Probability Density Evolution Research Articles

Point selection strategy based on minimizing <italic>GL</italic><sub>2</sub>-discrepancy and its application to multi-dimensional numerical integration

An optimal point selection strategy based on minimizing GL 2 -discrepancy, where the effects of assigned probabilities are involved, is proposed in the present paper for multi-dimensional numerical integration and stochastic dynamical response analysis. By doing so the optimal point selection is converted to a nonlinear optimization problem. In the case the coordinates of the points are prescribed, a two-step approach is suggested. The existence and uniqueness of the solution is proved. Through typical numerical examples, the GL 2 -discrepancy of different point sets and the accuracy of the numerical integration are compared, and the effectiveness and advantages of the proposed method is verified. Based on the investigation, the probability density evolution method is employed to implement the nonlinear stochastic dynamic response analysis of a concrete frame structure.

Nonlinear response and reliability analysis of tunnels under strong earthquakes

The damages of tunnels under earthquakes showed that soil–rock deposits around a tunnel have significant effects on its response to earthquake excitations, caused by the spatial variability and correlation of the soil properties, which may properly be modelled as a random field. This paper studied these effects through modelling the surrounding soil as a random field on the response of a tunnel and made some comparisons with the corresponding deterministic analysis. A two-dimensional finite element model of the tunnel-soil system was examined, and the corresponding random field was simulated by the spectral representation method. The results revealed that the corners of the tunnel are the critical locations for the axial force and bending moment. The comparisons showed that the results of the deterministic analysis usually yield higher results than the corresponding random field results. The reliabilities of a tunnel for two limit-state functions were considered: one is defined as exceeding the design limit according to the Chinese design code, whereas the other is exceeding the ultimate strength of the reinforced concrete tunnel lining. The equivalent extreme events were formulated for the respective limit-state functions, from which the tunnel system reliability can be evaluated. The probability density evolution method was used to calculate the stochastic response and reliability of the tunnel under a strong earthquake.

Nonlinear stochastic seismic analysis of buried pipeline systems

In this paper, a nonlinear stochastic seismic analysis program for buried pipeline systems is developed on the basis of a probability density evolution method (PDEM). A finite element model of buried pipeline systems subjected to seismic wave propagation is established. The pipelines in this model are simulated by 2D beam elements. The soil surrounding the pipelines is simulated by nonlinear distributed springs and linear distributed springs along the axial and horizontal directions, respectively. The joints between the segmented pipes are simulated by nonlinear concentrated springs. Thereafter, by considering the basic random variables of ground motion and soil, the PDEM is employed to capture the stochastic seismic responses of pipeline systems. Meanwhile, a physically based method is employed to simulate the random ground motion field for the area where the pipeline systems are located. Finally, a numerical example is investigated to validate the proposed program.

Response and reliability analysis of nonlinear uncertain dynamical structures by the probability density evolution method

The paper deals with the response and reliability analysis of hysteretic or geometric nonlinear uncertain dynamical systems of arbitrary dimensionality driven by stochastic processes. The approach is based on the probability density evolution method proposed by Li and Chen (Stochastic dynamics of structures, 1st edn. Wiley, London, 2009; Probab Eng Mech 20(1):33–44, 2005), which circumvents the dimensional curse of traditional methods for the determination of non-stationary probability densities based on Markov process assumptions and the numerical solution of the related Fokker–Planck and Kolmogorov–Feller equations. The main obstacle of the method is that a multi-dimensional convolution integral needs to be carried out over the sample space of a set of basic random variables, for which reason the number of these need to be relatively low. In order to handle this problem an approach is suggested, which reduces the number of basic random variables to merely a single one. Correspondingly, the response and reliability problems reduce to the solution of one-dimensional quadratures.

Liouville equation-based stochastic model for shoreline evolution

Long-term shoreline evolution due to longshore sediment transport is one of the key processes that need to be addressed in coastal engineering design and management. To adequately represent the inherent stochastic nature of the evolution processes, a probability density evolution model based on a Liouville-type equation is proposed for predicting the shoreline changes. In this model, the standard one-line beach evolution model that is widely used in coastal engineering design is reformulated in terms of the probability density function of shoreline responses. A computational algorithm involving a total variation diminishing scheme is employed to solve the resulting equation. To check the accuracy and robustness of the model, the predictions of the model are evaluated by comparing them with those from Monte Carlo simulations for two idealised shoreline configurations involving a single long jetty perpendicular to a straight shoreline and a rectangular beach nourishment case. The pertinent features of the predicted probabilistic shoreline responses are identified and discussed. The influence of the density distributions of the input parameters on the computed results is investigated.

