Radiation Configuration Factors Between Disks and a Class of Axisymmetric Bodies

1982 ◽  
Vol 104 (3) ◽  
pp. 426-431 ◽  
Author(s):  
M. H. N. Naraghi ◽  
B. T. F. Chung
Keyword(s):  
Numerical Integration ◽  
Analytical Solutions ◽  
Coaxial Cylinder ◽  
Present Approach ◽  
Shape Factors ◽  
Present Technique ◽  
Analytical Formulae ◽  
View Factors ◽  
Axisymmetric Bodies

A general formulation is developed for the radiation shape factors between a disk and a class of coaxial axisymmetric bodies such as cylinder, cone, ellipsoid, and paraboloid. In certain cases, the view factors can be derived analytically directly from the present technique, while in others, they can be generated from a single numerical integration. The analytical solutions for view factors from a disk to a coaxial cylinder based on the present approach are found to agree with those published earlier. The analytical formulae of view factors from disks or annular rings to circular cones, truncated cones, ellipsoid, paraboloids, and truncated praboloid are herein presented.

A Simpler Formulation for Radiative View Factors From Spheres to a Class of Axisymmetric Bodies

1982 ◽  
Vol 104 (1) ◽  
pp. 201-204 ◽  
Author(s):  
B. T. F. Chung ◽  
M. H. N. Naraghi
Keyword(s):  
Numerical Integration ◽  
View Factor ◽  
View Factors ◽  
New Formulation ◽  
Axisymmetric Bodies

A simpler formulation is developed for radiative view factor from a sphere to a class of axisymmetric bodies. The new formulation is semianalytical in nature and only requires a single numerical integration at most.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

On the extrication of large objects from the ocean bottom (the breakout phenomenon)

1982 ◽  
Vol 117 ◽  
pp. 211-231 ◽  
Author(s):  
Mostafa A. Foda
Keyword(s):  
Boundary Layer ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Time Dependent ◽  
Ocean Bottom ◽  
Leading Order ◽  
Boundary Layer Approximation ◽  
Slender Bodies ◽  
Axisymmetric Bodies ◽  
Time Dependent Analysis

An analytical theory is developed to describe how negative pressure, (or ‘mud suction’, as it is sometimes referred to) develops underneath a body as it detaches itself from the ocean bottom. Biot's quasistatic equations of poro-elasticity are used to model the ocean bottom, and a general three-dimensional time-dependent analysis of the problem is worked out first using the boundary-layer approximation recently proposed by Mei and Foda. Then, explicit leading-order analytical solutions are presented for the problems of extrication of slender bodies as well as axisymmetric bodies from the ocean bottom.

EXACT QUANTIZATION RULE AND ITS APPLICATIONS TO PHYSICAL POTENTIALS

2007 ◽  
Vol 16 (01) ◽  
pp. 189-198 ◽  
Author(s):  
SHI-HAI DONG ◽  
D. MORALES ◽  
J. GARCÍA-RAVELO
Keyword(s):  
Harmonic Oscillator ◽  
Analytical Solutions ◽  
Bound States ◽  
Energy Levels ◽  
Present Approach ◽  
Three Dimensions ◽  
One Dimension ◽  
Quantization Rule ◽  
Exact Quantization Rule ◽  
Pseudoharmonic Oscillator

By using the exact quantization rule, we present analytical solutions of the Schrödinger equation for the deformed harmonic oscillator in one dimension, the Kratzer potential and pseudoharmonic oscillator in three dimensions. The energy levels of all the bound states are easily calculated from this quantization rule. The normalized wavefunctions are also obtained. It is found that the present approach can simplify the calculations.

scholarly journals Numerical Evaluation of Coulomb Integrals For 1, 2 and 3-electron Distance Operators, RC1 -nRD1 -m , RC1 -n r12 -m and r12 -n r13 -m with Real (N, M) and The Descartes Product of 3 Dimension Common Density Functional Numerical Integration Scheme

2018 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Quantum Chemistry ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Density Functional ◽  
Integration Scheme ◽  
Functional Theory ◽  
Computational Quantum Chemistry ◽  
Numerical Integration Scheme ◽  
Analytical Integration ◽  
Coulomb Integrals

Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the n, m= 0 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|<b>r</b><sub>1</sub>|<sup>2</sup>), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|<b>r</b><sub>1</sub>|) as well.

scholarly journals Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ai-Min Yang ◽  
Yu-Zhu Zhang ◽  
Carlo Cattani ◽  
Gong-Nan Xie ◽  
Mohammad Mehdi Rashidi ◽  
...  
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Expansion Method ◽  
Cantor Sets ◽  
Present Technique ◽  
Klein Gordon ◽  
Series Expansion Method

We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

scholarly journals Highly accurate analytical solution for free vibrations of strongly nonlinear Duffing oscillator

2021 ◽  
pp. 146134842110340
Author(s):  
Gamal Mohamed Ismail ◽  
Mahmoud Abul-Ez ◽  
Mohra Zayed ◽  
Hijaz Ahmad ◽  
Maha El-Moshneb
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Perturbation Technique ◽  
Duffing Oscillator ◽  
Free Vibrations ◽  
Strongly Nonlinear ◽  
Present Technique ◽  
Analytical Perturbation

Based on the suggested parameter, a new analytical perturbation technique is presented to obtain highly ordered accurate analytical solutions for nonlinear Duffing oscillator with nonlinearity of high order. Comparing the obtained results with the numerical and other previously published results reveals the usefulness and correctness of the present technique. It is shown that the results are valid for small and large amplitudes. Indeed, it is found that our proposed technique produces more accurate and computationally results than the rival known methods. The obtained results show the efficiency and capability of the present perturbation technique to be applied to various strongly nonlinear differential equations.

scholarly journals Numerical Evaluation of Coulomb Integrals For 1, 2 and 3-electron Distance Operators, RC1 -nRD1 -m , RC1 -n r12 -m and r12 -n r13 -m with Real (N, M) and The Descartes Product of 3 Dimension Common Density Functional Numerical Integration Scheme

2018 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Quantum Chemistry ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Density Functional ◽  
Integration Scheme ◽  
Functional Theory ◽  
Computational Quantum Chemistry ◽  
Numerical Integration Scheme ◽  
Analytical Integration ◽  
Coulomb Integrals

Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the n, m= 0 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|<b>r</b><sub>1</sub>|<sup>2</sup>), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|<b>r</b><sub>1</sub>|) as well.

Sign in / Sign up

Export Citation Format

Share Document