A preconditioning strategy for microwave susceptibility in ferromagnets

Abstract

Abstract3D numerical simulations of ferromagnetic materials can be compared with experimental results via microwave susceptibility. In this paper, an optimised computation of this microwave susceptibility for large meshes is proposed. The microwave susceptibility is obtained by linearisation of the Landau and Lifchitz equations near equilibrium states and the linear systems to be solved are very illconditionned. Solutions are computed using the Conjugate Gradient method for the Normal equation (CGN Method). An efficient preconditioner is developed consisting of a projection and an approximation of an "exact" preconditioner in the set of circulant matrices. Control of the condition number due to the preconditioning and evolution of the singular value decomposition are shown in the results.

Stability of static walls for a three-dimensional model of ferromagnetic material

“…Existence results for the Landau-Lifschitz equation can be found in [26], [2], [6] and [18] for the weak solutions, and in [7], [8] and [9] for the strong solutions. Numerical simulations are performed in [3], [4], [19], [20] and [21].…”

Metastability of Wall Configurations in Ferromagnetic Nanowires

“…Partial regularity for particular weak solutions is obtained in [21,23]. Numerical simulations are described in [4,6,28,29,30,31,39]. The main difficulties are the quasilinear character of the equations, the preservation of the saturation constraint by the numerical schemes, and the non local character of the demagnetizing field in 3d.…”

Numerical simulations of ferromagnetic materials

“…It turns out that the system (4) is very ill-conditionned, and there is a crucial need for a suitable preconditioner. Such a preconditioning strategy is proposed in [9] where it is shown that the matrix M can be efficiently preconditioned by (h tot,h ) −1 . The cost of the inversion of the full operator h tot,h being too high, we use an approximationh tot,h of this operator obtained as the projection of h tot,h on skew matrices in the sense of the Froebenius norm.…”

“…In a second section, we focus on the discretization of the stray field (more details are given in [4,8] while its use is described in [7,10]). The last part is dedicated to the microwave susceptibility computation [9] and particularly to the strategy adopted to treat the ill-conditioned systems that need to be solved for each frequence.…”