SCDyn

1
SCDyn

• Paolo PieriniINFN Milano LASA
• Paolo.Pierini_at_mi.infn.it

2
Background TRASCO

• TRASCO design of a high beam power (gt30 MW) SC
  proton linac as a driver for a subcritical system
  for nuclear waste transmutation
• Funded accelerator activities (TRASCO project)
  include the design of the driver accelerator
  (from source to the high energy end) and
  prototypical activities on the main components
  (source, RFQ, DTL, superconducting cavities, see
  Carlo Pagani Talk)
• Our group is involved in the TTF/TESLA
  collaboration since its starting phase
• Procurement of SC cavities through Italian
  industries
• Design, fabrication and installation of the
  cryostats
• Our natural interest is the design of the high
  energy SC section of the proton linac (and
  related prototypical activities
• The basic lattice of the SC linac is that of a
  periodic doublet array with cavity cryomodules in
  the long drifts between the warm magnets. The
  linac is divided in three sections

3
Motivation of This Work

• After completing the linear beam dynamics design
  of the linac, multiparticle simulations are
  needed to assess the design and investigate its
  reliability
• A 3D space charge algorithm is needed to properly
  take into account the non axially symmetric
  nature of the beam envelopes.
• A z-dependent description of the cavity fields
  would allow to properly describe the dynamics in
  the reduced b cavities, where the field escapes
  out of the large bore at the end cells
• To simulate the SC linac, only few element types
  are needed RF cavities, quadrupoles and drifts
  (plus RF gaps)

TRACE3D has been a starting point for the linear
beam dynamics
On the basis of these considerations we started
developing an ad-hoc code for the dynamics in the
SC linac with a 3D space charge algorithm.
4
The Code - General Layout

• The F9x code is written in a modular approach for
  the data and program design
• Space charge calculations require knowing the
  position of all the particles at the same time
• advance by phase steps DfwDt
• checks are done for the element boundary crossing
  in the step Df (substepping)
• t code
• Z-dependent cavity field
• 3D space charge by multigrid scheme
• Simple second order integration scheme
• Space charge is evaluated every chosen number of
  steps, as input to the program
• Beam matching is performed with linear design
  codes Trace3D/TraceWin/DoLinac

5
Integration in the RF Cavities
11 doublet periods (2 cavities each) 100-190 MeV
The on-axis cavity field profile is expressed
analytically and a direct integration in the
field is performed for the particles
Analytical field
Off-axis expansion
6
Different Cavity Modeling

• We used 2 different cavity models
• RF gaps computed from the SUPERFISH files, that
  give exactly the right energy gain at each
  particle b, but a different transport matrix
  (both longitudinally and transversally)
• A "Sine-like" equivalent field computed from
  SUPERFISH files, that gives approximately the
  same energy gain in the same b range, and a
  closer transport matrix to the real field
• Implemented same modeling

Red triangles single RF gap, Blue dots
Sine-like, Blue line Superfish file
Longitudinal
Transverse
7
Transit Time Equivalence
8
Synchronous Phase Scan

• Doing a direct integration we need to know each
  cavity phase ?c
• For simplicity, we like to define the synchronous
  phase of the cavity as the relative phase at
  which the reference particle arrives at the
  center of the cavity
• This is done in a pre-processing iterative phase
• Launch the reference particle (only) along the
  beamline
• Push it until the cavity mid-position, with a
  guess cavity phase
• If the phase wt?c is not equal to the desired
  ?synch, ?c is reset and the cavity is restarted
  (until convergence)
• It is a convenient way to set a large number of
  cavity phases, taking a few seconds for hundreds
  of cavities

Start
synchronous phase scan At mid cavity with
wtphis 59.90448 Goal for wtphis is 60.00000
and distance is 0.9552E-01 Phis changed to
-285.72015 and element is restarted At mid
cavity with wtphis 59.99610 Goal for wtphis
is 60.00000 and distance is 0.3903E-02 Phis
changed to -285.71625 and element is restarted
At mid cavity with wtphis 59.99999 Goal for
wtphis is 60.00000 and distance is 0.6517E-05
Phis changed to -285.71608 and element is
restarted At mid cavity with wtphis 60.00000
Goal for wtphis is 60.00000 and distance is
0.2663E-06 Element 6, Phis changed to
-285.72 ... Element 622, Phis changed to
-292.81 End synchronous phase scan