A statistical model of information evaporation of perfectly reflecting black holes

We provide a statistical communication model for the phenomenon of quantum information evaporation from black holes (BHs). A BH behaves as a reflecting quantum channel in a very special regime, which allows for a receiver to perfectly recover the absorbed quantum information. The quantum channel of a perfectly reflecting (PR) BH is the probabilistically weighted sum of infinitely many qubit cloning channels. In this work, we reveal the statistical communication background of the information evaporation process of PR BHs. We show that the density of the cloned quantum particles in function of the PR BH's mass approximates a Chi-square distribution, while the stimulated emission process is characterized by zero-mean, circular symmetric complex Gaussian random variables. The results lead to the existence of Rayleigh random distributed coefficients in the probability density evolution, which confirms the presence of Rayleigh fading (a special type of random fluctuation) in the statistical communication model of BH information evaporation.

Dynamic Response Analysis of MSCSS Constructed with Rubber Vibration Isolator

The seismic responses of the mega-sub controlled structural system (MSCSS) can be reduced effectively based on the combination of the construction and control theory. One of the main construction features of the MSCSS is the connection between the additional column and the mega beam. The rubber vibration isolator is proposed as a new connection mode between the above two components according to the base isolation technique. The dynamic responses of the MSCSS subjected to the earthquake excitation are investigated by employing the probability density evolution method. The analysis results are compared with the seismic responses of the MSCSS constructed with other connection modes. The results show that the seismic responses of the MSCSS constructed with rubber vibration isolator can be reduced effectively, but the effect is not better than the other connection modes.

Nonlinear Response of Structures Subjected to Stochastic Excitations via Probability Density Evolution Method

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.

Dynamic Stability and Failure Probability Analysis of Dome Structures Under Stochastic Seismic Excitation

The intrinsic relationship between deterministic system and stochastic system is profoundly revealed by the probability density evolution method (PDEM) with introduction of physical law into the stochastic system. On this basis, stochastic dynamic stability analysis of single-layer dome structures under stochastic seismic excitation is firstly studied via incorporating an energetic physical criterion for identification of dynamic instability of dome structures into PDEM, which yields to sample stability (stable reliability). However, dynamic instability is not identical to structural failure definitely, where strength failure can be experienced not only in the stable structure but also when the structure is out of dynamic stability. It is practically feasible to decouple the stochastic dynamic response of dome structures to be a stable one and an unstable one according to the generalized density evolution equation (GDEE). Consequently, the global failure probability can be investigated separately based on the corresponding independent stochastic response. For unstable failure probability assessment, the failure probability is the unstable probability if the dome's failure is attributed to instability, whereas inverse absorbing is firstly implemented to get rid of the stochastic response before instability and a complementary process is filled in the safe domain immediately to finally assess the probability of strength failure after dynamic instability.

PDEM-based dimension-reduction of FPK equation for additively excited hysteretic nonlinear systems

Engineering structures usually exhibit hysteretic property when they are subjected to disastrous loadings such as earthquakes, strong wind and huge waves. The solution of FPK equation related to such complex systems are of great significance but is still an extremely challenging problem. In the present paper, the FPK equation for hysteretic systems is studied and the dimension is reduced to render an efficient numerical solution. The equivalence of probability flux in two different treatments, i.e., in the FPK equation which is based on the state space description and in the generalized density evolution equation which is mainly based on the random event description, are first examined. Then the FPK equation for additively excited multi-degree-of-freedom structures exhibiting nonlinear behaviors, including hysteresis together with strength degradation and stiffness degradation, is discussed. By invoking the basic idea of equivalence of probability flux, an equivalent probability flux due to drift could be constructed by the probability density evolution method (PDEM) instead of the direct coupled high-dimensional integral. Numerical implementation procedure is outlined. Two numerical examples are illustrated. Problems to be further studied are discussed.

Dimension Reduction of the FPK Equation via an Equivalence of Probability Flux for Additively Excited Systems

AbstractSolving 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...

