U.W. Bangor - School of Informatics - Mathematics Preprints 2001
Computational Applied Mathematics
01.01 : CANCE, G.V.
Numerical solution of a partial differential equation using wavelets
This thesis presents the application of several wavelet methods to the solution of Burgers equation, which is a one-dimensional non-linear partial differential equation. Wavelet methods are expected to perform well on Burgers equation in particular because the wavelet bases can decouple the localised details of the solution from its underlying globally smooth behaviour. Burgers equation is considered both with periodic boundary conditions and with homogeneous boundary conditions on the interval [0,1] and solved using a basis of wavelets and/or scaling functions satisfying the boundary conditions. The basis functions used are: Daubechies scaling functions, interpolating cubic splines as scaling functions, a biorthogonal system of cubic spline scaling functions and wavelets, and a biorthogonal system of scaling functions and wavelets generated by the lifting scheme. The flexibility of the lifting scheme is found to facilitate the modifications of the basis functions to suit the different boundary conditions. An adaptive algorithm originating from BMP92 is implemented to concentrate the computational effort around the stationary front which the solution develops with time. This adaptive algorithm is later modified to enable it to track a steep front moving laterally. The numerical results are compared to the analytical solution and to published results obtained by non-adaptive finite element methods Ali92 and found competitive, particularly for the solutions presenting steeper fronts. Parts of the algorithms are also compared with other adaptive wavelet algorithms CW96 and GC97.
Published in:University of Wales, Bangor, M.Phil. Dissertation (September 2000).
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01.16 : RIDLEY, P.H.W., ROBERTS, G.W. & CHANTRELL, R.W.
Simulation of the Micromagnetic Behaviour of Arrays of Interacting Nanoelements
In this work we investigate the interacting behavior of small arrays of permalloy particles at the sub micron level. Each individual particle is termed a nanoelement and is rectangular in form with varying elongation. The interest in such structures of magnetic material is increasing mainly due to the possible potential use in future high-density magnetic storage media applications. To carry out our investigations we have developed a dynamical micromagnetic model based on the use of the finite element method. For our results we investigate the effects of disordered and ordered anisotropy distributions on arrays with varying size and space of nanoelement. We observe that the reversal mechanism of the arrays is very sensitive to the disorder of the intrinsic material properties. In the case of aligned uniaxial anisotropy a highly symmetric cooperative switching mechanism is observed. The larger anisotropy has the effect of stabilizing states during the reversal process leading to distinctive switching along the hysteresis curve. A random anisotropy breaks this high symmetry sufficiently to reduce the cooperative switching leading to a relatively random reversal of individual nanoelements.
Finite element method; Interactions; Nanoelements.
J. Appl. Phys. 92(2) (2002)
(Selected for Virtual Journal of Nanoscale Science and Technology 8 July 2002)
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01.23 : SPARGO, A.W., RIDLEY, P.H.W., ROBERTS, G.W. & CHANTRELL, R.W.
Influences of Granular Microstructure on the Reversal Mechanism of Co Nanoelements.
A two dimensional micromagnetic simulation is developed using a spatial finite element method, with the dynamic Gilbert equations discretized by a Galerkin method. A novel algorithm is presented for the generation of granular micro structures via the Voronoi tessellation and rectangular nanoelements of 200nm x 40nm x 20nm are constructed. Hysteresis simulations are performed on isolated nanoelements in order to determine the extent to which experimentally observed variations in coercivity are attributable to the microstructural properties rather than magnetostatic interactions with neighboring members of an array. Dependence of coercivity on grain size is examined, as well as the role of grain-structure in the reversal process. Mean grain size is shown to dictate the mode of reversal whereas grain irregularity is observed to influence specific magnetization configurations. Low correlation between grain irregularity and coercivity indicates that the magnetization dynamics depend on a combination of physical microstructure and easy-axis distribution unique to each nanoelement.
Published in:J. App. Phys. 91 (2002) 6923-6925.
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