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University of Wales, Bangor - Mathematics Preprints 2003
Computational Applied Mathematics |
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This paper presents a scheme for the calculation of a numerical solution to a Poisson problem. The variational formulation of the equation is discretized with wavelets and expressed as a matrix equation of the type A u = b, where A is a sparse, symmetric, positive definite matrix. For this reason, the matrix equation is solved by an iterative method such as the method of steepest descent or the conjugate gradients method.
The properties of wavelets will have two main roles in this scheme. On the one hand, the condition number of the $A$ matrix will be improved by a diagonal wavelet preconditioner, with the effect of accelerating the convergence of the iterative scheme. On the other hand, the scheme will take advantage of the sparse wavelet representation of functions in order to express the right hand side of the equation and the solution as accurately as possible, given the number of degrees of freedom that can be afforded in practice.
Two simulations of the evolution of the magnetization of a rectangular nanoelement are presented. In both simulations the magnetization is initially uniform, parallel to the x_2-axis, # which is the direction of the length of the nanoelement, m_0=(0,1,0). The anisotropy vector is parallel to the x_1-axis, the direction of the width of the nanoelement, h_a = (1,0,0). The applied field, responsible for the initial magnetization, is assumed to be zero at all times > t_0.
We introduce the stochastic Langevin-Gilbert equation and describe a simple numerical integration scheme for a pointwise solution converging to the Stratonovich interpretation. Reversal of the magnetisation vector is illustrated at various temperatures. We then develop a Galerkin finite element discretisation of the problem. The zero-field relaxation of an isolated thin film cobalt grain at ambient temperature is examined. Despite thermal agitations which are uncorrelated in space or time the magnetisation configuration is observed to pass smoothly between low-energy states. Finally, hysteresis simulations are performed; frustration of the magnetisation configuration is shown to cause a reduction in calculated coercivity values at ambient temperature as compared with athermal simulations.
In this theoretical study we discuss the dynamical switching of magnetically soft cylindrical nanodots of width 10-200nm. The Landau Lifshitz Gilbert equation of motion is used to describe the time evolution of the model which is coupled with a finite element discretisation of the spatial magnetization and total effective field. The finite element method gives us a detailed micromagnetic model of such a system by allowing an accurate representation of the time dependent magnetic structure of any geometry. The transition between metastable configuration states occurs via nucleation of vortices in the larger particle size. When the particle size is decreased, single domain behavior is observed. By forming an array of exchange coupled nanodots we investigate their switching by examining the system dynamics. In the case of the 50nm and 10nm sizes there is little difference in coercivity. However the dynamics are dissimilar. The non interacting nanodot has a larger coercivity which decreases as the particle size increases and is able to support a vortex structure. A brief comparison of mid-point time integration methods used in the model will be discussed in terms of their overall stability and performance.
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