Fig. 1: We model a 3D computational domain with a spike-domain microstructure.

a The 3D computational domain is Ω embedded inside an oblate ellipsoid \({{{\mathcal{E}}}}\), which is several times larger in size than Ω. A spike-domain microstructure forms around a nonmagnetic defect that grows under an applied field. (b) Spike-domain microstructures are commonly found in magnetic materials around defects such as cavities and pores in Fe film and Fe–Si crystal, respectively^21,40. (Reprinted subfigure (b) with permission from ref. ⁴⁰. Copyright (2020) by the American Physical Society).