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Title: Uncertainty modeling, propagation, and quantification techniques with applications in engineering dynamics
Author: Zhang, Y.
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2017
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In the field of engineering dynamics, three of the main challenges associated with stochasticity relate to a) uncertainty modeling, b) uncertainty quantification, and c) uncertainty propagation. Addressing challenge a) relates to the development of methodologies for the interpretation/analysis of measured/available data, as well as for subsequent estimation of pertinent stochastic models, i.e. quantification of the underlying stochastic process/field statistics, while challenge b) relates to quantifying the error of those estimates in a priori, if possible, manner. However, in several engineering applications large amounts of data can be difficult to acquire for several reasons, such as cost, data loss or corruption, as well as limited bandwidth/storage capacity. Furthermore, available data can often be highly limited and irregularly sampled, and thus, standard techniques for spectral estimation, (e.g. Fourier decomposition), can demonstrate poor performance. Further, addressing challenge c) relates to the development of methodologies for determining complex system response/reliability statistics, i.e. development of analytical/numerical methodologies for solving nonlinear high-dimensional stochastic (partial) differential equations efficiently. In this regard, the Monte Carlo simulation (MCS) has been perhaps the most versatile tool. Nevertheless, there are cases, especially for large-scale systems, where the MCS can be computationally prohibitive. Thus, there is a need for developing efficient approximate analytical and/or numerical approaches. In this thesis, techniques are developed for addressing selected aspects of challenges a, b), and c). First, a general Lp norm (0 < p = 1) minimization approach is proposed for estimating stochastic process power spectra subject to realizations with incomplete/missing data. Specifically, relying on the assumption that the recorded incomplete data exhibit a significant degree of sparsity in a given domain, employing appropriate Fourier and wavelet bases, and focusing on the L1 and L1/2 norms, it is shown that the approach can satisfactorily estimate the spectral content of the underlying process. Finally, the effect of the chosen norm on the power spectrum estimation error is investigated, and it is shown that the L1/2 norm provides almost always a sparser solution than the L1 norm. Numerical examples consider several stationary, non-stationary, and multi-dimensional processes for demonstrating the accuracy and robustness of the approach, even in cases of up to 80% missing data. Second, the challenge of quantifying the uncertainty in stochastic process spectral estimates based on realizations with missing data is addressed. Specifically, relying on relatively relaxed assumptions for the missing data and on a Kriging modeling scheme, utilizing fundamental concepts from probability theory, and resorting to a Fourier based representation of stationary stochastic processes, a closed-form expression for the probability density function (PDF) of the power spectrum value corresponding to a specific frequency is derived. Next, the approach is extended for determining the PDF of spectral moments estimates as well. Clearly, this is of significant importance to various reliability assessment methodologies that rely on knowledge of the system response spectral moments for evaluating its survival probability. Further, it is shown that utilizing a Cholesky-like decomposition for the PDF related integrals the computational cost is kept at a minimal level. Several numerical examples are included and compared against pertinent Monte Carlo simulations for demonstrating the validity of the approach. Third, a Wiener path integral (WPI) technique based on a variational formulation is developed for nonlinear oscillator stochastic response determination and reliability assessment. This is done in conjunction with a stochastic averaging/linearization treatment of the problem. Specifically, first the nonlinear oscillator is cast into an equivalent linear one with time-varying stiffness and damping elements. Next, relying on the concept of the most probable path a closed-form approximate analytical expression for the oscillator joint transition probability density function (PDF) is derived for small time intervals. Finally, the transition PDF in conjunction with a discrete version of the Chapman-Kolmogorov (C-K) equation is utilized for advancing the solution in short time steps. In this manner, not only the non-stationary response PDF, but also the oscillator survival probability and first-passage PDF are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multi-dimensional integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators are considered in the numerical examples section. Comparisons with pertinent Monte Carlo simulation data demonstrate the reliability of the developed technique. Finally, an approximate analytical technique for assessing the reliability of a softening Duffing oscillator subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging treatment of the problem the oscillator time-varying survival probability is determined in a computationally efficient manner. In comparison with previous techniques that neglect the potential unbounded response behavior of the oscillator when the restoring force acquires negative values, the herein developed technique readily takes this aspect into account by introducing a special form for the oscillator non-stationary response amplitude probability density function (PDF). A significant advantage of the technique relates to the fact that it can readily handle cases of stochastic excitations that exhibit strong variability in both the intensity and the frequency content. Numerical examples include a softening Duffing oscillator under evolutionary earthquake excitation, as well as a softening Duffing oscillator with nonlinear damping modeling the nonlinear ship roll motion in beam seas. Comparisons with pertinent Monte Carlo simulation data demonstrate the efficiency of the technique.
Supervisor: Patelli, Edoardo ; Beer, Michael ; Kougioumtzoglou, Ioannis Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral