Effective Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design

The Proper Orthogonal Decomposition (POD) method has been employed to extract the important signatures of the random field presented in an engineering product or process. Our preliminary study found that coefficients of the signatures are statistically uncorrelated but may be dependent. In general, the statistical dependence of the coefficients is ignored in the random field characterization for probability analysis and design. This paper thus proposes an effective approach to characterize the random field for probability analysis and design while accounting for the statistical dependence among the coefficients. The proposed approach is composed of two technical contributions. The first contribution is to develop a generic approximation scheme of random field as a function of the most important field signatures 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. Second, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. There exist so many transformation sequences when the number of random field variables becomes large. It was found that an improper selection of a transformation sequence may introduce high nonlinearity into system responses, which causes inaccuracy in probability analysis and design. Hence, a novel procedure is proposed for determining an optimal transformation sequence that introduces the least degree of nonlinearity to the system response after the Rosenblatt transformation. The proposed random field characterization can be integrated with one of the advanced probability analysis methods, such as the Eigenvector Dimension Reduction (EDR) method, Polynomial Chaos Expansion (PCE) method, etc. Three structural examples including a Micro-Electro-Mechanical Systems (MEMS) bistable mechanism are used to demonstrate the effectiveness of the proposed approach. The results show that the statistical dependence in random field characterization cannot be neglected for probability analysis and design. Moreover, it is shown that the proposed random field approach is very accurate and efficient.