Test Case : Comsol - Pancake

1. Introduction

This is the test case of High-Temperature Superconductors with transport current using the T-A Formulation on a Pancake geometry surrounded by air in axisymmetric coordinates.

2. Run the Calculation

To run this case, compute the Study 1 solver on Comsol Multyphysics on the file TA_Axi.mph.

3. Data Files

The case data file is available on Github here.

4. Equation

The T-A Formulation in axysimmetric coordinates is :

T-A Formulation in axisymmetric coordinates
\[\text{(T-A Axis)} \left\{ \begin{matrix} -\frac{1}{\mu}\Delta A_\theta + \frac{1}{\mu r^2}A_\theta &=& \partial_z T_r & \text{on } \Omega^{axis} & \text{(A Axis)} \\ \begin{pmatrix} 0 & 0\\ 0 & -\rho \end{pmatrix}\Delta T_r &=& \frac{\partial(\partial_z A_\theta)}{\partial t} & \text{on } \Omega_c^{axis} & \text{(T Axis)} \end{matrix} \right.\]

With :

  • \(A_\theta\) : \(\theta\) component of potential magnetic field

  • \(T_r\) : \(r\) component of potential current density

  • \(\rho\) : electric resistivity \(\Omega \cdot m\) defined by the E-J power law on HTS : \(\rho=\frac{E_c}{J_c}\left(\frac{\mid\mid J \mid\mid}{J_c}\right)^{(n)}=\frac{E_c}{J_c}\left(\frac{\mid\mid \partial_z T_r \mid\mid}{J_c}\right)^{(n)}\)

  • \(\mu\) : electric permeability \(kg/A^2/S^2\)

5. Geometry

The geometry is a set of stacked tapes in axisymmetric coordinates \((r,z)\), surrounded by air. The geometry is created on Comsol, but in order to show the different physical names, this is a recreation on gmsh :

axi pancake1
Geometry
axi pancake2
Zoom on Tapes

The geometrical domains are :

  • Conducting domain : the tapes

    • Upper Edge : the top boundary of the tapes

    • Lower Edge : the bottom boundary of the tapes

    • Conducting domain boundary : the side boundaries of the tapes

  • Air : the air surrounding Conductor

    • Symmetry Line : the boundary for the symmetry in axisymmetric coordinates

  • Spherical Shell : the air surrounding Conductor

    • Exterior boundary : the Spherical Shell 's boundary

Symbol

Description

value

unit

\(W_{tapes} (\delta)\)

tapes width

\(1e-6\)

m

\(H_{tapes}\)

tapes height

\(4e-3\)

m

\(step\)

space between the tapes

\(0.15e-3\)

m

\(R_{inf}\)

radius of infty border

\(0.08\)

m

6. Parameters

The parameters of the problem are :

  • On Tapes :

Symbol

Description

Value

Unit

\(\mu=\mu_0\)

magnetic permeability of vacuum

\(4\pi.10^{-7}\)

\(kg \, m / A^2 / S^2\)

\(Wid\)

tapes width

\(1e-6\)

\(m\)

\(Hei\)

tapes height

\(4e-3\)

\(m\)

\(f\)

frequency

\(50\)

\(Hz\)

\(Imax\)

max current

\(120\)

\(A\)

\(Iapp\)

applied current

\(0.8*Imax*sin(2*\pi*f*t)\)

\(A\)

\(j_c\)

critical current density

\(3.10^{10}\)

\(A/m^2\)

\(e_c\)

threshold electric field

\(10^{-4}\)

\(V/m\)

\(n\)

material dependent exponent

\(40\)

\(\rho\)

electrical resistivity (described by the \(e-j\) power law)

\(\frac{e_c}{j_c}\left(\frac{\mid\mid \partial_z T_r \mid\mid}{j_c}\right)^{(n)}\)

\(\Omega\cdot m\)

  • On Air :

Symbol

Description

Value

Unit

\(\mu=\mu_0\)

magnetic permeability of vacuum

\(4\pi.10^{-7}\)

\(kg \, m / A^2 / S^2\)

On MPH file, the parameters are written :

Table 1. Parameters on MPH file
comsolaxiparam1
parameters
comsolaxiparam2
parameters

7. Boundary Conditions

For the Dirichlet boundary conditions, we want to impose the transport current :

The transport current is the difference of current potential between the top and the bottom of the tapes divided by the thickness of the tapes, so we can impose \(0\) at the bottom and \(Iapp/thickness\) at the top of the tapes.

