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Harmonic Wavelets Procedures and Wiener Path and Integral Methods for Response Determination and Reliability Assessment of Nonlinear Systems/StructuresJanuary 2011 (has links)
In this thesis a novel approximate/analytical approach based on the concepts of stochastic averaging and of statistical linearization is developed for the response determination of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems subject to evolutionary stochastic excitation. The significant advantage of the approach relates to the fact that it is readily applicable for excitations possessing even non-separable evolutionary power spectra (EPS) circumventing ad hoc pre-filtering and pre-processing excitation treatments associated with existing alternative schemes of linearization. Further, the approach can be used, in a rather straightforward manner, in conjunction with recently developed design spectrum based analyses for obtaining peak response estimates without resorting to numerical integration of the nonlinear equations of motion. Furthermore, a novel approximate/analytical Wiener path integral based solution (PIS) is developed and a numerical PIS approach is extended to determine the response and first-passage probability density functions (PDFs) of nonlinear/hysteretic systems subject to evolutionary stochastic excitation. Applications include the versatile Preisach hysteretic model, recently applied in modeling systems equipped with smart material (shape memory alloys) devices used for seismic hazard risk mitigation. The approach is also applied to determine the capsizing probability of a ship, whose rolling dynamics is captured by a softening Duffing oscillator. Finally, novel harmonic wavelets based joint time-frequency response analysis and identification approaches are developed capable of determining the time-varying frequency content of non-stationary complex stochastic phenomena encountered in engineering applications. Specifically, a harmonic wavelets based statistical linearization approach is developed to determine the EPS of the response of nonlinear/hysteretic systems subject to stochastic excitation. In a similar context, an identification approach for nonlinear time-variant systems based on the localization properties of the harmonic wavelet transform is also developed. It can be construed as a generalization of the well established reverse multiple-input/single-output (MISO) spectral identification approach to account for non-stationary inputs and time-varying system parameters. Several linear and nonlinear time-variant systems are used to demonstrate the reliability of the approach.
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Resolução numérica de EDPs utilizando ondaletas harmônicas / Numerical resolution of partial differential equations using harmonic waveletsPedro da Silva Peixoto 16 July 2009 (has links)
Métodos de resolução numérica de equações diferenciais parciais que utilizam ondaletas como base vêm sendo desenvolvidos nas últimas décadas, mas existe uma carência de estudos mais profundos das características computacionais dos mesmos. Neste estudo analisou-se detalhadamente um método espectral de Galerkin com base de ondaletas harmônicas. Revisou-se a teoria matemática referente às ondaletas harmônicas, que mostrou ter grande similaridade com a teoria referente à base trigonométrica de Fourier. Diversos testes numéricos foram realizados. Ao analisarmos a resolução da equação do transporte linear, e também de transporte não linear (equação de Burgers), obtivemos boas aproximações da solução esperada. O custo computacional obtido foi similar ao método com base de Fourier, mas com ondaletas harmônicas foi possível usar a localidade das ondaletas para detectar características de localidade do sinal. Analisamos ainda uma abordagem pseudo-espectral para os casos não lineares, que resultaram em um expressivo aumento de eficiência. Tendo em vista o uso das propriedades de localidade das ondaletas, usamos o método de Galerkin com base de ondaletas harmônicas para resolver um sistema de equações referente a um modelo de propagação de frentes de precipitação. O método mostrou boas aproximações das soluções esperadas, custo computacional ótimo e ainda a possibilidade de se obter espectralmente informações sobre a localização da frente de precipitação. / Numerical methods to solve partial differential equations based on wavelets have been developed in the last two decades, but there is a lack of studies on their computational characteristics. In this study a Galerkin spectral method using harmonic wavelets base has been thoroughly analyzed. We performed a review on the mathematics of harmonic wavelets, that showed a great similarity with Fourier basis. Several numerical experiments were made. Analyzing the use of the Galerkin method, with harmonic wavelets, on linear and non linear transport equations, we achieved good approximations in respect to the expected solution. The computational cost resulted to be similar to the same method with Fourier basis. On the other hand, employing harmonic wavelets we were able to obtain local information of the solution by simple inspection of the spectral coeffcients. We also analyzed a pseudo-spectral method based on harmonic wavelets for the non linear equations, resulting in a great improvement in efficiency. Looking towards using the locality propriety of harmonic wavelets, we tested the Galerkin method on a precipitation front propagation model. The method resulted in good approximations to the expected solution, optimal computational cost and the possibility of obtaining information on the locality of the precipitation fronts spectrally.
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Resolução numérica de EDPs utilizando ondaletas harmônicas / Numerical resolution of partial differential equations using harmonic waveletsPeixoto, Pedro da Silva 16 July 2009 (has links)
Métodos de resolução numérica de equações diferenciais parciais que utilizam ondaletas como base vêm sendo desenvolvidos nas últimas décadas, mas existe uma carência de estudos mais profundos das características computacionais dos mesmos. Neste estudo analisou-se detalhadamente um método espectral de Galerkin com base de ondaletas harmônicas. Revisou-se a teoria matemática referente às ondaletas harmônicas, que mostrou ter grande similaridade com a teoria referente à base trigonométrica de Fourier. Diversos testes numéricos foram realizados. Ao analisarmos a resolução da equação do transporte linear, e também de transporte não linear (equação de Burgers), obtivemos boas aproximações da solução esperada. O custo computacional obtido foi similar ao método com base de Fourier, mas com ondaletas harmônicas foi possível usar a localidade das ondaletas para detectar características de localidade do sinal. Analisamos ainda uma abordagem pseudo-espectral para os casos não lineares, que resultaram em um expressivo aumento de eficiência. Tendo em vista o uso das propriedades de localidade das ondaletas, usamos o método de Galerkin com base de ondaletas harmônicas para resolver um sistema de equações referente a um modelo de propagação de frentes de precipitação. O método mostrou boas aproximações das soluções esperadas, custo computacional ótimo e ainda a possibilidade de se obter espectralmente informações sobre a localização da frente de precipitação. / Numerical methods to solve partial differential equations based on wavelets have been developed in the last two decades, but there is a lack of studies on their computational characteristics. In this study a Galerkin spectral method using harmonic wavelets base has been thoroughly analyzed. We performed a review on the mathematics of harmonic wavelets, that showed a great similarity with Fourier basis. Several numerical experiments were made. Analyzing the use of the Galerkin method, with harmonic wavelets, on linear and non linear transport equations, we achieved good approximations in respect to the expected solution. The computational cost resulted to be similar to the same method with Fourier basis. On the other hand, employing harmonic wavelets we were able to obtain local information of the solution by simple inspection of the spectral coeffcients. We also analyzed a pseudo-spectral method based on harmonic wavelets for the non linear equations, resulting in a great improvement in efficiency. Looking towards using the locality propriety of harmonic wavelets, we tested the Galerkin method on a precipitation front propagation model. The method resulted in good approximations to the expected solution, optimal computational cost and the possibility of obtaining information on the locality of the precipitation fronts spectrally.
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