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Solving the Fokker-Planck-Kolmogorov (FPK) equation is one of the most important and challenging problems in high-dimensional nonlinear stochastic dynamics, which is widely encountered in various science and engineering disciplines. To date, no method available is capable of dealing with systems of dimensions higher than eight. The present paper aims at tackling this problem in a different way, i.e., reducing the dimension of the FPK equation by constructing an equivalent probability flux. In the paper, two different treatments for multidimensional stochastic dynamical systems, the FPK equation and the probability density evolution method (PDEM), are outlined. Particularly, the FPK equation is revisited in a completely new way by constructing the probability fluxes based on the embedded dynamics mechanism and then invoking the principle of preservation of probability. The FPK equation is then marginalized to reduce the dimension, resulting in a flux-form equation involving unknown probability flux caused by the drift effect. On the basis of the equivalence of the two treatments, this unknown probability flux could be replaced by an equivalent probability flux, which is available using the PDEM. An algorithm is proposed to adopt the data from the solution of the generalized density evolution equation in PDEM to construct the numerical equivalent flux. Consequently, a one-dimensional parabolic partial differential equation, i.e., the flux-equivalent probability density evolution equation, could then be solved to yield the probability density function of the response of concern. By doing so, a high-dimensional FPK equation is reduced to a one-dimensional partial differential equation, for which the numerical solution is quite easy. Several numerical examples are studied to verify and validate the proposed method. Problems to be further studied are discussed. [ABSTRACT FROM AUTHOR]

Stochastic response analysis plays an increasingly important role in assessing the performance and reliability of engineering structures subjected to disastrous dynamic loads such as earthquakes and strong winds. In the past decade, a family of probability density evolution method (PDEM) has been developed where a completely decoupled generalized density evolution equation (GDEE) is proposed. The dimension of GDEE could be arbitrary and in most cases a reduced one-dimensional equation is adequate. This provides an efficient solution for stochastic response of complex engineering structures. In the present paper, PDEM is incorporated with the efficient representation of stochastic processes to implement stochastic dynamic response analysis of multi-degree-of-freedom (MDOF) systems subjected to stochastic excitations. Stochastic dynamic responses of linear and nonlinear base-excited systems are investigated by PDEM, the pseudo-excitation method and the Monte Carlo simulations. The Sobol' sequence is employed to generate representative white noise process, and a high-performance stochastic harmonic function representation with fewer random variables is employed to generate representative time histories of a stochastic process with specified power spectral density model. The involved seismic excitations are modeled by banded white noise, Kanai-Tajimi filtered stationary process, Clough-Penzien filtered stationary process and Clough-Penzien filtered process with non-stationary modulation, respectively. Numerical results reveal that PDEM is feasible for stochastic response analysis of MDOF nonlinear systems subjected to stochastic excitations with fair accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Efficient and accurate stochastic response analysis of offshore structures subjected to ocean wave loads has been a challenging problem for several decades. This is especially true for multi-degree-of-freedom nonlinear structures when information about the probability density is desired. How to account for the random field of the wave loads, and how to deal with the coupling of high-dimensional partial differential equation governing the joint probability density functions are among the major challenges. To this end, in the present paper an approach is proposed by incorporating the filter approximation of random fields of sea waves with the recently developed Globally-evolving based generalized density evolution equation (GV-GDEE). To deal with the first issue regarding the random wave filed, the analog filter technique is adopted. In particular, a filter approximation of the wave kinematics field distributed over a vertical line along the ocean domain is proposed. It comprises a linear differential equation set with white noise input. An augmented system is thereby constructed by incorporating the additional differential equation set yielding wave excitation simulated by an analog filter into the equation of motion of a monopile offshore structure. To circumvent the issue of coupling of high-dimensional equations for this augmented system with even higher dimension, the GV-GDEE approach is employed. Specifically, by constructing the effective drift coefficients based on the high-dimensional drift coefficients in the associated Fokker-Planck-Kolmogorov (FPK) equation, a two-dimensional GV-GDEE in terms of the response quantity of interest is derived. Further, this GV-GDEE is solved by the path integral method. Furthermore, two numerical examples are included. In particular, the nonstationary stochastic responses of a 5 megawatt (MW) fixed monopile offshore wind turbine structure under two operating conditions are studied. Comparisons with pertinent Monte Carlo simulations (MCS) demonstrate the accuracy and efficiency of the proposed approach. • An analog approximation for random field of sea wave kinetics along depth. • The model is incorporated with GV-GDEE for response analysis of stochastic systems. • A monopile offshore wind turbine structure under wave loads is studied. • Probability densities of response are obtained with high efficiency and accuracy. [ABSTRACT FROM AUTHOR]

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation. [ABSTRACT FROM AUTHOR]

• A framework of TC wind hazard assessment based on the probability density evolution method integrated with the change-of-measure technique is proposed. • The high efficiency of the proposed framework in assessing the TC wind hazard at multiple locations is validated. • The sensitivity of the TC wind hazard to controlling random variables is analyzed. • The TC wind hazard on the southeast coast of China is assessed. The tropical cyclone (TC) wind hazard assessment formulates a change-of-random-variable problem, which maps the probability space of controlling parameters to that of the TC-induced wind. This study presents a framework for analyzing the TC wind hazard based on the probability density evolution method (PDEM) integrated with the change of probability measure (COM). It is advantageous in the problem of assessing the TC wind hazards at multiple locations over the traditional approach because a common set of representative deterministic solutions of physical equations can be re-used for all locations rather than solving a separate set of solutions at each location. The proposed method is applied to estimate the wind hazard on the Chinese southeast coastline. The common representative deterministic points are selected by the generalized-F-discrepancy-based method with the probability distributions of controlling parameters at Shantou. Based on the COM, the geographical dependence of probabilistic characteristics is represented by changing the assigned probability. The PDEM is incorporated to evaluate the propagation of randomness from the input-controlling parameters to the TC-induced surface wind. The comparison to the Monte Carlo simulation demonstrates the high efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper a recently developed technique relying on the globally-evolving-based generalized density evolution equation (GE-GDEE) is extended. It is applied to determining the response statistics of multi-dimensional nonlinear systems with fractional derivative elements subject to Gaussian white noise. In particular, the GE-GDEE is derived by a new approach, which enhances its applicability to general continuous processes. Thus, non-Markovian system responses can be treated directly and efficiently. Specifically, for a high-dimensional nonlinear system with fractional derivative elements, of which the responses are non-Markovian, a one- or two-dimensional GE-GDEE, in terms of the response quantity of interest, is obtained. Note that the associated effective drift coefficients involve no fractional terms, and are estimated numerically from data derived from a fairly small number of deterministic analyses. Then, the GE-GDEE is solved by path integration. The accuracy and efficiency of the proposed technique are assessed by applying it for specific nonlinear system and juxtaposing the derived results with pertinent Monte Carlo Simulation data. • The probability density functions of responses of high-dimensional nonlinear systems endowed with fractional derivative elements are considered. • The globally-evolving-based generalized density evolution equation (GE-GDEE) is introduced to solve the problem in a convenient and efficient manner. • With a small number of deterministic analyses, the obtained GE-GDEE is of only one-/two-dimensions, and with no fractional derivatives involved. • The transient numerical solutions of the PDFs and CDFs of the response quantities of interest at different time instants have been obtained. • Good agreements with pertinent Monte Carlo simulation results have been observed, even in probability tails. [ABSTRACT FROM AUTHOR]