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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
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 2
Outline
▪ Motivation
▪ Computational model
- Problem formulation in 3-D
- Problem formulation in 2-D (boundary condition)
▪ Numerical examples
- 1.3 GHz structure, single cavity
- 3.9 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
Outline
▪ Motivation
▪ Computational model
- Problem formulation in 3-D
- Problem formulation in 2-D (boundary condition)
▪ Numerical examples
- 1.3 GHz structure, single cavity
- 3.9 GHz structure, string of four cavities (preliminary, without bellows)
▪ Summary / Outlook
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 4
Motivation
▪Particle accelerators
- FLASH at DESY, Hamburg
http://www.desy.de
TESLA 1.3 GHz
TESLA 3.9 GHz
RF Gun
Laser
Bunch
Compressor
Bunch
Compressor
Diagnostics Accelerating Structures Collimator Undulators
250 m
19. Dezember 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 5
Motivation
▪ Linac: Cavities
- Photograph
- Numerical model http://newsline.linearcollider.org
CST Studio Suite 2012
upstream downstream
Motivation
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 6
▪Superconducting resonator 9-cell cavity
Downstream
higher order mode coupler
Input
coupler
Upstream
higher order mode coupler
Motivation
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 7
▪Superconducting resonator 9-cell cavity
Downstream
higher order mode coupler
Input
coupler
Upstream
higher order mode coupler
Variation:
Penetration depth
Variation:
Coupler orientation
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 8
Outline
▪ Motivation
▪ Computational model
- Problem formulation in 3-D
- Problem formulation in 2-D (boundary condition)
▪ Numerical examples
- 1.3 GHz structure, single cavity
- 3.9 GHz structure, string of four cavities (without bellows)
▪ Summary / Outlook
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 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 | 10
Computational Model
▪Problem formulation
- Local Ritz approach
continuous eigenvalue problem
+ boundary conditions
vectorial function
global index
number of DOFs
scalar coefficient
discrete eigenvalue problem
Galerkin
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 11
Computational Model
▪Eigenvalue formulation
- Fundamental equation
- Matrix properties
- Fundamental properties
Notation:
A - stiffness matrix
B - mass matrix
C - damping matrix
for proper chosen scalar and vector basis functions
or static dynamic
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 12
Computational Model
▪Fundamental properties
- Number of eigenvalues
- Orthogonality relation
Notation:
A - stiffness matrix
B - mass matrix
C - damping matrix Matrix B nonsingular:
• matrix polynomial is regular
• 2n finite eigenvalues
If the vectors and are no longer B-orthogonal:
Computational Model
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 13
▪Geometrical model
9-cell cavity
Beam
tube
Downstream
HOM
coupler
Input coupler
Upstream
HOM
coupler
High precision
cavity simulations
for closed structures
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 14
Outline
▪ Motivation
▪ Computational model
- Problem formulation in 3-D
- Problem formulation in 2-D (boundary condition)
▪ Numerical examples
- 1.3 GHz structure, single cavity
- 3.9 GHz structure, string of four cavities (preliminary, without bellows)
▪ Summary / Outlook
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 15
Computational Model
▪Port boundary condition
Port face, fundamental coupler
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 16
Computational Model
▪Port boundary condition
Port face, HOM coupler
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 17
Computational Model
▪Problem formulation
- Local Ritz approach
vectorial function
global index
number of DOFs
scalar coefficient
Port face
Mixed 2-D vector and scalar basis
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 18
Computational Model
▪Problem formulation
- Local Ritz approach
continuous eigenvalue problem, loss-free
+ boundary conditions
vectorial function
global index
number of DOFs
scalar coefficient
discrete eigenvalue problem
Galerkin
0 5 10 15 200
100
200
300
400
0 2 4 6 8 100
50
100
150
200
Computational Model
▪Wave propagation in the applied coaxial lines
- Main coupler
- HOM coupler
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 19
12.5 mm
60.0 mm
3.4 mm
16.0 mm
TEM
TE11 TE21
TEM
TE11 TE21
f0 = 1.3 GHz
f0 = 1.3 GHz
Dispersion relation
propagation
damping
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 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 | 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 …
Computational Model
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 22
▪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 | 23
Computational Model
▪Memory requirement for waveguide formulations
- Example 1: 1.119.219 tetrahedrons
- Example 2: 1.218.296 tetrahedrons
- General rule:
b = mean bandwidth of sparse system matrix
h = mean grid step size
TESLA 1.3 GHz
Refined port mesh area
NNZ = number of non-zero elements
Computational Model
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 24
▪Waveguide ports implementation options
- Standard approach:
Incorporate dense matrix blocks into sparse system matrix
- Memory-efficient approach:
Utilize that system matrix is dense but of low rank
Port mode series expansion
Dyadic product of modified port mode vectors
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 25
Outline
▪ Motivation
▪ Computational model
- Problem formulation in 3-D
- Problem formulation in 2-D (boundary condition)
▪ Numerical examples
- 1.3 GHz structure, single cavity
- 3.9 GHz structure, string of four cavities (preliminary, without bellows)
▪ Summary / Outlook
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 26
▪Problem definition
- Geometry
- Task
Numerical Examples
TESLA 9-cell cavity
PEC
boundary condition
Search for the field distribution, resonance frequency and quality factor
Port
boundary
conditions
Port boundary conditions
Numerical Examples
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 27
▪Computational model
9-cell cavity
Beam
cube
Input
coupler
Upstream HOM coupler
Distribute
computational load
on multiple processes
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 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 | 29
Numerical Examples
▪Simulation results
Accelerating mode
(monopole #9)
Higher-order mode
(dipole #37)
Beam tube
HOM coupler
Coaxial
input coupler
Coaxial line
Beam tube
HOM coupler
Coaxial
input coupler
Numerical Examples
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 30
100
200
500100020005000
Monopole
Dipole
Quadrupole
Sextupole
2.46 2.48 2.50 2.52 2.54 2.56 2.58 2.60
0.00
0.01
0.02
0.03
0.04
Real
Imag
inary
▪Controlling the Jacobi-Davidson eigenvalue solver
- Evaluation in the complex frequency plane
- Select best suited eigenvalues in circular region around
user-specified complex target
Real and imaginary part of the complex frequency
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 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 | 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 | 33
Outline
▪ Motivation
▪ Computational model
- Problem formulation in 3-D
- Problem formulation in 2-D (boundary condition)
▪ Numerical examples
- 1.3 GHz structure, single cavity
- 3.9 GHz structure, string of four cavities (preliminary, without bellows)
▪ Summary / Outlook
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 34
Numerical Examples (preliminary)
▪3.9 GHz structure (3rd harmonic cavity)
- String of four cavities
- Field distribution for selected modes
4 main coupler, 8 HOM coupler and 2 beam pipes = 14 ports
Bellows omitted for the time being
1.040.212 tetrahedrons
6.349.532 complex DOF
December 19, 2012 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 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