SC 002 - Twisted superconductor AC Loss

What you’ll learn

• How to import a complex twisted-filament STEP geometry and define regions for air, copper, and superconducting domains.
• How to set up the H-φ formulation with coupled Magnetism H and Magnetism φ physics, apply a Lump I/V cut current source, and configure the φ constraint (gauge condition).
• How to customize mesh element size and curvature enhancement, apply Newton-Raphson linearization in the script, and run a transient simulation with custom Joule loss outputs.

In this tutorial, AC loss in a high-temperature superconducting (HTS) wire is simulated using the H-φ formulation.

Model definition

The wire consists of twisted superconducting filaments embedded into a copper matrix. The whole modelling domain with an air cylinder around the wire is illustrated below.

Element	Dimension
Air cylinder diameter	10 mm
Copper cylinder diameter	535 μm
Filament diameter	350 μm
Domain length	10 mm

Output results

• Joule losses as a function of time in the copper and the superconducting filament regions. The losses over the volume of interest can be computed as

$$P(t)=\int\boldsymbol{E}(t)\cdot \boldsymbol{J}(t)~\rm{d} V.$$

Material Data

Magnetic permeability, $\mu$:

• all domains: $\mu_0$

Electric resistivity, $\rho$:

• Copper: $10^{-10} ~ \Omega \rm m$
• Superconducting filaments:
  • $\rho = \frac{E_{\rm c}}{J_{\rm c}} \left( \frac{||J||}{J_{\rm c}} \right) ^{n-1}$
    • Critical electric field strength, $E_{\rm c} = 100 ~ \rm μV/m$
    • Exponent, $n = 30$
    • Critical electric current, $I_{\rm c} = 100 ~ \rm A$
    • Total cross-section are of superconducting filaments, $A_{\rm sc} = 3.4541 \cdot 10^{-7} ~ \rm m^2$
    • Critical electric current density, $J_{\rm c} = I_{\rm c} / A_{\rm sc}$

Source

The problem is sourced by applying a total current of

$$I(t) = 0.8 ~ I_{\rm c} ~ \sin(2 \pi f t),$$

where the frequency $f$ is $50 ~ \rm Hz$.

Step-by-step guide

Here you’ll find a detailed step-by-step tutorial on how to simulate AC loss in a twisted filament HTS wire Quanscient Allsolve.

Step 1 - Import the geometry

1. Start with a new project and name it as

   SC twisted filament AC loss

2. Import the geometry as a .step file with default import options.

   File download link: twisted-superconductor.step

   

3. Confirm model changes.

Step 2 - Define regions

1. Go to the Common sidebar.

2. Define a region for air:

   Region name	Region type	Target
   air	Volume	Air cylinder

   

3. Define a region for copper:

   Region name	Region type	Target
   copper	Volume	Copper matrix

   

4. Define a region for the superconducting filaments:

   Region name	Region type	Target
   sc	Volume	SC filaments

   

Step 3 - Define materials

Adding materials

To add materials, go to the Physics section and click + next to Materials. Select a predefined material from the library, then assign it to the target volume or region.

1. Assign the Air material to the air region:

   

2. Assign the Copper material to the copper region:

   

3. Assign the YBCO superconductor material to the sc region:

   

   Note: YBCO power law

   When the YBCO material is added, a predefined function and variables related to the YBCO power law are automatically created.

Step 4 - Define variables

1. Define new variables:

   Name	Description	Expression
   f	Frequency [Hz]	50
   YBCO_Ic	Critical current [A]	100
   YBCO_Asc	Filament cross-section area [m^2]	3.4541e-7
   Iop	Operating current [A]	0.8 * YBCO_Ic * sin(2 * pi * f * t)

2. Modify existing variables:

   Name	Modified expression
   YBCO_Jc	YBCO_Ic / YBCO_Asc
   YBCO_n	30

Step 5 - Define physics and apply the current source

1. Go to the Physics section.

2. Add the Magnetism H physics:

   Physics	Target
   Magnetism H	Copper matrix and SC filaments

   

