Particle Accelerators1976, Vol. 7, pp. 213-222

© Gordon and Breach, Science Publishers Ltd.Printed in the United Kingdom

SUPERFISH-A COMPUTER PROGRAM FOR EVALUATION OFRF CAVITIES WITH CYLINDRICAL SYMMETRY

K. HALBACH

Nuclear Science Division, Lawrence Berkeley Laboratory, 0 niversity o.f Cal(fornia, Berkeley,California 94720, USA

and

R. F. HOLSINGER

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545, USA

(Received June 17, 1976)

The difference equations for axisymmetric fields are formulated in an irregular triangular mesh, and solved with a direct,noniterative method. This allows evaluation of resonance frequencies, fields, and secondary quantities in extremegeometries, and for the fundamental as well as higher modes. Finding and evaluating one mode for a 2000 point problemtakes of the order of 10 sec on the CDC 7600.

1 INTRODUCTION

Over the last 10 to 15 years, a number of computerprograms have been developed that find theelectromagnetic resonance frequency and evaluatethe axisymmetric fields in rf cavities with axi­symmetric symmetry. The codes that allow thisanalysis to be made in an essentially arbitraryaxisymmetric geometry (see for instance Refs. 1-3)have the following in common: For some geo­metries, like cavities that have a large diametercompared to their length, and/or for modeshigher than the fundamental mode, the convergencerate can be extremely small, or convergence maynot be achieved at all. Stated very briefly, thereason for these problems is the fact that in all thesecodes, an overrelaxation method is used to solvea set of homogeneous linear field equations. Theproperties of these equations are such that somewell developed methods for overrelaxation-factoroptimization are not applicable, and it might wellbe true that the eigenvalues of the matrices for someproblems are located in such a way that even anoptimized overrelaxation scheme would still resultin unacceptably low convergence rates.

To eliminate these problems, we developed thecode SUPERFISH that uses a direct, noniterativemethod to solve a set of inhomogeneous fieldequations. This code is a combination of some parts

213

of the code RFISH,4 some new ideas, and thedirect solution method used by Iselin in his magnetcode FATIMA. 5 In order to give a good overallunderstanding of SUPERFISH in a limited space,we do not present all detailed formulas, but doinclude the description of all parts that are con­ceptually significant, even at the expense ofreformulating and/or condensing parts of the citedliterature.

In Sections 2 and 3 we discuss separately thestructure of the difference equations, and the direct,noniterative method used to solve a set of in­homogeneous linear equations. In Section 4 thebasic structure of SUPERFISH is described, andthe remaining sections give some details of thetheory and of the program as it exists today, and anoutline of contemplated future developments.

2 STRUCTURE OF THE DIFFERENCEEQUATIONS IN AN IRREGULARTRIANGULAR MESH

Inspection of Maxwell's equations shows that foraE/a¢ == 0, aH/a¢ == 0, i.e., axisymmetric fields,two independent sets of solutions can exist: onehaving as only nonzero field components Ell>'

Hz, Hr; the other, H¢, Ez, Er. These two solutionsare, for equivalent boundary conditions, identical;

214 K. HALBACH AND R. F. HOLSINGER

we therefore talk only about the latter set.Assuming, without loss of generality, that themagnetic field is proportional to cos wt, and theelectric field is proportional to sin wt, and usingsuitable units, Maxwell's equations can be writtenas

with k = w/c, and Hand E representing the electricand magnetic fields divided by their respective timedependence.

We seek to find numerical solutions for some ofthe eigenvalues k and associated fields of Eqs. (1a)and (1 b) in cylindrical cavities of essentiallyarbitrary shapes, with H = 0 on the axis andpossibly some other parts of the boundary(Dirichlet boundaries), and the electric field perpen­dicular to the remaining boundaries (Neumannboundaries), implying infinitely conducting wallsthere.

curl H = kE,

curl E = kH,

(1a)

(1b)

the z, r coordinates of interior points, for a given setof boundary points, with an iterative process thatis similar to a numerical method used to solveLaplace's equation. Figure 2a shows a logicalmesh, and Figure 2b a physical mesh, for one halfof an Alvarez cavity. In Figure 2a, points on heavilydrawn logical lines represent those with assignedz, r coordinates. The two heavily drawn interiorlines are used to delineate zones with differentmesh point densities. Exterior mesh points, i.e.,points inside the drift tube, are not shown since theydo not affect the field calculations.

