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IEEE TRANSACTIONS ON EDUCATION, VOL. 32 , NO . 2, MAY 1989 85

A Simple Introduction to Finite ElementElectromagnetic Problems

MAT T HE W N . 0. SADIKU

Abstract-This paper is a tutorial introduction for an absolute be-ginner in finite element numerical analysis. The finite element methodis applied to Laplacian electrostatic field problems. Suggestions areoffered on how the basic concepts developed can be extended to finiteelement analysis of problem involving Poissons or wave equation. Astep-by-step procedure for coding the num erical method is presented;a useful, working Fortran program is also included.

I. INTRODUCTIONH E finite elem ent method has its origin in the field ofT tructural analysis. Although the earlier mathematicaltreatment of the method was provided by Courant [ l ] in1943, the method was not applied to electromagnetic ( EM )problems until 1968. With the increasing application ofthe method in the numerical solution of continuum prob-lems (of which EM problems are an integral part), an ef-fort to apply the method in both undergraduate and grad-uate EM courses has gained momentum [2]-[4].

So far, little attempt has been made to present finiteelement method i n simple terms which undergraduate stu-dents in electrical engineering can understand. It is dis-appointing to note that while a large amount of researchpapers and textbooks on the subject have been publishedby civil and mechanical engineers, it is hard for an elec-trical engineering student or a practicing electrical engi-neer, who is an absolute beginner in using finite elementmethod, to get a simple, brief introductory material. Civilor mechanical engineering materials on finite elementmethod are not easy for electrical engineers to under-stand. Th is is due to the fact that the terminology andnotation used in those materials are usually inconsistentand unfam iliar to electrical engin eers. It has been the gen-eral belief among electrical engineers that the method isno t so easy for undergraduates to grasp. The students aremerely asked to use an existing finite element program togenerate solutions to specific problem s assigned [3], [4].This is justified on the ground that numerical methodsconstitute a fragm ent of an EM course and that the finiteelement method involves calculus of variation which thestudents may not be familiar with at the undergraduatelevel. Asking students to perform finite element analysiswith either packaged comm ercial codes or faculty-devel-oped program s can be dangerous if care is not taken. It is

Manuscript received January 18, 1989.The author is w i t h the Department of Electr ical Engineering, TempleIEEE Lo g Number 8927100.University, Philadelphia, P A 19122.

like asking students to usecourse in networks. Although

Analysis of

SPICE in an introductorymost co mmercial finite ele-ment packages are easy to use, yet pitfalls, not widelyappreciated, exist. W hile, it is unreasonable to expect thatstudents armed with only an introductory knowledge offinite element method write comm ercial codes, a basic un-derstanding of the method and its limitations is vital.This paper presents a tutorial introduction to the finiteelement method without requiring the reader to be famil-iar with variational calculus. This not only helps the cu-rious user of an existing finite element program under-stand the mathematical basis of the method, it gives himthe ability to modify the program if necessary or develophis own cod e. It is hoped that the paper will help the readerwho is first being introduced to the finite element analy-sis.Most EM problems involve either partial differentialequations or integral equations. While partial differentialequations are usually solved using the finite differencemethod or finite element method, integral equations aresolved conveniently using moment method [5]. In con-trast to other m ethods, the finite element method accountsfor nonhomogeneity of the solution region. The system-atic generality of the method makes it a versatile tool fora wide range of problems. As a result, flexible general-purpose com puter programs can be constructed.The finite element analysis of any problem involves ba-sically four steps: 1 ) discretizing the solution region intofinite num ber of subreg ions or elements, 2) deriving gov-erning equations for a typical element, 3) assembling ofall elements in the solution region, 4) nd solving the sys-tem of equations obtained. Each of these steps will bediscussed briefly in the subsequent sections.

