Confidence Level Estimation and Design Sensitivity Analysis for Confidence-Based RBDO

2012 ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
Ikjin Lee ◽  
David Gorsich
Keyword(s):  
Confidence Level ◽  
Input Data ◽  
Surrogate Models ◽  
Design Sensitivity ◽  
Distribution Model ◽  
Conditional Probabilities ◽  
Probabilistic Constraint ◽  
Input Distribution ◽  
Distribution Parameters ◽  
New Formulation

In practical engineering problems, often only limited input data are available to generate the input distribution model. The insufficient input data induces uncertainty on the input distribution model, and this uncertainty will cause us to lose confidence in the optimum design obtained using the reliability-based design optimization (RBDO) method. Uncertainty on the input distribution model requires us to consider the reliability analysis output, which is defined as the probability of failure, to follow a probabilistic distribution. This paper proposes a new formulation for the confidence-based RBDO method and design sensitivity analysis of the confidence level. The probability of the reliability analysis output is obtained with consecutive conditional probabilities of input distribution parameters and input distribution types using a Bayesian approach. The approximate conditional probabilities of input distribution parameters and types are suggested under certain assumptions. The Monte Carlo simulation is applied to practically calculate the output distribution, and the copula is used to describe the correlated input distribution types. A confidence-based RBDO problem is formulated using the derived the distribution of output. In this new formulation, the probabilistic constraint is modified to include both the target reliability and the target confidence level. Finally, the sensitivity of the confidence level, which is a new probabilistic constraint, is derived to support an efficient optimization process. Using accurate surrogate models, the proposed method does not require generation of additional surrogate models during the RBDO iteration; it only requires several evaluations of the same surrogate models. Hence, the efficiency of the method is obtained. For the numerical example, the confidence level is calculated and the accuracy of the derived sensitivity is verified when only limited data are available.

Confidence-Based Method for Reliability-Based Design Optimization

2014 ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
David Lamb
Keyword(s):  
Design Optimization ◽  
Confidence Level ◽  
Probabilistic Model ◽  
Input Data ◽  
Mathematical Problem ◽  
Design Sensitivity ◽  
Conditional Probabilities ◽  
Probabilistic Constraint ◽  
Reliability Based Design Optimization ◽  
Reliability Based Design

An accurate input probabilistic model is necessary to obtain a trustworthy result in the reliability analysis and the reliability-based design optimization (RBDO). However, the accurate input probabilistic model is not always available. Very often only insufficient input data are available in practical engineering problems. When only the limited input data are provided, uncertainty is induced in the input probabilistic model and this uncertainty propagates to the reliability output which is defined as the probability of failure. Then, the confidence level of the reliability output will decrease. To resolve this problem, the reliability output is considered to have a probability distribution in this paper. The probability of the reliability output is obtained as a combination of consecutive conditional probabilities of input distribution type and parameters using Bayesian approach. The conditional probabilities that are obtained under certain assumptions and Monte Carlo simulation (MCS) method is used to calculate the probability of the reliability output. Using the probability of the reliability output as constraint, a confidence-based RBDO (C-RBDO) problem is formulated. In the new probabilistic constraint of the C-RBDO formulation, two threshold values of the target reliability output and the target confidence level are used. For effective C-RBDO process, the design sensitivity of the new probabilistic constraint is derived. The C-RBDO is performed for a mathematical problem with different numbers of input data and the result shows that C-RBDO optimum designs incorporate appropriate conservativeness according to the given input data.

A Maximum Confidence Enhancement Based Sequential Sampling Scheme for Simulation-Based Design

2013 ◽  
Author(s):  
Zequn Wang ◽  
Pingfeng Wang
Keyword(s):  
Confidence Level ◽  
Sequential Sampling ◽  
Surrogate Models ◽  
Design Sensitivity ◽  
Reliability Estimation ◽  
Sampling Scheme ◽  
Design Variables ◽  
Simulation Based Design ◽  
Simulation Based ◽  
Maximum Improvement

