A Two-Scale Adaptive Integral Method

Transcription

1 A Two-Scale Adaptive Integral Method Ali Yilmaz Department of Electrical & Computer Engineering University of Texas at Austin IEEE APS International Symposium USC/URSI ational Radio Science Meeting San Diego, CA, July 5-1, 008

2 Outline Motivation - Multi-scale Problems Background - Single-scale AIM (Pre-corrected FFT) Formulation - Two-scale AIM (Pre- & Post-corrected FFT) umerical Results - Validation & Verification Conclusions

3 MOTIVATIO - Multi-scale problems (demand side) - Multi-scale solvers (supply side)

4 Antennas on Platforms 1.5 m

5 Electromagnetic Compatibility & Interference 3.4 m m

6 Interconnects and Packaging z x 0.4 mm 0.55 mm 055mm mm T G T G T z y x

7 Integral Equations and Multi-Scale Analysis Challenges Orders-of-magnitude variation in geometrical features - Meshing Uniform meshes result in too many unknowns/loss of features - Kernel stability/preconditioning Classical IE kernels break down for sub-wavelength problems (lowfrequency breakdown) - Fast algorithms Most accelerators are effective for single-scale problems Related work Loop-tree/star/charge: Wilton and Glisson 1981, Mautz and Harrington 1984, Zhao and Chew 000. Calderon techniques: Contopaganagos et al. 00, Adams and Champagne 004, Andriulli and Michielssen 006. Multiscale algorithms: Brandt 1991, Cheng, Greengard and Rokhlin 1999, Boag et al. 00, Jiang and Chew 005, Shanker et al. 007.

8 BACKGROUD - EFIE and MOM - Single-scale AIM (Pre-corrected FFT) - Moment matching

9 EFIE and MOM EFIE for a PEC in free space inc sca E E jωa tans tans E () r = E () r = A () r + φ () r tans () μ ( ) Ar 0 G(, ) = ds φ Jr rr jωε () r J( r )/( ) S 0 e G(, rr ) = 4π jk 0 r r r r MOM J() r I S () r k k S k = 1 ds S () r i EFIE k Z I V = inc Computational complexity PEC S CPU cost per iteration: O ( ) Memory requirement: O ( )

10 Single-scale AIM and Pre-correction AIM steps Embed S in a uniform Cartesian grid with nodes C Project to, propagate on (via FFT), interpolate from the auxiliary grid Pre-correct near interactions FFT near ZI ( Z + Z ) I Block-Toeplitz O ( ) sparse Z Z =Λ G Λ FFT t c c c (, kk ) = 0 else FFT Z(, kk) Z (, kk) if S &S are near near k k Δ c Δ s Computational complexity CPU cost per iteration: O ( log ) Memory requirement: C C + O + near C near ( )

11 AIM Accuracy Moment matching m my m x z dxdydz x y z S r u k δ r r k k, u u= 1 M k [ ( ) Λ(, ) ( )] = 0 Essentially scale-independent accuracy at sub-wavelength scales Cell seperation d = 1 M = 3 Δc M = M = 6 jk R AIM e AIM d kk, k, k, d k, k, d k, k, d k, k k, k max / ; 0,, 0 e err = o o o o = o = Λ u k Λ u, k R R k uu ( ) ( ) k jk R u C u C uu

12 FORMULATIO - Two-scale AIM - Two different-scale grids - Pre- & Post-correction

13 Sample Two-Scale Problem: Classical AIM S S S L + L L S J () r I S () r + I S () r k k k k k = 1 k = + 1 inc VL Z Z I LL LS L inc V Z Z I S SL SS S ZI = = Computational complexity L CPU cost per iteration: O([ + ] ) Memory requirement: L S O ([ + ] ) L S

14 Sample Two-Scale Problem: Classical AIM S S S Δc Δs 1 L L + L L S J () r I S () r + I S () r k k k k k = 1 k = + 1 L ~ L inc near,1 V L Z Z I LL LS L FFT Z 0 LL IL inc 1 near,1 V Z Z I S SL SS S 0 Z I SS S ZI = = + Z I Δc Δs 1 L Computational complexity ~ S + O + C1 C1 near1 C 1 near1 CPU cost per iteration: O ( log ) Memory requirement: ( )

15 Sample Two-Scale Problem: Classical AIM S S S L Δc Δs S + L L S Jr () I S () r + I S () r k k k k k = 1 k = + 1 L inc near, V L Z Z I LL LS L FFT Z 0 LL IL inc near, V Z Z I S SL SS S 0 Z I SS S ZI = = + Z I Computational complexity, C S L Δc Δs S CPU cost per iteration: O ( log ) Memory requirement: O ( + ) + C C near C near

16 Two-scale AIM AIM steps If no near inter-scale interactions ti Z 0 I near,1 FFT LL L I S ZI Z I Z FFT 1-partial I S Post-correction 0 0 I FFT L + Z I + -partial S near, 0 Z I SS S Computational complexity CPU cost per iteration Memory requirement log + C1 C1 near1 + C 1 near1 O log + C1-partial C1-partial O + C 1-partial + log + C-partial C-partial near + + C -partial near Δc Δc Δs 1 L Δs S

17 Two-scale AIM AIM steps If near inter-scale interactions ti Option 1: Pre-correct the near interscale interactions near,1 near,1 FFT Z Z I LL LS L ZI Z I + 1 near,1 Z 0 SL I S FFT Z I 1-partial S 0 0 FFT I L + Z I + -partial S near, 0 Z SS I S Simple. Additional computational complexity: O ( ) L1 : L1 The number of unknowns at the large-scale structure that are near the small-scale structure. S Option : Post-correct Δc Δs 1 L near,1 FFT Z 0 I LL L ZI Z I I S I FFT L Z 1-partial I S near, I FFT L 0 ZLS IL + Z + -partial near, near, I S Z Z SL SS I S

18 UMERICAL RESULTS - Validation & Verification

19 Two-scale AIM Validation: RFID Tag Antenna Array Tag Antenna mm (~ λ / 3) Meshes 15.4 mm (~ λ / 0) Array Configuration λ y λ / x K. V. Seshagiri Rao, P. V. ikitin, and S. F. Lam, Antenna design for UHF RFID tags: A review and a practical application, IEEE Trans. Antennas Propagat., vol. 53, no. 1, Dec. 005, pp

20 RFID Tag Antenna Array Array Size Increase Mesh type Matrix-Fill Time Time per Iteration Memory Requirement coarsest log finest log

21 RFID Tag Antenna Array Mesh Refinement Array Size Matrix-Fill Time Time per Iteration Memory Requirement 1 1 log 1 log

22 RFID Tag Antenna Array Mesh Refinement Array Size Matrix-Fill Time Time per Iteration Memory Requirement log 4 4 log

23 Conclusions Two-scale AIM Multiple AIM grids at different scales (spacing, location) Post-correction as well as pre-correction stage Effectively reduces computational complexity for two-scale problems Multi-scale extensions under development Acknowledgements Fangzhou Wei for the RFID tag structure and meshes Texas Advanced Computing Center (TACC) for providing HPC resources SF for supporting this research through the grant CCF-07888