11. FIN ITE LEM ENT ISCRETIZA TIONWe divide the solution region into a number of -finiteelements as illustrated in Fig. 1 where the region is sub-divided into four nonov erlapping elements (tw o triangularand two quadrilateral) and seven nodes. We seek an ap-proximation for the potential Ve within an element e an dthen interrelate the potential distributions in various ele-ments such that the potential is continuous across inter-element boundaries. The approximate solution for the

whole region is NV ( x , Y ) = 2 V,(& Y ) ( 1 )e = I

0018-9359/89/0500-0085$01OO O 1989 IE E E

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86 IEEE TRANSACTIONS ON EDUCATION, VOL. 32, NO . 2, MAY 1989

1

II7

i node no.@ element no

Actual boundary

Approximate boundary6

Fig. 1 . A typical finite element subdivision of an irregular dom ain.where N is the numbe r of tr iangular elements into whichthe solution region is divided.The most common form of approximation for V withinan element is polynomial approximation, namely

V , ( x , y ) = a + bx + cyV , ( x , y ) = a + bx + cy + dxy

( 2 )

( 3 )for tr iangular element and

for quadrilateral element. The potential V , in general isnonzero within element e but zero outside e . In view ofthe fact that quadrilateral elements are in general noncon-forming elements (se e Fig. l) , we prefer to use tr iangularelements throughout our analysis in this paper. Notice thatour assumption of linear variation of potential within thetriangular element as in (2 ) is the same as assuming thatthe electric field is uniform within the elem ent, i. e. ,+E , = -VV, = -(b.',, + c. ' , ) . ( 4 )

111. EL EM ENT OV ER NIN GQUATIONSConsider a typical tr iangular element shown in Fig . 2 .The potential Vel, V e 2 ,an d V, , at nodes 1, 2 , an d 3 , re-spectively, are obtained using ( 2 ) , . e . ,

The coefficients a , 6, an d c are determined from ( 5 ) as

Substituting this into (2 ) gives1

2A, = [ I x y ] -

y f

I t XFig . 2 . Typical triangular element; the local node numbering 1-2-3 mustproceed counterclockwise as indicated by the arrow.or

( 7 )where

orA = L[( x2 - X I ) (Y 3 - Y l ) - (x3 - X I ) (YZ - Y l ) ] .

( 9 )The value of A is positive if the nodes a re numbered coun-terclockwise (starting from any node) as shown by the ar-row in Fig . 2 . Note that (7 ) gives the potential at any point( x , y ) within the element provided that the potentials atthe vertices are known. This is unlike finite differenceanalysis where the potential is known at the grid pointsonly. Also note that a, are linear interpolation functions.They are called the element sha pef inctions and they havethe following properties [7]:

3c f f i ( X , y ) = 1i = 1

The shape functions a I an d a2 , or example, are illus-trated in F ig . 3 .

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S ADI KU: ANALYSIS OF E L E C T R O M A G N E T I C P R O B L E M S

1i l2 1

(a ) (b )F i g . 3 . Shape functions a , an d a2 or a tr iangular element.

The energy per unit length associated with the elemente is given by 161, i.e. ,

where a two-dimensional solution region free of charge( p , = 0) is assumed. But from (7),

3VV, = c V,,Va,. ( 1 2 )

I = I

Substituting (12) into (1 1 ) gives3 3

W , = t c c t V c , , [ i V a l . V a J d SI = [ J = I

If we define the term in brackets asCf = Va, VaJd S , ( 1 4 )

we may write (13) in matrix form asw, = + t [ Vel [ c]V, ] ( 1 5 )

where the superscript T denotes the transpose of the ma-trix,

( 1 6 ) a

and c\1;)cy C\;CC) cy; c:;)[c] [ l Cll2 Cb ( 1 6 ) b

The matrix [ C] s usually called the element coeficienfmatrix (or stiffness matrix in structural analysis). Thematrix element Cf of the coefficient matrix may be re-garded as the coupling between nodes i andj; its value isobtained from (8) and (14). For example,

C\; = 1V a l . Vcu,dS1--2 ( Y ? - 43) ( Y 3 - Y I )1

4A- ( Y z - 4 3 ) 0 3 - Y I )+ (x3 - (X I - X ? ) ] . ( 1 7 ) a

87

Similarly,

IV . ASSEMBLINGF ALL ELEMENTSHaving cons idered a typical ele ment, the next step is toassemble all such elements in the solution region. Theenergy associated with the assemblage of elements isN