Reliability analysis of structures with complex limit state functions using probability density evolution method

In reliability-based structural analysis and design optimization, there exist some limit state functions exhibiting disjoint failure domains, multiple design points and discontinuous responses. This study addresses this type of challenging problem of reliability assessment of structures with complex limit state functions based on the probability density evolution method (PDEM). Probability density function (PDF) of stochastic structures under static and dynamic loads can be acquired, which is independent of the specific form of limit state functions. Numerical results of several typical examples illustrate that, the time-invariant and instantaneous PDF curves and failure probabilities of stochastic structures with disjoint failure domains, multiple design points and discontinuous responses are calculated effectively and accurately. Moreover, the PDEM is validated to be more efficient than the Monte Carlo simulation and the subset simulation, and is a feasible and general approach to tackle the reliability analysis of complicated problems. In addition, the influence of random design parameters of structures on uncertainty propagation is also scrutinized.

Submarine pressure hull butt weld fatigue life reliability prediction method

As stress intensity factor for submarine pressure hull butt weld is very complicated, it is difficult to use common structural reliability analysis methods such as first-order second-moment method to predict its fatigue life reliability under stochastic loading. In view of the circumstance mentioned above, the probability density evolution method is employed to predict submarine pressure shell weld fatigue life reliability in this paper. State equation of crack propagation rate model with random initial conditions constructed by introducing extended state vector, then crack propagation growth probability density evolution equation is derived. Numerical method is used to solve the crack size probability density function in order to predict submarine pressure hull butt weld fatigue life reliability. The example shows that the result form proposed method is in good agreement with form Monte-Carlo method and crack size probability density function evolves with loading cycle.

Dimension-reduction of FPK equation via equivalent drift coefficient

The Fokker—Planck—Kolmogorov (FPK) equation plays an essential role in nonlinear stochastic dynamics. However, neither analytical nor numerical solution is available as yet to FPK equations for high-dimensional systems. In the present paper, the dimension reduction of FPK equation for systems excited by additive white noise is studied. In the proposed method, probability density evolution method (PDEM), in which a decoupled generalized density evolution equation is solved, is employed to reproduce the equivalent flux of probability for the marginalized FPK equation. A further step of constructing an equivalent coefficient finally completes the dimension-reduction of FPK equation. Examples are illustrated to verify the proposed method.

On dynamic information theory

In this paper, the author presents an overview on his own research works. In recent ten years, we extended the present static statistical information theory to dynamic processes and established a dynamic statistical information theory whose core is the dynamic information evolution equation describing the evolution law of dynamic information. Starting from the idea that the state variable probability density evolution equations of the stochastic dynamic system, the classical and quantum nonequilibrium statistical physical systems obeying stochastic law and the electrodynamic system obeying decterministic law can be regarded as their information symbol evolution equations and the definitions of dynamic information and dynamic entropy, we derived the evolution equations of dynamic information and dynamic entropy that express the evolution laws of dynamic information. These show that for the dynamic systems obeying a stochastic law, the time rate of change of dynamic information densities originates from their drift, diffusion and dissipation in state variable space inside the systems and coordinate space in the transmission processes, and that the time rate of change of dynamic entropy densities is caused by their drift, diffusion and production in state variable space inside the systems and coordinate space in the transmission processes. For the dynamic systems obeying the deterministic law, the evolution equations of dynamic information and dynamic entropy are the same mathematical type as the former except that dynamic information (entropy) density only has drift in state variable space inside the systems. Information and entropy have been connected with the state and change law of the system. Information diffusion and information dissipation occur at the same time. When the space noise can be neglected, information wave will appear. If we only consider the information change inside the systems, the dynamic information evolution equations reduce to information equations corresponding to the dynamic equations which express evolution laws for the above dynamic systems. This reveals that the evolution laws of the respective dynamic systems can be expressed by information equations in a unified fashion. Furthermore, we have presented the formulas for drift and diffusion information flow, information dissipation rate, and entropy production rate and a unified information expression for degradation and self-organizing evolution. Obtained the dynamic mutual information and dynamic channel capacity reflecting the dynamic dissipative character in transmission process, in when in the limiting case the ratio of channel length to signal transmission rate approaches zero, reduces itself to the present static mutual information and static channel capacity. All these new theoritical formulas and results are derived from the dynamic information evolution equation.

