Complete monotonic expression of the fourth-moment normal transformation for structural reliability

Zhao, Yan‐Gang; Zhang, Xuanyi; Lu, Zhao–Hui

doi:10.1016/j.compstruc.2017.11.006

Cited by 52 publications

(16 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, it is noteworthy that the PNTF-based methods are accurate enough only in the applicable range of their corresponding PNTF, large errors may be introduced for problems with statistical moments outside the applicable range. In this case, the complete formula recently proposed in Zhao et al (2018a) might be an option to bypass the limitation of the present work, which is an aspect worth for further investigation.…”

Section: Discussionmentioning

confidence: 95%

Hermite polynomial normal transformation for structural reliability analysis

Wang

Aldosary

Cen

et al. 2021

View full text Add to dashboard Cite

Purpose Normal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables. Design/methodology/approach The new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies. Findings Comprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems. Originality/value This study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

show abstract

Section: Discussionmentioning

confidence: 95%

Hermite polynomial normal transformation for structural reliability analysis

Wang

Aldosary

Cen

et al. 2021

View full text Add to dashboard Cite

show abstract

“…(1) The marginal PDF of A l ( t ) can be firstly estimated as pAlfalse(a;tfalse) by the maximal entropy model [48] or the fourth-moment normal transformation [49] according to the sample data of the responses at each time step. For example, if the first fourth moments of A l ( t ) are estimated as ms,Alfalse(hfalse), for s = 1,…, 4, according to the samples aqfalse(hfalse) at each time step t h , then based on the maximal entropy model, the PDF of A l ( t ) can be expressed as right left right left right left right left right left right left3pt0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0emtruepAl(a,t)=eη0false(tfalse)+η1false(tfalse)a+η2false(tfalse)a2+η3false(tfalse)a3+η4false(tfalse)a4,where η s ( t ), s = 1,…, 4, are undetermined parameters, and can be estimated as ηsfalse(hfalse) by solving the following nonlinear equations at each time step t h : right left right left right left right left right left right left3pt0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0emtrue∫…”

Section: Implementation Procedures For Probabilistic Response Analysi...mentioning

confidence: 99%

Globally-evolving-based generalized density evolution equation for nonlinear systems involving randomness from both system parameters and excitations

Chen

Lyu

2022

Proc. R. Soc. A.

View full text Add to dashboard Cite

It has long been one of the main challenges in science and engineering to capture the probabilistic response of high-dimensional nonlinear stochastic dynamic systems involving double randomness, i.e. randomness in both system parameters and excitations. For this purpose, a globally-evolving-based generalized density evolution equation (GE-GDEE) is established. Generally, for a multi-dimensional nonlinear system involving double randomness, if one single physical quantity as a response of the system is of interest, a GE-GDEE, as a two-dimensional partial differential equation (PDE) governing the probability density function (PDF), can be derived. The effective drift coefficients, which represent the physically driving force function in the GE-GDEE, can be determined based on the data from some representative deterministic dynamic analyses of the underlying physical system. A new estimator for effective drift coefficients is developed based on the vine copulas. Once the effective drift coefficients are determined, the GE-GDEE can be solved to capture the probability distributions of the quantities of interest. Several numerical examples, including linear and nonlinear multi-degree-of-freedom (MDOF) systems subjected to white noise or non-stationary earthquake ground motions, are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

show abstract

“…It reduces the calculation difficulty of design under uncertainty [18,19]. As one of the basic methods based on the moment method, the first order second moment (FOSM) is effective, but not accurate [20,21]. Therefore, it is necessary to use higher-order moments to improve the calculation accuracy.…”

Section: Introductionmentioning

confidence: 99%

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

Guo¹,

Lv²

2023

Computer Modeling in Engineering &Amp; Sciences

View full text Add to dashboard Cite

In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.

show abstract

Complete monotonic expression of the fourth-moment normal transformation for structural reliability

Cited by 52 publications

References 32 publications

Hermite polynomial normal transformation for structural reliability analysis

Hermite polynomial normal transformation for structural reliability analysis

Globally-evolving-based generalized density evolution equation for nonlinear systems involving randomness from both system parameters and excitations

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

Contact Info

Product

Resources

About