# getDP model for computing the sheath losses in a twisted three-core AC submarine cable
## In a nutshell:
This finite-element formulation transforms a Cartesian 3-D helicoidally symmetric problem into a problem in terms of helicoidal coordinates, where it is solvable in 2-D.
![3-D model of twisted submarine cable including sheaths][def]
## Problem setting:
The submarine cable has 3 twisted inner conductors, carrying each a known, $2\pi/3$-phase shifted, AC current at $50$ Hz. The generated magnetic fields are penetrating the twisted sheaths and hence induce currents, which cause Ohmic losses within the sheaths. As conductors and sheaths are twisted, a magnetic field component in logitudinal direction is present ($H_z \neq 0$), which would be neglected using a standard 2-D formulation (e.g., formulated in terms of a $z$-component of a magnetic vector potential $\mathbf{A}$).
## FEM formulation:
Besides the coordiante transformation, the eddy current problem is solved using the $\mathbf{H}-\phi$ formulation, in which cuts have to be introduced within the non-conducting domain $\Omega_i$, for allowing the magnetic field to be expressed as the gradient of a magnetic scalar potential $\phi$. To each cut, a cohomology basis function is assigned, which allows a non-zero circulation of $\mathbf{H}$ around the corresponding hole, or cycle $c$, i.e. $\oint_c \mathbf{H} \cdot \mathrm{d}\mathbf{s} \neq 0$, which is (obviously) needed if we want to model current carrying conductors. For this model, a challange arises to track the holes/cycles of each conductor/conducting part separately (homology), s.t. the coefficient of the corresponding cohomology basis function has the interpretation of the net current flowing through that hole/cycle.
This is based on the fact, that $\Omega_i$ posses holes within holes (sheath surrounding inner conductor). It is possible though, to assign the desired cohomology basis functions for the inner conductors, where the total current is known. The coefficients of the three other cohomology basis functions (part of the solution of the FEM formulation) correspond here to an integer combination of the true sheath currents, i.e. not directly to the total current in each sheath. These total currents though, can be computed by $\int_{A_{\mathrm{sheath}}} \mathbf{J} \cdot \mathrm{d}\mathbf{A}$ in a post-processing step.
## Further information:
The model had been developed by Albert Piwonski (Technische Universität Berlin).
Here you can find further information about the ansatz:
- for $\mathbf{H}-\phi$ formulation:
-[IEEE Transactions on Magnetics paper version](https://ieeexplore.ieee.org/document/10006756)
-[arXiv paper version](https://arxiv.org/abs/2301.03370)
- for $\mathbf{A}-v$ formulation:
-[IEEE Transactions on Magnetics paper version](https://ieeexplore.ieee.org/document/10261253)
-[arXiv paper version](https://arxiv.org/pdf/2307.00814)
More details about the submarine cable model will be published as a paper called:
- _Efficient computation of sheath losses in
three-core AC submarine cables_
## Files in this directory:
- getDP solver file: 3_p_sea_cable.pro (extensively documented and close to the paper notation as possible)
→ open this file using Gmsh
- Gmsh mesh file: 3_p_sea_cable.msh
I.e., the .msh file was not generated using a .geo scripting file, but using the [Julia API for Gmsh](https://gmsh.info/). This is because the calculation of the conductors' cross-sections was performed in Julia as well. Also, the .pro file may not look like a _typical_ getDP .pro file, since [Julia's string interpolation functions](https://docs.julialang.org/en/v1/manual/strings/#string-interpolation) were used extensively to create it automatically.
## Credits:
Special thanks to Julien Dular (TE-MPE-PE, CERN) who gave useful implementation tips.
