Reliability Based Design Optimization With Correlated Input Variables Using Copulas

2007 ◽  
Author(s):  
Kyung K. Choi ◽  
Yoojeong Noh ◽  
Liu Du
Keyword(s):  
Reliability Analysis ◽  
Random Variables ◽  
Performance Measure ◽  
Gaussian Copula ◽  
Joint Probability Distribution ◽  
Random Input ◽  
Marginal Distributions ◽  
Wide Range ◽  
Input Variables ◽  
Nataf Transformation

For the performance measure approach (PMA) of RBDO, a transformation between the input random variables and the standard normal random variables is necessary to carry out the inverse reliability analysis. For reliability analysis, Rosenblatt and Nataf transformations are commonly used. In many industrial RBDO problems, the input random variables are correlated. However, often only limited information such as the marginal distribution and covariance could be practically obtained, and the input joint probability distribution function (PDF) is very difficult to obtain. Thus, in literature, most RBDO methods assume all input random variables are independent. However, in this paper, it is found that the RBDO results can be significantly different when the input variables are correlated. Thus, various transformation methods are investigated for development of a RBDO method for problems with correlated input variables. It is found that Rosenblatt transformation is impractical for problems with correlated input variables due to difficulty of constructing a joint PDF from the marginal distributions and covariance. On the other hand, Nataf transformation can construct the joint CDF using the marginal distributions and covariance, and thus applicable to problems with correlated random input variables. The joint CDF is Nataf model, which is called a Gaussian copula in the copula family. Since the Gaussian copula can describe a wide range of the correlation coefficient, Nataf transformation can be widely used for various types of correlated input variables. In this paper, Nataf transformation is used to develop a RBDO method for design problems with correlated random input variables. Numerical examples are used to demonstrate the proposed method. Also, it is shown that the correlated random input variables significantly affect the RBDO results.

Uncertainty Quantification in Metamodel-Based Reliability Prediction

2016 ◽  
Author(s):  
Saideep Nannapaneni ◽  
Zhen Hu ◽  
Sankaran Mahadevan
Keyword(s):  
Design Optimization ◽  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Point Load ◽  
System Model ◽  
Robust Design Optimization ◽  
Statistical Uncertainty ◽  
Input Variables ◽  
Metamodel Uncertainty

Optimization under uncertainty has been studied in two directions — (1) Reliability-based Design Optimization (RBDO), and (2) Robust Design Optimization (RDO). One of the crucial elements in an RBDO problem is reliability analysis. Reliability analysis is affected by different types of epistemic uncertainty, due to inadequate data and modeling errors, along with aleatory uncertainty in input random variables. When the original physics-based model is computationally expensive, a metamodel has often been used in reliability analysis, introducing additional uncertainty due to the metamodel. This work presents a framework to include statistical uncertainty and model uncertainty in metamodel-based reliability analysis. Inadequate data causes uncertainty regarding the statistics (distribution types and distribution parameters) of the input variables, and regarding the system model parameters. Model errors include model form errors, solution approximation errors, and metamodel uncertainty. Two types of metamodels have been considered in literature for reliability analysis: (1) metamodels that compute the system model output over the desired ranges of the input random variables; and (2) metamodels that concentrate only on modeling the limit state. This work focuses on the latter type, using Gaussian process (GP) metamodels for performing both component reliability (single limit state) and system reliability (multiple limit states) analyses. A systematic procedure for the inclusion of model discrepancy terms in the limit-state metamodel construction is developed using an auxiliary variable approach. An efficient single-loop sampling approach using the probability integral transform is used for sampling the input variables with statistical uncertainty. The variability in the GP model prediction (metamodel uncertainty) is also included in reliability analysis through correlated sampling of the model predictions at different inputs. Two mechanical systems — a cantilever beam with point-load at the free end and a two-bar supported panel with point load at its center, are used to demonstrate the proposed techniques.

Reliability Analysis Using Second-Order Saddlepoint Approximation and Mixture Distributions

2018 ◽  
Author(s):  
Dimitrios I. Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Gaussian Mixture ◽  
Second Order ◽  
State Function ◽  
Gaussian Random Variables ◽  
Non Gaussian ◽  
Nataf Transformation

This paper proposes a new second-order Saddlepoint Approximation (SOSA) method for reliability analysis of nonlinear systems with correlated non-Gaussian and multimodal random variables. The proposed method overcomes the limitation of current available SOSA methods which are applicable to problems with only Gaussian random variables, by employing a Gaussian Mixture Model (GMM). The latter is first constructed using the Expectation Maximization (EM) method to approximate the joint probability density function of the input variables. Expressions of the statistical moments of the response variables are then derived using a second-order Taylor expansion of the limit-state function and the GMM. The standard SOSA method is finally integrated with the GMM to effectively analyze the reliability of systems with correlated non-Gaussian random variables. The accuracy of the proposed method is compared with existing methods including a SOSA based on Nataf transformation. Numerical examples demonstrate the effectiveness of the proposed approach.

