Numerical analysis of nonlinear models of ferromagnetic materials.

摘要

Micromagnetic simulation has been an active research area for engineers, physicists and applied mathematicians due to the importance of applications of magnetic materials, for example, to data recording, in modern industry. In this thesis, we derive some analytical properties and numerical schemes for two nonlinear ferromagnetic models: the micromagnetic model and the eddy current micromagnetic model.; Using mass-lumping, we apply a finite element method to discretize the LL equation in space and prove the convergence of the semi-discrete scheme in the case where the effective field contains no exchange contribution. To discretize the LLG equation in time, we choose an explicit/implicit time stepping scheme which preserves the norm of the magnetization. We also prove a stability result when the effective field contains only the exchange energy. The discretization of our model ends up with a large sparse matrix problem to be solved at each time step. To efficiently solve the linear problem from computation of the stray field in the micromagnetic model, the AMG method is investigated. We propose an adaptive coarsening algorithm for the AMG which has good performance on non-uniform grids, especially applied to larger problems.; In order to fully capture the dynamic effects of a disk writer, we investigate a coupled eddy current and micromagnetic model. The existence of a weak solution for the model is proved. Energy conservation for both continuous and semi-discrete cases is shown. Ultimately we discretize the model by a simple finite difference method derived from a mass-lumped finite element method that is adapted to conserve energy. We prove the convergence in the semi-discrete case and show some numerical results.