analysis and simulation of magneto-thermal dynamics in
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This work is supported by:
Graduate School CE within the Centre for Computational Engineering at Technische Universität Darmstadt.
The Gentner program of the German Federal Ministry of Education and Research (grant no. 05E12CHA).
Lorenzo Bortot
Analysis and Simulation of Magneto-Thermal Dynamics
in High-Temperature Superconducting Magnets
19th Gentner Day, 28 April 2021, CERN
https://indico.cern.ch/event/1002599/
Supervisors:
Prof. Dr. rer. nat. Sebastian Schöps (TU Darmstadt)
Dr. Matthias Mentink (CERN)
Acknowledgments
I.C. Garcia, H. de Gersem
A. Verweij, J. van Nugteren, C. Petrone, G. Deferne, T.
Koettig, J.C. Perez, F.O. Pincot, G. de Rijk, S. Russenschuck,
G. Kirby, M. Pentella, M. Maciejewski
B. Auchmann
2
Contents from research activity over the past three years.
The Gentner program of the German
Federal Ministry of Education and Research
(grant no. 05E12CHA).
Outline
I. Dynamic Phenomena in High-Temperature Superconductors (HTS)
II. Coupled Field Formulation
III. Numerical Analysis
IV. Experimental Work
V. Summary and Outlook
3
HIGH TEMPERATURE SUPERCONDUCTORS
Copper oxides (CuO2) doped with rare earths (La, Bi-Sr-Ca, Y-Ga-Ba etc.)
Higher critical temperature and upper critical field with respect to the traditional
low-temperature superconductors (LTS), such as Nb-Ti or Nb3Sn
4
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40
(kA
mm
-2)
Temperature (K)
J(T) @ 5T
Nb-TiNb3SnReBCO
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40
(kA
mm
-2)
Magnetic flux density (T)
J(B) @ 1.9K
Nb-Ti
Nb3Sn
ReBCO
High-thermal load applications High-field applications
SCREENING CURRENTS
HTS tape in a time-dependent magnetic field 𝜕t𝐁:
𝜕t𝐁 Screening currents 𝐉screenρ → 0 Persistent magnetization 𝐁screen
Large filament size (5-12 mm) large 𝐁screen
Magnetic field quality and thermal behavior, as principal Joule loss contribution
Inhomogeneous current density distribution Detrimental impact on field quality
5
■ Copper
■ ReBCO
■ Substrate
~mm
~μm
ρ → 0
𝐉screen
𝜕t𝐁 𝐁screen
QUENCH
Local transition from superconducting to normal conducting state
Energy dissipated in the resistive zone
Potentially irreversible effects for high energy-density devices (accelerator magnets)!
HTS characteristics:
• low heat diffusivity
• low Ԧ𝑣quench, small resistive zone, difficult to detect
• high Thot−spot, potential damage in short time (tens of ms)
6
HTS tape
T
x
Thot−spot
Ԧ𝑣quench
resistive zone
DYNAMIC EFFECTS IN ACCELERATORS
Circuit (e.g. LHC mains)
7 km long, 154 Magnets
Topological: Switches, thyristors, diodes
Electrical: Propagative phenomena,
dynamic impedances (magnets) ,EM
crosstalk
• …
Magnets (e.g. MB magnet)
~102 coil turns, ~104 strands
Magnetic: magnetization, coupling currents, EM crosstalk…
Thermal: magnetization, coupling losses, quenches, coolant…
Mechanic: Lorentz forces, thermal strain…
7
Field-Circuit Coupling of multi-physics, multi-rate and multi-scale Systems!
