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COMBI Improved Field Solver and Space Charge Adrien Florio 22nd - PowerPoint PPT Presentation

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge Adrien Florio 22nd February 2016 COMBI Improved Field Solver and Space Charge F2M2 FPPS F2M2, FPPS

  1. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge Adrien Florio 22nd February 2016 COMBI Improved Field Solver and Space Charge

  2. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion New numerical solver § Aim: Lessen numerical noise for head-on simulations COMBI Improved Field Solver and Space Charge

  3. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Fixed Fast Multipole Method Fast Polar Poisson Solver F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  4. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Fixed Fast Multipole Method Fast Polar Poisson Solver F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  5. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion § Far-field and nearest-neighbours field decomposition § Efficient far-field computation based on multipole expansion COMBI Improved Field Solver and Space Charge

  6. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  7. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  8. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  9. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  10. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  11. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  12. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  13. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  14. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  15. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  16. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  17. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  18. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  19. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  20. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  21. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  22. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  23. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  24. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  25. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  26. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  27. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  28. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  29. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion ρ p r q9 exp p ´ r 2 2 σ 2 q 0 . 4 0 . 2 E p r q 0 N “ 256 p “ 8 ´ 0 . 2 Analytic. ´ 0 . 4 ´ 4 ´ 2 0 2 4 r σ Errors 1 ¨ 10 ´ 4 0 . 5 ∆ 0 ´ 0 . 5 ´ 1 ´ 4 ´ 2 0 2 4 r σ COMBI Improved Field Solver and Space Charge

  30. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Fixed Fast Multipole Method Fast Polar Poisson Solver F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  31. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion x “ r cos p θ q r P r 0 , r max s y “ r sin p θ q B r B r φ p r , θ q ` 1 r B r φ p r , θ q ` 1 r 2 B θ B θ φ p r , θ q “ ´ ρ Ó Fourier expansion r B r φ n p r q ´ n 2 B r B r φ n p r q ` 1 r 2 φ n p r q “ ´ ρ n COMBI Improved Field Solver and Space Charge

  32. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion x “ cos p θ q f : r 0 , 1 s Ý Ñ r 0 , 8r u ÞÝ Ñ f p u q y “ sin p θ q B r B r φ p r , θ q ` 1 r B r φ p r , θ q ` 1 r 2 B θ B θ φ p r , θ q “ ´ ρ Ó Fourier expansion r B r φ n p r q ´ n 2 B r B r φ n p r q ` 1 r 2 φ n p r q “ ´ ρ n COMBI Improved Field Solver and Space Charge

  33. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion x “ f p u q cos p θ q f : r 0 , 1 s Ý Ñ r 0 , 8r u ÞÝ Ñ f p u q y “ f p u q sin p θ q f 1 p u q 2 B u B u φ p u , θ q ` f 1 p u q 1 f p u q B u φ p u , θ q ´ ¯ ´ f 2 p u q 1 1 ` f 1 p u q ` B θ B θ φ p u , θ q “ ´ ρ p u , θ q f 1 p u q 2 f p u q 2 Ó Fourier expansion f 1 p u q 2 B u B u φ n p u q ` f 1 p u q 1 f p u q B u φ n p u q ´ ¯ n 2 ´ f 2 p u q 1 ´ f 1 p u q ` φ n p u q “ ´ ρ n p u q f 1 p u q 2 f p u q 2 COMBI Improved Field Solver and Space Charge

  34. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Finite difference scheme for u dependence. g p´ r , θ q “ g p r , θ ` π q 8 8 g n p´ r q e in θ “ ÿ ÿ p´ 1 q n g n p r q e in θ ´8 ´8 ñ g n p´ r q “ p´ 1 q n g n p r q ù COMBI Improved Field Solver and Space Charge

  35. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Finite difference scheme for u dependence. g p´ r , θ q “ g p r , θ ` π q 8 8 g n p´ r q e in θ “ ÿ ÿ p´ 1 q n g n p r q e in θ ´8 ´8 ñ φ n p u 0 q “ p´ 1 q n φ n p u 1 q ù COMBI Improved Field Solver and Space Charge

  36. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion ρ p x , y q9 exp p´ x 2 x ´ y 2 y q with σ x σ y “ 5 2 σ 2 2 σ 2 0 . 1 E p x , 0 q 0 . 05 0 0 10 20 30 40 50 60 70 80 x σ x COMBI Improved Field Solver and Space Charge

  37. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion ρ p x , y q9 exp p´ x 2 x ´ y 2 y q with σ x σ y “ 5 2 σ 2 2 σ 2 0 . 1 E p x , 0 q 0 . 05 0 0 1 2 3 4 5 x σ x COMBI Improved Field Solver and Space Charge

  38. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion ρ p x , y q9 exp p´ x 2 x ´ y 2 y q with σ x σ y “ 5 2 σ 2 2 σ 2 0 . 1 E p x , 0 q 0 . 05 0 0 1 2 3 4 5 x σ x COMBI Improved Field Solver and Space Charge

  39. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Fixed Fast Multipole Method Fast Polar Poisson Solver F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion COMBI Improved Field Solver and Space Charge

  40. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Conv. rates: FFT=1.95, F2M2=1.91, FPPS=1.96 FFT ´ 2 F2M2 log 10 p max p ∆ qq FPPS ´ 3 ´ 4 ´ 5 4 5 6 7 8 9 log 2 p N q COMBI Improved Field Solver and Space Charge

  41. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Timing 10 4 10 2 t r s s 10 0 10 ´ 2 10 ´ 4 0 200 400 600 800 1 , 000 N COMBI Improved Field Solver and Space Charge

  42. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Timing 10 4 10 2 t r s s 10 0 10 ´ 2 10 ´ 4 0 200 400 600 800 1 , 000 N COMBI Improved Field Solver and Space Charge

  43. F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion Timing 10 4 10 2 t r s s 10 0 10 ´ 2 10 ´ 4 0 200 400 600 800 1 , 000 N COMBI Improved Field Solver and Space Charge

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