TWO DIMENSIONAL ANALYSIS OF RF ELECTROMAGNETIC FIELDS
BY FINITE ELEMENT METHOD
M. Hara, T. Yada, T. Fukasawa* and F. Kikuchi**
The Institute of Physical and Chemical Research* Nihon University** College of General Education, The University of Tokyo
Abstract
to solvecodes
computerdeveloped
alreadyhaveWeelectromagnetic field problems which reduce to two dimensional
scalar Helmholtz equations. We improved these codes, and also
developed bandwidth minimization code.
1. Introduction
In the design of RF resonators of accelerators, it isresonancefields,electromagnetic
calculatetonecessary
frequencies, Q value, power loss, and other pertinent quantities
in the cavities surrounded by metals. To get these, we have
(H2DB and HAX) based on
already developed two computer codes
finite element method. H2DB code is for the fields in thestraight resonator with arbitrary cross section and HAX code isfor the axisymmetric fields. In both calculations, we must solve
the boundary-value problems for two dimensional scalar Helmholtz
equations. Automatic mesh generation codes MESHGEN2 and MESH2D,field plot code H2DPLOT, and Q value and power loss calculation
code HAXQE also have been developed1 >. We improved these codes.
Bandwidth minimization code BNDMIN was also developed.
2. Field calculation H2DB and HAX
We added new functions. One of them is a restart function. The
calculation which is terminated by time limit can be restarted
any number of times reading the preceding results from disk file.
minimization BNDMIN and mesh generation MESH2DBNDMIN minimizes the bandwidth of band matrices in finite
element method in order to save memory size and computing time.
We use connection array to minimize it2). Minimization procedure
is useful especially for complicated shape of cavity. The code
MESH2D is also improved in numbering the nodal points to minimize
the bandwidth.

~Bandwidth

4. Field plot H2DPLOT and Q value calculation HAX~In H2DPLOT and HAXQE, it is necessary to find boundary forplotting or calculation. But it takes much time to find boundaryby the previous method. In the new method, we use connection
array to find boundary. This extremely reduces computing time.
Example of calculations
Four-vane RFQ resonator of quadrangular shape is analyzed by
H2DB code. We have already analyzed the quarter of four-vane RFQ
resonator imposing symmetry conditions. At this time, we analyze
full shape of this resonator. Figure 1 shows the calculated
cut-off frequencies and the electric lines of force.
5-271-References
1) M. Hara, T. Wada, A. Toyama and F. Kikuchi : "Calculation ofRF Electromagnetic Field by Finite Element Method" ScientificPapers IPCR, 1Q (1981) 143.
2) E. Cuthill and J. Mckee : "Reducing the Bandwidth of Sparse
Symmetric Matrices", PROC. 24th Nat. Conf., Assoc. Comput. Mach.,
ACM Publ. p-69, 1122 Ave. of the Americas, New York, N.Y. (1969)

~~~

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2nd mode TE110
Freq. = 232.94 MHz

3rd mode TE110
Freq. = 233.48 MHz

4th mode TE210
Freq. = 238.48 MHz

5th mode
Freq. = 1602.9 MHzFig. 1. Cut-off frequencies and electric lines of force forfour-vane RFQ resonator of quadrangular shape. (Side length = 200mm, vane width = 15 mm, bore radius = 5 mm.) The eigenvalue ofthe 1st mode is zero. It has not significant physical meaning.
Frequencies of the 2nd and the 3rd modes are expected to have thesame value analytically, but are not equal in this calculation.
The reason is that the approximate function used in thiscalculation is not symmetric.

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