The Ginzburg-Landau Equation Solved by the Finite Element Method

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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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.

Type II superconductor


Magnetic flux penetration and two critical magnetic fields.

Phase transition model

Gibbs energy:

 
1 0
n s
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 field in terms of the magnetic potential A:

Introduction


Superconducting state. Cooper pairs:

Crirical temperature :
Critical magnetic field:


London magnetic penetration depth:

Coherence length:


Absolute temperature: Ginzburg Landau parameter:

The Ginzburg Landau equations

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.

Numerical method

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:

0
45
)
,
(



k
k
c
T
c
B


1
c
B
B

The magetic field is
expelled for

2
1
c
c
B
B
B


Penetration of
magetic fluxes

B
B
c

2
Normal conducting
state

4
2
2







n
s
G
G
The
α

parameter controls the phase transition from the
normal state to the superconducting state
:

)
/
1
)(
0
(
)
(
C
T
T
T




T
Ref.: T. Schneider and J.M. Singer,

Phase Transition Approach to High Temperature Superconductivity. Imperial College Press,
London, (2000).




























2
2
2
2
1
2


qA
i
m
q
i
t
mD



A
B
i



The electric potential is denoted:









t
A
E
The electric field is given by:

The order
parameter is

)
,
,
(
t
y
x

The equation for the magnetic vector potential.

A
A
m
q
mi
q
t
A


























0
2
2
*
*
1
)
(
2



Ref.: W.D. Gropp et al. Numerical simulations of vortex dynamics in type
-
II superconductors.

Jour. Of Comp. Phys. 123, p254
-
266 (1996).

The boundary conditions are on

0












n
qA
i

0













n
t
A
a
i
B
B




F
t
u
d
a







in


Auxiliary equation is needed for fulfilling the BC:

5
,
2
,
1
,
2
,
1
,
5
0
u
A
A
A
A
u
y
x
y
x
t






)
,
(
2
1
5
A
A


G
n




BC:

on



Note that
Г

is a 5
-
vector.

Numerical simulations and results

Dynamics of penetrating magnetic vortices into a type II
superconductor. Defects can result in formation of giant vortices.
Complex pattern formation.

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.


Circular superconductor with a defect.

)
,
,
(
t
y
x

8
.
0

a
B
4


20

t
100

t
15000

t
Supercurrent

2
*
*
)
(
2






A
i
j
s





Phase








)
Re(
)
Im(
tan



Arc
200

t
We acknowledge financial support from the Danish
Center for Scientific Computing (DCSC).




/