The panel is symmetrical about both x- and y- axes. The central strip (strip1) of width 2h1 contains a central transverse crack of width 2a on x-axis. The two strips (strip2) contain transverse cracks of width c-b also on x-axis. The panel is subjected to axial loads with uniform intensities p1 and p2 in strip1 and strip2 , respectively at . Materials of all strips are assumed to be linearly elastic and isotropic. Due to double symmetry, only one quarter of the problem and will be considered. The solutions are obtained by using Fourier transforms both in x and y-directions. Summing several solutions is due to the necessity for sufficient number of unknowns in general expressions in order to be able to satisfy all boundary conditions of the problem. The conditions at the edges of the strips and at the interfaces are satisfied and the general expressions for a three layer panel become expressions for the panel with free edges. Use of remaining boundary conditions leads the formulation to a system of two singular integral equations. These equations are converted to a system of linear algebraic equations which is solved numerically

ABSTRACT CONTACT MECHANICS OF GRADED MATERIALS WITH TWODIMENSIONAL MATERIAL PROPERTY VARIATIONS G&ouml / kay, Kemal M.S., Department of Mechanical Engineering Supervisor: Asst. Prof. Dr. Serkan Dag September 2005, 62 pages Ceramic layers used as protective coatings in tribological applications are known to be prone to cracking and debonding due to their brittle nature. Recent experiments with functionally graded ceramics however show that these material systems are particularly useful in enhancing the resistance of a surface to tribological damage. This improved behavior is attributed to the influence of the material property gradation on the stress distribution that develops at the contacting surfaces. The main interest in the present study is in the contact mechanics of a functionally graded surface with a two &ndash / dimensional spatial variation in the modulus of elasticity. Poisson&rsquo / s ratio is assumed to be constant due to its insignificant effect on the contact stress distribution [30]. In the formulation of the problem it is assumed that the functionally graded surface is in frictional sliding contact with a rigid flat stamp. Using elasticity theory and semi-infinite plane approximation for the graded medium, the problem is reduced to a singular integral equation of the second kind. Integral equation is solved numerically by expanding the unknown contact stress distribution into a series of Jacobi polynomials and using suitable collocation points. The developed method is validated by providing comparisons to a closed form solution derived for homogeneous materials. Main numerical results consist of the effects of the material nonhomogeneity parameters, coefficient of friction and stamp size and location on the contact stress distribution.

This thesis concerns the study of geometric numerical integrators and how they preserve phase space structures of Hamiltonian ordinary differential equations. We examine the invariant sets of differential equations and investigate which numerical integrators preserve these sets, and under what conditions. We prove that when periodic orbits of Hamiltonian differential equations are discretized by a symplectic integrator they are preserved in the numerical solution when the integrator step size is not resonant with the frequency of the periodic orbit. The preservation of periodic orbits is the result of a more general theorem which proves preservation of lower dimensional invariant tori from dimension zero (fixed points) up to full dimension (the same as the number of degrees of freedom for the differential equation). The proof involves first embedding the numerical trajectory in a non-autonomous flow and then applying a KAM type theorem for flows to achieve the result. This avoids having to prove a KAM type theorem directly for the symplectic map which is generally difficult to do. We also numerically investigate the break up of periodic orbits when the integrator's step size is resonant with the frequency of the orbit. We study the performance of trigonometric integrators applied to highly oscillatory Hamiltonian differential equations with constant frequency. We show that such integrators may not be as practical as was first thought since they suffer from higher order resonances and can perform poorly at preserving various properties of the di fferential equation. We show that, despite not being intended for such systems, the midpoint rule performs no worse than many of the trigonometric integrators, and indeed, better than some. Lastly, we present a numerical study of a Hamiltonian system consisting of two magnetic moments in an applied magnetic field. We investigate the effect of both the choice of integrator and the choice of coordinate system on the numerical solutions of the system. We show that by a good choice of integrator (in this case the generalised leapfrog method) one can preserve phase space structures of the system without having to resort to a change of coordinates that introduce a coordinate singularity.

