The Ginzburg-Landau Equation Solved by
the Finite Element Method
Tommy S. Alstrøm and Mads Peter Sørensen, DTU Mathematics.
Niels Falsig Pedersen and Søren Madsen, DTU Electrical Engineering.
Introduction
The Ginzburg Landau equations
Numerical simulations and results
Superconducting state. Cooper pairs: ( k , k )
Crirical temperature : Tc
Critical magnetic field:
In the following we present the time dependent Ginzburg Landau
equation coupled to the magnetic field in order to model the dynamics of
flux penetration into complex geometry mesoscopic type II
superconductors.
Dynamics of penetrating magnetic vortices into a type II
superconductor. Defects can result in formation of giant vortices.
Complex pattern formation.
London magnetic penetration depth:
Coherence length: 
Absolute temperature:
T
Bc

Bi    A
Magnetic field in terms of the magnetic potential A:
   /
Ginzburg Landau parameter:
Type II superconductor

The electric potential is denoted:
E
The electric field is given by:
A
 
t
The equation for the order parameter ψ(x,y,t). We consider
superconductors with two space dimensions and one time
dimension. Spatial region is dentoed: 
Magnetic flux penetration and two critical magnetic fields.
 
q 
1 
2

  i    
   qA        
2mD  t  
2m  i

2
2
Penetration of
magetic fluxes
Normal conducting
state
B  Bc1
Bc1  B  Bc 2
Bc 2  B
Phase transition model
Gibbs energy: Gs  Gn    
2

2
4
The order
parameter is
 ( x, y, t )
The α parameter controls the phase transition from the
normal state to the superconducting state:
 (T )   (0)(1  T / TC )
( x, y, t )
t  20
Ba  0.8
2

A
q

q
1
2


*
*
     
(      ) 
 A  A
m
0
 t
 2mi
The boundary conditions are on 
 A

     n  0
 t



   qA   n  0
i

t  15000
t  100
Bi  Ba
Ref.: W.D. Gropp et al. Numerical simulations of vortex dynamics in type-II superconductors.
Jour. Of Comp. Phys. 123, p254-266 (1996).
Numerical method

Circular superconductor with a defect.
 4
The equation for the magnetic vector potential.
The magetic field is
expelled for
Future work on modelling high-Tc superconducting electric
generators for windmills. (http://www.superwind.dk). Industrial
mathematics, nonlinear dynamics and scientific computing as a tool
for design and development of superconducting generators.
Supercurrent
The Ginzburg Landau model consists of 4 coupled partial
differential equations . The real and imaginary parts of the order
parameter ψ(x,y,t) plus two components of the vector field
A(x,y,t). The model is implemented in the COMSOL finite element
programme using quadratic Lagrange elements. In general form
the COMSOL implementation reads:
u
da
   F
t
in
i
js 
( *  * )  A
2
 Im( ) 
  Arc tan 

 Re( ) 
t  200
Phase
2

Auxiliary equation is needed for fulfilling the BC:

1 n
0 s
Ref.: T. Schneider and J.M. Singer,
Phase Transition Approach to High Temperature Superconductivity. Imperial College Press,
London, (2000).

0  u5,t  A1, x  A2, y  A1, x  A2, y  u5
BC:
 n  G
on

5  ( A1 , A2 )
Note that Г is a 5vector.
We acknowledge financial support from the Danish
Center for Scientific Computing (DCSC).