This paper presents a maximum confidence enhancement based sequential sampling approach for simulation-based design under uncertainty. In the proposed approach, the ordinary Kriging method is adopted to construct surrogate models for all constraints and thus Monte Carlo simulation (MCS) is able to be used to estimate reliability and its sensitivity with respect to design variables. A cumulative confidence level is defined to quantify the accuracy of reliability estimation using MCS based on the Kriging models. To improve the efficiency of proposed approach, a maximum confidence enhancement based sequential sampling scheme is developed to update the Kriging models based on the maximum improvement of the defined cumulative confidence level, in which a sample that produces the largest improvement of the cumulative confidence level is selected to update the surrogate models. Moreover, a new design sensitivity estimation approach based upon constructed Kriging models is developed to estimate the reliability sensitivity information with respect to design variables without incurring any extra function evaluations. This enables to compute smooth sensitivity values and thus greatly enhances the efficiency and robustness of the design optimization process. Two case studies are used to demonstrate the proposed methodology.

scholarly journals Sensitivity Developments for RBDO With Dependent Input Variable and Varying Input Standard Deviation

2017 ◽  
Vol 139 (7) ◽  
Author(s):  
Hyunkyoo Cho ◽  
K. K. Choi ◽  
David Lamb
Keyword(s):  
Standard Deviation ◽  
Random Variable ◽  
Design Sensitivity ◽  
Statistical Correlation ◽  
Mean Value ◽  
Distribution Model ◽  
Target Point ◽  
Probabilistic Constraint ◽  
Reliability Based Design ◽  
Standard Normal

In reliability-based design optimization (RBDO), dependent input random variables and varying standard deviation (STD) should be considered to correctly describe input distribution model. The input dependency and varying STD significantly affect sensitivity for the most probable target point (MPTP) search and design sensitivity of probabilistic constraint in sensitivity-based RBDO. Hence, accurate sensitivities are necessary for efficient and effective process of MPTP search and RBDO. In this paper, it is assumed that dependency of input random variable is limited to the bivariate statistical correlation, and the correlation is considered using bivariate copulas. In addition, the varying STD is considered as a function of input mean value. The transformation between physical X-space and independent standard normal U-space for correlated input variable is presented using bivariate copula and marginal probability distribution. Using the transformation and the varying STD function, the sensitivity for the MPTP search and design sensitivity of probabilistic constraint are derived analytically. Using a mathematical example, the accuracy and efficiency of the developed sensitivities are verified. The RBDO result for the mathematical example indicates that the developed methods provide accurate sensitivities in the optimization process. In addition, a 14D engineering example is tested to verify the practicality and scalability of the developed sensitivity methods.

scholarly journals Multiple causes of nonstationarity in the Weihe annual low-flow series

2018 ◽  
Vol 22 (2) ◽  
pp. 1525-1542 ◽  
Author(s):  
Bin Xiong ◽  
Lihua Xiong ◽  
Jie Chen ◽  
Chong-Yu Xu ◽  
Lingqi Li
Keyword(s):  
Frequency Analysis ◽  
Driving Forces ◽  
Potential Evapotranspiration ◽  
Aridity Index ◽  
Base Flow ◽  
Low Flow ◽  
Distribution Model ◽  
Flow Index ◽  
Explanatory Variables ◽  
Distribution Parameters

Abstract. Under the background of global climate change and local anthropogenic activities, multiple driving forces have introduced various nonstationary components into low-flow series. This has led to a high demand on low-flow frequency analysis that considers nonstationary conditions for modeling. In this study, through a nonstationary frequency analysis framework with the generalized linear model (GLM) to consider time-varying distribution parameters, the multiple explanatory variables were incorporated to explain the variation in low-flow distribution parameters. These variables are comprised of the three indices of human activities (HAs; i.e., population, POP; irrigation area, IAR; and gross domestic product, GDP) and the eight measuring indices of the climate and catchment conditions (i.e., total precipitation P, mean frequency of precipitation events λ, temperature T, potential evapotranspiration (EP), climate aridity index AIEP, base-flow index (BFI), recession constant K and the recession-related aridity index AIK). This framework was applied to model the annual minimum flow series of both Huaxian and Xianyang gauging stations in the Weihe River, China (also known as the Wei He River). The results from stepwise regression for the optimal explanatory variables show that the variables related to irrigation, recession, temperature and precipitation play an important role in modeling. Specifically, analysis of annual minimum 30-day flow in Huaxian shows that the nonstationary distribution model with any one of all explanatory variables is better than the one without explanatory variables, the nonstationary gamma distribution model with four optimal variables is the best model and AIK is of the highest relative importance among these four variables, followed by IAR, BFI and AIEP. We conclude that the incorporation of multiple indices related to low-flow generation permits tracing various driving forces. The established link in nonstationary analysis will be beneficial to analyze future occurrences of low-flow extremes in similar areas.

