Fomel, B M

Karliner, M M

Lysyansky, P B

Yakovlev, V P

CERN Document Server

Particle Accelerators1981 Vol.ll pp.173-1790031-2460/81/1103-0173$06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaLANS-A NEW CODE FOR EVALUATION OF THEELECTROMAGNETIC FIELDS AND RESONANCE FREQUENCIESOFAXISYMMETRICAL RF CAVITIESB. M. FOMEL, v. P. JACKOWLEV, M. M. KARLINER, and P. B. LYSYANSKYUSSR Academy of Sciences, Siberian Division, Institute of Nuclear Physics, Novosibirsk 90, USSR(Received August 1, 1980)A new code for calculation of axisymmetric cavities is developed. The mathematical basis of this code is the methodof inverse iterations with a shift, which is the most adequate for the problem of finding the eigenfrequencies andfields for the cavities. This code has some advantages compared with SUPERFISH; it requires a smaller numberof operations necessary for calculations and it gives better resolution of resonance modes with close frequencies.Eliminating one of the fields, E, from Eqs. (1),one can get a wave equation for another com-2. THE PROBLEM OF EIGENVALUESAND EIGENFUNCTIONS. THEMETHOD OF INVERSE ITERATIONS.Electromagnetic fields in a cavity are describedby Maxwell's equations, which in the case ofharmonic oscillations have the formwhere 0 is a vector normal to the surface. An-other kind of boundary conditions (Dirchletboundary), usually used on the symmetry planeof a cavity, is(2)(1)(3)E x 0 = 0, H·o = 0E·o = 0, H x 0 = 0curl E = - iWfLH - Jm,curl H = iweE + Je,where J,n and Je are the densities of magnetic andelectric currents, respectively. In addition toEqs. (1), the fields E and H should also satisfythe boundary conditions. Usually, in these cal-culations, the boundary conditions are taken tobe ideal. The boundary conditions on the idealconducting surface (Neumann boundary) havethe formI. INTRODUCTIONRF cavities are important components of linearand cyclic particle accelerators, as well as of rfgenerators and amplifiers. Naturally, the possi-bility of using computers for calculations of theresonance frequencies and electromagnetic fieldsof cavities facilitates significantly the design andproduction of cavities with optimized character-istics. Of the known programs for calculations ofaxisymmetric cavities, the most perfect is ap-parently the program SUPERFISH.]We have developed a new program LANS2 themathematical basis of which is the so-calledmethod of inverse iterations (with a shift) ,3 whichis the most adequate for the problem of findingthe eigenfrequencies and fields for the cavities.This program has some advantages comparedwith SUPERFISH; namely, it requires a smallernumber of operations for calculations; it alsogives better resolution of resonance modes withclose frequencies.A mathematical formulation of the method ofinverse iterations (with a shift) in application tothe evaluation of electromagnetic fields in cavi-ties is given in Section 2. In Section 3 an algo-rithm is discussed for the calculation of the mag-netic field and resonance frequencies. In Section4 a direct noniterative method is described forsolution of a set of inhomogeneous linear equa-tions. The main features of the program LANSinteresting for users are given in Section 5.173174 B.M. FOMEL ET AL.ponent of the electromagnetic fieldcurl curl H - K 2H = curl Je - iK ~~ Jmwhere K 2 = W2J.LE = w2/e 2The right-hand side of this equation can bewritten in a simpler waycurl curl H - K 2H = KJ, (4)The fields H(m) and H(m + 1) can be expandedinto series over the eigenfunctions of Eq. (5)H(m) = L A/m)Hi ; H(In + I) == L A/rn + I)Hi (9)i iSubstitution of Eq. (9) into Eq. (8) will give(10)Solving the equationJ being a vector proportional to the magnetic cur-rent density.It is known that Eq. (4) with boundary condi-tions (2) and (3) has a unique solution if K 2 doesnot coincide with any of the eigenvalues K n 2 ofthe homogeneous equationcurl curl H(I) - K~oH(1) == K~oH(O) (7)we find the first iteration H(I). Since by assump-tion, K~o is close to K n2 , the field H(1) will beclose