Methods and Simulation Tools
for Cavity Design
Prof. Dr. Ursula van Rienen, Dr. Hans-Walter Glock
SRF09 Berlin - Dresden
Dresden 18.9.09
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
• Finite Integration Technique (FIT)
• Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
• Some generalities
• Some selected examples
2
U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
• Finite Integration Technique (FIT)
• Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
• Some generalities
• Some selected examples
3
U. van Rienen, H.-W. Glock
Superconducting accelerator cavities
www.kek.jp/intra-e/press/2005/image/ilc1.jpg
[Podl], IAP Frankfurt/M.
http://www.linearcollider.org/newsline/images/2008/
20080501_dc_1.jpg
... and some other types
Jefferson Lab /
http://irfu.cea.fr/Images/astImg/2407_2.jpg
http://www.physics.umd.edu/courses/Phys263/Kelly/cavity.jpg
=> Often chains of repeated structures, combined with flanges / coupling devices.
4
U. van Rienen, H.-W. Glock
What do you need to know?
I) Accelerating Mode:
• Do v part
( 2fres )
and Lcell match?
• How much energy does the particle gain?
ΔEkin = q part ∫
z= Lcav
z=0
Ez (z) ⋅ cos(2π fres z / v part + ϕ 0 ) dz
Jefferson Lab
• How much energy is stored in the cavity?
Wstored =
ε0
2
∫∫∫
Vcav
JG
2
μ0
E amplitude dV =
2
∫∫∫
JJG
2
H amplitude dV
Vcav
• How big is the power loss in the wall? And how big is the unloaded quality factor?
Ploss =
Rsurface
2
w
∫∫
cavity
JJG 2
H tan dA =
JJG 2
ωresWstored
1 ωres μ
⋅
=
H
Q
tan dA;
0
w
∫∫
Ploss
2
2σ cavity
• Where are the maxima of electric (field emission) and magnetic (quench) field
strength at the surface? Which values do they reach?
5
U. van Rienen, H.-W. Glock
What do you need to know?
II) Fundamental passband:
• All resonant frequencies! Which frequency spread
does the fundamental passband have? How close is
the next neighbouring mode to the accelerating
fmode/MHz
mode? Æ Cell-to-Cell coupling Æ filling time
• How strong is the beam interaction of all modes?
Æ Look at “R over Q”:
R ⎛ ΔEkin ⎞
=⎜
Q ⎝ q part ⎟⎠
2
(ω resWstored )
cell-to-cell phase advance ϕ
III) Higher Order Modes (HOM):
• same questions as for fundamental passband
• wake potential Æ kick factor
• special field profiles strongly confined far away
from couplers: "trapped modes"
6
U. van Rienen, H.-W. Glock
What do you need to know?
IV) Input and HOM-coupler/absorber:
• Qloaded of all beam-relevant modes
• Field distortions due to coupler?
C
t:
plo
AD
S
DE
Y
• Field distribution within the coupler
V) Multipacting
VI) Mechanical stability wrt. Lorentz Forces
7
U. van Rienen, H.-W. Glock
What do you need to know?
I) Accelerating mode
II) Fundamental passband
III) HOMs
IV) Input and HOM-coupler/absorber
V) Multipacting
VI) Mechanical stability wrt. Lorentz Forces
Jefferson Lab
Eigenmodes provide most of the information needed:
100% of I)
100% of II)
80% of III)
25% of IV)
25% of V)
50% of VI)
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U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
•
Mode Matching Technique
•
Finite Integration Technique (FIT)
•
Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
•
Some generalities
•
Some selected examples
9
U. van Rienen, H.-W. Glock
Need for Numerical Methods
Most practical electrodynamics problems cannot be
solved purely by means of analytical methods, see e.g.:
• radiation caused by a mobile phone near a human
head
• shielding of an electronic circuit by a slotted metallic
box
• mode computation in accelerator cavities, especially in
chains of cavities
In many of such cases, numerical methods can be
applied in an efficient way to come to a satisfactory
solution.