2.2 The Difference Equations for Interior Points

We use the quantity H = He/> to describe the rffields. This somewhat unconventional choice(usually r . He/> is used) has the ~dvantage of notrequiring any special treatment of the region closeto the· axis, since H will be proportional to r there,whereas rH4J ~ rZ for small r. From Eqs. (1a) and

2.1 The Mesh

To solve the differential Eqs. (1a) and (1b), weintroduce an irregular triangular mesh6 in thez-r plane. Figure 1 shows the logical mesh, withmesh points identified by labels K and L, assumingthe integer values 1 through K max = K z, and 1through L z . To establish a mesh that can be usedto solve the field equations for a particular geom­etry, defined by its boundaries, the user firstassigns boundary coordinates z, r to an arbitrarybut reasonable selection of logical points K, L.The mesh generator, described in Ref. 6, thengenerates a mesh of triangles that is topologicallyidentical to the logical mesh, but has all boundariesdefined by mesh lines. The mesh generator finds

L-i

II

FIGURE 1 Logical triangular mesh. FIGURE 2a Logical mesh for I/2-Alvarez cavity.

SUPERFISH-A COMPUTER PROGRAM FOR RF CAVITIES 215

FIGURE 3 Irregular triangular mesh with secondary do­decagon.

The integrals in Eq. (3) can therefore be expressedin terms of the value of H at the "center-meshpoint" of the dodecagon and its six nearest logicalneighbors, giving a relationship of the followingkind

(4)

6

2

3

2.3 The Treatment of Boundary Points

Turning now to mesh points on the boundariesof the problem, it is clear that no difference equa­tions are needed for H at boundary points when theboundary conditions require H == 0 there. Never­theless, we have to explore whether or not thedifference equations for such points are satisfied.To this end, we consider first Dirichlet-boundarypoints that are not on the problem axis. This kindof boundary condition can obviously only beimposed as a symmetry condition along a planedefined by z == const. This implies that in the realworld a point on one side of this line has an H-valueof the same magnitude, but opposite sign, as thesymmetrically located point, and the differenceequation, Eq. (4), is clearly satisfied for every suchboundary point.

This argument cannot be applied withoutelaboration for points on axis (r == 0)., and thedifference equations resulting from Eq. (3) arein fact not satisfied for those points. To see howthis can be interpreted, we can introduce on the

6

L Hn(v" + k2~) == 0,o

with v" and ~ depending only on the coordinatesz, r of the seven mesh points involved.

Identifying each difference equation with its"center-point," we therefore get one differenceequation for H at every interior mesh point.

Rt

-g~~-.z

FIGURE 2b Physical mesh for I/2-Alvarez cavity.

(1 b) we obtain as the differential equation for H:

curl(curl H) == k2H. (2)

To derive difference equations for H., we use theprocedure described by Winslow6

: we first intro­duce a secondary mesh by drawing connectinglines between the "center of mass" of every triangleand the center of each of the three sides of thetriangle. As a consequence, every mesh point isnow surrounded by a unique twelve-sided polygon.This secondary mesh of dodecagons covers com­pletely the whole problem area, and Figure 3shows the dodecagon surrounding just one meshpoint. The difference equations for H are nowobtained by integrating Eq. (2) over the area(in the z-r plane) of one dodecagon at a time.This yields

fcurl(curl H) . da = f curl H . ds = k2 fH . ds.

(3)

Assuming that H behaves like a linear function ofz and r within every triangle, H inside everytriangle is uniquely determined by the values of Hat the three corner-mesh points of the triangle.

216 K. HALBACH AND R. F. HOLSINGER

the coordinates of the other mesh points contrib­uting to Eq. (4) differ from K o and Lo by not morethan ± 1.

