Fast Fourier Transform at Nonequispaced Nodes and Applications

The direct computation of the discrete Fourier transform at arbitrary nodes requires O(NM) arithmetical operations, too much for practical purposes. For equally spaced nodes the computation can be done by the well known fast Fourier transform (FFT) in only O(N log N) arithmetical operations. Recently, the fast Fourier transform for nonequispaced nodes (NFFT) was developed for the fast approximative computation of the above sums in only O(N log N + M log 1/e), where e denotes the required accuracy. The principal topics of this thesis are generalizations and applications of the NFFT. This includes the following subjects: - Algorithms for the fast approximative computation of the discrete cosine and sine transform at nonequispaced nodes are developed by applying fast trigonometric transforms instead of FFTs. - An algorithm for the fast Fourier transform on hyperbolic cross points with nonequispaced spatial nodes in 2 and 3 dimensions based on the NFFT and an appropriate partitioning of the hyperbolic cross is proposed. - A unified linear algebraic approach to recent methods for the fast computation of matrix-vector-products with special dense matrices, namely the fast multipole method, fast mosaic-skeleton approximation and H-matrix arithmetic, is given. Moreover, the NFFT-based summation algorithm by Potts and Steidl is further developed and simplified by using algebraic polynomials instead of trigonometric polynomials and the error estimates are improved. - A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on hard thresholding complex ridgelet coefficients combined with total variation minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the NFFT and then applying a dual-tree complex wavelet transform. - A new robust local scattered data approximation method is introduced. It is an advancement of the moving least squares approximation (MLS) and generalizes an approach of van den Boomgard and van de Weijer to scattered data. In particular, the new method is space and data adaptive

Computational and numerical analysis of differential equations using spectral based collocation method.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods, both modified and new, and apply them to solve differential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of differential equations defined over simple geometries. Procedurally, these methods entail transforming the gov erning differential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deficiency, the present study explores the development of numerical algorithms for solving ordinary and partial differential equations defined over simple geometries. The solutions of the differential equations considered are approximated using interpolating polynomials that satisfy the given differential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve differential equations defined on large computational domains much effort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con firms the superiority of these multiple domain techniques in terms of accuracy and computational efficiency over the single domain approach when applied to problems defined over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across different dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary differential equations. Subsequent chapters then build on solutions to partial differential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical algorithms. The numerical results of the study confirm that incorporating domain decomposition techniques in spectral collocation methods work effectively for all dimensions, as we report highly accurate results obtained in a computationally efficient manner for problems defined on large do mains. The findings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter

Splines and local approximation of the earth's gravity field

Bibliography: pages 214-220.The Hilbert space spline theory of Delvos and Schempp, and the reproducing kernel theory of L. Schwartz, provide the conceptual foundation and the construction procedure for rotation-invariant splines on Euclidean spaces, splines on the circle, and splines on the sphere and harmonic outside the sphere. Spherical splines and surface splines such as multi-conic functions, Hardy's multiquadric functions, pseudo-cubic splines, and thin-plate splines, are shown to be largely as effective as least squares collocation in representing geoid heights or gravity anomalies. A pseudo-cubic spline geoid for southern Africa is given, interpolating Doppler-derived geoid heights and astro-geodetic deflections of the vertical. Quadrature rules are derived for the thin-plate spline approximation (over a circular disk, and to a planar approximation) of Stokes's formula, the formulae of Vening Meinesz, and the L₁ vertical gradient operator in the analytical continuation series solution of Molodensky's problem

Modelling damage in turbine blades and their coatings

In the process of modelling damage in turbine blades and their coatings, four studies were carried out in this thesis. The first two chapters focus on modelling the damage in thermal barrier coatings or TBCs which are commonly used for Ni-based turbine blades. The second two chapters focus on development of meshfree methods for modelling micro-cracks in woven SiC_f/SiC_m composites which are a potential future material to be used for turbine blades. A brief summary of the main achievements from each chapter is given here as an insight into what is expected from each work. Swelling as the Main Source of Rumpling in TBC, chapter 2: Previously it was believed that the main source of rumpling growth in TBC systems is from phenomenon such as phase transformation and thermal mismatch that occur during the heating and cooling processes of a thermal cycle. However, the findings from an experimental work by Tolpygo & Clarke could not be explained with the previously suggested theories, where no difference in the rumpling amplitude was observed as the lower temperature of the thermal cycles was changed, except for the isothermal case. This behaviour was puzzling because it mitigated the effects of phase transformation and thermal mismatch. In this work, the existing analytical model of rumpling by Balint et al. was modified to include a relatively new phenomenon known as swelling, and used to reproduce and explain the experimental results. The analysis of the data from the developed model revealed that most of rumpling occurs during the dwell which is caused by swelling; its effects are also apparent during heating and cooling processes. Therefore, swelling proves to be the main source of rumpling growth. Lateral Growth in the Bond Coat and Inter-diffusion Layers, chapter 3: In the process of understanding the puzzling outcome from the experimental work of Chen et al. on measuring the lateral growth of the bond coat/inter-diffusion layers of TBC system after 50 hours of isothermal heat treatment at 1150 C, two finite element models of the system were produced; one with the coating modelled as two layers and another with the coating modelled as four layers. Chen et al. experimental results showed a large lateral deformation for the bond coat and almost none for the inter-diffusion layer, which was surprising because swelling effect which is a volumetric phenomenon, was observed in both layers, hence, it should have led to lateral swelling for the inter-diffusion layer as well. In this work, it was shown that because of the non-uniform nature of the Ni/Al inter-diffusion the two-layered model is not detailed enough to capture the real behaviour of the system, hence, the four-layered model is introduced which more closely matches the experimental results. This outcome indicated that modelling this system with two layers can create implications when modelling rumpling, therefore, a multi-layered coating system, such as the four-layered model shown in this work, is needed for modelling rumpling more accurately. MQ-RPIM Optimisation for Engineering Single Body Problems, chapter 4: Multi-quadrics radial point interpolation meshfree (MQ-RPIM) method is one of the common mehsfree methods currently used. However, the shape parameters involved in the generalised multi-quadric method have a strong influence on the accuracy of the solutions. In addition to the shape parameters, there are variables related to the integrations involved in the MQ-RPIM method that affect its accuracy. In this work a novel systematic algorithm was introduced which produces the best values for the variables involved in the MQ-RPIM method, including the integration and MQ shape parameters, for any engineering problem. For demonstration, this method is applied to three solid mechanic problems in both two- and three-dimensional forms. MQ-RPIM Model of Plain Woven Composite with Frictionless Contact, chapter 5: In the process of developing an explicit model for the micro-cracks at the yarn-matrix interface of a woven SiC_f/SiC_m composite using meshfree methods, the first three-dimensional MQ-RPIM frictionless contact code is developed from scratch and successfully applied to the preliminary model of a plain woven composite unit cell for two limiting conditions; i) full-stick (0% delamination) and ii) full-slip (100% delamination). As part of the development for this contact model, a two-dimensional frictionless contact model is also produced, where both two- and three-dimensional forms of the contact code are verified against analytical and finite element results for two Hertzian contact problems. The MQ-RPIM results for the Hertzian examples made a use of optimisation algorithm introduced in chapter 4, confirming the use of this algorithm and flexibility of the MQ-RPIM method compared to the FEM for models with non-uniform distribution of nodes, particularly at contact regions.Open Acces