3. Add the Magnetism φ physics:

   Physics	Target
   Magnetism φ	Air cylinder (air region)

   

4. Add a Lump I/V cut interaction to Magnetism φ.

   Interaction name	Interaction type	Target	Value
   Current source	Lump I/V cut	a counter-clockwise loop at the top edge of the copper matrix	Iop

   

5. Add a Constraint interaction to Magnetism φ:

   Interaction name	Interaction type	Target	Value
   Gauge	Constraint	a point at the external boundary of the air domain	0

   Any point on the external boundary of the air domain can be selected. This constraint is the gauge condition for the scalar potential φ.

   

6. Add the H-φ coupling interaction to Magnetism H.

7. Before moving on, check that your physics tree matches the one below:

   

Step 6 - Generate a mesh

1. Go to the Simulations section.

2. Add a new mesh.

3. Set Autorefine to Disabled.

4. Open the collapsible menu for Mesh element size.

5. Set Curvature enhancement to 25.

6. Generate the mesh and check the preview.

   Meshing time

   This geometry may take approximately 2 minutes to mesh. You can monitor progress in the mesh Logs.

   

   

Step 7 - Select simulation settings

1. Add a new simulation.

2. Set Analysis type to Transient.

3. Select Transient settings:

   Timestep algorithm	Start time [s]	End time [s]	Timestep size [s]
   Implicit Euler	0	0.01	0.0001

4. Set Solver mode to Iterative solver.

5. Set Node count to 50.

6. Select Mesh 1 as the mesh for your simulation.

7. Define custom value outputs for computing Joule losses in the filaments and copper:

   Output name	Output type	Output expression
   SC loss	Custom value output	integrate(reg.sc, transpose(E) * j, 4)
   Cu loss	Custom value output	integrate(reg.copper, transpose(E) * j, 4)

   In the expressions above, E and j are 3 x 1 column vectors. transpose(E) creates a 1 x 3 row vector from E so that the multiplication with j produces a dot product $\boldsymbol{E} \cdot \boldsymbol{J}$. The last argument 4 is the integration order.

   Tip

   Use the built-in AI Assistant to help you write output expressions and formulations, or get explanations for demo project implementations.

8. Open the Script for your simulation.

9. Enable Scripting mode.

10. Replace the first autogenerated line under # Magnetism H with the following Newton-linearization [4]:

    rho = 1/par.sigma(df.j)
    dedj = rho*qs.eye(3) + (expr.YBCO_n-1.0)*rho/qs.max(df.j*df.j, 1e-40) * df.j * qs.transpose(df.j)
    dofe = rho*df.j + dedj * (qs.curl(qs.dof(fld.H))+var.curl_dof_Hs - qs.curl(fld.H)-var.curl_Hs)
    form += qs.integral(reg.sc, dofe * (qs.curl(qs.tf(fld.H)) - var.curl_tf_Hs))
    form += qs.integral(reg.copper, qs.inverse(par.sigma(df.j)) * (qs.curl(qs.dof(fld.H)) + var.curl_dof_Hs) * (qs.curl(qs.tf(fld.H)) - var.curl_tf_Hs))

    

Step 8 - Run the simulation and see results

1. Run the simulation.

2. To follow the simulation progress, open Logs.

3. The SC and Cu loss results can be seen in Plotting, even while the simulation is running:

   

References

[1] H-φ Formulation in Sparselizard Combined With Domain Decomposition Methods for Modeling Superconducting Tapes, Stacks, and Twisted Wires. https://doi.org/10.1109/TASC.2023.3240389

[2] Allsolve demo project of Superconductor AC losses. https://allsolve.quanscient.com/#/projects/demo/8fed82d1-5bf0-4c02-835b-94e65a60f847

[3] Youtube tutorial of Superconductor AC losses. https://youtu.be/B9QZZ5y7RpQ

[4] Newton Linearization. https://en.wikiversity.org/wiki/Nonlinear_finite_elements/Newton_method_for_finite_elements