Numerical ElectroMagnetics & Semiconductor Industrial Applications Ke-Ying Su Ph.D. National Central University Department of Mathematics 12 2D-NUFFT & Applications

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3/67 Part II : 2D-NUFFT

4/67 Outline : 2D-NUFFT 1.Introduction 2.2D-NUFFT algorithm 3.Approach 4.Results and discussions 5.Conclusion

5/67 I. Introduction Finite difference time domain approach (FDTD) Spectral domain approach (SDA) Finite element method (FEM) Integral equation (IE) Mode matching technique Numerical methods Widely used technique Method of Moment (MoM)

6/67 Disadvantages 1) Slow convergence of Green’s functions 2) A large number of basis functions Used solutions Using the 2D discrete fast Fourier Transform (FFT) 1)  2D-FFT + Using the first few resonant modes’ current distributions 1,2)  Nonuniform meshs for mixed potential integral equation (MPIE)

7/67 NUFFT : 1D  2D The q+1 nonzero coefficients. The core idea of the 1D-NUFFT: SDA + 2D-NUFFT + Nonuniform meshs New solution

8/67 The (q+1) 2 nonzero coefficients. The square 2D-NUFFT Some of these 2D coefficients approach to zero rapidly. NUFFT : 1D  2D coefficients remove directly least square error accuracy  accuracy  The nonsquare 2D-NUFFT

9/67 Our aim 2D-NUFFT 2D-FFT II. 2D-NUFFT Algorithm The key step (3.1) (3.2) (3.3)

10/67 Regular Fourier Matrix For a given ( x t, y s ), for m = -M/2,…, M/2-1 and n = -N/2,…, N/2-1 Ar(x t,y s ) = b(x t,y s ) r(x t,y s ) = [A * A] -1 [A * b(x t,y s )] = F r -1 P r where F r is the regular Fourier matrix with size (q 2 /2+3q+1) 2 F r & P r : closed forms   A: (MN)  (q 2 /2+3q+1) b : (MN)  1 (3.5)

11/67 Solution Extract F r and P r from F f and P f where F f is the regular Fourier matrix with size (q+1) 2 [a 1, a 2, …, a m ]  [b 1, b 2, …, b n ] = [a 1 b 1, a 2 b 1, …, a m b 1, …, a 1 b n, a 2 b n, …, a m b n ] 1) Define a vector product  as Let V p and V g be the (p+1) th and (g+1) th row of the regular Foruier matrix for 1D problem p and g = 0, 1, …, q. The [g(q+1)+(p+1)] th row of F f equals V p  V g.  (3.6)

12/67 Let  = e i2  /cM and  = ei2  /cN, (3.7) (3.8)

13/67 2) Choose  mn = cos(m  /cM)cos(n  /cN). The [g(q+1) +(p+1)] th element of P f where {x} = x - [x].  (3.9)

14/67  = [1, 2, …, (q+1) 2 ] = [q/2, q/2+1, q/2+2, 3q/2, …, q 2 +3q/2, q 2 +3q/2+1, q 2 +3q/2+2] F r (i, j) = F f ( (i), (j)) P r (i) = P f ( (i)) For square grid points: For octagonal grid points: 3) Fill F r and P r from F f and P f

15/67 The relation between F r and F f, and P r and P f Example: Let q = 8 Index = [4, 5, 6, 12, 13, 14, 15, 16, …, 76, 77, 78] Index  = [1, 2, 3, …, 79, 80, 81]

16/67 1~3) F f, P f  F r, P r 4) 2D-FFT: 5) r r = F r -1 P r 2D-NUFFT  If M = N = 2 10 and c = 2, then a 2D-FFT with size cM  cN uses 3.02 seconds (CPU:1.6GHz). (3.5) (3.4) (3.3)

17/67 III. Approach substrate thickness t box dimension a  b  c The Green’s functions k xm = m  /a, k yn = n  /b is the spectral domain Green’s function where (3.10)

18/67 Spectral domain Green’s functions

19/67 Solution procedure Asymmetric rooftop functions and the nonuniform meshs source J(x, y) =  a x  J x  (x, y) +  b y  J y  (x, y) load terminal (3.11)

20/67 Asymmetric rooftop function J x  = J xx (x, x  )J xy (y, y  ) (3.12) (3.14b) (3.14a)

21/67 Galerkin’s procedure Final MoM matrix Trigonometric identities (3.15) (3.16)

22/67 Procedure for evaluating the MoM matrix

23/67 III. Numerical Results Hairpin resonator  r = 10.2, L 1 = 0.7, L 2 = 1.01, L 3 = 2.74, L 4 = 8, L 5 = 6, w 1 = 1, w 2 = 1.19, g 1 = 0.2 and g 2 = 0.8. All dimensions are in mm.

24/67 Comparison of CPU Time and L 2 error of One Call of the 2D-NUFFT in Analysis of a Hairpin Resonator Comparison of Analyses of The Hairpin Resonator with Uniform and Nonuniform Grids Table 3.1 Table 3.2.1

25/67 The measured and calculated S parameters of the hairpin resonator.

26/67 Normalized magnitudes of the current distribution on the hairpin resonator at GHz. (a) |J x (x,y)| (b) |J y (x,y)|

27/67 Normalized magnitudes of the current distribution on the hairpin resonator at GHz. (c) |J x (x,y)| (d) |J y (x,y)|

28/67 Interdigital capacitor Comparison of Analyses of The Ingerdigital capacitor with Uniform and Nonuniform Grids  r = 10.2, L 1 = 8, L 2 = 1.6, L 3 = 0.8, L 4 = 1.2, L 5 = 7.9, d = 0.4, e = 0.4, g = 0.2 and s = 0.2. The thickness of substrate is All dimensions are in mm. Table 3.2.2

29/67 The measured and calculated S parameters of the interdigital capacitor.

30/67 Normalized magnitudes of the current distribution on the interdigital capacitor at 5 GHz. (a) |J x (x,y)| (b) |J y (x,y)|

31/67 Wideband filter Comparison of Analyses of The Wideband filter with Uniform and Nonuniform Grids L 1 = 8, L 2 = 0.56, L 3 = 0.576, L 4 = 0.69, L 5 = , L 6 = 0.125, L 7 = 0.125, L 8 = 0.125, L 9 = 5.19, L 10 = 4.88, L 11 = 0.38, L 12 = 2.06, L 13 = 1.9, L 14 = 7.75, L 15 = 11.3, t = 0.635,  r = All dimensions are in mm. Table 3.2.3

32/67 The measured and calculated S parameters of the wideband filter.

33/67 Normalized magnitudes of the current distribution at 6 GHz (a) |J x (x,y)| (b) |J y (x,y)|

34/67 VI. Conclusion A 2D-NUFFT algorithm with octagonal interpolated coefficients are used to enhance the Computation. The octagonal 2D-NUFFT uses less CPU time than the square 2D-NUFFT. The L 2 error of the octagonal 2D-NUFFT is the same as that of square 2D-NUFFT. The scattering parameters of the hairpin resonator, an interdigital capacitor and a wideband filter are calculated and validated by measurements.

35/67 THE END Thank You for your Participation !