Computational models, Computational models, algorithms & computer codes in algorithms & computer codes in

accelerator physicsaccelerator physics

Valentin IvanovStanford Linear Accelerator Center

23 February 2006

ContentsContentsIntroductionThe areas of activityTypes of the problemsStationary problems (Scalar Fields)F/T-domain Electromagnetism (Vector Fields)Beam Dynamics & Image Electron OpticsOptimization & Structural Synthesis

Physical ElectronicsPhysical Electronics

,ii i i ii

ci

f f f fFv St M tr v

∂ ∂ ∂ ∂ + + = + ∂ ∂∂ ∂

urr

r r

Fokker-Plank equation

Particle Tracking model

1/ 22

2 2

1( , ) ( , ) ( , , ) ,

1 .

i i i i

i ii

i i

P q E r t v H r t U r v t Uc

P PRM M c

−

= + × −∇ +∇

= +

r r rr r r r r& &

r& r

Plasma Physics Accelerator Physics (vacuum electronics)

0

0 0

,

,

1 ,

0.

i i ii

i ii

Brot EtDrot H j q nVt

div D q n

div B

ρε ε

∂= −

∂∂

= + +∂

= +

=

∑

∑

rr

rr rr

r

r

0.div jρ + =r

&

Laser Physics (non linear optics) Semiconductor micro electronics

The areas of activityThe areas of activityStationary fields Electrodynamics Electron optics Plasma physics Hydrodynamics

Poisson equation F-domain T-domain

Magnetostatics

Thermalphysics

HelmholtzEq.+PIC

MaxwellEq.+PIC

Poisson-2

Poisson-3

Maxwell-2

Maxwell-3

Optics-2

Optics-3

Track-3p

Electrostatics

Beamdynamics

Imageoptics

Beam-plasma

interaction

SemiconductorCrystal growthin low gravity

Navier-Stokes,Heat & MassTransfer eqs.

Stressanalysis

Navier-2Elast-2

HYDMIG-3Elast-3

Poisson-C Maxwell-T

Types of the Design ProblemsTypes of the Design Problems

••Field & particle AnalysisField & particle Analysis

••Parametric OptimizationParametric Optimization

••Structural SynthesisStructural Synthesis

••Tolerance AnalysisTolerance Analysis

Comparative features of different Comparative features of different numerical methodsnumerical methods

Method FDM FEM BEMMatrix Solver Iterative Iterative, Direct Direct Matrix dimension

2D 3D

104 106

103

104 - 105

102 102-103

Advantages • Uniform operations; • Simple struc-

ture of the mesh

• Flexible boundary-fitted mesh • Analytical

integration over theelement’s volume

1. High accuracy; 2. Open boundary;3. High-order derivatives; 4. Edge singularity5. External

asymptotic; Disadvantages Less accuracy

for complex- shape regions

Complex mesh- generator in 3D;

1. Dense matrix; 2. More complex analytics

.,,,)(4

1)(4

1)( rrRVrSrdVRrdS

Rrr

VS

′−=∈′′∈′′′

+′

= ∫∫rrrr

rrr ρ

πεσ

πεϕ

Applying the Dirichlet, Neumann conditions and continuity of electric induction at the boundary gives aset of integral equations for unknown function σ

An integral representation for the electrostatic potential ϕ at observation point r is a sum of the surface sources with the density σ and space charge density ρ :

Boundary Element Method for Boundary Element Method for Stationary ProblemsStationary Problems

0 01 2( ) , ,

S S S Sr U E

n n nϕ ϕ ϕϕ ε ε+ −+ −

∂ ∂ ∂= − = =

∂ ∂ ∂r

r r r

Linear System after discretization

.),(),(

),(1,4

1

ητητητητψ

ητ

ddJrrhh

GFGXm

mij ∑∫∫

= ′−== rr

Parametric representation of the boundaryParametric representation of the boundary

