih, G. C.↗
NASA NTRS1973

Application of Papkovich-Neuber potentials to a crack problem.

The problem of an elastic solid containing a semi-infinite plane crack subjected to concentrated shears parallel to the edge of the crack is considered in this paper. A closed form solution using four harmonic functions is found to satisfy the finite displacement and inverse square root stress singularity at the edge of the crack. Explicit expressions in terms of elementary functions are given for the distribution of stress and displacement in the solid. These are obtained by employing Fourier and Kontorovich-Lebedev integral transforms and certain singular solutions of Laplace equations in three dimensions. The variations of the intensity of the local stress field along the crack border are shown graphically.

Kassir, M. K.↗
NASA NTRS1973

Stability of matrix equations arising in electromagnetics

Many problems in electromagnetics can be formulated as a Fredholm equation of the first kind. In these problems it is desired to determine some function f(x) from its integral transform g(y). Except for special cases for which an explicit inverse operator is known, the equation is usually discretized by the method of moments into a matrix equation Af = g, which is then solved on a computer. A method of calculating the sensitivity of the solution of Af = g due to errors in either the vector g or the matrix A is developed. The usefulness of this method is demonstrated with problems from antenna theory, scattering theory, and remote sensing.

Klein, C.↗
NASA NTRS1974

Viscoelastic structures

Numerical analysis of viscoelastic problems may be achieved by either a step-by-step solution procedure or by the integral transform approach. However, for complicated loading and material property relationships, the latter method proves ineffective. Programs specifically developed for the analysis of viscoelastic structures are considered along with multipurpose programs with specific viscoelastic analysis capabilities.

Gupta, K. K.↗
NASA NTRS1976

Dynamic ductile fracture of a central crack

A central crack, symmetrically growing at a constant speed in a two dimensional ductile material subject to uniform tension at infinity, is investigated using the integral transform methods. The crack is assumed to be the Dugdale crack, and the finite stress condition at the crack tip is satisfied during the propagation of the crack. Exact expressions of solution are obtained for the finite stress condition at the crack tip, the crack shape, the crack opening displacement, and the energy release rate. All those expressions are written as the product of explicit dimensional quantities and a nondimensional dynamic correction function. The expressions reduce to the associated static results when the crack speed tends to zero, and the nondimensional dynamic correction functions were calculated for various values of the parameter involved.

Tsai, Y. M.↗
NASA NTRS1976

Structure of the atmosphere of Jupiter from Pioneer 10 and 11 radio occultation measurements

A technique for incorporating effects of the oblateness of Jupiter's atmosphere into the data analysis procedure for radio-occultation measurements is described which makes use of a spherical harmonic representation of the gravity field to compute the shape of the planet. With this technique, the center of refraction is located by the radius of curvature and the normal direction at the closest approach point of the ray. The present technique, a subsequent approximation involving the use of a constant center of refraction, and the Abel integral transform inversion method for spherical planets are applied to Pioneer 10 and 11 data. All the intermediate results obtained show a temperature inversion between 10 and 100 mb, with temperatures from 130 to 170 K at 10 mb and from 80 to 120 K at 100 mb. Comparison of the radio-occultation profiles with radiative-convective equilibrium models and the temperature profile based on Pioneer 10 IR radiometer data indicates good agreement between the models and the occultation results.

Kliore, A. J.↗
NASA NTRS1977

Transient wave propagation in composite plates due to impact

Stress-wave propagation in a composite plate due to an impact has been examined using anisotropic elasticity theory. The plate is modelled as a number of identical anisotropic layers whose elastic moduli are given. Mindlin's (1959) approximate theory of plates is then applied to each layer to obtain a set of difference-differential equations of motion by using the interlaminar stresses and displacements as explicit variables. Dispersion relationships for harmonic waves are found when traction-free boundary conditions are applied to both surfaces of the plate. The difference-differential equations are reduced to difference equations via integral transforms. With given impact boundary conditions these equations are solved for an arbitrary number of layers in the plate, and the transient propagation of a stress wave is calculated by means of a fast Fourier transform algorithm.

Sung Kim, B.↗
NASA NTRS1977

Unsteady linearized transonic flow analysis for slender bodies

An unsteady linearized formulation based on Oswatitsch-Keune's parabolic method is developed to analyze transonic flow past oscillating slender bodies. In contrast to the widely used integral transform method, it is shown that all solutions can be derived by a simpler method directly in the physical plane. By various expansion procedures, low-frequency solutions then are derived according to two clearly defined frequency ranges. Adams-Sears' iteration is employed to account for the second-order effects. Stability derivatives are compared with available theories and data. It is found that the derivatives depend more sensitively on thickness than on the reduced frequency. Finally, a critical assessment of the present method is given.

Liu, D. D.↗
NASA NTRS1977

Impact on multilayered composite plates

Stress wave propagation in a multilayer composite plate due to impact was examined by means of the anisotropic elasticity theory. The plate was modelled as a number of identical anisotropic layers and the approximate plate theory of Mindlin was then applied to each layer to obtain a set of difference-differential equations of motion. Dispersion relations for harmonic waves and correction factors were found. The governing equations were reduced to difference equations via integral transforms. With given impact boundary conditions these equations were solved for an arbitrary number of layers in the plate and the transient propagation of waves was calculated by means of a Fast Fourier Transform algorithm. The multilayered plate problem was extended to examine the effect of damping layers present between two elastic layers. A reduction of the interlaminar normal stress was significant when the thickness of damping layer was increased but the effect was mostly due to the softness of the damping layer. Finally, the problem of a composite plate with a crack on the interlaminar boundary was formulated.