10
U. van Rienen, H.-W. Glock
Numerical Methods
Semi-analytical Methods
• Methods based on Integral Equations
• Method of Moments (MoM)
Discretization Methods
• Finite Difference Method (FD)
• Boundary Element Method (BEM)
• Finite Element Method (FEM)
• Finite Volume Method (FVM)
• Finite Integration Technique (FIT)
Possible Problems
• Need for geometrical simplifications Æ MoM
• Violation of continuity conditions Æ FD
• Unfavorable matrix structures Æ BEM
• Unphysical solutions Æ FEM if not mixed FEM
• Etc.
U. van Rienen,
Discretization (I) of Solution itself
Discretization error; 1D and 2D
Picture source:
http://www.integra.co.jp/eng/whitepapers/inspirer/inspirer.htm
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U. van Rienen, H.-W. Glock
Discretization (II) of Boundary of Solution Domain
In general: geometrical error
Structured „boundary-fitted“ grid
Picture source:
http://www.sri.com/poulter/crash/crown_victoria/crvic_figures/fig_cvic2.html
13
U. van Rienen, H.-W. Glock
Discretization (III) – Spatial Grid Types
Unstructured 2D grid:
Picture source:
http://www.uni-karlsruhe.de/RZ/Dienste/GVM/DIENSTE/CAE-ANWENDUNGEN/
FIDAP/erfahrung/node46.html
14
U. van Rienen, H.-W. Glock
Discretization (V) – Space and Time
space ⊗ time → grid space × grid time
R
3
⊗
R+
→
G
×
T
H = 4,96 A/m
H =0,085 A/m
Laptop with
WLAN card
f = 2, 4 GHz
Computed with CST MWS
U. van Rienen,
Grid Time Æ Example
Wave propagation in rectangular waveguide
Energy density of „wake fields“ in the TESLA cavity
16
U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
• Finite Integration Technique (FIT)
• Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
• Some generalities
• Some selected examples
17
U. van Rienen, H.-W. Glock
FIT Grid
18
U. van Rienen, H.-W. Glock
FIT Discretization of Induction Law
G G
∂ G G
∫∂A E ⋅ ds = − ∂t ∫∫A B ⋅ dA
ek
el
bn
ei
ej
∂
ei + e j − ek − el = − bn
∂t
^
=
∂ Ce = − b
∂t
⎛ ei ⎞
⎜ ⎟
⎜ . ⎟
⎛.⎞
. . .
⎛
⎞ ⎜ ej ⎟
∂ ⎜ ⎟
⎜
⎟⎜ ⎟
⎜1 . 1 . −1 . −1⎟ ⎜ . ⎟ = − ∂ t ⎜ bn ⎟
⎜.⎟
⎜
⎟ ⎜e ⎟
.
.
.
⎜ ⎟
⎝
⎠ ⎜ k ⎟
⎝N⎠
⎜.⎟
C
⎜ e ⎟
b
l ⎠
⎝N
e
U. van Rienen,
FIT Discretization of Gauss Law
B ⋅ dA = 0
w
∫∫
∂
^
=
V
bm
bi
bk
bl
bn
bj
bi + bj + bk − bl − bm − bn = 0
Sb = 0
⎛ . ⎞
⎜ ⎟
⎜ bi ⎟
⎜ ⎟
⎜ bj ⎟
. .
⎛
⎞ ⎜ ⎟
⎜
⎟ ⎜ bk ⎟
.
1
1
1
1
1
1
−
−
−
.
⎜
⎟ ⎜ ⎟ = 0
⎜
⎟ ⎜ bl ⎟
. .
