Modelling I - The Cockcroft Institute · 2011-09-15 · Broad approaches FFAG School 2011,...

Modelling I

David KelliherASTeC/RAL/STFC

FFAG School, FFAG11, Daresbury

Accelerator codes - why we need them

• Design accelerator

• Establish likely machine performance

• Develop commissioning strategies

• Diagnose problems in operating machine and suggest corrective measures.

FFAG School 2011, Daresbury, UK

Broad approaches


Lorentz Force BvqqEF

• Track motion of single particle through simulated magnets and cavities. (e.g ray-tracing codes such as Zgoubi)

• Create map that represents the accelerator – matrix/map based codes. Basis of beam optics codes (MAD-X, SAD etc)

• Deal with collective effects – space charge, wakefields etc. Beam dynamics codes (Orbit, Simpson, Impact, ...)

qqm

H Ap2

1Hamiltonian

Beam Optics vs Beam Dynamics codes

• Beam Optics codes– Fast, good for initial design and tuning

– Matrix based, usually to first order

– Hard-edge field approximation

– Space charge forces approximated (if included at all)

• Beam dynamics codes– Slower, good for detailed studies

– Particle tracking, all orders included

– 3D field maps

– Solve Poisson equation at every step


MAP APPROACHMAD, MARYLIE, PTC , COSY INFINITY...


Map approach

• In general, can construct a one-turn map of an accelerator

• To third order may write

• To follow perturbative method, truncate Taylor Series at some order. However, map will no longer be symplectic, i.e. is not fully consistent with Hamilton’s equation.


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Hamiltonian

• Hamiltonian for particle motion in a curved reference system with bending radius ρ

• Hamilton’s equations of motion

where the following are canonical pairs of variables

• Normalise variables with respect to ideal momentum ps and apply

canonical transformation so that E1,t1 are expressed as deviations w.r.t. fixed reference frame. Can then apply perturbative approach.


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Symplecticity

• Jacobian

• Symplectic condition

• Symplectic map conserves phase space volume – Liouville

• Lack of symplecticity leads to unphysical growth or damping of phase space volume


JJMM T

0I-

I0J

Lie Algebra

• Lie algebra used in studying differentiable manifolds

• Poisson Bracket on differentiable functions f(q,p), g(q,p)

• Lie Operator

• Lie Transformation – operation on phase space vector z


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Lie Algebraic Map

• Dragt-Finn factorisation theorom (1976) - A symplectic map can be written as an infinite product of Lie transformations.

• Can truncate at any stage, still left with a symplectic map.

• Lie representatinon minimises number of parameters up to a given order.


...:::::: 432 fff

eeeM

Transfer Matrix Corresponding Lie Transformation

R exp(f2)

T exp(f3)

U exp(f3)exp(f4)

Putting one-turn map together

• Build up library of (f2, f3, f4) for each lattice element.

• Concatenate successive maps using Lie algebraic tools

• May still need to approximate calculation of f3, f4. Calculation will still be symplectic to some order.

• Much faster operating on map than tracking through many elements.


:::::: 432 fff

f eeeM

:::::: 432 ggg

g eeeM

:::::: 432 hhh

gfh eeeMMM

A. Chao

Normal form analysis

• Identifies a transformation A that recasts the one-turn map Minto a block diagonal matrix N.

• Extract tune from the invariant quantity N

• Extract s-dependent parameters such as linear lattice functions from transformation A.

• Can generalise to higher order maps and obtain non-linear effects such as tune variation with amplitude, coupled lattice functions etc


1 ANAM

Differential Algebra

• Map not just particle coordinates z but all its derivatives.

• DA can be used to efficiently compute maps to arbitrarily high order.

• Implementation easiest in an object oriented programming language


Integration methods (I)


xp

px

Equations for simple harmonic oscillator

Euler’s method

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hxpp

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1Error of order h => first order method

2nd order Runge-Kutta (RK2)(Midpoint method)

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Error of order h2

4th order Runge-Kutta (RK4) Trade off between accuracy and computational cost means RK4 is highest order used in practice.

Integration methods (II)


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2

3

2

11

nnn

nnn

xhFvv

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Leapfrog integration (Second order integrator)

JJMM T Can show that the leapfrog method is symplectic

Integration methods (III)

• Forest-Ruth fourth order symplectic integrator (1990)

• Yoshida produced 6th and 8th order integrators, but with many more substeps.