to the resonance field with a frequency W n •For the (m + 1)-th iteration the equation has theform(12)Kn2 = (1 + ~J K~o2 2Kno K n K~+1 2I • KLocation of the eigenvalues neighboring K n 2 •FIGURE 1whenceThus with this iteration process one can find allnecessary characteristics in addition to the ei-genfunction Hn , i.e., the magnetic field and alsothe value of the resonance frequency wn •Discrete approximation of Eq. (4) is performedAn(rn + 1) K~oA (rn) == '"'i == K 2 _ K 2 ' (11)n n noLet us assume that the approximation K~o is soselected that in absolute value the difference K 22 . n- K no IS the smallest among the differences ofthe kind K i 2 - K~o. Such a choice is illustratedby Fig. 1, which shows location of the eigenval-ues neighbouring K n 2 • Then with each succeedingiteration, the relative value of the factor An in-creases compared with the remaining coefficientsof expansion. In other words, with the growth ofiteration number, the field H(rn) tends to its ei-genfunction Hn • Since the absolute value of thefield H(rn) increases because of a resonant factor'"'in == K~o/(Kn 2 - K~o), it is reasonable to nor-malize the field to unity after each iteration.The number of iterations necessary for ap-proximation to Hn with a given accuracy de-pends, first, on how close the chosen approxi-mation K~o is to K n 2 and, second, on the zerothapproximation of the field H(O).Note, however, that K~o should not be tooclose to K n 2 since, in this case, the method offinding a solution for Eq. (8) becomes incorrectdue to approaching the resonance. After carryingout the required number of iterations the eigen-value K n 2 can easily be found from the relation(6)curl curl H - K 2H = 0 (5)with the same boundary conditions.Calculation for a cavity consists first of defin-ing resonant frequencies wn 2 = c2K n 2 and eigen-functions Hn i.e., the fields at resonance. Usu-ally, one confines oneself to the solution of apartial problem of eigenvalues, finding only someof the eigenvalues and their corresponding ei-genfunctions. For example, the design of accel-erator cavities starts with the calculation of themode with the lowest resonant frequency. Often,though, it is necessary to calculate in addition anumber of higher modes.The program LANS is based on the use of aninverse iteration method. 3 The essence of themethod is as follows. Instead of the homogeneousEq. (5) one solves the inhomogeneous Eq. (4)with the boundary conditions (2) and (3). Let usassume, that (with any arbitrary method) we haveobtained an approximation of the eigenvalues Kn 2to be equal to K no 2 and also a zeroth approxi-mation of the field H(O). In the right hand side ofEq. (4), let us take the current to beNEW ELECTROMAGNETIC FIELD CODE 175th iteration we obtainSubstituting these expansions into Eq. (17), weobtainThe vectors 'Je(m) and 'Je(m + I) in Eq. (17) can berepresented as expansions over eigenvectors ofthe matrix B(K2 ).(18)(19)N'Je(m) = L A/m + 0'Je;.;=1N'Je(m) = L A/m)'Je;;=1(W - K~oV)'Je(m+l) = K~oV'Je(m). (17)Let 'Je; be eigenvectors of the matrix B(K2 ) andlet K;2 be its eigenvalues, i.e.,b 0L 'Je;p(w;p - K 2v ,P ) = L KJ;qv;q, (14)p=O q=Owhere the coefficients Vip and w;p depend on thecoordinates of the triangular mesh.by a method similar to that given in Ref. 1. It istherefore not presented in this paper.We should note that the only difference is thathere Eq. (4) has a right-hand side. Therefore afterintegration of Eq. (4) over the internal area of adodecahedron (see Fig. 1) and use of the Stokes'theorem, one obtainsf curl H<p de - K 2 f H<p dS = K f J<p dS (13)And, finally, taking into account the linear de-pendence on the coordinates inside every trian-gle, one has3. MAGNETIC FIELD AND RESONANTFREQUENCY CALCULATION A ,.(m+1) = K~o A.(m) = -A.mK .2 _ K 2 I ~l I •I no(20)If Eq. (14) is written for all mesh points, one canobtain the relation Let us assume that the vectors 'Je; are normalizedto unity, i.e.,Substituting the vector 'Je(I) obtained from Eq.