3 DIRECT, NONITERATIVE SOLUTIONOF A SET OF INHOMOGENEOUSLINEAR EQUATIONS

oft 1 G1

oft2 G2

oft 3 G3(5)x

:YfL2-1 GL2 - 1

:YfL2 GL2

a L2 - 1. L2

a L2 ,L2

a L2 - 1, L2 - 2 a L2 - 1, L2 - 1

aL2, L2-1

all al2

a21 a22 a23

a32 a3 3 a34

In this matrix equation, oft 1 represents a columnvector with the components H K 1, K == 1 - K 2;oft2,avectorwithcomponentsHK2~K == 1 - K 2 ,

etc, and the Gn represent the correspondinginhomogeneous terms. The matrix on the left sideof Eq. (5) has all zeroes except for the blockmatrices aij of size K 2 x K 2' These blocks arealso sparse, the diagonal blocks containing onlythree nonzero elements in every row, and theoff-diagonal blocks not having more than three inany row. Deferring to Section 4 the discussion ofhow we can cast our field evaluation problem inthe form of Eq. (5), involving all points of thelogical. mesh as well as inhomogeneous terms onthe right side of Eq. (5), we discuss now the methodused to solve Eq. (5).

We first transform all diagonal blocks into unitymatrices,· and remove all blocks to the left of thediagonal blocks, with the Gaussian block elimina­tion process: we multiply the equations representedby the first row of blocks from the left by all, and

If one wrote down difference Eq. (4) for all (i.e.,including all boundary and exterior mesh points)H KL of the logical mesh by rows from left to right;and if one also had some inhomogeneous terms, onecould write the resulting system of equations in thefollowing form:

3

1FIGURE 4 Boundary mesh point with neighboring interiormesh points, and secondary polygon.

f curl H . ds == O.4'-0-1'

This means that the difference equation for H atsuch a boundary point is identical to that forinterior points except that only contributions frominterior triangles are taken into account.

A very important property of the differenceequation, Eq. (4), is the fact that if we use the logicalcoordinates (indices) K, L to identify mesh points,and if K o, Lo are the coordinates of any specificmesh point for which we write down Eq. (4), then

right-hand side of Eq. (1b) a (magnetic) currentdensity term j that has only an azimuthal com­ponent. This gives 9n the right-hand side of Eq. (3)the additional term kI, where I represents the totalcurrent associated with the point under considera­tion, and assumed to be concentrated there. Inour case of axis points, an azimuthal current onthe axis is, of course, without consequences, andthis whole argument could also be used to lendlegitimacy to the application of the symmetryconsideration to axis points.

When E is required to be perpendicular toboundary-mesh lines, we consider that part of thedodecagon surrounding a boundary point thatgoes through inside problem triangles, i.e., thepolygon 0-1'-2' - 3'-4'-0 in Figure 4. Since E ==curl H/k is required to be perpendicular to lines4-0 and 0-1,

SUPERFISH-A COMPUTER PROGRAM FOR RF CAVITIES 217

then subtract from the second row the new firstrow after multiplication from the left by a21.

The new set of equations is then the same as theoriginal one, except that a21 == 0; all == I; anda12' Gb a22' and G2 are now modified. Thisprocess is repeated, involving rows 2 and 3, then3 and 4, etc. The very last step in this process is themultiplication of the last row (modified by theprevious step) from the ·left with the modifiedblock matrix aL2!' L2 •

Having Eq. (5) rewritten in this form, the lastrow now represents directly the solution for YeL2 •

Using this now numerically known vector in rowL z - 1 yields directly the solution for YeLz - b

and continuing this back-substitution processyields the numerical values of all components ofall block vectors Yen.

It is important to recognize the fact that thisparticular fast direct method to solve inhomo­geneous linear equations can be used only if theycan be cast in the form of Eq. (5), and if the matrixon the left side of Eq. (5) is nonsingular.

4 CALCULATION OF FIELDS ANDRESONANCE FREQUENCIES INSUPERFISH

In order to allow application of the direct linearequation solution described in Section 3, we haveto include in an artificial way in the system ofequations also those points that are part of thelogical mesh, but are external to the actual fieldsolution problem. How this is done, and thetreatment of points on Dirichlet boundaries, isdiscussed in Section 4.1; the creation of the in­homogeneous terms is discussed in Section 4.2,and the resonance frequency determination isdiscussed in Section 4.3.