,),(),,(),,(),,(,1

ητητητητητ ddJdSzzyyxxSS kkkk

N

kk =====

=U

Polynomial approximation for surface sources

[ ] [ ] ,)()()()()()(

),(

1,111,1,11 ητηηττσττσηηττσττσ

ητσ

hhjijiijijijiiij −−+−+−−+−

=

−−−−−−−

Linear system G σ = F with matrix elements

.),(),(

),(1 4

1

ητητητητψ

ητ

ddJrrhh

Gm

mij ∑∫∫

= ′−= rr

MultiMulti--flow Gun for Klystronflow Gun for Klystron

Gun schematic sketch Iron geometry

Simulation of multi beam gun by TOPAZ 3D

File for beam envelope and trajectories: F:\miram\170kV new\final with washers\no washes\15 kV without washesTrajectory data file: F:\miram\170kV new\final with washers\no washes\15 kV without washes\testf2.trlInput file: ~\testf2.tsk -> do not forget crosssectionFile for magnetic field correction for anode-cathode area: CCR-1xxxx.dwgFiles of external off-axis B(x,y,z): nx.prn, ny.prn, nz,prn dated by 11-12-02Files of external on-axis B(x,y,z):nxon.prn, nyon.prn, nzon.prn dated by 07-07-02Bz_max= 922G, Bb= 369G, Bz_cath= 24GIb=1.46A, V0=15 kVAdjusted coefficients for TOPAZ input file:dx=0.063 m, dy = 0.0 m, dz=0.046 m, #4=0.0381, #12=0.0377

Original & optimized beam envelopesOriginal & optimized beam envelopes

The results of sheetThe results of sheet--beam gun simulationbeam gun simulation 2 1 3

1. Focusing electrode

2. Emitter

3. Anode

Diode, triode & grid gunsDiode, triode & grid guns

Cathode Assembly

Computational Models Gun Prototype, Calabazas Creek, Inc.

Non Elastic Collisions inNon Elastic Collisions in Gas Filled DiodeGas Filled Diode

2D model of collisions*

0/i e iM M dnα σ=

02468

101214161820

0 2 4 6 8 10 12 14 16alpha

J/Jb

Current multiplication. Blue – 1D model,

green – ionization; red – ionization+rechargingReaction trees

FrequencyFrequency--domain Electrodynamicsdomain ElectrodynamicsMaxwell equations

0 0 0 0 0, ,rotE i B rotH i D Jω ω= − = +

Integral representation,

.

1( ) ( ) ( , )2

1( ) ( ) ( , )2

eQn

S

mQn

S

E P J Q G P Q dS

H P J Q G P Q dS

β αβ

β αβ

π

π

=

= −

∫

∫Green functions [ ]

( )2

2 2

, cos( ) / ,

1 ( )( ) 1

e

m

G kR R

R RG kkR R kR R R R

αβ

αβ

α φ β φ

φ α β φ φαβ φ

= ∇ =

∂ ∂ ∂ = + + − ∂ ∂ ∂ Integral equations1( ) ( ) ( , ) ( ) ( , ) ,

41( ) ( ) ( , ) ( ) ( , ) .

4

m mQ

S

m mQ

S

H P H Q G P Q H Q G P Q dS

H P H Q G P Q H Q G P Q dS

τ τ τη η ττ

η τ ηη η ητ

π

π

= − Ω−

= − Ω−

∫

∫

EigenEigen modes ofmodes ofRF cavitiesRF cavities

The codes MAXWELL-2 and MAXWELL-3 can simulate azimuthally harmonics and 3D modes of oscillations

TimeTime--domain Electrodynamicsdomain Electrodynamics

0,

.

k k

k kk

B Et g

D H Jt g

αβ

α β

αβ

α β

ε

ε

∂+ ∇ =

∂

∂− ∇ =

∂

Maxwell equations

Material equations

0 ( , ), ( ) , ( ) .i i i i k i kik ikJ E J t r B H H D E Eσ µ ε= + = =

Conservation law

0.cdiv Jtρ∂+ =

∂

RFRF--gun design for LCLS. The codegun design for LCLS. The code MAXWELLMAXWELL--TTUV Laser Light

