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!


Well, in the Ampère 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 field…This gives something like this in your weak form in freefem

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

hope this helps,

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 “+on()” construct is quite flexible and apparently
supports an “N” for normal with components available as N.x etc.

While I’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 aren’t
used to it :slight_smile:

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 :slight_smile:

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

[for high school? ](

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 Ginzburg–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 :slight_smile: I use my own tool called “TooBib” 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 …
[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