Benchmark on the 3D Numerical Modeling of a
Superconducting Bulk
Kévin Berger, Loïc Quéval, Abelin Kameni, Lotfi Alloui, Brahim Ramdane,
Frédéric Trillaud, Ludovic Makong Hell, Gérard Meunier, Philippe Masson,
Jean Lévêque
To cite this version:
Kévin Berger, Loïc Quéval, Abelin Kameni, Lotfi Alloui, Brahim Ramdane, et al.. Benchmark on the
3D Numerical Modeling of a Superconducting Bulk. 2016. hal-01548728v4
HAL Id: hal-01548728
https://hal.archives-ouvertes.fr/hal-01548728v4
Preprint submitted on 15 Nov 2017
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Benchmark on the 3D Numerical
Modeling of a Superconducting Bulk
Kévin Berger1, Loïc Quéval2, Abelin Kameni2, Lotfi Alloui2,3, Brahim Ramdane4,
Frédéric Trillaud5, Ludovic Makong Hell2,6, Gérard Meunier4, Philippe Masson6, Jean Lévêque1
GREEN, Université de Lorraine, 54506 Vandœuvre-lès-Nancy, France
Group of electrical engineering - Paris (GeePs), CNRS UMR 8507, CentraleSupélec, UPSud, UPMC, Gif-sur-Yvette, France
3
Laboratoire de Modélisation des Systèmes Energétiques LMSE, Université de Biskra, BP 145, 07000 Biskra, Algeria
4
University Grenoble Alpes / CNRS, G2Elab, 38042 Grenoble, France
5
Instituto de Ingenieria, Universidad Nacional Autonoma de Mexico, CDMX, 04510 Mexico
6
University of Houston, Houston, TX, USA
1
2
So far, various numerical models have been developed to simulate high temperature superconductor in 3D. However, the lack of
analytical solutions in the 3D case and the scarcity of experimental data make it difficult to evaluate the accuracy of the proposed models.
In the present work, a benchmark on the 3D numerical modeling of a superconducting bulk is introduced. The problem is tackled by
five independent teams using six different numerical models. After a detailed description of the models and their implementations, the
results obtained are compared and discussed.
Index Terms— 3D modeling, AC losses, FEM, FVM, high temperature superconductors.
I. INTRODUCTION
igh Temperature Superconductors (HTS) are promising for
applications requiring high power densities, e.g.
superconducting motors [1] or bearings [2]. Driven by practical
and commercial applications, various numerical models have
been developed to simulate 2D and 3D problems involving
HTS [3]. Most of them have been developed to assess the AC
losses in thin wires and tapes. Indeed AC losses are one of the
key factors to size properly the cryogenic system [4].
Different formulations have been proposed to deal with the
nonlinearity of superconductors. The most popular one is
probably the H-formulation [5]–[8]. But others formulations
have been used too such as A-V and T- [9]–[11]. The latter
formulation proved to be more suitable to model 3D eddy
current problems due to a better numerical stability and
convergence even though the former may lead to a smaller
problem to solve [12]. In addition, mixed and alternative
formulations as well as topological approach have also been
considered [13]–[16].
These formulations were readily available or have been
implemented or modified to incorporate the nonlinearity of the
superconducting material in various softwares. The following
commercial finite element method (FEM) softwares have been
used to model superconductors: ANSYS, Opera-3D, Flux2D,
Flux3D, COMSOL Multiphysics and FlexPDE [17]–[21].
Alternatively, open-source FEM codes have been tried out with
a special mention of ONELAB and FreeFem++ [22], [23]. It
should be noted that even though FEM has proved to be a
powerful method to carry out modeling of superconductors,
other approaches have been developed based on finite
difference method, finite volume method, variational principle
or integral methods either to cross-check results or as an
alternative approach to lower the computational load [24]–[29].
In this vast pool of options tackling diverse problems, it is
H
difficult to point out the advantages and disadvantages of one
method compared to the other.
Over the years, partial attempts to address this issue have
been conducted [11]. The present work is a step further in that
direction. Methods and formulations used by five independent
teams involved in HTS numerical modeling are compared on
the specific problem of the magnetization of a superconducting
cube [30], [31]. This problem is of interest because of the
specific current density distribution that may be induced in the
cube.
In section II, we describe in details the benchmark problem.
Section III gives an overview of the numerical models used by
the different teams. The main results are summarized in section
IV: the AC losses in partial and complete penetrations, and the
current density distributions.
