lencova.pdf

7/30/2019 lencova.pdf

1/4

1107B E E TRANSACTIONS ON MAGNETICS,VOL. 8, N0.2, ARCH 1992LINEAR FINITE ELEMENT METHOD IN AXISYMMETRICMAGNETOSTATIC PROBLEMS

B. Lencov6 )Particle Optics Group

Department of Applied PhysicsTU Delft, Lorentzweg 1NL-2628 J Delft, The Netherlands

Ab-- If there is a long iron cylinder on the axisof rotation al symmetry, theFEMwith linear shape functionfor the computation of the angular componentofthe vectorpotential yields an i n c o m t solution. Based on a suitableweight factor in the expression for the z omponent of theflux density, new ormulas for the coefficien ts of the linearequations have been derived, which give correct answers.

I. IIWRODUCIIONIn electron optics, we wish to determine the electron

opticalPropert i esof lensesand other elementswithhigh a m %this demands a high accuracy of field evaluation. Thishas contributed to the widespread appreciation of the finiteelement method (FEM), introduced into electron optics byMunro [l],and reviewed also in [2,3]. It was therefore anunpleasant surprise that all the computation methods usedup to now for the angular component of the vector potentialsuffer from an unacceptable inherent error, which manifestsitself stronglyin the case of cylindricalhigh permeabilityrodson the axis, as n case of the BH tester of Tahir and Mulvey[4]or the pot transformer of Melissen and Simkin [5].To overcome this problem, several solutions modifying thelinear E M ere proposed. Boglietti et al [6 ] proposeda technique which uses weighted linear shape functionsA=I"fftgzthr); the weighting factorn changes from the valuen=0 at the rotational ax i s to the value n = I at the outerradial boundary. Melissen and Simkin [SI proposed to usethe fluxV =2xrA instead ofA and the linear shape functionsfor F= f t g z t h u , U=?

a) On leave from Inst. Sci. Instr., Bmo.Manuscript receivedJuly 8,1991.Thisworkwa s done as a partofFOMproject IOP-IGDTN 45.006.

M. LencInstitute of Scientific Instruments

Czechoslovak Academy of SciencesKr6lovopolski 147CS-61264 Brno, Czechoslovakia

In thspaper we point out the source of the problem. Furthera short description of FEM in electron optics is given, anda simpleand,we believe, more suitablesolution is put forward.

11. DESCRIFTIONF THE PROBLEMThe energy functional in cylindrical coordinates

The integration of the energy functional for A, namely

can be performedon individual triangles of the finite elementmesh with the linear shape functionA (cz)= g zthr. Herewe have, for counter clockwise notation of the triangle vertices,

1 1f =- a i A i , g =- b i A i ,D i D iwithai=rkzrzk'i, i=?qh Ci'Zk-Zp (i,j,k) being a @C ~ " . h t i O nof (1,2,3), nd D = b l c 2 - b s 1 s twice the area of the triangle.

The linear shape function in the triangular finite elementguarantees the continuityof the normal component of thefluxdensityB, between elements containing materials withdifferent permeabilityp,but not the continuity of the tangentialcomponent of the field intensity Hr This effect is mostpronounced where B, =0 in theimportant parts of the magneticcircuit. One-dimensional or quasi-one-dimensional problemsare the most striking examples, as revealed in [4 - 61, wherethe solution is determined by the tangential component Ht(i.e. by H,) only.

Onthecylindricalboundayatr=ro between two materialswith relative permeabilities p l and p,, the local error ofA

0018-9464/92$03.00 1992 IEEE

Authorized licensed use limited to: IEEE Xplore. Downloaded on February 9, 2009 at 05:00 from IEEE Xplore. Restrictions apply.


2/4

1108canbe evaluated from the known analyticalsolution in a regionwith constantHz t the mesh points i=I and i=2above andbelow the mesh point at r=ro, namely

ri A i=roA, + p i Hz ri - ro) ri /2 . (3)Expression of coejjicients of FEM equations

The expression of coefficients of FEM equations dependson the variant of the finte element method, particularlyonthe choiceof mesh, and on themethod chosen for the integrationof (1).The meshes used in electron optics are made ofquadrilateral elements. The mesh is topologically equivalentto a square mesh. The geometry of the axially symmetric lensisspecified with quadrilateralsof general shape, defined withthe help of a coarse mesh of horizontal and vertical lines. Asmooth fine mesh with variable step size is made with the

help of an automeshing procedure, built into an interactiveprogram for input ofdata and displayofresults This is i l l ustratedin Fig. 1on the coarse and fine mesh used later for the testcomputations.

