solution of axisymmetric potential problem in oblate spheroid

Research ArticleSolution of Axisymmetric Potential Problem in Oblate SpheroidUsing the Exodus Method

O D Momoh1 M N O Sadiku2 and S M Musa2

1 College of Engineering Technology and Computer Science Indiana University-Purdue University Fort Wayne IN 46805 USA2 Roy G Perry College of Engineering Prairie View AampM University Prairie View TX 77446 USA

Correspondence should be addressed to O D Momoh momohdipfwedu

Received 7 November 2013 Accepted 15 February 2014 Published 17 March 2014

Academic Editor Quan Yuan

Copyright copy 2014 O D Momoh et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 systemAn explicit finite difference method for solving Laplacersquos equation in oblate spheroidal coordinate systems for an axially symmetricgeometry was developedThis was used to determine the transition probabilities for the Exodus method A strategy was developedto overcome the singularity problems encountered in the oblate spheroid pole regions The potential computation results obtainedcorrelate with those obtained by exact solution and explicit finite difference methods

1 Introduction

An oblate spheroid is the surface generated by the rotationof an ellipse about its minor axis and depending upon theellipsersquos eccentricity the spheroid will be flattened about theminor axis [1] An oblate spheroidal shell for instance isconsidered as a continuous system constructed from twospherical shell caps by matching the continuous boundaryconditions [2]

Oblate and prolate spheroidal coordinates are widelyused in many fields of science and engineering such aspotential theory fluid mechanics heat and mass transferthermal stress and elastic inclusions For example oblateand prolate spheroids being surfaces of revolution can bemore easily conformed to most districts of human body (egextremities) which is of interest for dedicated MRI systems[3] Oblate spheroidal coordinates are the natural choice forthe translation of any ellipsoid parallel to a principal axis[4] There is a more recent improvement in the lightningground tracking systems based on the time-of-arrival (TOA)technique because of the refinement in the mathematics tomore accurately accommodate the oblate shape of the earthspheroid Approximating the earth as a perfect sphere affects

not only the accuracy of time clock offset calculations butalso the accuracy of stroke coordinate computation givenreceiver time differences Oblate solution mathematics canprovide a substantial systematic error reduction of up to 50percent [5]

In this paper Exodusmethod is used to compute potentialdistribution inside conducting oblate spheroidal shells main-tained at two potentials This work is a continuation of ourprevious work in which a fixed random walk Monte Carlomethod (MCM) was used for numerical computation ofpotential distribution with two conducting oblate spheroidalshells maintained at two potentials [6] An explicit Neumannboundary condition was imposed at the pole regions (120578 =

minus1205872 1205872) of the oblate spheroid to treat the presence ofsingularities in those regions

The Exodus method is one of Monte Carlo methodswhich are nondeterministic (probabilistic or stochastic)numerical methods employed in solving mathematical andphysical problems The fixed random walk and Exodusmethods are the most frequently used Monte Carlo methodsfor solving heat conduction and potential problems TheExodus method is however preferred to fixed random walkmethod because of its computational efficiency It yieldsmore

Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2014 Article ID 126905 6 pageshttpdxdoiorg1011552014126905

2 Journal of Computational Engineering

120585 = constant

120578 = constant

120578

120578 = 0

120601 = 0

120601 = constant

x

120578 = 120587

120601 = 1205872

z

y0 c998400c998400

Figure 1 The oblate spheroidal coordinate system [9]

accurate results with less computing time as compared to theoriginal Monte Carlo method [7 8]

2 Oblate Spheroidal Coordinate Systems

The geometry of an oblate spheroid showing surfaces of con-stant oblate spheroidal coordinates is illustrated in Figure 1The oblate spheroidal coordinates are related to the rectangu-lar coordinates as follows

119909 = 1198881015840 cosh 120585 sin 120578 cos120593

119910 = 1198881015840 cosh 120585 sin 120578 sin120593

119909 = 1198881015840 sinh 120585 cos 120578

(1)

where 1198881015840 is the focal length of the oblate spheroidAn arbitrary grid point 119903 on the oblate spheroid is given

by

119903119894119895119896

= 120585119894119886120585+ 120578119895119886120578+ 120601119896119886120601 (2)

where 119886120585 119886120578 and 119886

120601are the unit vectors in the direction of 120585

120578 and 120601 coordinates respectivelyThe respective scale factor for each of the three coordi-

nates (120585 120578 120601) is

ℎ120585= 1198881015840radicsinh2120585 + sin2120578

ℎ120578= 1198881015840radicsinh2120585 + sin2120578

ℎ120601= 1198881015840 cosh 120585 cos 120578

(3)