II. BENCHMARK MODEL
A. Geometry
The geometry is a superconducting cube having an edge
length d of 10 mm. The cube is surrounded by a 100 mm box of
air, as shown in Fig. 1.
B. Material properties
The electrical behavior of the superconductor is modeled
with a power law linking the electrical field E to the current
density J as:
n 1
Ec J
2
2
2
(1)
J with J J x J y J z
Jc Jc
The parameters Jc = 2.5×106 A/m2 and n = 25 have been
chosen close to values obtained from experimental
characterization on cylindrical Bi-2223 samples with
Ec = 1 µV/cm [32]. This opens the possibility to perform later
experiments to validate this benchmark. The magnetic property
of the superconductor can be fully described by the nonlinear
E
dependence of its electrical resistivity upon current which is
directly inferred form the power law. Therefore, it is sensible to
use the magnetic permeability of vacuum µ0 to describe the BH curve for magnetic flux densities ranging from a few milliTesla, corresponding to the lower critical magnetic field, to the
upper critical magnetic field of type-II superconductors.
C. Case studies
A uniform sinusoidal external magnetic flux density Ba(t) of
amplitude Bmax, and frequency 50 Hz, is applied to the cube
along the z axis.
The full penetration flux density Bp can be estimated
according to Bp = µ0 Jc d / 2 = 15.7 mT [33]. Two cases will be
therefore investigated:
Partial penetration: Bmax = 5 mT < Bp. In this case, a
current-free region exists in the center of the superconductor
region.
Complete penetration: Bmax = 20 mT > Bp. In this case, the
current penetrates the entire superconductor region.
The variables of interest are the instantaneous AC losses pAC:
pAC (t )
E J dv
(2)
HTS cube
and the steady-state average AC losses P computed as [8]:
T
P
2
p AC (t ) dt
T T/ 2
(3)
III. NUMERICAL MODELS
A. General description
Table I summarizes the methods, the formulations, the
software and the contributors for each numerical model.
minimum time step, which can lead to long computation times
especially in 3D.
GetDP is an open-source finite element solver using mixed
elements to solve partial differential equations in 1D, 2D and
3D. The computation of AC losses in superconductors has been
recently reported [38].
The Finite Volume Method (FVM) can be regarded as a special version of the weighted residuals method. It consists in dividing the domain into a number of non-overlapping subdomains or control volumes such that there is one control volume
surrounding each computing node. The weighting function is
setting to be unity over one subdomain and zero everywhere
else. This method proved to be numerically stable in the case of
highly nonlinear problems. For this reason, this approach has
been implemented in a 3D computation code, under the
MATLAB environment, dedicated mainly to solving electromagnetic problems with superconducting materials [24].
Although there is no validation shown in this paper, all the
mentioned methods have been tested on 1D and/or 2D cases,
and successfully compared with the results of analytical formulas, e.g. [33].
B. Mesh
Simulations have been carried out using the same reference
mesh for the models (B.1)-(B.4). It consists of 27 982
tetrahedrons with 8 688 tetrahedrons in the HTS cube region,
which leads to 33 612 DOF for 1st order elements. It was chosen
as the best compromise between the estimated AC losses and
the computation. Fig. 1 shows the mesh (left) with the HTS
cube at the center and its quality in a y-z cut-plane at x = 0 (right).
TABLE I
NUMERICAL MODELS
Label
Method
Formulation
Software / Code
(B.1)
FEM
H
Daryl-Maxwell
Task leader
B. Ramdane &
G. Escamez
(B.2)
FEM
COMSOL(1)
K. Berger
H
(B.3)
FEM
COMSOL(2)
K. Berger
H
(B.4)
FEM
COMSOL(3)
L. Quéval
H
(B.5)
FEM
GetDP
A. Kameni
H
(B.6)
FVM
A-V
MATLAB
L. Alloui
In COMSOL 5.0, the H-formulation can be implemented: (1) by using the
available mfh physics, (2) by substituting H into A in the A formulation of the
mf physics, (3) by manually implementing the differential equations with the pde
physics.
Daryl-Maxwell is a homemade FEM software developed at
Polytechnique Montréal. It is devised specifically for solving
large 3D nonlinear electromagnetic problems in the time domain. It was initially developed to tackle problems involving
superconductors [34] and ferromagnetic materials. Daryl-Maxwell uses an H-formulation with fixed and/or adaptive time
stepping.