4.1 Treatment of Exterior Points and Points onDirichlet Boundaries

The simplest and most practical way to includeexterior points without affecting the actual fieldequations, and without causing the matrix on theleft side of Eq. (5) to become singular, is to let theequation for every exterior point read H exterior == 0,and to make all couplings to other equations zeroby setting the corresponding coefficients equal tozero also. In other words, if no is the index identify­ing an exterior point (not a block!) in the overallH-vector on the left side of Eq. (5), one simply sets

all elements of row no and column no of the matrixin Eq. (5) equal to zero, with the exception of theno, no diagonal element, which is set equal to one.The no-element of the inhomogeneous contribu­tion vector on the right side of Eq. (5) is set equalto zero also. The logic of the equation-solvingroutine is arranged in such a way that the thus­introduced zeroes are actually never used inmultiplications,just as the other zeroes in the sparsematrices are never used as multipliers either. Pointson Dirichlet boundaries are treated in exactly thesame way. However, in contrast to exterior points,their z-r coordinates do enter into the expressionsfor J!;" ~ in Eq. (4) involving the other point(s) ofthe triangles that have one or more Dirlchlet­boundary points at their corners.

4.2 Generation ofInhomogeneous Termsfor Eq. (5)

In order to turn the set of homogeneous differenceequations [Eq. (4)] into a well-posed set of in­homogeneous field equations, one could be temptedto introduce at one mesh point a driving (magnetic)current, as discussed in Section 2.3. That wouldbe an unwise procedure when one is close to aresonance, since the matrix in Eq. (5) is singularfor every resonance frequency, leading, as it must,to infinite fields. Instead, we prescribe that anappropriately chosen off-axis mesh point has. thefield value one and in effect remove the differenceequation for that point from the system ofdifferenceequations. To do this without destroying the struc­ture of the field equations, we can proceed in oneof the following two ways:

1) If the chosen point is identified by its indexnl in the overall H vector, we set all matrix elementsin row nl of the matrix in Eq. (5) equal to zero,except for the diagonal element n1, nb which isset equal to 1. In the vector G on the right-handside of Eq. (5), all elements are set equal to zero,except the n1-element is set equal to one. Column n1

of the matrix is left unchanged.2) Every matrix element in row nl and in column

n 1 is set equal to zero, except the n1, n1 diagaonalelement is set equal to one. The vector on the right­hand side of Eq. (5) is set to equal minus theoriginal column nl of the matrix, except for ele­ment nb which is set equal to one.

The second procedure treats the point with thefixed field value in the same way as exterior pointsand points on Dirichlet boundaries, and in

218 K. HALBACH AND R. F HOLSINGER

4.3 Resonance Frequency Determination

The driving current II introduced above dependson k2 through the coupling coefficients in thedifference Eq. (4), and the resonance condition ischaracterized by

since then there is no difference between the valueof H n1 as calculated from the difference equationfor that point, and the prescribed value used thereto solve for the fields; i.e., the difference equationsare 'satisfied for all points of consequence. Tofind the value(s) of k2 for which Eq. (6) is satisfied,

5 PROPERTIES OF I 1(k2

) ANDINTRODUCTION AND PROPERTIESOF D(k 2

)

To obtain an understanding of some of the proper­ties of the function I 1(k 2

), and later D(k2), we will

go back to the differential equations, Eqs. (la) and(lb), with Eq. (lb) amended on the right side by themagnetic current density i, assumed to be constantover a small area surrounding the driving point.In the process of deriving some formulas, we haveto evaluate integrals like JH . i1 . dv, and set thisequal 2nr1 . h1 . 11 where 11 is the total drivingcurrent; rb the distance of the dr~ving point fromthe problem axis; and hl' the magnetic field averagedover the region where j 1 =I O. The associationbetween h1 resulting from this continuum theoryand the value of H at a mesh point in the repre­sentation by the difference equations is complicatedby the fact that h1 has a logarithmic singularitywhen the area where j =I 0 is reduced to zero (forfixed I 1)' While it seems reasonable to set h1

equal to Hat the driving point, or the value resultingfrom averaging H over the dodecagon associatedwith the driving point, it is clear that the quantita­tive relationships developed below will describeonly approximately the relationships between thequantities derived from the difference equations.However, it is also clear that the general behaviorof the functions of interest is correctly describedby the results derived from the continuum theorybelow.

Adding the term i1 to the right side of Eq. (lb)gives

curl E = kH + i1. (7)

Forming the scalar product of both sides of thisequation with H, and subtracting from thatEq. (la), after being multiplied by E, yields:

H curl E - Ecurl H == div(E x H)

= kH2+ j1H - kE2.

we can combine the above-described "functiongenerator" for J 1(k 2

) with a numerical root-findingalgorithm, such as the secant method, or a parabolafit method. The latter method is used in the presentstage of code development. But we expect that itwill be useful to use a root-finding algorithm thattakes into account some of the properties ofI 1(k 2

) that are described in Section 5. Figure 5depicts a flow diagram of the major parts ofSUPERFISH.

