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

2 Francesco Grilli

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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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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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

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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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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

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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

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