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The development of permanent magnets containing less or no rareearth elements is linked to profound knowledge of the coercivity mechanism. Prerequisites for a promising permanent magnet material are a high spontaneous magnetization and a sufficiently high magnetic anisotropy. In addition to the intrinsic magnetic properties the microstructure of the magnet plays a significant role in establishing coercivity. The influence of the microstructure on coercivity, remanence, and energy density product can be understood by using micromagnetic simulations. With advances in computer hardware and numerical methods, hysteresis curves of magnets can be computed quickly so that the simulations can readily provide guidance for the development of permanent magnets. The potential of rare-earth reduced and free permanent magnets is investigated using micromagnetic simulations. The results show excellent hard magnetic properties can be achieved in grain boundary engineered NdFeB, rare-earth magnets with a ThMn 12 structure, Co-based nano-wires, and L1 0 -FeNi provided that the magnet's microstructure is optimized.

Conjugate gradient methods for energy minimization in micromagnetics are compared. When the step length in the line search is controlled, conjugate gradient techniques are a fast and reliable way to compute the hysteresis properties of permanent magnets. The method is applied to investigate demagnetizing effects in NdFe12 based permanent magnets. The reduction of the coercive field by demagnetizing effects is µ0H = 1.4 T at 450 K. I. INTRODUCTIONThe computation of hysteresis properties of large ferromagnetic systems such as sensor elements or permanent magnets require fast and reliable solvers. Hysteresis simulations are based on the theory of micromagnetics "Brown (1963)". The primary purpose of these simulations is to understand the influence of the microstructure on magnetization reversal. In this work we are focusing on the role of demagnetizing fields in platelet shaped grains of permanent magnets. We also describe the key elements of a micromagnetic solver suitable for simulating large magnetic systems.After discretization of the total Gibbs free energy with finite elements or finite differences the states along the demagnetization curve can be computed by subsequent minimization of the energy for decreasing applied field as outlined in "Kinderlehrer (1997)". The system is in a metastable state. A change of the applied field shifts the position of the local energy minimum. At a critical field, the magnetization becomes unstable. An irreversible switching occurs which is seen as a kink in the demagnetization curve. Then the system either accesses a different metastable state or if fully reversed the magnetization is in a stable state. A reliable numerical method for energy minimization must track all local minima along the demagnetization curve. The resulting algebraic minimization problem is large. Typically the number of unknowns is in the order of 10 to 50 million for a model magnet consisting of around 10 grains. Therefore fast numerical methods are required to obtain results in a Electronic

We use a machine learning approach to identify the importance of microstructure characteristics in causing magnetization reversal in ideally structured large-grained Nd 2 Fe 14 B permanent magnets. The embedded Stoner-Wohlfarth method is used as a reduced order model for determining local switching field maps which guide the data-driven learning procedure. The predictor model is a random forest classifier which we validate by comparing with full micromagnetic simulations in the case of small granular test structures. In the course of the machine learning microstructure analysis the most important features explaining magnetization reversal were found to be the misorientation and the position of the grain within the magnet. The lowest switching fields occur near the top and bottom edges of the magnet. While the dependence of the local switching field on the grain orientation is known from theory, the influence of the position of the grain on the local coercive field strength is less obvious. As a direct result of our findings of the machine learning analysis we show that edge hardening via Dy-diffusion leads to higher coercive fields.

Finite element micromagnetic simulations are used to compute the temperature-dependent hysteresis properties of Nd2Fe14B permanent magnets in order to assess the influence of a hard (Dy,Nd)2Fe14B shell. The simulations show that the 4 nm thick shell cancels out the reduction in coercivity from an outer defect layer, which is known to exist at the grain boundaries in NdFeB permanent magnets. Activation volumes are computed and shown to depend on the structure's configuration as well as the temperature.