Advanced Engineering Electromagnetics, ECE750 LECTURE 11 THE FDTD METHOD PART III

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1 Advanced Engineering Electromagnetics, ECE750 LECTURE 11 THE FDTD METHOD PART III 1

2 11. Yee s discrete algorithm Maxwell s equations are discretized using central FDs. We set the magnetic loss equal to zero. Then, i σ m = 0, J m = 0 E E E E y z n n n n n+ 0.5 n 0.5 t zi, j+ 1, zi, j, yi, j, + 1 yi, j, x = i, j, x i, j, H H µ E E E E z x n n n n n+ 0.5 n 0.5 t xi, j, + 1 xi, j, zi+ 1, j, zi, j, y = i, j, y i, j, H H µ E E E E x y n n n n n+ 0.5 n 0.5 t yi+ 1, j, yi, j, xi, j+ 1, xi, j, z = i, j, z i, j, H H µ

3 11. Yee s discrete algorithm cont. H n+ 0.5 n 0.5 n 0.5 n z H + i, j, z H + i, j 1, y H + n E n E i, j, yi, j, 1 in Ex = i, j, E Ex + i, j, H Jex i, j, y z n+ 0.5 n+ 0.5 n+ 0.5 n+ 0.5 H 1 x H i, j, x H i, j, 1 z H n E n E i, j, zi 1, j, in Ey = i, j, E Ey + i, j, H Jey i, j, z x n+ 0.5 n+ 0.5 n+ 0.5 n+ 0.5 H 1 y H i, j, y H i 1, j, x H n E n E i, j, xi, j 1, in Ez = i, j, E Ez + i, j, H Jez i, j, x y σ e t t 1 E ε E E = σ e t H = ε σ e 0 σ e t ε ε 3

4 11. Yee s discrete algorithm cont. The above coefficients are obtained by averaging the E- field, which appears in the loss term. For example, E x H H z y i ε + σeex = Jex t y z n+ 1 n n+ 1 n Ex E i, j, x E i, j, x + E i, j, xi, j, ε + σe = t n+ 0.5 n+ 0.5 n+ 0.5 n+ 0.5 H H H H y z zi, j, zi, j 1, yi, j, yi, j, 1 i J ex i, j, The discretization steps in time and in space, as well as the numerical constant α = c t / h are determined as for the wave equation. 4

5 11. Yee s discrete algorithm cont. Yee s cell E y ( i+ 1, j, + 1) E x (, i j, ) E z ( i + 1, j, ) H y (, i j, ) E x (, i j, + 1) E z (, i j, ) H x ( i + 1, j, ) E y ( i + 1, j, ) H z (, i j, ) H x (, i j, ) H z (, i j, + 1) E y (, i j, + 1) E x (, i j + 1, ) E z ( i+ 1, j+ 1, ) H y (, i j + 1, ) E z (, i j + 1, ) E y (, i j, ) E x (, i j + 1, + 1) ( ) x i z ( ) y ( j ) 5

6 11. Yee s discrete algorithm cont. -D problems and their discretization The -D TE z mode The -D TM z mode µ H E z y E x mh z t σ + = x y ε E z E H H J t σ + = x y E x H z i x z ε + σeex = Jex µ H + σmh x = E t y t y E y H z i y z + eey = Jey µ H + σmh E y = t x t x ε σ Ey y x i e z ez E z H x Ez E x H E z x H y H y () x i y ( j ) Ey () x i E z H x E z y ( j ) 6

7 1. Absorbing (radiation) boundary conditions ABCs constitute a special type of BCs, which simulate reflection-free propagation out of the computational domain. ABCs are necessary in open (radiation/scattering) problems, as well as in guided-wave problems where matched port terminations are needed. The simplest to implement ABCs are associated with various approximations of a one-way plane wave propagation. One-way wave equation (Mur s ABC) Liao extrapolation Perfectly Matched Layers basics Others: Higdon operator, Bayliss-Turel annihilating operators, etc. 7

8 1. ABCs cont. A.The one-way wave equation (B.Engquist and A.Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation, vol. 31, 1977, pp ) This is an equation which permits wave propagation in only one direction. Consider the 3-D scalar wave equation f f f 1 f + + = x y z c t The partial derivative operator is defined as 1 L= + + = + + c x y z c t 0 Lf = 0 x y z t We wish to simulate propagation along x at x=0. 8

9 1. ABCs cont. The partial differential operator L can be factored, i.e., represented as sequentially applied two operators: + Lf = L L f = 0 1 = x t 1 L c S + 1 = x + t 1 L c S S y z = 1 1 c t + c t The partial differential operators L + and L - are pseudodifferential operators. They cannot be applied directly to a function. Formally, the equation L f = 0 represents a wave traveling along x, while the equation + L f = 0 represents a wave traveling along +x. 9