w = c w, = ; t [ V fe = Iwhere

[ V I = [[I5n is the number of nodes, N is the number of elements,and [C ] is called the over-all or global coeficient matrixwhich is the assemb lage of individual elem ent coefficientmatrices.The process by which individual element coefficientmatrices are assembled to obtain the global coefficientmatrix is best illustrated with an example. Consider thefinite element mesh consisting of three finite elements asshown in F ig . 4. Observe the numberings of the mesh.The numbering of nodes as 1, 2 , 3 , 4, and 5 is calledglobal numbering. The numbering i-j-k is called localnumbering and it correspon ds with 1-2-3 of the elementin Fig. 2. For example, for element 3 in Fig. 4, the localnumbering 3-5 -4 corresponds with 1-2-3 of the elementin Fig . 2 . Note that the local numbering must be i n coun-terclockwise sequence starting from any node of the ele-ment. For element 3, we could choose 4-3-5 instead of3-5-4 to be correspond with 1-2-3 of the element in Fig.2 . Thus, the numbering i n Fig. 4 is not uniqu e. Assumingthe particular numbering in Fig. 4, the global coefficient

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88 IEEE TRANSACTIONS ON EDUCATION, VOL. 32 , NO . 2 , MAY 1989

Fig. 4. (1-2-3) of the element in Fig . 2 .matrix is expected to have the form

[ C l =

c 1 2 c13 c14 c15c21 c22 c23 c24 c2Sc31 c32 c33 c34 c3.5c 4 l c 4 2 c 4 3 c 4 4 c4.5

-G I c s 2 c s 3 cs4 c5s-

2) Since C, = 0 if no coupling exists between nodes iandj, i t is ev ident that for a large number of elements[ C ] becomes sparse and banded.3) It is singular. Although this is not so obvious, it ca nbe shown using the element coefficient matrix of (16)b.Adding columns 2 and 3 to column 1 yields ze ros in col-um n 1 in (16)b.numbering

which is a 5 X 5 matrix since five nodes ( n = 5 ) ar einvolved. Again, C, is the coupling between nodes i an dj . We obtain C, by using the fact that the potential distri-bution must be continuous across interelement bounda-ries. The contribution to the i , j position in [ C ] comesfrom all elements containing nodes i and j . For example,in Fig. 4, lements 1 and 2 have node 1 in common ; henceCI, = cl! + Cg .

Node 2 belongs to elem ent 1 only; hencec,, = cg. ( 2 2 ) b

( 2 2 ) cNode 4 elongs to elem ents 1, 2, and 3; hence

c, = cp + cp + c gNodes 1 and 4 elong simultaneously to elemen ts 1 and2; hence

Cl4 = c,, = cl: + c\:). ( 2 2 ) dSince there is no coupling (or direct link) between nodes2 and 3,

c 2 3 = c32 = 0 . ( 2 2 ) eContinuing in this manner, we obtain all the terms in theglobal coefficient matrix by inspection of Fig . 4 s

IV . SOLVINGH E RESULTING Q UATIO NSIt can be shown that Laplaces (or Poissons) equationis satisfied when the total energy in the solution region isminimum. Thus, we require that the partial derivatives ofW with respect to each nodal value of the potential bezero , i .e . ,

- 0aww awav, av2 a v n. . . . . . . . . -_ _ -or

For example, to get a W / a V , = 0 for the finite elementmesh of Fig. 4,we substitute (21) into (19) and take thepartial derivative of W with respect to V I . We obtain

awav,= - = 2 1 / IC11 + v 2 c 1 2 + v3c13 + v 4 c 1 4 + v5c15+ v 2 c 2 1 + v3c31 + v4c41 + v5c51

or0 = VICII + V 2 C 1 2 + V3CI-j + I/,C14 + VSCl,. ( 2 5 )In general, dW / aV, = 0 leads to

I1

0 = c KC;, ( 2 6 )i = Iwhere n is the number of nodes in the mesh. By writing(26) for all nodes k = 1 , 2 , . . . . . , n , we obtain a setof simultaneous equations from which the solution of[ V ] T = V I , v 2 , , V I , ] an be found. T his can bedone in two ways similar to those used in solving finite

Note that element coefficient matrices overlap at nodesshared by elements and that there are 27 terms (9 for eachof the 3 elements) in the global coefficient matrix [ C ] .Also note the following properties of the matrix [ C]:1) It is symmetric (C , = Cji)ust a s the element coef-

difference equations obtained from Laplaces (or Pois-sons) equation [8] .A ) I teration Method: Suppose node 0 is connected tom nodes as shown in Fig . 5 . Using the idea in (25),

ficient matrix. 0 = v,,c,,+ v, ,,+ v, , ,2 + * * * - - * * * + VI,c,,,,

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SADIKU: ANALYSIS OF ELECTROMAGNETIC PROBLEMS 89

rnFig. 5 . Node 0 connected to m othe r nodes.

