RADIA – a CPU-Efficient 3D Magnetostatics Computer Code

1. RADIA –a CPU-Efficient 3D Magnetostatics Computer Code O. Chubar, P. Elleaume, J. Chavanne (ESRF, France)

2. Topics • Motivation • Previous Codes • Approach: Volume (/ Magnetization) Integrals • Examples: Undulators, Accelerator Magnets,Inverse Problems • Possible Evolution

3. H(r) SR e- Motivation  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

4. Computer Codes Using Similar Approach  Solving 3D Magnetostatics Problemsvia 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-…)

5. 3D MagnetostaticsMagnetic Field created by Uniformly Magnetized Volumes Poisson equation for scalar magnetic potential: Solution through volume and surface integrals: Magnetic field created by uniformly magnetized volume:

6. 3D Magnetostatics Uniformly Magnetized Polyhedron , s=1,2,…,Ns , - coord. of vertex points of the face s ns- external normal to the faces Nf- number of faces (x0,y0,z0) – coord. of observation point Magnetic field:

7. 3D Magnetostatics Space Transformations and Symmetries  Space Transformations V TV   Symmetries (multiplicity m > 1) V TV T2V T3V Treatment of Symmetries reduces memory requirements and speeds up computation

8. 3D Magnetostatics Relaxation  Interaction Matrix and Material Relations Hi - total field strength in the center of objecti Hex i - external field the center of the objecti Mk - magnetization in the objectk 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 objecti Relaxation Scheme - local susceptibility tensor for the material of the objecti Mri - remnantg magnetization in the objecti for nonlinear isotropic material: This scheme is robust: no “relaxation parameter” required (!)

9. 3D Graphics Relaxation Space Transf. Relaxable Material “Rec. Mag.” Extr. Polygon Polyhedron Nonlin. Isotr. Linear Unisotr. 3D Magnetostatics RADIA Implementation: C++ Field Source  Main Classes Permanent Container Coils Backgr. Field “Rec. Current” “Arc. Current” “Filament Cur.”  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)

10. 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3 0 0.5 1 1.5 2 2.5 0.4 0.4 0.2 0.2 0 0 - 0.2 - 0.2 B [T] Bz Bx - 0.4 - 0.4 0 20 40 60 80 0 20 40 60 80 A B C D RADIA ExamplesAPPLE-II PPM Undulator Circular/Elliptical Polarization (parallel displacement of 2 magnet arrays) Tilted Linear Polarization (anti-parallel displacement of 2 magnet arrays) Gap: 17 mm Bz Bx Bz Bx B [T] s [mm] s [mm] Peak Field vs Shift bw Magnet Arrays Magnet Dimensions: 40 mm x 20 mm Mr = 1.1 T Invented by Sh.Sasaki et. al.

11. RADIA ExamplesHybrid Undulator U20 (SOLEIL) Optimization by C.Benabderrahmane

12. RADIA / SRW ExamplesHybrid Wiggler 3T Hybrid Wiggler Magnetic design by J.Chavanne, ESRF Magnetic Field and Electron Trajectory Spectrum vs Phot. Energy at diff. obs. points Magnets are designed using RADIA SR computations are done using SRW Spectral Fux per Unit Surface vs Horizontal and Vertical Positions

13. RADIA ExamplesElectromagnet Elliptical Undulator HU640 (SOLEIL) Pure coil structure (no iron parts) Consumed electrical power: ~100 kW Vertical and Horizontal Magnetic Fields Magnetic design by O.Marcouille

14. RADIA Examples Electromagnet Elliptical Undulator HU256 (SOLEIL-BINP) The Structure (A.Dael - P.Vobly) Magnetic Fields at Max. Currents (Radia) Iz max= 180 A, Ix max= 250 A Calculated Spectra at Maximal Currents Specifications: Circular Polar.: 1 min < 10 eV Linear Hor. Polar.: 1 min < 10 eV Linear Vert. Polar.: 1 min < 20 eV Aperture: 0.7 mr x 0.7 mr

15. RADIA Examples: Fast-Switching Elliptical Undulator for XMCD Preliminary Design, derived from ESRF EMPHU Original design by J.Chavanne 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 model with reduced number of periods (coil layer changes are not taken into account) Magnetic Field “Roll-Off” On-Axis Magnetic Field

16. RADIA ExamplesAccelerator Magnets Edge Field (vs longitudinal position) Transverse Field A low excitation A: dipole B: quadrupole with integrated sextupole component simulations by L-J.Lindgren high excitation MAX-II Magnets B Discrepancy between Radia simulations and measurements: < 1% (peak field)

