accuracy and performance upgrade of pic and hybrid space-charge solvers

CERN, Space Charge 2013, April 17 L.G.Vorobiev 1Muons, Inc.

Accuracy and Performance Upgrade of PIC and Hybrid Space-Charge Solvers

Leonid Vorobiev

Muons, Inc., Batavia, IL 60510, USA

SPACE CHARGE 2013

16-19 April, CERN

ABSTRACT

For the high luminosity LHC (HL-LHC), space charge study requires high quality modeling. In this talk the sources of numerical errors in space charge algorithms are analyzed, and their mitigation is suggested for both conventional space charge grid solvers (based on PIC formalism) and hybrid solvers. The implementation of these improvements may significantly increase the accuracy, speed and physical validity of numerical results and contribute to achieve the required beam parameters in the LHC accelerator complex.


Overview

General SC: Grid(PIC), HybridComplex boundaries, Image Forces,Wakes, Beam Loading, Halo, Clusters ...

Works fine for many Works fine for many applications at Macro-levelapplications at Macro-level

High-Current Beam Simulation = Tracker & SC Solver

• Symplectic maps

• Integrators, high orderTracking enginesTracking engines =

&

SCSC

Zero/Non-Zero Frozen SCFree space, Pre-assigned Distribution: KV, Gaussian,…,A starting point

EXACT, Not General APPROXIMATE, General

~EXACT

Recent findings: Frank Schmidt, et al. Micro-scale effects ARE persistent.

E.G.: ORBIT code J.Galambos, J.Holmes, D.Olsen, A.Luccio and J.Beebe-Wang, 1999.


Some History

Micro-scale unphysical effects: H-detuning (left), fake tuneEvolutions, artificial dipole oscillations,…

May jeopardize the validity of beam modeling

WANTED: Consistency between ~Exact Trackers and approximate SC Solvers

Search for remedies

Fake Tune Evolution PTC-ORBITH-DETUNING PTC-ORBIT

F.SchmidtAPC Seminarat Fermilab,November 2012


Outline

1. PIC Formulation. Clouds & Grid Functions. Errors & Accuracy Upgrades. Limitations.

2. Hybrid Space Charge Solvers.

3. SC Templates. Back to PIC - Split operator revisit. Hierarchy of SC Templates

4. Discussion.

5. Conclusion. Credits.

6. References.

7. SC Solvers vs Applications.


1. PIC Formulation

The Green’s function in Differential form, for electrostatic approximation, leads to

n

uand/or , :conditionsBoundary

),,( ),,,( ,4

u

zyxzyxuuu

STEPS: A. Build a Grid. Charge density on the grid.B. Potential/Fields on the grid ( FFT-series, Multi-Grid, Iterations

with accel., Boundaries/Symmetries.C. Interpolation between nodes.

Refs.: Birdsall and Langdon [1], Hockney and Eastwood [2]

Poisson equation:


1.A Grid Density. Known-Unknown?

IF SC density is known (Gaussian,…exact)

(Multi-) Grid solvers may approach a solution with machine accuracy, exact SC Fields

Refs. www.nr.com [3], Saraniti [4]

IF SC density is approximate, - a starting

point in SC errors


1.A Grid Density. Clouds.

Contour lines (left), 3D plot (right). Simplest Cloud. Strong Numerical noise

PIC Clouds:Size, Shape, Ramp Redistribution of SC

over the grid

From: Ref. [5], Vorobiev and Hirata.


1.A Grid Density. Num. Noise Movie

Contour lines (left), 3D plot (right). Noise – suppressed. What about Accuracy?

100K Part.256x256


1.A SC Density. Grid Dim & Npart

1M, 256x256 4M, 256x256 4M, 512x512

More part.More grid

dimensions

Slow Down

Viciouscircle


1.A Grid Density. Num. Noise Movie

Contour lines (left), 3D plot (right). Noise – suppressed. What about Accuracy?

4M Part.512x512


1.B Grid Potential and Fields

Num. Noise concealed,but persists in dynamics

y

uE

x

uE

ugrad

y

x

E

2

2

2

2

is y

u

x

u


1.B Field Errors Suppression

Field Ex,y errors, depending on SC Clouds

What is ?max,

*,,

,yx

yxyxerryx E

EEE

*

, yxE


1.B Field Errors, Snapshots


1.B Field Errors. Choice of

Field errors, depending on SC Clouds and

*, yxE

*, yxE


Field errors, dependingon SC Cloud shape/size

*0

*0

200

function goal ,2

function goal ,2

4)(1

y

x

Eyy

u

Exx

u

rur

ur

rr

*, yxEmax

,

*,,

,yx

yxyxerryx E

EEE

min. But what is ?