FORMULATION
Magnet domain decomposition
ΩH = ΩH,s ∪ ΩH,c source region (coils):
• ΩH,s superconductors (𝜎 → +∞)
• ΩH,c normal conductors
ΩA = ΩA,c ∪ ΩA,i passive region (Iron, air):
• ΩA,c normal conductors
• ΩA,i insulators (𝜌 → +∞)
Field equations
8
𝛻 × ρ𝛻 × 𝐇 + μ𝜕t𝐇 + 𝛻 × 𝛘 us = 0 𝐇 formulation in ΩH
𝛻 × ν 𝛻 × 𝐀⋆ + σ𝜕t𝐀⋆ = 0 𝐀⋆ formulation in ΩA
ρmCp𝜕tT − 𝛻 ⋅ k𝛻T − 𝐉 ⋅ ρ𝐉 = 0 Heat balance equation T in Ω
ΩH𝛘 ⋅ 𝛻 × 𝐇dΩ = is Current constraint
𝛘 = −𝛻ξ, ξ ∶ 𝛻 ⋅ σ𝛻ξ = 0 Voltage distribution function
+
ΩA,c
𝐧
Ω ∈ ℝ3
isus−
ΩH,c
𝐧
ΓHA
ΓE
ΓJ
ΩH,s
ΓA
ΩA,i
SEMI-DISCRETE PROBLEM
Semi-discrete representation for numerical implementation (FE method)
Finite material properties, 𝜌 in (super)conductors, 𝜎 in insulators Solver stability
Electric ports ΓJ and ΓE as connections with the external circuit
𝐮𝐬 = 𝑓(𝐢𝐬) magnet as one-port component
9
Kν+Mσd
dt−𝐐 𝟎 𝟎
𝐐𝐓 K𝝆+M𝝁d
dt−𝚾 𝟎
𝟎 𝚾𝐓 𝟎 𝟎
𝟎 𝟎 𝟎 Kκ+Mρd
dt
a
h
us
t
𝟎
𝟎
is
q ⋅
=
𝐀⋆ form
𝐇 form
Field coupling
Heat Balance
Circuit coupling
⋅
𝐮𝐬 ≈ 𝐑𝐢𝐬 t + 𝐋d
dt𝐢𝐬 t
COSIMULATION FRAMEWORK
10
Middle Layer
• Communication bus,
common interface
• Standard set of rules for data
exchange
Bottom Layer
• Interpreter (framework vs.
simulation tools)
• Tool-oriented (single adapter
for multi-models)
Top Layer
• Hierarchical co-simulation
algorithm
• Waveform relaxation, Gauss-
Seidel scheme
Lean, Modular, Expandable
Waveform Relaxation, Gauss Siedel
Communication Bus
Tool Adapter𝑗
API𝑗
Tool𝑗CoupledProblem
Tool Adapter𝑘
API𝑘
Tool𝑘
Hierarchical Co-Simulation
User Input Output
Model1
Model𝑛
Model𝑚
(*) https://espace.cern.ch/steam
(*)
ANALYSIS: FIELD QUALITY (1/3)
11
2D FEM model
Iron
Wing deckCentral deck
Feather-M2 HTS insert dipole magnet
Main features:
• Aligned coil concept
• Coil made of two central and wing decks
• Roebel cable, fully transposed tapes
2D model:
• Layer jumps 4-quadrant
• 48 turns, 720 tapes
Cable cross section
Photos courtesy of J. Van Nugteren
ANALYSIS: FIELD QUALITY (2/3)
Computational time
0.5 h (*) for about 120k DoF
Current density plots: same source current, different time steps
12
Normalized current density in the coil (first quadrant)
a) b) c)
(*) CPU: Intel Core i7-3770 @ 3.40GHz. RAM: 32 Gb. OS: Win 10
ANALYSIS: FIELD QUALITY (3/3)
Magnetic field quality
Multipole expansion series as function of current, for different temperatures
13
Very good agreement with measurements
HTS solenoid protected by quench heaters
Electro-magneto-thermal coupling between the solenoid and the quench heaters!
Solenoid supply Heaters supply
ANALYSIS: QUENCH (1/3)
14
(*) Material properties available at https://gitlab.cern.ch/steam/steam-material-library
(**) Circuit parameters in appendix
Iron core
Holder
Coil
Insulation
Heater strips
Rendering of the HTS solenoid (*) Electrical layout (**)
0
20
40
60
80
100
0
500
1000
1500
0 0.05 0.1 0.15I_
str
ip(A
)
I_co
il(A
)
time (s)
Coil cosim Coil mono Heater
ANALYSIS: QUENCH (2/3)
Circuital currents Peak temperature
15
Current decay in the coil (left) and current
discharge in the heater strips (right)
Induced quench
100
200
300
400
0 5 10 15 20
T_hots
pot(K
)
Quench detection time (ms)
Max. Temperature
Max. detection time
Adiabatic hotspot temperature in the
coil, as a function of the quench
detection time
≈ 15 msTime windows
ANALYSIS: QUENCH (3/3)
Temperature in the coil cross section
16
100 ms 150 ms50 ms20 ms 30 ms10 ms
Temperature distribution in the superconducting coil, as a function of time
Tem
pera
ture
(K)
Quench Heat diffusionHeater strips
EXPERIMENT (1/3)
Working principle
HTS tape geometry:
• Thin superconducting layer,≈ 1 μm
• High aspect ratio, ≈ 1 × 103
Thin shell approximation
HALO
Harmonics-Absorbing Layered Object
Key features:
1. Magnetic field lines shaped by
persistent screening currents
2. Cancellation only of undesired field
components (static and dynamic)
3. Passive and self-regulating
17
HTS ReBCO
σ → +∞
𝜕t𝐁𝜕t𝐁∥
𝜕t𝐁⊥
𝐁screen
𝜕t𝐁⊥ canceled
𝜕t𝐁∥ unchanged
[T]
Quasi-perfect electric wallSelective field-cancellation!
