关键词: micromagnetic simulation, standard problem No. 3, dipolar interaction

Abstract: Cubic-shaped magnetic particles subjected to a dimensionless uniaxial anisotropy ($Q = 0.1$) aligned with one of the crystallographic axes provide an ideal system for investigating magnetic equilibrium states. In this system, three fundamental magnetization configurations are identified: (i) the flower state, (ii) the twisted flower state, and (iii) the vortex state. This problem corresponds to standard problem No. 3 proposed by the NIST Micromagnetics Modeling Group, widely adopted as a benchmark for validating computational micromagnetics methods. In this work, we approach the problem using a computational method based on direct dipolar interactions, in contrast to conventional techniques that typically compute the demagnetizing field via finite difference-based fast Fourier transform (FFT) methods, tensor grid approaches, or finite element formulations. Our results are compared with established literature data, focusing on the dimensionless parameter $\lambda=L/l_{\rm ex}$, where $L$ is the cube edge length and $l_{\rm ex}$ is the exchange length of the material. To analyze equilibrium state transitions, we systematically varied the size $L$ as a function of the simulation cell number $N$ and intercellular spacing $a$, determining the critical $\lambda$ value associated with configuration changes. Our simulations reveal that the transition between the twisted flower and vortex states occurs at $\lambda \approx 8.45$, consistent with values reported in the literature, validating our code (Grupo de Física da Matéria Condensada - UFJF), and shows that this standard problem can be resolved using only interaction dipolar of a direct way without the need for sophisticated additional calculations.

Key words: micromagnetic simulation, standard problem No. 3, dipolar interaction