der etc; it is advantageous to formulate FDTD in cylindrical coordinates. But, exceedingly large computer resources are required when a high circumferential resolution imposes a small time step. The problems associated with a short time step are particularly pronounced at low frequencies, where time cycles are long. The solution described interfaces two methods: the PEE and the classical FDTD in cylindrical co-ordinates, which results in a new, efficient modelling tool. esign using genetic algorithm J.S. P a n We wish to thank V, Delport for making us aware of his Letter [l, 21. Although he uses a genetic algorithm, it differs from ours considerably [3]. The main differences are as follows: Delport uses the codebook indices of the training data as the coding string, the length of the coding string is thus the number of the training data points in the training set. In our algorithm, we use the codebook vectors as the coding string, the length of the coding string is thus equivalent to the number of codewords. This means that the string is much shorter in our algorithm. Theoretically and practically, it is difficult to converge to better optimal value if the coding string is too long. In addition, in our algorithm, the coding strings of the initial population can be assigned randomly from the training data, because it can be converged to optimal value easily in any initial condition. Delport uses the binary splitting method to obtain the better initial population to improve his algorithm. Finally, we employ a sorting technique based on the central value of the training data to facilitate convergence to better optima. This is unique to our algorithm. To sum up, Delport applies a genetic algorithm to adapt the codebook index of points in the training data, whereas we apply a genetic algorithm to adapt the value of the codebook vector. 31 October 1995 0 IEE 1996 Electronics Letters Online No: 19960138 J.S. Pan (Centre for Communication Interface Research, The University of Edinburgh, 80 Soutlz Bridge, Edinburgh EH1 IHN, Scotland, United Kingdom) /= FDTD Ar Fig. 1 Schematic diagram of method of analysis Analysis; The geometry of the structure is shown in Fig. 1. The PEE algorithm described in [l] is used inside the radius R,where the field components are expanded in a Fourier series with respect to 4.A typical component has a form N E~(T4 , ,z)= (1) e T n ( T , z)ejnd n=l References 1 DELPORT, v.: ‘Comment: VQ codebook design using genetic algorithm’, Electron. Lett., 1996, 32, (3), p. 193 2 DELPORT, v., and KOSCHORRECK, M.: ‘Genetic algorithm for codebook design in vector quantisation’, Electron. Lett., 1995, 31, (2), pp. 84-85 3 PAN, J.s., MCINNES, F.R., and JACK, M.A.: ‘VQ codebook design using genetic algorithms’, Electron. Lett., 1995, 31, (17), pp. 1418-1419 D method for efficient field lindrical co-ordinates M. Mrozowski, M. Okoniewski and M.A. Stuchly Between R, and R,, an FDTD algorithm in cylindrical co-ordinates is used. The transition between the two algorithms (interior PEE and exterior FDTD) is carried out as follows: fields calculated from the PEE on the surface r = R, are directly imported to the FDTD algorithm. In the opposite direction, the fields given by the FDTD at R,+ Ar are used to obtain the expansion coefficients of the PEE by taking the inner product of each function [l]. In cylindrical co-ordinates, and media independent of 4, this algorithm resembles the PEE-FDTD [2]. However, there are two important differences between the two algorithms. The PEE computations are carried with several expansion terms simultaneously, while the BOR-FDTD considers each mode separately. Only axisymmetric structures can be analysed with the BOR-FDTD method, while the hybrid PEE-FDTD method can be applied to any inhomogeneuity in the FDTD region of the computational space. A relative stability criterion of the hybrid method compared to the FDTD can be derived following the method given in [3] as Indexing terms: Numerical methods, Finite-dijference time domain method, Electromagnetic waves Two algorithms are combined in the cyhndrical co-ordinates for accelerated computations of electromagnetic fields in the time domain. The partial eigenfunction expansion (PEE) algorithm is used close to the co-ordinate system origin, and the FDTD algorithm is used in the remaining region. This eliminates the use of a prohibitively short time step if high circumferentialresolution is required away from the origin. A significant acceleration of computations is obtained, owing to the superior stability and higher efficiency of the PEE method. Introduction: As time domain techniques, particularly the finite difference time domain (FDTD), are increasingly used to simulate electromagnetic fields in more complex and larger structures, there is a need for new algorithms which require fewer computer resources, and are stable. Recently, an efficient hybrid partial eigenfunction expansion (PEE) - FDTD method was developed for shielded structures in the Cartesian co-ordinates [l]. For some structures; e.g. a corrugated waveguide, antenna arrays on a cylin- 194 where A is a constant dependent on b and R,, and N is the number of terms in the expansion series. Thus, the smaller the A$ and the lower the number of terms in the expansion required, the greater the mcrease in the tme step. Additionally, the dispersion of the PEE algorithm is inherently lower, because of a lack of discretisation in the I$ direction, so more accurate results are to be expected. The increase in speed of the hybrid method compared with the FDTD is caused by two factors: (i) as the required number of terms in the expansion is usually small, a larger time step can be used resulting in fewer computations, and (ii) fewer unknowns are used to evaluate the fields close to the origin. The gain in speed - R,)for a caused by the second factor is proportional to R,I(R,xox large number of discretisation points along Y and small N. The overall acceleration is a product of both factors, and large gains in speed are expected for large R,IRmax.The gain in speed is also accompanied by savings in the computer memory. ELECTRONICS LETTERS 1st February 1996 Vol. 32 No. 3

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