Calculation of Eigenmodes in Superconducting Cavities
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1 Calculation of Eigenmodes in Superconducting Cavities W. Ackermann, C. Liu, W.F.O. Müller, T. Weiland Institut für Theorie Elektromagnetischer Felder, Technische Universität Darmstadt Status Meeting December 17, 2012 DESY, Hamburg December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 1
2 Outline Motivation Computational model - Problem formulation in 3-D - Problem formulation in 2-D (boundary condition) Numerical examples GHz structure, single cavity GHz structure, string of four cavities (preliminary,without bellows) Summary / Outlook December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 2
3 Outline Motivation Computational model - Problem formulation in 3-D - Problem formulation in 2-D (boundary condition) Numerical examples GHz structure, single cavity GHz structure, string of four cavities (preliminary, without bellows) Summary / Outlook December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 3
4 Motivation Particle accelerators - FLASH at DESY, Hamburg RF Gun Diagnostics Accelerating Structures Collimator Undulators Laser Bunch Compressor Bunch Compressor 250 m TESLA 1.3 GHz TESLA 3.9 GHz December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 4
5 Motivation Linac: Cavities - Photograph - Numerical model upstream downstream CST Studio Suite Dezember 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 5
6 Motivation Superconducting resonator 9-cell cavity Input coupler Upstream higher order mode coupler Downstream higher order mode coupler December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 6
7 Motivation Superconducting resonator 9-cell cavity Input coupler Variation: Penetration depth Variation: Coupler orientation Upstream higher order mode coupler Downstream higher order mode coupler December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 7
8 Outline Motivation Computational model - Problem formulation in 3-D - Problem formulation in 2-D (boundary condition) Numerical examples GHz structure, single cavity GHz structure, string of four cavities (without bellows) Summary / Outlook December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 8
9 Computational Model Problem formulation - Time domain Efficient algorithms available (explicit method, coupled S-parameter) Field excitation to discriminate unknown mode polarizations - Frequency domain (driven problem) Efficient algorithms available (coupled S-parameter) Field excitation to discriminate unknown mode polarizations - Frequency domain (eigenmode formulation) Expensive algorithms Field distributions available including mode polarization Localized mode patterns with weak coupling naturally included December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 9
10 Computational Model Problem formulation - Local Ritz approach + boundary conditions continuous eigenvalue problem Galerkin vectorial function scalar coefficient global index number of DOFs discrete eigenvalue problem December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 10
11 Computational Model Eigenvalue formulation - Fundamental equation - Matrix properties Notation: A - stiffness matrix B - mass matrix C - damping matrix - Fundamental properties for proper chosen scalar and vector basis functions static or dynamic December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 11
12 Computational Model Fundamental properties - Number of eigenvalues Matrix B nonsingular: matrix polynomial is regular 2n finite eigenvalues Notation: A - stiffness matrix B - mass matrix C - damping matrix - Orthogonality relation If the vectors and are no longer B-orthogonal: December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 12
13 Computational Model Geometrical model 9-cell cavity Upstream HOM coupler Downstream HOM coupler Beam tube Input coupler High precision cavity simulations for closed structures December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 13
14 Outline Motivation Computational model - Problem formulation in 3-D - Problem formulation in 2-D (boundary condition) Numerical examples GHz structure, single cavity GHz structure, string of four cavities (preliminary, without bellows) Summary / Outlook December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 14
15 Computational Model Port boundary condition Port face, fundamental coupler December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 15
16 Computational Model Port boundary condition Port face, HOM coupler December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 16
17 Computational Model Problem formulation - Local Ritz approach Port face vectorial function scalar coefficient global index Mixed 2-D vector and scalar basis number of DOFs December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 17
18 Computational Model Problem formulation - Local Ritz approach + boundary conditions continuous eigenvalue problem, loss-free Galerkin vectorial function scalar coefficient global index number of DOFs discrete eigenvalue problem December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 18
