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RADIA – a CPU-Efficient 3D Magnetostatics Computer Code. O. Chubar, P. Elleaume, J. Chavanne (ESRF, France). Topics. Motivation Previous Codes Approach: Volume (/ Magnetization) Integrals Examples: Undulators, Accelerator Magnets, Inverse Problems Possible Evolution. H ( r ). SR. e -. - PowerPoint PPT Presentation
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RADIA –RADIA –a CPU-Efficient 3D Magnetostatics a CPU-Efficient 3D Magnetostatics
Computer CodeComputer Code
O. Chubar, P. Elleaume, J. Chavanne (ESRF, France)
TopicsTopics
Motivation Previous Codes Approach: Volume (/ Magnetization) Integrals Examples: Undulators, Accelerator Magnets,
Inverse Problems Possible Evolution
MotivationMotivation
Computation of stationary Magnetic Field produced by Permanent Magnets, Coils and Iron Blocks in 3D space
Optimized for the design of Insertion Devices (Undulators and Wigglers) for SR Sources, as well as Accelerator Magnets
e-
H(r) SR
Computer Codes Using Similar ApproachComputer Codes Using Similar Approach
Solving 3D Magnetostatics ProblemsSolving 3D Magnetostatics Problems via Volume (/ Magnetization) Integrals via Volume (/ Magnetization) Integrals
– GFUN (Trowbridge et. al., Rutherford Laboratory, 1970s)
– B3D (P.Elleaume, J.Chavanne, ESRF, 1989-95)
– RADIA (O.Chubar, P.Elleaume, J.Chavanne, ESRF, 1996-…)
3D Magnetostatics3D MagnetostaticsMagnetic Field created by Uniformly Magnetized VolumesMagnetic Field created by Uniformly Magnetized Volumes
;)(
4
1)(
, )()(
: Si
, )(
4
1 )(
4
1)(
,
4
1
4
1)(
,
,
,
,0
3
33
0
Sd
SdVd
SdVd
S
S
S
S
V
S
S
V
rr
nrrrQ
MrQrH
constM
rr
nMrr
rr
MrrrH
rr
nM
rr
Mr
M
H
MHB
B
Poisson equation for scalar magnetic potential:
Solution through volume and surface integrals:
Magnetic field created by uniformly magnetized volume:
)()( bcacba
3D Magnetostatics 3D Magnetostatics Uniformly Magnetized PolyhedronUniformly Magnetized Polyhedron
, s=1,2,…,N , - coord. of vertex points of the face n- external normal to the face Nf - number of faces(x0,y0,z0) – coord. of observation point
1111
121
112
0
0
0
11
1
11
1
1
1
1
~~ ,~~
, )1()1(
)1()1(
; ~
~
~
,
,)()(
,
),,,,(),,,(),,,(
),,,(),,,(
),,,(),,,(
4
1
,)(
yyxx
nnn
nnnnnnn
nnnnnnn
zz
yy
xx
z
y
x
xayb
xxyya
zbaxxzbaxfzbaxf
zbaxfzbaxf
zbaxfzbaxf
NN
zyx
yzzxzyx
xzyxzzy
s
s
s
s
s
ssss
sssss
N
s
ssssssszsssz
sssysssy
sssxsssx
N f
TT
F
TkFTQ
(~ , ~ ,~ )x y zs s s
MQH Magnetic field:
010
20-5
051015
0
10
20
-5051015
-5 0 5
-50
51015
-10
0
10
20
-50
51015
- 5- 2.5
02.55
X - 10- 50510 Y
- 10
- 5
0
5
10
Z
- 5- 2.5
02.55
X- 10
- 5
0
5
10
Z
3D Magnetostatics 3D Magnetostatics Space Transformations and SymmetriesSpace Transformations and Symmetries
),(),( 1 VV rTTHTrH
1
0
1
0
),(),()(m
i
iim
i
itot VV rTHTTrHrH
Treatment of Symmetries reduces memory requirements and speeds up computation
Space Transformations
Symmetries (multiplicity m > 1)
V TV
V TV
T2V T3V
3D Magnetostatics 3D Magnetostatics RelaxationRelaxation
Interaction Matrix and Material Relations
H Q M H
M f H
exi ik kk
N
i
i i i i N
1
1 2
,
( ), , ,..., ,
Relaxation Scheme
Hi - total field strength in the center of object i
Hex i - external field the center of the object i
Mk - magnetization in the object k
Qik - component of the Interaction Matrix (being itself a 3 x 3 matrix)
fi(H) - magnetization vs. field strength law for the material of the object i
),(
),~
(][
,~
,,
,1
1,,
11,
1
1,,
)(
piipi
iiipipiiiipi
N
ikpkki
i
kpkikipi
HfM
MQHHχQEH
MQMQHH
rex
exex
- local susceptibility tensor for the material
of the object iMr i - remnantg magnetization in the object i
i ( )H
,0|| , (0)
,0|| , ]||)||([)(
HE
HEHHHχ
i
ii f
fThis scheme is robust:no “relaxation parameter” required (!)
