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Calculation of Eigenfields for the
European XFEL Cavities
Wolfgang Ackermann, Erion Gjonaj, Wolfgang F. O. Müller, Thomas Weiland
Institut Theorie Elektromagnetischer Felder, TU Darmstadt
Status MeetingDecember 21, 2010DESY, Hamburg
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 1
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Overview
▪ Task
- Calculation of fields for the European XFEL cavities in 3D
considering coupling ports as well as non-ideal geometries
- Coupling ports:
• Modeling of ports
• Include ports in the eigenvalue formulation
• Implementation for large scale applications
- Non-ideal geometries
• Support flexible geometry description in 3D
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 2
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Motivation
▪ Particle accelerators
- Linear accelerator at DESY, Hamburg
http://www.desy.de
Cavity 1.3 GHz
Cavity 3.9 GHz
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 3
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Computational Model
▪ Superconducting resonator
- Geometry
9-Cell Cavity Beam Tube
Upstream
Higher Order Mode
Coupler
Downstream Higher Order Mode Coupler
Input Coupler
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 4
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Computational Model
▪ Available grid structures
‚Staircase‘-grid partially filled cells tetrahedral mesh
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 5
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▪ Modeling using CST Studio Suite
- 3d.tet:NodeID, X, Y, Z
EdgeID, NodeID0, NodeID1
FaceID, EdgeID0, EdgeID1, EdgeID2
ElemID, FaceID0, FaceID1, FaceID2, FaceID3
ElemID, NodeID0, NodeID1, NodeID2, NodeID3
Object3D GroupID, #Elems <immediately followed by> ElemID List
Object2D GroupID, #Faces <immediately followed by> FaceID List
Object1D GroupID, #Edges <immediately followed by> EdgeID List
Object0D GroupID, #Nodes <immediately followed by> NodeID List
- bc3d.tetObj3D_ID, MediaCode
Obj2D_ID, BCCode
Obj1D_ID, BCCode
Computational Model
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 6
PEC, PMC and port
boundary conditions
can be extracted
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▪ Modeling using CST Studio Suite
- 3d.tet:NodeID, X, Y, Z
EdgeID, NodeID0, NodeID1
FaceID, EdgeID0, EdgeID1, EdgeID2
ElemID, FaceID0, FaceID1, FaceID2, FaceID3
ElemID, NodeID0, NodeID1, NodeID2, NodeID3
Object3D GroupID, #Elems <immediately followed by> ElemID List
Object2D GroupID, #Faces <immediately followed by> FaceID List
Object1D GroupID, #Edges <immediately followed by> EdgeID List
Object0D GroupID, #Nodes <immediately followed by> NodeID List
Computational Model
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 7
CST 3d.tet
modify point locations but maintain the topology
Modify 3d.tet
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Computational Model
▪ Modeling using CST Studio Suite
- 3d.tet:
linear element curvilinear element
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 8
only linear geometry transformation available
standard points
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Computational Model
▪ Modeling using CST Studio Suite
- 3d.tet:
linear element curvilinear element
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 9
insert additional control points (at the surface)
additional points
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Computational Model
▪ Modeling using CST Studio Suite
- 3d.slim:
linear element curvilinear element
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 10
available in CST but not yet used here… (ToDo)
control points
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Motivation
▪ Input coupler and coupler to extract unwanted modes
Beam Tube
Downstream Higher Order Mode Coupler
Coaxial Input Coupler
Coaxial Line
Antennas
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 11
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Motivation
▪ Input coupler and coupler to extract unwanted modes
Antennas
in linear scale
Beam TubeCoaxial Line
Downstream Higher Order Mode Coupler
Coaxial Input Coupler
f0 = 1.300 GHz
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 12
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Motivation
▪ Input coupler and coupler to extract unwanted modes
Antennas
in linear scale
Beam TubeCoaxial Line
Downstream Higher Order Mode Coupler
Coaxial Input Coupler
f0 = 1.709 GHz
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 13
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▪ Problem definition
- Accelerating field
Motivation
Determine the accelerating
-mode with high precision
9-Cell Cavity
including couplers
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 14
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Computational Model
▪ Problem formulation
- Fundamental equations
- Boundary conditions
Maxwell‘s equations
Material relations
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 15
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0 5 10 15 200
100
200
300
400
0 2 4 6 8 100
50
100
150
200
Motivation
▪ Wave propagation in the applied coaxial lines
- Main coupler
- HOM coupler
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 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 16
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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 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 17
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Computational Model
▪ Numerical formulation
- Function definition
Pär
Ingels
tröm
,
A N
ew
Se
t o
f H
(cu
rl)-
Co
nfo
rmin
g H
iera
rch
ica
l
Ba
sis
Fu
nctio
ns fo
r Te
tra
he
dra
l Me
sh
es,
IEE
E T
RA
NS
AC
TIO
NS
ON
MIC
RO
WA
VE
TH
EO
RY
AN
D T
EC
HN
IQU
ES
,
VO
L. 5
4, N
O.