RF Photoinjector 10 ps, 1 nQ bunch dynamics Electric field distribution

Transverse normalized emittance vs. z Space charge distribution Space charge profile

Transverse charge distributionTransverse charge distribution

At the cathode At the exit

Beam OpticsBeam OpticsKlystron gun design using different numerical methods

Non Stationary problem. FDM + PIC. The code “MAXWELL-T”, Mesh: 400x500. Np=3500, T= 65”

U = 500 kV

I = 266 A

Stationary problem. BEM.

The code “POISSON-2”. T=5”

TimeTime--ofof--flight Massflight Mass--SpectrometerSpectrometerPortable high resolution TOF MS for analysis of Green-house gases

Temporal Resolution for Green-house gases

Spiral-Quadruple LensDetectors

Lens

Dark Current simulation modelDark Current simulation modelField emission Secondary emission

( )9 1.54.52 6.53 1026

( , ) 1.54 10 ,Ef

EJ r t e

ϕβϕ β

ϕ

− ×− + = ×

σ = Isec / Ipri = δ + η+ r

1

, ( ) 1 ,nn zm

mm m

g z g z z egδ εδ

ε+− − = = −

Secondary emission spectrum

Lye-Dekker model

SLAC XSLAC X--band TWband TW--structurestructure

30-cell structure, 11.424GHz, G=62MV/m

Distributed mesh for parellellMawxell solver

Transient EM-field. Electric field overshoot for rise-time 15 ns

Dark current evolution. Red-primary part., green –secondaries

NLC Dark Current PulseNLC Dark Current Pulse

10 ns rise time

15 ns rise time

20 ns rise time

Track3P

Dark current @ 3 pulse risetimes

Dark current pulses were obtained for the FIRST TIME from Track3P simulation for direct comparison with measurement

Experimental data by J.Wang

-- 10 nsec-- 15 nsec-- 20 nsec

XX--Ray spectrum simulation for Ray spectrum simulation for Square bend waveguideSquare bend waveguide

Electric field

Magnetic field

X-Ray Attenuation, Cu - 5 mm

00.10.20.30.40.50.60.70.80.9

1

10 100 1000 10000 100000

En, keV

Simulation

02468

101214161820

25000 50000 75000 100000 125000 150000E, eV

N

Square bend waveguide & EM fields

X-ray energy spectrum. Experimental

data by C. Adolphsen, simulation by

V. Ivanov

MultipactoringMultipactoring in super conducting cavityin super conducting cavityMultipactoring effect is the main factor limiting field gradient in SC cavities

Track3P simulationMultiPac simulations

Meshing for SNS Cavity G = 59 MV/m

Aberrational approachAberrational approachTrajectory equation

* *2 *

* *

1 1 (2 ) 0, .2 2

r r r q r rr r r r r r x iyz r m z r rϕ ϕ ψ ψ ψ

ε ϕ ε ϕ′ ′ ′ ′+ ∂ ∂ + ∂ ∂ ∂ ′′ ′ ′ ′ ′+ − + + − + = = + + ∂ ∂ + ∂ ∂ ∂

* * 2( )( , ) ( ) ( ) ( )4 64

IVrr rrr z z z zϕ ′′= Φ − Φ + Φ −LField expansion

Paraxial equation ( )0( ) ( ) / / 2 ( )( ) 0, .( ) ( )

i zz

z z

z z q mB zz uu u u rez z

ξρ εε ε

′′′ ′ Φ + +Φ′′ + + = =Φ + Φ +

0 0 0 0( ) 0, ( ) 1, ( ) 1/ , ( ) 1/ .z cz w z z w z Rυ υ ε′ ′= = = =Principal trajectories

Aberrational expansion3/ 2

0 0

2 (2 ) (2 ) 30 0 0 0 .

