A Geometrically Exact Incremental Variational Formulation for Phase Field Models in Micromagnetics

Abstract

Magnetic materials have been finding increasingly wide areas of application. We focus here on the continuum modeling of such materials and present an incremental variational principle for a dissipative micro-magneto-elastic model. It describes the quasi-static evolution of both magnetically as well as mechanically driven magnetic domains, which also incorporates the surrounding free space. Furthermore, the algorithmic preservation of the geometrical nature of the variables is an important challenge from the numerical perspective and to this end we present a novel FE discretization whereby the geometric property of the magnetization director is pointwise exactly preserved by nonlinear rotational updates at the nodes. Functionals in micro-magneto-elasticityFerromagnetic materials develop magnetic domains which may evolve due to applied magnetic fields or stresses. The time evolution of the magnetization is governed by the Landau-Lifschitz-Gilbert equation. We propose a new incremental variational principle similar to [2] and inspired by [3], that is consistent with the inherent geometric nature of the problem. Introduction of primary variable fields: displacement, electric potential, and polarizationLet Ω ⊂ R d denote a vacuum free space box with dimension d ∈ [2, 3] and B ⊂ Ω the domain occupied by the material solid, as depicted in Figure 1 . Ω is considered to be large enough such that the magnetic field induced by the magnetization of the body B is decayed at its surface ∂Ω ⊂ R d−1 . We study the deformation and the magnetization of the body under quasi-static, magneto-mechanical loading in the time interval T ⊂ R + . Hence, we focus on a three field problem with the primary variableswhere we identify the mechanical displacement u, the magnetic potentialφ induced by the magnetization, and the magnetization director m with the important geometric property |m| = 1.1.2 Construction of energy-enthalpy, dissipation, and loading functionals The three field problem is primarily governed by the constitutive energy-enthalpy functional and the rate-type dissipation functional which, for a body B embedded in free space Ω, readdescribing both elastic and magnetic energy storage and the dissipative nature of the microstructure due to domain wall motion. The material free energy Ψ mat is composed of 3 contributions, the elastic energy, the anisotropy energy and the

A projection method for phase field models in micromagnetics

“…A particular challenge in the modeling of such materials is the algorithmic preservation of the geometric constraint on the magnetization field, that remains constant in magnitude [4]. In earlier works, [5] and [6], we presented a phase field model within a geometrically exact incremental variational framework where the geometric property of the magnetization director is exactly preserved pointwise by nonlinear rotational updates at the nodes. In the current work however, we present an alternative approach that involves an operator split along with a projection step for the magnetization vector.…”