Beam Dynamics: From Theory to Simulation The …Beam Dynamics: From Theory to Simulation The MSU...

Beam Dynamics:From Theory to Simulation

The MSU Program

Martin BerzKyoko Makino

Michigan State University

Martin Berz, MSUHEPAP AARD SubpanetFNAL, February 15-17, 2006

Current Ph.D. Students

Supported by DOE, and Other Sources

• Shashikant Manikonda DOE and GSI• Youn-Kyung Kim DOE• Johannes Grote DOE and StuSti• Pavel Snopok FNAL/ILC• Alexey Poklonskiy FNAL/Muon• Gabi Weizman GSI• Stephen Weathersby SLAC

Support leveraging: 2-3 DOE students --> 7 total


Former Ph.D. Students

• Georg Hoffstätter Assoc. Prof. Cornell• Weishi Wan Staff Scientist LBNL• Kyoko Makino Assoc. Prof. UIUC / MSU• Khodr Shamseddine Assist. Prof. WIU• Jens Hoefkens GeneData• Bela Erdelyi Assist. Prof. NIU/ANL


Map Methods

• Understanding dynamics– Numerically integrate – Look at the flow: initial coordinates -> final coordinates

• Perturbation theories for flows– Transfer matrix methods: Transport (2nd order), GIOS (3rd

order), Trio (3rd order), COSY 5.0 (5th order)– Lie algebraic methods: MaryLie (3rd order)

• In principle, arbitrary order – but mostly in principle:– Arbitrary order here implies arbitrarily painful– Even Chinese graduate students don’t have the patience and

stamina to do it (German graduate students have no chance…) – Even formula manipulators don’t. Those used for COSY 5.0

produce >40,000 lines of code for ONE element.

The Difference between Theory and Practice Is greater in Practice than in Theory


Differential Algebras

• Allow to solve the flow computation problem – Arbitrary order– Very straightforward, even lazy German grad students can do it– Converting existing integrators with TPSA (our first idea): easy

to understand– Much more efficient: use differential algebraic structure of

Picard Operator of ODEs, PDEs, …• Allows arbitrary order for Normal Form problems

– Resonances, tune shifts– Long-term stability

• Allows arbitrary order for Generating Functions– General theory of extended generators, infinite dimensional

family containing common F1 – F4– Symplectic tracking, “Optimal” generators, very stable


COSY INFINITY

• First Arbitrary Order Beam Dynamics Code– Based on Differential Algebraic and TPSA methods, both

introduced by us. Can handle LHC to order 13.– Normal forms, generating functions, symplectic tracking– Rigorous treatment of remainder errors– Large library of standard elements, DIY elements (give fields)

• Features of the Code – About 50,000 lines of code– Versions for Unix, Windows, Cray– Parallel Version for MPI, installed at NERSC, uses up to 2000

processors • User Base

– As of December 2004, more than 1000 registered users– Two-volume user manual (downloadable – 150 pages)– User questions are a major effort for our group


Courses Offered at MSU

• PHY861 – Introduction to Beam Physics– Linear Theory, ODEs and Maps, Intro to Nonlinearities– Basic Types of Accelerators and their Components

• PHY961 – Nonlinear Beam Dynamics– Nonlinearities, DA methods, Lie methods, Symplectic Tracking– Normal Forms, Resonances, Tunes, Stability

• PHY962 - The World of Accelerators– Guest lectures from all major accelerator labs – Videoconference format

• PHY963 - Participation in USPAS– Students sent to participate in semi-annual courses

• PHY964 - Special Topics in Beam Physics– Advanced research topics– Structured online programs exist


The VUBeam Program

• Participation in individual courses– About 80 student-courses per year– Largest MSU Physics graduate specialization field

• Remote M.Sc. Degree– Accumulate 30 credits through online courses, USPAS, …– Pass departmental Qualifying Exam– Can earn up to 12 credits with thesis via local mentor

• Remote Ph.D. Degree– All of remote M.Sc., plus:– Departmental Subject Exams in four core areas– Dissertation