1, JA
NU
AR
Y 2
00
6
FEM06: lowest order approximation
(edge elements, Nedelec)
scala
rvecto
r
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 18
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Computational Model
▪ Numerical formulation
- Function definition
Pär
Ingels
tröm
,
A N
ew
Se
t o
f H
(cu
rl)-
Co
nfo
rmin
g H
iera
rch
ica
l
Ba
sis
Fu
nctio
ns fo
r Te
tra
he
dra
l Me
sh
es,
IEE
E T
RA
NS
AC
TIO
NS
ON
MIC
RO
WA
VE
TH
EO
RY
AN
D T
EC
HN
IQU
ES
,
VO
L. 5
4, N
O.
1, JA
NU
AR
Y 2
00
6
scala
rvecto
r
FEM12: higher order approximation
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 19
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Computational Model
▪ Numerical formulation
- Function definition
Pär
Ingels
tröm
,
A N
ew
Se
t o
f H
(cu
rl)-
Co
nfo
rmin
g H
iera
rch
ica
l
Ba
sis
Fu
nctio
ns fo
r Te
tra
he
dra
l Me
sh
es,
IEE
E T
RA
NS
AC
TIO
NS
ON
MIC
RO
WA
VE
TH
EO
RY
AN
D T
EC
HN
IQU
ES
,
VO
L. 5
4, N
O.
1, JA
NU
AR
Y 2
00
6
scala
rvecto
r
FEM20: higher order approximation
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 20
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Computational Model
▪ Numerical formulation
- Implementation
contribution of
element-matrices
ready availabe
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 21
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Computational Model
▪ Eigenvalue formulation
- Fundamental equation
- Matrix properties
- Fundamental properties
Notation:
A - stiffness matrix
B - mass matrix
C - damping matrix
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 22
for proper chosen scalar and vector basis functions
orstatic dynamic
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Computational Model
▪ Fundamental properties
- Number of eigenvalues
- Orthogonality relation
Notation:
A - stiffness matrix
B - mass matrix
C - damping matrixMatrix B nonsingular:
• matrix polynomial is regular
• 2n finite eigenvalues
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 23
If the vectors and are no longer B-orthogonal:
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Computational Model
▪ Fundamental properties
- Orthogonality relation
- Scalar product
1)
2)
3)
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 24
currently not available
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Computational Model
▪ Eigenvalue formulation
- Fundamental equation
- Companion notation
Notation:
A - stiffness matrix
B - mass matrix
C - damping matrix
real symmetric
matrices
real asymmetric
matrices
A, B, C: real symmetric
conjugate complex eigenvalues
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 25
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Computational Model
▪ Eigenvalue solution
- Fundamental equation
- Subspace projection method
- Companion notation for the projected system
Notation:
A - stiffness matrix
B - mass matrix
C - damping matrix
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 26
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▪ Problem definition
- Geometry
Numerical Examples
TESLA 3rd harmonic
9-cell cavity
including couplers
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 27
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▪ Problem definition
- Geometry
- Task
Numerical Examples
Distribution
on 64 nodes
ParMeTiS, VTK and
CST - Studio Suite®
Search for the - mode field distribution
9-Cell Cavity
including couplers
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 28
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Numerical Examples
▪ Efficient solution of large problems
- Domain composition
parallel computing
domain #2
domain #1
cavity model
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 29
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▪ Fields along the axis of an accelerator cavity
