A Vine Copula-Based Hierarchical Framework for Multiscale Uncertainty Analysis

Uncertainty analysis is an effective methodology to acquire the variability of composite material properties. However, it is hard to apply hierarchical multiscale uncertainty analysis to the complex composite materials due to both quantification and propagation difficulties. In this paper, a novel hierarchical framework combined R-vine copula with sparse polynomial chaos expansions is proposed to handle multiscale uncertainty analysis problems. According to the strength of correlations, two different strategies are proposed to complete the uncertainty quantification and propagation. If the variables are weakly correlated or mutually independent, Rosenblatt transformation is used directly to transform non-normal distributions into the standard normal distributions. If the variables are strongly correlated, the multidimensional joint distribution is obtained by constructing R-vine copula, and Rosenblatt transformation is adopted to generalize independent standard variables. Then, the sparse polynomial chaos expansion is used to acquire the uncertainties of outputs with relatively few samples. The statistical moments of those variables that act as the inputs of next upscaling model can be gained analytically and easily by the polynomials. The analysis process of the proposed hierarchical framework is verified by the application of a 3D woven composite material system. Results show that the multidimensional correlations are modeled accurately by the R-vine copula functions, and thus uncertainty propagations with the transformed variables can be done to obtain the computational results with consideration of uncertainties accurately and efficiently.

A Novel Hierarchical Framework for Uncertainty Analysis of Multiscale Systems Combined Vine Copula With Sparse PCE

Uncertainty analysis is an effective methodology to acquire the variability of composite material properties. However, it is hard to apply hierarchical multiscale uncertainty analysis to the complex composite materials due to both quantification and propagation difficulties. In this paper, a novel hierarchical framework combined R-vine copula with sparse polynomial chaos expansions is proposed to handle multiscale uncertainty analysis problems. According to the strength of correlations, two different strategies are proposed to complete the uncertainty quantification and propagation. If the variables are weakly correlated or mutually independent, Rosenblatt transformation is used directly to transform non-normal distributions into the standard normal distributions. If the variables are strongly correlated, multidimensional joint distribution is obtained by constructing R-vine copula, and Rosenblatt transformation is adopted to generalize independent standard variables. Then the sparse polynomial chaos expansion is used to acquire the uncertainties of outputs with relatively few samples. The statistical moments of those variables that act as the inputs of next upscaling model, can be gained analytically and easily by the polynomials. The analysis process of the proposed hierarchical framework is verified by the application of a 3D woven composite material system. Results show that the multidimensional correlations are modelled accurately by the R-vine copula functions, and thus uncertainty propagations with the transformed variables can be done to obtain the computational results with consideration of uncertainties accurately and efficiently.

Alternative Environmental Contours for Marine Structural Design—A Comparison Study1

A new approach to estimate environmental contours in the original physical space by direct Monte Carlo simulations rather than applying the Rosenblatt transformation has recently been proposed. In this paper, the new and the traditional approach to estimating the contours are presented and the assumptions on which they are based are discussed. The different results given by these two methods are then compared in a number of case studies. Simultaneous probability density functions are fitted to the joint distribution of significant wave height and wave period for selected ocean locations and environmental contours are estimated for both methods. Thus, the practical consequences of the choice of approach are assessed. Particular attention is given to mixed sea systems. In these situations, the two approaches to environmental contours may be very different while for other wave conditions the contours are similar.

A New Method for Environmental Contours in Marine Structural Design

The environmental contour concept is often applied in marine structural design in conjunction with the Inverse First Order Reliability Method (IFORM). It allows for the great advantage of considering the environmental loads independently of the structural response. In this way, design sea states may be identified along the contour and time consuming response calculations are only needed for a limited set of design sea states. The traditional way of establishing such environmental contour lines is by applying the Rosenblatt transformation and identify the circle (in two dimensions) with radius equal to the reliability index βr The points along this circle are then transformed back to the original environmental space, specifying the closed contour. In this paper, an alternative approach for establishing the environmental contour lines in the original environmental space is proposed, eliminating the need for any transformations. This approach utilizes Monte Carlo simulations of the joint environmental model and is generally found to perform well. Advantages are that it yields a more precise interpretation and allows for more flexible modelling of the environmental parameters. This makes it easier to modify the environmental models to account for effects such as climate change if this is desired. In addition, possible over- or underestimation of failure probabilities due to the Rosenblatt transformation inherent in the traditional approach can be avoided with the proposed method.

Analysis of Non-Normal Process Capability Based on Rosenblatt Transformation System

Assuming that a process is subject to the normal distribution when calculating the process capability index traditionally. A main flaw lies in the process capability index is very sensitive to such changes if normal hypothesis was not satisfied. In order to obtain accurate process capability index of quality characteristics under this circumstance, this paper adopts Rosenblatt transformation to compare with Box-Cox transformation and Johnson transformation with type 6110 Connecting Rod Bush radius as an example, the results show that Rosenblatt transformation proposed performs better and implement is simple.

Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design

The proper orthogonal decomposition method has been employed to extract the important field signatures of random field observed in an engineering product or process. Our preliminary study found that the coefficients of the signatures are statistically uncorrelated but may be dependent. To this point, the statistical dependence of the coefficients has been ignored in the random field characterization for probability analysis and design. This paper thus proposes an effective random field characterization method that can account for the statistical dependence among the coefficients for probability analysis and design. The proposed approach has two technical contributions. The first contribution is the development of a natural approximation scheme of random field while preserving prescribed approximation accuracy. The coefficients of the signatures can be modeled as random field variables, and their statistical properties are identified using the chi-square goodness-of-fit test. Then, as the paper’s second technical contribution, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, this paper proposes a novel procedure of determining an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. The proposed random field characterization can be integrated with any advanced probability analysis method, such as the eigenvector dimension reduction method or polynomial chaos expansion method. Three structural examples, including a microelectromechanical system bistable mechanism, are used to demonstrate the effectiveness of the proposed approach. The results show that the statistical dependence in the random field characterization cannot be neglected during probability analysis and design. Moreover, it is shown that the proposed random field approach is very accurate and efficient.