19 Computational Model Wave propagation in the applied coaxial lines - Main coupler 60.0 mm 12.5 mm Dispersion relation 200 f 0 = 1.3 GHz 150 TEM TE 11 TE HOM coupler 16.0 mm 3.4 mm propagation damping f 0 = 1.3 GHz TEM 100 TE 11 TE December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 19
20 Computational Model Problem formulation - Determine propagation constant for a fixed frequency algebraic eigenvalue problem eigenvector and eigenvalue December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 20
21 Computational Model Problem formulation - Determine propagation constant for a fixed frequency algebraic eigenvalue problem eigenvector and eigenvalue Mode 1 Mode 2 Mode 3 Mode 4 December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 21
22 Computational Model Memory requirement for waveguide formulations - Ports with 2-D eigenmodes - Impedance boundary condition Port mode series expansion Projection of the port fields on the set of 3-D basis functions results in a dense matrix block Same population pattern as PMC (natural boundary condition) December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 22
23 Computational Model Memory requirement for waveguide formulations - Example 1: tetrahedrons - Example 2: tetrahedrons TESLA 1.3 GHz Refined port mesh area - General rule: NNZ = number of non-zero elements b = mean bandwidth of sparse system matrix h = mean grid step size December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 23
24 Computational Model Waveguide ports implementation options - Standard approach: Incorporate dense matrix blocks into sparse system matrix Port mode series expansion - Memory-efficient approach: Utilize that system matrix is dense but of low rank Dyadic product of modified port mode vectors December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 24
25 Outline Motivation Computational model - Problem formulation in 3-D - Problem formulation in 2-D (boundary condition) Numerical examples GHz structure, single cavity GHz structure, string of four cavities (preliminary, without bellows) Summary / Outlook December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 25
26 Numerical Examples Problem definition - Geometry TESLA 9-cell cavity Port boundary conditions PEC boundary condition - Task Port boundary conditions Search for the field distribution, resonance frequency and quality factor December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 26
27 Numerical Examples Computational model 9-cell cavity Input coupler Beam cube Distribute computational load on multiple processes Upstream HOM coupler December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 27
28 Numerical Examples Simulation results - Accelerating mode (monopole #9) - Higher-order mode (dipole #37) December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 28
29 Numerical Examples Simulation results Accelerating mode (monopole #9) Beam tube Higher-order mode (dipole #37) Beam tube Coaxial line Coaxial input coupler Coaxial input coupler HOM coupler HOM coupler December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 29
30 Imaginary Numerical Examples Controlling the Jacobi-Davidson eigenvalue solver - Evaluation in the complex frequency plane - Select best suited eigenvalues in circular region around user-specified complex target Monopole Dipole Real and imaginary part of the complex frequency 0.02 Quadrupole 0.01 Sextupole Real December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 30
31 Numerical Examples Quality factor versus frequency Calculations kindly performed by Cong Liu December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 31
32 Numerical Examples Field distribution of selected mode - First/second dipole passband (mode 17) E B E B December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 32
33 Outline Motivation Computational model - Problem formulation in 3-D - Problem formulation in 2-D (boundary condition) Numerical examples GHz structure, single cavity GHz structure, string of four cavities (preliminary, without bellows) Summary / Outlook December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 33
34 Numerical Examples (preliminary) 3.9 GHz structure (3 rd harmonic cavity) - String of four cavities Bellows omitted for the time being 4 main coupler, 8 HOM coupler and 2 beam pipes = 14 ports - Field distribution for selected modes tetrahedrons complex DOF December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 34
35 Summary / Outlook Summary: Request for precise modeling of electromagnetic fields within resonant structures including small geometric details - Geometric modeling with curved tetrahedral elements - Port boundary conditions with curved triangles - Memory-efficient implementation to evaluate the port fields now available Outlook: - Application to 1.3 GHz and 3.9 GHz structure and strings December 19, 2012 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Wolfgang Ackermann 35
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