or. mIV, = -- c VkC,,.CO, k = l

Thus, if the potentials at nodes conn ected to 0 are known,we can determine V , using (27). Th e iteration process be-gins by setting the potentials at the free nodes (where thepotential is unknown) equal to zero or to the average po-tential [9]

where V,,, an d V,,, are the minimum and maximum val-ues of V at the$x ed nodes (wh ere the potential V is pre-scribed or known). With these initial values, the poten-tials at the free nodes are calculated using (27). At theend of the first iteration, when the new values have beencalculated for all the free nodes, they become the old val-ues for the second iteration. Th e proce dure is repeateduntil the change between subsequent iterations is neglibleenough.B ) Band Matrix Method: If all free nodes are num-bered first and the fixed nodes last, (19) can be writtensuch that

where subscripts and p , respectively, refer to nodes withfree and fixed (or prescribed) potentials . Since VP is con-stant (it consists of known, fixed values), we only differ-entiate with respect to Vf so that applying (24) to (29)yields [7]

This equation can be written as

where [ V I = [ V f l , [ A I = rcff1, B I = -[Cfp1 [V, I .Since [ A is, in general, nonsingular, the potential at thefree nodes can be found using (31). W e can solve for [ V Iin (3 )a using Gaussian elimination technique. We canalso solve for [ V I in (31)b using matrix inversion if thesize of the matrix to be inverted is not large.Notice that as from (1 1) onward, the solution has beenrestricted to two-dimensional problem involving La-place's equation, V V = 0. Th e basic concepts developedin this p aper can be extended to finite element analysis ofproQIems inyolving P oisson's equation ( v 2 = - p ( , / ~ ,V 2 A = - p J ) or wave equation ( V 2 + - y2+ = 0) .

V I. IL L UST RAT IVEX A M P L E SImplementation of the concepts discussed in the pre-ceeding sections is illustrated with tw o exam ples. Th e firstexample is a necessary preparation fo r the second wherethe numerical process is developed into a Fortran pro-gram. The code developed is simple and self-explanatorysince it is intended for instructional purposes. The nota-tions used in the program s are as close as possible to thoseused in previous sections; some are defined wh erever nec-essary.Example 1: Consider the two-element mesh shown inFig. 6( a). Using finite element method , determine the po-tentials within the mesh.Solution: Th e elem ent coefficient matrices can be cal-culated using (17) and (18). However, ou r calculationswill be easier if we definePI = (Y2 - Y3>,p2 = (Y3 - Yl), p3 = (Yl - Y 2 )QI = ( ~ 3 x2)3 Q2 = ( X I - x3), Q3 = (x2 - xi> .

( 3 2 )With Pi an d Qi ( i = 1, 2, 3 are the local node numbers),each term in the element coefficient matrix is found as

1C$') = 4~ P;P, + QiQ,] ( 3 3 )where A = : ( P 2 Q 3 - P 3 Q 2 ) . It is evident that (33) ismore convenient to use that (17) and (18). For e lement 1consisting of nodes 1-2-4 corresponding to the local num-bering 1-2-3 as in Fig. 6(b ),

PI = -1 . 3 P2 = 0 . 9 P3 = 0 . 4 ,QI = -0.2 Q2 = -0.4 Q3 = 0 . 6 ,A = i ( 0 . 5 4 + 0 . 1 6 ) = 0 . 3 5 .

Substituting all these into (33) gives[C" ' ] = -0.7786 0.6929 0.0857 . ( 3 4 )

[ V I = P I ( 3 1 b I0.4571 0.0857 0.3714Ir

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IEEE TRANSACTIONS ON EDUCATION, V O L . 32 . NO . 2. MAY 1989

follows:C 2 2 = C$: + C\: = 0.6929 + 0.5571 = 1.25C 2 4 = C$\ )+ C::) = 0.0857 - 0 .1 = -0.0143Node: ( x , y )1 (0.8.1.8)2 (1.4,1.4)

3 (2.1,Z.l)v = lg C4, = Cii) + Ci:) = 0.3714 + 0.4667 = 0.8381Of 4 (1.2J.7)C2l = C$l = -0.7786C23 = C\i) = -0.4571

lTv - 0 2

, X(a )

(b )Fig. 6. For example 1 : (a) two-element mesh, (b) local and global num-bering of the elements.

