Integral series solution of the Schroedinger equation for the helium atom.
Rapidly converging analytic solution by integral series of nonrelativistic Schroedinger equation for He atom
TWO-ELECTRON SYSTEM↗
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Results for “analytical solution”
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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Dissipative MHD solutions for resonant Alfven waves in 1-dimensional magnetic flux tubes
The present paper extends the analysis by Sakurai, Goossens, and Hollweg (1991) on resonant Alfven waves in nonuniform magnetic flux tubes. It proves that the fundamental conservation law for resonant Alfven waves found in ideal MHD by Sakurai, Goossens, and Hollweg remains valid in dissipative MHD. This guarantees that the jump conditions of Sakurai, Goossens, and Hollweg, that connect the ideal MHD solutions for xi(sub r), and P' across the dissipative layer, are correct. In addition, the present paper replaces the complicated dissipative MHD solutions obtained by Sakurai, Goossens, and Hollweg for xi(sub r), and P' in terms of double integrals of Hankel functions of complex argument of order 1/3 with compact analytical solutions that allow a straight- forward mathematical and physical interpretation. Finally, it presents an analytical dissipative MHD solution for the component of the Lagrangian displacement in the magnetic surfaces perpen- dicular to the magnetic field lines xi(sub perpendicular) which enables us to determine the dominant dynamics of resonant Alfven waves in dissipative MHD.
Goossens, Marcel↗
An approximate solution of the equations of motion for arbitrary rotating spacecraft
Approximate analytical solution for motion of arbitrary rotating spacecraft with variable disturbance functions
ROTATING BODY↗
Asymptotic solution to the tangential low thrust energy increase trajectory
A approximate analytic solution to the low thrust constant acceleration energy increase trajectory using a tangential steering program is presented. The solution is based on a two-variable asymptotic expansion of the equations of motion formulated in regularized variables. The high accuracy of the second order solution derived is demonstrated by comparing it with numerically integrated trajectories.
Schwenzfeger, K. J.↗
The general solution to the classical problem of finite Euler Bernoulli beam
An analytical solution is obtained for the problem of free and forced vibrations of a finite Euler Bernoulli beam with arbitrary (partially fixed) boundary conditions. The effects of linear viscous damping, Winkler foundation, constant axial tension, a concentrated mass, and an arbitrary forcing function are included in the analysis. No restriction is placed on the values of the parameters involved, and the solution presented here contains all cited previous solutions as special cases.
Hussaini, M. Y.↗
Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 1: Technical discussion
Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical solution to the problem on N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The derivation of the second-order solution is summarized by showing the essential steps, some in functional form. The general asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-earth, and interplanetary solutions. The results show that the accuracies of the asymptotic solutions range from an order of magnitude better than conic approximations to that of numerical integration itself. Also, since no iterations are required, the asymptotic boundary value solutions are obtained in a fraction of the time required for comparable numerically integrated solutions. The subject of minimizing the second-order error is discussed, and recommendations made for further work directed toward achieving a uniform accuracy in all applications.
Lancaster, J. E.↗
Shape determination and control for large space structures
An integral operator approach is used to derive solutions to static shape determination and control problems associated with large space structures. Problem assumptions include a linear self-adjoint system model, observations and control forces at discrete points, and performance criteria for the comparison of estimates or control forms. Results are illustrated by simulations in the one dimensional case with a flexible beam model, and in the multidimensional case with a finite model of a large space antenna. Modal expansions for terms in the solution algorithms are presented, using modes from the static or associated dynamic mode. These expansions provide approximated solutions in the event that a used form analytical solution to the system boundary value problem is not available.
Weeks, C. J.↗
Static shape determination and control for a large space antenna
An integral operator approach is used to derive solutions to static shape determination and control problems associated with large space structures. Problem assumptions include a linear self-adjoint system model, observations and control forces at discrete points, and quadratic performance criteria for the comparison of estimates or control forces. Results are illustrated by simulations with a finite element model of a large space antenna. Modal expansions for terms in the solution algorithms are presented, using modes from the static or associated dynamic model. These expansions provide approximate solutions in the event that a closed form analytical solution to the system boundary value problem is not available.
Weeks, C.↗
A decomposition method for optimization of large-system reliability
This paper considers a decomposition approach for optimization of the reliability of a large system with a general network structure. A 3-level methodology is developed for optimal allocation of available resources among subsystems in order to ensure maximization of system-reliability. The decentralized nature of this methodology greatly reduces the complexity of the large problem and facilitates seeking the optimal solution. Two examples show that the complexity of a large system can be greatly reduced by solving several smaller-dimensional sub-problems iteratively. Sub-problems whose dimensions are small can be efficiently solved by any existing nonlinear programming method. Another important feature of this approach is the possible simplification of the objective function during the solution. This leads in some cases to an analytic solution for the lower-level optimization problems in a 3-level decomposition solution.
Li, Duan↗
A massively parallel computational approach to coupled thermoelastic/porous gas flow problems
A new computational scheme for coupled thermoelastic/porous gas flow problems is presented. Heat transfer, gas flow, and dynamic thermoelastic governing equations are expressed in fully explicit form, and solved on a massively parallel computer. The transpiration cooling problem is used as an example problem. The numerical solutions have been verified by comparison to available analytical solutions. Transient temperature, pressure, and stress distributions have been obtained. Small spatial oscillations in pressure and stress have been observed, which would be impractical to predict with previously available schemes. Comparisons between serial and massively parallel versions of the scheme have also been made. The results indicate that for small scale problems the serial and parallel versions use practically the same amount of CPU time. However, as the problem size increases the parallel version becomes more efficient than the serial version.
