Subspace projection schemes for stochastic finite element analysis
This research is concerned with the development of subspace projection schemes for efficiently solving large systems of random algebraic equations obtained by finite element discretizations of stochastic partial differential equations. Reduced basis projections schemes employing the preconditioned stochastic Krylov subspace are compared with the polynomial chaos approach in terms of their accuracy and computational efficiency. For the class of problems considered, it is shown that stochastic reduced basis methods can be up to orders of magnitude faster, while providing results of comparable accuracy. Reduced basis projections schemes are further improved by hybridizing them with polynomial chaos expansions. The hybrid formulation ensures better accuracy by enabling the efficient application of reduced basis schemes even when a large number of basis vectors is used to approximate the response process. It also extends the application of reduced basis methods to non-Gaussian uncertainty models. A new scheme referred to as the strong Galerkin projection scheme is introduced which imposes orthogonality in a more strict sense compared to the conventional Galerkin projection scheme. It is shown that the proposed formulation is a generalization of stochastic reduced basis projection schemes and gives more accurate results than the polynomial chaos projection scheme, while incurring significantly lower computational cost. Finally, a software framework which utilizes existing deterministic finite element codes is developed for stochastic analysis of linear elastic problems. Numerical studies are presented for two-dimensional and three-dimensional problems to illustrate the capabilities of this framework.