Maximum Entropy Method-Based Reliability Analysis With Correlated Input Variables via Hybrid Dimension-Reduction Method

2019 ◽  
Vol 141 (10) ◽  
Author(s):  
Wanxin He ◽  
Gang Li ◽  
Peng Hao ◽  
Yan Zeng
Keyword(s):  
Dimension Reduction ◽  
Reliability Analysis ◽  
Maximum Entropy Method ◽  
Reduction Method ◽  
Entropy Method ◽  
Statistical Moments ◽  
Dimension Reduction Method ◽  
Performance Function ◽  
Input Variables ◽  
Nataf Transformation

The estimation of the statistical moments is widely involved in the industrial application, whose accuracy affects the reliability analysis result considerably. In this study, a novel hybrid dimension-reduction method based on the Nataf transformation is proposed to calculate the statistical moments of the performance function with correlated input variables. Nataf transformation is intrinsically the Gaussian copula, which is commonly used to transform the correlated input variables into independent ones. To calculate the numerical integration of the univariate component function in the proposed method, a normalized moment-based quadrature rule is employed. According to the statistical moments obtained by the proposed method, the probability density function of the performance function can be recovered accurately via maximum entropy method. Six examples are tested to illustrate the accuracy and efficiency of the proposed method, compared with that of Monte Carlo simulation, the conventional univariate dimension-reduction method, and the bivariate dimension-reduction method. It is confirmed that the proposed method achieves a good tradeoff between accuracy and efficiency for structural reliability analysis with correlated input variables.

Reliability Analysis Using Second-Order Saddlepoint Approximation and Mixture Distributions

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Dimitrios I. Papadimitriou ◽  
Zissimos P. Mourelatos ◽  
Zhen Hu
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Gaussian Mixture ◽  
Second Order ◽  
State Function ◽  
Gaussian Random Variables ◽  
Non Gaussian ◽  
Nataf Transformation

This paper proposes a new second-order saddlepoint approximation (SOSA) method for reliability analysis of nonlinear systems with correlated non-Gaussian and multimodal random variables. The proposed method overcomes the limitation of current available SOSA methods, which are applicable to problems with only Gaussian random variables, by employing a Gaussian mixture model (GMM). The latter is first constructed using the expectation maximization (EM) method to approximate the joint probability density function (PDF) of the input variables. Expressions of the statistical moments of the response variables are then derived using a second-order Taylor expansion of the limit-state function and the GMM. The standard SOSA method is finally integrated with the GMM to effectively analyze the reliability of systems with correlated non-Gaussian random variables. The accuracy of the proposed method is compared with existing methods including a SOSA based on Nataf transformation. Numerical examples demonstrate the effectiveness of the proposed approach.

scholarly journals Optimal Estimation of Shear Strength Parameters Based on Copula Theory Coupling Information Diffusion Technique

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Xinlong Zhou ◽  
Guang Zhang ◽  
Shaohua Hu ◽  
Junzhe Li
Keyword(s):  
Shear Strength ◽  
Reliability Analysis ◽  
Information Diffusion ◽  
Gaussian Copula ◽  
Shear Strength Parameters ◽  
Strength Parameters ◽  
Distribution Models ◽  
Marginal Distributions ◽  
Copula Theory ◽  
Diffusion Technique

In geotechnical reliability analysis, random volatility in marginal distributions of shear strength parameters has been rarely considered. Unfortunately, conventional marginal distribution models cannot characterize real probability distribution accurately, leading to considerable dispersion with incomplete probabilistic information. In this paper, an estimation methodology is proposed based on copula theory coupling information diffusion technique. Firstly, information diffusion distribution is extended to represent one-dimensional marginal distributions of shear strength parameters. Secondly, copula theory is employed to characterize the dependence structures among the parameters. Eventually, equivalent sample is yielded by information diffusion distribution that has been already established. A case study in Singapore is implemented to enunciate and validate the competence of the proposed method. The performances of the candidate copulas coupling different marginal distributions are further discussed. Results indicate that information diffusion distribution can efficiently capture the random volatility of real distributions of shear strength parameters and hold remarkable superiority in modeling marginal distributions. The equivalent sample, estimated by information diffusion technique in conjunction with Gaussian copula, has considerable consistency with original data. The proposed method can provide a reference to reliability analysis in geotechnical engineering.

scholarly journals Predictive modelling of soils’ hydraulic conductivity using artificial neural network and multiple linear regression

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Charles Gbenga Williams ◽  
Oluwapelumi O. Ojuri
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
Hydraulic Conductivity ◽  
Linear Regression ◽  
Multiple Linear Regression ◽  
Coarse Grained ◽  
Soil Types ◽  
Wide Range ◽  
Artificial Neural ◽  
Input Variables

