Sensitivity analysis framework for micromagnetism with application to the optimal shape design of magnetic random access memories

The paper concerns the Landau–Lifshitz (LL) equation with space variable gyromagnetic factor γ ⩾ 0. This allows for a flexible description of the magnetic properties of materials. The area of a domain where the material exhibits ferromagnetic behaviour is simply compact support of γ. Outside this region, material is non-magnetic (γ = 0). First, a stability analysis of the LL equation and the corresponding sensitivity equation is carried out. We have to deal with a loss of strong ellipticity due to the fact that we allow γ to vanish in a part of a domain. Therefore we introduce a regularized problem where γ > for a constant positive and we investigate the limit process → 0. Next, we apply previous theoretical results to the optimal shape design of a magnetic random access memory (MRAM) core. A primal-dual approach is employed to tackle this problem. In the last section, numerical results for the standard model of MRAM are presented.