(16) into the left-hand side of Eq. (15), one cancontinue the process of iteration. For the (m + 1)-where B(K2) = W - K 2V, Wand V are matricesformed of the coefficients wand v respectively,~, J are column vectors formed of values of themagnetic field and magnetic current densities inthe mesh points having dimension N = K maxx L max (the mesh contains K max points on thez coordinate and L max points on the R coordi-nate). A square matrix B(K2) is a three-diagonalmatrix, the elements of which are small matricesof dimension Lmax x Lmax . Small matrices on thediagonal of the matrix B(K2) have three nonzerodiagonals but each of the remaining small mat-rices have two nonzero diagonals.Let W n = KnC be an eigenfrequency of the re-quired mode with number nand W no = KnoC asits initial approximate value. Also, let the distri-bution of magnetic currents with density J =Kno'Je(O) be given in a cavity. Then from Eq. (15)follows(23)with coefficients a;(m+ I) equal toIf the result of every iteration is normalized tounity, for the normalized vector eat(m + I) we ob-tainN'Je(m + 1) = L a/m + 1)eat;,;=1A·(m+1)a (m + I) - I (22)i - L~l (A/rn + ll?r/2 •Substituting the latter equality into the expres-sion (20) for A/m + 1) and squaring it, we obtain[A/m)]2[a/m + 1)]2 = -N--(-K-2--K-2-)~2--- .L ;2 - ;0 [A~ (m)]2s=l K s -KnoAssuming that normalization is carried out afterevery iteration, in eq. (23) one can substitute a/m )(15)176 B.M. FOMEL ET AL.for A;(m) to find genvalue Kn 2(Kn (1'11»2 = K~o (1 + _1_)'Yn (1'11) • (30)the denominator in eq. (25) is then less than unity.Thusunless [an (In + I)] < 1. Consequently, under thiscondition we have the inequality(31)I (1'11) \. _"~_(m_+_I)_11'Yn II ~(m) II 'To improve accuracy in the program LANS, thecoefficient 'Yn is evaluated with the aid of a normtaking into account that II ~(,n + 1) II is determinedprior to normalizing ~(ln + I) to unity. The sign for'Yn can be found from the equality (20).Note that the accuracy of approximation ofthe final iteration ~(m) to the eigenvector ~n doesnot determine entirely the accuracy of approxi-mation of the obtained solution to the true ei-genfunction Un. Indeed, the vector ~n ap-proaches the Un mode with an accuracy definedby the degree of approximation of the differentialEq. (4) by the set of difference Eqs. (15).The result of the iteration process is a conven-ient sequence of vectors ~(,n), the limit of whichare the vectors ~n. At the same time, a sequence(Kn (In))2 can be obtained with the formula (30).The convergence rate for sequence (Kn (In»2 ishigher than that for ~(ln) because small variationsof field do not affect the eigenvalues accordingto their well-known property. Therefore, the cal-culation of eigenvalues with desired accuracy re-quires much smaller number of iterations thanthat needed for calculation of a field.The program LANS includes the completioncriterion for iterations which is based on ap-proaching the limit over the field norm. The it-eration process is considered to be completedwhen the following inequality is satisfied(27)(24)(25)[an (m+I)J2--(-)- > 1;an I'll~ 1In particular, at i = n[an(m+ I)J2an(m)NL [a s (m)]2 = 1,s=1Let us assume that the difference I Kn2 - K~o Iis less than any of the similar differences I K;2- K~o I for i -:/= n. Sincei.e., after every iteration, the coefficient an (m + I)increases in absolute value, remaining less thanunity. Hence, one can draw the conclusion thatwheree(m+1) <eo[l - 'Yl(m+1)], (32)lim [an (1'11)]2 = 1In~oolim [a/m )]2 = 0 at i -:/= n.,n~oo(28)(29)Thus, making a certain number of iterationsone can, with sufficient accuracy, make the so-lution close enough to the eigenvector ~n. Afterthat, from the equality (20), one can find an ei-NEW ELECTROMAGNETIC FIELD CODE 177oIhKmax -11x (~~~a.) = (P~~a.) (35)where hj = dj -I CjPj = dj -IFj = dj -IGj -d; -l ejF j _ 1 , or otherwiseP j = fjG.i - gjPj - l ,where fj = dj,-l, gj = fjej. Then ~Kmax =P Kmax ; ~j = P j - hj Pj - 1 (1 ~j < K max ). Thus,'JC is calculated through the matrices fj, hj , gj.For finding