Simplest benchmarking:

1.B Field Errors. Choice of *, yxE


1.B Clusters

All this works similarlyfor Symmetries,Boundary constraints,as well as forclustered beams

Contour lines, 3D shape, Potential (top), Fields (bottom)


1.C Interpolation.

SC Fields are solved on the Grid

Particles are in between Grid nodes:

Bi-Tri-Piecwise Interpolation - WrongBi-Tri-Linear Interpolation - InsufficientBi-Tri-Cubic Interpolation: Splines - Best

Ref. Vorobiev and Hirata [5]


1. Grid Solvers. Limitations

PIC Grid Solvers: solid/clustered beams, OkFilaments, Sheet beams, Halo, Stray particles not OkMesh refinement, • Density errors, irregular Clouds, use more particles• Errors due to interpolation on irregular grid• Slowing down speed…

Aside from PICs:• Particle-Core, Envelope (linear) & non-linear model ?• Integral representation of the Green’s function: OkComplex Boundary 3D, Sheet Beam (Photo-Injector) Ok• Hybrid SC Solvers


2. Hybrid SC Solvers. Outline.Hybrid Space Charge Solvers = Green’s function

in Integral form with Tricks

The Best ones for photo-injectors, el.optics, complex boundaries:VERY accurate, SLOW

Refs: Roger Harrington [6], Miklosh Szilagyi [7], Valentin Ivanov [8],In part: Fikera, Quang, Ryan, Kapin, Hess, Krassilnikov, Vorobiev

This Presentation:A. Space charge templates. Instead of integrated GF, use

Library of EXACTLibrary of EXACT SC potential & fields within boundariesSC potential & fields within boundariesB. From SC Templates to PIC: Split Operator RevisitSplit Operator RevisitC. Hierarchy of Templates


Disk-Template, LibraryDisk-Template, Library

Ring-Template = SuperpositionRing-Template = Superpositionof positive (BLUE) & negative of positive (BLUE) & negative (RED) derived from Disc-(RED) derived from Disc-TemplateTemplates

3D beam (non-elliptical!)Within conductingBoundary (not shown)3D Templates withinthe same boundary

NO GRID 3D Poisson Equation, INSTEAD: Convolve Template Potentials/Fields

)()(

, )(4)(

xx

xxx

beambeam

beambeam

Uu

u

)()(

, )(4)(

xx

xxx

tmptmp

tmptmp

Uu

u

zdzzSzzyxuzyxu yxtmpbeam~))~(),~(,~,,(),,( ,

3.A Hybrid Solvers, SC Templates

Ref. Vorobiev and York [9]


Library of TemplatesPre-Assigned densities

Ring Disc Template

• General • Halo

• Hollow Beam

EXP

ARB

-r_m 0 +r_m

3.A Hybrid Solvers, SC Templates

Private Communications:

A. Friedman, D. Grote, I. Hofmann, M. Reiser, J. Struckmeier and R. York


3.A Template Solvers, Machinery

SC Templates - Machinery

Template PotsFree space (green)Cond. pipe (blue)Image Dens (red)

Beam, Cond.Pipe Template Pots Template Fields Template uzz

Beam Pot Beam Field Beam uzz


3.B Hybrid Solvers. From Templates Back to PICs

BEAM NOT A RODBEAM NOT A RODE.G.

FNAL Booster, MI,… x84,x588

Acceleration withtransition crossing

Squeeze Head Tail

2.5D – series of 2D solutions, as if

BEAM BEAM == ROD ROD“BEAM=ROD” model - Good for PSR, SNS,… 2D beam, 1 long bunch 80% of circumference)


3.B Hybrid Solvers,Split-Operator Revisit

2

2

2

2

2

2

4z

u

y

u

x

u

corry

u

x

u 42

2

2

2

4/)/(),,( 22 zuzyxcorr

Ref: Vorobiev and York, Phys. Rev. ST Accel. Beams 3, 114201 (2000)

2

22

4

1),(

z

u

H

yxTZ

D

-0.2 -0.1 0.0 0.1 0.2

Z [m]

-10000

-6000

-2000

2000

Off-

axis

dE

z/dz

[V

/m2 ]

Sub-3D


3.B Hierachy of Space Charge Templates

SC Templates - Flowchart


4. Discussion. Existing SC Classes.Project X,Project X, Main Injector, H- Multiturn Injection

Transverse Painting (x,x’),(y,y’), Longitudinal Painting

FNAL BoosterTransition Crossing:RF cavities - narrow

SC issues

Ref. Vorobiev [11-12]