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Applications in celestial mechanics!

CODES USED FOR FFAG MODELLINGPTC, Zgoubi, S-code...


MAD8/MAD-X

• Not an ideal code for FFAG modelling – e.g. assumes reference beam travels through centre of elements.


beam; periodL:=0.394481;periodN:=42; qf: quadrupole,l=0.058782,k1:=+15;qd: quadrupole,l=0.075699,k1:=-11.8;ld: drift,l=0.21; sd: drift,l=0.05; kink: yrotation, angle=-2*pi/periodN; cell: line=(ld,qf,sd,qd,kink);ring: line=(42*cell);

FFAGs – modelling issues

• Particle design orbit need not follow the layout of magnet elements

• Non-linear fields in magnet body

• Fringe field effects may be more significant than in a synchrotron

• Complicated edges

• Space charge – beam not at centre of vacuum vessel, influence of one beam on another


PTC (E. Forest)

• Creates Taylor map using FPP package. Makes use of Lie Algebra, DA package and Normal Form.

• Symplectic integrator (several methods)

• Lattice constructed from a linked list of fibres, that in turn point to magnets. Allows construction of non-standard accelerator topologies – recirculators, colliders etc.


PTC and EMMA


• PTC can handle the displaced quadrupole doublets that make up EMMA• Analytic form of quadrupole fringe fields included• Currently, PTC has no analytic representation of scaling FFAGs.

S-code (S. Machida)

• Layout of magnets defined in a global coordinate system.

• Field profile of each magnet defined in local coordinates.

• Magnetic field derived from vector potential of each element, using an Enge funcion to describe the fringe field fall off.

• Elements include wedge and rectangular shaped scaling FFAG magnets.

• Integrate using a drift-kick scheme.


S-code integration method

• Drift-kick scheme – at each slice update momentum. Proceed in straight line to next slice.


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Zgoubi (F. Méot)

• Ray-tracing code which allows tracking in field maps and analytic models

• A wide variety of optical elements available, including a wedge shape and spiral shape scaling FFAG magnet

• Integrate Lorentz equation via truncated Taylor expansion of parameters

• Calculate u0 and its derivatives from Lorentz then find (R1, u1)


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Zgoubi and FFAGs


KEK 150 MeV radial sector FFAG

J. Fourrier

RACCAM 180 MeV spiral FFAG

PyZgoubi (S. Tygier)

• Python interface that minimises the effort involved in creating a Zgoubi data file and processing the results.


COLLECTIVE EFFECTS


Space-Charge

• Collective effects : charged particles mutually interact and with conductive walls of vacuum chamber.

• Space-charge effect due to Coulomb interactions. Particle experience self-field of bunch.

• Leads to emittance blow-up, halo formation, tune shift etc. Effect decreases with 2.

• For bunch charge distribution , code solves Poisson’s equation over a mesh with boundary conditions


2

0

2

Space-Charge calculations

• Space-charge calculations time consuming. Include limited number of space-charge kicks per element. Non-SC tracking in between.

• Boundary often approximated by rectangle or circle.

• Poisson’s equation solved using– Finite difference methods

– Finite element methods

– FFT approach

– SOR, LU decomposition

– Wavelet methods


Split-operator method


Code timeline


PyZgoubi and space charge (S. Tygier)

• Interleave tracking and space-charge steps.

• Bunch extracted from Zgoubi and passed to separate space charge solver, resulting bunch passed back to Zgoubi .


Summary

• Many advances in accelerator modelling codes in the last decades

• Latest tools should be used in our FFAG codes.

• Increasingly sophisticated code required to model space-charge and other collective effects in FFAGs.


Bibliography


• E. Forest, “Geometric integration for particle accelerators”, J. Phys. A: Math. Gen. 39 (2006) 5321–5377

• S. Machida, “S-code for Accelerator Design and Particle Tracking”, Int J Mod Phys A 26 (10-11) 1794-1806 (2011)

• MaryLie Documentation (http://www.physics.umd.edu/dsat/dsatmarylie.html)

• Zgoubi (sourceforge.net/projects/zgoubi)

• PyZgoubi (www.hep.man.ac.uk/u/sam/pyzgoubi)

http://www.physics.umd.edu/dsat/dsatmarylie.html

Modelling I - The Cockcroft Institute · 2011-09-15 · Broad approaches FFAG School 2011,...

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