0

z

y

1205852

1205851

100V

50V

Figure 2 Two oblate spheroidal shells

3 Potential Distribution Computation

31 Finite Difference Transformation of Oblate SpheroidLaplacersquos Equation The Laplacian equation in oblatespheroidal coordinate systems is

0 = nabla2119881 = (

1

sinh2120585 + sin2120578)

lowast [(sech2120585 tan2120578 + sec2120578 tanh2120585) 1205972119881

1205971206012

+1205972119881

1205971205852+ tanh 120585120597119881

120597120585+1205972119881

1205971205782minus tan 120578120597119881

120597120578]

(4)

The term outside the square bracket may be ignored Alsothe first term inside square bracket is ignored due to therotational symmetry about the vertical (119911) axisTherefore (4)reduces to

1205972119881

1205971205852+ tanh 120585120597119881

120597120585+1205972119881

1205971205782minus tan 120578120597119881

120597120578= 0 (5)

Equation (5) governs potential distribution in an axisymmet-ric oblate spheroid potential problem

Two oblate spheroidal shells made up of two constantconducting surfaces 120585

1and 120585

2are shown in Figure 2 The

two equipotential surfaces are maintained at 50V and 100V(Dirichlet boundary conditions) respectively The choiceof these Dirichlet boundary conditions is arbitrary Anypotential value can be assigned Also the value of the constantoblate spheroidal surfaces equipotential that constitutes thetwo conducting shells are arbitrarily chosen as 120585

1= 10 and

1205851= 20 respectively

Journal of Computational Engineering 3

The explicit finite difference transformation of (5) is

119881 (119894 119895)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 119895) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 119895) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 119895 + 1) [1

(Δ120578)2minustan (120578

119895)

2Δ120578]

+119881 (119894 119895 minus 1) [1

(Δ120578)2+tan (120578

119895)

2Δ120578]]

(6)

Figure 3 shows one-quarter of the constant oblate spheroidalsurfaces The figure exhibits symmetry with respect to the120578 coordinate Therefore two lines of symmetries will beencountered in this range of 120578 They are 120578 = 90∘ and 120578 = 0∘On these lines of symmetries the condition 120597119881120597120578 = 0

is imposed This strategy eliminates the singularity causingterm (tan(120578

119895)2Δ120578) at the oblate spheroid poles as seen in (6)

Consequently the finite difference equations along thetwo lines of symmetries become as follows

Along 120578 = 90∘ 119895 = 119895max

119881 (119894 119895max)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 119895max) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 119895max) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+2119881 (119894 119895max minus 1) [1

(Δ120578)2]]

(7)

Along 120578 = 0∘ 119895 = 0

119881 (119894 0)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 0) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 0) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+2119881 (119894 1) [1

(Δ120578)2]]

(8)

32 Transition Probabilities Determination In a more com-pact form (6) can be rewritten as

119881 (119894 119895) = 119875120585+119881 (119894 + 1 119895) + 119875

120585minus119881 (119894 minus 1 119895)

+ 119875120578+119881 (119894 119895 + 1) + 119875

120578minus119881 (119894 119895 minus 1)

(9)

where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10b)

119875120578minus= 05(

[(1(Δ120578)2

) + (tan (119895Δ120578) 2Δ120578)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10c)

119875120578minus= 05(

[(1(Δ120578)2

) minus (tan (119895Δ120578) 2Δ120578)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10d)

Note that 119875120585++ 119875120585minus+ 119875120578++ 119875120578minus= 1 Equations (10a) (10b)

(10c) and (10d) serve as the transition probabilities for theExodusmethod used in this work If119873 particles are dispersedat node (120585

0 1205780) they have probabilities 119875

120585+ 119875120585minus 119875120578+ and 119875

120578minus

of moving to points (120585 + Δ120585 120578) (120585 minus Δ120585 120578) (120585 120578 + Δ120578)and (120585 120578 minus Δ120578) respectively The direction of movement ofthe particles is determined by generating a random number119880 0 lt 119880 lt 1 The particles are instructed to walk as follows

(1205850 1205780) 997904rArr (120585

0+ Δ120585 120578

0) if (0 lt 119880 lt 119875

120585+)

(1205850 1205780) 997904rArr (120585

0minus Δ120585 120578

0) if (119875

120585+lt 119880 lt 119875

119879)

