Benchmark on the 3D Numerical
Modeling of a Superconducting Bulk
K. Berger1, L. Quéval2, A. Kameni2, L. Alloui2,3,
B. Ramdane4, F. Trillaud5, L. Makong Hell2,6,
G. Meunier2, P. Masson6, and J. Lévêque1
1- GREEN, Université de Lorraine, 54506 Vandœuvre-lès-Nancy, France, 2- 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.
AC losses are one of the key parameters in sizing High Temperature Superconducting (HTS) large scale devices. Therefore, their estimation has to be
realistic in order to properly design the cryogenic system. So far, various 3D numerical models have been developed to that estimate AC losses. However,
the lack of analytical solutions in the 3D case and the scarcity of the experimental data make it difficult to evaluate their accuracy. Here, a benchmark on
the 3D numerical modeling of a superconducting bulk is introduced. After a detailed description of the methods and their implementations, the results
obtained by various independent teams are compared and discussed.
☑A benchmark for 3D modeling of superconductors has been solved
with 6 different numerical models implemented in 4 different
environments (Daryl-Maxwell, COMSOL, GetDP and MATLAB).
☑Differences between the models helped us understanding the
impact of various modeling decisions: resistivity of the air domain,
resistivity vs. conductivity based model, etc.
☑The impact of the mesh needs to be analyzed.
everywhere, 1 m,
and ( ) in the HTS cube
The studied geometry is an HTS cube of side d=10 mm,
The air domain is delimited by a100 mm side cube.
Equations, solvers and methods used
In COMSOL 5.0, several methods can be used to implement an Hformulation: (1) with the
developed package called MFH physic, (2) by analogy between Hinto Ain the A
formulation of the MF physic, (3) by manually implementing the constitutive equations
with the PDE physic.
2.5 10 /m and 25 have been extrapolated
from experimental results obtained on Bi-2223 samples
with 1 µV/cm
Superconducting and air domains are discretized with tetrahedral elements.
The mesh “cube_12”is composed of 27 982 volume elements including
8688 elements in the HTS cube.
cube_12 is equivalent in terms of mesh density to cube_12r, based on a
regular mapped mesh with 12 elements on each edge of the HTS cube.
cube_12 was chosen as the best compromise between the AC losses
calculation and the computation time based on parametric studies made on
various mesh sizes.
Formulation Code Task leader
B. Ramdane &
42 min 20 s
06 min 39 s
11 min 06 s
with and 10
A E J
H formulation A-Vformulation (B.6)
All methods use an implicit time discretization scheme,
FEM methods are based on edge elements (Nédélec) of order 1,
(B.1) uses PARDISO solver and fixed time steps (but a variant of DASSL is also possible),
(B.2), (B.3), (B.4) and (B.5) use MUMPS as direct solver,
In (B.2), (B.3), and (B.4), linearization is obtained through a Newton-Raphson method,
In (B.2), (B.3), and (B.4), time stepping adaptive is performed with a DASPK solver,
In (B.5), a first order linearized expression of E(J)is used,
(B.1), (B.5) and (B.6) used fixed time steps with usually 400-500 steps per period,
In (B.6), solutions are obtained thanks to a Gauss-Seidel iterative method.
in (B.3), we used:
1 with 10
(B.3) Normalized current density Jx/Jcfor Bmax = 5 mT at t= ¾T
In a bulk superconductor, current
flows following a square path
Current density can only take
values close to ±Jc,
Another particular feature for
superconductors is that induced
currents flow with a non-null Jz
component close to the corners!
Comparison of the instantaneous AC losses for the 6 methods
•Applied field case
A uniform sinusoidal external magnetic flux density at
50 Hz with an amplitude Bmax,is applied to the cube along the z-axis.
( ) sin(2 )B t B f t