EXPERIMENT (2/3)
Proof of concept design Experimental setup
18
[T]
2
3
1
4 4
5
1. Iron yoke
2. Coil
3. Iron bar (field error)
4. HTS screen
5. Rotating probe
3
77 K
296 K
Experimental setup at cold
HTS screen Screen holder
EXPERIMENT (3/3)
Experimental and numerical result
Magnetic field quality (total harmonic
distortion) in the magnet aperture, without
and with HALO
19
0
10
20
30
40
50
60
70
1 2 3 4Fi
eld
err
or
[un
its]
Scenarios
Magnetic field quality
MEAS 300K SIM 300K MEAS 77K SIM 77K
2~4 factor in field quality improvement
(*)
(*) L. Bortot et al., https://arxiv.org/abs/2103.14354 preprint.
Field quality for the four scenarios
1)
2)
3)
4)
No HALO HALO
CONCLUSIONS AND OUTLOOK
Conclusions
1. Transient effects in accelerators: multi physics, scale and rate phenomena
2. Specialized models + Co-simulation (divide et impera)
3. A-H coupled field formulation Field quality and quench protection simulations
4. HALO as a potential game-changer for magnet design
Outlook
1. Field quality optimization
2. Quench protection studies
3. HALO demonstrator in a full-scale magnet
20
Thank you for your attention!Contact: lorenzo.bortot@cern.ch
Ohmic loss distribution
in a HTS solenoid
W
m2
normalized induced
current in a HTS bulk
Persistent magnetization
in a block-coil dipole
21
BACKUP
OUTPUT
Publications
1. L. Bortot, M. Mentink, C. Petrone, J. Van Nugteren, G. Deferne, T. Koettig, G. Kirby, M. Pentella,J. Perez, F. Pincot, et al. “Proof of Concept of High-
Temperature Superconducting Screens for Magnetic Field-Error Cancellation in Accelerator Magnets.” arXiv preprint arXiv:2103.14354. Apr.2021.url:https://arxiv.org/pdf/2103.14354.pdf.
2. [A1]L. Bortot, M. Mentink, C. Petrone, J. V. Nugteren, G. Kirby, M. Pentella, A. Verweij, and S. Schöps. “Numerical Analysis of the Screening Current-Induced
Magnetic Field in the HTS Insert Dipole Magnet Feather-M2.1-2”. In:Superconductor Science and Technology33.12 (Oct. 2020). issn: 0953-2048.doi:10.1088/1361-6668/abbb17. arXiv:2005.09467.
3. [A2]L. Bortot, B. Auchmann, I. Cortes Garcia, H. De Gersem, M. Maciejewski, M. Mentink, S. Schöps, J. Van Nugteren, and A. Verweij. “A Coupled A-H
Formulation for Magneto-Thermal Transients in High-Temperature Superconducting Magnets”. In: IEEE Transactions on Applied Superconductivity30.5 (Aug.2020). issn: 1051-8223.doi:10.1109/TASC.2020.2969476. arXiv:1909.03312.
4. [A11]L. Bortot, B. Auchmann, M. Maciejewski, M. Prioli, S. Schöps, I. Cortes Garcia, and A. Verweij. “A 2-DFinite-element Model for Electrothermal Transients inAccelerator Magnets”. In: IEEE Transactions on Magnetics54.3 (Mar. 2018), pp. 1–4.issn: 0018-9464.doi:10.1109/TMAG.2017.2748390.arXiv:1710.01187.
5. [A12]L. Bortot, M. Maciejewski, M. Prioli, A. M. Fernandez Navarro, J. B. Ghini, B. Auchmann, and A. Verweij. “A Consistent Simulation of Electro-thermalTransients in Accelerator Circuits”. In: IEEE Transactions on Applied Superconductivity27.4 (June 2017).issn: 1051-8223.doi:10.1109/TASC.2016.2639585.
