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pagesResearch ArticleSolution of Axisymmetric Potential Problem in Oblate Spheroid Using the Exodus Method,1 ,2 and 21College of Engineering, Technology, and Computer Science, Indiana University-Purdue University, Fort Wayne, IN 46805,
USA2Roy G. Perry College of Engineering, Prairie View A&M University, Prairie View, TX 77446,
USAReceived 7 November 2013; Accepted 15 February 2014; Published
16 March 2014Academic Editor: Quan Yuan Copyright © 2014 O. D. Momoh et al. This is an open access article distributed under the , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper presents the use of Exodus method for computing potential distribution within a conducting oblate spheroidal system. An explicit finite difference method for solving Laplace’s equation in oblate spheroidal coordinate systems for an axially symmetric geometry was developed. This was used to determine the transition probabilities for the Exodus method. A strategy was developed to overcome the singularity problems encountered in the oblate spheroid pole regions. The potential computation results obtained correlate with those obtained by exact solution and explicit finite difference methods.1. IntroductionAn oblate spheroid is the surface generated by the rotation of an ellipse about its minor axis, and depending upon the ellipse’s eccentricity, the spheroid will be flattened about the minor axis []. An oblate spheroidal shell, for instance, is considered as a continuous system constructed from two spherical shell caps by matching the continuous boundary conditions [].Oblate and prolate spheroidal coordinates are widely used in many fields of science and engineering, such as potential theory, fluid mechanics, heat and mass transfer, thermal stress, and elastic inclusions. For example, oblate and prolate spheroids being surfaces of revolution can be more easily conformed to most districts of human body (e.g., extremities) which is of interest for dedicated MRI systems []. Oblate spheroidal coordinates are the natural choice for the translation of any ellipsoid parallel to a principal axis []. There is a more recent improvement in the lightning ground tracking systems based on the time-of-arrival (TOA) technique because of the refinement in the mathematics to more accurately accommodate the oblate shape of the earth spheroid. Approximating the earth as a perfect sphere affects not only the accuracy of time clock offset calculations, but also the accuracy of stroke coordinate computation given receiver time differences. Oblate solution mathematics can provide a substantial systematic error reduction of up to 50% percent [].In this paper, Exodus method is used to compute potential distribution inside conducting oblate spheroidal shells maintained at two potentials. This work is a continuation of our previous work in which a fixed random walk Monte Carlo method (MCM) was used for numerical computation of potential distribution with two conducting oblate spheroidal shells maintained at two potentials []. An explicit Neumann boundary condition was imposed at the pole regions
  of the oblate spheroid to treat the presence of singularities in those regions.The Exodus method is one of Monte Carlo methods which are nondeterministic (probabilistic or stochastic) numerical methods employed in solving mathematical and physical problems. The fixed random walk and Exodus methods are the most frequently used Monte Carlo methods for solving heat conduction and potential problems. The Exodus method is however preferred to fixed random walk method because of its computational efficiency. It yields more accurate results with less computing time as compared to the original Monte Carlo method [, ].2. Oblate Spheroidal Coordinate SystemsThe geometry of an oblate spheroid showing surfaces of constant oblate spheroidal coordinates is illustrated in Figure . The oblate spheroidal coordinates are related to the rectangular coordinates as follows:
  is the focal length of the oblate spheroid.Figure 1: The oblate spheroidal coordinate system [].An arbitrary grid point
  on the oblate spheroid is given by
  are the unit vectors in the direction of
  coordinates, respectively.The respective scale factor for each of the three coordinates
3. Potential Distribution Computation3.1. Finite Difference Transformation of Oblate Spheroid Laplace’s EquationThe Laplacian equation in oblate spheroidal coordinate systems is
The term outside the square bracket may be ignored. Also the first term inside square bracket is ignored due to the rotational symmetry about the vertical
  axis. Therefore, () reduces to
Equation () governs potential distribution in an axisymmetric oblate spheroid potential problem.Two oblate spheroidal shells made up of two constant conducting surfaces
  and
  are shown in Figure . The two equipotential surfaces are maintained at 50 V and 100 V (Dirichlet boundary conditions), respectively. The choice of these Dirichlet boundary conditions is arbitrary. Any potential value can be assigned. Also, the value of the constant oblate spheroidal surfaces equipotential that constitutes the two conducting shells are arbitrarily chosen as
  and