⎝
⎠ ⎜ ⎟
⎜ bm ⎟
S
⎜ ⎟
⎜ bn ⎟
⎜ . ⎟
⎝N⎠
b
U. van Rienen,
Dual FIT-Grid
introduction of dual grid
dual grid
grid
d
h
dual FIT cell
d : electric flux
h : magnetic voltage
U. van Rienen,
FIT Discretization of Ampère‘s and Coulomb‘s Law
FIT equations on the dual grid
⎛ ∂D ⎞
v∂∫A H ⋅ ds = ∫∫A ⎝⎜ ∂t + J ⎠⎟ ⋅ dA
w
∫∫ D ⋅ dA = ∫∫∫ ρ ⋅ dV
∂V
∂ = d+ j
Ch
∂t
=q
Sd
d
V
h
U. van Rienen,
Maxwell‘s „Grid Equations“ (MGE)
∂B
curl E = −
∂t
∂D
+J
curl H =
∂t
div D = ρ
div B = 0
D=εE
B=μH
J = κ E + Je
FIT
T. Weiland, 1977, 1985
∂ Ce = − b
∂t
∂
h=
C
d+ j
∂t
S d = q
Sb = 0
d = Mε e
b = Mμ h
j = M κ e + je
U. van Rienen,
Shape Approximation on Cartesian Grids
Modelling
Curved Boundaries
Staircase FDTD/FDFD
(standard) :
poor convergence
FIT with diagonal filling:
better convergence
Weiland 1977
FIT on
non-orthogonal
grids
2nd order accuracy
FIT with
non-equidistant step size:
2nd order convergence
not always applicable
Weiland 1977
Conformal FIT / PBA:
2nd order convergence
always applicable
R.Schuhmann,
Krietenstein, Schuhmann, Thoma,
Weiland 1998
1998
24
U. van Rienen, H.-W. Glock
FIT on Triangular Grids
Dual grid
Grid
Æ URMEL-T
U. van Rienen 1983
U. van Rienen,
FIT on Non-Orthogonal Grids
Hilgner, Schuhmann, Weiland
TU Darmstadt 1998
TESLA 9 cell struct.
(Grid representation,
N=13x69x13=11.661)
Field in model of 1 cell
26
U. van Rienen, H.-W. Glock
Maxwell‘s „Grid Equations“ (MGE) Æ Curl Curl Equation
∂B
curl E = −
∂t
∂D
+J
curl H =
∂t
div D = ρ
div B = 0
D=εE
B=μH
J = κ E + Je
FIT
T. Weiland,
1977, 1985
∂ Ce = − b
∂t
∂
h=
C
d+ j
∂t
S d = q
Sb = 0
d = Mε e
b = Mμ h
j = M κ e + je
In full
analogy
to
analytical
derivation
of
wave
equation
U. van Rienen,
Curl-Curl-Eigenvalue Equation
jS ≡ 0
σ = 0, ε , µ real
(no current excitation )
(loss-free )
2
Mε CMµ−1C e = ω e
−1
eigenvalue problem
Ax = λ x
A CC = Mε−1 CM
µ −1C
ε −1 curl µ -1 curl E = ω 2E
28
U. van Rienen, H.-W. Glock
Curl-Curl-Eigenvalue Equation
1/2 e ' = Mε e
Use transformation
to derive an equation with
symmetric system matrix
−1/2 2 Mε CMµ−1 CMε e ' = ω e '
1/2
system matrix
−1/2
A ' = M1/2
A
M
ε
CC
ε
−1/2
= Mε−1/2CM
CM
µ −1
ε
(
−1/2
= Mε
)(
−1/2 M−1/2CM
−1/2
CM
µ
ε
µ
29
)
T
is symmetric
U. van Rienen, H.-W. Glock
Solution Space of Eigenvalue Problem
system matrix
−1/2
A ' = M1/2
A
M
ε
CC
ε
−1/2
= Mε−1/2CM
CM
µ −1
ε
(
−1/2
= Mε
)(
−1/2 M−1/2CM
−1/2
CM
µ
ε
µ
)
T
is symmetric
Eigenvalues of A‘
(and thus of ACC as well) are
1. real
Eigenfrequencies ωi are
2. non-negative
real and non-negative
because
as it should be for a loss-less
1. A‘ is symmetric
resonator
2. A‘ = matrix · (matrix)T
30
U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
• Finite Integration Technique (FIT)
• Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
• Some generalities
• Some selected examples
31
U. van Rienen, H.-W. Glock
Coupled S-Parameter Calculation
Coupled S-Parameter Calculation
H.-W. Glock, K.Rothemund, UvR 98
32
U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