these matrices, one should reversethe matrices dj only once. These matrices canreplace matrices aj, bj , Cj in the computer mem-ory and then be used for solution of Eq. (15) withdifferent right-hand sides. As can be seen, forstoring matrix B(n2), a memory of (3Kmax -- 2)x L~ax words is required.5. CALCULATION OF ELECTRIC FIELDAND INTEGRAL CHARACTERISTICSOF CAVITIESAfter the magnetic-field distribution is found andresonance frequency of the desired mode is eval-uated, one can determine in every triangle theelectric-field components (radial and axial) fromMaxwell's equations curl Hn = iwnEn by the as-sumption that the magnetic field inside every tri-angle depends linearly on the coordinates Randz. Electric fields at the mesh points are calculatedas the average over the adjacent triangles.In addition to electromagnetic fields and ei-genvalues, the program provides for calculationof the following values, which characterize a cav-ity:a) overvoltage, i.e., the ratio of maximumvalue of electric field on the cavity surface to theeffective value of accelerating field on the axis:K max ); can be transformed into the formwherewhere ai, bi, Ci are matrices of dimension L maxx L max and G = KVJ is the right-hand side ofEq. (15). Following the Gaussian elimination al-gorithm, one can transform Eq. (33) into the form4. SOLUTION OF A SET OF LINEAREQUATIONS.The iteration process described above providesa repeated solution of an inhomogeneous set oflinear equations (15) with the matrix B(K2) = W- K 2V the same for all iterations and only theright-hand sides of the equations changed. There-fore, it seems more reasonable to transform thematrix B(K2) into a modified LU-form4 and thenuse it in further calculation rather than to solvethe whole equation (15) many times.Equation (15) can be written in the formSummation is carried out over all the meshpoints. Usually, the value Eo is selected in therange 10- 6 to 10- 7 •or otherwise(J' = E surf. max/Eeff ,F;. = Gj - ejFj _ 1(here ej = bj dj__11 , j varies within the limits 2 to2 (L/2Eeff = LJoEz(z) cos(Kz) dz178 B.M. FOMEL ET AL.Dwhere L is the length of accelerating gap and Ez(z)is an axial component of electric field on the cav-ity axis.b) transit time factor for relativistic particlesl L/2 / (L/2T = 0 Ez(z) cos(Kz)d Jo £z(z) dzc) quality factorQ = ~ Iv I H 12 dV / L1 H 12 ds,where 8 is the skin-depth thickness.d) characteristic impedanceI(L/2 12P = Jo E z(7) dz IwfJ. Iv 1 H 12 dVe) shunt impedanceR sh = QpIn the program LANS it is also possible to cal-culate the resonance frequency variation withsmall changes in shape, which is important fordesign purposes. To achieve this, the programLANS calculates the so-called influence factorsthat indicate the variation of resonance frequencydue to unit wall deformation at a given point, theunit deformation being the axisymmetric defor-mation of a wall section with unit length (alonga cross section) by a shift of unit length in thedirection normal to the surface. The influencefactors are calculated through differences of mag-netic and electric energy densities in the vicinityof the surface.6. ACCURACY AND OTHERCHARACTERISTICS OF THE PROGRAMErrors in calculation of electromagnetic fieldsand eigenfrequencies for cavities depend on thenumber of mesh points as a/N2 , where N is thetotal number of points. The coefficient "a" de-pends on the cavity geometry.The more complex is the cavity geometry, thelarger is the number of mesh points that are re-,quired for achieving the necessary accuracy. Ex-perience shows that 500 to 2500 mesh points arerequired for achieving errors of the level ~ 10- 4in the calculation of the eigenfrequency for themain mode. The error of magnetic-field evalua-tion under this condition is greater and is about10- 3 • For testing the program and checking theaccuracy of evaluation fields and resonance fre-quencies, as well as integral characteristics ofcavities, calculations were performed for cylin-drical and spherical cavities. The error in cal-culation of