1. 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. 3/67 Part II : 2D-NUFFT

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

5. 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. 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. 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. 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. 9/67 Our aim 2D-NUFFT 2D-FFT II. 2D-NUFFT Algorithm The key step (3.1) (3.2) (3.3)

10. 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. 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. 12/67 Let  = e i2  /cM and  = ei2  /cN, (3.7) (3.8)

13. 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. 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. 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. 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. 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. 18/67 Spectral domain Green’s functions

19. 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. 20/67 Asymmetric rooftop function J x  = J xx (x, x  )J xy (y, y  ) (3.12) (3.14b) (3.14a)

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

22. 22/67 Procedure for evaluating the MoM matrix

23. 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. 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. 25/67 The measured and calculated S parameters of the hairpin resonator.

26. 26/67 Normalized magnitudes of the current distribution on the hairpin resonator at 2.473 GHz. (a) |J x (x,y)| (b) |J y (x,y)|

27. 27/67 Normalized magnitudes of the current distribution on the hairpin resonator at 2.397 GHz. (c) |J x (x,y)| (d) |J y (x,y)|

28. 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 1.27. All dimensions are in mm. Table 3.2.2

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

30. 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. 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 = 0.3605, 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 = 10.8. All dimensions are in mm. Table 3.2.3

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

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

34. 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. 35/67 THE END Thank You for your Participation !