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**Unformatted text preview: **1 EE256 Numerical Electromagnetics H. O. #15 Marshall 24 June 2008 Summer 2008 LECTURE 10 10.1 ABSORBING BOUNDARY CONDITIONS The necessarily finite nature of any FDTD spatial grid is a most important limitation in problems for which the FDTD algorithm must be run for durations larger than the propagation time from a scattering object to the grid boundaries and back to the scatterer. The field components at the outer edge of a finite FDTD space are not completely surrounded by their counterparts as are the interior points. Accordingly, there is not enough information to correctly update these components during the implementation of the FDTD algorithm. An Absorbing Boundary Condition (ABC) is a means to approximately estimate the missing field components just outside the FDTD grid. Such an approximation typically involves assuming that a locally plane wave is incident on the boundary and estimating the fields just outside the boundary by using the fields just inside the boundary. However, this cannot be done without error, since in most cases the wave arriving at the boundary is not exactly a plane wave and is not normally incident. The ABCs are thus in general approximations, and reflect some of the waves back into the FDTD space. First order ABCs are those which estimate the value of the fields outside the boundary by looking back one step in time and one grid cell in space, whereas higher order ABCs may look back over more steps in time and more grid cells in space. In general, different ABCs are better suited for different applications and the choice of the particular ABC is also made by considering its numerical efficiency and stability properties. Among the different types of absorbing boundary conditions, we shall limit our attention in this course to two particular ones, namely (i) those based on one-way wave equations, and (ii) those based on surrounding the FDTD domain with a layer of absorbing material, essentially creating a numerical anechoic chamber . 1 10.2 ABCS BASED ON THE ONE-WAY WAVE EQUATION The first category of ABCs are based on the fact that the solution of Maxwells two coupled curl equations via an FDTD algorithm is equivalent to the solution of the second-order wave equation for any one of the field components. Although the wave equation naturally supports waves propagating in both forward and backward directions, it can be factored into two one-way wave equations each of which support waves in only one direction, providing the basis for an algorithmic method by which the fields can be propagated out of the FDTD domain, minimizing reflections back into the numerical space. A first-order scheme of this type is first discussed below, using the 1-D wave equation, for simple one-way wave equations. This 1 Microwave dark rooms, or anechoic chambers are constructed with walls completely covered with absorbing materials, often in corrugated form (for tapered impedance matching), designed to minimize reflections and thus make the rooms suitable for testing...

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