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Unformatted text preview: 1 EE256 Numerical Electromagnetics H. O. #17 Marshall 08 July 2008 Summer 2008 LECTURE 11 11.1 RADIATION OPERATORS AS ABSORBING BOUNDARY CONDITIONS The use of various so-called radiation operators as the basis for Absorbing Boundary Conditions for FDTD calculations is discussed extensively in Sections 6.2, 6.4 and 6.6 of the textbook, Taflove and Hagness, Computational Electrodynamics . In fact, the use of the one-way wave equation as described in Lecture 10 (and in Section 6.3 of the textbook) is a special case of the general class of these types of analytical ABCs based on the use of a linear partial differential operator, which is particularly useful for cartesian FDTD grids. In this Lecture, we thus provide a brief summary, relying largely on the more extensive discussion provided by Taflove and Hagness. In general, the idea in using radiation operators is to construct a linear partial differential operator from a weighted sum of three types of derivatives of the field components: (i) spatial derivatives in the direction outward from the FDTD space, (ii) spatial derivatives in the direction transverse to the outward direction, and (iii) temporal derivatives. 11.1.1 Bayliss-Turkel Operators The so-called Bayliss-Turkel 1 operators are based on the time-domain expressions for the radiating (far- field) solutions of the wave equation, which have been derived in series expansion form, for both spherical 2 and cylindrical 3 coordinates. Here we briefly describe the two-dimensional (cylindrical) case. The two-dimensional cylindrical wave equation, for example for the wave component E z for a 2D TM mode, is given as: 2 E z- 1 v 2 p 2 E z t 2 = 0 2 E z r 2 + 1 r E z r + 1 r 2 2 E z 2- 1 v 2 p 2 E z t 2 = 0 [11.1] 1 Bayliss, A., and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. , Vol. 23, pp. 707-725, 1980. 2 Friedlander, F. G., On the radiation field of pulse solutions of the wave equation, Proc. Royal Soc. London Ser. A, Vol. 269, pp.53-69, 1962; Wilconx, C. H., An expansion theorem for electromagnetic fields, Comm. Pure Appl. Math. , Vol. 9, pp. 115-134, 1956. 3 Karp, S. N., A convergent far-field expansion for two-dimensional radiation functions, Comm. Pure Appl. Math. , Vol. 14, pp. 427-434, 1961. 2 since we assume no variations in z in this 2D system. Note that v p = ( )- 1 / 2 . Those solutions of [11.1] which propagate in directions outward from the origin (i.e., in the r direction for a cylindrical system) can be expanded in a convergent series as: E ( r,,t ) = X m =1 E m ( v p t- r, ) r m- 1 / 2 = E 1 ( v p t- r, ) r 1 / 2 + E 2 ( v p t- r, ) r 3 / 2 + [11.2] where we have dropped the subscript z for convenience. Here E m are functions of r and , but are all propagating in the radial direction, i.e., have time dependences of the form ( v p t- r ) . It is clear from the r- ( m- 1 / 2) dependence that the leading terms in [11.2] will dominate at large distances from the origin. Wedependence that the leading terms in [11....
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