• Finite Integration Technique (FIT)
• Coupled S-Parameter Simulation (CSC)
• Simulation Tools – not exhaustive
• Practical Examples
• Some generalities
• Some selected examples
33
U. van Rienen, H.-W. Glock
CST STUDIO Suite
CST MICROWAVE STUDIO®
-Transient Solver
- Eigenmode Solver
- Frequency Domain Solver
- Resonant: S-Parameters and Fields
- Integral Equations Solver
- Predecessors for eigenmode calculation: MAFIA,
URMEL, URMEL-T
http://www.cst.com/
34
U. van Rienen, H.-W. Glock
HFSS
http://www.ansoft.com/products/hf/hfss/
35
U. van Rienen, H.-W. Glock
ACE3P Suite
ACE3P - Advanced Computational Electromagnetic Simulation Suite
developed at SLAC by the group around Kwok Ko, Cho NG et al.
Æ See examples in next part of the talk
https://confluence.slac.stanford.edu/display/AdvComp/Omega3P
36
U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
•
Mode Matching Technique
•
Finite Integration Technique (FIT)
•
Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
•
Some generalities
•
Some selected examples
37
U. van Rienen, H.-W. Glock
Workflow of eigenmode computation
define geometry / import CAD-data
take care of your grid
define material properties
define symmetries and boundaries
check solver parameters
run solver (and maybe: wait)
frequency range ok ???
precision ok ???
(quasi) mode degeneration ???
post process: visualization
post process: derived quantities
38
U. van Rienen, H.-W. Glock
Workflow of eigenmode computation
define symmetries and boundaries
39
U. van Rienen, H.-W. Glock
Passband Fields with Etan = 0 in middle of the cavity
Make use of symmetry Æ compute only ½ of the cavity
Monopole-type E- and Hfield, common to all
modes and cells
Computed with
CST MWS
TM010- π/5 (pseudo 0)
40
TM010-3π/5
TM010-π (accelerating)
U. van Rienen, H.-W. Glock
Passband Fields with Htan = 0 in middle of the cavity
Use of
symmetry Æ
compute only
½ of the
cavity
Needs of
course 2
(shorter) runs!
Visualization
of full cavity
provided by
CST MWS
Even 1/8 part
might be
enough for
computation
TM010-2π/5
TM010-4π/5
41
Computed with
CST MWS
U. van Rienen, H.-W. Glock
Passband Fields altogether
fmode/MHz
TM010-π/5
700.40098 MHz
TM010-2π/5
702.21446 MHz
TM010-3π/5
704.52333 MHz
cell-to-cell phase advance ϕ
TM010-4π/5
706.34471 MHz
... which seems to obey some rule ?!
TM010-π
706.97466 MHz
Computed with
CST MWS
42
U. van Rienen, H.-W. Glock
Passband fields altogether
fmode/MHz
TM010-π/5
700.40098 MHz
TM010-2π/5
702.21446 MHz
TM010-3π/5
704.52333 MHz
cell-to-cell phase advance ϕ
TM010-4π/5
706.34471 MHz
In fact:
TM010-π
706.97466 MHz
Computed with
CST MWS
fmode
f0 + fπ ⎡ κ cc
⎤
≈
1−
cos (ϕ )⎥
⎢
2 ⎣
2
⎦
κcc : cell-to-cell coupling;
compare K. Saito´s talk
43
U. van Rienen, H.-W. Glock
So, what are passbands?
Cavities built by chains of identical cells show resonances in certain
frequency intervalls, called passbands, determined only by the shape of
the elementary cell.