Shia, David↗
Parsec-Scale Obscuring Accretion Disk with Large-Scale Magnetic Field in AGNs
A magnetic field dragged from the galactic disk, along with inflowing gas, can provide vertical support to the geometrically and optically thick pc (parsec) -scale torus in AGNs (Active Galactic Nuclei). Using the Soloviev solution initially developed for Tokamaks, we derive an analytical model for a rotating torus that is supported and confined by a magnetic field. We further perform three-dimensional magneto-hydrodynamic simulations of X-ray irradiated, pc-scale, magnetized tori. We follow the time evolution and compare models that adopt initial conditions derived from our analytic model with simulations in which the initial magnetic flux is entirely contained within the gas torus. Numerical simulations demonstrate that the initial conditions based on the analytic solution produce a longer-lived torus that produces obscuration that is generally consistent with observed constraints.
galaxies↗
Fast and Reliable Approximations for Interplanetary Low-Thrust Transfers
A three-step process bridges the gap between lower-fidelity solutions that ignore optimal dynamics and fully optimized solutions that are computationally expensive to generate. First, analytic solutions for transfers with free time and angle characterize the evolution of the shape and orientation of the orbit. Next, optimal control theory supplies the thrust vector with variable specific impulse while satisfying flight time and transfer angle constraints. Transfers with the additional constraint of constant specific impulse then provide a more realistic thruster model for preliminary trade studies. These approximations deliver a hundredfold improvement in run time at the expense of a few percent error in mass.
Landau, Damon↗
An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations
The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner in using the method of pseudocompressibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme based on flux-difference splitting is used to compute the convective terms. The upwind differencing is biased based on the sign of the local eigenvalue of the Jacobian matrix. Third-order or fifth-order spatial accuracy is maintained throughout the interior grid points. The equations are solved using an implicit line-relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudotime, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at -90 deg angle-of-attack is also computed.
Rogers, Stuart E.↗
Polar decomposition for attitude determination from vector observations
This work treats the problem of weighted least squares fitting of a 3D Euclidean-coordinate transformation matrix to a set of unit vectors measured in the reference and transformed coordinates. A closed-form analytic solution to the problem is re-derived. The fact that the solution is the closest orthogonal matrix to some matrix defined on the measured vectors and their weights is clearly demonstrated. Several known algorithms for computing the analytic closed form solution are considered. An algorithm is discussed which is based on the polar decomposition of matrices into the closest unitary matrix to the decomposed matrix and a Hermitian matrix. A somewhat longer improved algorithm is suggested too. A comparison of several algorithms is carried out using simulated data as well as real data from the Upper Atmosphere Research Satellite. The comparison is based on accuracy and time consumption. It is concluded that the algorithms based on polar decomposition yield a simple although somewhat less accurate solution. The precision of the latter algorithms increase with the number of the measured vectors and with the accuracy of their measurement.
Bar-Itzhack, Itzhack Y.↗
An exact solution for thermal conduction through a two-dimensional eccentric constriction.
Exact analytic solution for thermal conductance of two dimensional eccentric constriction in dimensionless numbers
Burkhard, D. G.↗
A Model of Radiative and Conductive Energy Transfer in Planetary Regoliths
The thermal regime in planetary regoliths involves three processes: propagation of visible radiation, propagation of thermal radiation, and thermal conduction. The equations of radiative transfer and heat conduction are formulated for particulate media composed of anisotropically scattering particles. Although the equations are time dependent, only steady state problems are considered in this paper. Using the two-stream approximation, solutions are obtained for two cases: a layer of powder heated from below and an infinitely thick regolith illuminated by visible radiation. Radiative conductivity, subsurface temperature gradients, and the solid state greenhouse effect all appear intrinsically in the solutions without ad hoc additions. Although the equations are nonlinear, approximate analytic solutions that are accurate to a few percent are obtained. Analytic expressions are given for the temperature distribution, the optical and thermal radiance distributions, the hemispherical albedo, the hemispherical emissivity, and the directional emissivity. Additional applications of the new model to three problems of interest in planetary regoliths are presented by Hapke.
Hapke, Bruce↗
Energy transfer in volume-reflecting heat shields
An approximate analysis of radiative transfer in highly scattering materials was developed based on the Kubelka-Munk differential equations--a set of two differential equations representing the spatial rate of change of radiative half-fluxes within the scattering media. These approximate solutions of the Kubelka-Munk equations together with analytic solutions for the steady state temperature distribution for two types of boundary conditions are given. These solutions show the influence of back surface reflectance, scattering power, incident radiative flux parameter and boundary conductive flux parameter on overall reflectance and temperature distributions. This radiation field analysis, adapted to spherical geometry, was applied to the evaluation of the thermal performance of teflon and fritted quartz as heat protection materials for entry into the atmosphere of Jupiter.
Weston, K. C.↗
Analytic theory of orbit contraction
The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by gravitational and aerodynamic forces. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. A rigorous analytic solution is used to integrate this equation with a high degree of accuracy, using Poincare's method of small parameters and Lagrange's expansion to explicitly express the major axis as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. Numerical solutions were generated to display the accuracy of the analytic theory.
Vinh, N. X.↗