Similarly, for element 2 consisting of nodes 2-3-4 corre-sponding to local numbering 1-2-3 as in Fig. 6(b),PI = -0.6 P 2 = 1.3 P3 = -0.7,Q , = -0.9 Q2 = 0. 2 Q, = 0.7,A = i ( 0 . 9 1 + 0.14) = 0.525.

Hence,

I.5571 -0.4571 -0.10.8238 -0.3667 . (35 )-0.3667 0.4667Applying (30) gives

This can be written in a more convenient form as

or[C l [ V I = P I . ( 3 7 ) b

The terms of the global coefficient matrix are obtained as

C41 = Ci:) = -0.4571C43 = Cii = -0.3667.

Note that we follow local numbering for the elemen t coef-ficient matrix and global numbering for the global coef-ficient matrix. Thus, the square matrix [ C ] is obtained as

LO -0.0143 0 0 .8381Jand the matrix [ B ] on the right-hand side of (37)a is ob-tained as

[ B ] = [1i::71].3.667

(39)

By inverting matrix [ C ] in (38), we obtain

Thus, V , = 0, V 2 = 3.708, V 3 = 10, and V4 = 4.438.Once the values of the potentials at the nodes are known,the potential at any point within the mesh can be deter-mined using (7).Example 2: Write a Fortran program to solve the La-places equation using finite element method. Apply theprogram to the two-dimensional problem shown in Fig .7 (a ) .Solurion: The solution region is divided into 25 three-node triangular eleme nts with total num ber of nodes being21 as shown in Fig. 7 (b). This is a necessary s tep in orderto have input data defining the geometry of the problem.Based on the discussions in Sections 11-IV, a general For-tran program for solving problems involving Laplacesequation using three-node triangular elements is devel-oped as i n the Appendix. T he development of the program

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S A D I K U : A N A L Y S IS OF E L E C T R O M A G N E T I C PROBLEMS

'1.91

T A B L E IN O D A LC O O R D I N A T E SF T H E FINITE ELE M EN TE SH F FIG. 7.

Node Y Y Node r Y123456789

I O1 1

0. 0 0.00.2 0.00 . 4 0.00.6 0.00 . 8 0.01 . o 0.00. 0 0 .20.2 0 .20 . 4 0 . 20. 6 0. 20 . 8 0 . 2

12131415161718192021

0.0 0. 40 . 2 0.40 .4 0. 40. 6 0 . 40.0 0. 60 . 2 0.60 .4 0. 60.0 0.80 . 2 0. 80.0 1 .o

T A B L E I1E L E ME Y T - N O D EDEYTIFICATIO?IYzlk9 90

2 3

(b )Fig . 7. For example 2: (a) two-dimensional electrostatic pro blem, ( b ) so-lution region divided into 25 tr iangular elements.

basically involves four steps indicated in the program andexplained as follows.Step 1: This involves inputting the necessary data de-fining the problem. This is the only step that depends onthe geometry of the problem at hand. Th rough a data file,we input the number of elements, the number of nodes,the number of fixed nodes, the prescribed values of thepotentials at the fixed nodes, the x an d y coordinates ofall nodes, and a list identifying the nodes belonging toeach element in the order of the local numbering 1-2-3.For the problem in Fig. 7, the three sets of data for co-ordinates, element-node relationship, and prescribed po-tentials at fixed nodes are shown in Tables I , 11, an d 111,respectively. The data are inputted according to the cor-responding form at in the program in the App endix.Step 2: This step entails finding the element coefficientmatrix [ C ' " ] for each element and the global matrix [ C 1 .The procedure explained in the previous example is ap-plied. The matrix [ B ] n the right-hand side of (37) is

Local Node no. Local Node no.Element 1 2 3 Element I 2 3

~

also obtained at this stage. ysis, the potentials at the free nodes are obtained as

