New flexible nonlinear correction with feeddown of RDTs for IR - - PowerPoint PPT Presentation

New flexible nonlinear correction with feeddown of RDTs for IR correctors

OMC Meeting

Joschua Dilly

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Theory Feeddown RDTs Nonlinear Correction Implementation Tests & Outlook

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Theory

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Feeddown

If the beam does not go through magnet center, the field looks different: Taylor expansionf or x, y → x, y + ∆x, y H = −ℜ ∞

• n=2

(Kn + iJn)(x + iy)n n!

• T

= −ℜ    

∞

• n=1

   

∞

• q=0, if n≥2

q=1, if n=1

(Kn+q + iJn+q)(∆x + i∆y)q q!     (x + iy)n n!     which means: Kn + iJn

w/ feeddown

→

Q

• q=0

(Kn+q + iJn+q)(∆x + i∆y)q q! .

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Nonlinear Correction

LHC

10 10 X [mm] B1X B2X 200 150 100 50 50 100 150 200 Distance from IP5 [m] 10 10 Y [mm] B1Y B2Y

IP Quadrupole Dipole

HL-LHC

10 10 X [mm] B1X B2X 200 150 100 50 50 100 150 200 Distance from IP5 [m] 10 10 Y [mm] B1Y B2Y

IP Quadrupole Dipole

• in the triplets orbit can be 5 − 10 mm due to the crossing scheme

(In misalignment study it was only ±0.4 mm). ⇒ higher order terms should be corrected in the IRs! ⇒ I developed and implemented a correction scheme, based on the previous correction scheme of specfic sextupole and octupole RDTs1, for any RDT if correctors of that order are present.

• including feed-down contributions as derived earlier.
• only for simulations as all magnetic fields need to be known

1 Br¨ uning et al., Dynamic aperture studies for the LHC separation dipoles., LHC Project Note 349, (2004)

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Nonlinear Correction Outline

Idea: Minimize RDT locally in IR with the corrector.

• a simulation based local correction
• minimize two RDTs, as ther are two correctors (L/R)
• shown only numerator of RDT

⇒ extract terms of the ”optimization variable” corrector-strength

f IR

jklm

=

• IR

ℜ  

∞

• q=0

(Kn+q(s) + iJn+q(s)) (∆x(s) + i∆y(s))q q!   il+mβx (s)

j+k 2

βy (s)

l+m 2

eiπnθ(s−sIP )

• ds

thin lens

=

• w∈W

ℜ  

∞

• q=0

(Kn+qL(sw ) + iJn+qL(sw )) (∆x(sw ) + i∆y(sw ))q q!   il+mβx (sw )

j+k 2

βy (sw )

l+m 2

eiπnθ(sw −sIP )

• correction

=

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Nonlinear Correction Outline

⇒ simple linear equation system:

• B(cl)

jklm

B(cr)

jklm

KnL (scl) KnL (scr)

• = −Ijklm

, for even l + m

• B(cl)

jklm

B(cr)

jklm

JnL (scl) JnL (scr)

• = i Ijklm

, for odd l + m For multiple rdt/beams:       B(cl,B1)

jklm

B(cr,B1)

jklm

B(cl,B1)

j′k′l′m′

B(cr,B1)

j′k′l′m′

B(cl,B2)

jklm

B(R,B2)

jklm

B(cl,B2)

j′k′l′m′

B(cr,B2)

j′k′l′m′

     

• KnL (scl)

KnL (scr)

• = −

      I (B1)

jklm

I (B1)

j′k′l′m′

I (B2)

jklm

I (B2)

j′k′l′m′

      Status: Correction script ready and working. Simulation tests missing.

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Implementation

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Input Parameters

• accel: defines accelerator

⇒ same script for all, easily extendable (only corrector names hardcoded)

• optics, errors: optics and error files

⇒ no predifined names, can take two files each for correcting both beams

• rdts, rdts2: RDTs in the format ’F####’, ’J####’ or

’K####’ ⇒ can correct for any RDTs, recognizes order and orientaion automatically, can have different RDTs for second optics

• solver: which solver to use ’lstsq’, ’inv’ or ’linear’

⇒ can optimize overdetermined equation systems (e.g. both beams, more RDTs)

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Input Parameters

• feeddown: feeddown up to which order to include
• update optics: update optics with new corrector strength

⇒ takes new strength into account for feeddown (always corrects high orders first)

• ips: define IPs to correct
• ignore missing columns: Missing KL-columns assumed 0

Other advantages:

• written in python, can handle pandas-dataframes, returns

madx-code and dataframes

• can be called from python and commandline

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Example (python)

correction_cmd , correction_df = irnl_correct ( accel=’LHC ’,

• ptics=( df_twiss_nominal_for_correction ,),

errors=( df_errors_for_correction ,), rdts=(’J3000 ’, ’J0003 ’, # correct a3 ’K2001 ’, ’K1002 ’, # correct b3 ’J1003 ’, ’J3001 ’, # correct a4 ’K4000 ’, ’K0004 ’, # correct b4 ’K6000 ’, ’K0006 ’, # correct b6 ), feeddown=2, ips=[1, 2, 5, 8], solver=’lstsq ’, update_optics =False , ignore_missing_columns =False ,

• utput=’correction_mcx .madx ’,

) # Apply correction madx.input( correction_cmd )

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Example (python)

• rrection_cmd , correction_df = irnl_correct (

accel=’HLLHC ’,

• ptics=( df_twiss_nominal_correct ,),

errors=(df_errors_correct ,), rdts=(’J0003 ’, ’J3000 ’, # correct a3 ’K1002 ’, ’K2001 ’, # correct b3 ’J1003 ’, ’J3001 ’, # correct a4 ’K0004 ’, ’K4000 ’, # correct b4 ’J0005 ’, ’J5000 ’, # correct a5 ’K5000 ’, ’K0005 ’, # correct b5 ’J5001 ’, ’J1005 ’, # correct a6 ’K6000 ’, ’K0006 ’, # correct b6 ), feeddown=2, ips=[1, 2, 5, 8], # in which IPs to correct solver=’lstsq ’, update_optics =False ,

• utput=’correction_mcx .madx ’,

)

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Example (Mad-X)

system, ’python3 irnl_rdt_correction.py

• -accel HLLHC
• -optics temp/optics0_inser.mad
• -errors temp/tripD1D2.errors
• -output MCX_setting.mad
• -ignore_missing_columns
• -rdts J0003 J3000 K1002 K2001 J1003 J3001 K0004 K4000

J0005 J5000 K5000 K0005 J5001 J1005 K6000 K0006’

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Tests & Outlook

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Tests & Outlook

• Tested vs. fortran and Ewens python code (with feeddown = 0)

⇒ up to tfs-precision the same results

• rewritten masks in cpymad ⇒ No need to write any files for running

corrections → Test inlfuence of feeddown correction → Influence on amplitude detuning (PTC) and DA (Sixtrack) ⇒ Have some talks about SixTrack(Lib)/(py)SixDesk

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