Test Case : Comsol - Magnetostatic HTS with erf
1. Introduction
This is the test case of High-Temperature Superconductors with the A-V Formulation and Gauge Condition on a cylinder geometry surrounded by air in 2D coordinates. The formulation used here is A-V Formulation in 2D coordinates with erf.
2. Run the Calculation
To run this case, compute the Study 1 solver on Comsol Multyphysics on the file aform_erf.mph.
3. Data Files
The case data file is available in Github here.
4. Equation
With :
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\(A_{z}\) : \(z\) component of potential magnetic field
-
\(J=J_c \text{erf}\left(\frac{-A_z}{A_r}\right)\) : current density \(A/m^2\)
-
\(\mu\) : electric permeability \(kg/A^2/S^2\)
5. Geometry
The geometry is a circle in 2D coordinates \((x,y)\) representing a bulk cylinder, surrounded by air. The geometry is created with a .geo file on GMSH, exported as a .bdf mesh and imported on Comsol.
Geometry
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The geometrical domains are :
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Conductor: the cylinder -
Air: the air surroundingConductor-
Infty: theAir's boundary
-
Symbol |
Description |
value |
unit |
\(R\) |
radius of cylinder |
\(0.001\) |
m |
\(R_{inf}\) |
radius of infty border |
\(0.01\) |
m |
6. Parameters
The parameters of the problem are :
-
On
Conductor:
Symbol |
Description |
Value |
Unit |
\(\mu=\mu_0\) |
magnetic permeability of vacuum |
\(4\pi.10^{-7}\) |
\(kg \, m / A^2 / S^2\) |
\(A_r\) |
parameter resulting from the comination of (!!lien!!)\(E_0\) and the time it takes to reach the peak of AC excitation |
\(1.10^{-7}\) |
\(Wb/m\) |
\(Bmax\) |
external applied field |
\(0.02\) |
\(T\) |
\(Jc\) |
critical current density |
\(1.10^8\) |
\(A/m^2\) |
The error function erf can be used on Comsol:
|
-
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 :
parameters
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7. Boundary Conditions
For the Dirichlet boundary conditions, we want to impose the applied magnetic field :
We have \(B_{max}=0.02 T\), and in 2D Cartesian coordinates, \(B=\nabla\times A\) becomes \(B_x=\partial A/\partial y\) and \(B_y=-\partial A/\partial x\). So in order to obtain a magnetic field \(B_{max}\) along \(y\), it is sufficient to impose \(A_z=-xB_{max}\).
Finally we have :
-
On
Infty: \(A = -x B_{max}\)
On MPH file, the boundary conditions are written :
Dirichlet
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8. Weak Formulation
9. Coefficient Form PDEs
The Comsol library Coefficient Form PDEs is used here. The coefficients associated to the Weak Formulation are :
-
On
Conductor:
Coefficient |
Description |
Expression |
\(c\) |
diffusion coefficient |
\(1\) |
\(f\) |
source term |
\(\mu J_c \text{erf}\left(\frac{-A}{A_r})\right)\) |
On MPH file, the coefficients are written :
CFPDE
|
-
On
Air:
Coefficient |
Description |
Expression |
\(c\) |
diffusion coefficient |
\(1\) |
CFPDE
|
10. Results
The results that we obtain with this formulation with Comsol are compared to the results of the article A numerical model to introduce student to AC loss calculation in superconductors where the solver FreeFEM is used.
10.1. Electric current density
The electric current density \(J\) is defined by :
Figure 1. "Electric current density \(J (A/m^2)\)
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We compare the current density profiles with Comsol and FreeFEM on the \(O_r\) axis, on the diameter of the cylinder, for a maximum applied field of 0.02 T.
L2 Relative Error Norm : \(3.3 \%\) |
10.2. Magnetic flux density
The magnetic flux density \(B\) is defined by:
Therefore, \(B_y\), the y-component of the magnetic flux density is defined as \(-\partial_x A\) :
Figure 2. y-component of the magnetic flux density \(B_y (T)\), caption=
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