The distribution of resonances in the band depends on the number of
cells in the chain:
fmode/MHz
12-cell-chain
resonances
5-cell-chain
resonances
Computed with
CST MWS
cell-to-cell phase advance ϕ
44
U. van Rienen, H.-W. Glock
Periodic Boundary Conditions
In fact, it is possible, to calculate the spectrum of an infinite chain by
discretizing a single cell (exploiting other symmetries as well) ...:
Δϕ
... and to preset the cell-to-cell phase advance by application of an
appropriate longitudinal boundary condition.
Computed with
CST MWS
45
U. van Rienen, H.-W. Glock
Periodic Boundary Conditions
This needs only one single run for each Δϕ, but gives eigenmode
frequencies of several passbands with a very small grid (here 19,000
mp), e.g:
fmode/MHz
cell-to-cell phase advance ϕ
Computed with
CST MWS
46
U. van Rienen, H.-W. Glock
What are Monopole-, Dipole-, Quadrupole-Modes?
Consider structures of axial circular symmetry. Then all fields belong
to classes with invariance to certain azimuthal rotations:
Quadrupole, ϕ =90°
Monopole, any ϕ
Oktupole, ϕ =45°
Dipole, ϕ =180°
Sextupole, ϕ =60°
Computed with
CST MWS
47
Dekapole, ϕ =36°
U. van Rienen, H.-W. Glock
Trapped Mode Analysis
H_18: 2037.1746 MHz
Computed with
CST MWS
E_18: 2039.66 MHz
Etan = 0
Etan = 0
H_19: 2040.81 MHz
Htan = 0
Htan = 0
Search for strongly confined field distributions by simulating same
structure with different waveguide terminations at beam pipe ends.
Compare spectra! Small frequency shifts indicate weak coupling.
E_17: 2037.1373 MHz
Remark: TE11-cut off of beam pipe at 1953 MHz
48
U. van Rienen, H.-W. Glock
Overview
• Introduction
• Methods in Computational Electromagnetics (CEM)
• Examples of CEM Methods:
•
Mode Matching Technique
•
Finite Integration Technique (FIT)
•
Coupled S-Parameter Simulation (CSC)
• Simulation Tools
• Practical Examples
•
Some generalities
•
Some selected examples
49
U. van Rienen, H.-W. Glock
TESLA 9-Cell Structure
Niobium; acceleration at 1.3 GHz
Ez (normiert)
Field on axis
z/m
Foto: DESY
Computed with
MAFIA
2D-simulation of upper half; only azimuthal symmetry exploited
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U. van Rienen, H.-W. Glock
Filling of TESLA
“Superstructure“- Semi-analytical calculation
9λ/2 3λ/2
Structure of CDR
Ez /(V/m)
2500
2000
1500
1000
500
0
0
1
2
3
„Superstructure“, J. Sekutowicz 1997
Ez /(kV/m)
t=T
4
λ/2
7λ/2
300
200
100
0
-100
-200
z/m -300
0
1
Ez /(MV/m)
t = 1000 T
2
3
4
t = 106 T
40
20
0
-20
z/m -40
z/m
0
1
2
3
4
rf period T = 0.7688517112 ns
H.-W. Glock; D. Hecht; U. van Rienen; M. Dohlus. Filling and Beam Loading in TESLA Superstructures. Proc.
Computed with MAFIA
of the 6th European Particle Accelerator Conference EPAC98, (1998): 1248-1250.