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Choice of the Multigrid Algorithm

• Space charge evaluation is performed by solving
  the 3D Poisson equation in the beam rest frame
• CIC scheme used for charge density reconstruction
  
• Linear interpolation for force assignment on the
  particles
• The Poisson solver is based on a simple (almost
  straightforward) implementation of a multigrid
  scheme (V cycle and F cycle)
• Multigrid schemes are iterative schemes based on
• Standard (and simple) relaxation techniques
  (weighted Jacobi or Gauss-Seidel)
• Discretization of the problem on a series of
  nested grids levels (obtained, for example, by
  step doubling)
• Relaxation is used at each grid level to
  eliminate the fast oscillating errors
• When convergence deteriorates (after few
  relaxation cycles, say 3-4) due to the presence
  of smooth errors, the estimate of the solution is
  restricted to a coarser grid, where some of the
  errors are no longer smooth.
• Then the solution on the coarse grid is used as a
  correction

10
The Multigrid V-cycle (2D)
S is the standard Gauss-Siedel (normal or
red-black) smoothing algorithmR is the coarse to
fine restriction operatorP is the fine to coarse
prolongation operator Makes use of the spectral
properties of the normal relaxation algorithms
(very efficient with fast oscillating error
components, bad for smooth components), combined
with the discretization of the problem on a
series of grids.
11
Implementation in the Code
All sorts of cycling strategies are possible

• V-cycling and FMV-cycling strategies are
  implemented
• V-cycling is preferable since it allows to use
  the potential map
• at the previous space charge step as a better
  initial guess
• (allowed 1/3 CPU time reduction in most cases)
• Restriction operator is implemented as a
  trilinear averaging
• (full weighting) of the 27 neighboring nodes of
  the fine grid
• Prolongation operator is trilinear interpolation
  of the coarse grid
• The smoothing operator is the gauss-seidel
  operator
• (no mesh ordering, which is needed for a parallel
  version)
• ! Associates g to the nth level grid
• G gt Grid(n)
• GU(i,j,k) (GU(i-1,j ,k ) gU(i1,j ,k )
  
• GU(i ,j-1,k ) gU(i ,j1,k )
  

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Test Case for the Poisson Solver
As a test case consider the potential and fields
generated by a uniformly charged sphere
(generated depositing a set of 10 k particles
uniformly on a sphere with radius a0.1). The
Poisson equation is solved on a 65x65x65 cubical
mesh extending from five times the sphere radius
in each direction. Here the contour and density
plots of the potential are shown in the plane z0
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Analytical Check - Uniform Sphere
Mesh values for the radial field (derived from
the potential values), compared with the
analytical expression
Mesh is 653 points, extending to five times the
sphere radius in each direction
Field interpolated on the particle positions
10 k particles
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Interpolation of Er Along Circles

• The only interpolated curve that show strong
  asymmetries is the curve at the charge radius due
  to
• the square mesh
• a low order interpolation scheme (trilinear
  interpolation)

15
Scaling with grid and particle
It is possible to simulate long beamlines with
100 k to 1 M particles
16
Simulations of the TRASCO Linac

• The code has been used to simulate the
  Superconducting high energy end of the TRASCO
  linac, from 85-100 to 2000 MeV
• The rms beam emittance growth is limited to below
  6 along the whole 350 m linac, for the matched
  case and in the absence of misalignment and
  errors
• Calculations with a direct (point-to-point) space
  charge routine give the same results (with a
  limited resolution on the number of particles and
  a higher noise)
• The beam dynamics of the 30 mA nominal current
  agrees noticeably with the TRACE-3D and TRACEWin
  beamlines that we routinely use to determine
• the matched input beam
• the quad setting for the chosen transverse phase
  advance behavior
• the choice of the quadrupole gradients and cavity
  phases at the three section interfaces (necessary
  for section matching)
• Very confident that the SpaceCharge-MultiGrid
  scheme works reliably (made some tests with
  Nicolas)

