Sensitivity Analysis Of Correlated Variables And Research On Model Validation

Different uncertainties widely exist in aeronautics and astronautics industry and mechanical engineering.Studying the effects of the uncertainties on the output performance of the structures,i.e.,sensitivity analysis,is of great significance for structural behavior prediction,risk evaluation and guidance of engineering optimization design.Besides,with the rapid development of technology,the engineering systems tend to be more complicated and larger.The computational models have become commonplace for engineering system design and evaluation because of the complex and expensive experiments.However,due to the inadequate knowledge of the system principles and limited amount of data,the discrepancy inevitably exists between the physical experiments and the computational models,therefore,establishing the validation criteria for the computational models is of great importance for system design and policy decision.This paper mainly develops the theory of global sensitivity analysis and model validation with various output forms in the presence of various uncertainties.The detailed contents are listed as follows:(1)For the correlated input variables,the correlated inputs are firstly transformed into independent ones based on Mara's method,then the sensitivities of the main variance contribution and the total variance contribution of the input with respect to each distribution parameter are derived analytically.From the analytical solutions,the basic laws of how distribution parameters affect the variance contribution are developed and some explanations are given.Besides,the structural contribution and the correlative one of the correlated normal inputs are derived on the existing importance measure method.By comparing the presented indices with the existing indices,the differences between them are identified and some general conclusions are drawn.(2)For the structural system with correlated input variables in the presence of aleatory and epistemic uncertainties,the effect of the inputs' distribution parameters with the epistemic uncertainty to the model output is studied.Then equality between the each order variance contributions of the distribution parameters to the output before and after separating uncertainties is validated.In addition,the model validation is carried out for the inputs with aleatory and epistemic uncertainties.A validation metric,the G metric,is established on the concept of the model-free sampling.The G metric allows a direct comparison of the physical and computational responses,and it avoids information loss,neither the increase of the input dimensions nor the consideration of the different kinds of uncertainty will add difficulty in its estimation.(3)Considering the problem of ignoring the partial information in the existing model validation methods,the partial model validation is studied on the whole model validation to discuss the difference between the physical observations and computational model.Based on the theory of high dimensional model representations(HDMR)of independent input variables,because the conditional expectations are component functions of model output,the conditional expectations reflect partial information of model output.Therefore,the model validation based on the conditional expectations can reveal the discrepancy between the partial information of the computational model output and that of the observations.Then a calibration of the conditional expectations is carried out to reduce the difference between the computational model and the experiments.Besides,when an input is fixed in an interval,the concept of regional validation is proposed to study the partial difference.The area metric and the G metric are applied to make regional validation specific and easily understandable.The contribution to whole validation(CWV)metric is proposed to depict the relationship between the proposed regional validation and the initial whole validation.(4)For the structural system with multivariate responses,the model validation is studied,and a new distance ratio metric for models is proposed on the Mahalanobis distance of the cumulative distribution functions.Besides,the use of the spearman rho makes the new distance ratio metric not limited to the distribution type of the data.In addition,to consider both the uncertainty and the correlation of the multiple responses and validation sites,a new method which combines the factor analysis method with the area metric is proposed in the process of model validation.The factor analysis has a favorable ascendency,dimensionality reduction,which makes multiple responses validation easy to be implemented.