, respectively.Figure 2: Two oblate spheroidal shells.The explicit finite difference transformation of () is
shows one-quarter of the constant oblate spheroidal surfaces. The figure exhibits symmetry with respect to the
  coordinate. Therefore, two lines of symmetries will be encountered in this range of
. They are
  and
. On these lines of symmetries, the condition
  is imposed. This strategy eliminates the singularity causing term
  at the oblate spheroid poles as seen in ().Figure 3: Oblate spheroidal surface path used in computing the charge enclosed.Consequently, the finite difference equations along the two lines of symmetries become as follows.Along
3.2. Transition Probabilities DeterminationIn a more compact form, () can be rewritten as
. Equations (), (), (), and () serve as the transition probabilities for the Exodus method used in this work. If
  particles are dispersed at node
, they have probabilities
  of moving to points
, respectively. The direction of movement of the particles is determined by generating a random number
. The particles are instructed to walk as follows:
For the random walk computation along the line of symmetry (
), () becomes
  along this line of symmetry. If a random-walking particle is instantaneously at the point
, it has probabilities
  and
  of moving to points
, respectively. The particles are instructed to walk as follows:
For the random walk computation along the line of symmetry
, () becomes
The radial parts of the transition probabilities (
  and
) in () are the same as those in () and (). The transition probability for the
  coordinate component of () is
Note also that
. For dispersed particles at point
, they have probabilities
  of moving to points
, respectively. The particle walking pattern of the particles is as follows:
The potential at a specific point (
) is to be determined. We define the transition probability
  as the probability that a random walk starting at the point of interest (
) ends at the boundary node (
) with prescribed potential (
).Large particles are dispatched at the free node (
). The application of Exodus method begins by setting particles
  at all other nodes (both fixed and free), except at the free node (
), where we assume a large value
. By scanning the mesh as in finite difference analysis (FDM), the particles are dispatched at each free node to its neighboring nodes according to the random walk transition probabilities described above. Detailed description of this process is given in []. If
  and
  designate the total number of particles dispersed and the number of particles that have reached the boundary
, respectively, the probability that a random walk terminates on the boundary is
If there are
  boundaries or fixed nodes (excluding the corner points), the potential at the specified node (
Since there are just two boundaries (the inner and outer oblate shell surfaces) in this potential problem, () simplifies to
  and
  are the prescribed potentials at the inner and outer oblate spheroidal shells, respectively.The equation for the exact solution of the potential computation in oblate spheroidal shells is []
  and
  represent the nonzero potential (Dirichlet boundary conditions) at the inner and outer conducting oblate spheroidal shells, respectively, as shown in Figure . Also,
  denotes the Gudermannian function [] and is represented as
4. ResultsEquations ()–() and ()-() are used to implement the computation of potential distributions within two conducting oblate spheroidal shells numerically and analytically, respectively. The results obtained are as shown in Table . The results obtained from exact solution (analytical), the Exodus method, and the explicit finite difference solutions are compared in Table . Same step size was used for both the Exodus method and the FDM and this accounted for the closeness in the computed results obtained. The Exodus solution results were very close to the results obtained from exact solution because of the fact that though the Exodus method is a probabilistic method, its operation does not depend on random number generation which ultimately depends on the computation accuracy of the machine involved.Table 1: Comparing the Exodus solution with finite difference and exact solution.5. ConclusionThe use of the Exodus method to compute potential distribution inside two conducting oblate spheroidal shells maintained at two potentials has been implemented in this paper. The results obtained agreed with those obtained using finite difference (FDM) solution and the exact solution method. The Exodus method employed in this work can be said to be almost as accurate as the exact method when compared to the fixed random walk Monte Carlo method because the results obtained were as accurate as the exact method.Conflict of InterestsThe authors declare that there is no conflict of interests regarding the publication of this paper.References
W. L. Gates, “Derivation of the equations of atmospheric motion in Oblate Spheroidal Coordinates,” Journal of the Atmospheric Sciences, vol. 61, pp. , 2004. A. M. Al-Jumaily and F. M. Najim, “An approximation to the vibrations of oblate spheroidal shells,” Journal of Sound and Vibration, vol. 204, no. 4, pp. 561–574, 1997.
· V. Punzo, S. Besio, S. Pittaluga, and A. Trequattrini, “Solution of Laplace equation on non axially symmetrical volumes,” IEEE Transactions on Applied Superconductivity, vol. 16, no. 2, pp. , 2006.
· R. F. Tuttle and S. K. Loyalka, “Gravitational collision efficiency of nonspherical aerosols II: motion of an oblate spheroid in a viscous fluid,” Nuclear Technology, vol. 69, no. 3, pp. 327–336, 1985.
· P. W. Casper and R. B. Bent, “The effect of the earth's oblate spheroid shape on the accuracy of a time-of-arrival lightning ground strike locating system,” in Proceedings of the International Aerospace and Ground Conference on Lightning and Static Electricity, vol. 2, pp. 81.1–81.8, The National Aeronautics and Space Administration, The National Interagency Coordination Group (NICG), and Florida Institute, 1991. O. D. Momoh, M. N. O. Sadiku, and S. M. Musa, “A fixed random walk Monte Carlo computation of potential inside two conducting oblate spheroidal shells,” in Proceedings of the IEEE Southeastcon, pp. 196–200, Nashville, Tenn, USA, March 2011.
· O. D. Momoh, M. N. O. Sadiku, and C. M. Akujuobi, “Solution of axisymmetric potential problem in spherical coordinates using Exodus method,” in Proceedings of the PIERS Conference, pp. , Cambridge, Mass, USA, July 2010. M. N. O. Sadiku, Numerical Techniques in Electromagnetics with MATLAB, CRC Press, Boca Raton, Fla, USA, 3rd edition, 2009. R. S. Alassar and H. M. Badr, “Analytical solution of oscillating inviscid flow over oblate spheroids with spheres and flat disks as special cases,” Ocean Engineering, vol. 24, no. 3, pp. 217–225, 1997.
· O. D. Momoh, M. N. O. Sadiku, and C. M. Akujuobi, “Potential distribution computation in conducting prolate spheroidal shells using the exodus method,” IEEE Transactions on Magnetics, vol. 47, no. 5, pp. , 2011.内容介绍：
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相关资源：追加200分，求一本比较好的关于图像处理的matlab电子书_百度知道
追加200分，求一本比较好的关于图像处理的matlab电子书
这本书，最好是讲述关于图像处理的，讲解详细点，并且有对应matlab程序的例题讲解~~我的邮箱，如果好多人提供的都是比较满意的资料的话，我会马上开贴每人追加200分的，注：&&数字图像处理第二版中文版（冈订常斥端俪得筹全船户萨雷斯）&&我已经有了，就不要发给我了 谢谢！《matlab函数速查手册》电子版，邓薇编著，如果您有这本书的电子版，我将开5个帖子追加您1000分的，嘿嘿
本人还有大约5天时间就回家了，希望有好电子书的朋友尽快！ 给我发邮件的都不能白发啊，不管有没有订常斥端俪得筹全船户用，我放假走人之前都会根据您所提供书的价值给你们每人相应的分的！
提问者采纳
函数速查手册这些地方都有一些。你去下载一下吧。希望能满订常斥端俪得筹全船户意。那本书不好意思，没找到。
提问者评价
虽然不是我想要的，但是已经很不错了!谢谢
其他类似问题
图像处理的相关知识
其他5条回答
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我倒有几本，你看看吧，还有个关于图像函数的列表及简单解释，发过去了。
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- Steven T. Karris.pdfDigital Image Processing Using Matlab - Gonzalez Woods & Eddins.pdfDigital Signal and Image Processing Using MATLAB - Gerard Blanchet & Maurice Charbit.pdfDigital Signal Processing - Computer Based Approach - Sanjit K. Mitra.pdfDigital Signal Processing Using Matlab V4 - Ingle and Proakis.pdfDynamic Simulations of Electric Machinery - Chee Mun Ong.djvuElectronic Devices and Amplifier Circuits with MATLAB Applications - Steven T. Karris.pdfElectronics and Circuit Analysis Using MATLAB - John O. Attia.pdfElem. Math. and Comp. Tools for Engineers using MATLAB - J. Manassah.pdfEmbedded Control Systems in C C++ Using MATLAB - Jim