COMSOL Multiphysics is a commercial multiphysics FEM
software [18]. In the superconductor modeling community, it
has been intensively used for 2D and 3D modeling of superconductors, mainly using the H-formulation implemented with the
PDE physics [7], [8], [35]–[37]. It provides only adaptive time
stepping for the temporal resolution with little control on the
Fig. 1. (left) Mesh with the HTS cube at the center. (right) Quality of the mesh
in a y-z cut-plane at x = 0.
Due to tag identification problems, the mesh used in GetDP
(B.5) has been built with Gmsh [39]. Because the solver is very
sensitive to the quality of the mesh and its size, a mesh denser
than the reference mesh was selected. This mesh has been
optimized to get the best results in a reasonable computation
time. It consists of a total of 59 617 tetrahedrons with 25 930
tetrahedrons in the HTS cube.
The FVM used by (B.6) uses a cubic mesh with 20 elements
per edge of the HTS cube which leads to 8 000 hexahedrons in
the superconducting domain.
C. Technical information about the methods
Table II summarizes the methods used for each model. Note
that:
All methods used an implicit time discretization scheme,
FEM methods are based on edge elements (Nédélec) of
order 1,
(B.1) uses PARDISO as direct solver and fixed time steps
but a variant of DASSL is also possible.
In COMSOL, absolute and relative tolerances have been
set at 10-3 and 10-6, respectively.
TABLE II
METHODS USED FOR EACH NUMERICAL MODEL
Time dependent solver
DASPK
Fixed time steps
Direct solver MUMPS
Direct solver PARDISO
Gauss Seidel iterative
method
Newton Raphson
method
(B.1)
X
X
(B.2)
(B.3)
(B.4) (B.5) (B.6)
X
X
X
X
X
X
X
X
X
X
X
X
X
X
D. Treatment of the non-linearity arising from E(J)
All H-formulations except (B.3) solve:
IV. RESULTS AND DISCUSSION
A. AC losses
The instantaneous AC losses pAC for partial and complete
penetration cases, are shown in Fig. 2. Since these calculations
are based on a time transient nonlinear simulation with a
superconductor initially in a virgin state, it is necessary to run
the simulation until the initial transient dies out and the steady
state is established. In the present cases, the steady state regime
is reached after half a period. The results of the different
numerical models are in good agreement with each other with
the exception of (B.6). One can see a slight overshoot, for the
partial penetration case, of the instantaneous losses at the
maximum value with some discrepancies during the increasing
portion of the losses. The A-V formulation presents additional
challenges compared to the H-formulation especially in terms
of convergence and stability.
n 1
E J
H
0
J H 0 with J c
(4)
Jc Jc
t
Since (B.3) is based on an A formulation where A H, then
and . Consequently, (J) is needed as an input of
the model and not (J). In order to avoid instabilities when the
conductivity of the HTS material tends to infinity,
corresponding to the case of null current density, the following
expression was used:
1
E J n 1
J c 0 with 0 1014 m (5)
Jc Jc
To avoid the evaluation of zero at a negative power in
COMSOL, (B.3) and (B.4) computed the norm of current
density using:
J J x2 J y2 J z2 with 2.22 1016 A2 m4 (6)
The A-V formulation employed by (B.6) involves a Coulomb
gauge (div A = 0) and the conservation of the current density
(div J = 0) which leads to the following set of equations:
A
1
1
0 A 0 A E t V 0
(7)
E A V 0
t
1
E E ( n 1)/ n
(8)
with E c
0
J c Ec
The conductivity of the HTS material is also required but the
function is different, indeed (B.6) uses (E) compared to (J)
for (B.3). Previous studied have shown that the use of (E)
made the convergence of the solver much more difficult [18],
[40].
To ensure the stability and convergence of the solver, (B.5)
requires a linearization at each time step of the nonlinear function in (4) which is achieved as shown in [41].
Fig. 2. Instantaneous AC losses pAC for the six different models with (a)
Bmax = 5 mT and (b) Bmax = 20 mT.
When the applied field amplitude increases, we can notice
that the instantaneous AC losses peaks shift towards t = T/2 and
t = T, corresponding to the moment when the derivative of the
applied field is maximum.
The average AC losses P and the computation times are given
in Table III.
