BELFEM is a special-purpose finite element framework for modeling the electromagnetic and thermal behavior of HTS tapes, the cables wound from them, and the coils and magnets that contain them.
These examples cover geometries and physics modeled by BELFEM, from magnet cross-sections to complete 3D cable and coil windings. Every case is solved with the h-φ formulation. Superconducting tapes use thin-shell elements; bulk conductors, iron, and air are meshed as volumes.
Longitudinal section of a superconducting undulator: two rows of REBCO tape coils on an iron yoke, driven by a user-defined current waveform at 15 K. The plot shows the flux density in the yoke and in the beam gap, where the field alternates along the beam axis.
A quarter-section model of a cos-theta dipole contains 68 conductor blocks (outlined at lower left) inside an iron yoke with a nonlinear B-H curve. One symmetry and one antisymmetry plane reduce the magnet to a quarter. The current ramps to 8.5 kA; the plot shows the flux density at full current.
Upper half of a proton-therapy gantry dipole with 464 Bi-2223 thin-shell tapes in its coils and an iron yoke. The plot combines the flux density magnitude in the yoke and aperture with the normalized current density j/jc in each tape.
A Conductor on Round Core cable consists of REBCO tapes wound helically around a round core and modeled as thin shells. The plot shows the normalized current density j/jc and its direction along each tape under transport current. A periodic condition over one twist pitch, with translation or twist, represents the infinite cable.
A stack of REBCO tapes carries transport current past two local defects. The current density drops in the damaged regions and redistributes into the undamaged width of the tape.
A racetrack coil is wound from a stack of tapes and has two current leads. The normalized current density j/jc varies along the winding, and the arrows show the surrounding flux density.
BELFEM solves the magnetodynamic h-φ formulation: the magnetic field h is an edge-element unknown in the conducting regions, and a scalar potential φ carries the field through the non-conducting air. Superconducting tapes are far thinner than they are wide, so BELFEM represents them as thin shells whose layers (superconductor, substrate, stabilizer) are resolved inside the shell without a volume mesh of the tape. BELFEM automatically generates the cohomology cuts required by multiply connected conductors from the mesh topology.
In addition to the field problem, BELFEM couples:
BELFEM is a collaboration between the Berkeley Center for Magnet Technology at Lawrence Berkeley National Laboratory and the Department of Electrical Engineering at Polytechnique Montréal, where the h-φ and thin-shell formulations it implements originate. The publications page lists the papers behind the code.
BELFEM ships its own material database. Every property is a function of temperature. For the pure metals and alloys the residual resistivity ratio is a free input, and Kohler tables give the resistivity and the thermal conductivity their dependence on both temperature and magnetic field, so a deck states copper with an RRR and gets consistent transport properties from cryogenic to room temperature.
| Pure metals |
|
|---|---|
| Solders and alloys |
Mixed by formula from the pure metals, in mass percent, with the RRR of your choice:
|
| Other materials |
|
| B-H curves |
|
| HTS conductors |
Measured jc(B, θ, T) and n(B, θ, T) tables from a few kelvin to above 90 K. The REBCO tables are built from the measurements of the Robinson Research Institute's HTS database:
|
| Superconductor models |
|
Every tabulated property is stored as a spline: the conductor tables are quadratic B-splines in temperature, field, and angle, and the metal and B-H curves are splines as well. Their first derivatives are therefore continuous, so the Newton-Raphson iteration sees no jumps in the tangent. That is what lets a nonlinear transient converge cleanly on these materials.
Anything not in the database comes in through a user material plugin, as the racetrack and tape-quench examples in the source tree show.
BELFEM is written in C++17 with Fortran numerical kernels and runs on distributed meshes with MPI. It links to the major sparse direct solvers (STRUMPACK, MUMPS, PETSc, SuperLU, PARDISO) and to either Armadillo or Blaze as its dense linear algebra backend. Memory management is explicit, with no hidden allocations on the assembly and solve paths, and a test suite ships with the build.
Meshes are generated with gmsh, and results are written in Exodus format for ParaView. BELFEM's numerical dependencies are packaged as SCLS, the Scientific Core Library Stack, so a single package-manager command installs a consistent set of compilers, MPI, BLAS/LAPACK, and solvers.
BELFEM 0.9.0 was released on September 4, 2026 under the Lawrence Berkeley National Laboratory BSD variant (BSD-3-Clause-LBNL). The source is hosted on the BELFEM GitLab at Berkeley Lab and mirrored on GitHub, and the rendered documentation covers the input file, the module guides, and the API. The source distribution includes eighteen input decks covering the cases above, from a two-dimensional dipole that runs in minutes to a coupled magneto-thermal tape stack with periodic boundaries.
To build BELFEM, you need CMake, a C++17 compiler, a Fortran compiler, and Open MPI:
git clone https://belfem.lbl.gov/gitlab/codes/belfem.git
cd belfem && mkdir build && cd build
cmake .. && make -j8
make checkThe GitHub mirror serves the same tree: git clone https://github.com/cmesse/belfem.git.
If you use BELFEM in academic work, please cite Messe et al., BELFEM: a special purpose finite element code for the magnetodynamic modeling of high-temperature superconductor tapes, Supercond. Sci. Technol. 36 (2023), doi:10.1088/1361-6668/acf7f9.