Notice of Retraction Mean-value first order saddlepoint approximation based collaborative optimization for multidisciplinary problems under aleatory uncertainty

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. For example, the random load of a bridge may have two peaks, with a distribution consisting of a weighted sum of two normal distributions, suggested by traffic load data. When binomial variables are involved, traditional reliability methods, such as the First Order Second Moment (FOSM) method and the First Order Reliability Method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation for bimodal variables and then employs a mean value reliability method to accurately predict the reliability. A limit-state function is at first approximated with the first order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The saddlepoint approximation is then applied to estimate the reliability. Examples show that the new method is more accurate than FOSM and FORM.

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Probabilistic uncertainty analysis by mean-value first order Saddlepoint Approximation

Reliability Engineering & System Safety ◽

10.1016/j.ress.2006.10.021 ◽

2008 ◽

Vol 93 (2) ◽

pp. 325-336 ◽

Cited By ~ 111

Author(s):

Beiqing Huang ◽

Xiaoping Du

Keyword(s):

Uncertainty Analysis ◽

Saddlepoint Approximation ◽

Mean Value ◽

Probabilistic Uncertainty ◽

First Order

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Some Important Issues on First-Order Reliability Analysis With Nonprobabilistic Convex Models

Journal of Mechanical Design ◽

10.1115/1.4026261 ◽

2014 ◽

Vol 136 (3) ◽

Cited By ~ 4

Author(s):

C. Jiang ◽

G. Y. Lu ◽

X. Han ◽

R. G. Bi

Keyword(s):

Reliability Analysis ◽

Reliability Index ◽

Probability Model ◽

Mean Value ◽

Convex Model ◽

First Order ◽

Reliability Method ◽

Complex Engineering ◽

Mean Value Method ◽

Form Type

Compared with the probability model, the convex model approach only requires the bound information on the uncertainty, and can make it possible to conduct the reliability analysis for many complex engineering problems with limited samples. Presently, by introducing the well-established techniques in probability-based reliability analysis, some methods have been successfully developed for convex model reliability. This paper aims to reveal some different phenomena and furthermore some severe paradoxes when extending the widely used first-order reliability method (FORM) into the convex model problems, and whereby provide some useful suggestions and guidelines for convex-model-based reliability analysis. Two FORM-type approximations, namely, the mean-value method and the design-point method, are formulated to efficiently compute the nonprobabilistic reliability index. A comparison is then conducted between these two methods, and some important phenomena different from the traditional FORMs are summarized. The nonprobabilistic reliability index is also extended to treat the system reliability, and some unexpected paradoxes are found through two numerical examples.

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TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000098 ◽

2013 ◽

Vol 95 (1) ◽

pp. 76-128 ◽

Cited By ~ 17

Author(s):

VALENTINO MAGNANI

Keyword(s):

Implicit Function Theorem ◽

Group Structure ◽

Mean Value ◽

Implicit Function ◽

Inverse Mapping ◽

Natural Group ◽

Stratified Groups ◽

First Order ◽

Inverse Mapping Theorem ◽

Rank Theorem

AbstractWe study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

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Second-Order Reliability Method With First-Order Efficiency

Volume 1: 36th Design Automation Conference, Parts A and B ◽

10.1115/detc2010-28178 ◽

2010 ◽

Cited By ~ 1

Author(s):

Xiaoping Du ◽

Junfu Zhang

Keyword(s):

Limit State ◽

Saddlepoint Approximation ◽

Quadratic Function ◽

Second Order ◽

State Function ◽

Limit State Function ◽

First Order ◽

Reliability Method ◽

Order Of Magnitude ◽

Univariate Function

The widely used First Order Reliability Method (FORM) is efficient, but may not be accurate for nonlinear limit-state functions. The Second Order Reliability Method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second order reliability analysis method with first order efficiency. The method first performs the FORM and identifies the Most Probable Point (MPP). Then the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated as a quadratic function, which is created with the gradient information at the MPP and one more point near the MPP. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied for computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.

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Unified Uncertainty Analysis by the First Order Reliability Method

Journal of Mechanical Design ◽

10.1115/1.2943295 ◽

2008 ◽

Vol 130 (9) ◽

Cited By ~ 129

Author(s):

Xiaoping Du

Keyword(s):

Probability Theory ◽

Interval Analysis ◽

Probabilistic Analysis ◽

Epistemic Uncertainty ◽

Evidence Theory ◽

Black Box ◽

Aleatory Uncertainty ◽

First Order Reliability Method ◽

First Order ◽

Reliability Method

Two types of uncertainty exist in engineering. Aleatory uncertainty comes from inherent variations while epistemic uncertainty derives from ignorance or incomplete information. The former is usually modeled by the probability theory and has been widely researched. The latter can be modeled by the probability theory or nonprobability theories and is much more difficult to deal with. In this work, the effects of both types of uncertainty are quantified with belief and plausibility measures (lower and upper probabilities) in the context of the evidence theory. Input parameters with aleatory uncertainty are modeled with probability distributions by the probability theory. Input parameters with epistemic uncertainty are modeled with basic probability assignments by the evidence theory. A computational method is developed to compute belief and plausibility measures for black-box performance functions. The proposed method involves the nested probabilistic analysis and interval analysis. To handle black-box functions, we employ the first order reliability method for probabilistic analysis and nonlinear optimization for interval analysis. Two example problems are presented to demonstrate the proposed method.

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Risk Based Inspection Methodology for Components Subject to High-Temperature Creep

Volume 6B: Materials and Fabrication ◽

10.1115/pvp2017-66022 ◽

2017 ◽

Author(s):

Carl E. Jaske ◽

Panos Topalis ◽

Wong Sin Loong ◽

Azura Sharina Md Sidek

Keyword(s):

Limit State ◽

Creep Damage ◽

Mean Value ◽

Pressure Vessels ◽

State Function ◽

Limit State Function ◽

First Order ◽

First Order Second Moment ◽

Time In Service

Risk-based inspection (RBI) methodologies are widely used by industry to develop effective inspection programs for pressure vessels and piping. The RBI approach use data on equipment design, maintenance, and operation along with inspection history to evaluate both the likelihood and consequences of failure. RBI results provide a basis for selecting inspection methods and establishing inspection intervals and coverage. API RP 580 provides guidance on developing a RBI program for fixed equipment and piping, while API RP 581 provides quantitative procedures for establishing RBI methodology. Appendix J of the first edition (2000) of API RP 581 contained procedures for application to creep damage of furnace tubes. However, the second (2008) and third (2016) did not contain any procedures for application to creep damage of equipment, including furnace tubes. DNV GL undertook a RBI project for a coal-fired power plant in Malaysia that required evaluation of components subject to creep damage. As part of this project, a detailed likelihood of failure (LoF) model for creep was developed. This paper reviews the creep LoF model that was developed and discusses a case study of its application. The LoF is estimated using a limit state function where the resistance is characterized using Larson-Miller parameter creep-rupture expressions for the materials of interest and the load is characterized by the time in service. A mean value first order second moment (MVFOSM) method is employed to numerically compute LoF. Guidelines for including metallurgical replication results in the LoF estimate and for assigning inspection effectiveness for creep damage also are discussed.

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