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

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

Sep 10, 2022 129 likes •845 views

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

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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

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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

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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

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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

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

´4 ´2 2 4 ´0.4 ´0.2 0.2 0.4

r σ

Eprq ρprq9 expp´r2

2σ2 q

N “ 256 p “ 8

Analytic. ´4 ´2 2 4 ´1 ´0.5 0.5 1 ¨10´4

r σ

∆ Errors

COMBI Improved Field Solver and Space Charge

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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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

x “ r cospθq y “ r sinpθq r P r0, rmaxs BrBrφpr, θq ` 1

r Brφpr, θq ` 1 r2 BθBθφpr, θq “ ´ρ

Ó Fourier expansion BrBrφnprq ` 1

r Brφnprq ´ n2 r2 φnprq “ ´ρn

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

x “ cospθq y “ sinpθq f : r0, 1s Ý Ñ r0, 8r u ÞÝ Ñ f puq BrBrφpr, θq ` 1

r Brφpr, θq ` 1 r2 BθBθφpr, θq “ ´ρ

Ó Fourier expansion BrBrφnprq ` 1

r Brφnprq ´ n2 r2 φnprq “ ´ρn

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

x “ f puq cospθq y “ f puq sinpθq f : r0, 1s Ý Ñ r0, 8r u ÞÝ Ñ f puq

1 f 1puq2 BuBuφpu, θq ` f 1puq f puq Buφpu, θq

`

1 f 1puq2

´ ´f 2puq

f 1puq ` 1 f puq2

¯ BθBθφpu, θq “ ´ρpu, θq Ó Fourier expansion

1 f 1puq2 BuBuφnpuq ` f 1puq f puq Buφnpuq

´

n2 f 1puq2

´ ´f 2puq

f 1puq ` 1 f puq2

¯ φnpuq “ ´ρnpuq

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Finite difference scheme for u dependence. gp´r, θq “ gpr, θ ` πq

8

ÿ

´8

gnp´rqeinθ “

8

ÿ

´8

p´1qngnprqeinθ ù ñ gnp´rq “ p´1qngnprq

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Finite difference scheme for u dependence. gp´r, θq “ gpr, θ ` πq

8

ÿ

´8

gnp´rqeinθ “

8

ÿ

´8

p´1qngnprqeinθ ù ñ φnpu0q “ p´1qnφnpu1q

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

10 20 30 40 50 60 70 80 0.05 0.1

x σx

Epx, 0q ρpx, yq9 expp´ x2

2σ2

x ´ y2

2σ2

y q with σx

σy “ 5

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

1 2 3 4 5 0.05 0.1

x σx

Epx, 0q ρpx, yq9 expp´ x2

2σ2

x ´ y2

2σ2

y q with σx

σy “ 5

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

1 2 3 4 5 0.05 0.1

x σx

Epx, 0q ρpx, yq9 expp´ x2

2σ2

x ´ y2

2σ2

y q with σx

σy “ 5

COMBI Improved Field Solver and Space Charge

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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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

4 5 6 7 8 9 ´5 ´4 ´3 ´2 log2pNq log10pmaxp∆qq

  • Conv. rates: FFT=1.95, F2M2=1.91, FPPS=1.96

FFT F2M2 FPPS

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

200 400 600 800 1,000 10´4 10´2 100 102 104 N t rss Timing

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

200 400 600 800 1,000 10´4 10´2 100 102 104 N t rss Timing

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

200 400 600 800 1,000 10´4 10´2 100 102 104 N t rss Timing

COMBI Improved Field Solver and Space Charge

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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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Yokoya factor λ: ∆Qπ´σ “ λξ Round beam, exact: λ « 1.21 Round beam, soft Gauss: λ « 1.10

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

´1.4 ´1.2 ´1 ´0.8 ´0.6 ´0.4 ´0.2 0.2 0.5 1 λ A Centroid’s spectra

FPPS HFMM Gauss

  • Th. Yok. factor

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

2 4 ¨106 0.5 1 ¨10´2 # macropart. Relative precision

HFMM FPPS Soft-Gauss

1 ? N

2 4 ¨106 0.5 1 # macropart. Execution time rss

COMBI Improved Field Solver and Space Charge

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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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Amdhal Law: SpN, αq “

1 α` 1´α

N

S: Speed-up α: % of unparallelised code N: Number of threads

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 Number of threads Speed-up Parallelised code performance Amdhal law, α “ 0.03 Time

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Outlooks

§ Optimising the change of coordinates?

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Outlooks

f puq “ au 1 ´ u

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Outlooks

§ Optimising the change of coordinates? § Could be extended to 3D.

COMBI Improved Field Solver and Space Charge

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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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Some beam dynamics

x” “ ´ 1 m0γv2 pCx ¨ xq y” “ ´ 1 m0γv2 pCy ¨ yq

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Some beam dynamics

x” “ ´ 1 m0γv2 pCx ¨ x ` Fxscq y” “ ´ 1 m0γv2 pCy ¨ y ` Fyscq

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Model

´4 ´2 2 4 ´4 ´2 2 4 x y Gaussian distribution ηprq “ NQ 2πσ2∆L exp ˆ ´ r2 2σ2 ˙ Frprq “ NQ2 2πǫ0γ2∆L 1 r p1 ´ exp ´r2 2σ2 q

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

SPS

0.05 0.1 0.15 0.2 0.1 0.15 0.2 0.25 0.3 kx rs´1s ky rs´1s SPS

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

LHC

Source: E. M´ etral and F. Ruggiero, Stability diagrams for Landau damping with two- dimensional betatron tune spread from both

  • ctupoles and non-linear space charge

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

LHC

0.3090 0.3095 0.3100 0.3185 0.3190 0.3195 0.3200 kx rs´1s ky rs´1s LHC

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

LHC and octupoles, preliminary work

0.3060 0.3080 0.3100 0.3120 0.3140 0.3160 0.3180 0.3200 0.3220 0.3240 kx rs´1s ky rs´1s LHC

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

LHC and octupoles, preliminary work

0.3060 0.3080 0.3100 0.3120 0.3140 0.3160 0.3180 0.3200 0.3220 0.3240 kx rs´1s ky rs´1s LHC

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

PIC code, preliminary work

0.1 0.2 0.3 0.05 0.1 0.15 0.2 kx rs´1s ky rs´1s SPS

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Open-problems

§ Need to properly modelise the arcs in COMBI § Need to refine octupoles/space-charge case

COMBI Improved Field Solver and Space Charge

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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

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

§ New Poisson solver in COMBI, more accurate and parallelised. § Space-charge module

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

For more informations

§ A. Florio, X. Buffat, and T. Pieloni, “Fast Poisson Solvers for

Self-Consistent Beam-Beam and Space-Charge Field Computation in Multiparticle Tracking Simulations,” Nov

  • 2015. CERN-ACC-NOTE-2015-0038, (CERN, Geneva,

Switzerland).

§ adrien.florio@epfl.ch for course reports

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Acknowledgements

§ Hannes Bartosik, for space-charge checks

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Acknowledgements

§ Hannes Bartosik, for space-charge checks § Claudia Tambasco and Javier Barranco

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Acknowledgements

§ Hannes Bartosik, for space-charge checks § Claudia Tambasco and Javier Barranco § Tatiana Pieloni and Xavier Buffat

COMBI Improved Field Solver and Space Charge

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F2M2 FPPS F2M2, FPPS & FFT COMBI, FPPS & HFMM OpenMP FPPS Space Charge Conclusion

Thank you for your attention!

COMBI Improved Field Solver and Space Charge