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Unformatted text preview: 1 EE256 Numerical Electromagnetics H. O. #6 Marshall 24 June 2008 Summer 2008 LECTURE 4 4.1 THE FDTD GRID AND THE YEE ALGORITHM After a brief exposure to different finite difference algorithms and methods, we now focus our attention on the so-called FDTD algorithm, or alternatively the Yee algorithm, for time-domain solutions of Maxwell’s equations. In this algorithm, the continuous derivatives in space and time are approximated by second-order accurate two point centered-difference forms, a staggered spatial mesh is used for interleaved placement of the electric and magnetic fields, and leap frog integration in time is used to update the fields. Note that the cell locations are defined so that the grid lines pass through the electric field components 1 and coincide with their vector directions. y y x z x x y z y z x x z y x y ( i,j,k ) ( i-1 ,j +1 ,k) ( i-1 ,j +1 ,k +1) ( i,j +1 ,k ) ( i,j,k +1) z z Figure 4.1: Placement of electric and magnetic field components in a three-dimensional staggered mesh, known as the Yee cell. The small vectors with thick arrows are placed at the point in the mesh at which they are defined and stored. For example, E y is defined/stored at mesh points ( i + m 1 ,j + 1 2 + m 2 ,k + m 3 ) , where m 1 , 2 , 3 = 0 , ± 1 , ± 2 ,... , while H y is de- fined/stored at mesh points ( i- 1 2 + m 1 ,j + m 2 ,k + 1 2 + m 3 ) , where m 1 , 2 , 3 = 0 , ± 1 , ± 2 ,... . 1 The choice here of the electric field rather than the magnetic field is somewhat arbitrary. However, in practice, boundary conditions imposed on the electric field are more commonly encountered than those for the magnetic field, so that placing the mesh boundaries so that they pass through the electric field vectors is more advantageous. Note also that the Yee cell depicted in Figure 3.1 of the textbook, Computational Electrodynamics by Taflove and Hagness, has the cell boundaries to be aligned with the magnetic field components, rather than with the electric field components as in Figure 4.1 above. This choice of the particular way of associating the spatial indices i , j , and k with the field quantities is obviously arbitrary, and should not matter in the final analysis, as long as one is consistent. Please note this difference between the Lecture Notes and the text in formulating your FDTD algorithms. Sorting out the differences will actually enable you to fully understand the inner workings of the FDTD algorithm. 2 H n +1/2 Leapfrog in Time E n E n +1 H n +3/2 t Figure 4.2: Leapfrog time marching of electric and magnetic field vectors. With the Yee cell as defined in Figure 4.1, the spatial derivatives of various quantities are evaluated using a simple two-point centered difference method. For example, the z derivative of any given field component G evaluated at time n Δ t and at the mesh point ( i,j,k ) is given as ∂ G ∂z n i,j,k = G n i,j,k +1 / 2- G n i,j,k- 1 / 2 Δ z...
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This note was uploaded on 01/24/2011 for the course EE 256 at Stanford.