After successful completion of the course, students are able to understand some mathematical models of magnetic materials, which involve nonlinear partial differential equations (PDEs), nonlocal effects, nonconvex energies and constraints. The analytical and numerical techniques presented in the lecture can be applied to treat other mathematical models.
Magnetic phenomena have been known for millennia, since in ancient times people noticed that lodestones (magnetite) could attract iron. Nowadays, the use of magnetic materials in technological processes is ubiquitous (e.g., energy transformation and data storage). Moreover, they play an essential role in many devices (e.g., magnetic sensors and actuators, electric motors and generators, microphones, loudspeakers, telephones, and hard disk drives).
Magnetic processes are multiscale and multiphysics phenomena and their modeling involves nonlinear partial differential equations (PDEs), nonlocal effects, nonconvex energies and constraints.
In this lecture, we give an overview on the mathematics behind magnetic materials, touching on several topics, mostly in the fields of mathematical modeling, analysis, and numerics.
Topics and keywords:
- Modeling: magnetic moment, type of magnetism, atomistic vs. continuum theories, micromagnetics, hysteresis, Maxwell equations, Landau-Lifshitz-Gilbert (LLG) equations.
- Analysis: micromagnetic energy minimization, thin-film limits, existence and (non)uniqueness results for LLG equations.
- Numerics: numerical treatment of Maxwell and LLG equations, finite element methods, boundary element methods, unconditional stability and convergence.
Further topics could be also addressed depending on students' interests.
