Advances of Probability Density Evolution Method for Nonlinear Stochastic Systems

Li, Jie; Chen, Jianbing; Sun, Weiyuan

doi:10.3850/978-981-08-7619-7_p042

Cited by 19 publications

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“…(4) and (5) are the so-called generalized density evolution equations (GDEEs) [12,15] and the corresponding initial condition of Eq. (5) …”

Section: Fundamentals Of Probability Density Evolution Methods (Pdem)mentioning

confidence: 99%

“…Mathematically equivalently, we have the following partial differential equation [12,15], which governs the evolution of the joint PDF…”

Section: Fundamentals Of Probability Density Evolution Methods (Pdem)mentioning

confidence: 99%

“…are selected and the corresponding assigned probabilities are calculated as the following equation [12,18] so that the probabilistic information of the subdomain q  could be represented by q θ , i.e.…”

Section: The Generalized F-discrepancy (Gf-discrepancy)mentioning

confidence: 99%

“…Two possible paths, i.e., the ensemble evolution path and the point evolution path, could be employed in the numerical solution of PDEM [12]. In the point evolution based approach, a smart selection of representative points is of paramount importance.…”

Section: Introductionmentioning

confidence: 99%

“…From the perspective of physical stochastic systems, a family of probability density evolution method (PDEM), which is capable of capturing the instantaneous probability density functions of stochastic responses of nonlinear structures, has been developed by Li and Chen in the past decade [11]. Based on the random event description of principle of preservation of probability, a generalized density evolution equation (GDEE), of which the dimension is free from the dimension of the original dynamical system, is derived and solved by incorporating the embedded physical equations [12].Two possible paths, i.e., the ensemble evolution path and the point evolution path, could be employed in the numerical solution of PDEM [12]. In the point evolution based approach, a smart selection of representative points is of paramount importance.…”

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confidence: 99%

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Probability Density Evolution Analysis of Seismic Response of Nonlinear Structures With Random Parameters

Yang¹,

Chen²,

Li³

2015

Proceedings of the 1st International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Abstract. The performance and reliability of engineering structures under extreme disastrous dynamic excitations (e.g. earthquakes) is of great concern due to the strong nonlinear mechanical behaviors as well as the uncertainty involved in both structural properties and external excitations. Great efforts have been devoted into this field in the past INTRODUCTIONDue to the randomness of the time and magnitude of earthquakes, it is possible for all the practical engineering structures to enter the phase of exhibiting nonlinear behaviors [1,2]. Therefore, it is of great importance to take into account the coupling of nonlinearity and randomness in assessing the performance and reliability of a practical structure rationally. In the past few decades, though the stochastic dynamics of structures, including the random vibration and the stochastic structural analysis, was extensively investigated, yielding a variety of approaches such as the Monte Carlo simulation (MCS) [3], the random perturbation method [4,5], and the orthogonal polynomials expansion [6,7], ect., great challenges are still encountered in the stochastic dynamical analysis of nonlinear structures of practical engineering interest [8]. Moreover, these methods are mainly aimed at obtaining the first second order statistics of structural responses rather than higher order statistics or probability density functions, which are of crucial importance in the reliability evaluation of nonlinear structures [9,10]. From the perspective of physical stochastic systems, a family of probability density evolution method (PDEM), which is capable of capturing the instantaneous probability density functions of stochastic responses of nonlinear structures, has been developed by Li and Chen in the past decade [11]. Based on the random event description of principle of preservation of probability, a generalized density evolution equation (GDEE), of which the dimension is free from the dimension of the original dynamical system, is derived and solved by incorporating the embedded physical equations [12].Two possible paths, i.e., the ensemble evolution path and the point evolution path, could be employed in the numerical solution of PDEM [12]. In the point evolution based approach, a smart selection of representative points is of paramount importance. Great endeavors have been devoted into the optimal selection of representative point set [13,14]. In the present paper, the generalized F-discrepancy (GF-discrepancy) is employed in the determination of representative points. A numerical example of a nonlinear frame-shear structure is presented. The stochastic seismic responses and the corresponding probabilistic information, including probability density functions, are obtained and analyzed. Then the evaluation of reliabilities under different threshold levels are carried out so that the accuracy and efficiency of PDEM could be verified. Problems to be further studied are discussed.