The fine quadrilateral mesh, defined above, is for thecomputation of coefficients subdivided into two tr i angl es;mostlyan average from both subdi vi si onsa r e d A different eqnsbnfor the coefficients of the resulting linear equations canbe obtained, depending on which integration method is usedeither forBz= (I/?)qA)/&,hez component of the fluxdensity,or for B: in (1).Munro originally used an average value ofBz in the triangle.In[3]we presented three methods of integrating eq. (l),

in particular the critical term rB:, which provide differentexpressions for the coefficients of the FEM equations, al lof which have the same order of numerical error O(h2) -see Fig. 2. The first method, using the gravity center of thetriangle, provides identical expressions o those of Munro[l].The second method takes the integrand values at the verticesof the triangle, and gives, for rectangular meshes, exactlythe coefficients of the five point finite difference method.The third method, based onusing the values at the midpointsof the triangle sides, gives mostly the best accuracy and itis not sensitive to the change in the mesh step; it is thus themost suitable one for the computation of coefficients,althoughit requires about U)% increase of the computation effort.

Unfortunately, all three methods have the same behaviorof the local error on the cylindrical boundary in the basicallyone dimensional situation, e.g.

Fig. 1. Coarse and fine mesh in a magnetic lens with long polepieces. Thez axis isdrawn horizontal; the plane at z=O mm is a symm etry plane. Thegap between the polepieces and the diameter of the lens bore are 5 mm.Fine mesh is used with 55 points in each direction.

Fig. 2. Trianglesused or the integrationofeq. 1). Squaresdenote he pointswhere the funct ion value is evaluated: in the left triangle the value of theenergy functional is taken at the center of the triangle, in the right triangle

at the mid points of the sides.



3/4

1109of the term A /r and that contains, in the case of a rightangle triangle, the potentials of the nodal points on the sidealong the r axis only. We have chosen for an average valueof B, an expression

withp i= Irl+ol.Eq.(4) was obtainedby subst i tu t i ng the analyticalfor the last computation method for coefficients. The erroris proportional to A , and thus it reaches quite a largevaluewithahighpermeabiUymatexialnearthez axis. Analogousresults have been obtained in [7l, see also [5] , or differentformulas for coefficients.

solution (3) into theexpmion for cleffi&nts of FEMequations

Effect of the erriorWith the geometry of Fig. 1we have made computations

of lenswithMerent perm- of the polepieax A rectangularh e mesh was used having constant step with 5 points betweenthe symmetry plane and the polepiece face, and variable stepelsewhere,allowing expansion by 5%.The relative permeabilityprelwas changedbetween 10and lo6, the relative permeabilityof the outside part of the circuit was lo6.Taking as a criterionof the error the percentage of the input excitation calculatedas the integral ofH, long the axis, the error is about 1.5%.If we now close the circuit, and eventually put the polepieceup to the optical axis,the amount of lost excitation is increaseddramatically - see Fig. 3.

111. SOLUTION OF THE PROBLEMIn the one-dimensional E M ormulation for A(r) , the

integrationmethod using the center values provides exact resultsboth forA and rA [5].Therefore we have tried to find and o p y n twodimensions hat would providecorrect ntegration

I 1 I

y. ...........+....____.....................*;..........- --00-+ -Lens - s t a n d .+Red - s t a n d .W Z o .-+.. en s & r o d - new me t h o d

0 I I I I J0 1 2 3 4 5 6

Fig. 3. Excitation on the axis for the lens from Fig. 1, and for the case ofrod with 20 mm diameter nstead of the polepieces, in dependenceon herelative permeability of the rod or polepieces.

while for B,=-g we have the usual form

with both expressions in the denominator of (6) being equalto D in (2). The difference in the r coordinates of the centerof gravity of the triangle and the interval in the r directioncan be cancelled by using both possible divisions of thequadrilaterals nto triangles. With the help of (5) and(6),newexpressions of coefficients for FEM equations canbe derived.They also give the same order of error O(h2),and differ fromthe other methods only by terms of higher order in meshsize h . On the cylindrical boundary, the local error in thenew FEM approximation can be written as

This error is no longer proportional to A , and it isindependent of the relative permeability of the material ontheaxis.A furtherfeatureof thenewmethod is that inrectangularmeshes the coefficients of the FEM equations depend onlyon the nearest neghbors inr andz andnot on the other cornerpotentials, as it also occurs in the method using potentialsat the vertices and in the five point finite difference formulas.The computation effort for the coefficients is less than thatfor the previously chosen default method using the mid pointsof the sides.