Errata inserted inside back end-paper. / Bibliography: p. 243-249. / xvi, 249 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis presents a brief overview of microstrip antenna analysis and describes the connections between spectral and spatial domain periodic Green's functions in integral equation methods. A hybrid formulation is applied to a variety of problems from simple metal strip dipoles to more complicated microstrip geometries. A further development to finite array analysis is described. An improvement in the accuracy of this approximative technique is explored. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 1997

Errata inserted inside back end-paper. / Bibliography: p. 243-249. / xvi, 249 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / This thesis presents a brief overview of microstrip antenna analysis and describes the connections between spectral and spatial domain periodic Green's functions in integral equation methods. A hybrid formulation is applied to a variety of problems from simple metal strip dipoles to more complicated microstrip geometries. A further development to finite array analysis is described. An improvement in the accuracy of this approximative technique is explored. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 1997

SOLA International, a company which manufactures optical lenses, attended the 2000 Mathematics-in-Industry Study Group (MISG) with a wish list. Topping this list was the creation of a mathematical model of a lens, which given the lens geometry and material properties, could predict the deformation of the lens when it was subjected to an impact, such as that experienced in the fracture tests lenses must pass before being approved for sale. The first steps towards such a model were taken at MISG. At MISG, a lens was modelled simply as a thin uniform thickness plate, undergoing small, linear deformations. In the first section of my thesis I extend this model by considering variable thickness plates and larger, nonlinear deformations. For this extended model I have confirmed that the result obtained at MISG, that the contact between a plate and a spherical indentor occurs at a single point, still holds. The second part of this thesis looks at the dynamic deformation, or vibration, of plates. I have developed numerical solution methods for the large amplitude vibration equations with and without the in-plane inertia terms, based on a finite difference scheme. A comparison of these solutions confirms the often used assumption that the in-plane inertia may be neglected. I have also implemented a number of solution methods from the literature, which use separation of variables techniques. Comparing these with the numerical solutions, we find that the numerical solutions better capture the multi-modal nature of the vibration - showing multiple cycles in the approximate period. Having achieved an understanding of the types of forces involved in plate deformation and vibration I consider shell theory in the final section of my thesis. While time constraints meant no dynamic results could be obtained, general nonlinear deep shell equations have been derived. The static version of these equations has then been solved, with the development of a new solution technique which combines a Taylor expansion to approximate the behaviour at the shell centre with a numerical shooting method. Various shallow shell simplifications of the deep shell equations are then discussed and solved. By comparison of the solutions obtained for the deep and shallow, linear and nonlinear equations I have been able to determine which theories apply to which geometries. A complete model of a lens needs to take into account the shape, its thickness and curvature and the material from which it is made. From the work done in this thesis we have been able to determine that a lens model would require the nonlinear theory. Whether the deep shell theory is necessary is debatable as the geometry of a typical lens falls in the grey area, where either theory could be used depending on the accuracy required. For a very accurate model, deep shell theory would be necessary; if an approximate solution obtained quickly was more useful then I suggest the use of a particular set of shallow shell equations. A full lens model would require variable thickness shell theory and the solution of the dynamic equations, neither of which has been achieved here, but the solution techniques I have developed would be applicable to these theories.

Thesis (Ph.D.)--McMaster University (Canada), 1995. / Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1839. Adviser: A. Peirce.

A força hidrodinâmica em termos dos coeficientes de massa adicional e amortecimento, para obstáculos aproximadamente circulares, finos e submersos sob uma superfície livre aquática, é calculada numericamente usando um método espectral. Primeiramente, é apresentado um modelo matemático para ondas aquáticas de superfície e em seguida, o problema de difração de ondas devido à presença de um obstáculo é descrito. Quando o obstáculo é submerso e fino, o problema pode ser formulado em termos de uma equação integral hipersingular. Usando um mapeamento conforme sobre um disco circular, é mostrado que a solução pode ser obtida através de um método espectral onde a hipersingularidade é avaliada analiticamente em termos de polinômios ortogonais. Os coeficientes da força hidrodinâmica, em função do número de onda, são obtidos para obstáculos quase circulares. A ocorrência de frequências ressoantes ´e observada para submersões suficientemente pequenas e subpicos de ressonância aparecem para valores moderados da submersão, em comparação com o caso do disco circular. / The hydrodynamic force, in terms of the added mass and damping coefficients, for thin and submerged nearly circular obstacles below a water free surface is computed by a spectral method. Firstly, a mathematical model for surface water waves is presented. Next, the diffraction problem of waves due to the presence of an obstacle is described. When the body is thin and submerged, the problem can be formulated in terms of a hypersingular integral equation. Using a conformal mapping over a circular disc, it is shown that the solution can be obtained by means of a spectral method where the hipersingularity is analytically evaluated in terms of orthogonal polynomials. The hydrodynamics coefficients, in function of the wavenumber, are computed and shown for nearly circular obstacles. The occurrence of resonant frequencies is observed for sufficiently small submergences and subpeaks of resonances appear for moderate values of the submergence, in comparison with the case of a circular disc.