(1205850 1205780) 997904rArr (120585

0 1205780+ Δ120578) if (119875

119879lt 119880 lt 119875

119879119879)

(1205850 1205780) 997904rArr (120585

0 1205780minus Δ120578) if (119875

119879119879lt 119880 lt 1)

(11)

where

119875119879= 119875120585++ 119875120585minus 119875

119879119879= 119875120585++ 119875120585minus+ 119875120578+ (12)

For the random walk computation along the line ofsymmetry (120578 = 90∘) (7) becomes

119881 (119894 119895max) = 119875120585+119881 (119894 + 1 119895max) + 119875120585minus119881 (119894 minus 1 119895max)

+ 119875120578minus119881 (119894 119895max minus 1)

(13)


(120585 = 18 120578 = 87∘)

(120585 = 17 120578 = 84∘)(120585 = 16 120578 = 81∘)(120585 = 15 120578 = 78∘)

(120585 = 14 120578 = 75∘)(120585 = 13 120578 = 72∘)(120585 = 12 120578 = 69∘)

(120585 = 10 V0 = 50V)

120578 = 90∘

120578 = 0∘

(120585 = 2 V0 = 100V)

120585 = constant120578 = constant

Figure 3 Oblate spheroidal surface path used in computing the charge enclosed

where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14b)

119875120578minus= (

[(1(Δ120578)2

)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14c)

Note that 119875120585++ 119875120585minus+ 119875120578minus

= 1 along this line of symmetryIf a random-walking particle is instantaneously at the point(1205850 90∘) it has probabilities 119875

120585+ 119875120585minus and 119875

120578minusof moving to

points (120585+Δ120585 90∘) (120585minusΔ120585 90∘) and (120585 90∘minusΔ120578) respectivelyThe particles are instructed to walk as follows

(1205850 90∘) 997904rArr (120585

0+ Δ120585 90

∘) if (0 lt 119880 lt 119875

120585+)

(1205850 90∘) 997904rArr (120585

0minus Δ120585 90

∘) if (119875

120585+lt 119880 lt 119875max)

(1205850 90∘) 997904rArr (120585

0 90∘minus Δ120578) if (119875max lt 119880 lt 1)

(15)

where

119875max = 119875120585+ + 119875120585minus (16)

For the random walk computation along the line ofsymmetry (120585

0 0∘) (8) becomes

119881 (119894 0) = 119875120585+119881 (119894 + 1 0) + 119875

120585minus119881 (119894 minus 1 0)

+ 119875120578+119881 (119894 1)

(17)

The radial parts of the transition probabilities (119875120585+

and 119875120585minus)

in (17) are the same as those in (14a) and (14b)The transitionprobability for the 120578 coordinate component of (17) is

119875120578+= (

[1(Δ120578)2

]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (18)

Note also that 119875120585++ 119875120585minus+ 119875120578+

= 1 For dispersed particlesat point (120585

0 0∘) they have probabilities 119875

120585+ 119875120585minusand 119875

120578+of

moving to points (120585 + Δ120585 0∘) (120585 minus Δ120585 0∘) and (120585 Δ120578)

respectively The particle walking pattern of the particles isas follows

(1205850 0) 997904rArr (120585

0+ Δ120585 0) if (0 lt 119880 lt 119875

120585+)

(1205850 0) 997904rArr (120585

0minus Δ120585 0) if (119875

120585+lt 119880 lt 119875

1198790)

(1205850 0) 997904rArr (120585

0 Δ120579) if (119875

1198790lt 119880 lt 1)

(19)

where

1198751198790= 119875120585++ 119875120585minus (20)

The potential at a specific point (1205850 1205780) is to be determined

We define the transition probability 119875119899as the probability that

a random walk starting at the point of interest (1205850 1205780) ends at

the boundary node (120585119899 120578119899) with prescribed potential (119881

119887(119899))

Large particles are dispatched at the free node (1205850 1205780)

The application of Exodus method begins by setting particles119875(120585119894 120578119895) = 0 at all other nodes (both fixed and free) except

at the free node (1205850 1205780) where we assume a large value119873 By

scanning the mesh as in finite difference analysis (FDM) theparticles are dispatched at each free node to its neighboringnodes according to the random walk transition probabilitiesdescribed above Detailed description of this process is givenin [10] If 119873 and 119873

119899designate the total number of particles

dispersed and the number of particles that have reached theboundary 119899 respectively the probability that a random walkterminates on the boundary is