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“…(4) and (5) are the so-called generalized density evolution equations (GDEEs) [12,15] and the corresponding initial condition of Eq. (5) …”

Section: Fundamentals Of Probability Density Evolution Methods (Pdem)mentioning

confidence: 99%

“…Mathematically equivalently, we have the following partial differential equation [12,15], which governs the evolution of the joint PDF…”

Section: Fundamentals Of Probability Density Evolution Methods (Pdem)mentioning

confidence: 99%

Section: The Generalized F-discrepancy (Gf-discrepancy)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Probability Density Evolution Analysis of Seismic Response of Nonlinear Structures With Random Parameters

Yang¹,

Chen²,

Li³

2015

Proceedings of the 1st International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Error estimate of point selection in uncertainty quantification of nonlinear structures involving multiple nonuniformly distributed parameters

Chen

Chan

2019

Numerical Meth Engineering

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Summary The error analysis for the selection of representative point sets (RPSs) in multidimensional random‐variate space assigned with arbitrary nonuniform distribution with compact support for uncertainty quantification is developed by extending the Koksma‐Hlawka inequalities, which bound the worst error with some kind of discrepancy of the RPS and the variation of the integrand. The novel concepts of the EF‐discrepancy and the GF‐discrepancy are introduced, and the connection and equivalence between them are inquired into. Based on such theoretical basis, the error estimate of selecting RPSs in the standardized space instead of that in the original physical space is studied, showing that the standardization of the input random variables does not increase, but usually reduce, the error bound by GF‐discrepancy. The extended Koksma‐Hlawka inequality also establishes the theoretical basis for the error estimate of the probability density evolution method. Besides, the closed‐form expressions for the EF‐discrepancy are given in the Appendix. A numerical example involving a complex nonlinear engineering structure modeled by the finite element method is studied, showing the accuracy of the proposed approach.

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A Newton iteration‐based interval analysis method for nonlinear structural systems with uncertain‐but‐bounded parameters

Qiu

Zhu

2021

Numerical Meth Engineering

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In this article, the propagating effect of uncertainties in nonlinear structural systems is investigated. Via considering uncertain-but-bounded parameters as interval variables, a Newton iteration-based interval analysis method (NI-IAM) is proposed to quantify the uncertainty of the nonlinear structural response, namely predicting the upper and lower bounds of the response. In the proposed method, a nonlinear structural system with uncertain interval parameters is described as a series of interval nonlinear equations. With Taylor series expansion, an interval iteration scheme is established to successively approximate the upper and lower bounds of the response based on the Newton method. The structural response bounds in every iteration step are updated by solving linear interval increment determination equations via the Lagrangian multiplier method. Thereout, an uncertain nonlinear problem is simplified to a series of uncertain linear issues. The convergence of the proposed method is further discussed in this article. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed method by contrast with several existing methods. Moreover, the compatibility between numerical results of the perturbation-based probabilistic method and the proposed method is then studied and verified.

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Advances of Probability Density Evolution Method for Nonlinear Stochastic Systems

Cited by 19 publications

References 0 publications

Probability Density Evolution Analysis of Seismic Response of Nonlinear Structures With Random Parameters

Probability Density Evolution Analysis of Seismic Response of Nonlinear Structures With Random Parameters

Error estimate of point selection in uncertainty quantification of nonlinear structures involving multiple nonuniformly distributed parameters

A Newton iteration‐based interval analysis method for nonlinear structural systems with uncertain‐but‐bounded parameters

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