IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 20103445
A 2-D Robust FE-FV Mixed Method to Handle Strong
Nonlinearities in Superconductors
Abelin Kameni?, Smail Mezani?, Frédéric Sirois?, Denis Netter?, Jean Lévêque?, and Bruno Douine?
GREEN-UHP-INPL, Faculté des Sciences, 54506, BP. 239, Vandoeuvre-lès-Nancy, France
Ecole Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada
A robust numerical method based on 2-D mixed finite-elements-finite volumes (FE-FV) allows the solution of diffusion problems in
superconducting (SC) materials. The proposed approach handles the strong nonlinearity of the E(J) constitutive power law of high-tem-
perature superconductors (HTS). The method is tested for a SC cylinder submitted to a sinusoidal transport current or to a transverse
sinusoidal external field. The current density distributions as well as the AC losses are computed. Comparisons to a FE analyses that use
the magnetic field as state variable show the validity of the proposed approach. It can be seen that the proposed method is very stable
even for large n-values for which the FE method does not converge.
Index Terms—Finite elements, finite volumes, superconducting materials.
HE CONSTITUTIVE power law is widely used to char-
acterizehightemperaturesuperconductors. Itis writtenas
represents the critical state model suggested by Bean.
Finite-element methods are widely used to compute the mag-
lations , . Unfortunately, the use of (1) leads to numerical
oscillations and convergence problems for large n-values (typi-
In order to solve the nonlinear diffusion-convection-reaction
equations in porous media, a coupled finite-elements-finite vol-
umes (FE-FV) method has been successfully used . In this
paper, we propose to use a similar approach to solve the electric
field diffusion equation in HTS materials. Applications to a su-
perconducting cylinder under transport current or submitted to
conventional approaches based on FE methods do not converge.
correspondstoa normal conductorand
II. PRINCIPLE OF THE METHOD
The superconductor has a vacuum magnetic permeability
In 2-D problems,
andare scalar quantities having only
one component in the
direction. The SC domain is noted
and its external boundary. Using Maxwell equations and the
constitutive law (1), the problem to solve is
Manuscript received December 21, 2009; revised February 11, 2010;
accepted February 16, 2010. Current version published July 21, 2010. Corre-
sponding author: S. Mezani (e-mail: email@example.com).
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TMAG.2010.2044025
magnetic field according to Faraday’s law
is the outward normal vector on
The discretization methods (FEM or FVM) are applied to the
weak formulation of (2) in the time interval
components oftheflux den-
is a test function,
the time step.
The method exposed here is based on the FE-FV coupling
as the footbridge operators between FE and FV approximation
spaces, and combines FE and FV discretizations terms of (4).
is the solution at instant , and
A. FE-FV Coupling Principle
The FE-FV coupling is widely used in fluid mechanics to
solve convection-diffusion type equations. The computation
procedure consists in successively solving the diffusion equa-
tion on the FE mesh and the convection equation on the FV
cell. In so doing, it is necessary to calculate the projection of
the FE solution on the FV mesh and vice versa.
Starting from a solution
using the FF-FV coupling scheme is determined as
i) calculate the solution
ii) determine its projection
to compute the solution
iv) the needed solution
The problem treated here is a convection-diffusion equation.
The objective is to combine the advantages of each method (FE
in (2). In so doing, we have to ensure that:
— the two approaches give the same solution;
— the diffusion on a FE node is equivalent to the convection
on the corresponding FV cell.
, the solutionat
atof the FV approach;
on the FE mesh;
atof the FE
is the projection ofon the
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3446IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010
Fig. 1. Association of each node of the FE mesh to the cells of the FV mesh:
we link the centers of gravity of two contiguous triangles through the vertex of
their common edge. (a) The nodes Ni correspond to the middle of the edges.
(b) The nodes Ni correspond to the center of the cells ? .
B. Footbridge Operators
approaches. Thefootbridge operators
imation spaces are well described in . They are defined as
a regular triangulation of
is obtained from
FV cell, Fig. 1. The basis functions of the FV approximation
space are obtained from the indicatrix functions of each cell of
the FV mesh. We remark that each first order FE base functions
constructed on the nodes
prolonged on the corresponding FV cell.
anda FV partition of.