Made available in DSpace on 2015-01-26T13:21:24Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-21Bitstream added on 2015-01-26T13:30:32Z : No. of bitstreams: 1 000797296.pdf: 529415 bytes, checksum: 44fe9b1364fdd4b4728b6ba26737aca8 (MD5) / Neste trabalho, apresentamos algumas definições de funções inerentes ao Cálculo Fracionário bem como as definições para Derivada e Integral Fracionárias. Como um dos objetivos primordiais deste trabalho é solucionar problemas reais, foi dado um enfoque à derivada fracionária segundo Caputo, uma vez que esta definição é mais pertinente a este tipo de problema, como vamos ver mais adiante. Apresentamos o modelo exponencial que descreve o crescimento bacteriano em um meio ideal e propomos uma generalização do mesmo via Cálculo Fracionário. Com o intuito de refinar a solução dada pela clássica equação logística e ampliar o seu campo de aplicações no estudo de dinâmicas tumorais, propomos e resolvemos uma generalização para a mesma, utilizando o Cálculo Fracionário, isto é, substituímos a derivada de ordem 1 presente na equação ordinária por uma derivada de ordem não inteira 0 < ≤ 1. Em ambos os casos, a solução da equação fracionária tem, como caso particular, a solução do modelo clássico. Por fim, apresentamos a parte original deste trabalho, i.e., analisamos a aplicabilidade do modelo Logístico Fracionário para a descrição do crescimento de tumores de câncer, isto é, sabendo os modelos de crescimentos tumorais presentes na literatura, mostramos graficamente que o comportamento do modelo proposto é, em diversos casos, mais conveniente para descrever o crescimento de tumores de câncer do que os modelos usualmente utilizados / This work presents the definitions of some important functions inherent to Fractional Calculus as well as the definitions for Fractional Integral and Fractional Derivative. One of the main goals of this work is to solve real problems, that is why focus was given on fractional derivatives, in accordance with Caputo, once this definition is more pertinent to this kind of problem. It was introduced the exponential model wich describes bacterial growth in an ideal way and it was proposed its generalization through Fractional Calculus. In order to refine the solution given by the classical logistic equation and expand its application range in the study of tumor dynamics, we propose and solve its generalization, using the Fractional Calculus , i. e., we replace the derivative of order 1 in the ordinary equation by a non-integer order derivative 0 < ≤ 1. In both cases, the solution of the fractional equation has as a special case the solution of the classic model. Finally, we present the original part of this work, i.e., we analyse the applicability of the fractional logistic model to describe the growth of cancer tumor, that is, we compare the model with some presented in the literature and showed graphically that in several cases our model is more convenient than the usual ones

Cette thèse présente plusieurs nouvelles techniques pour la convergence rapide des solutions aux éléments de frontière de problèmes électromagnétiques. Une attention spéciale a été dédiée aux formulations pertinentes pour les solutions aux problèmes électromagnétiques dans les tissus biologiques à haute et basse fréquence. Pour les basses fréquences, de nouveaux schémas pour préconditionner et accélérer le problème direct de l'électroencéphalographie sont présentés dans cette thèse. La stratégie de régularisation repose sur une nouvelle formule de Calderon, obtenue dans cette thèse, alors que l'accélération exploite le paradigme d'approximation adaptive croisée (ACA). En ce qui concerne le régime haute fréquence, en vue d'applications de dosimétrie, l'attention de ce travail a été concentrée sur l'étude de la régularisation de l'équation intégrale de Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) à l'aide de techniques hiérarchiques. Le travail comprend une analyse complète de l'équation pour des géométries simplement et non-simplement connectées. Cela a permis de concevoir une nouvelle stratégie de régularisation avec une base hiérarchique permettant d'obtenir une équation pour les milieux pénétrable stable pour un large spectre de fréquence. Un cadre de travail propédeutique de discrétisation et une bibliothèque de calcul pour des thèmes de recherches sur les techniques de Calderon en 2D sont proposés en dernière partie de cette thèse. Cela permettra d'étendre nos recherches à l'imagerie par tomographie. / This thesis presents several new techniques for rapidly converging boundary element solutions of electromagnetic problems. A special focus has been given to formulations that are relevant for electromagnetic solutions in biological tissues both at low and high frequencies. More specifically, as pertains the low-frequency regime, this thesis presents new schemes for preconditioning and accelerating the Forward Problem in Electroencephalography (EEG). The regularization strategy leveraged on a new Calderon formula, obtained in this thesis work, while the acceleration leveraged on an Adaptive-Cross-Approximation paradigm. As pertains the higher frequency regime, with electromagnetic dosimetry applications in mind, the attention of this work focused on the study and regularization of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation via hierarchical techniques. In this effort, a complete analysis of the equation for both simply and non-simply connected geometries has been obtained. This allowed to design a new hierarchical basis regularization strategy to obtain an equation for penetrable media which is stable in a wide spectrum of frequencies. A final part of this thesis work presents a propaedeutic discretization framework and associated computational library for 2D Calderon research which will enable our future investigations in tomographic imaging.