YES

(6)

K2

COUVERGENCENO

SECONMRYQUANTITIESDERIVEDFROM FIELDS

FIGURE 5 Flow diagram of SUPERFISH.

addition nonzero terms are generated on the right­hand side ofEq. (5). We therefore use that procedurein the code.

In contrast to the explicit introduction of adriving current, with procedures (1) and (2) thematrix on the left side ofEq. (5) is well conditionedeven for resonance frequencies.

If we take the original difference equation for thepoint with the prescribed field value and solve forthe field value at that point, using the solutionvalues of the field at the neighbor points, we willget a value different from the prescribed value,except at resonance. This difference can be inter­preted as being proportional to the current I 1

necessary at that point to drive the cavity to theprescribed amplitude at the point with the pre­scribed field value. For this reason we will refer tothis point as the driving point.

SUPERFISH-A COMPUTER PROGRAM FOR RF CAVITIES 219

(9)

(10)

Integrating this over the whole problem volumegIves

5div(E x H)dv == f(E x H) . da

= 2nr1h1I1 - k 5(£2 - H 2 )dv.

(8)

Since E x H is either zero on the problem bound­ary, or perpendicular to the boundary normal,f (E x H)da == 0, and we get

D(e) == 2nr1h1k11 = R(e) - k2f H 2 dv ' ' ,

R(e) = Jk2£2dv = J(curl Hf dv

f H 2 dv f H 2 dv .

The new function D(k 2) has the property that its

value does not depend on the scaling of h1, or 11

if that is the quantity that one wants to consideras the primary variable.

To obtain more information about the behaviorof 11(k

2), we now calculate dI 1/d(k2) == 1'1' To thisend, we take the derivatives with respect to k2

of Eqs. (la) and (7). Indicating derivatives withrespect to k 2 by primes, we get

curl H' == kE' + E/2k (11)

curl E' == kH' + H/2k + fl' (12)

It should be noted that for our procedure of fieldevaluation, H' == 0 on Dirichlet boundaries, be­cause H == 0 there for all k 2

• Similarly E' is perpen­dicular to Neumann boundaries since thecomponent of E parallel to a Neumann boundaryis zero for all k 2

. We now consider

div(E x H' - E' x H) == H' . curl E - E· curl H'

- H . curl E' + E' curl H.

Using for the curl expressions the appropriateright sides of Eqs. (la), (7), (11) and (12) yields

div(E x H' - E' x H) == H' . jl - H . fl- (£2 + H2)/2k.

Integrating this over the problem volume gives,as in Eq. (8), zero on the left side, yielding

2nr1k(h'l I 1 - hI1'1) = 5(£2 + H 2 )dv/2. (13)

We intentionally made no a priori assumptionswhether we consider h1 or 11 fixed when k2 ischanged. However, for the case considered so

far, h'l == 0, and we can immediately deduce thefollowing conclusions from Eq. (13):

h1I'1 < 0 (Foster's theorem). (14)

This means that for fixed hI, between every tworesonances (I 1(k 2

) == 0) 11(k 2) must have a singular­

ity such that the sign of 11(k 2), and therefore also

of D(k 2), changes.

At a resonance, f £2 dv == f H 2 dv [see Eqs. (9)and (10)], giving f H 2 dv on the right side ofEq. (13).We therefore get from Eqs. (13), (9), and (10) at aresonance (I 1 == 0):

2nrlh kI'l--::---_1_ == D'(k 2 ) == R'(k 2

) - 1 == -1 (15)f H 2 dv . .

Since 11 (k 2) has a singularity between resonances,

it is more convenient to study D(k 2) in the vicinity

of these singularities. To this end, we first considerR(k 2

). According to Eq. (9), R(k2) == k2 at every

resonance, and R' == 0 at resonance follows fromEq. (15). Since R cannot be negative, R(k2

) mustlook qualitatively as indicated in Figure 6 andR - k 2

- == D(k 2) as shown in Figure 7. An important

consequence is that between resonances, D(k2)

goes through zero, and this sign change must takeplace where I 1(k 2

) has a singularity.To study D(k 2

) in the vicinity of this root ofD(k2

) that does not represent a resonance, we takeadvantage of the fact that D(k 2

) is independent ofthe scaling of the field and current quantities. Wecan therefore consider 11 as given and keptconstant, and consider h1 as the k2-dependent

y

resona nee k2

~--JIV-----+--------~__~ k2

FIGURE 6 Graphical representation of properties of R(k 2 ).

220 K. HALBACH AND R. F. HOLSINGER

resonance k2

dD/dk2: -1

FIGURE 7 Graphical representation of properties of D(k2)

quantity that causes D(k2) = o. At this "between­

resonallce root," it follows from Eqs. (9) and (10)that JE2 dv = JH 2 dv, giving again JH 2 dv as theright side of Eq. (13). We therefore get from thatequation

2nr kI· h'1 1 1 = D'(k2 ) = R'(k2 ) - 1 = 1 (16)JH 2 dv .

A possible use of Eqs. (15) and (16) will be brieflydescribed at the end of Section 6.3.

6 PROGRAM STATUS AND FUTUREDEVELOPMENT

The computer program was originally written forthe CDC 7600 operating under the LivermoreTime-Sharing System (LTSS), and we describehere that particular version.

6.1 Computer Time and Storage Requirements

The CPU time required for a field evaluation isdominated by the tjme required to invert theblock matrices. For the system of equationsdescribed at the beginning of Section 3, the timeused for inversion of the block matrices is pro­portional to K~ . L 2 • When K 2 > L 2 , the differenceequations are arranged -along columns of thelogical mesh, leading to this expression for theCPU time

T=T1N2

8, (17)

with N representing the total number of logicalmesh points, and 8 the smaller of the two numbersK 2/L 2 , L 2/K 2 • For the CDC 7600 under LTSS,T1 ~ 0.75 j.1sec.

With the present system to find the roots ofI 1(k 2

), it takes 3 to 6 field iterations to determinea resonance frequency accurately.

The storage requirements for the program areapproximately 11· N exclusive of the memoryrequired for the modified off-diagonal blockmatrices, needed for the back-substitution. Thesematrices represent N 3

/2

• 8 1/2 words, too much to be

accommodated in core for N > 1500. For largerproblems, the disk has to be used. However,S. B. Magyary (LBL) has pointed out that oneneeds to store only two such matrices when onehas to calculate only I 1(k 2

) (and not the completefield map), provided the driving point is associatedwith the last row of block matrices on the left sideof Eq. (5). In that case, the large amount of storageis not needed until one has a converged resonancefrequency.

6.2 Accuracy_

Since we know from our experience with theRFISH code and the magnet code POISSON thatthe program is unlikely to have problems relatedto curved boundaries, we have made analyticallytestable runs so far only for empty pill-box cavities.

To see whether this code has any problems withextreme geometries, we ran an empty box of 5 cmlength and 150 cm radius with 1267 points. Withoutany difficulty, the code returned the fundamentalfrequency correct to all five printed digits.

Much more extensive runs were made for anempty box 60 cm long and a radius of 88 cm. Themesh point separation was 2 cm in both the axialand radial direction, giving a total of 1395 points.The fundamental frequency of this cavity is130.389 MHz and is reproduced by the code withan error of 1 part in 10,000, while the stored energyis reproduced' to an accuracy of 1 part in 3000. Amuch more severe test is the evaluation of highermodes. Resonance frequency number eight is582.44 MHz, and is returned by the code as583.59 MHz;· the stored energy calculated by thecode is 3%smaller than the correct value. Figure 8shows the pattern of electrical field lines (rH =const) for this mode. It should be noted that thedistance between an extreme value of rH and thenext axial node is only 7.5 mesh spacings. Modes29 and 30 represent an even more extreme testThe analytical frequencies are 1179.9 MHz and1186.3 MHz, and the code~produced frequenciesare 1183.0 MHz and 1196.6 MHz, while the energJof these modes is off by approximately 10 %Considering the fact that mode 29 has six radiaand one axial nodes, and mode 30 has three radia

SUPERFISH-A COMPUTER PROGRAM FOR RF CAVITIES 221

l------------------.l~z

FIGURE 8 Electric field lines (rH = canst) for mode No.8in test cavity.

and four axial nodes, these numbers are sur­prisingly good. The closeness of the two resonancesdid not cause any problems. "Turning on" thepartial and complete pivoting of the matrixinversion routines, or iterating on the field re­siduals of the solution of the difference equations,did not change any of these numbers. However,increasing the number of mesh points caused amarked improvement of the accuracy of thefrequency and the stored energy, indicating that thenumerical errors are due to mesh size, and notround-off errors.