Confidence-Based Design Optimization Under Data Uncertainty Using Most Probable Point-Based Approach

2019 ◽  
Author(s):  
Yongsu Jung ◽  
Hyunkyoo Cho ◽  
Ikjin Lee
Keyword(s):  
Statistical Model ◽  
Design Optimization ◽  
Performance Measure ◽  
Cumulative Distribution ◽  
Data Uncertainty ◽  
Normal Space ◽  
Most Probable Point ◽  
Input Distribution ◽  
Design Variables ◽  
Distribution Parameters

Abstract An accurate input statistical model has been assumed in most of reliability-based design optimization (RBDO) to concentrate on variability of random variables. However, only limited number of data are available to quantify the input statistical model in practical engineering applications. In other words, irreducible variability and reducible uncertainty due to lack of knowledge exist simultaneously in random design variables. Therefore, the uncertainty in reliability induced by insufficient data has to be accounted for RBDO to guarantee confidence of reliability. The uncertainty of input distributions is successfully propagated to a cumulative distribution function (CDF) of reliability under normality assumptions, but it requires a number of function evaluations in double-loop Monte Carlo simulation (MCS). To tackle this challenge, reliability measure approach (RMA) in confidence-based design optimization (CBDO) is proposed to handle the randomness of reliability following the idea of performance measure approach (PMA) in RBDO. Input distribution parameters are transformed to the standard normal space for most probable point (MPP) search with respect to reliability. Therefore, the reliability is approximated at MPP with respect to input distribution parameters. The proposed CBDO can treat confidence constraints employing the reliability value at the target confidence level that is approximated by MPP in P-space. In conclusion, the proposed method can significantly reduce the number of function evaluations by eliminating outer-loop MCS while maintaining acceptable accuracy.

Treating Epistemic Uncertainty Using Bootstrapping Selection of Input Distribution Model for Confidence-Based Reliability Assessment

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Min-Yeong Moon ◽  
K. K. Choi ◽  
Nicholas Gaul ◽  
David Lamb
Keyword(s):  
Physical System ◽  
Epistemic Uncertainty ◽  
Bootstrap Method ◽  
Reliability Assessment ◽  
Assessment Method ◽  
Reliability Estimation ◽  
Distribution Model ◽  
Hierarchical Bayesian ◽  
Distribution Models ◽  
Input Distribution

Accurately predicting the reliability of a physical system under aleatory uncertainty requires a very large number of physical output testing. Alternatively, a simulation-based method can be used, but it would involve epistemic uncertainties due to imperfections in input distribution models, simulation models, and surrogate models, as well as a limited number of output testing due to cost. Thus, the estimated output distributions and their corresponding reliabilities would become uncertain. One way to treat epistemic uncertainty is to use a hierarchical Bayesian approach; however, this could result in an overly conservative reliability by integrating possible candidates of input distribution. In this paper, a new confidence-based reliability assessment method that reduces unnecessary conservativeness is developed. The epistemic uncertainty induced by a limited number of input data is treated by approximating an input distribution model using a bootstrap method. Two engineering examples and one mathematical example are used to demonstrate that the proposed method (1) provides less conservative reliability than the hierarchical Bayesian analysis, yet (2) predicts the reliability of a physical system that satisfies the user-specified target confidence level, and (3) shows convergence behavior of reliability estimation as numbers of input and output test data increase.

Engine Failure Data Analysis Method Based on Weibull Distribution Model

2011 ◽  
Vol 128-129 ◽  
pp. 850-854
Author(s):  
Ying Kui Gu ◽  
Jing Li
Keyword(s):  
Weibull Distribution ◽  
Distribution Model ◽  
Parallel Model ◽  
Analysis Method ◽  
Failure Data ◽  
Rod System ◽  
Distribution Parameters ◽  
Engine Failure ◽  
Engine Reliability ◽  
Drawing Method

The failure data of crank rod system was analyzed by using weibull parallel model on the base of the simple weibull method. The distribution parameters of the weibull parallel model were estimated by using drawing method. The solving process of WPP drawing method was given in detial. Results show that the fitting degree of the failure data in the weibull parallel model is higher than that of the simple weibull distribution model, and it can more accurately described the failure distribution curve of the system in life cycle, which can provide necessary information for engine reliability indexes computation.

