Research On Uncertainty Propagation And Computational Inverse Methods Based On Probability

Uncertainty exists widely in engineering practice. It has became a fundamental topic of uncertainty analysis, optimization design and identification based on uncertainty by accurately and quantificationally evaluating the influence of uncertain parameters on the responses or the influence of uncertain responses on the unknown parameters. Sometimes even the small uncertainty yields to large deviation of responses or unknown parameters. The uncertain variables are assumed to be deterministic in traditional mathematics treatment method, dramatically reducing the computation. But the obtained analysis results only have relative mathematical significance because of neglect of inherent uncertainty in system. Therefore, it is helpful to master the degree of influence of the inevitable uncertainty on the results, thus to lay the foundation for further eliminating the influence of uncertainty by studying the uncertain forward or inverse problems and developing effective uncertainty propagation methods and computational inverse methods. In order to resolve these problems, aiming at the different advantages of different uncertainty propagation methods in dealing with problems, this thesis adopts reliability analysis based method, dimension reduction integration and polynomial chaos expansion respectively to research the theory of uncertainty propagation and computational inverse methods under uncertainty. The work of this thesis is as follows:(1) Considering the real probability distribution of random variables in practical problems, the reliability analysis based method is used to obtain the uncertain responses for aiming at dealing with the system models whose degree of nonlinearity are weak. Based on the uncertainty propagation method, a novel computational inverse method is proposed, which combines maximum entropy principle with MPP based method. Maximum entropy theory is employed to model the PDFs of unknown parameters in the inner layer of the computational inverse method. Then the system model with random parameters which follows maximum entropy distributions is solved sequentially through MPP based method for obtaining the PDFs of computational responses. In the outer layer, the uncertain inverse problems can be translated into the deterministic optimization problems by minimizing the probability values of the measured and computational responses. The intergeneration projection genetic algorithm is implemented to find the means, standard deviations, coefficients of skewness and coefficients of kurtosis of unknown parameters. At last the probability distribution can be obtained by the maximum entropy theory. The method can identify real probability distribution forms of the unknown parameters and has a high computational efficiency.(2) A new uncertainty propagation method based on ?-PDF and dimension reduction integration is proposed to aim at dealing with the systems which have high degree of nonlinearity and enhancing the computational efficiency. Random parameter which has mono-peak probability density function(PDF) is approximated through the bounded random variables with ?-PDF or the deuterogenic PDF, which not only avoids the extremity of random parameters, but also avoids the improvement of degree of nonlinearity, which is resulted from the transformation form abnormal distributions into normal distributions, of system model. The original system is transformed into the combinations of single random parameter subsystems by the dimension reduction method. Gauss-Gegenbauer integration is adopted for single random parameter subsystem to obtain each statistical moments of response. Then ?-PDF of random response can be acquired. For each subsystem, the acquisition of the first four-order statistical moments just requires a small number of integral nodes. For the identification problem of unknown parameters, the least-squares objective function is established by the residual of each moment of computed and measured responses. The inner layer of the method nests the uncertainty propagation method. In the outer layer, the optimization algorithm is used to gain the means, standard deviations, coefficients of skewness and coefficients of kurtosis of unknown parameters. The method can deal with the system which has high degree of nonlinearity and possesses very high accuracy and efficiency.(3) A novel uncertainty propagation method and computational inverse method based on orthogonal polynomial chaos is presented to aim at dealing with the system model whose cross terms have great influence on the responses. In the process of uncertainty propagation, the adopted uncertainty modeling method is the same as the method in the preceding chapter. Then some samples are taken by the Latin hypercube sampling method and the responses of theses samples are computed through the system model. The response of original model can be expressed as the optimal standard Gegenbauer polynomial model which contains each random parameter through the structure-selection technique based on error reduction ratio. Because of the weighted orthogonality of the polynomials under ?-PDF, the means and variances of random responses can be obtained by analyzing the coefficients of the optimal polynomial chaos expansion. Based on the above theory, high-order statistical moments of random response can be acquired by dealing with the derivatives of the original system model using the same method. Finally the PDFs of responses can be gained on the basis of probabilistic approximation approach. For uncertain inverse problems, the inner layer of the method nested the uncertainty propagation method. In the outer layer, the PDF of unknown parameter can be got by analyzing the deterministic optimization problems which are translated from the uncertain inverse problems.