Stochastic Finite Element Methodbased On Orthogonal Decomposition Of Random Field

Stochastic finite element method is widely used to tackle uncertain systems governed by stochastic partial differential equations. In the stochastic FEM, uncertainties are represented by random fields. To enable a computational treatment of this problem, the random fields are discretized to represent them in terms of a finite number of random variables.The perturbation method offers a computationally efficient technique to compute the first two statistical moments of the response quantities. The major drawback of such local approximation technique is that the results become highly inaccurate when the coefficients of variation of the input random variables are increased. Therefore, this dissertation focuses on the stochastic finite element method based on the Karhunen-Loève expansion for the random field. According to the different presentation of solution process, the theory is classified to spectral stochastic finite element method (SSFEM) and stochastic finite element method based on stochastic reduced basis (SRBSFEM).In the SSFEM, the solution process is approximated by its projection onto a finite subspace spanned by orthogonal Hermite polynomial chaos expansions (PCE). The coefficients in the expansion can be uniquely computed using Galerkin scheme, which involves the solution of a deterministic system of equations with increased dimensionality. Once the coefficients have been computed, the stochastic response in the form of PCE can be used to generate statistical moments and probability distributions. It has been shown that unlike perturbation-based method, good accuracy can be achieved even when the coefficients of variation of the input random variables are increased. But the major drawback of this method is that together with the proposed fine mesh makes it computationally prohibitive to attain a converged solution to the lager number degree of freedom.In the SRBSFEM, the solution process is approximated using basis vectors spanning the preconditioned stochastic Krylov subspace. A Galerkin projection scheme is used to convert the original stochastic PDEs into a set of coupled reduced-order deterministic PDEs which are used to compute the undetermined coefficients. Therefore, the stochastic response in the form of stochastic reduce basis can be used to generate statistical moments and probability distributions. For the class of problems considered, we find that the SRBSFEM can be up to orders of magnitude faster than the SSFEM which employs PC projection schemes, while providing results of comparable or better accuracy particularly for large-scale problems with many random variables.