• Done in collaboration with remote mentors in National Labs• Committee Meetings at MSU or teleconference


The VUBeam Logistics

• Live Lectures – Local and Remote Lecturers– IP-based videoconferencing– RealVideo streaming– (Classroom – how old fashioned)

• Downloadable Lectures and Material– RealVideo and RealAudio– Lecture notes as pdf, entire books, weblinks …

• Online Homework System– Computational and Multiple Choice (interactive) – Extended narrative problems (FAX or scan)

• Send Students to USPAS– Rich assortment of classes, taught by world leaders– A concept that cannot be duplicated by any university


Former VUBeam Students

• Mohamed Nasr MSc 2002, now Riyad U.• Mandi Meidlinger MSc 2002, now NSCL• David Meidlinger MSc 2002, now NSCL• Jason Ong MSc 2002• Reiko Taki MSc 2003, now KEK• Pavel Snopok MSc 2003, now FNAL• Alexey Poklonskiy MSc 2003, now FNAL• Stephen Weathersby MSc 2004, now SLAC• Andrew Steere MSc 2005, now Sidney• Mahuya Sengupta MSc 2006, now ANL


DOE-Supported Output• Publications

– >100 refereed papers– >100 invited talks– Many non-refereed conference proceedings contributions

• Books– M. Berz, Modern Map Methods in Particle Beam Physics,

Academic Press, 1999– M. Berz, W. Wan and K. Makino, An Introduction to Beam

Physics, Taylor&Francis, 2006– High-Order Numerical Methods, Springer (in preparation,

expected 2006)• Three Volumes of Conference Proceedings

– Computational Differentiation 1996 (MSU / SIAM)– Computational Accelerator Physics 2002 (MSU)– Computational Accelerator Physics 2004 (St. Petersburg)


External Collaborations

• Fermilab– Tevatron Dynamics – Muon Collider Simulations– ILC

• Argonne– Aberrations in Spectrographs– Dynamics in LINACs, RIA– High-Performance Computing

• Verified Computing– DA with Remainder Bounds– Significantly reduces the main curse of verified methods,

the so-called dependency problem – Verified Integration and Global Optimization

Fringe Field Effects in a Muon Storage RingNo Fringe Field Effects, With Fringe Field Effects.

With Fringe Field Effects but Kinematic Correction.30GeV. Frames: 50mm x 6mrad.

Kinematic Correction in a Muon Storage RingNo Correction, Full Correction, 30GeV. Frames: 500mm x 60mrad. Lowest Order Correction, -- Symplectified.

Kinematic Correction and Fringe Field Effectson Tune Footprint in a Muon Storage Ring

1) No Kinematic Correction. No Fringe Field Effects. (0.7189796 - 0.7189797) x (0.2185736 - 0.2185738)

2) With Kinematic Correction. No FF.(0.71898 - 0.71904) x (0.21857 - 0.21860)

3) With Kinematic Correction.With Fringe Field Effects.(0.719 - 0.722) x (0.218 - 0.223)

1)

2) 3)

LOCAL THEORY AND APPLICATIONS OF... 271

Figure 8: 1000 turn symplectic tracking of a lattice of the pro-posed Neutrino Factory with the conventional generating func-tions (F1 through F4)

Figure 9: 1000 turn symplectic tracking of a lattice of the pro-posed Neutrino Factory with the generator type associated toS = 0

References

[1] D. Abell, Ph.D. Thesis, University of Maryland (1995).

The HGQ return end

The HGQ lead end

Pictures :

G. Sabbi, “Magnetic Field Analysis of HGQ Coil Ends”

Quad Cooling Cell (4m Cell)

RFRFRFRF RFRF RF

60cm 60cm40cm17.5cm

30cm

• Incoming Muons: 180 MeV/c to 245 MeV/c• Magnetic Quadrupoles: k=2.88• 35cm Liquid H Absorber: Energy loss ≈ 12 MeV.

The same design as Study II 2.75m sFOFO cell.• RF Cavity: Energy gain to compensate the loss.