AbstractAs a result of heterogeneity nature of soils and variation in its hydraulic conductivity over several orders of magnitude for various soil types from fine-grained to coarse-grained soils, predictive methods to estimate hydraulic conductivity of soils from properties considered more easily obtainable have now been given an appropriate consideration. This study evaluates the performance of artificial neural network (ANN) being one of the popular computational intelligence techniques in predicting hydraulic conductivity of wide range of soil types and compared with the traditional multiple linear regression (MLR). ANN and MLR models were developed using six input variables. Results revealed that only three input variables were statistically significant in MLR model development. Performance evaluations of the developed models using determination coefficient and mean square error show that the prediction capability of ANN is far better than MLR. In addition, comparative study with available existing models shows that the developed ANN and MLR in this study performed relatively better.

scholarly journals An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties

Algorithms ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 229
Author(s):  
Fangyi Li ◽  
Yufei Yan ◽  
Jianhua Rong ◽  
Houyao Zhu
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Random Variables ◽  
Equivalent Model ◽  
Independent Random Variables ◽  
Problem Analysis ◽  
Analysis Method ◽  
Calculation Algorithm ◽  
Reliability Problem ◽  
Interval Variables

In practical engineering, due to the lack of information, it is impossible to accurately determine the distribution of all variables. Therefore, time-variant reliability problems with both random and interval variables may be encountered. However, this kind of problem usually involves a complex multilevel nested optimization problem, which leads to a substantial computational burden, and it is difficult to meet the requirements of complex engineering problem analysis. This study proposes a decoupling strategy to efficiently analyze the time-variant reliability based on the mixed uncertainty model. The interval variables are treated with independent random variables that are uniformly distributed in their respective intervals. Then the time-variant reliability-equivalent model, containing only random variables, is established, to avoid multi-layer nesting optimization. The stochastic process is first discretized to obtain several static limit state functions at different times. The time-variant reliability problem is changed into the conventional time-invariant system reliability problem. First order reliability analysis method (FORM) is used to analyze the reliability of each time. Thus, an efficient and robust convergence hybrid time-variant reliability calculation algorithm is proposed based on the equivalent model. Finally, numerical examples shows the effectiveness of the proposed method.

Statistical Dependence of Input Variables in Doproc Method / Statistická Závislost Vstupních Veličin V Metodě Popv

2012 ◽  
Vol 12 (2) ◽  
pp. 48-58 ◽  
Author(s):  
Petr Janas ◽  
Krejsa Martin
Keyword(s):  
Reliability Assessment ◽  
Random Variables ◽  
Statistical Dependence ◽  
Software System ◽  
Dependent Variables ◽  
Computational Procedures ◽  
Input Variables

Abstract In probabilistic tasks, input random variables are often statistically dependent. This fact should be considered in correct computational procedures. In case of the newly developed Direct Optimized Probabilistic Calculation (DOProC), the statistically dependent variables can be expressed by the socalled multidimensional histograms, which can be used e.g. for probabilistic calculations and reliability assessment in the software system ProbCalc.

Transnational Corporate Governance: The State of the Art and Twenty-First-Century Challenges

2021 ◽  
pp. 614-646
Author(s):  
Dionysia Katelouzou ◽  
Peer Zumbansen
Keyword(s):  
Corporate Governance ◽  
Institutional Investors ◽  
Shareholder Wealth ◽  
Performance Measure ◽  
The State ◽  
First Century ◽  
Access To Health Services ◽  
Wide Range ◽  
Shareholder Wealth Maximization ◽  
Twenty First Century

This chapter explores corporate governance as a transnational regulatory field. Mirroring the rise in importance of the idea of shareholder wealth maximization as a firm’s definitive performance measure, corporate governance became a hotly contested field of competing visions of firms’ institutional and normative infrastructure in search of creating the most advantageous conditions to attract capital in volatile markets. This shift occurred at the same time that regulatory transformations in Western postindustrial societies since the early 1980s had begun to significantly shift public service provision and state-organized frameworks for old-age security guarantees and access to health services. Today’s corporate governance laboratory is a transnational force field, fought over by a host of different state and nonstate actors and also by private actors such as institutional investors. Meanwhile, following the financial crises in 2001, 2008 and 2020 and the simultaneously growing pressure on corporations from human rights, gender equality, and environmental groups, the corporate governance debate again is shifting. This time, a diversity of issues are being discussed under the corporate governance rubric, indicating a more comprehensive engagement with the firm’s purpose and functions and its societal obligations and responsibilities. Given the crucial role of firms as the residual claimants of a wide-ranging retreat of the state from its role in guaranteeing and providing a wide range of social functions, corporate governance is a mirror for the transformation of public and private power, and it has to address the twenty-first-century challenges, including global value chains and the proliferation of institutional investors, unfolding on a planetary scale.

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