Uncertainty Analysis And Dimension Reduction Method Based On Polynomial Chaos Expansion

There are many uncertainties in the process of modeling and simulation,which inevitably leads to the deviation between the prediction results of the model and the experimental data.Therefore,identifying and quantifying uncertainty is critical for improving the reliability and robustness of simulation models.Polynomial chaos expansion,as a fast and low-cost method of uncertainty quantification,has developed rapidly,but it has been plagued by the curse of dimensionality.Based on polynomial chaos expansion,the thesis firstly reviews the global sensitivity analysis and active subspace dimensionality reduction method,and then studies the problems in them.Finally,a method is proposed to quantify time-variant system with hybrid uncertainty.The thesis presents the following contributions:1.The Sobol’ indices with hybrid uncertainty is studied.An efficient algorithm based on polynomial chaos expansion is proposed,and it is extended to the global sensitivity analysis problem of random vibration power spectral density response of structures.Keeping the double-loop idea,Monte Carlo is used to quantify epistemic uncertainty in the outer loop and considers a non-intrusive polynomial chaos which avoids a substantial number of samples and greatly improves the calculation efficiency in the inner loop.In addition,the existing global sensitivity analysis method is not suitable for the systems with power spectral density response.In this paper,an example of frame structure is used to calculate the Sobol’ indices of a few sample points near the resonance peak to analyze the importance of the input variables.This approach providing guidance for dimension reduction and uncertainty analysis.2.Performing polynomial chaos expansion directly on the high-dimensional model will inevitably encounter the curse of dimensionality.Hence,a dimension reduction approach based on active subspace is used to identify important directions with respect to the output,and these important directions constitute active variables for defining a new input space.In the following,a surrogate model of high-dimensional system is built by polynomial chaos expansion based on active variables.Finally,the validity of the method is verified by two numerical examples and one the HyShot II scramjet’s performance prediction example,and the research results show that,compared with the global sensitivity analysis that directly fixes the value of the secondary variable to reduce the dimension,the active subspace method is more reasonable and economical.3.Based on the probability box,a new probability evolution method is developed to quantify the hybrid uncertainty of the system in the time domain.Combined with the time-varying law of the cumulative distribution function of the system response,a method named nested Monte Carlo and non-intrusive polynomial chaos expansion is improved to solve the problem of probability boxes of the response at different times,which in a form a time-variant probability box.Two circuit examples are used to verify the feasibility of the proposed method.The results show that the time-variant probability box comprehensively evaluates the time-varying law of the hybrid uncertainty of the system response at different times,and also shows the evolution of the system response over time.It overcomes the defect that the single time probability box at a single time can only quantify the hybrid uncertainty at a specific time.In addition,in order to quickly collect the failure samples of time-variant system with hybrid uncertainty,the surrogate model is constructed by polynomial chaos expansion,and a large number of qualified initial samples can be fast filtered out based on the evaluation results of the surrogate model,leaving a bit of candidate failure samples.Then,the original model is used to confirm whether the candidate failure samples are really invalid.Comparing with Monte Carlo,this method greatly reduces the computational expense for the backward propagation of hybrid uncertainties based on the failure threshold of time-variant system.