119875119899=119873119899

119873 (21)

If there are 119872 boundaries or fixed nodes (excluding thecorner points) the potential at the specified node (120585

0 1205780) is

119881 (1205850 1205780) =

119872

sum119899=1

119875119899119881119887(119899) =

sum119872

119899=1119873119899119881119887(119899)

119873 (22)

Since there are just two boundaries (the inner and outeroblate shell surfaces) in this potential problem (22) simplifiesto

119881 (1205850 1205780) =

(1198731119881119887(1) + 119873

2119881119887(2))

119873 (23)


Table 1 Comparing the Exodus solution with finite difference and exact solution

Potential point Exact potential (V) Exodus methodpotential (V)

Finite difference (FDM)potential (V)120585 120578

11 66∘ 571513 571542 57153612 69∘ 637529 637577 63756613 72∘ 698269 698328 69831314 75∘ 753997 754060 75404315 78∘ 805005 805067 80505016 81∘ 851601 851656 85164217 84∘ 894096 894141 89412918 87∘ 932798 932830 93282219 87∘ 968004 968021 968017

where 119881119887(1) and 119881

119887(2) are the prescribed potentials at the

inner and outer oblate spheroidal shells respectivelyThe equation for the exact solution of the potential

computation in oblate spheroidal shells is [9]

119881 = 119881119887(1) + (119881

119887(2) minus 119881

119887(1))

lowast ([gd (120585) minus gd (120585

1)]

gd (1205852) minus gd (120585

1))

(24)

where 119881119887(1) and 119881

119887(2) represent the nonzero potential

(Dirichlet boundary conditions) at the inner and outerconducting oblate spheroidal shells respectively as shown inFigure 3 Also gd(sdot) denotes the Gudermannian function [6]and is represented as

gd (120585) = sinminus1 (tanh (120585)) (25)

4 Results

Equations (10a)ndash(23) and (24)-(25) are used to implementthe computation of potential distributions within two con-ducting oblate spheroidal shells numerically and analyticallyrespectively The results obtained are as shown in Table 1The results obtained from exact solution (analytical) theExodus method and the explicit finite difference solutionsare compared in Table 1 Same step size was used for boththe Exodus method and the FDM and this accounted for thecloseness in the computed results obtainedThe Exodus solu-tion results were very close to the results obtained from exactsolution because of the fact that though the Exodus methodis a probabilistic method its operation does not depend onrandom number generation which ultimately depends on thecomputation accuracy of the machine involved

5 Conclusion

The use of the Exodus method to compute potential distri-bution inside two conducting oblate spheroidal shells main-tained at two potentials has been implemented in this paperThe results obtained agreed with those obtained using finitedifference (FDM) solution and the exact solution method

The Exodus method employed in this work can be said to bealmost as accurate as the exactmethodwhen compared to thefixed random walk Monte Carlo method because the resultsobtained were as accurate as the exact method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W LGates ldquoDerivation of the equations of atmosphericmotionin Oblate Spheroidal Coordinatesrdquo Journal of the AtmosphericSciences vol 61 pp 2478ndash2487 2004

[2] A M Al-Jumaily and F M Najim ldquoAn approximation to thevibrations of oblate spheroidal shellsrdquo Journal of Sound andVibration vol 204 no 4 pp 561ndash574 1997

[3] V Punzo S Besio S Pittaluga and A Trequattrini ldquoSolutionof Laplace equation on non axially symmetrical volumesrdquo IEEETransactions on Applied Superconductivity vol 16 no 2 pp1815ndash1818 2006

[4] R F Tuttle and S K Loyalka ldquoGravitational collision efficiencyof nonspherical aerosols II motion of an oblate spheroid in aviscous fluidrdquo Nuclear Technology vol 69 no 3 pp 327ndash3361985

[5] P W Casper and R B Bent ldquoThe effect of the earthrsquos oblatespheroid shape on the accuracy of a time-of-arrival lightningground strike locating systemrdquo in Proceedings of the Interna-tional Aerospace and Ground Conference on Lightning and StaticElectricity vol 2 pp 811ndash818 The National Aeronautics andSpace Administration The National Interagency CoordinationGroup (NICG) and Florida Institute 1991

[6] O D Momoh M N O Sadiku and S M Musa ldquoA fixedrandom walk Monte Carlo computation of potential inside twoconducting oblate spheroidal shellsrdquo in Proceedings of the IEEESoutheastcon pp 196ndash200 Nashville Tenn USA March 2011

