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.
Top Related