Mau-Sen Chiu2012/03/14

Beam Dynamics Group, NSRRC

Optimization of dynamic aperture by TESLA

for double mini-y lattice in TPS

Outline

• Introduction - Double mini-y lattice in TPS.

• Multi-objective genetic algorithm• Optimization of dynamic aperture (DA) by TESLA - Check 268 sets of solutions after completion: DA, tune shift with momentum and tune shift with amplitude. - Pick up the best one to calculate frequency map and Touschek lifetime.

• Conclusion

TPS Storage Ring Parameters

Purpose: Install two small gap IDs in tandem to obtain 4 times brightness.

Locations of the double mini-y lattice

Introduction

TPS Storage ring

Circumference 518.4 (m)

Nominal energy 3.0 GeV

Betatron tune 26.18/13.28

Natural chromaticity -75/-26

RF frequency 499.654

Harmonic number 864

Natural emittance 1.6 nm-rad

Energy spread 8.86E-04

Energy loss per turn 853 keV

7m X 1812m X 6

Double mini-y lattice

Q1 Q1

Q2 Q2

Q3 Q3Q4 Q4

Q54.9 m 4.9 m

(m)

1. Use 240 quadrupoles to match a lattice (x/y =26.18/12.82). 13.26-0.442. Add three sets of quadrupole triplet at the center of three long straights, respectivel

y.3. Apply trick to match the double mini-y lattice (x/y = 26.18/14.26).

　 L (m) K1

Q1 0.3 -1.217349 Q2 0.6 1.384064 Q3 0.3 -1.526270 Q4 0.5 -1.431159 Q5 0.6 1.681944

Purpose: Install two small gap IDs in tandem to obtain 4 times brightness.

y = 14.26y = 12.82

Sextupole integral strength: for each family.

Multiobjective Optimization

A general multiobjective optimization problem consists of a number of objectives and is associated with a number of inequality and equality constraints. Mathematically, the problem can be written as follows

Minimize/Maximize fk(x) k = 1, 2, …, K Subject to gj(x) j = 1, 2, …, J hm(x) m= 1, 2, …, K with i = 1, 2, …, N

Uii

Li xxx

The variable vector x represents a set of variables xi, i = 1, 2, …, N

00

Ex: Optimization of dynamic aperture (DA) for double mini-y lattice in TPS

636 Lb

Chromaticity are corrected at fixed values with SF and SD through out DA optimization.

Constraints:

f1(S1, S2, …, S6, SA, SB): DA area in (x-) plane.f2(S1, S2, …, S6, SA, SB): Tune shift with amplitude terms.

Objectives:

A genetic algorithm (GA) is routinely used to generate useful solutions to optimization and search problems using techniques inspired by natural evolution, such as inheritance, crossover,mutation, and selection.

Genetic Algorithm

Crossover during meiosis Mutation of gene

Type of mutation:point mutation, substitution,insertion, deletion,

This diagram labels a region of only 50 or so bases as a gene. In reality, most genes are hundreds of times larger.

What’s a gene?

A gene is a segment of DNA neededto contribute to a function.

Multi-Objective Genetic Algorithm (MOGA)

1: Initialize population (first generation, random)2: for ( int i = 2; i <= gen; i++) { - crossover: Within crossover probability, apply crossover to two parents to generate two children. These two parents are randomly chosen from the survivals of the last generation. Otherwise, copy parents to children.

- mutation : Within mutation probability, apply mutation to parents to generate children. Otherwise, do nothing.