for nonlinear isotropic material:
3D Magnetostatics 3D Magnetostatics RADIA Implementation: C++RADIA Implementation: C++
Main Classes
Field Source
“Rec. Current” “Arc. Current” “Filament Cur.”
Container
Space Transf.
General Methods for Field Sources– computation of Magnetic Field Strength, Vector Potential, Field Integral, Energy, Force, Torque– subdivision (segmentation)
RADIA is interfaced to:- Mathematica (Wolfram Research)- IGOR Pro (Wavemetrics)
Exists on Windows, Linux, Mac OS platforms Available for download from ESRF and SOLEIL Web sites (Insertion Devices pages)
Permanent
Coils Backgr. Field
3D GraphicsRelaxation
MaterialRelaxable
“Rec. Mag.” Extr. Polygon Polyhedron Nonlin. Isotr. Linear Unisotr.
RADIA ExamplesRADIA ExamplesAPPLE-II PPM UndulatorAPPLE-II PPM Undulator
0 20 40 60 80
0.4
0.2
0
0.2
0.4
0 20 40 60 80
0.4
0.2
0
0.2
0.4
Circular/Elliptical Polarization(parallel displacement of 2 magnet arrays)
Tilted Linear Polarization (anti-parallel displacement of 2 magnet arrays)
Gap: 17 mm
B [T]
Bz
Bx Bz Bx
s [mm] s [mm]
Invented by Sh.Sasaki et. al.0 0.5 1 1.5 2 2.5 3
0.1
0.2
0.3
0.4
0.5
0.6
0.7B [T]
BxBz
A B C D
Peak Field vs Shift bw Magnet Arrays
Mr = 1.1 T
Magnet Dimensions: 40 mm x 20 mm
The integral field Ibz for differents gaps
-20
-15
-10
-5
0
5
10
15
20
5,5 7 8 9 10 11 12 13 14 15 16 17 18 19
Gap (mm)
IBz (
G.c
m)
RADIA ExamplesRADIA ExamplesHybrid Undulator U20 (SOLEIL)Hybrid Undulator U20 (SOLEIL)
Optimization by C.Benabderrahmane
RADIA / SRW ExamplesRADIA / SRW ExamplesHybrid WigglerHybrid Wiggler
3T Hybrid WigglerMagnetic design by J.Chavanne, ESRF
Magnetic Field and Electron Trajectory
Spectrum vs Phot. Energyat diff. obs. points
Spectral Fux per Unit Surfacevs Horizontal and Vertical Positions
Magnets are designed using RADIA
SR computations are done using SRW
RADIA ExamplesRADIA ExamplesElectromagnet Elliptical Undulator HU640 (SOLEIL)Electromagnet Elliptical Undulator HU640 (SOLEIL)
Vertical and Horizontal Magnetic Fields
Magnetic design by O.Marcouille
Pure coil structure (no iron parts)Consumed electrical power: ~100 kW
The Structure (A.Dael - P.Vobly)The Structure (A.Dael - P.Vobly) Magnetic Fields at Max. Currents (Radia)Magnetic Fields at Max. Currents (Radia)IIz maxz max= 180 A, I= 180 A, Ix maxx max= 250 A= 250 A
Specifications:Specifications:
Circular Polar.: Circular Polar.: 1 min1 min < 10 eV < 10 eVLinear Hor. Polar.: Linear Hor. Polar.: 1 min1 min < 10 eV < 10 eVLinear Vert. Polar.: Linear Vert. Polar.: 1 min1 min < 20 eV < 20 eV
