Globally-Evolving-Based Generalized Probability Density Evolution Equation For Response Determination And Reliability Evaluation Of High-Dimensional Stochastic Dynamical Systems

The response determination and reliability analysis of high-dimensional nonlinear stochastic dynamical systems have long been significant challenges in science and engineering.Many methods for complex systems have been developed recently,but it is still difficult to accurately estimate the small failure probability under rare catastrophic dynamic actions.With the development of the 3^rd Generation of Structural Design Theory,this problem is becoming more and more important and urgent.The development of probability density evolution method（PDEM）and its ensemble evolution path have provided a new perspective for the accurate and efficient solution of this problem.To this end,in the present thesis,the globally-evolution-based generalized density evolution equation（GE-GDEE）is developed following this path.A unified formalism of the GE-GDEE for generic continuous stochastic processes is established.Based on this theoretical formalism,the classical probability density evolution equations,including the Liouville and FPK equations,can be regarded as special forms of the GE-GDEE for some specific physical systems.In particular,Markov property is not a requisite of the GE-GDEE.In the present thesis,the GE-GDEE is firstly applied to the probabilistic response determination of high-dimensional nonlinear systems involving randomness from both parameters and excitations,in which the parameters can be regarded as independent or correlated random variables,whereas the excitations can be considered as Gaussian white or non-stationary non-white noise.If a single response quantity is of interest,the GE-GDEE with respect to its instantaneous probability density function（PDF）can be established as a one-or two-dimensional partial differential equation.The intrinsic drift coefficients in the GE-GDEE are the physically driving force for the evolution of the PDF of the response.They are the conditional expectation function of the drift force in the equation of motion of the original high-dimensional system,and determined by the physical mechanism of the system.For general high-dimensional nonlinear systems,the intrinsic drift coefficients can be analytically or numerically constructed via data given by representative deterministic analyses,namely solving the physical equations,e.g.,by using the locally weighted smoothing scatterplots or the vine copulas-based parametric approach.Then,solving the GE-GDEE with the obtained intrinsic drift coefficients substituted yields the instantaneous PDF of quantity of interest.The reliability for stochastic dynamical systems is investigated based on the extreme value distribution（EVD）and the absorbing boundary methods,respectively,in the thesis.For this purpose,the probability evolution integral equation for the time-variant EVD of one-dimensional continuous Markov processes is firstly derived.Meanwhile,the augmented Markov vector（AMV）method for the determination of EVD of low-dimensional Markov systems under various excitations,such as Gaussian or Poisson white noise,is also proposed.Combined with the AMV method and the globally-evolution-based dimension reduction,the GE-GDEE with respect to the time-variant extreme value process of the high-dimensional Markov systems can be established,so as to obtain the numerical solutions of the EVD and time-variant reliability under various thresholds for high-dimensional systems.Numerical examples show that the proposed methods are of high accuracy and effectiveness for the determination of the EVDs for low-and high-dimensional Markov systems.As an alternation,if an absorbing boundary process（ABP）of the response quantity of interest in a specific safety domain is constructed,the GE-GDEE with respect to the instantaneous PDF of the ABP can be established,and the first-passage reliability analysis of high-dimensional stochastic dynamic systems can be realized numerically.Meanwhile,the eligibility of imposing an absorbing boundary for the first-passage reliability is rigorously proved in the thesis.The proof of non-exchangeability of imposing absorbing boundary condition and dimension reduction lays the theoretical foundation for solving the reliability based on the dimension-reduced ABPs.Examples compared with Monte Carlo simulation show the accuracy and efficiency of response and reliability results based on the GE-GDEE,especially for the tail efficiency of the PDF and the small probability of failure under rare events.The GE-GDEE is further applied to engineering practice.A high-rise reinforced concrete structure with random parameters involved containing around 280000 degrees of freedom is invesgated,and the stochastic response and seismic reliability of the structure subjected to the stochastic ground motion is analyzed in the thesis,which is of great significance for further guiding disaster prevention and reliability-based design of engineering structures.Problems for further exploration are discussed in the finality.