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Nominal Linac Beam Dynamics (1 GeV TRASCO linac)
Beam Emittances
Control of the particle distribution
From EPAC 2000 Vienna, June 2000
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Beam Envelopes
Transverse rms envelopes (redx, bluey) mm
Longitudinal rms envelopesdeg
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Emittance Growth
egxy mm mrad
enxy mm mrad
ez deg MeV
Ek MeV
20
Phase advances
In this case zero current matched beam from
Trace3D
SCDyn, numerically computed ltIgt 20 mA beam
Trace3D ltIgt 20 mA beam
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Tune depression Tune ratio
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Beam Size Fractions
99.99
99.9
99
90
sx /sxrms
sy /syrms
sz /szrms
st /strms
23
Emittance Fractions
100
eny mm mrad
enx mm mrad
90
rms
ez deg MeV
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General Linac Design Matching

• Collected all rules used to constrain the linac
  design in a single tool
• Choices of cavity parameters
• RF Frequency
• b values
• Determined mainly by the machine energy range and
  number of sections
• Number of cells
• Determined by efficient use
• How many cavities per cryomodule?
• Many good use of space (beware of too much
  acceleration in one lattice cell!)
• Few waste of space
• Cavity design provides constraints
• Max B surface field (and Bpeak/Eacc)
• Max E surface field (and Epeak/Eacc)
• Longitudinal beam dynamics constraints
• Longitudinal phase advances
• Longitudinal focussing (synch. phase)
• Beamline space inventory

25
DoLinac Design Matching
Transverse Phase Advance Shaping
Matching along each section
Wrote a code for D M (with linear space charge
forces, as Trace3D) Cavity design Limits on
mz ?Long. Phase advance law Choose (GUI, as
TraceWin) a transverse phase advance shape ?Quad
field settings Adiabatic matching within each
section Matching at section interface
Setting of quadrupoles
Matching between sections
26
DoLinac (INFN, PPMN) builds linac from simple
rules, with control of longitudinal and
transverse phase advances
TraceWin (CEA, Uriot Pichoff)Longitudinal
plane in inputLattice cell (Maille) based, very
convenient to useVery elaborate matching with
great control over the transverse phase advance
lawsCan directly give input and control
multiparticle simulations with different codes
Linac design life cycle 1/2
TRACE3D (originally by LANL)Standard beam optics
codeAwkward to use in this context, since it
ignores the lattice periodicityDifficult to
implement transverse phase advance laws as
TraceWin (so the two codes need to talk)
From Workshop on RD on SC Proton
Linac Saclay, October 19-20, 2000
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SCDyn (INFN/PP) Multigrid space charge
codeCavities as on-axis sin-like
fieldt-dependent
From Workshop on RD on SC Proton
Linac Saclay, October 19-20, 2000
Code Comparisons INFN/CEA/IN2P3 Meeting (by
NP) Saclay, Oct 18-19, 1999
Linac design life cycle 2/2
PARMILA/(PARTRAN?)(CEA/Pichoff)
2D(scheff)/3D(PICNIC) space charge
routinesError studiesz-dependentStandard code
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Multigrid Solver Is a PRIVATE MODULE With PUBLIC
Interfaces

• Type GRID_STRUCTURE
• Real (Kind8), Pointer, Dimension(,,)
  U, Rho, EPS
• Real (Kind8) Delta
• Real (Kind8), Dimension(13) Xi
• Integer NPT
• End Type GRID_STRUCTURE
• Type (GRID_STRUCTURE), Dimension(), Allocatable,
  Target Grid ! Array storing all grids
• ! Ng is the number of grid levels in the
  structure
• ! Presmooth and postsmooth are the
  pre-post-smoothing cycles
• Integer, PRIVATE NG,PRESMOOTH,POSTSMOOTH,MAXIT,
  NVCYCLES
• Real (Kind8), PRIVATE EPS ! eps is the
  desired convergence ratio
• Real (Kind8), Dimension(3), PRIVATE LOW,HIGH
  ! low and high are the grid ranges
• Private Grid, GRID_STRUCTURE
• PRIVATE Restrict,Prolongate,Correct,Gauss_Siede
  l,
• Evaluate_Residual,Evaluate_Norm,Zero_So
  lution, Direct_Solve