Kim, B. S.↗
NASA NTRS1977

Recovery of refractivity profiles and pressure and temperature distributions in the lower atmosphere from satellite-to-satellite radio occultation data

The feasibility of recovering parameters from one-way range rate between two earth orbiting spacecraft during occultation of the tracking signal by the earth's lower atmosphere. The tracking data is inverted by an integral transformation (Abel transform) to obtain a vertical refractivity profile above the point of closest approach of the ray connecting the satellites. Pressure and temperature distributions can be obtained from values of dry refractivity using the hydrostatic equation and perfect gas law. Two methods are investigated for recovering pressure and temperature parameters. Results show that recovery is much more sensitive to satellite velocity errors than to satellite position errors. An error analysis is performed. An example is given demonstrating recovery of parameters from radio occultation data obtained during satellite-to-satellite tracking of Nimbus 6 by the ATS 6 satellite.

Murray, C. W., Jr.↗
NASA NTRS1976

Calculation methods for compressible turbulent boundary layers

Calculation procedures for non-reacting compressible two- and three-dimensional turbulent boundary layers were reviewed. Integral, transformation and correlation methods, as well as finite difference solutions of the complete boundary layer equations summarized. Alternative numerical solution procedures were examined, and both mean field and mean turbulence field closure models were considered. Physics and related calculation problems peculiar to compressible turbulent boundary layers are described. A catalog of available solution procedures of the finite difference, finite element, and method of weighted residuals genre is included. Influence of compressibility, low Reynolds number, wall blowing, and pressure gradient upon mean field closure constants are reported.

Bushnell, D. M.↗
NASA NTRS1981

An extended classical solution of the droplet growth problem

Problems of applying the classical kinetic theory to the growth of small droplets from vapor are examined. A solution for the droplet growth equation is derived which is based on the assumption of a diffusive field extending to the drop surface. The method accounts for partial thermal and mass accommodation at the interface and the kinetic limit to the mass and heat fluxes, and it avoids introducing the artifact of a discontinuity in the thermal and vapor field near the droplet. Consideration of the environmental fields in spherical geometry utilizing directional fluxes yields boundary values in terms of known parameters and a new Laplace transform integral.

Anderson, B. J.↗
NASA NTRS1980

Transmitted sound field due to an impulsive line acoustic source bounded by a plate followed by a vortex sheet

The propagation of sound due to a line acoustic source in the moving stream across a semiinfinite vortex sheet which trails from a rigid plate is examined in a linear theory for the subsonic case. A solution for the transmitted sound field is obtained with the aid of multiple integral transforms and the Wiener-Hopf technique for both the steady state (time harmonic) and initial value (impulsive source) situations. The contour of inverse transform and hence the decomposition of the functions are determined through causality and radiation conditions. The solution obtained satisfies causality and the full Kutta conditions. The transmitted sound field is composed of two waves in both the stady state and initial value problems. One is the wave scattered from the edge of the plate which is associated with the bow wave and the instability wave. These waves exist in the downstream sectors. The other is the wave transmitted through the vortex sheet which is also associated with the instability wave. Regional divisions of the transmitted sound field are identified.

Miura, T.↗
NASA NTRS1981

A hybrid method for improving MCSCF convergence

It has been found that the convergence problems for many ill conditioned single-configuration SCF calculations arise from mixing among only a small number of orbitals. This orbital set includes the highest closed, the partially filled, and (possibly) a few of the lowest virtual orbitals. The fact that convergence problems can be, in very large measure, linked to a small orbital set is used to design a hybrid MCSCF procedure in which these orbitals are treated using a second-order MCSCF method, while other mixings are treated with a lower-order method which avoids the time consuming integral transformation. Tests on BeO show that the hybrid method yields convergence even when the simple lower-order treatment diverges. The method is expected to facilitate determination of MCSCF wave functions for large basis problems and for the construction of potential energy surfaces.

Bauschlicher, C. W., Jr.↗
NASA NTRS1985

The Control of Float Zone Interfaces by the Use of Selected Boundary Conditions

Since the float zone process involves two solid-melt interfaces, possible gas interfaces, heat and mass transfers, various driving forces and complex heating sources, an analysis of the entire process would be very complex. For an initial investigation, a more feasible approach is to examine each component of the process separately. The three principal components are: (1) the shapes of the melt and solid-melt interfaces, (2) the heat and mass transfers, and (3) the heating and cooling sources. This study combined facets of all three components. The purpose of this effort was to study and compute the surface boundary conditions required to give flat (float zone) FZ solid-melt interfaces. This study was undertaken in two phases. The first phase was to investigate the solid zones surface boundary conditions required for flat solid-melt interfaces when given the melt zone surface boundary conditions. The second phase complemented the first and was to investigate the melt zone surface boundary conditions required for flat solid-melt interfaces if given the solid zones surface boundary conditions. Dual integral transform methods were used in both phases; in addition, the use of various numerical methods for differential equations and linear systems of equations were required.

Foster, L. M.↗
NASA NTRS1985

Bounding solutions of geometrically nonlinear viscoelastic problems

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Stubstad, J. M.↗
NASA NTRS1986

Bounding solutions of geometrically nonlinear viscoelastic problems

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Stubstad, J. M.↗