6.3 Secondary Quantities, and Near FutureDevelopments

The program calculates now, or will calculate inthe very near future, the following secondaryquantities: stored energy; transit time factors;energy dissipated on designated surfaces; IH Imax,IE Imax on designated surfaces; shunt impedance;Q; and frequency perturbation by drift tube stems.

We also plan to calculate and print out co­efficients that indicate how the movement ofdesignated surfaces perturbs the resonance fre­quency. These quantities were calculated by RFISHand proved extremely valuable.

To simplify the work on high-order modes, weintend to generate printout plots of node lines(i.e., H == O-lines) and/or plots of points with localextrema of r H, and II (k 2

) and D(k2) plots.

To simplify the design of cavities that have tohave a predetermined resonance frequency, weintend to run the code with that fixed frequency(possibly modified by drift tube stems) and toaccomplish I 1 == 0 by moving or deforming adesignated boundary. The techniques necessaryto do this are already used in the magnet designcode MIRT7 and can easily be incorporated inSUPERFISH.

To reduce the number of iterations necessaryto find a resonance frequency, we plan to employa root-finding routine that uses the properties ofD(k2

) expressed by Eqs. (15) and (16). If this code isused extensively to find high-order modes, itmight also be profitable to attempt to develop amode pattern analysis and prediction routine.t

6.4 Advantages of SUPERFISH

The main advantage of the code is the capabilityto solve problems that other codes cannot solveat all, or only with great expenditure of computertime. In addition, the code is quite fast, requiringonly about 1 sec per iteration on the frequency forthe test problem discussed above. With fiveiterations and the time used to calculate miscel­laneous other quantities, one has a completesolution in 6 sec. The irregular triangular mesh,while not allowing as many mesh points as asquare mesh, has the advantage of allowing thedefinition of boundaries by mesh lines, and toproduce a mesh with a large density of meshpoints in regions where the problem requires highresolution.

6.5 Disadvantages of SUPERFISH

The drawback associated with the irregular tri­angular mesh is the fact that one has to generatesuch a mesh. This extra step can slow down the

t Since submission of this report for printing, the moresophisticated root-finding routine has been developed and isworking very well. Work on the mode prediction algorithm hasstarted and looks very promising.

222 K. HALBACH AND R. F. HOLSINGER

total process of receIvIng the desired answers.This problem has been partly reduced by thecreation of the code AUTOMESH, developedby one of us (R.F.H.) while 'at CERN. This codeoptimizes automatically the coordination betweenspace-boundary coordinates and logical coordi­nates, provided that one is satisfied with a uniformmesh point density in a limited number of distinctregions. At the time of writing this paper, aneffort is being undertaken at LASL by D. Swenson,w. Jule, and one of us (R.F.H.) to improve thewhole process of d~ta input and mesh generation.

There is one basic drawback associated with thenecessity of having a driving point in the problem:if one happend to choose its location such that it ison a H == 0 line for the problem under considera­tion computational problems would result. Forthat reason, it is advisable to put the driving pointon a Dirichlet boundary. When .the code detectsthe computational difficulty, it can switch thedriving point to a more favorable neighboringpoint on the boundary, thus eliminating theproblem.

ACKNOWLEDGEMENTS

We thank Dr. T. Elioff, Mr. A. Faltens, Dr. L. J. Laslett, Mr. S.B. Magyary at LBL, and Dr. R. A. Jameson, Ms. S. Johnson,Dr. W. E. Jule; Dr. E. A. Knapp, and Dr. D. A. Swenson atLASL, for discussions and/or support of this work. This workwas done with support from the U.S. Energy Research andDevelopment Administration. Any conclusions or opinions ex­pressed in this report represent solely those of the authors andnot necessarily those of The Regents of the University ofCalifornia, the Lawrence Berkeley Laboratory or the U.S.Energy Research and Development Administration.

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