[7] O D Momoh M N O Sadiku and C M Akujuobi ldquoSolutionof axisymmetric potential problem in spherical coordinatesusing Exodus methodrdquo in Proceedings of the PIERS Conferencepp 1110ndash1114 Cambridge Mass USA July 2010


[8] M N O SadikuNumerical Techniques in Electromagnetics withMATLAB CRC Press Boca Raton Fla USA 3rd edition 2009

[9] R S Alassar and H M Badr ldquoAnalytical solution of oscillatinginviscid flow over oblate spheroids with spheres and flat disksas special casesrdquo Ocean Engineering vol 24 no 3 pp 217ndash2251997

[10] O D Momoh M N O Sadiku and C M Akujuobi ldquoPotentialdistribution computation in conducting prolate spheroidalshells using the exodus methodrdquo IEEE Transactions on Magnet-ics vol 47 no 5 pp 1426ndash1429 2011

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120585 = constant

120578 = constant

120578

120578 = 0

120601 = 0

120601 = constant

x

120578 = 120587

120601 = 1205872

z

y0 c998400c998400

Figure 1 The oblate spheroidal coordinate system [9]

accurate results with less computing time as compared to theoriginal Monte Carlo method [7 8]

2 Oblate Spheroidal Coordinate Systems

The geometry of an oblate spheroid showing surfaces of con-stant oblate spheroidal coordinates is illustrated in Figure 1The oblate spheroidal coordinates are related to the rectangu-lar coordinates as follows

119909 = 1198881015840 cosh 120585 sin 120578 cos120593

119910 = 1198881015840 cosh 120585 sin 120578 sin120593

119909 = 1198881015840 sinh 120585 cos 120578

(1)

where 1198881015840 is the focal length of the oblate spheroidAn arbitrary grid point 119903 on the oblate spheroid is given

by

119903119894119895119896

= 120585119894119886120585+ 120578119895119886120578+ 120601119896119886120601 (2)

where 119886120585 119886120578 and 119886

120601are the unit vectors in the direction of 120585

120578 and 120601 coordinates respectivelyThe respective scale factor for each of the three coordi-

nates (120585 120578 120601) is

ℎ120585= 1198881015840radicsinh2120585 + sin2120578

ℎ120578= 1198881015840radicsinh2120585 + sin2120578

ℎ120601= 1198881015840 cosh 120585 cos 120578

(3)

0

z

y

1205852

1205851

100V

50V

Figure 2 Two oblate spheroidal shells

3 Potential Distribution Computation

31 Finite Difference Transformation of Oblate SpheroidLaplacersquos Equation The Laplacian equation in oblatespheroidal coordinate systems is

0 = nabla2119881 = (

1

sinh2120585 + sin2120578)

lowast [(sech2120585 tan2120578 + sec2120578 tanh2120585) 1205972119881

1205971206012

+1205972119881

1205971205852+ tanh 120585120597119881

120597120585+1205972119881

1205971205782minus tan 120578120597119881

120597120578]

(4)

The term outside the square bracket may be ignored Alsothe first term inside square bracket is ignored due to therotational symmetry about the vertical (119911) axisTherefore (4)reduces to

1205972119881

1205971205852+ tanh 120585120597119881

120597120585+1205972119881

1205971205782minus tan 120578120597119881

120597120578= 0 (5)

Equation (5) governs potential distribution in an axisymmet-ric oblate spheroid potential problem

Two oblate spheroidal shells made up of two constantconducting surfaces 120585

1and 120585

2are shown in Figure 2 The

two equipotential surfaces are maintained at 50V and 100V(Dirichlet boundary conditions) respectively The choiceof these Dirichlet boundary conditions is arbitrary Anypotential value can be assigned Also the value of the constantoblate spheroidal surfaces equipotential that constitutes thetwo conducting shells are arbitrarily chosen as 120585

1= 10 and

1205851= 20 respectively



119881 (119894 119895)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 119895) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 119895) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 119895 + 1) [1

(Δ120578)2minustan (120578

119895)

2Δ120578]

+119881 (119894 119895 minus 1) [1

(Δ120578)2+tan (120578

119895)

2Δ120578]]

(6)





Along 120578 = 90∘ 119895 = 119895max

119881 (119894 119895max)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 119895max) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 119895max) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+2119881 (119894 119895max minus 1) [1

(Δ120578)2]]

(7)