I 1 2 72 8 73 2 3 84 3 9 85 3 4 96 4 I O 94 5 I O8 5 I I I O9 5 6 I 17 8 12O

1 1 8 13 1212 8 9 139 14 133

141516171819202122232425

9 I O 14I O 15 14I O 1 1 1512 13 1613 17 1613 14 1714 18 1 714 15 1816 17 1917 20 1917 18 2019 20 21

T A B L E I11PRESCRIBED POTENTIALS AT F I X E D ODESPrescribedNode Potential PrescribedNode Potential

1 0.02 0.03 0.04 0.05 0. 06 50.0

I 1 100.015 100.0

18 100.02 0 100.021 50.019 0.016 0.012 0.07 0.0

Step 3: The global matrix obtained in the previous stepis inverted by calling subroutine INVERSE. The valuesof the potentials at all nodes are obtained by matrix mul-tiplication as in (31)b. Instead of inverting the global ma-trix, it is also possible to solve for the potentials at thenodes using Gaussian elimination technique.Step 4: This involves outputting the result of t h e com-putation. The output data for the problem in Fig. 7 ar epresented in Table IV .The validity of the result in Table IV is checked usingfinite difference method. From the finite difference anal-

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92TABLE IV

O U T P U T D A T A F TH E P R O G R A M N FIG. 7No. of Nodes = 21No . of Elements = 25No . of Fixed Nodes = 15Node X Y Potential

123456I89101112131415161718192021

0.000 .2 00 .4 00 .600 .801 oo0.000 .2 00 .4 00 .600 .800.000.200 .400 .600.000 .2 00 .4 00.000.200.00

0.000.000.000.000.000.000.200.200 .2 00.200 .200 .400 .400 .4 00 .4 00 .6 00 .6 00 .6 00.800 .801 .oo

0.0000.0000.0000.0000.00050.0000.00018.18236 .36459.091100.0000.00036.36468.182100.0000.00059.091

100.0000.000100.00050.000

Vs = 15.41 V9 = 26.74 VI0 = 56.69VI3 34.88VI4 = 65.41 VI, = 58.72 V .

Although the result obtained using finite difference isconsidered more accurate in this problem, increased ac-curacy of finite element analysis can be obtained by di-viding the solution region into greater number of trian-gular elements. As referred to earlier, the finite elementmethod has a major advantage over the finite differencemethod. While in finite difference method, field quantitiesare obtained only at discrete positions in the solution re-gion, they can be obtained at any point in the solutionregion in finite element method.

C ONC LUSIONSThe finite elem ent method was originally expounded forcivil engineering applications but has recently seen in-creasing applications in EM . A brief, tutorial introductionto the method has been presented. The method is appliedspecifically to Laplacian electrostatic field problems. Sug-gestions are offered on how the concepts developed canbe extended to other types of EM problems.It should be noted that one problem associated with thefinite element method is inputting the data defining the

geometry of the solution region. Since a large amou nt ofdata is involved, a mistake can easily be made in en teringthis data. This problem can be attacked in three differentways. One is by having an in-built subroutine for dataverification. Another method is using computer graphicsto plot the mesh before analyzing the data. These two

IEEE TRANSACTIONS ON EDUCATION, VOL. 32, NO. 2 , MAY 1989

methods enable recognition of mistakes. A third methodis using an automatic mesh generating scheme, e .g. , [111,[121; his labor-saving method of discret izing the solutionregion into triangular elements eliminates the problem.Another major problem in finite element analysis is therelatively large amount of computer memory and time re-quired. With the advent of batch processing and time-sharing, the problem has been alleviated to som e degree.In spite of the limitations and problems, the finite ele-ment method has successfully been app lied to various EMproblems. For such applications, the reader is referred tothe literature, e.g. , [4], [7].