About 200 MHz, φ = 30◦.

COSY Simulation of the Quad Channel

• Transfer Map Method using Differential Algebra (DA)

• Realistic Field Treatment using DA PDE Solver

• Treatment of Deterministic Effects in Transfer Map andNondeterministic Random Effects by Monte Carlo Kicks

2

Figure 4. Plots of the beta functions for the quadrupole cooling cell at p = 155 MeV/c, 200 MeV/c and 300 MeV/c. Constancy of the peak beta function implies that the physical dimension and divergence of the beam is maintained independent of energy; i.e. a constant geometrical admittance. By contrast, in the solenoidal channel, the normalized admittance is constant because the beta function increases with beam energy. Due to rising beta functions (also indicated by a rapid change in phase advance) the solenoidal channel displays a sharp cutoff at 245 MeV/c due to physical apertures. Because of the sharp rise as a function of momentum, increasing the solenoid’s apertures further (they are already comparable to the quadrupole bores), does not significantly improve the momentum performance. From the table it is evident that the quadrupole channel is still responsive at 300 MeV. The exact cooling limits of the channel will be explored later in this work.

Simulations Simulations were performed with the program COSY[9] on the quadrupole cooling cell described in the last section:

• using full nonlinear terms; • with/without quadrupole fringe fields[10,11] (including different models); • including multiple scattering (hydrogen absorber + windows); • energy loss, dE/ds, as a function of energy; • full energy loss simulation: straggling and spin; • a 200 MHz sinusoidal rf cavity.

Fringe-fields Simulations were initiated using a standard Enge function fall-off[12] for the quadrupole end-field profile which is known to represent fringe field effects to an accuracy sufficient for the initial analysis.

)5())/(...)/(exp(1

1)( 5621 DzaDzaa

zF+++

=

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Phase Space Distribution in COSY Coordinates: After Matching Section

x - ay - bt - dE

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Phase Space Distribution in COSY Coordinates: After Cell #1

x - ay - bt - dE

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1


x - ay - bt - dE

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0 10 20 30 40 50 60 70 80 90 100

Quad Cell Number

COSY-ECALC9 computed Transversal Emittance (m)

EM_t

C:\Berz\Docs\Slides\TM\FactFiction\Sources\Emails\RE YES !.txt

Page 1

From: Ramon Edgar Moore [[email protected]]Sent: Saturday, June 27, 1998 6:17 PMTo: Martin BerzCc: 'Makino, Kyoko'Subject: RE: YES !

Martin,

YES !

I now understand your method completely. It SOLVES the "wrappingeffect" problem, something I and many others (Krueckeberg, Lohner, et al)have tried off and on for three decades and failed to do.

I do not have words for the exhilaration I feel at your solution,for which I have waited so long. I'm chagrined I did not find out about itsooner, but never mind. Now is just fine. It will be the centerpiece of mytalk in Toronto.

There is one more thing I need to know, now or in Toronto if youcan't get back to me before then. Have you applied it yet to asteroid orbitcalculations? I mean to determine the distance of closest approach, takinginto account observational errors, and following their effect on orbitalelements, and subsequent calculations? I have been in contact with PaulChodas and Donald Yeomans at JPL concerning possibilities for using intervalmethods for such purposes. Your method would be ideal, and by far the bestapproach. Have you already done it? If not, it HAS TO be put on the "mustdo" list. With astonishment and gratitude,

