# Question about boundary condition related to vector potential

Dear all

Hi. I am trying to solve time-dependent ginzburg landau equation of 2D superconductor. When I try to solve this problem, I encounter some problem.

Basically, the boundary condition for the vector potentials is as follows;

I am not sure how can I deal with this type of boundary condition in FreeFEM++.

Thank you!

Sincerely
Daehan

Well, in the AmpĂ¨re Maxwell equation, you integrate by part vA\cdot\nabla\times\nabla\times A (here vA is the test function), to get the boundary term. There replace the \nabla\times A by the desired value of the magnetic fieldâ€¦This gives something like this in your weak form in freefem

-int1d(Th,boundaryLBL)( (vAxN.y-vAyN.x)*(Bext) )

hope this helps,
julien

1 Like

Thank you so much! It would work!

Excuse me, but is there some reason for using matching of normal components of curl(A) with external field as a boundary condition, instead of matching the tangential components of it (which corresponds to the vanishing of the superconducting current through the boundary)?

Did he indicate z is normal to gamma ? lol. Actually I think the issue is expressing
the constraint in FF. In this case, the â€ś+on()â€ť construct is quite flexible and apparently
supports an â€śNâ€ť for normal with components available as N.x etc.

While Iâ€™m here is there some reason the linear and bilinear pieces need their own
int2d() expressions? It sounds like whining but it takes a while to find if you arenâ€™t
used to it

Yeah. Because the model I considered had translational symmetry along the z-direction. Also, the direction of the magnetic field was along the z-direction. In this circumstance, the corresponding boundary condition for the vector potential is what I wrote. By the way, what is the boundary conditions for matching the tangential component of the vector potential?

Are you just after the Meissner effect ?
Just on a quick google search I would guess the BC is a research topic

I guess if you could avoid BC in favor of microscopic analysis or
materials properties that may help.

[1]Microscopic derivation of superconductor-insulator boundary conditions for Ginzburg-Landau theory revisited: Enhanced superconductivity at boundaries with and without magnetic field, Samoilenka , Albert and Babaev , Egor; Phys. Rev. B. 2021

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.224516

[for high school? ](https://iris.unipa.it/retrieve/handle/10447/98009/127187/The%20explicative%20power%20of%20the%20vector%20potential%20for%20superconductivity-%20a%20path%20for%20high%20school.pdf)



Dear Mike. Thank you for your comment.

It was already done in the past, but for the record, I write the full boundary conditions I used here;
You can find the boundary condition in the following reference;
(\frac{i}{\kappa}\nabla\psi +\vec{A}\psi) \cdot \hat{n} = 0
\nabla\times\vec{A} = \vec{H}, \text{on the boundary}

[1] Numerical approximations of the Ginzburgâ€“Landau models for superconductivity
Q. Du, J. Math. Phys. 46, 095109 2005
[2] Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity
Q.Du, Comput. Math. with. 27. 119


I could not get your links to work but may look into this once I get past classical
diffusion I use my own tool called â€śTooBibâ€ť for extracting a bibtex entry from a
URL and creating a citation for various purposes such as posting here.
Google my last name marchywka and toobib or see this,

see marchywka TooBib

Is this[1] of any interest? They use COMSOL I guess if I can find a tractable worked
example in the literature I may try this for my next exercise â€¦

`
[1]Time-dependent Ginzburg-Landau treatment of rf magnetic vortices in superconductors: Vortex semiloops in a spatially nonuniform magnetic field, Oripov , Bakhrom and Anlage , Steven M.; Physical Review E. 2020 2020