CCCCCCCCCCCCCCCCCCC

CCCC10

20304 0cCCC

50C

60

C

70C

C

80

C

90

100110

A P P E N D I X :C O M P U T E RR OGR AMOR E X A M P L EFINITE ELEMENT SOLUTION OF LAPLACE'S EQUATION FORTWO-DIMENSIONAL PROBLEMSTRIANGULAR ELEMENTS ARE USEDND = NO. OF NODESNE = NO. OF ELEHENTSNP = NO. OF FIX ED NODES ( WHERE POTENTIAL IS PRESCRIBED)NDPIII = NODE NO. OF PRESCRIBED POTENTIAL, I = 1,2,...NPVAL(I1) = VALUE OF PRESCRIBED POTEKPIAL AT NODE NDPII)NL(1.J) = LIST OF NODES FOR EACH ELEMENT I, WHEREJ = 1. 2, 3 IS TH E LOCAL NODE NUMBERCE(1.J) = EL= COEFFICIENT MATRIXC1I.J) = GLOBAL COEFFICIENT MATRIXBIII = RIGHT-HAND SIDE MATRIX IN THE S Y S T M O FSIMULTANEOUS EQUATIONS; SEE EQ.131) OR EQ.137)X(1). YII) = GLOBAL COORDINATES OF NODE IX L I J ) , YLIJ) = LOCAL COORDINATES OF NODE J = 1,2,3VII) = POTENTIAL AT NO DE IMATRICES PII) AND QIII ARE DEFI NED IN EQ.132)DIMENSION XflOO), YflOO ), CflOO.100). CEf100,lOO)DIMENSION B1100). NLl100.3). NDPflOO ), VAL( 100)DIMENSION VIlOO), P(31, Q(3) , XL(3). YLl3). . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . .

FIRST STEP - INPUT DATA DEFINING GEOMFPRY ANDBOUNDARY CONDITIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .RERD15.10) NE,ND. NPFORMATl313)READf5.20)( I, ( NL1I.J). J=1,3),1=1,NE)FORMAT(413)RERD(5.301 11, XII), YII) , I=l. ND)FORMATfI 3, 2F6.21READ(5.40) ( NDPIII, VALII), I=l,NP)FORMATlI3,F6.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .SECOND STEP - EVALUATE COEFFICIENT MATRIX FOR EACH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .GO 50 H =1, NDELEMENT AND ASSEMBLE GLOBALLY

BfM)=O.ODO 50 N=l.ND.C1M.N) = 0.0CONTINUEDO 140 I = 1. NEFIND LOCAL COORDINATES XL(J), YL(J1 FOR ELEMENT IDO 60 J=1.3K=NL I, ~XLfJ) = X(K1YLIJ) = YIK)CONTINUEPf1) = YLl2) - YL(31P(2) = YL(3) - YLl1)Pl3) = YLi1) - YLf2)~Q l l ) = XLf3) - XLIZIQl2) = XLl1) - XL(31Ql3) = XLf2) - XLf1)AREA = 0.5*ABSl P(2)*Ql3) - Ql2)*Pl3) )DETERMINE COEFFICIENT MATRIX FOR EL= IDO 70 M=1,3DO 7 0 N.1.3CE(t4.N) = I PlMl*PlN) + QlM)*QfN) IIfI.O*AREA)CONTINUEASSEMBLE GLOBALLY - FIND Cl1,J) AND BII)DO 130 5.1.3IR=NLfI.J)CHECK IF ROW CORRESPONDS TO FIXED NODEDO 80 K=l,NP1FfIR.EQ.NDPlK)) O TO 120CONTINUEDO 110 L=1.3IC = NLI1.L)CHECK IF-COLUMNCORRESPONDS TO FIXED NODEDO 90 K=l,NPIF( IC.EQ.ND PIK1 GO TO 100CONTINE-C(IR,IC) = ClIR,IC) + CE1J.L)GO TO 110BIIRI = BIIR) - CElJ,Ll*VAL(K)CONTINUEGO TO 130

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SADIKU120

130140CCCC

150CCCC160

17 0180

CCCCC

102 030

40

50

ANALYSIS O F ELECTROMAGNETIC PROBLEMSCONTINUEC(IR,IRI = 1.0B(IR1 = VAL(K1CONTINECONTINUE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .THIRD STEP - SOLVE THE RESULTING SYSTEM OF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .NMRX=100CALL INVERSE(C.ND.NMRX1DO 150 I=l,NOV(I1 = 0.000 150 J=l,NDVI11 = VI11 + ClI,J)*B(JlCONTINUE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .FOURTH STEP - OUTPUT THE RESULTS, THE POTENTIALV ( I 1 AT NODE I, I = 1.2.. .. . D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .WRITE(6.1601 ND,NE,NPFORMAT(2X,N O. OF NODES = ,13,2X,NO. F ELDlENT.5 = WRITE(6.1701

