Simulação da dinâmica do micromagnetismo de vórticesiImplementados em plataformas de programação em paralelo

Abstract

Modelling ferromagnetic materials is a relevant global issue, for most of our electronic devices and equipment, for all kinds of purposes (be it medicinal, industrial, etc.), rely intrinsically on the effects of their local magnetization to properly function, and this is true of both commonplace and developing technology. The established model in analyzing such materials is based on the Landau-Lifshitz-Gilbert(LLG) equation, which describes the time evolution of the vector field Magnetization in a solid ferromagnet. The equation is non-linear, and as such the use of analytical solutions is limited, even as approximations, to all but the simplest cases, requiring a numerical treatment. However, owing to the "long range" nature of the interactions in a general ferromagnet (we cannot neglect the interaction between any two elements, no matter how far apart), even a numerical treatment cannot be done to structures of relevant sizes using appropriate discretizations, for these interactions imbue a high order in the number of operations required to numerically approximate the model, translating into elevated computational costs. A previously proposed solution is to introduce a cut radius for every interaction in a ferromagnetic system, but for sizes of technological interest (10−9 to 10−6 meters), truncating the interactions at a distance severely modifies the resulting dynamics and boundary conditions. Other approximations such as Fast Multipole Methods (FMM) consider long range interactions to be short range interactions among “cells” covering parts of the full system, by introducing an accuracy parameter ∈ representing the method’s disagreement in relation to considering all interactions fully, with ∈ being inversely proportional to the number of operations in a simulation run. In this work, in order to solve this issue, without introducing approximations for the long range interactions acting, we use the parallel programming platform CUDA to perform dynamics simulations of vortexes in ferromagnetic materials modeled with exchange interactions (short range) and dipolar interactions (long range), which have potential applications in devices. We developed an algorithm using the theoretical model as starting point, modifying its structure in order to fit it in the functioning mechanism of the platform. Once the model is restructured and the code is developed, we present as results comparisons between our method and a matured and optimized serial platform algorithm, which also does not consider approximation for long range dipolar interactions. The results indicate an average factor of 35 times in diminishing the total run time of each simulation compared to the optimized serial version, in the interesting size scales regarding applications, representing not only a tremendous gain in computational cost, but also an increase on the size of the Sample Space of structures which we can simulate. We also present the model’s validation comparing it to experimental and theoretical results, using as sample material the Permalloy(Py). As application of the computational tool, we also present the proposition of an original numerical integration method, in order to better preserve certain properties of the physical dynamical model, allowing for the analysis of systems undergoing adverse conditions (i.e high external fields and sharp variations between regions of the vector field). The author acknowledges support from CAPES, FAPEMIG and CNPq in the realization of this work.