IV.RESULTSThe new method of coefficient computation has been

implemented in the program LENS 3,8]using the boundaryconditionA=O.Merent methods for cOmputationof coefficientscanbe realisedby exchangingone subroutine where theevaluationof eq. (1) n each triangle is performed. The differenceis thenonly in 15 lines of code.



4/4

i i i o

--_- - - e - - ~ i d points of s i d e s -%_-x _ _ _ _ _---e--ew m e t h o d x---.-----

In the variant of LENS with the homogeneous Neumannboundary condition we canmodel cylindrical rods of inf ii telength with just 2 lines in the z direction.The results obtainedwith different coefficient computationmethods re illustratedinFig. for a cylindrical rod with permeability prez= o00withthe spme geometry as in the pot transformer of [q, .e. rodradius 1 mm, a coil with constant current density up to r = 8mm, and a radial boundary at r=12 mm at the positionof outer shield. Constant stepsize was used between the axkand the line at r = l mm, i.e. for hr=O.O1mm there were 100mesh points in the first 1 mm radius, and a variable step meshwith expansion by 10%with 50 more points up to the outsideradial boundary at 12 mm. In this case, the new method ofcomputation of coefficients gives very good agreement withthe expected value of B, as given by the coil excitation.

Results obtained with the new method are also given inFig. 3: the error in excitation is less than 0.1 %for al lvaluespmz both for lens and rod examples.

V. CONCLUSIONSThe new method for the computation' of the coefficients

of theFEM equations in anaxisymmetric case for the vector

0 . 8

0.6p1.-mv 0 . 4ct:

,L .*...; ..................... 1.........*........:......*.............1

I / N = h, [mm]

Fig. 4. The depen dence of the relative axial flux density in a cylindricalrod of 1 mm radius with relative permeability km,=lo00 [SI, alculated byFEMwith different coefficientcomputations,on the mesh step n the radial

direction h,.

potentialdoesnot s u t e r h heinherenterrorofothermethodswhen modellingcasessimilar tothatonewithhigh permabilitycylindrical rod on he axis. We prefer our solution as comparedwith the solutions proposed in [q and [6], as we s t i l l useas he axkymmetric magnetic circuits are usuallycomposednot only from cylindrical, but also from conical parts.

In order to decide ifthisnew method shouldbeimplementedas a default coefficient method for FEM computations ofmagnetic lenses, further considerations are s t i l l necessary.In magnetic axisymmetric lenses the geometry issuch, thatthe continuity of the normal component of the flux densityat the material boundaries is crucial for the results.

shapefimdionslinearmrandzcoordinatesThisisimportant,

E M u m " p i m ~ & p o f e l e c a n h b y t h ? ~ i t e $ e n a umethod",in"ImageprocessingadcomputerAided Des@ mEkctroaOptics",edited by P.W. a de s. London: AcademicPress, 1973,pp. 284-323.E.Munro et al: 'Field computation techniques in electmn optics",IEEETransactionsonMagnetics,Vol.MAG-26,No.,March1990,Pp.1019-1022.B.L e n d , M. Lcnc: "Afinite ekment method i r the computationofmagnetic electron lensesn, Scanning Electron Mi-1986/III, pp. 897-915. EM Inc., AMF O'H an, Chicago, 1986.KTahir,T.Muhrcy: "Pitfalrsinthecalcukuionfthe&kidisIrib&nof magnetic electron kttsesby t h e f ethod",Nucl.Instr.Meth. in Phys. Research,Vol.A298, ecember 1990,pp. 383-388.J. B.M. Melissen,J. Simkin: 'M ew coordinate tran sfom for thefinite element solution of a t i y " e a i C problems inmagnetostm'cs", EEE TransactionsonMagnetics,Vol.MAG-26,No. , March 1990,pp. 391-394.A.Ebglietti, M. Chiampi,D. Ckraba@o,M. Tartagiia: 'F*elemeuapproximation in ar@"@ical domain",EEE T ransactions onMagnetics,Vol.MAG-26, No. , March 1990,pp. 395-398.J. B. M. Melissen, "Cyhdn 'cal coordMates in elecmmagneticcomputations .. a onedimendonal rrorstudyA versusrA"PhilipsCPS Technical note Nr.010 UDR/MSW/O89/HMO92/hm (1989)B.Len&, G. Wisselink '!Program ackage for the computotiOnof lenses and defletoors': Nucl. Instr. Meth. in Phys. R esearch,Vol.A298,Decemb er 1990, pp. 5666.

Authorized licensed use limited to: IEEE Xplore Downloaded on February 9 2009 at 05:00 from IEEE Xplore Restrictions apply

lencova.pdf

Documents