SIMULTANEOUS EQUATIONS

1 13,2X,NO. OF FIXED NODES = .13,IlFORMAT(ZX,NODE,SX,X Y.7X.POTVTIAL,/lwRITE(6,1801 (I , (I1, Y(I1. V(I1, I-1.NDF0RMAT12X,13.2X,F6.2,2X,F6.2,2X,F10.4,/1STOPEN0

SUBROUTINE INVERSESX IS TH E MATRIX TO BE INVERTED; IT IS DESTROYEDIN THE COMPUTATION AND REPLACED BY THE INVERSEN IS THE ORDER OF SXIDM IS THE DIMENSION OF SXSUBROUTINE INVERSlSX,N,IDMlDIMENSION SXlIOM,IDMlESP=l.OE-500 50 K.1.NDO 30 J.1.NIF(J.EQ.Kl CO TO 30IF1 ABSlSX(K,Kll I 20,10,20SXlK.Kl=ESPSX( J =SX( , /SXl KlCONTINUESX( ,K = ? .0 X ( , lDO 40 I.1.NIF(1.EQ.K) GO TO 4 0DO 40 J=l,NIF(J.EQ.K) O TO 4 0SX(I,Jl=SX!I.Jl SX(K.JI*SX(I,KIC NT NUED O 50 I=l,N1FII.EQ.K) O TO 50S:

A C K N O W L E D G M E N TThe author would like to express his appreciation to Y.Kim and L . Agba for their review and helpful com mentson the manuscripts. Th e corrections and com ments of the

reviewers are also appreciated.

93R EFER ENC ES

R. Courant, Variational methods for the solution of Problems ofEquilibrium and vibrations, Bull. Am. Math. Soc.. vol . 49 , 1943.J. F. H oburg and J . L. Davis, A student-oriented f inite element pro-gram for electrostatic potential prob lems, IEEE Trans. Educ. . vol.- Enhanced capabi l i t ie s for a student-oriented finite elementelectrostatic potential prog ram, IEEE Trans. Educ. , vol. E-28, pp.25-28, Feb. 1985.S . R. H. Hoole and P. R. P. Hoole, Finite element programs forteaching electromagnetics, IEEE Trans. Educ. , vol. E-29, Feb.1986.R . F . Har r ington , Field Computation by Moment Methods. Malabar:Kriger , 1968.W . H . H a y t , Engineering Elecrromagnerics. New York: McGraw-Hil l , 1981, pp . 121, 184-192.P. P. S ilvester and R. L. Ferrari, Finite Elements for electrical En-xineers.

E-26, pp . 138-142, NOV . 1983.

Cambridg e: Cambrid ge Univ. Press, 1983, pp. 1 -32.[8 ] C. R. Paul and S . A.Nasar , Introduction to Electromagneric Fie lds.New York: McGraw-Hil l , 1982, pp . 465-472.[9 ] 0 . W . Andersen , Laplacian electrostatic field calculations by f initeelements with automatic grid generation, IEEE Trans. Power Ap-para tus Sys t . , vo l . PAS-92 , no . 5 , Sept./O ct. , pp. 1485-1492, 1973[IO] D. R. J . Owen and E. Hinton , A Simple Guide to Finite ElementsSwansea: Pineridge Press, 1980, pp. 47-79.1111 E. Hinton and D. R . J . Ow en, An Introduction to Finite ElemenrComputations.[ I21 W . R. Buell and B. A. Bush, Mesh generation-A surve y, TransASME, J . Engng. Ind . , pp. 332-338, Feb. 1973.Swans ea: Pineridge Press, 1985, pp. 328-326.

networks, and supercc

Matthew N. 0. Sadiku graduated from AhmaduBello University, Z aria (Nig eria) and received theM.S . and Ph.D. degrees f rom Tennessee Tech.Unive r s i ty , Cookevi l le .He is currently an Assistant Professor of Elec-tr ical Engineering at Temple University, Phila-delphia. PA. He was formerly at Florida AtlanticUniversity, Boca Raton. He is the author of Ele-ments of Electromagnetics (HRW, 1989) . Hiscurrent research interests are in numerical tech-niques in electromagnetics, local area computern d u c t i v i t y .