Ramon

At 06:47 PM 6/27/98 -0400, you wrote:>Ramon,>>sorry for the delay, I am currently on travel in Europe and on the way to >Russia. To do the example with the sine function you asked for by hand is >probably too involved (at least with my level of patience), but when Kyoko >and I met here today, we could run it on the notebook. I have included the >results below. They show the polynomials describing the final values of x >and y as they depend on the three variables x_i, y_i, and t. The notation >is such that the first number is the Taylor coefficient, and the subsequent >four integers are the order and the exponents of x_i, y_i, and t, >respectively.>>Although the result is now more complicated, the basic approach is really >just the same as for the oscillator, except that now one has to propagate >any occurring three-variable Taylor model through the intrinsic function >"sin", and then as before integrate with respect to the time variable and >iterate. For the reference, how the intrinsics are done on Taylor models is >discussed in some detail in the Makino-Berz paper in the Santa Fe >Computational Differentiation book, which I believe I sent you.>>After plugging in the value for the time step, one obtains a (here >nonlinear) polynomial dependence of final conditions on initial conditions, >which then describes how the original box is twisted and nonlinearly >distorted as it runs through the system. Conceptually this is actually very >similar to the second example in the integrator paper, which treats a more

COSY INFINITY - Muon Simulation Mini School

Martin Berz, Kyoko Makino, Pavel Snopok,Alexey Poklonskiy, Shashikant Manikonda, Youn-Kyung Kim

Department of Physics and Astronomy, Michigan State University

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Makino

Makino

Motion in Regular Coordinates

Makino

Makino

Tracking Phase Space Motion of 5 Particles in Regular Coordinates

Makino

Tracking Phase Space Motion of 5 Particles in Normal Form Coordinates

Makino

Example of Phase Space Motion

The Beale function. f = [1.5-x(1-y)]^2 + [2.25-x(1-y^2)]^2 + [2.625-x(1-y^3)]^2

-4 -3 -2 -1 0 1 2 3 4x -4-3

-2-1

01

23

4

y

020000400006000080000

100000120000140000160000180000200000

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

COSY-GO with LDB/QFB. The Beale function

0

10

20

30

40

50

60

0 200 400 600 800 1000 1200

Num

ber

Step (Number of Boxes Studied)

COSY-GO The Beale Function (w SQR): Number of Boxes -- CF

To Be StudiedSmall Boxes ( < 1e-6 )

0

5

10

15

20

25

30

35

0 50 100 150 200 250

Num

ber


COSY-GO The Beale Function (w SQR): Number of Boxes -- LDB/QFB

To Be StudiedSmall Boxes ( < 1e-6 )

1e-030

1e-025

1e-020

1e-015

1e-010

1e-005

1

0 50 100 150 200 250

Cut

off V

alue


COSY-GO The Beale Function (w SQR): Cutoff Value -- LDB/QFB

3.1 The Rosenbrock Function

As a first example, we consider a relatively benign-looking function of twovariables that contains a single local minimum; but the minimum occurs alonga long and very shallow parabolic valley. This function originally proposed byRosenbrock has the form

fR(x, y) = 100 · (y − x2)2 + (1− x)2

and it apparently assumes its minimum of 0 at (x, y) = (1, 1).

We study the function over the domain [−1.5, 1.5]2, where its values rangefrom the minimum 0 to more than 1400 near the point (−1.5,−1.5). Fig-ure 5 shows a three-dimensional rendering of the function and the seeminglyinnocent parabolic valley generated by the first term, as well as a set of loga-rithmic contour lines for function values 0.1 · 10i/2 for i = 1, ..., 13, revealingthe substructure of the parabolic valley generated by the second term.

-1.5-1

-0.50

0.51

1.5x -1.5

-1-0.5

00.5

11.5

y

0

300

600

900

200

-1.5 -1 -0.5 0 0.5 1 1.5-1.

-1

-0.

0

0.5

1

1.5

Fig. 5. The Rosenbrock function; threedimensional rendering (left) and contour lines0.1 · 10i/2 for i = 1, ..., 9This function causes difficulties even for powerful conventional local minimiz-ers because as soon as the optimizer is probing not exactly inside the valley, thedirection of steepest descent is nearly perpendicular to the valley and hence inthe almost completely wrong direction. Table 1 shows the performance of var-ious tried-and-true optimizers in COSY[4][6], the LMDIF algorithm based onsteepest descent with various enhancements, the Simplex algorithm which isparticularly powerful for non-smooth problems, as well as the Anneal approachbased on stochastic search of the global minimum by simulated annealing. Thestarting point was chosen at (−1.2, 1.0), which is near the valley but on theopposite side of the true minimum. For LMDIF and Simplex, an accuracy tol-erance of 10−12 and a maximum number of steps of 100, 000 was specified, andAnneal was given a total number of 100, 000 annealing steps. Table 1 showsthe performance of the three non-validated optimizers. It can be seen that the