A guide to least-squares reverse time migration for subsalt imaging: Challenges and solutions

2017 ◽  
Vol 5 (3) ◽  
pp. SN1-SN11 ◽  
Author(s):  
Chong Zeng ◽  
Shuqian Dong ◽  
Bin Wang
Keyword(s):  
Least Squares ◽  
Confidence Level ◽  
Input Data ◽  
Convergence Rates ◽  
Depth Image ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Area Of Interest ◽  
Time Migration ◽  
Subsalt Imaging

Least-squares reverse time migration (LSRTM) overcomes the shortcomings of conventional migration algorithms by iteratively fitting the demigrated synthetic data and the input data to refine the initial depth image toward true reflectivity. It gradually enhances the effective signals and removes the migration artifacts such as swing noise during conventional migration. When imaging the subsalt area with complex structures, many practical issues have to be considered to ensure the convergence of the inversion. We tackle those practical issues such as an unknown source wavelet, inaccurate migration velocity, and slow convergence to make LSRTM applicable to subsalt imaging in geologic complex areas such as the Gulf of Mexico. Dynamic warping is used to realign the modeled and input data to compensate for minor velocity errors in the subsalt sediments. A windowed crosscorrelation-based confidence level is used to control the quality of the residual computation. The confidence level is further used as an inverse weighting to precondition the data residual so that the convergence rates in shallow and deep images are automatically balanced. It also helps suppress the strong artifacts related to the salt boundary. The efficiency of the LSRTM is improved so that interpretable images in the area of interest can be obtained in only a few iterations. After removing the artifacts near the salt body using LSRTM, the image better represents the true geology than the outcome of conventional RTM; thus, it facilitates the interpretation. Synthetic and field data examples examine and demonstrate the effectiveness of the adaptive strategies.

Reliability Analysis of a Small Pneumatic Cylinder Life

2011 ◽  
Vol 110-116 ◽  
pp. 4240-4245
Author(s):  
Jun Zhao Zhang ◽  
Cong Ling Wang ◽  
Xue Fa Fang
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Maximum Likelihood Method ◽  
Likelihood Method ◽  
Distribution Model ◽  
Pneumatic Cylinder ◽  
Life Test ◽  
Life Distribution ◽  
Median Rank ◽  
Distribution Parameters

The reliability of the pneumatic cylinder was investigated by routine life test. The results show that the failures of the pneumatic cylinder can be described as a Weibull distribution and fatigue fracture of the aluminum end cap and the head of install bolt is the major failure for the pneumatic cylinder. The pneumatic cylinder life distribution parameters were estimated by the median rank method in combination with maximum likelihood method. The distribution model for the reliability of the pneumatic cylinder was also proposed here.

Foreground Detection by Competitive Learning for Varying Input Distributions

2018 ◽  
Vol 28 (05) ◽  
pp. 1750056 ◽  
Author(s):  
Ezequiel López-Rubio ◽  
Miguel A. Molina-Cabello ◽  
Rafael Marcos Luque-Baena ◽  
Enrique Domínguez
Keyword(s):  
Neural Network ◽  
Computer Vision ◽  
Input Data ◽  
State Of The Art ◽  
Background Modeling ◽  
Cluster Detection ◽  
Foreground Detection ◽  
Learning Mechanism ◽  
Input Distribution ◽  
Computer Vision Applications

One of the most important challenges in computer vision applications is the background modeling, especially when the background is dynamic and the input distribution might not be stationary, i.e. the distribution of the input data could change with time (e.g. changing illuminations, waving trees, water, etc.). In this work, an unsupervised learning neural network is proposed which is able to cope with progressive changes in the input distribution. It is based on a dual learning mechanism which manages the changes of the input distribution separately from the cluster detection. The proposal is adequate for scenes where the background varies slowly. The performance of the method is tested against several state-of-the-art foreground detectors both quantitatively and qualitatively, with favorable results.

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