10 1. ABCs cont. This becomes obvious in the case of a plane wave propagating along +/- x. 0, 0 0 z y S = = = L = x c t, L = x + c t f 1 f L f = = x c t + f 1 f L f = + = x c t 0 0 Solution: f( x+ ct) Solution: f( x ct) The radical appearing in L + and L - can be expanded using Taylor series 1/ 1 4 (1 S ) = 1 S + O( S ) 10

11 1. ABCs cont. If S is very small, then 1/ (1 S ) 1 The above is a first-order approximation of S. This means that the partial derivatives with respect to y and z are very small when compared with the partial derivative with respect to time scaled by the velocity of propagation c. S y z = c t c t This happens when the wave is incident upon the x=const. plane almost normally. The L - operator then becomes 1 1 = x t 1 L f x f c t f 0 L c = = 11

12 1. ABCs cont. When the wave, however, impinges upon the x=0 boundary wall at larger angles, the 1 st order approximation is very inaccurate. At grazing angles, S is large! For better accuracy, the second-order approximation can be used 1/ 1 y z (1 S ) 1 S S = 1 1 c t + c t 1 1 The L- operator now becomes L = x c t 1 S 1 1 y 1 z L = x c t c t c t t 1 c ( L = ) x + yy + zz c Multiplying by t t 1

13 1. ABCs cont. 1 c ( L f = xt f tt f + ) 0 yy f + zz f = at x = 0 c + 1 c ( L f = xt f + tt f ) 0 yy f + zz f = at x = xmax c 1 c ( L f = yt f tt f + ) 0 xx f + zz f = at y = 0 c + 1 c ( L f = yt f + tt f ) 0 xx f + zz f = at y = ymax c etc. 13

14 1. ABCs cont. Mur s ABC of nd order (G. Mur, Absorbing boundary conditions for the finitedifference approximation of the time-domain electromagnetic field equations, IEEE Trans. Electromagnetic Compatibility, vol. 3, 1981, pp Mur implemented the above approximate expressions into finite-difference equations. Mur expands the partial derivatives in the L + /L - operators using central finite differences of the field component about an auxiliary grid point displaced half a step along the direction of absorption and along time. Consider propagation along x, at the x=0 boundary. We assume that the scalar function U is evaluated at integer spatial grid positions (i,j,) and time positions n. n+ 1 n+ 1 n 1 n 1 n 1 f1, j, f0, j, f1, j, f0, j, xt f = 1/, j, t x x 14

15 1. ABCs cont. Now, the nd order time derivative has to be evaluated ½ step from the boundary as well. Mur averages the time derivatives at x=0 and x=1. n+ 1 n n 1 n+ 1 n n 1 1 f0, j, f0, j, + f0, j, f1, j, f1, j, + f1, j, tt f = + t t Now, the nd order y- and z- derivatives also have to be evaluated ½ step from the boundary. Mur averages those as well. n n n n n n 1 f0, j 1, f0, j, + f0, j+ 1, f1, j 1, f1, j, + f1, j+ 1, yy f = + y y n n n n n n 1 f0, j, 1 f0, j, + f0, j, + 1 f1, j, 1 f1, j, + f1, j, + 1 zz f = + z z 15

16 1. ABCs cont. Substitute all the FD approximations above in 1 c xt tt ( L f = f f + ) yy f + zz f = c 0 The result is n+ n n+ n n n f = f + f + f + f + f + 1 ( ) ( ) , j, 0, j, 1, j, 0, j, 0, j, 1, j, = ( n n n n n n ) + f f + f + f f + f + 3 y 0, j 1, 0, j, 0, j+ 1, 1, j 1, 1, j, 1, j+ 1, ( n n n n n n ) + f f + f + f f + f 3 z 0, j, 1 0, j, 0, j, + 1 1, j, 1 1, j, 1, j, + 1 c t x c t + x = x c t + x = 3 y = 3 z ( c t ) x y ( c t+ x) ( c t ) x z ( c t+ x) 16

17 1. ABCs cont. Mur s ABC of 1 st order To obtain Mur s approximation of L f = 1 x f c t f = 0 simply remove the nd order y- and z-derivatives from the formula above. 1 ( ) ( ) f = f + f + f + f + f n+ 1 n 0,, 0,, 1 n+ 1 1,, 1 n 0,, 1 n n j j j j 0, j, 1, j, = c t x c t + x = x c t + x In Yee s algorithm, the E-field components tangential to the boundary are evaluated at this boundary. For example, at an x=0 boundary wall, the E y and E z field components define the boundary values of the EM field problem. Mur s ABC is applied to them. 17