, the duality between the FE and FV
of the FE triangulation can be
C. Numerical Scheme
established using the FE method whereas it is not adapted to
the strong nonlinearity of the
term. Sinceis continuous, its
is the surface of a FV cell andis the mean value of
Hence, using the footbridge operators, we combine FE-FV
In the FV method, the flux
andmust be continuous. The mixed numerical scheme is
justified if the diffusion on a FE node is equal to the convection
in the FV cell. This leads to
between two adjacent cells
To ensure the equality in (8), the
the stiffness matrix coefficient
are constructed from
of the FE discretization as
If we define
erty of the FV formulation is ensured.
The discrete problem is formed using (7) and (9). For the
internal FV cells, the discrete problem is written as
, the local conservationprop-
and for the cells on the boundary
, we write
The existenceof thediscrete solution is provedby the saddle-
point theorem. The numerical computation uses the unknowns
. Sinceis continuous and differentiable,
we can use a Newton–Raphson algorithm
III. APPLICATION EXAMPLES
We consider a superconducting cylinder having a radius
and characterized by
. Two values ofare considered;
tors (BiSCCO or YBCO materials) and the second corresponds
to low temperature superconductors (NbTi for example). For
this last case, the Bean model can be used to validate the pro-
A. Cylinder Under Self Field (Transport Current)
The cylinder carries a sinusoidal transport current
used in the boundary condition (3) are obtained by
Fig. 2 presents the current density distribution on the
(T is the period) for the two considered frequencies.
 where the magnetic field is used as state variable is also
given. A good agreement is observed.
We have also compared in Fig. 3, the instantaneous self field
losses obtained by the FE-FV method and those issued from
the FE method. Once again, satisfactory results are obtained.
These results validate the proposed FE-FV approach for weak
. Two frequency values are
KAMENI et al.: A 2D ROBUST FE-FV MIXED METHOD TO HANDLE STRONG NONLINEARITIES IN SUPERCONDUCTORS 3447
Fig. 2. Comparison of current density profiles for ? ? ??. (a) ? ? ??? ?? at
? ? ??? ? ??? ?. (b) ? ? ? ??? at ? ? ??? ? ???? ??.
Fig. 3. Comparison of instantaneous self field losses for ? ? ??,? ? ?????.
In the case where
verge. The average AC losses obtained by our method are 0.024
W/mm. The Bean formula  gives 0.028 W/mm which also
validate the FE-FV method in the case of large
, the FE method does not con-
Fig. 4. Current density distribution ??? at ? ? ??? for ? ? ?? ??
?? ???. (a) ? ? ? ???. (b) ? ? ??? ??.
Fig. 5. Current density distribution ??? at ? ? ??? for ? ? ??? ??
?? ???. (a) ? ? ? ???. (b) ? ? ??? ??.
B. Cylinder Submitted to an External Magnetic Field
The cylinder is submitted to a sinusoidal transverse magnetic
field in the
quency values are considered,
Thecurrentdensity distributioninthecylinderis presentedin
(when the applied field reaches its maximum value) and
show that the penetration of the current density at
is more important than the penetration at
thecurrent density reachesa value
and for. This shows the influence of the
speed variation (i.e., the frequency) of the applied field.
, it can be seen that the penetration of the current
density is almost the same for both frequencies. This is in ac-
cordance with the Bean’s model for which the speed variation
of the applied field does not play any role when
Fig. 6 presents the magnetization versus applied field. In
is equal to 50 mT, a value for which the
cylinder is in a complete penetration state.
, Fig. 6(a), the maximum value of the magne-
tization is higher for
. Again, this shows the influence of the speed varia-
tion (i.e., the frequency) of the applied field.
, the influence of the frequency is practically
inexistent. The maximum value of the magnetization given
by the Bean model can be computed as
which leads to a value
example. Using the proposed method, Fig. 6(b), we obtain
. Again, this validates our method in the
case of large
. Two fre-
than the value obtained at
for the considered
3448IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010 Download full-text
Fig. 6. Magnetization against applied field ??
(b) ? ? ???.
? ?? ???. (a) ? ? ??.
In this paper, we proposed a mixed finite elements-finite vol-
umes numerical scheme to solve the 2-D scalar diffusion of the
electric field in superconducting materials. It has been shown
that the proposed approach can successfully handle the strong
nonlinearity of the E-J power law. Comparisons to purely FE
analyses show the validity of our method in the case of weak
n-values. In the case of large n-values for which the FE method
does not converge, it has been shown via the comparisons to
Bean’s model that the proposed approach is very robust to treat
the strong nonlinearities.
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