Ledin.chmEmbedded Image Processing with DSP Examples in MATLAB - Shehrzad Qureshi.pdfEngineering Analysis Interactive Methods and Programs with MATLAB - Y. C. Pao.pdfEngineering and Scientific Computations Using MATLAB - Sergey E. Lyshevski.pdfEnvironmental Modeling Using MATLAB - Ekkehard Holzbecher.pdfEssential MATLAB for Engineers and Scientists - Brian D. Hahn & Daniel T. Valentine.pdfExploratory Data Analysis with MATLAB - Martinez and Martinez.pdfFundamentals of Electromagnetics with Matlab - Lonngren & Savov.pdfGraphics and GUIs with MATLAB - Patrick Marchand and O. Thomas Holland .pdfIntroduction to Fuzzy Logic using MatLab - Sivanandam Sumathi and Deepa.pdfIntroduction to MATLAB - Sikander M. Mirza.pdfIntroduction to Simulink with Engineering Applications - Steven T. Karris.pdfIntuitive Probability and Random Processes Using MatLab - Steven M. Kay.pdfKalman Filtering Theory and Practice Using MATLAB - Grewal and Andrews.pdfMathWorks Documentation - MATLAB V7 Function References.pdfMathWorks Documentation - MATLAB V7 Introductory and Programming.pdfMATLAB Guide - Desmond J. Higham & Nicholas J. Higham.djvuMATLAB Primer (6th Ed) - Kermit Sigmon & Timothy A. Davis.pdfMATLAB Primer (7th Ed) - Timothy A. Davis & Kermit Sigmon.pdfMATLAB Programming - David Kuncicky.pdfMATLAB Recipes for Earth Sciences - M.H.Trauth.pdfMATLAB Simulations for Radar Systems Design - Bassem R. Mahafza & Atef Z. Elsherbeni.pdfMechanics of Composite Materials with MATLAB - George Z. Voyiadjis & Peter I. Kattan.pdfNumerical Analysis Using MATLAB and Excel - Steven T. Karris.pdfNumerical Analysis Using MATLAB and Spreadsheets - Steven T. Karris.pdfNumerical Computing with MATLAB - Cleve Moler.pdfNumerical Methods in Engineering with MATLAB - Jaan Kiusalaas.pdfNumerical Methods in Finance & Economics A MATLAB based Introduction - Paolo Brandimarte.pdfNumerical Methods using MATLAB - Mathews and Fink.pdfNumerical Techniques for Chemical & Biological Engineers Using MATLAB - Elnashaie & Uhlig.pdfOptical Scanning Holography with MATLAB - Ting Chung Poon.pdfOptics Learning by Computing with Examples using MATLAB - K.D. Moller.pdfOrdinary and Partial Differential Equation Routines in Matlab - H.J. Lee & W.E. Schiesser.pdfRadar Systems Analysis and Design Using MatLab - Mahafza Bassem R.pdfRobust Control Design with Matlab - Gu Petkov and Konstantinov.pdfScientific Computing with Matlab - Alfio Quarteroni & Fausto Saleri.djvuScientific Computing with Matlab and Octave - Alfio Quarteroni & Fausto Saleri.pdfSignals and Systems with MATLAB Applications - Steven T. Karris.pdfSignals and Systems with MATLAB Computing and Simulink Modeling - Steven T. Karris.pdfSolving ODEs with MATLAB - Shampine Gladwell Thompson.pdfSolving ODEs with Matlab Instructors Manual - L.F. Shampine.pdfSpectral Methods in MATLAB - Lloyd N. Trefethen.pdfThe Finite Element Method using MATLAB - Kwon and Bang.pdfVibration Simulation Using MATLAB and ANSYS - Michael R Hatch.pdf
倒，直接买实体书 吧，又不贵
Intuitive Probability and Random Processes using MATLABby Steven Kay该书我在看，很细有许多例子，可以一边帮你复习知识一般学习matlab外国教材一般都很好
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