11

LMDIF Simplex Anneal

Number of Steps 100, 000 225 100, 000

Error in f(x, y) 1 · 10−10 2 · 10−13 3 · 10−4

Error in (x, y) 2 · 10−5 4 · 10−7 6 · 10−3Table 1Performance of various local optimizers for finding the minimum of the Rosenbrockfunction in [−1.5, 1.5]2 from the starting point (−1.2, 1.0).

IN CF COSY-GO

Total box processing steps 1325 1325 143

Max number of active boxes 47 47 9

Retained small boxes (< 10−6) 15 15 1

LDB domain reduction steps − − 43

Table 2Performance of various validated global optimizers for finding the minimum of theRosenbrock function in [−1.5, 1.5]2.usually quite powerful LMDIF algorithm performs rather poorly and is in thisexample significantly outperformed by the Simplex algorithm.

Now we utilize various validated global optimization tools to study the prob-lem. In particular, we use the common Moore-Skelboe algorithm and deter-mine ranges of the function by the mere interval evaluation method, and alsothe usually more accurate centered form method. From the perspective of in-terval methods, the function has a rather benign form with little dependency,and the squares that appear in the two terms can even be handled exactly.We compare these methods with the COSY-GO optimizer. Table 2 shows theperformance of these three methods. It is apparent that for COSY-GO, thenumber of processing steps, i.e. the number of boxes considered, comparesvery favorably even with the performance of the best non-validated optimizer,aside from the fact that it of course determines a rigorous global optimum. Onthe other hand, the Interval and Centered Form methods each require approx-imately one order of magnitude more steps, but still compare rather favorablyto LMDIF and Anneal. It is also interesting to note that in a significant num-ber of cases, LDB could reduce the size of a box under consideration withoutactually fully eliminating it.

We now study in more detail the performance of the validated global optimiz-

ers. To this end we show all boxes that were studied in the process within theoriginal domain box. Figure 6 shows these for both the interval- and COSY-GO methods. Apparently both methods successfully eliminate relatively large

12

Fig. 9. Projection of the normal form defect function. Dependence on two anglevariables for the fixed radii r1 = r2 = 5 · 10−4

Region Boxes studied CPU-time Bound Transversal Iterations

[0.2, 0.4] · 10−4 82, 930 30, 603 sec 0.859 · 10−13 2.328 3 · 108

[0.4, 0.6] · 10−4 82, 626 30, 603 sec 0.587 · 10−12 3. 407 2 · 107

[0.6, 0.9] · 10−4 64, 131 14, 441 sec 0.616 · 10−11 4.870 1 · 106

[0.9, 1.2] · 10−4 73, 701 13, 501 sec 0.372 · 10−10 8.064 5 · 105

[1.2, 1.5] · 10−4 106, 929 24, 304 sec 0.144 · 10−9 2.083 3 · 105

[1.5, 1.8] · 10−4 111, 391 26, 103 sec 0.314 · 10−9 0.95541 · 105Table 8Global bounds obtained for six radial regions in normal f orm space for the Tevatron.Also computed are the guaranteed minimum transversal iterations.

London, 1992. IOP Publishing.

[3] M. Berz. Modern Map Methods in Particle Beam Physics. Academic Press, SanDiego, 1999. Also available at http://bt.pa.msu.edu/pub.

[4] M. Berz, J. Hoefkens, and K. Makino. COSY INFINITY Version 8.1 -programming manual. Technical Report MSUHEP-20703, Department ofPhysics and Astronomy, Michigan State University, East Lansing, MI 48824,2002. see also http://cosy.pa.msu.edu.

[5] M. Berz and K. Makino. Verified integration of ODEs and flows using differential

18

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