i i i i i ir r z z r r z

i i i i iz r r

r e r we He r Ke Be Pe

r Qe r Ce De r Fe Ge r E

α β α β α α

β β α α α β β

ε υ ε ε ε ε ε ε

ε ε ε− −

= + + + + + +

+ + + + + + L

HighHigh--speed streak cameraspeed streak camera Infrared image converterInfrared image converter

Night vision glassesNight vision glasses XX--ray light amplifierray light amplifier

Optimization ProblemOptimization Problem 0 ( ), ( ), ( ) min( )F r u E u H u u U⇒ ∈

r rr r r r rObjective functional

( ), ( ), ( ) 0, 1, ;

( ), ( ), ( ) 0, , ;

i

i

F r u E u H u i n

F r u E u H u i n N

< =

= =

r rr r r r

r rr r r r

Constraints

Optimization domain U: u u , 1, .j j ju j m− +< < =

.Q PQ Q P PS

G dS Uσ λσ+ =∫Original equation

Equation in variations

.Q PQ Q P P Q PQ QS S

G dS U G dSδσ λδσ δ σ δ+ = −∫ ∫

Types of the geometry Types of the geometry perturbationsperturbations

Boundary problemsBoundary problemsFourier expansion

Helmholtz equations

with the boundary conditions

( , , ) ( ) ( , )cos[ ( )]m mm

r z z r z mϕ θ ϕ θ θ= Φ + +∑

2 2 2

2 2 21m m m

m mm

r z r r rϕ ϕ ϕ ϕ ρ∂ ∂ ∂

+ + − =∂ ∂ ∂

12

( , ); ( , ).mm m mS

S

U r z E r znϕϕ ∂

= =∂

rr

Algorithms of BEMAlgorithms of BEM

Integral equation

Kernels

, ,1

, ,2

( ) ( , ) ( ) ( ) ( , ) ,

( ) ( ) ( , ) ( ) ( ) ( , ) .2

m m m m mS V

mm m m m mS V

Q G P Q dS U P t G P Q dV

P Q G P Q dS E P t G P Q dV

σ ρ

σ ρ

σ ρ

σ σ ρ

= −

− = −

∫ ∫

∫ ∫2

, 2 2 20

cos( ) ,( ) 2 cos ( )

mm dG r

r r rr z z

π

σθ θ

θ′=

′ ′ ′+ − + −∫

2

, 2 2 2 3/ 20

[( )sin( ) ( cos )cos( )]cos( ) .[ ( ) 2 cos ( ) ]

z zm

z z ne r r ne m dG rr r rr z z

π

νθ θ θθ

′ ′− − −′=′ ′ ′+ − + −∫

rr rr

Boundary Variations for the Boundary Variations for the Electron Optical DeviceElectron Optical Device

a1 – change of the tube diameter; a2 – change curvature radius for the cathode ; a3 – change of the radius for the anode hole; a4 –change of the outer radius for the anode cone; a5 – change of the cathode-anode distance; a6 – change of the cathode-anode distance

Non perturbed potential and Non perturbed potential and derivativesderivatives

-30

-20

-10

0

10

20

30

0 5 10 15 20 25 30 35 z, mm

VV'V"V'"V""

Variation of the cathode radius (m=1)Variation of the cathode radius (m=1)

Perturbed field for varying Rc

-25

-20

-15

-10

-5

0

5

10

15

0 5 10 15 20 25 30 35

z, mm

VV'V"V'''V""

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

z, mm

VV'V"V'''V""

Tilt of the axis (m=1)Tilt of the axis (m=1) Elliptical deformation (m=2)Elliptical deformation (m=2)

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

z, mm

VV'V"V'''V""

Numerical results for the tolerant Numerical results for the tolerant defects defects aaii, mm, mm