Finally we have :

  • On Top : \(T_r = Iapp/\delta\)

  • On Bottom : \(T_r = 0\)

On MPH file, the boundary conditions are written :

Table 2. Boundary conditions on MPH file
comsolaxidir1
Dirichlet
comsolaxidir2
Dirichlet

8. Weak Formulation

Weak formulation of T-A Formulation in axisymmetric coordinates
\[\scriptsize{ \text{(Weak TA Axis)} \left\{ \begin{eqnarray*} \int_{\Omega^{axis}}{ \frac{r}{\mu} \tilde\nabla A_\theta \cdot \tilde\nabla \phi \ drdz } + \int_{\Omega^{axis}}{\frac{1}{\mu r} A_\theta \cdot \phi \ drdz} = \int_{\Omega^{axis}_c}{ \frac{\partial T_r}{\partial z} \cdot\phi \ rdrdz } \\ \int_{\Omega^{axis}_c}{ \begin{pmatrix} 0 & 0 \\ 0 & \rho \end{pmatrix} \tilde\nabla T_r \cdot \tilde\nabla \varphi \ drdz } = \int_{\Omega^{axis}_c}{ \frac{\partial (\partial_z A_\theta}{\partial t} \cdot\varphi \ rdrdz }\\ \text{for } \phi \in H^1(\Omega) \text{ and } \varphi \in H^1(\Omega_c) \end{eqnarray*} \right. }\]

9. Ampere’s law & Coefficient Form PDE

For the A Formulation, the physic Magnetic Field (mf) is used on Comsol. It allows using the Ampère’s Law, which defined the A formulation :

comsolaxiampere
Ampère’s Law

The physic Coefficient Form PDE is used for the T Formulation. The coefficients associated to the Weak Formulation are :

  • On Conducting domain :

Coefficient

Description

Expression

\(c\)

diffusion coefficient

\(\begin{pmatrix} 0 & 0\\ 0 & -\rho \end{pmatrix}\)

\(f\)

source term

\(-\mu_0\frac{\partial H}{\partial t}\)

with \(H=\frac{1}{\mu_0}B\)

On MPH file, the coefficients are written :

Table 3. CFPDE coefficients on MPH file
comsolaxicfpde
CFPDE

10. Numeric Parameters & Mesh

  • Time

    • Initial Time : \(0s\)

    • Final Time : \(0.02s\)

    • Time Step : \(2e-4s\)

  • Mesh

comsoltaaxi mesh
Mesh

11. Results

The results that we obtain with the T-A formulation with Comsol are compared to the results of the article COMSOL implementation of the H-φ-formulation with thin cuts for modeling superconductors with transport currents, where the solver Comsol is used with the H-φ Formulation.

11.1. Electric current density

The electric current density \(J\) is defined by :

\[ J_\theta = \frac{\partial T}{\partial z}\]
Electric current density \(J_\theta (A/m^2)\)

We compare the current density profiles with the T-A formulation and the H-φ formulation on Comsol on the \(O_z\) axis, on the middle tape of the pancake, at time \(t=0.02s\) for a maximum applied current of 150 T and \(n=40\).

L2 Relative Error Norm : \(5.68 \%\)

11.2. Magnetic flux density

The magnetic flux density \(B\) is defined by:

\[ B=\nabla\times A =\begin{pmatrix}-\partial_z A_\theta\\ 0\\ \frac{1}{r}\partial_r (rA_\theta)\end{pmatrix}\]
Magnitude of the magnetic flux density \(B (T)\)

We can see that when the transport current is applied the tapes are generating a magnetic field, that stay trapped in the superconductor once the applied current is 0.

We compare the distribution of the r-component of the magnetic flux density with the T-A formulation and the H-φ formulation on Comsol on the \(Oz\) axis, in the middle of the pancake \(t=0.01s\).

L2 Relative Error Norm : \(0.94 \%\)

12. References

  • Real-time simulation of large-scale HTS systems: multi-scale and homogeneous models using the T–A formulation, Edgar Berrospe-Juarez et al 2019 Supercond. Sci. Technol. 32 065003, PDF

  • COMSOL Implementation of the H-ϕ-Formulation With Thin Cuts for Modeling Superconductors With Transport Currents, A. Arsenault, B. d. S. Alves and F. Sirois, in IEEE Transactions on Applied Superconductivity, vol. 31, no. 6, pp. 1-9, Sept. 2021, Art no. 6900109, doi: 10.1109/TASC.2021.3097245. PDF