Efficient micromagnetic finite element simulations using a perturbed Lagrange multiplier method

High performance magnets play an important role in critical issues of modern life such as renewable energy supply, independence of fossile resource and electro mobility. The performance optimization of the established magnetic material system relies mostly on the microstructure control and modification. Here, finite element based in‐silico characterizations, as micromagnetic simulations can be used to predict the magnetization distribution on fine scales. The evolution of the magnetization vectors is described within the framework of the micromagnetic theory by the Landau‐Lifshitz‐Gilbert equation, which requires the numerically challenging preservation of the Euclidean norm of the magnetization vectors. Finite elements have proven to be particularly suitable for an accurate discretization of complex microstructures. However, when introducing the magnetization vectors in terms of a cartesian coordinate system, finite elements do not preserve their unit length a priori. Hence, additional numerical methods have to be considered to fulfill this requirement. This work introduces a perturbed Lagrangian multiplier to penalize all deviations of the magnetization vectors from the Euclidean norm in a suited manner. To reduce the resulting system of equations, an element level based condensation of the Lagrangian multiplier is presented.