Cracked beam problem is an elliptic boundary value problem with singularity. It is often used as a testing model for numerical methods. We use numerical and symbolic boundary approximation methods and boundary collocation method to compute its extremely high accurate solution with global error $O(10^{-100})$. This solution then can be regarded as the exact solution. On the other hand, we vary the boundary conditions of this problem to obtain several related models. Their numerical solutions are compared to those of cracked beam and Motz problems, the prototypes of singularity problems. From the comparison we can conclude the advantage of each model and decide the best testing model for numerical methods.

It is reasonable to assume that an aircraft might experience some in-flight discrete source damage caused by various incidents. It is, thus, necessary to evaluate the impact of such damage on the performance of the aircraft. This study is focused on evaluating the effect of a simple discrete damage in an aircraft wing on its static and dynamic response. The damaged wing is modeled by a thin-walled beam with a longitudinal crack the response of which can be obtained analytically. As uncertainties are present in the location and size of the crack as well as in the applied loads, their effects are incorporated into the framework consisting of structural response, crack propagation and aeroelasticity. The first objective of this study is to examine the effect of damage represented by a crack on the wing flexibility that influences its deformation and aero-elastic divergence characteristics. To study this, the thin-walled beam is modeled by Benscoter thin-walled beam theory combined with Gunnlaugsson and Pedersen compatibility conditions to accurately account for the discontinuity at the interface of the cracked and uncracked beam segments. Instead of conducting a detailed finite element analysis, the solution is obtained in an exact sense for general distributed loads representing the wind pressure effects. This analytical approach is shown to provide very accurate values for the global beam response compared with the detailed finite element shell analysis. This analytical solution is, then, used to study the beam response probabilistically. The crack location and size are assumed to be uncertain and are, thus, characterized by random variable. For a specified limit state, the probability of failure can be conveniently calculated by the first order second moment analysis using the safety index approach. The same analytical solution is also used to study the aero-elastic divergence characteristics of a wing, the inner structure of which is represented by a thin-walled beam with a crack of uncertain size and position along the beam. The second objective of this study is to examine the time growth of a crack under dynamic gust type of loading to which a wing is likely to be exposed during flight. Damage propagating during operation further deteriorates the safety of the aircraft and it is necessary to study its time growth so that its impact on the performance can be evaluated before it reaches its unstable state. The proposed framework for the crack growth analysis is based on classical fracture mechanics where the remaining flight time is obtained by Monte Carlo simulation in which various uncertainties are taken into account. To obtain equivalent cyclic loading required for crack growth analysis, random vibration analysis of the thin-walled beam is conducted for stochastic wind load defined by a gust load spectral density function. The probability of failure represented by the crack size approaching the critical crack size within the flight duration or the remaining flight time before a crack reaches its limiting value are obtained. This study with a simple representation of a wing and damage is anticipated to provide initial guidance for future studies to examine the impact of discrete source damage on the in-flight performance of the aircrafts, with the ultimate goal of minimizing the adverse effect and enhancing the safety of aircrafts experiencing damage. / Ph. D.

For Laplace's eigenvalue problems, this thesis presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use a iteration process to yield approximate eigenvalues and eigenfunctions. The new iteration method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.

Neste estudo, o novo algoritmo hibrido autoconfigurado PSOS (Particle Swarm Optimization - Simplex) para avaliação da integridade estrutural a partir de respostas dinâmicas é apresentado. A formulação da função objetivo para o problema de minimização definido emprega funções de resposta em freqüência e/ou dados modais do sistema. Uma nova estratégia para o controle dos parâmetros do algoritmo Particle Swarm Optimization (PSO), baseada no uso do método de Nelder - Mead é desenvolvida; conseqüentemente, a convergência do PSO fica independente dos parâmetros heurísticos e sua estabilidade e precisão são melhoradas. O método híbrido proposto teve melhor desempenho, nas diversas funções teste analisadas, quando comparado com os algoritmos simulated annealing, algoritmos genéticos e o PSO. São apresentados diversos problemas de detecção de dano, levando em conta os efeitos do ruído e da falta de dados experimentais. Em todos os casos, a posição e extensão do dano foram determinadas com sucesso. Finalmente, usando o PSOS, os parâmetros de um oscilador não linear (oscilador de Duffing) foram identificados. / In this study, a new auto configured Particle Swarm Optimization - Simplex algorithm for damage detection has been proposed. The formulation of the objective function for the minimization problem is based on the frequency response functions (FRFs) and the modal parameters of the system. A novel strategy for the control of the Particle Swarm Optimization (PSO) parameters based on the Nelder-Mead algorithm (Simplex method) is presented; consequently, the convergence of the PSOS becomes independent of the heuristic constants and its stability and accuracy are enhanced. The formulated hybrid method performs better in different benchmark functions than the Simulated Annealing (SA), the Genetic Algorithm (GA) and the basic PSO. Several damage identification problems, taking into consideration the effects of noisy and incomplete data, were studied. In these cases, the damage location and extent were determined successfully. Finally, using the PSOS, a non-linear oscillator (Duffing oscillator) was identified with good results.