Aperture: 0.7 mr x 0.7 mrAperture: 0.7 mr x 0.7 mr
Calculated Spectra at Maximal CurrentsCalculated Spectra at Maximal Currents
RADIA Examples RADIA Examples Electromagnet Elliptical Undulator HU256 (SOLEIL-BINP)Electromagnet Elliptical Undulator HU256 (SOLEIL-BINP)
On-Axis Magnetic Field Magnetic Field “Roll-Off”
General Parameters u 65 mm; Lu 1.6 m
Min. gap: 16 mm Kx1 max Kz1 max 1.5
Coils Imax < 300 A
Conductor cross-section: 7 x 7 mm2
Number of Layers: 8
Yoke Pole transv. size: 40 mm Pole longit. size: 12 mm Material: iron (laminated )
Permanent Magnets Block dimensions: 30 x 17 x 15 mm3
Transv. distance bw blocks: ~ 5 mm Material: Sm2Co17
RADIA Examples: RADIA Examples: Fast-Switching Elliptical Undulator for XMCDFast-Switching Elliptical Undulator for XMCDPreliminary Design, derived from ESRF EMPHUPreliminary Design, derived from ESRF EMPHU
RADIA model with reduced number of periods(coil layer changes are not taken into account)
Original design by J.Chavanne
RADIA ExamplesRADIA ExamplesAccelerator MagnetsAccelerator Magnets
A: dipoleB: quadrupole with integrated sextupole componentsimulations by L-J.Lindgren
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-300 -200 -100 0 100 200mm
tesl
a
0.53
0.54
0.55
-30 -20 -10 0 10 20 30mm
tesl
a
1.47
1.48
1.49
1.5
1.51
1.52
-30 -20 -10 0 10 20 30mm
tesl
a
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-80 -60 -40 -20 0 20 40 60
mm
tesl
a
0
0.1
0.2
0.3
-600 -500 -400 -300 -200 -100 0 100mm
tesl
a
A
B
Edge Field (vs longitudinal position) Transverse Field
low excitation
high excitation
Discrepancy between Radia simulations and measurements: < 1% (peak field)
MAX-IIMagnets
RADIA Examples RADIA Examples Accelerator Magnets: SOLEIL DipolesAccelerator Magnets: SOLEIL Dipoles
Central Field vs Longitudinal Position
400 450 500 550 600 650 700 7500
0.25
0.5
0.75
1
1.25
1.5
0 100 200 300 400 500
1.64
1.66
1.68
1.7
1.72
B [T]
y [mm]
B [T]
y [mm]
Edge Field vs Longitudinal Position
Radia Examples Radia Examples Comparison with a commercial FEM CodeComparison with a commercial FEM Code
Chamfer Optimization of ESRF Quadrupolesimulation by J.Chavanne
FLUX3D200 000 Tetrahedrons200 MB of Memory3 Hours of CPU
RADIA1500 Polyhedrons200 MB of Memory1 Hour of CPU
Radia Examples Radia Examples Radia vs commercial FEM Code FLUX3DRadia vs commercial FEM Code FLUX3D
Hybrid Wiggler Simulation ComparisonCase A: Solution for 1% accuracy in peak field