0.70.50.40.20.1a6

0.10.080.060.040.01a5

0.040.030.030.020.007a4

0.170.020.170.020.080.07804018,2518-8018-80Cathode dia, mm

0.20.10.04a3

0.20.10.04a2

0.050.030.01a1

506070Resolution mm-1

Structural SynthesisStructural Synthesis

Classical approachesClassical approaches1. Scherzer series

Numerical instability;

2. Conformal mapping Mapping singularity;

3. Finite-difference approximation for the Cauchy problem. Numerical instability;

4. No additional requirements (like constraint functionals Fj);

5. No tolerance analysis.

),0()(),(2)!(

)1(),( )2(2

02 zzzr

izr i

i

i

i

ϕϕ ≡ΦΦ

−

= ∑∞

=

∫ +=π

ααπ

ϕ2

0

.)cos(21),( dirzfzr

The ideal analytical model and The ideal analytical model and the practical Pierce gunthe practical Pierce gun

z

+=

rarctgzrCzr cos)(),( 3/222ϕ

1- heater; 2 – cathode; 3 – focusing electrode; 4 – anode; 5 –ceramic insulator; 5 – collector.

The results of classical approachThe results of classical approachThe absence of technical constraints in classical synthesis produces the solutions unrealizable in practical manufacturing.

New formulation of the synthesis New formulation of the synthesis problem is suggestedproblem is suggested

1. Obtain the Cauchy data Φ(z) by minimizing the objective functional F0;

2. Introduce 3 types of surfaces: a) fixed-shape electrodes with the boundary conditions; b) surface with the Cauchy data and c) “skeleton”surface with the field sources (charges, dipoles).

3. Determine the field source values from the solution of mixed boundary/Cauchy problem;

4. Obtain the electrode shape from the map ϕi=c;5. Obtain the real shape in the “tolerance range”.

Algorithms of BEMAlgorithms of BEMBoundary conditions

Cauchy data on the axis

Integral equations (mixed formulation)

),(sn S

ψϕβαϕ =

∂∂

+

),(0

τϕ Φ=S

).(2),()(),()()(

),(2),()(

)(2),()()(

sdtsGtdtsGt

sdtsGn

t

sdtsGn

ts

S

S

πννστ

πνβαν

πσβασψ

νσ

ν

σ

+Γ+Γ=Φ

+Γ

∂∂

+

++Γ

∂∂

+=

∫∫

∫

∫

Γ+Γ

Γ

+Γ

Approximation and linear systemApproximation and linear system

Source approximation with cubic spline

Green functions (single & double layer potentials)

Linear Problem in weak formulation

.)6

()6

(6

)(6

)()( 122

111

31

3

1i

iiii

i

iiii

i

ii

i

ii h

tthMXhtthMX

httM

httMtX −

−−−

−+−

−−

= −−−

−

−−

.)cos()()(

)()()cos()()()()(2

1

,)()(,)(

22

222

0

222

0

′−+′−′−′

+

−

′−+−′′=

′−+′+=′

=

zr enzzrrkEzzrenkKkEzzrr

rrG

zzrrkKrG

rrrr

δδπε

δδπε

ν

σ

min)()( *** →−−=ℜ FGXDFGX

,,

Φ

≡

≡

ψνσ FX

The Synthesis of Matching LensesThe Synthesis of Matching Lenses

Ideal lens Aperture hole a=0.3 Aperture hole a=0.4

.4463.0)1(3/2 00 ≈−= λaCritical aperture:

The Synthesis of Single LensThe Synthesis of Single Lens

Ideal Lens Aperture hole a=0.25

The Butler Lens Forming an The Butler Lens Forming an Ideal Cylindrical BeamIdeal Cylindrical Beam

W = 0.09 W = 0.2

Beam parameters : U=4KeV; P=0.55µA/V3/2

SummarySummaryNew approaches in formulation for complex problems of analysis, optimization and synthesis are suggested;Set of precision methods and algorithms are implemented. Their effectiveness was demonstrated for different application areas;The numerical tools and computer codes for 2D and 3D problems are successfully used in practical design during more than 30 years. Many unique devices have been constructed using numerical design.