Experimental and analytical studies on stochastic seismic response control of structures with MR dampers

The magneto-rheological (MR) damper contributes to the new technology of structural vibration control. Its developments and applications have been paid significant attentions in earthquake engineering in recent years. Due to the shortages, however, inherent in deterministic control schemes where only several observed seismic accelerations are used as the trivial input and in classical stochastic optimal control theory with assumption of white noise process, the derived control policy cannot effectively accommodate the performance of randomly base-excited engineering structures. In this paper, the experimental and analytical studies on stochastic seismic response control of structures with specifically designed MR dampers are carried out. The random ground motion, as the base excitation posing upon the shaking table and the design load used for structural control system, is represented by the physically based stochastic ground motion model. Stochastic response analysis and reliability assessment of the tested structure are performed using the probability density evolution method and the theory of extreme value distribution. It is shown that the seismic response of the controlled structure with MR dampers gain a significant reduction compared with that of the uncontrolled structure, and the structural reliability is obviously strengthened as well.

Orthogonal Expansion Method of Random Processes for Fluctuating Pressure of Water Flow

Combining with its auto-correlation function, the stochastic process of water flows fluctuating pressure is decomposed on the trigonometric bases by employing an expansion method based on normalized orthogonal bases which is prescribed, and thus establish an incentive model for the random dynamic response analysis of structures vibrated by water flows fluctuating pressure. By using the model, main probabilistic characters of the flows stochastic process are captured with only a few random variables, and therefore laid the foundation for further random response and reliability analysis from the point of probability density evolution. By using it in an instance, the validity of the model is tested from the aspect of second order statistics such as sampling ensemble power spectrum and sample mean square error. The effects of simulation duration and expansion numbers needed on the simulations efficiency and accuracy are also discussed.

Analysis of structural reliability from time response using fully probabilistic

The probability density evolution of structural dynamic prediction models under several times random loads are proposed in this paper. Firstly, the probability density of several times stochastic loads can be obtained from the probability density of single time stochastic load. The prediction models of structural reliability from time responses under several times stochastic loads with and without strength degeneration over time are derived using the stress-strength interference theory and probability density evolution method. The differential equation in the models can be solved using the forward finite difference method. The probability density functions of the structural strength redundancy at any time can be obtained. In the end, structural dynamic reliability can be calculated using integral method. The efficiency of the proposed method is demonstrated numerically through a speed reducer. The results have shown that the proposed method is practicable, feasible and gives reasonable accurate prediction.

Failure Analysis of Wind Turbines by Probability Density Evolution Method

The aim of this study is to present an efficient and accurate method for estimation of the failure probability of wind turbine structures which work under turbulent wind load. The classical method for this is to fit one of the extreme value probability distribution functions to extracted maxima of the response of wind turbine. However this approach may contain high amount of uncertainty due to arbitrariness of the data and the distributions chosen. Therefore less uncertain methods are meaningful in this direction. The most natural approach in this respect is the Monte Carlo (MC) simulation. This however has no practical interest due to its excessive computational load. This problem can alternatively be tackled if the evolution of the probability density function (PDF) of the response process can be realized. The evolutionary PDF can then be integrated on the boundaries of the problem, i.e. the exceedance threshold of the response, which results in the accurate values of the failure probability. For this reason we propose to use the probability density evolution method (PDEM). PDEM can alternatively be used to obtain distribution of the extreme values of the response process by simulation. This approach requires less computational effort than integrating the evolution of the PDF; but may have less accuracy. In this paper we present the results of failure probability estimation by the PDEM. The results will then be compared to the extrapolated values from the extreme value distribution fits to the samples response values.

Simulation of the Stochastic Wave Loads Using a Physical Modeling Approach

In analyzing stochastic dynamic systems, analysis of the system uncertainty due to randomness in the loads plays a crucial role. Typically time series of the stochastic loads are simulated using traditional random phase method. This approach combined with fast Fourier transform algorithm makes an efficient way of simulating realizations of the stochastic load processes. However it requires many random variables, i.e. in the order of magnitude of 1000, to be included in the load model. Unfortunately having too many random variables in the problem makes considerable difficulties in analyzing system reliability or its uncertainty. Moreover applicability of the probability density evolution method on engineering problems faces critical difficulties when the system embeds too many random variables. Hence it is useful to devise a method which can make realization of the stochastic load processes with low, say less than 20, number of random variables. In this article we introduce an approach, so-called "physical modeling of stochastic processes", and show its applicability for simulation of the wave surface elevation.