Research On Solution Methods, Error Estimations For Stochastic Finite Element Equation And Morphology Descriptions Of Random Media

Stochastic finite element method(SFEM) is one of the most important tools for uncertainty quantification. At present, how to model the random media and solve the SFEM equations are two main related research directions. For modeling random media,the introduction of morphology functions, such as the lineal path functions and the twopoint cluster functions, can significantly improve the accuracy of reconstructed samples.However, it is still short of theoretical basis for discriminating microstructures by using morphology functions. And for solving SFEM equations, the solution scheme with high computational e?ciency, good convergence criteria and error estimation is hardly found.This dissertation focuses on these two problems in SFEM, and tries to develop e?ective solution schemes. The main contributions are as follows:I. A new e?cient method with adjustable accuracy and based on the Neumann expansion, named by generalized Neumann expansion(GNE) method, is proposed for solving SFEM equations. This GNE method exhibits the same high computation e?ciency as the perturbation methods, and possesses a posteriori necessary and su?cient convergence criterion, a priori su?cient but not necessary convergence criterion and a posteriori error estimation. In the cases of Gaussian random variables, the expectations and covariances of the GNE solutions can be directly calculated without sample analysis, and the expressions of the first three order solutions are derived. The problem scale that GNE can process is also suggested by analyzing the computational consumptions and storage requirements. It is showed that the GNE method, the perturbation method and the Neumann expansion method are mathematically equivalent for solving linear SFEM equations with deterministic loadings. But for the general SFEM equations, only the GNE and the Neumann expansion methods are equivalent. The proposed GNE method combines the advantages of both perturbations and Neumann expansions, and overcomes their most deficiencies.II. A priori error estimation system is established for the perturbation type solution methods of SFEM equations, including the GNE method, the perturbation method and the Neumann expansion method. A new vector norm and a consistent matrix norm are both defined. Two mathematical theorems and five corollaries, concerning the eigenvalues of the sum of matrices and stochastic matrices, are proposed and proved. Furthermore, two theorems regarding the spectral radius of the stochastic iteration matrix, the spectral radius of the stochastic sti?ness matrix, are also presented. Based on these theorems,corollaries and norms, five prior error estimations are proposed for the perturbation type solution methods of SFEM equations, in which the elastic moduli are stochastic fields.The error estimations can be applied to priori estimate the upper limits of errors for the solutions, the expectations and covariances of the solutions. Moreover, the error estimations can be applied to determine the expansion order according to the accuracy requirements.The error estimations indicate that the relative errors of the solution vectors induced by the homogenization of elastic moduli are not greater than the absolute maximum relative perturbation values of the elastic moduli. If the absolute maximum relative perturbation values of the elastic moduli are 50%, then the relative errors of the third order solution vectors will not be larger than 6.25%.III. General computational theories for the morphology functions of convex particles are proposed. The theoretical expressions of lineal path functions and two point cluster functions for regular triangles, regular squares, regular hexagons, circles and spheres with di?erent size distributions and di?erent orientation distributions, are established.This work provides theoretical reference for the reconstruction of microstructures. Furthermore, an unbiased and e?cient numerical extraction method of lineal path functions for periodic material samples is also suggested to simultaneously extract the particle and matrix lineal path functions in any direction.