frequencies for modes £0]0 and Eo]]of a cylindrical resonator is about 10 - 4, the num-ber of mesh points being equal to 400. The errorin calculation of magnetic fields is of order 10- 3.For the £0]0 mode of a spherical cavity, the errorin calculation of frequency (at the same numberof mesh points) is 4 x 10- 4 , and that of magneticfield is 2 x 10- 3 •Time. consumption is square dependent on themesh-point number, but depends weakly on thenumber of iterations, since the major part of thetime is spent in modifying the matrix B(K2) to theLU-form at a given approximate value K~o. Itrequires o.N2 multiplicative operations. Everyiteration requires f3N 3/2(P < 0.) of such opera-tions, i.e., at N ~ 1 a much smaller number thanrequired for transformation of the matrix B(K2).Note that the convergence rate of iterations isthat of a geometric progression with the denom-inator (Kn 2 - K~o)/(K~+] - K~o, where K~+]is the eigenvalue which is nearest to the eigen-value K n 2 that is sought. If the eigenvalue K n 2 isclose to K~ +] then the denominator is close tounity. In this case, the solution of a problem willrequire a few tens of iterations, but, this leads toa small increase in time of calculation. If the ei-genvalues Kn 2 and K~ +] are well separated, thencalculations require less than ten iterations.The properties of the program mentionedabove make the program LANS especially con-venient for calculations of periodic acceleratingstructures having several close resonance fre-quencies due to coupling between neighbouringcavities.A CI II II II II II II II I II I I---l-- .--1- ---'--__.--lBFIGURE 2 The configuration of the biperiodical acceler-ating structure.REFERENCESFIGURE 3 The cell of the accelerating structure.NEW ELECTROMAGNETIC FIELD CODE 179CII,IIIIIIII--. ----l .._-DI,IIII-LBA1. K. Halbach and R.F. Holsinger, Particle Accelerators, 7,213 (1976).2. M.M. Karliner, P.B. Lysyansky, B.M. Fomel and V.P.Jackowlev. Preprint INF 79-59, Novosibirsk, 1979.3. N.N. Kalitkin, Numerical Methods (Nauka, Moscow,1978), p. 166.4. G.E. Forsythe and C.B. Moler, Computer Solution of Lin-ear Algebraic Systems, (Nauka, Moscow, 1967), p. 65.same signs, but for the 7T mode with oppositesigns.Figure 2 shows schematically the configurationof a biperiodic accelerating structure for a su-perconducting accelerator which was calculatedwith the program LANS. The structure has ac-celerating and coupling cavities and it is designedfor operation on a standing wave of 7T/2 type. Thecalculation comprises the evaluation of the fre-quency of the 7T/2 mode and also the couplingfactor.For determining the coupling factor it was nec-essary to calculate the frequencies of the 0 and·7T modes. For calculations of these frequencies,the cell shown in Fig. 3 was cut on the symmetryplanes. Resonance frequencies were then cal-culated with Neumann conditions (curlL H = 0)being given on these planes of symmetry. Tworesonance frequencies were found under theseconditions, namely the frequencies of the 0 and7T modes. For calculation of the 7T/2 mode fre-quency, Dirichlet boundary conditions (H~ = 0)were given on the plane AB and Neumann con-ditions were given on the CD plane, as in theprevious case. Checking on symmetry of locationfor frequencies of 0 and 7T modes with respect toa frequency of the 7T/2 mode allows one to makecorrections for coupling cavity in such a way thatits resonance frequency should be equal to thefrequency of the accelerating cavity. The fre-quencies calculated for this example differ by 1to 3%. The program LANS enabled us to cal-culate frequencies with an accuracy level of10 - 4. The number of iterations used was not verylarge due to an appropriate choice of initial ap-proximation for the magnetic field c;jf(O). For ex-ample, for calculation of the O-mode magneticfields in adjacent cavities are selected with the

Lans - A New Code For Evaluation Of The Electromagnetic fields And resonance Frequencies Of Axisymmetrical Rf Cavities

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