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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 socalled 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 oneway 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 BaylissTurkel Operators The socalled BaylissTurkel 1 operators are based on the timedomain 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 twodimensional (cylindrical) case. The twodimensional 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 φ E z ∂φ 2 1 v 2 p ∂ 2 E z ∂t 2 = 0 [11.1] 1 Bayliss, A., and E. Turkel, Radiation boundary conditions for wavelike equations, Comm. Pure Appl. Math. , Vol. 23, pp. 707725, 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.5369, 1962; Wilconx, C. H., An expansion theorem for electromagnetic fields, Comm. Pure Appl. Math. , Vol. 9, pp. 115134, 1956. 3 Karp, S. N., A convergent farfield expansion for twodimensional radiation functions, Comm. Pure Appl. Math. , Vol. 14, pp. 427434, 1961. 2 since we assume no variations in z in this 2D system. Note that v p = ( μ ) 1 / 2 . Those solutions of [11.1] which propagate in directions outward from the origin (i.e., in the ˆ r direction for a cylindrical system) can be expanded in a convergent series as: E ( r,φ,t ) = ∞ X m =1 E m ( v p t r,φ ) r m 1 / 2 = E 1 ( v p t r,φ ) r 1 / 2 + E 2 ( v p t r,φ ) r 3 / 2 + ··· [11.2] where we have dropped the subscript z for convenience. Here E m are functions of r and φ , 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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 '09
 Derivative, Partial Differential Equations, Electromagnet, The Land, Partial differential equation, wave equation

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