Case B: Solution for 10 G-cm abs. accuracy in on-axis field integral
RADIA FLUX3D CPU time A
B 10 s
100 s 200 s
Number of 3D elements A B
400 980
~15 000
Memory required A B
7 MB 27 MB
11 MB ?
Accuracy of the field inside magnet blocks
poor* same as in the air
* accuracy is high only in centers of 3D elements
Evaluation:
1 2 3 4 5 6 7 8
5 4 8 1 7 2 6 3 7 2 6 35 4 8 1
1 3 5 7
2 4 6 8
Possible Variation Operators for Permutations (i.e. for Sorting):
Mutation : -- e.g. e.g. swap magnets at two randomly chosen positionsswap magnets at two randomly chosen positions -- [ 5 4 8 1 7 2 6 3 ]
Crossover : -- e.g. e.g. «order 1» «order 1» - - [ 1 2 3 4 5 6 7 8 ]
[ 3 5 6 8 1 2 7 4 ][ ? ? ? 4 5 6 7 ? ] [ 8 1 2 4 5 6 7 3 ]
Goals:Goals: Find optimal distribution of individual magnets (or magnet displacements) which would allow to compensate existing magnet imperfections and to reach required ID magnetic field characteristics
Ordered Magnet Ordered Magnet Sequence(s)Sequence(s)
«Decoded» «Decoded» Undulator Undulator StructureStructure
Magnetic Measurements DataMagnetic Measurements Dataon Individual Magnets (/ Modules)on Individual Magnets (/ Modules)& Partly Assembled Undulator& Partly Assembled Undulator
Mathematical Model Mathematical Model / Total Field Calc. Method / Total Field Calc. Method
Undulator Undulator Magnetic Field Magnetic Field (/ Field Integrals)(/ Field Integrals)
CharacteristicsCharacteristics/ Fitness Terms/ Fitness Terms
Electron Trajectory Electron Trajectory StraightnessStraightness
Radiation Phase ErrorRadiation Phase Error
Field Integral Field Integral deviation from zerodeviation from zero
Integrated MultipolesIntegrated Multipoles
Fitness Fitness
WeightsWeights
. . .
Genetic Genetic AlgorithmsAlgorithms
[ 5 4 6 1 7 2 8 3 ]
ID Magnet Sorting was pioneered by A.Cox and B.Youngman (1985)
ID Magnet Shimming was pioneered at ESRF and ELETTRA (199x)
Radia Examples / Inverse Problems Radia Examples / Inverse Problems SortingSorting and and ShimmingShimming Insertion Device Magnets Insertion Device Magnets
RADIARADIA
Shim Signatures of In-Vacuum Hybrid Undulator U20Shim Signatures of In-Vacuum Hybrid Undulator U20RADIA Model for Central Part RADIA Model for Central Part
(A-, B- Modules and Poles)(A-, B- Modules and Poles)RADIA Model for Extremities RADIA Model for Extremities
(E- Modules)(E- Modules)
Variation of On-Axis FieldVariation of On-Axis Fielddue to 25 due to 25 m displacement of m displacement of
A-, B-, E-Modules and PolesA-, B-, E-Modules and Poles
Variation of Field IntegralsVariation of Field Integralsdue to 25 due to 25 m displacement of m displacement of
A-, B-, E-Modules and PolesA-, B-, E-Modules and Poles
On-Axis Single-Electron Spectra Before and After ShimmingOn-Axis Single-Electron Spectra Before and After Shimming (10 m from source)(10 m from source)
Evolution of 11Evolution of 11thth Harmonic Harmonic of Single-Electron Spectrumof Single-Electron Spectrum
On-Axis Spectrum Taking into account On-Axis Spectrum Taking into account E-Beam Emittance and Energy SpreadE-Beam Emittance and Energy Spread
x = 3.7 nm; E/E = 10-3RMS Radiation Phase Error after Shimming ~2.6RMS Radiation Phase Error after Shimming ~2.6
““Spectral” Shimming of In-Vacuum Hybrid Undulator U20Spectral” Shimming of In-Vacuum Hybrid Undulator U20
APPLE-II Undulator HU80-PLEIADES: APPLE-II Undulator HU80-PLEIADES: Evolution of Electron Trajectory and Field Integrals Evolution of Electron Trajectory and Field Integrals
(Min. Gap, Zero Phase)(Min. Gap, Zero Phase)Horizontal First Field IntegralHorizontal First Field Integral
Vertical First Field IntegralVertical First Field Integral
Horizontal Trajectory Horizontal Trajectory (Periodic Mode)(Periodic Mode)
Q.-P. Mode realized by 11 mm displacement of some Q.-P. Mode realized by 11 mm displacement of some longitudinally-polarized magnets.longitudinally-polarized magnets.