18 1. ABCs cont. B. Liao s extrapolation (Z.P. Liao, H.L. Wong, B.P. Yang, and Y.F. Yuan, A transmitting boundary for transient wave analyses, Scientia Sinica (series A), vol. XXVII, 1984, pp ) The ABC nown as Liao s ABC is easily explained as an extrapolation of the wave in space-time using Newton s bacward-difference polynomial. It is an order less reflective than Mur s nd order ABC and does not depend strongly on the angle of incidence. We now consider a boundary wall at x max. We assume that the field values are now for points located along a straight line perpendicular to the boundary. The objective is to find an approximation of the field at the boundary at the next time step f( xmax, t+ t). 18

19 1. ABCs cont. The field values used for the approximation are obtained by a simultaneous shift in space-time: ( δ ) ( δ ) ( δ ) m= 1 f1 = f xmax c t, t m= f = f xmax c t, t t m= 3 f3 = f xmax 3 c t, t t m= N f = f x N c t t n t N ( max δ, ( 1) ) Notice that such representation corresponds to a wave propagating in the +x direction: f( x ct) We aim at finding f0 = f( xmax, t+ t) 19

20 1. ABCs cont. We now define bacward finite-difference approximation of p th order at the point ξ1 = ( xmax αc t, t). 1 1 ( ξ 1) 1 = ξ 1 1 = ξ 1 1 = 1 D f f f f D f( ) f f f, D f( ) f f f, N N N N D f( ) f f f 1 1 ξ 1 1 = 1 1 f = f f f = f f3 1 f 3 = f 3 f 4 ( max ) f = f( ξ ) = f x mδc t, t ( m 1) t m m 0

21 1. ABCs cont. The N-th bacward difference can be written in terms of the function values as N + 1 N m 1 N 1 m 1 m = 1 = f ( 1) C f, where the Newton binomial coefficients are m! C N N N m 1 = m 1 = ( N m+ 1)!( m 1)! Alternatively, a function can be expressed (interpolated) in terms of the bacward finite differences at f 1 m β( β + 1) β( β + 1)( β + )! 3! β f f + f + f + f + β( β + 1) ( β + N ) N + ( N 1)! 1 f 1, 1 m N, β = 1 m 1

22 1. ABCs cont. We now use the above formula to extrapolate the function values, and we set m = 0. Then, β = N 1 0 = ( +, max) f f t t x f f f f f This is Liao s ABC. Liao et al. showed that for a sinusoidal plane wave of unit amplitude and wavelength λ, the maximum error is given by Assuming that N f max = N sin N ( πc t/ λ) h = c t and h = λ /3 the error is estimated at 0.1%. Liao s ABC is robust and depends little on the angle if incidence. With orders higher than N=3, however, it sometimes causes instabilities.

23 1. ABCs cont. 1 3 N 1 f = f( t+ t, x ) f + f + f + f + f Liao s ABC is simple to implement. 0 max ! ABC in MAIN ix=nt-((nt-1)/)* ix3=nt-((nt-1)/3)*3 ix4=nt-((nt-1)/4)*4 F0=>AX(:,:,n);F1=>AX(:,:,n-1)! front, ABC F=>A_F(:,:,ix);F3=>A3_F(:,:,ix3);F4=>A4_F(:,:,ix4) call LIAO(ni,nj)! before end of cycle! history A_B(:,:,ix)=AX(:,:,3) A3_B(:,:,ix3)=AX(:,:,4) A4_B(:,:,ix4)=AX(:,:,5) ******************************************************************************************** subroutine LIAO(dim1,dim) D1=F1-F;D=F-F3;D3=F3-F4 DD1=D1-D;DD=D-D3 DDD1=DD1-DD F0=F1+D1+DD1+DDD1 return end subroutine LIAO! define variables real(8),dimension(ni,nj,),target:: A_F,A_B real(8),dimension(ni,nj,3),target:: A3_F,A3_B real(8),dimension(ni,nj,4),target:: A4_F,A4_B real(8),dimension(:,:),pointer:: F0,F1,F,F3,F4 3

24 Important topics not discussed in this overview lecture Perfectly Matched Layer ABC FDTD on curvilinear grids FDTD in dispersive and anisotropic media FDTD in nonlinear and gain materials Integrating lumped elements with the FDTD full-wave analysis Near-to-Far-Field transformation for antenna radiation patterns Modified implicit FDTD schemes the FDTD-ADI 4