TABLE III
COMPUTING TIME AND AC LOSSES
Bmax = 5 mT
Bmax = 20 mT
Comp. time
P (mW)
Comp. time
P (mW)
(1)
(B.1) (1) 134 min 00 s
0.8354
318 min 00 s
14.3399
(2)
(2)
(B.2)
42 min 20 s
0.7933
931 min 38 s
13.0074
(2)
(2)
6 min 39 s
10 min 55 s
(B.3)
0.7933
13.0090
(2)
(2)
(B.4)
11 min 06 s
0.7969
17 min 17 s
13.0088
(B.5)
0.8396
13.9087
(B.6)
0.8303
13.8453
The computers used are: (1) Intel® Core™ i5-2540M CPU @ 2.60 GHz
(Turbo Boost 3.30 GHz), 8 GB RAM, Mac OS X 10.10, and (2) Intel® Core™
i7-6700 CPU @ 3.40 GHz (Turbo Boost 4.00 GHz), 32 GB RAM, Windows 7
64 bits.
Label
Values of losses are below 1 mW for Bmax = 5 mT, while they
grow up to 13 mW for Bmax = 20 mT. While the instantaneous
AC losses predicted by (B.6) were different from the other
models, the average AC losses are similar due to the integration
(3). Choosing (B.3) as the reference, the maximum deviation is
5.8 % and 10.2 % for Bmax = 5 mT and 20 mT, respectively.
Although the results are very similar, these values reflect clearly
some discrepancies arising from differences in the
implementation of the methods.
(B.3) seems to be the fastest model. It is interesting to notice
that methods (B.2), (B.3) and (B.4) use the same software with
exactly the same solvers settings and tolerances. However, (B.2)
is six times slower than (B.3). This is surprising because (B.2)
is the formulation proposed by COMSOL for solving problems
with superconductors since the version 4.3b. This is probably
the reason why (B.4) is still widely being used. Therefore, some
questions remain pending and the answers will be reported as
soon as we can get them from the COMSOL support.
Fig. 3 shows the solver outputs of (B.2) with the number of
iterations (a) and the stepsize taken by the adaptive solver (b).
We observe that the higher the number of iterations, the smaller
the stepsize, which leads to large computing times. Excluding
the transient, the stepsize of (B.2) are roughly 20 times larger
than the ones observed with (B.3) and (B.4) with 5 times more
iterations. This explains the differences in computing times reported in Table III between (B.2) and (B.3) but not between
(B.3) and (B.4).
corners of the cube. This Jz component is a consequence of the
self-field that, for square loops of current, is higher near the
corners. This tends to bend the current lines as shown with the
streamlines in Fig. 4 (left). The current flowing in square loops
cannot fully cancel the self-field that is canceled thanks to the
Jz components. For applied fields well above the penetration
field, the self-field is not relevant, and the Jz components tend
towards zero everywhere. This explanation is consistent with
those given in [43].
Jx / Jc
Jx / Jc
Fig. 4. Normalized current density Jx / Jc from (B.3) at t = 15 ms for
Bmax = 5 mT (top) and 20 mT (bottom). y-z cut planes are located at x/d = ± 1/3
and x = 0. The current flows following the shape of the borders as shown with
the cones in the x-y plane at z = 0.
V.
Fig. 3. (a) Number of iterations vs. time and (b) stepsize taken by the adaptive
solvers (B.2)-(B.4).
B. Current density
Fig. 4 shows the normalized current density Jx / Jc from (B.3)
at t = 15 ms for the partial penetration (left) and the complete
penetration case (right). The y-z cut planes used for the
representation are located at x/d = ± 1/3 and x = 0. The induced
currents are flowing in closed square loops in a plane
perpendicular to the applied field. In a conventional ohmic
conductor, the current loops would be more circular close to the
center of the cube. This particularity of superconductors is due
to the nonlinear relationship between E and J. This
phenomenon has been seen from the magneto-optic images
performed on thin films [42].
Another particular feature of superconductors is that the
induced currents flow with a non-zero Jz component close to the
CONCLUSION AND PERSPECTIVES
A benchmark on 3D superconductor modeling has been
proposed. It has been solved using six different numerical
models developed by five independent teams. This is also the
first time that three different methods for solving
superconducting problems are implemented and compared
using COMSOL software. Even if all answers are not yet given,
we clearly state their differences and advantages. The results
show good agreement between the models when considering
the AC losses, despite large difference in terms of
computational load. The comparison of a model implemented
using the open-source code GetDP with other commonly used
models is a significant contribution of this article. Explanations
have been proposed to clarify the square shape of current loops
observed in the HTS cube, and the existence of the Jz
component near the corner of the cube.
This work will be pursued and an experimental setup is
planned to be built to strengthen the obtained numerical results.
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