A New Method For Solving Stochastic Finite Element Equations And Its Applications

There are various kinds of uncertainties in nature.Based on related mathematical knowledge,improving deterministic methods or establishing new methods for uncertainties can better reflect the true information of random problems.Although some existing methods have been applied to the practical random problems,a lot of difficulties need to be overcome.In this thesis,we deeply study the numerical algrithm for solving stochastic solutions and its applications,which opens up a new way for the analysis of large-scale and complex random problems.The main research contents are as following:1.In order to overcome the low efficiencies and accuracies of currently existing simulation methods of general non-Gaussian,non-stationary and high-dimensional random fields.this thesis proposes a general algrithm for simulating the samples of random fields.For the obtained samples of random fields,an efficient algrithm based on KL expansion is proposed to discretize random fields.Corresponding random variables are characterized by random samples.The advantages are that random samples can be used in subsequent analysis.For some practical problems,explicit expressions of random fields are needed,thus based on PC expansion,we propose an explicit algrithm for simulating random fields.For actual calculations,only a small number of deterministic iteration operations can effectively obtain the explicit representation of the random field.2.In order to develop efficient linear and nonlinear algrithms for solving stochastic finite element equations,spectral decomposition method and iterative technique are used to solve stochastic finite element equations.The proposed algorithm transforms the stochastic problem into deterministic problems.The size of derived deterministic equation is consistent with the original stochastic problem and will not produce a larger equation,which can be determined by use of mature finite element softwares.At the same time,the derived decomposition is applicable to linear,nonlinear and time-varying stochastic problems.3.The proposed linear and nonlinear algorithms for solving stochastic finite element equations can be directly extended to high-dimensional stochastic problems.The computational complexity does not increase sharply with the increase of stochastic dimensions and the computational cost becomes more insensitive with the increase of stochastic dimensions.It is expected to skillfully avoid so called ”Curse of Dimensionality”,thusthis paper provides a unified algorithm framework for solving stochastic finite element equations.4.The stochastic finite element method proposed in this paper is further applied to structural reliability analysis and stochastic inverse problem analysis.For the classical reliability analysis,there are different theories for different reliability annalysis.In order to overcome this shortcoming,this paper utilizes the stochastic solutions from proposed stochastic finite element algrithms to compute limit state functions,which is not necessary to consider different failure points and failure modes.The proposed method provide a unified analysis framework for various kind of reliability analysis.Overall failure probability nephogram can be readily obtained from the computed results,which simplifies the reliability analysis process.For the large-scale and high-dimensional stochastic inverse problem,computational costs of the forward problem and posterior probability distribution function are very high.In this thesis,the proposed stochastic finite element method is used to compute the forward problem,then the numerical estimation of the posterior probability distribution function can be directly obtained from random samples.The size of derived stochastic finite element equation is the same as the original stochastic problem and computational cost is insensitive to stochastic dimensions,which provides an efficient algorithm for large-scale and high-dimensional stochastic inverse problems.