17. RADIA Examples Accelerator Magnets: SOLEIL Dipoles Central Field vs Longitudinal Position Edge Field vs Longitudinal Position B [T] B [T] y [mm] y [mm]

18. Radia Examples Comparison with a commercial FEM Code Chamfer Optimization of ESRF Quadrupole simulation by J.Chavanne FLUX3D 200 000 Tetrahedrons 200 MB of Memory 3 Hours of CPU RADIA 1500 Polyhedrons 200 MB of Memory 1 Hour of CPU

19. Radia Examples Radia vs commercial FEM Code FLUX3D  Hybrid Wiggler Simulation Comparison Case A: Solution for 1% accuracy in peak field Case B: Solution for 10 G-cm abs. accuracy in on-axis field integral * accuracy is high only in centers of 3D elements

20. Fitness 1 2 3 4 5 6 7 8 5 4 8 1 7 2 63 Genetic Algorithms 4 8 1 5 2 6 3 7 1 3 5 7 2 4 6 8 Radia Examples / Inverse Problems Sorting and Shimming Insertion Device Magnets 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 Evaluation: Magnetic Measurements Dataon Individual Magnets (/ Modules) & Partly Assembled Undulator Undulator Magnetic Field (/ Field Integrals) Characteristics / Fitness Terms Weights «Decoded» Undulator Structure Ordered Magnet Sequence(s) Electron Trajectory Straightness Radiation Phase Error Field Integral deviation from zero RADIA Integrated Multipoles Mathematical Model / Total Field Calc. Method . . . ID Magnet Sorting was pioneered by A.Cox and B.Youngman (1985) ID Magnet Shimming was pioneered at ESRF and ELETTRA (199x) Possible Variation Operators for Permutations (i.e. for Sorting): Mutation :-e.g. swap magnets at two randomly chosen positions-[5 4 8 1 7 2 63 ] [5 4 6 1 7 2 8 3 ] [ 1 2 3 4 5 6 7 8 ] [ 3 5 6 8 1 2 7 4 ] Crossover :-e.g. «order 1» - [ ? ? ?4 5 6 7? ] [ 8 1 24 5 6 73 ]

21. Shim Signatures of In-Vacuum Hybrid Undulator U20 RADIA Model for Central Part (A-, B- Modules and Poles) RADIA Model for Extremities (E- Modules) Variation of Field Integrals due to 25 m displacement of A-, B-, E-Modules and Poles Variation of On-Axis Field due to 25 m displacement of A-, B-, E-Modules and Poles

22. “Spectral” Shimming of In-Vacuum Hybrid Undulator U20 On-Axis Single-Electron Spectra Before and After Shimming(10 m from source) On-Axis Spectrum Taking into account E-Beam Emittance and Energy Spread Evolution of 11th Harmonic of Single-Electron Spectrum x = 3.7 nm; E/E = 10-3 RMS Radiation Phase Error after Shimming ~2.6

23. APPLE-II Undulator HU80-PLEIADES: Evolution of Electron Trajectory and Field Integrals (Min. Gap, Zero Phase) Horizontal Trajectory (Periodic Mode) Horizontal First Field Integral Vertical First Field Integral Horizontal Trajectory (Quasi-Periodic Mode) Q.-P. Mode realized by 11 mm displacement of some longitudinally-polarized magnets.

24. RADIA “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 (?)

25. Acknowledgements  J.-L. Laclare, J.-M. Filhol,D. RaouxO. Rudenko, A. Dael  All Users of RADIA and SRW

27. In-Vacuum Hybrid Undulator U20 (-SWING): GA-Based Module Sorting and Magic Finger Adjustment 97 full periods, 390 magnet modules Vertical Field Integrals Horizontal Field Integrals at 5.5 mm Gap at 10 mm Gap

28. HU80 Field Integrals at Minimal Gap (15.5 mm) Horizontal Vertical TEMPO FOC (after extra shimming) PLEIADES

29. APPLE-II Undulator HU80-Foc: Extra Shimming to Reduce Multipoles Field Integrals at Minimal Gap and Different Phases Vertical Before Extra Shimming After Extra Shimming Phase-Dependent Normal Quad Variation Horizontal Before Extra Shimming After Extra Shimming Phase-Dependent Skew Quad Variation HU80-Foc was originally supplied by ELETTRA. Extra Shimming of the HU80-Foc was performed at SOLEIL. The Extra Shimming was based on Genetic Optimization and it took into account Phase-Dependent Magn. Interaction Effects. Only ELETTRA “native” spare parts (mechanical shims and “magic fingers”) were used.