Along 120578 = 0∘ 119895 = 0

119881 (119894 0)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 0) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 0) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+2119881 (119894 1) [1

(Δ120578)2]]

(8)


119881 (119894 119895) = 119875120585+119881 (119894 + 1 119895) + 119875

120585minus119881 (119894 minus 1 119895)

+ 119875120578+119881 (119894 119895 + 1) + 119875

120578minus119881 (119894 119895 minus 1)

(9)

where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10b)

119875120578minus= 05(

[(1(Δ120578)2

) + (tan (119895Δ120578) 2Δ120578)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10c)

119875120578minus= 05(

[(1(Δ120578)2

) minus (tan (119895Δ120578) 2Δ120578)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10d)




120585+ 119875120585minus 119875120578+ and 119875

120578minus


(1205850 1205780) 997904rArr (120585

0+ Δ120585 120578

0) if (0 lt 119880 lt 119875

120585+)

(1205850 1205780) 997904rArr (120585

0minus Δ120585 120578

0) if (119875

120585+lt 119880 lt 119875

119879)

(1205850 1205780) 997904rArr (120585

0 1205780+ Δ120578) if (119875

119879lt 119880 lt 119875

119879119879)

(1205850 1205780) 997904rArr (120585

0 1205780minus Δ120578) if (119875

119879119879lt 119880 lt 1)

(11)

where

119875119879= 119875120585++ 119875120585minus 119875

119879119879= 119875120585++ 119875120585minus+ 119875120578+ (12)



+ 119875120578minus119881 (119894 119895max minus 1)

(13)


(120585 = 18 120578 = 87∘)

(120585 = 17 120578 = 84∘)(120585 = 16 120578 = 81∘)(120585 = 15 120578 = 78∘)

(120585 = 14 120578 = 75∘)(120585 = 13 120578 = 72∘)(120585 = 12 120578 = 69∘)

(120585 = 10 V0 = 50V)

120578 = 90∘

120578 = 0∘

(120585 = 2 V0 = 100V)



where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14b)

119875120578minus= (

[(1(Δ120578)2

)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14c)



120585+ 119875120585minus and 119875



(1205850 90∘) 997904rArr (120585

0+ Δ120585 90

∘) if (0 lt 119880 lt 119875

120585+)

(1205850 90∘) 997904rArr (120585

0minus Δ120585 90

∘) if (119875

120585+lt 119880 lt 119875max)

(1205850 90∘) 997904rArr (120585


(15)

where

119875max = 119875120585+ + 119875120585minus (16)


0 0∘) (8) becomes

119881 (119894 0) = 119875120585+119881 (119894 + 1 0) + 119875

120585minus119881 (119894 minus 1 0)

+ 119875120578+119881 (119894 1)

(17)


and 119875120585minus)


119875120578+= (

[1(Δ120578)2

]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (18)




120585+ 119875120585minusand 119875

120578+of



(1205850 0) 997904rArr (120585

0+ Δ120585 0) if (0 lt 119880 lt 119875

120585+)

(1205850 0) 997904rArr (120585

0minus Δ120585 0) if (119875

120585+lt 119880 lt 119875

1198790)

(1205850 0) 997904rArr (120585

0 Δ120579) if (119875

1198790lt 119880 lt 1)

(19)

where

1198751198790= 119875120585++ 119875120585minus (20)





119887(119899))







119875119899=119873119899

119873 (21)


0 1205780) is

119881 (1205850 1205780) =

119872

sum119899=1

119875119899119881119887(119899) =

sum119872

119899=1119873119899119881119887(119899)

119873 (22)


119881 (1205850 1205780) =

(1198731119881119887(1) + 119873

2119881119887(2))

119873 (23)





11 66∘ 571513 571542 57153612 69∘ 637529 637577 63756613 72∘ 698269 698328 69831314 75∘ 753997 754060 75404315 78∘ 805005 805067 80505016 81∘ 851601 851656 85164217 84∘ 894096 894141 89412918 87∘ 932798 932830 93282219 87∘ 968004 968021 968017

where 119881119887(1) and 119881




119881 = 119881119887(1) + (119881

119887(2) minus 119881

119887(1))


1)]

gd (1205852) minus gd (120585

1))

(24)

where 119881119887(1) and 119881



gd (120585) = sinminus1 (tanh (120585)) (25)

4 Results


5 Conclusion





References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119881 (119894 119895)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 119895) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 119895) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 119895 + 1) [1

