About `Z-mag`¶

Z-mag is an open-source finite-element library for micromagnetics, developed in the Tape Research group at IBM Research, Zurich. It is designed for large-scale, MPI-parallel simulations of ferromagnetic systems and is built on top of DOLFINx / FEniCSx for mesh management, parallel linear algebra, and parallel input/output.

Stereographic parametrization¶

The magnetization at each point in space is a unit vector, so its evolution must stay on the unit sphere. Standard ways of enforcing this constraint can involve trade-offs that affect accuracy or robustness, especially in long simulations (see Magnetization dynamics).

Z-mag instead parametrizes the magnetization through a two-chart stereographic atlas. The unit-norm constraint is built into the parametrization, which removes the need for any explicit enforcement mechanism. The mathematical development is in Magnetization dynamics and Stereographic formulation.

Solver variants¶

Two variants are provided, distinguished by which energy contributions are active:

demag — exchange, anisotropy, Zeeman, and demagnetizing energy. The demagnetizing field is computed at every time step via the Fredkin–Koehler hybrid FEM–BEM method, which handles the open exterior domain exactly without domain truncation. See Magnetostatics and Solver algorithms.
nodemag — exchange, anisotropy, and Zeeman only, without magnetostatics. Appropriate for problems where the demagnetizing contribution is negligible or intentionally excluded; computationally lighter than demag.

Both variants share the same implicit BDF time integrator with adaptive timestep control and a multi-phase acceptance logic that combines algebraic, geometric, and thermodynamic criteria. The full acceptance algorithm is documented in Solver algorithms.

Repository¶

Source code, example drivers, and the C++ BEM assembler are published under the MIT License at:

https://github.com/IBM/Z-mag

Stable releases follow Semantic Versioning (v1.0.0, v1.1.0, …) and are described in CHANGELOG.md at the repository root. The environment.yml shipped with each tag pins the exact DOLFINx / FEniCSx stack required by that release. Installation instructions are in Installation.

How to cite¶

If you use Z-mag in published work, please cite:

Stella Brach, Federico Soldà, Kathrin Aebischer, and Mark A. Lantz. Z-mag: A two-chart stereographic formulation for finite-element micromagnetic simulations. Computer Methods in Applied Mechanics and Engineering 462 (2026) 119290. doi:10.1016/j.cma.2026.119290

BibTeX (also in CITATION.cff and the repository README.md):

@article{brach2026zmag,
  title     = {{Z-mag}: A two-chart stereographic formulation for finite-element micromagnetic simulations},
  author    = {Brach, Stella and Sold{\`a}, Federico and Aebischer, Kathrin and Lantz, Mark A.},
  journal   = {Computer Methods in Applied Mechanics and Engineering},
  volume    = {462},
  pages     = {119290},
  year      = {2026},
  issn      = {0045-7825},
  doi       = {10.1016/j.cma.2026.119290},
  url       = {https://doi.org/10.1016/j.cma.2026.119290}
}

The same entry is [Brach et al., 2026] in this manual. For a specific software snapshot, cite the git tag or commit as well.

License and contributors¶

The library was developed by:

Stella Brach, Staff Research Scientist — IBM Research, Zurich
Kathrin Aebischer, Research Scientist — IBM Research, Zurich
Federico Soldà, former intern

About Z-mag¶