- evaluate (children): calculate objective functions - merge ( parents, children): - non-dominated sort (rank): - select half of (parents, children) for next generation. }

TESLA•Author: Dr. LingYun Yang, NSLS-II, Brookhaven National Laboratory•Algorithm: NSGA-II (Non-dominated Sorting Genetic Algorithm II)•Parameters: -Number of individual: 2000. -Number of generation: 50 -Number of sextupole family: 8 -Lower and Upper bound of each sextupole family: 3 33 # L H SA -33 -3 # L H SB 0 30 # L H S1 -30 0 # L H S2 18 48 # L H S3 -48 -18 # L H S4 18 48 # L H S5 -48 -18 # L H S6 -Crossover probability: 0.8 -Mutation probability: 0.9 -Distribution index for crossover: 3 -Distribution index for mutation: 0.3

Reference:Tracking code development for beam dynamics optimization, L. Yang, BNL, PAC11A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II, K. Deb et al. IEEE, 2002

Flow of MOGA in TESLA

1. Master prepare the input for each individual in the child population then sent it to slaves2. Each slave : Execute DA tracking (128 turns) in (x-) plane, calculate the DA area and tune shift with amplitude terms. Chromaticity are corrected to a fixed value before DA tracking. After completion, send the results back to the master.3. Once master receive the results sent by some slave. This slave will receive another individual.4. This process will continue till the whole population are distributed to slaves completely.

GA loop

evaluate_pop (child)

GA loop

PC Cluster in NSRRC

Hardware for running TESLAnode28 ~ node37: CPU: Intel Xeon X5550, quad core 2.66 GHz * 2RAM: 16G ECC DDR2 RAMnode38 ~ node40: CPU: Intel Xeon X5550, quad core 2.66 GHz * 2RAM: 18G ECC DDR3 RAM

Total: 104 cores / 26 CPUs

2000 individuals and 50 generations takes about 10 days.

Hardware for running TRACY 2.6node19 ~ node23: CPU: Intel Xeon X5550, quad core 2.66 GHz * 2 RAM: 18G ECC DDR3

Total: 40 cores / 10 CPUs

Objective functions in TESLA

])1,(1/[11

umySf

222

2

y

y

y

x

x

x

JJJf

f1: The inverse of the sum of the on- and off-momentum DA area S in (x-) plane. (To transfer the maximum problem to minimum problem.)

f2: Square sum of tune shift with amplitude.

)3sin(

)3|3cos(|

)sin(

)|cos(|3)()(

16

1 ,,

1 1

2/32/333

x

xxkj

x

xxkjN

j

N

kxkxjkj

x

x LbLbJ

)2(sin

)]2(|2cos[|

)2(sin

)]2(|2cos[|

)sin(

)|cos(|4

)()(16

1

,,,,,

1 133

yx

yxykjxkj

yx

yxykjxkj

x

xxkj

N

j

N

kykyjxkxjkj

y

y LbLbJ

)2(sin

)]2(|2cos[|

)2(sin

)]2(|2cos[|

)sin(

)|cos(|2

)()(8

1

,,,,,

1 133

yx

yxykjxkjyk

yx

yxykjxkjyk

x

xxkjxk

N

j

N

kyjxkxjkj

x

y

y

y LbLbJJ

The Sextupole Scheme for the Swiss Light Source (SLS), 1997, Johan Bengtsson

Total number of points: 2000Each point represents a set of solution of sextupole strength.

Objective functions after generation 2na

ture

log

1. 286 points inside 3 regions are checked: DA ( =0, 3%, -3%), tune shift with momentum, tune shift with amplitude (horizontal and vertical direction). Total number of figures: 286 * 6 = 1716.2. Pick up the best solution by inspection for further analysis by frequency map and Touschek lifetime.

Objective functions after generation 50na

ture

log

DA, Tune shift with momentum and amplitude

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

━ ━ ━ ━

calculated at x = 0, y = 0

calculated at the long straight center

calculated at y = 0, = 0

calculated at x = 0, = 0Horizontal

Vertical

DA

Frequency map analysis (x – )

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ━ ━

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ━ ┿

DA & FMA (dp/p = 0)

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ━

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ┿

3x + y = 933y = 43

3x + y = 93

DA & FMA (dp/p = 3%)

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ━

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ┿

4x = 105

DA & FMA (dp/p = -3%)

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ━

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ┿

3y = 43 3x – 2y = 50

Ex: 1.58E-9Ey: 1.59E-11

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ━

T: 18.72/ Tp: 12.86/ Tn: 34.39 (hrs)

Bunch current: 400 mA / 800 bunchesBunch length: 2.86 mm

Momentum Acceptance & Touschek lifetime

Bruck’s formula:

Ex: 1.58E-9Ey: 1.60E-11

1% Emittance coupling

Multipole errors

Chamber limit

ID kick map

┿ ┿ ┿ ┿

T: 18.64/ Tp: 12.84/ Tn: 34.03 (hrs)

Bunch current: 400 mA / 800 bunchesBunch length: 2.86 mm

Momentum Acceptance & Touschek lifetime

Bruck’s formula:

Conclusions1. After DA optimization by TESLA, you still use tracking code to do DA tracking,

then pick up the best one. Finally, plot frequency map to decide whether the solution could be accepted or not.

2. We can see that the frequency map show resonance line when ID kick maps are included.

It should slightly move working point to avoid resonance line.

3. Maybe it should add the tune optimization in the near future to search ,

Appendix

1

12

11 ]

)(21[2

c

pp

Lp

xx

xx

Crossover: Simulated Binary Cross-Over (SBX)

))((5.0 121211 ppqppc xxxxx

)( 21U

ppL xxxx

1

12

22 ]

)(21[2

c

pp

pU

xx

xx

))((5.0 122212 ppqppc xxxxx

Comput. Methods Appl. Mech. Energ. 186 (2000) 311-338, K. Deb

, assume

)1/(1

1

)1/(11

1 )2

1(

)(

c

c

u

u

q

)1/(1

2

)1/(12

2 )2

1(

)(

c

c

u

u

q

if u <= 1/1 ,

otherwise,

if u <= 1/,

otherwise,

u : a random number between [0, 1].

Application multiobjective genetic algorithm in accelerator physics, ICAP09, L. Yang, et al.

distribution index for crossover): It control the shape of probability distribution function of crossover.

Within the crossover probability, apply crossover,Otherwise, copy the parents to children.

SBX is used to create child solutions xc1 , xc2 from parents xp1 , xp2.

Polynomial Mutation

)1/(1

)1/(1

)]1()5.0(2)1(2[1

)]1()21(2[

m

m

LU

pU

LU

Lp

q

xx

xxuu

xx

xxuu

Polynomial mutation is used to create a child solution xc in the vicinity of a parent solution xp.

)( LUqpc xxxx

if u <= 0.5,

otherwise,

Comput. Methods Appl. Mech. Energ. 186 (2000) 311-338, K. Deb

u = a random number between [0, 1]

Application multiobjective genetic algorithm in accelerator physics, ICAP09, L. Yang, et al.

mdistribution index for mutation): It control the shape

of probability distribution function of mutation. The smaller the m, the far away from the parents the child.

Within the mutation probability, apply mutation,Otherwise, do nothing.

Multipole errors

SR multipole tolerances

DM QM SM

Bn/BM @ 25mm (*E-4)

B1/B0 5 A2/B1 3 B0/B2 ±5B2/B0 -5±2 B0/B1 ±5 B1/B2 ±10B3/B0 ±2 B2/B1 ±2 B3/B2 ±2B4/B0 5±2 B3/B1 ±3 B4/B2 ±3B5/B0 ±1 B4/B1 ±1 B5/B2 ±0.5B6/B0 -2±0.2 B5/B1 0±1 B6/B2 ±0.5B8/B0 -0.6±0.6 B9/B1 0±1 B7/B2 ±0.1

Each rest term

±0.1 B13/B1 0±1 B8/B2 0±1

B17/B1 0±1 B14/B2 0±1 B21/B1 0±1 B20/B2 0±1 Each rest term ±0.1 B26/B2 0±1 Each rest term ±0.1Note: n=0 is dipole term, n=1 is quadrupole term and so on.

Bn is normal term, An is skew term.

Min(Chamber size, ID gap) (Aperture)IU22

mm5.6

IU22 IU22 IU22 EPU48

mm9.3

EPU46

mm5.3

Horizontal:

mm34

Vertical:

IU22

EPU48

EPU46

mm34

mm34

Beam Pipe mm15 mm34

S (m)

Y(m

m)

Injection point (80 cm, down stream of the long straightcenter)