(Δ120578)2minustan (120578

119895)

2Δ120578]

+119881 (119894 119895 minus 1) [1

(Δ120578)2+tan (120578

119895)

2Δ120578]]

(6)





Along 120578 = 90∘ 119895 = 119895max

119881 (119894 119895max)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 119895max) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 119895max) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+2119881 (119894 119895max minus 1) [1

(Δ120578)2]]

(7)

Along 120578 = 0∘ 119895 = 0

119881 (119894 0)

= 05 [

[

1

[(1(Δ120585)2

) + (1(Δ120578)2

)]

]

]

lowast [119881 (119894 + 1 0) [1

(Δ120585)2+tanh (119894Δ120585)2Δ120585

]

+ 119881 (119894 minus 1 0) [1

(Δ120585)2minustanh (119894Δ120585)2Δ120585

]

+2119881 (119894 1) [1

(Δ120578)2]]

(8)


119881 (119894 119895) = 119875120585+119881 (119894 + 1 119895) + 119875

120585minus119881 (119894 minus 1 119895)

+ 119875120578+119881 (119894 119895 + 1) + 119875

120578minus119881 (119894 119895 minus 1)

(9)

where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10b)

119875120578minus= 05(

[(1(Δ120578)2

) + (tan (119895Δ120578) 2Δ120578)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10c)

119875120578minus= 05(

[(1(Δ120578)2

) minus (tan (119895Δ120578) 2Δ120578)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (10d)




120585+ 119875120585minus 119875120578+ and 119875

120578minus


(1205850 1205780) 997904rArr (120585

0+ Δ120585 120578

0) if (0 lt 119880 lt 119875

120585+)

(1205850 1205780) 997904rArr (120585

0minus Δ120585 120578

0) if (119875

120585+lt 119880 lt 119875

119879)

(1205850 1205780) 997904rArr (120585

0 1205780+ Δ120578) if (119875

119879lt 119880 lt 119875

119879119879)

(1205850 1205780) 997904rArr (120585

0 1205780minus Δ120578) if (119875

119879119879lt 119880 lt 1)

(11)

where

119875119879= 119875120585++ 119875120585minus 119875

119879119879= 119875120585++ 119875120585minus+ 119875120578+ (12)



+ 119875120578minus119881 (119894 119895max minus 1)

(13)


(120585 = 18 120578 = 87∘)

(120585 = 17 120578 = 84∘)(120585 = 16 120578 = 81∘)(120585 = 15 120578 = 78∘)

(120585 = 14 120578 = 75∘)(120585 = 13 120578 = 72∘)(120585 = 12 120578 = 69∘)

(120585 = 10 V0 = 50V)

120578 = 90∘

120578 = 0∘

(120585 = 2 V0 = 100V)



where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14b)

119875120578minus= (

[(1(Δ120578)2

)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14c)



120585+ 119875120585minus and 119875



(1205850 90∘) 997904rArr (120585

0+ Δ120585 90

∘) if (0 lt 119880 lt 119875

120585+)

(1205850 90∘) 997904rArr (120585

0minus Δ120585 90

∘) if (119875

120585+lt 119880 lt 119875max)

(1205850 90∘) 997904rArr (120585


(15)

where

119875max = 119875120585+ + 119875120585minus (16)


0 0∘) (8) becomes

119881 (119894 0) = 119875120585+119881 (119894 + 1 0) + 119875

120585minus119881 (119894 minus 1 0)

+ 119875120578+119881 (119894 1)

(17)


and 119875120585minus)


119875120578+= (

[1(Δ120578)2

]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (18)




120585+ 119875120585minusand 119875

120578+of



(1205850 0) 997904rArr (120585

0+ Δ120585 0) if (0 lt 119880 lt 119875

120585+)

(1205850 0) 997904rArr (120585

0minus Δ120585 0) if (119875

120585+lt 119880 lt 119875

1198790)

(1205850 0) 997904rArr (120585

0 Δ120579) if (119875

1198790lt 119880 lt 1)

(19)

where

1198751198790= 119875120585++ 119875120585minus (20)





119887(119899))







119875119899=119873119899

119873 (21)


0 1205780) is

119881 (1205850 1205780) =

119872

sum119899=1

119875119899119881119887(119899) =

sum119872

119899=1119873119899119881119887(119899)

119873 (22)


119881 (1205850 1205780) =

(1198731119881119887(1) + 119873

2119881119887(2))

119873 (23)