Mu2e Exp. In Fermilab:Extinction interval for mu2e detector:NO STRAY PROTONS

Long. dynamics w/o (left), with (cntr) beam loading. Histograms (right)

4. Discussion. Existing SC Classes

Ref. Balbekov and Vorobiev [10]


4. Discussion: Done,…, TBD

Accuracy upgrades• SC Density. Clouds. Size. Shape. +• Interpolation. Splines… +

• 3D SC Templates for Ez +

• Revisit Split Operator, Sub-3D +• 2D Templates, Envelope & Images +• Halo TBD

Memory,

Speed• No Grid

• Much lesser part

• Lesser Memory demands due to parameterization

Beam Loading & Wakes, 1++ Kicks/turn? TBD

Existing SC modules: Multi-turn H- injection (Project-X), Beam Loading (mu2e), Long. Dynamics (Transition Xing, Head-Tail…

On the way to full Synchro-Betatron dynamics studies


4. Discussion. HL-LHC Computational Needs

HL-LHC high intensity mode of operation

Linacs → Series of Synchrotrons → LHC

Linacs, Multiturn Injection, Acceleration, Trans. Crossing, Circulation

Tune shift, Image forces, Wakes, B. Loading Halo, Losses…

Wanted: SC Solvers – consistent to Symplectic Trackers and Integrators

SC Performance - Overall Accuracy & Speed Upgrade, Reasonable Memory Capacities, Physical Validity


5. Conclusion. Credits.

Charles Ankenbrandt, Thomas Roberts, Jeffrey Holmes

Richard York, Pavel Zenkevich

ANDRolland Johnson, Elias Metral and Frank Schmidt

Optional Space Charge Solvers in existing code

Cross-checking between codes, Stress/Regression Tests

Keep development of different codes by different authors – MUST

GOAL: reliable, insightful modeling for HL-LHC

Credits to:


6. ReferencesPIC1. C.K.Birdsall and A.B. Langdon (1985). Plasma Physics via Computer Simulation. McGraw-Hill. 2. R.W.Hockney and J.W. Eastwood (1988). Computer Simulation Using Particles. CRC Press.

Multi-Grid Poisson Solvers3.W.H. Press, S.A.Teukolsky, W.T.Vetterling, B.P.Flannery, “Numerical recipes”, www.nr.com .4. M.Saraniti, et al. IEEE Transactions on CAD, Volume: 15 Issue: 2, Page(s): 141-150 (1996).

Symplectic SC solver5. Vorobiev and Hirata, Report KEK 95-12; http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?199524012\

Green’s Function6. R.Harrington, “Field Computation by Moment Methods”, Macmillan, New York (1968)7. M.Szilagyi, “Electron and Ion Optics”, Plenum, New York (1988)8. V.Ivanov “Green’s Function Techniques in Forming Intense Beams”, (1989, 2009), Int. J.Mod.Phys A,

Vol.24, No. 5, 869–878,…In part: Quang, et al. (2006), Kapin (2002), Hess, et al.(2007), Vorobiev (1999).

SC Templates:9. Vorobiev and York: Phys.Rev. STAB 3, 114201 (2000); MSUCL-Report 1117, 1998;PAC’1999, pp. 2781-2783; Springer: http://link.springer.com/chapter/10.1007%2F3-540-47789-6_33PAC’2001 pp. 3075-3078; EPAC 2002, pp. 1679-1681; PAC’2003 pp. 3533-3535;Fermilab Report 08-236, http://lss.fnal.gov/archive/2008/pub/fermilab-pub-08-236-apc.pdf

BL + Transition crossing: 10. Balbekov and Vorobiev, http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=407011. Vorobiev, https://indico.fnal.gov/conferenceDisplay.py?confId=4497

http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=413412. Vorobiev, http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=4258


7. SC Solvers vs ApplicationsApplications to

Single-pass systemsSingle-pass systems

• (Photo-) Injectors, Electron Optics• Linacs: LBT, Acceleration• Transport, Final Focusing

Multi-pass systemsMulti-pass systems

• Rings: Multi-Turn Injection+ Acceleration

• Storage Rings, Colliders

?(maybe/no) Symplecticity (maybe/yes)? ?(maybe/yes) Complex Boundary (no)?

?(maybe/yes) Multi-step SC (no/maybe UMd ERing)??(maybe/no) SC Kicks (yes)?

?(maybe) Beam Loading, Wakes (yes)?…

Yes / Maybe / Not Yes / Maybe / Not

accuracy and performance upgrade of pic and hybrid space-charge solvers

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