11 th International Computational Accelerator Physics Conference (ICAP) 2012

Transcription

1 Eigenmode Computation For Ferrite-Loaded Cavity Resonators Klaus Klopfer*, Wolfgang Ackermann, Thomas Weiland Institut für Theorie Elektromagnetischer Felder, TU Darmstadt 11 th International Computational Accelerator Physics Conference (ICAP) 2012 = THACC2 *Supported by GSI, Darmstadt August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 1 / 20

2 Contents Motivation Computational Model Fundamental Relations Implementation Parallel Computing Numerical Examples Biased Cylinder Resonator Biased Cavity With Ferrite Ring Cores Summary August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 2 / 20

3 Motivation GSI Helmholtzzentrum für Schwerionenforschung SIS 18 Heavy Ion Synchrotron UNILAC Linear Accelerator ESR Experiment Storage Ring August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 3 / 20 [gsi.de]

4 Motivation GSI SIS18 Ferrite Cavity Main benefits of ferrite cavities: reduction of wavelength more compact cavity modification of resonance frequency in a wide range SIS 18 cavity: 0.6 MHz 5 MHz August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 4 / 20

5 Motivation GSI SIS18 Ferrite Cavity: Main components = ferrite ring cores bias current winding beam pipe y x z RF winding ceramic gap copper (cooling) August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 5 / 20

6 Computational Model: Fundamental Relations Assumptions: Hd Hbias, effect of hysteresis negligible ( ) B(t) = µ 0 µ biashbias + µ 0 µ dhd (t) = µ 0 µ biashbias + µ 0 µ d Re Hd e iωt Linearization at working point B B bias P Hd 0 H bias H August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 6 / 20

7 Computational Model: Fundamental Relations Assumptions: Hd Hbias, effect of hysteresis negligible ( ) B(t) = µ 0 µ biashbias + µ 0 µ dhd (t) = µ 0 µ biashbias + µ 0 µ d Re Hd e iωt Linearization at working point B bias B P Modification of bias current Modification of differential permeability Adjustment of eigenfrequency 0 H bias H August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 6 / 20

8 Computational Model: Fundamental Relations Assumptions: Hd Hbias, effect of hysteresis negligible ( ) B(t) = µ 0 µ biashbias + µ 0 µ dhd (t) = µ 0 µ biashbias + µ 0 µ d Re Hd e iωt Linearization at working point B bias B P 0 H bias H Eigensolutions determined by: ( ɛ 1 µ 1 0 µ 1 ) d E = ω 2 E Boundary condition: n E = 0 on cavity boundary August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 6 / 20

9 Computational Model: Fundamental Relations [D. Polder, Phil. Mag., 40 (1949)] Properties of the differential permeability tensor µ d : Fully occupied (3 3)-tensor, for H = Hbias e z reduces to the Polder tensor µ 1 iµ 2 0 ( ) µ d = iµ 2 µ 1 0 with µ 1,2 = µ Hbias 1,2, ω If magnetic losses are included: Im(µ 1,2 ) 0 Non-Hermitian August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 7 / 20

10 Computational Model: Implementation ( ɛ 1 µ 1 0 µ 1 ) d E = ω 2 E August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 8 / 20

11 Computational Model: Implementation ( ɛ 1 µ 1 0 µ 1 ) d E = ω 2 E Discretization by Finite Integration Technique (FIT): Mɛ 1 CM 1 d,µ C e = ω 2 e August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 8 / 20

12 Computational Model: Implementation ( ɛ 1 µ 1 0 µ 1 ) d E = ω 2 E Discretization by Finite Integration Technique (FIT): Mɛ 1 CM 1 d,µ C e = ω 2 e permeability tensor M d,µ : non-diagonal dependend on Hbias and ω August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 8 / 20

13 Computational Model: Implementation ( ɛ 1 µ 1 0 µ 1 ) d E = ω 2 E Discretization by Finite Integration Technique (FIT): Mɛ 1 CM 1 d,µ C e = ω 2 e permeability tensor M d,µ : non-diagonal dependend on Hbias and ω if magnetic losses included: requirements on eigensolver: nonlinear non-hermitian August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 8 / 20

14 Computational Model: Implementation Concept Magnetostatic solver M 1 d,µ Eigensolver calculation of bias magnetic field determination of permeability matrix M 1 d,µ calculation of eigenmodes General requirements: support of nonlinear material support of lossy material parallel computation with distributed memory (scalability) August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 9 / 20

15 Computational Model: Implementation Magnetostatic Solver C / C++ (PETSc ( * ) ) M 1 d,µ Eigensolver ( * ) Portable, Extensible Toolkit for Scientific Computation August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 10 / 20

16 Computational Model: Implementation Magnetostatic Solver C / C++ (PETSc ( * ) ) M 1 d,µ Eigensolver M µ 1 M ɛ 1 H, B E, D CAD tool Modeling and Meshing Visualization ( * ) Portable, Extensible Toolkit for Scientific Computation August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 10 / 20

17 Computational Model: Implementation Magnetostatic Solver C / C++ (PETSc) M 1 d,µ Eigensolver H i -algorithm Helmholtz decomposition H = Hi + Hh with Hi = J and Hh = ϕ Solution of nonlinear equation: successive substitution or Newton method August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 11 / 20

18 Computational Model: Implementation Magnetostatic Solver C / C++ (PETSc) M 1 d,µ Eigensolver H i -algorithm Helmholtz decomposition H = Hi + Hh with Hi = J and Hh = ϕ Solution of nonlinear equation: successive substitution or Newton method Jacobi-Davidson algorithm harmonic Ritz-values for computation of interior eigenvalues Solution of nonlinear problem: successive substitution August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 11 / 20