11 66∘ 571513 571542 57153612 69∘ 637529 637577 63756613 72∘ 698269 698328 69831314 75∘ 753997 754060 75404315 78∘ 805005 805067 80505016 81∘ 851601 851656 85164217 84∘ 894096 894141 89412918 87∘ 932798 932830 93282219 87∘ 968004 968021 968017

where 119881119887(1) and 119881




119881 = 119881119887(1) + (119881

119887(2) minus 119881

119887(1))


1)]

gd (1205852) minus gd (120585

1))

(24)

where 119881119887(1) and 119881



gd (120585) = sinminus1 (tanh (120585)) (25)

4 Results


5 Conclusion





References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(120585 = 18 120578 = 87∘)

(120585 = 17 120578 = 84∘)(120585 = 16 120578 = 81∘)(120585 = 15 120578 = 78∘)

(120585 = 14 120578 = 75∘)(120585 = 13 120578 = 72∘)(120585 = 12 120578 = 69∘)

(120585 = 10 V0 = 50V)

120578 = 90∘

120578 = 0∘

(120585 = 2 V0 = 100V)



where

119875120585minus= 05(

[(1(Δ120585)2

) + (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14a)

119875120585minus= 05(

[(1(Δ120585)2

) minus (tanh (119894Δ120585) 2Δ120585)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14b)

119875120578minus= (

[(1(Δ120578)2

)]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (14c)



120585+ 119875120585minus and 119875



(1205850 90∘) 997904rArr (120585

0+ Δ120585 90

∘) if (0 lt 119880 lt 119875

120585+)

(1205850 90∘) 997904rArr (120585

0minus Δ120585 90

∘) if (119875

120585+lt 119880 lt 119875max)

(1205850 90∘) 997904rArr (120585


(15)

where

119875max = 119875120585+ + 119875120585minus (16)


0 0∘) (8) becomes

119881 (119894 0) = 119875120585+119881 (119894 + 1 0) + 119875

120585minus119881 (119894 minus 1 0)

+ 119875120578+119881 (119894 1)

(17)


and 119875120585minus)


119875120578+= (

[1(Δ120578)2

]

[(1(Δ120585)2

) + (1(Δ120578)2

)]) (18)




120585+ 119875120585minusand 119875

120578+of



(1205850 0) 997904rArr (120585

0+ Δ120585 0) if (0 lt 119880 lt 119875

120585+)

(1205850 0) 997904rArr (120585

0minus Δ120585 0) if (119875

120585+lt 119880 lt 119875

1198790)

(1205850 0) 997904rArr (120585

0 Δ120579) if (119875

1198790lt 119880 lt 1)

(19)

where

1198751198790= 119875120585++ 119875120585minus (20)





119887(119899))







119875119899=119873119899

119873 (21)


0 1205780) is

119881 (1205850 1205780) =

119872

sum119899=1

119875119899119881119887(119899) =

sum119872

119899=1119873119899119881119887(119899)

119873 (22)


119881 (1205850 1205780) =

(1198731119881119887(1) + 119873

2119881119887(2))

119873 (23)





11 66∘ 571513 571542 57153612 69∘ 637529 637577 63756613 72∘ 698269 698328 69831314 75∘ 753997 754060 75404315 78∘ 805005 805067 80505016 81∘ 851601 851656 85164217 84∘ 894096 894141 89412918 87∘ 932798 932830 93282219 87∘ 968004 968021 968017

where 119881119887(1) and 119881




119881 = 119881119887(1) + (119881

119887(2) minus 119881

119887(1))


1)]

gd (1205852) minus gd (120585

1))

(24)

where 119881119887(1) and 119881



gd (120585) = sinminus1 (tanh (120585)) (25)

4 Results


5 Conclusion





References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













11 66∘ 571513 571542 57153612 69∘ 637529 637577 63756613 72∘ 698269 698328 69831314 75∘ 753997 754060 75404315 78∘ 805005 805067 80505016 81∘ 851601 851656 85164217 84∘ 894096 894141 89412918 87∘ 932798 932830 93282219 87∘ 968004 968021 968017

where 119881119887(1) and 119881




119881 = 119881119887(1) + (119881

119887(2) minus 119881

119887(1))


1)]

gd (1205852) minus gd (120585

1))

(24)

where 119881119887(1) and 119881



gd (120585) = sinminus1 (tanh (120585)) (25)

4 Results


5 Conclusion





References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









solution of axisymmetric potential problem in oblate spheroid

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