51
U. van Rienen, H.-W. Glock
Higher Order Modes in TESLA Structure
Dipole mode 28, f = 2. 574621 GHz
1.041⋅10 −17 VAs/m3
2.196 ⋅10 −12 VAs/m3
Dipole mode 29, f = 2. 584735 GHz
1.272 ⋅10 −19 VAs/m3
2.662 ⋅10 −12 VAs/m3
Dipole mode 30, f = 2. 585019 GHz
Computed with
MAFIA
9.109 ⋅10 −18 VAs/m3
2.578 ⋅10−12 VAs/m3
52
U. van Rienen, H.-W. Glock
CSC - 9-Cell Resonator with Couplers
Input coupler
C
HOM
coupler
o
-p l
D
A
E
t: D
SY
HOM
coupler
• Resonator without couplers: N ~ 29,000 (2D)
N ~ 12·106 (3D)
• Resonator with couplers:
N ~ 15·106 (3D)
⇒ N increases by ~ 500
• CSC: „Coupled S-Parameter Calculation“ allows for
combination of 2D- and 3D-simulations
K. Rothemund; H.-W. Glock; U. van Rienen. Eigenmode Calculation of Complex RF-Structures using SParameters. IEEE Transactions on Magnetics, Vol. 36, (2000): 1501-1503.
53
U. van Rienen, H.-W. Glock
CSC - Resonator Chain – Variation of Tube Length
Identical sections
HOM1
HOM2
⏐SHom1Hom2 ⏐/dB
Variation of
coupler position
1,000 frequency points
31 lengths
resulting S-matrix: 16 x 16
intern 84 x 84,
1h 12min, Pentium III, 1 GHz
Weak dependence on position
54
Computed with MAFIA, CST MWS and
our own Mathematica Code for CSC
U. van Rienen, H.-W. Glock
Effect of Changed Coupler Design*
R=-1
R=-1
L2,u
L1
L1 = 45.0 mm
L2,u = 101.4 mm
L2,d = 65.4 mm
L1
L2,d
S-parameters of TESLA cavity:
Modal analysis**
S-parameters of HOM- & HOM-input-coupler:
CST MicrowaveStudio®
CSC to determine S-parameters of various object combinations
*New concept: M. Dohlus, DESY;
** Modal coeff. computed by M. Dohlus
H.-W. Glock, K. Rothemund
55
U. van Rienen, H.-W. Glock
Comparison: HOM(original) – HOM(mirrowed)
HOM (down)
m
Computed with MAFIA, CST MWS and
our own Mathematica Code for CSC
ed
w
irro
HOM (up)
Q
Q
107
107
106
106
105
105
104
104
2,46 GHz
2,5 GHz
2,54 GHz
2,58 GHz
In cooperation with M. Dohlus, DESY
2,46 GHz
2,5 GHz
2,54 GHz
2,58 GHz
H.W. Glock; K. Rothemund; D. Hecht; U. van Rienen. S-Parameter-Based Computation in Complex
Accelerator Structures: Q-Values and Field Orientation of Dipole Modes. Proc. ICAP 2002
56
U. van Rienen, H.-W. Glock
Some of his slides …
Courtesy to Martin Dohlus
57
U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
58
U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
59
U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
60
U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
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U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
62
U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
63
U. van Rienen, H.-W. Glock
Courtesy of M. Dohlus, DESY – ICAP 2009
64
U. van Rienen, H.-W. Glock
Courtesy of Cho Ng, Lixin Ge
65
U. van Rienen, H.-W. Glock
CLIC HDX Simulation
HDX Surface E & B Fields
Hs/Ea=0.0040 A/m/(V/m)
E
H
Es/Ea=2.80
Es/Ea=2.26
Hs/Ea=0.0049
A/m/(V/m)
Hs/Ea=0.0044
A/m/(V/m)
Courtesy of Cho Ng, Lixin Ge
Fields enhanced around slot rounding
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Electron Trajectory & Impact Energy
Courtesy of Cho Ng, Lixin Ge
emitted
here
Particles emitted from one of the irises
At 85 MV/m gradient, energy of
dark current electrons can reach
~0.4 MeV on impact
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At the end ....
... some other "cavity" type. Hope, you feel better.
Wilhelm Busch: Max und Moritz, sometimes in the 19th century. Widely published
http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Max_und_Moritz_(Busch)_026.png/800px-Max_und_Moritz_(Busch)_026.png
These are greetings of my co-author Hans-Walter Glock ….
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