Horizontal Trajectory Horizontal Trajectory (Quasi-Periodic Mode)(Quasi-Periodic Mode)
RADIA “ToDo” ListRADIA “ToDo” List Short-Term Tasks
- Simplifying definition of 3D geometries, by importing files from CAD / 3D meshing software (R. Carr)- Releasing RADIA for Python- Compiling and testing 64-bit versions for Windows and Linux- Updating all existing versions for Windows, Linux and Mac OS- Updating RADIA distribution pages at SOLEIL web site
Longer-Term Tasks- Further improving relaxation procedure (cases of many sub-volumes, non-linear mater.)- Implementing coils / conductors with polygonal cross-section- Addressing inverse problems of 3D magnetostatics (software for magnet sorting and shimming was the first step in this direction)- Implementing solvers of time-dependent direct problems (e.g. Eddy currents)- Supporting other similar direct problems of electrodynamics (e.g. electrostatics, potential current flow)- Fast 2D version of RADIA (?)
Acknowledgements Acknowledgements
J.-L. Laclare, J.-M. Filhol, D. RaouxO. Rudenko, A. Dael
All Users of RADIA and SRW
In-Vacuum Hybrid Undulator U20 (-SWING): In-Vacuum Hybrid Undulator U20 (-SWING):
GA-Based Module Sorting and Magic Finger GA-Based Module Sorting and Magic Finger AdjustmentAdjustmentHorizontal Field IntegralsHorizontal Field Integrals Vertical Field IntegralsVertical Field Integrals
at 5.5 mm Gapat 5.5 mm Gap
at 10 mm Gapat 10 mm Gap
97 full periods, 390 magnet modules
HU80 Field Integrals at Minimal Gap (15.5 mm)HU80 Field Integrals at Minimal Gap (15.5 mm)Horizontal Vertical
TEMPO
PLEIADES
FOC(after extra shimming)
APPLE-II Undulator HU80-APPLE-II Undulator HU80-Foc: Foc: Extra Shimming to Reduce MultipolesExtra Shimming to Reduce Multipoles
Field Integrals at Minimal Gap and Different PhasesField Integrals at Minimal Gap and Different PhasesVerticalVertical Phase-Dependent Normal Quad VariationPhase-Dependent Normal Quad Variation
HU80-HU80-Foc was originally supplied by ELETTRA. Extra Shimming of the HU80-Foc was originally supplied by ELETTRA. Extra Shimming of the HU80-Foc was performed at SOLEIL. Foc was performed at SOLEIL. The Extra Shimming was based on The Extra Shimming was based on Genetic OptimizationGenetic Optimization and it took into account and it took into account Phase-Dependent Magn. Interaction EffectsPhase-Dependent Magn. Interaction Effects. . Only ELETTRA “native” spare parts (mechanical shims and “magic fingers”) were used.Only ELETTRA “native” spare parts (mechanical shims and “magic fingers”) were used.
Before Extra ShimmingBefore Extra Shimming After Extra ShimmingAfter Extra Shimming
HorizontalHorizontal Phase-Dependent Skew Quad VariationPhase-Dependent Skew Quad VariationBefore Extra ShimmingBefore Extra Shimming After Extra ShimmingAfter Extra Shimming
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