6. [A9]L. Bortot, B. Auchmann, I. Cortes Garcia, A. M. Fernando Navarro, M. Maciejewski, M. Mentink, M. Prioli, E. Ravaioli, S. Schöps, and A. Verweij. “STEAM:
A Hierarchical Co-Simulation Framework for Superconducting Accelerator Magnet Circuits”. In: IEEE Transactions on Applied Superconductivity28.3 (Apr.2018). issn: 1051-8223.doi:10.1109/TASC.2017.2787665.
Awards
22
IEEE ASC Viktor Keilin Memorial Prize - Large Scale Applications (2020)
Best student paper for innovations in magnet science and technology
IEEE CSC Graduate Study Fellowship (2019)
Recognition of academic excellence and achievement in the area of applied superconductivity
(A) SIMULATION SETUP
23
Rcrow + Rm discharge the
current in the solenoid, limiting
the peak temperature in the
superconducting coil
Rm determined by the quench
dynamics within the solenoid
A. B.
C.
0t (ms)
5 10 30
Quench
detected
A.
Crowbar
resistance
inserted
B.
Quench
heaters
powered
C.
Solenoid
quenchedSolenoid at
nominal conditions
…
Quench
(A) TIME STEPPING
24
Maximum time step for each solver, for each time window
Crowbar
Quench
Quench heaters
Small steps required for:
1. Quench heater powering
2. Quench transition
HTS SOLENOID: CIRCUIT PARAMETERS
25
CONVERGENCE RATE
26
Definitions, at iteration 𝑖:
• 𝑥𝑖 signal (current in the magnet)
• 𝜀𝑎𝑏𝑠 𝜀𝑟𝑒𝑙 absolute & relative error
• 𝜀𝑖 convergence error
• Fconv convergence flag
Enforcement of at least three
iterations per time window
𝜀𝑖 = 1
Quench
𝜀𝑎𝑏𝑠 = 0.001𝜀𝑟𝑒𝑙 = 0.25
Quench as abrupt change in resistivity
High influence on the solenoid current
More iterations needed!
𝜀𝑖 = max𝑥𝑖 − 𝑥𝑖−1
𝜀𝑎𝑏𝑠 + 𝑥𝑖 𝜀𝑟𝑒𝑙, 𝑖 ≥ 2
Fconv = ቊ0, if 𝑖 < 2𝜀𝑖 < 1, if 𝑖 ≥ 2
DYNAMIC EFFECTS IN ACCELERATORS (2/2)
Simulation challenges
• Complex components
• Cutting-edge technologies
• Physical size of devices
• Extreme ambient conditions
• Extremely fast phenomena
(e.g. beam dynamics)
• …
• Mutual electro-thermo-dynamic
interaction between circuit, power
converters, magnets and protection
systems
27
Thermal and fluid dynamics of coolant
Electrodynamics of circuits,
switches, crowbars,
energy extractors
Time (s)
Scale (m)
10−6 10−3 100 103
10−6
10−3
100
103
Thermal dynamics
of diodes
Electro-thermal dynamics
mechanical dynamics
of magnets
Thermal
dynamics of
quench
phenomena
Power
converters
control
multi-physics multi-scale and multi-rate
FIELD-CIRCUIT COUPLING
Interface derivation
Schur complement applied in the semi-discrete problem
Assumption Kν+𝝀Mσ positive-definite (true for gauged 𝐀⋆) Invertible
Interface derived as optimal Schwarz transmission condition for linear systems
𝐙 jω = 𝐗T 𝐊ρ + jω𝐌μ + 𝐐T 𝐊ν + jω𝐌σ −1 𝐐 −1𝐗 −1
Approximation of derivatives in time domain (e.g. Taylor expansion series)
𝐙 jω ≈ 𝐙 0 + jω ቤ𝜕𝐙 jω
𝜕jωjω=0
𝐮𝐬 ≈ 𝐑𝐢𝐬 t + 𝐋d
dt𝐢𝐬 t
28
resistive term 𝐇-flux 𝐀⋆-flux eddy currents term
Multirate systems
• Fast System small steps, slow System large steps
• Different time steps for efficiency
• Different solvers since different nature of the problem
TIME-DOMAIN COUPLING
29
tendt t
tendttend
t
tend
S1
S2
One-way coupling / Parameter extraction Strong coupling
Weak coupling Waveform relaxation / Dynamic iteration
S1
S2
S1
S2
S1
S2
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