19 Contents Motivation Computational Model Fundamental Relations Implementation Parallel Computing Numerical Examples Biased Cylinder Resonator Biased Cavity With Ferrite Ring Cores Summary August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 12 / 20

20 Parallel Computing Aim: Efficient distributed computing Structure of system matrix standard arrangement August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 12 / 20

21 Parallel Computing Aim: Efficient distributed computing Structure of system matrix standard arrangement August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 12 / 20

22 Parallel Computing Aim: Efficient distributed computing Structure of system matrix standard arrangement August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 12 / 20

23 Parallel Computing Aim: Efficient distributed computing Structure of system matrix standard arrangement re-arrangement of DOFs higher computation to communication ratio August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 12 / 20

24 Parallel Computing Aim: Efficient distributed computing Structure of system matrix standard arrangement re-arrangement of DOFs remove pseudo DOFs higher computation to communication ratio better load balancing August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 12 / 20

25 Contents Motivation Computational Model Fundamental Relations Implementation Parallel Computing Numerical Examples Biased Cylinder Resonator Biased Cavity With Ferrite Ring Cores Summary August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 13 / 20

26 Numerical Examples Biased Cylinder Resonator test model: ( Hext = 2750 A ; µr = 7) m lossless, ferrite-filled cylindrical cavity resonator longitudinally biased by homogeneous magnetic field semi-analytical solution available [Chinn, Epp and Wilkins] Hext z L = 2 m R = 1m y x log 10` ω ω 0 /ω O( 1 ) 1 st 2 nd 3 rd 4 th degrees of freedom / 10 6 August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 13 / 20

27 Numerical Examples Biased Cylinder Resonator test model: lossless, ferrite-filled cylindrical cavity resonator longitudinally biased by homogeneous magnetic field y H ext L = 2 m R = 1m y x z Hext e z : H ext µ 1 iµ 2 0 µ x d = iµ 2 µ z August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 14 / 20

28 Numerical Examples Biased Cylinder Resonator test model: lossless, ferrite-filled cylindrical cavity resonator longitudinally biased by homogeneous magnetic field y H ext L = 2 m R = 1m y x z H ext e z : H ext α β x µ x,x µ x,y µ x,z µ d = µ y,x µ y,y µ y,z µ z,x µ z,y µ z,z z August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 14 / 20

29 Numerical Examples Biased Cylinder Resonator test of construction of permeability tensor for different orientations of cylinder axis to coordinate axes example shown for α = 45 and cos β = 2/3 y log 10` ω ω 0 /ω O( 1 ) 1 st 2 nd 3 rd 4 th z α β x degrees of freedom / 10 6 August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 15 / 20

30 Numerical Examples Biased Cavity With Ferrite Ring Cores Model description ferrite ring cores beam pipe Ferrite material is characterized by B(H) = µ tanh ( H 10 2 m A ) A m + µ 0H and ɛ r = 1 August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 16 / 20

31 Numerical Examples Biased Cavity With Ferrite Ring Cores Model description ferrite ring cores bias current winding beam pipe Ferrite material is characterized by B(H) = µ tanh ( H 10 2 m A ) A m + µ 0H and ɛ r = 1 bias magnetic field excited by current winding (2 ka) August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 16 / 20

32 Numerical Examples Biased Cavity With Ferrite Ring Cores Results of fully nonlinear computation Spectrum of the lowest nine eigenmodes frequency / MHz degrees of freedom / 10 6 # eigenmode 8 th and 9 th 5 th, 6 th and 7 th 3 rd and 4 th 2 nd 1 st August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 17 / 20

33 Numerical Examples Biased Cavity With Ferrite Ring Cores Results of fully nonlinear computation Spectrum of the lowest nine eigenmodes frequency / MHz degrees of freedom / 10 6 # eigenmode 8 th and 9 th 5 th, 6 th and 7 th 3 rd and 4 th 2 nd 1 st August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 17 / 20

34 Numerical Examples Biased Cavity With Ferrite Ring Cores Results of fully nonlinear computation # eigenmode 2 nd frequency / MHz degrees of freedom / st August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 18 / 20

35 Numerical Examples Biased Cavity With Ferrite Ring Cores Comparison nonlinear linear computation Linear approach 1. Compute lowest eigenfrequency ω 1,NL Nonlinear solver ω 1,NL 2. Set µ r homogeneously such that ω 1,Lin ω 1,NL Linear solver August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 19 / 20

36 Numerical Examples Biased Cavity With Ferrite Ring Cores Comparison nonlinear linear computation Linear approach Nonlinear solver ω 1,NL Linear solver /ω 1,NL (%) `ω1,lin ω1,nl # eigenmode August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 19 / 20

37 Contents Motivation Computational Model Fundamental Relations Implementation Parallel Computing Numerical Examples Biased Cylinder Resonator Biased Cavity With Ferrite Ring Cores Summary August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 20 / 20

38 Summary Goal: calculation of eigenvectors for biased ferrite cavities 1. Magnetostatic solver (nonlinear material): permeability tensor µ d Magnetostatic solver µ 1 d 2. Eigensolver: nonlinear complex eigenvalue problem Eigensolver August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 20 / 20

39 Summary Goal: calculation of eigenvectors for biased ferrite cavities 1. Magnetostatic solver (nonlinear material): permeability tensor µ d Magnetostatic solver µ 1 d 2. Eigensolver: nonlinear complex eigenvalue problem Eigensolver Status: Functionality of fully nonlinear solver for Hermitian problems demonstrated August 23, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Klaus Klopfer 20 / 20