Numerical Examples
9-Cell Cavity
including couplers
z
Ez
Longitudinal
electric field strength
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 30
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▪ Simulation results
Numerical Examples
0.5 0.4 0.3 0.2 0.1 0.0 0.10.015
0.010
0.005
0.000
0.005
0.010
0.015
Longitudinal coordinate, z m
Field
component,
Ex
Ez0
607 576 cells
reduced linear
full linear
reduced quadratic
hierarchical set of
basis functions
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 31
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0.5 0.4 0.3 0.2 0.1 0.0 0.10.015
0.010
0.005
0.000
0.005
0.010
0.015
Longitudinal coordinate, z m
Field
component,
Ex
Ez0
▪ Simulation results
Numerical Examples
2 064 944 cells
reduced linear
full linearreduced quadratic
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 32
hierarchical set of
basis functions
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Numerical Examples
▪ Transversal grid information
- Cut plane plots
Iris
Equator
unsymmetric mesh generation
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 33
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Numerical Examples
▪ Symmetric mesh generation
Coaxial coupler
Higher order mode coupler (HOM)
HOM
Beam tube
TESLA 9 cell cavity
Idea:
1) Meshing performed only
on ¼ of the model
2) Copy mesh to assemble
the full information
CST – Microwave Studio
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 34
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Numerical Examples
▪ Transversal grid information
- Cut plane plots
Iris
Equator
symmetric mesh generation
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 35
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Numerical Examples
▪ Simulation results
- Transverse mesh properties
arbitrary distribution
of tetrahedra
tetrahedra faces aligned
along coordinate faces
symmetric distribution
of tetrahedra
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 36
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0.5 0.4 0.3 0.2 0.1 0.0 0.1
0.010
0.008
0.006
0.004
0.002
0.000
Longitudinal coordinate, z m
Field
component,
Ex
Ez0
▪ Simulation results
Numerical Examples
3 282 467 cells
reduced quadratic set of basis functions
1 904 470 cells
652 742 cells
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 37
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▪ Simulation results
Numerical Examples
Symmetric mesh:
1 904 470 cells
11 780 962 DOFs
Symmetric mesh:
3 282 467 cells
20 370 322 DOFs
c0 Bxc0 By
c0 Bz
(no contribution)
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 38
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Numerical Examples
▪ Simulation results
Ex
z zFx
Fy
Ey
c0 By
c0 Bx
Contributions to the force for
a particle moving with speed
of light along the axis
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 39
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Computational Model
▪ Eigenvalue distribution
0f/MHz
f/MHz
target frequency
search direction
desired mode
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 40
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Motivation
▪ Superconducting Resonator
9-Cell Cavity Beam Tube
Upstream
Higher Order Mode
Coupler
Downstream Higher Order Mode Coupler
Input Coupler
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 41
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Numerical Examples
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 42
in linear scale
f 0 =
1.3
00 G
Hz
f 0 =
1.7
09 G
Hz
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Numerical Examples
December 21, 2010 | TU Darmstadt | Fachbereich 18 | Institut Theorie Elektromagnetischer Felder | Wolfgang Ackermann | 43
in linear scale
f 0 =
1.8
02 G
Hz
f 0 =
1.8
90 G
Hz