Research On Structural Uncertainty Propagation And Inverse Method Based On Evidence Theory

In practical engineering applications,uncertainties inevitably exist in various stages of structural design,manufacturing,operation and maintenance,which generally include geometric size,material characteristic,errors of installation and measurement,boundary conditions and some assumptions about engineering problems,etc.These uncertainties are presented in various forms,and the system performance may fluctuate dramatically and even failure because of the influence of uncertainties.Therefore,it has a great significance for the reliability,robustness and economy of the practical engineering structures to effectively quantify,control and even reduce the uncertainties.At present,many theories and models can be used to quantify the uncertainties,such as the probability theory,the interval model,the fuzzy set and evidence theory.As evidence theory can flexibly deal with various uncertainties under insufficient information in engineering problems through discrete evidence focal elements,it is considered to be a general choice for uncertainty quantification and analysis,and has a very considerable application prospect.However,the research and application of evidence theory in engineering structures are still in its preliminary stage,and many key problems need to solve further,especially the explosion of computational cost seriously hinders the application of evidence theory on the practical engineering problems.This dissertation is mainly focused on the computational efficiency,the quantification and modeling of correlation and the inverse problems based on evidence theory.The primary contributions of this research are summarized as follows:(1)An evidence uncertainty propagation method based on uncertainty domain analysis is proposed.In traditional evidence theory propagation analysis,because of the extreme analysis for each focal element,the computational cost is high.Additionally,and evidence theory only can employ the probability interval composed of the belief and plausibility functions to measure the uncertainty of a proposition due to the discrete characteristics of focal element.In this paper,the propagation analysis is transformed into the uncertain domain by the limit state equation of the system.By dividing the sub uncertain domains,a multistage and explicit limit state equation is constructed.The explicit multi-stage limit state equation can accurately determine the inclusion relationship between focal element and even the sub uncertain domain and the original event domain,thus effectively improving the computational efficiency.In addition,due to the use of the limit state equation,the probability of the focal element partly supporting the original proposition can be obtained by solving the approximate volume ratio,and then the maximum entropy measure within the belief and plausibility measures can be obtained directly,which provides an unambiguous measurement result for practical engineering problems.(2)An evidence uncertain propagation method based on marginal interval analysis is proposed for the combinatorial explosion problem with high-dimensional uncertainties.Firstly,the system performance function is transformed into the combination of the univariate sub-functions through dimension reduction decomposition,then the response of any point can be predicted by the responses of corresponding marginal collocation nodes.Hence,by collocating marginal nodes for each evidence variable,the responses of all joint collocation nodes can be easily obtained in whole uncertainty domain.This method can greatly reduce the call of the original system function in the focal element extreme analysis.On this basis,the marginal interval analysis method is further deduced.By the response of the marginal collocation point,the marginal interval of the sub-function can be obtained directly,and then the response extremums of the focal element can be obtained quickly through the interval operation of the marginal interval of each sub-function.It overcomes the inefficiency of subsequent numerical calculation especially for the high-dimensional problems.The examples show that the marginal interval analysis method can effectively improve the efficiency of evidence propagation,and because of the use of the original function at the marginal collocation point,the proposed method also has good calculation accuracy.(3)The traditional evidence theory model only can deal with the uncorrelated evidence variables,hence a new parallelotope-formed evidence theory(PFET)model is developed to quantify the correlated evidence variables.Firstly,a parallelotope-formed frame of discernment(FD)is constructed to quantify the uncertainty and correlation of evidence variables by the proposed a concept of evidence correlation coefficient(ECC).Secondly,the parallelotope-formed joint focal elements are further constructed in the established framework.No matter how correlation changes,the shape of joint focal elements is always consistent with the parallelotope-formed FD,that means they always maintain the same evidence correlation.Thus,the PFET model is constructed by combining the established parallelotope-formed FD,joint focal elements and their joint BPAs,and then unifies the correlated and uncorrelated evidence variables into the same framework.In view of that,in structural uncertainty propagation,the whole calculation process can be transformed into uncorrelated evidence space through establishing the PFET model,and then conveniently compute the belief and plausibility measures.Finally,two numerical examples and one engineering application are utilized to demonstrate the validity of the proposed PFET model.(4)For solving the inverse problem with uncertain structural parameters,an efficient evidence inverse method based on similar system principle is proposed.The proposed method can efficiently improve the computational efficiency of evidence inverse propagation from two levels.Firstly,the evidence inverse propagation is transformed into the finite deterministic inverse calculation at the collocation points of uncertain parameters by using the marginal interval analysis method,and then the number of deterministic inverse calculation is greatly reduced in evidence inverse propagation.Secondly,due to the small variation of uncertain parameters at adjacent collocation points,the system equations in the corresponding deterministic inverse calculation are similar.Then,a deterministic inverse method based on the similar system principle is further developed to reduce the forward model calls in deterministic inverse calculation.The example analyses show that the proposed evidence inverse method can efficiently and accurately calculate the cumulative belief and cumulative plausibility function of the identified parameters,and achieve the parameters identification under the uncertain structure.(5)For handle the inverse problem with uncertain structural response,a new uncertainty inverse method based on evidence theory is proposed for handle the inverse problem with uncertain structural response.The method can identify the evidential structure of unknown parameters according to the cumulative belief and cumulative possibility functions of the measured responses.Firstly,according to the upper and lower bounds of measurement responses,the evidential framework of identified parameters can be determined by classical interval inverse problem.Secondly,by adding linear constraints,the identification of evidence structure is transformed into the inverse of focal element node parameters,which reduces the structural parameters and decouples the nested optimization in the solution.Finally,the number combination of focal elements with small error of objective function is chosen as the optimal solution,and the corresponding calculated evidence structure in this case is regarded as the evidence BPA structure of the identified parameters.The numerical results also show that the proposed method can effectively characterize the uncertainty of the identified parameters derived from the epistemic uncertainty of the measurement responses.