INSTITUTE OF TECHNICAL PHYS
Numerical Modeling of Thin Superconducting Tapes
Francesco Grilli1, F. Sirois2 and R. Brambilla3
KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association
INSTITUTE OF TECHNICAL PHYS
www.kit.edu
Francesco Grilli1, F. Sirois2 and R. Brambilla3
1Karlsruhe Institute of Technology, Karlsruhe, Germany2Ecole Polytechnique Montréal, Montréal, Canada
3Ricerca sul Sistema Energetico - RSE S.p.A., Milano, Italy
Presented at the COMSOL Conference 2010 Paris
Present issues of numerical modeling of thin superconductors
Show results obtained with different mesh types and elements
Goal of this presentation
thinsuperconductors
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Superconducting filmWidth: 4-12 mm
Thickness: ~1 µm
Aspect ratio: up to 10,000!
Superconductor material
Non-linear behavior
Power-law current-voltage
characteristics
current density
ele
ctr
ic f
ield
JcEc
E/Ec=(J/Jc)n
Faraday’s law
Material properties for HTS
Constitutive equations
µ∂B
∂t+ ∇ × E = 0
ρ J( ) =Ec
Jc
J
Jc
n −1
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Current density
Comsol Conference, 17-19 November 2010, Paris, France
c c
B =µ0H
J = ∇ × H
Faraday’s law
Equations in 2D
µ∂Hx
∂t+
∂E
∂y= 0
µ∂Hy
∂t+
∂E
∂x= 0
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Expressions for J and E
Comsol Conference, 17-19 November 2010, Paris, France
∂t ∂x
J =∂Hy
∂x−
∂Hx
∂y
E = ρ J( )J
Implementation in general PDE system
ea
∂ 2u
∂t2
+ da
∂u
∂t+ ∇⋅ Γ = F
u =Hx
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
µ∂Hx
∂t+
∂E
∂y= 0
µ∂Hy
∂t+
∂E
∂x= 0
u =x
Hy
Physics: we want div(B)=0
B=µ0H � div(H)=0
Comsol’s curl elements shcurl
Not only do they impose div(H)=H1x+H2y=0
They also impose H1x=0 and H2y=0
Much more stringent condition than that obtained with Lagrange elements
Very important feature: edge elements
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
µ∂Hx
∂t+
∂E
∂y= 0
µ∂Hy
∂t+
∂E
∂x= 0
Take x-derivative
+Take y-derivative
µ∂
∂t
∂Hx
∂x+
∂Hy
∂y
= 0
∇ ⋅ H = constant=0
“Drift” of divH with Lagrange elements
Lagrange 12190 DOFs, 193 s
Lagrange 28698 DOFs, 3143 s
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Shcurl 13254 DOFs, 223 s
Aspect ratio 1,000-10,000
Large number of mesh elements, even with a ‘coarse’ mesh
Mesh issues
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Increase the thickness, keep Ic constant
Justification: flux penetration as in infinitely thin tape
Tape behaves as a 1-D object
Possible solution: increase the thickness
Expand thickness by
a factor N
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First International Workshop on Numerical Modeling of High-Temperature Superconductors
Lausanne, Switzerland, May 5-7, 2010
Critical current density=Jc
Width=w
Thickness=d
Ic=Jc*w*d
Critical current density=Jc/N
Width=w
Thickness=d*N
Ic=Jc/N*w*d*N=Jc*w*d
It works well for an isolated tape
What happens in case of interacting conductors?
Top/bottom losses become important (depend on actual thickness)
Expanded thickness may become comparable with tape separation
Possible solution: increase the thickness
10 Francesco Grilli
KIT-ITEP
First International Workshop on Numerical Modeling of High-Temperature Superconductors
Lausanne, Switzerland, May 5-7, 2010
Figures taken from Hong and Coombs, J. Supercond. Nov.Magn., 2010
Use elongated quadrangular elements (Rodriguez-Zermeno, # 6765)
Elongation allows reducing number of DOFs
For a 1-D description of the tape, 1 element along the thickness is enough
Successfully applied to the simulation of a Roebel cable
Another solution
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Magnetization losses
Simulation of a Roebel cable
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Magnetic field distribution in 2 stacks of 7 tapes
Representative of the (2-D) cross section of a Roebel cable
Simulation of a Roebel cable
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Magnetic field distribution in 2 stacks of 7 tapes
Representative of the (2-D) cross section of a Roebel cable
Simulation of a Roebel cable
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Standard triangular mesh
280,000 elements, 450,000 DOFs, several days
Elongated quadrilateral elements
23,000 elements, 36,000 DOFs, a few hours
Very important for design optimization
Simulation of a Roebel cable
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KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Magnetic field distribution in 2 stacks of 7 tapes
Representative of the (2-D) cross section of a Roebel cable
Simulation of a Roebel cable
16 Francesco Grilli
KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
Coupled
I1+ I2+…+I14=0
Uncoupled
I1=0
I2=0
…
I14=0
H-formulation implemented in COMSOL’s PDE General Form module to compute J and H profile and ac losses in HTS
Very flexible
Use 1st order curl elements
Ensure divB=0
Keep the number of DOFs at a reasonable level
Use elongated quadrilateral elements for thin conductors
Conclusion
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Use elongated quadrilateral elements for thin conductors
Lower number of mesh nodes with respect to standard triangular elements
Method applied to simulate Roebel cable
Comsol Conference, 17-19 November 2010, Paris, France
How many cycles?
Ac loss calculation
P(t) = J⋅ EdSS
∫T
∫
18 Francesco Grilli
KIT-ITEPComsol Conference, 17-19 November 2010, Paris, France
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