Boundary conditions at holes

If the domain is multiply connected, at the hole boundary the same boundary conditions as before apply, namely zero perpendicular supercurrent and magnetic field continuity. In what concerns the order parameter, one proceeds exactly as for the exterior boundary (Eqs. (22) and (23)). However, the magnetic field inside the hole is not known a priori. Our methodology has been implemented for the case of one rectangular hole, but it is easily generalized. Let H_i be the magnetic field inside the hole. Since there are no currents in the hole, H_i is uniform and only depends on time.

Consider the hole shown in Fig. 2. It is delimited by the vertex lines i=W, i=E, j=S and j=N. Since the adjacent cells must also have B_z=H_i, the magnetic field is uniform in the subdomain $\omega$ depicted in Fig. 2. The key remark is that the link variables corresponding to the boundary of $\omega$ can be calculated using Eqs. (19) and (20).

Figure 2: Scheme of a computational mesh with a hole.

The algorithm to update H_i and the link variables at the boundary of the hole is (with a notation similar to that in the program):

• Update all interior link variables according to (19)-(20).
• Compute
  

  $${\cal P}_{x1} {\overline U^y_{W,S-1}} \left ( \prod_{i=W}^{E-1} U^x_{i,S-1} \right ) U^y_{E,S-1}$$ = (26)

  $${\cal P}_{x2} { U^y_{E,N}} \left ( \prod_{i=W}^{E-1} {\overline U^x_{i,N+1}} \right ) {\overline U^y_{W,N}}$$ = (27)

  $${\cal P}_{y1} { U^x_{W-1,S}} \left ( \prod_{j=S}^{N-1} {\overline U^y_{W-1,j}} \right ) {\overline U^x_{W-1,N}}$$ = (28)

  $${\cal P}_{y2} { U^x_{E,S}} \left ( \prod_{j=S}^{N-1} { U^y_{E+1,j}} \right ) {\overline U^x_{E,N}}$$ = (29)

  
  
  Notice that ${\cal P}={\cal P}_{x1}{\cal P}_{x2}{\cal P}_{y1} {\cal P}_{y2}$ approximates $\exp (-\imath \int {\bf A}\cdot {\bf ds})$ where the circulation is calculated around $\omega$. Consistently with the previous approximations, we take ${\cal P}= \exp (- \imath M a_x a_y H_i )$ where M is the number of cells in $\omega$,
  
  M=(N-S)(E-W)+2(N-S)+2(E-W)
  
  

• Update H_i using, as immediate from above,
  

  $$\partial_t H_i = \frac{\imath}{Ma_xa_y} {\overline {\cal P}} \partial_t {\cal P}$$ (30)

  
  

• Update all link variables at the hole boundary using (24), with H_i in the place of H_e.

The implementation uses the logical array Bulk (i,j) to determine if cell (i,j) is in the superconductor or in the hole.