Temporal upscaling in micromagnetism via heterogeneous multiscale methods

Abstract

We consider a multiscale strategy addressing the disparate scales in the Landau-Lifschitz equations in micro-magnetism. At the microscopic scale, the dynamics of magnetic moments are driven by a high frequency field. On the macroscopic scale we are interested in simulating the dynamics of the magnetisation without fully resolving the microscopic scales.The method follows the framework of heterogeneous multiscale methods and it has two main ingredients: a micro-and a macroscale model. The microscopic model is assumed to be known exactly whereas the macro model is incomplete as it lacks effective quantities. The two models use different temporal and spatial scales and effective parameter values for the macro model are computed on the fly, allowing for improved efficiency over traditional onescale schemes.For the analysis, we consider a single spin under a high frequency field and show that effective quantities can be obtained accurately with step-sizes much larger than the size of the microscopic scales required to resolve the microscopic features. Numerical results both for a single magnetic particle as well as a chain of interacting magnetic particles are given to validate the theory.

Modelling long-range interactions in multiscale simulations of ferromagnetic materials

“…Recently, an HMM approach has been formulated in application to multiscale problems arising in micromagnetism. First, in [7], a multiscale method has been proposed to simulate the coarse-scale dynamics of a chain of atomistic spins. The atomistic spins were subjected to a high-frequency external field and a mathematical investigation of the convergence rates in relation to the coupling/upscaling errors, originating from a micro-macro coupling, was given for a simplified setting of a single spin.…”

“…In this subsection, the basic mathematical tools and notations for the HMM algorithms in subsections 3.2.2 and 3.2.3 are introduced. The HMM algorithms developed in [7] and [6] are based on the notion of upscaling, where a local average of small scale features in the atomistic solution (2.1) is computed and used in a macroscale model. The local averaging takes place in small domains in space and time.…”

“…HMM at non-zero temperature. An extension of the zero temperature algorithm from [7] to non-zero temperature, was introduced in [6]. Modelling the long-range interactions requires yet another set of modifications to the algorithm from [6].…”

“…Therefore, these interactions should be handled in a conceptually different way -using a continuum approach. The aim of this paper is to present an efficient strategy to include the long-range interactions in the partitioned-domain strategy based on [23,24] and the upscaling formalism developed in [7,6]. This paper is organised as follows.…”

Heterogeneous multiscale methods for the Landau-Lifshitz equation

“…For a simplified Landau-Lifshitz problem with a highly oscillatory external field and no spatial interaction, a possible HMM setup was introduced in [3] and extended to a non-zero temperature scenario in [4].…”