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

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 & 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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