THE SOLUTION OF STOCHASTIC LINEAR PARTIAL DIFFERENTIAL EQUATIONS USING SFEM THROUGH NEUMANN AND HOMOGENEOUS CHAOS EXPANSIONS

Abstract

Handling stochastic partial differential equations is an important issue due to their wide and practical applications in all engineering fields. For this purpose, this thesis introduces spectral stochastic finite elements method (SSFEM) for the solution of this type of. equations. The system of equations is assumed to have stochastic linear operator and subjected to stochastic excitation function with deterministic boundary and initial conditions.

To get the solution of the described system, two alternative expansions are introduced in the framework of the deterministic finite elements method namely ; Neumann and Homogenous Chaos expansions. Using Neumann expansion, the resolvent of the stochastic operator is evaluated to get an explicit expressions for the solution process. This solution is obtained in a set of uncorrelated random variables where the mathematical expressions for the first two statistical moments only can be practically obtained. On the other hand, Homogenous Chaos expansion is proven to have relative advantages in evaluating the statistical moments of any order beside the probability distribution function (p.d.f.) of system response. Also , it can handle those systems that have high level of variability when Neumann expansion fails. By using this expansion, the p.d.f. of the solution process is obtained as a multiplication of random functionals with deterministic constants. Mathematical expressions for the p.d.f. as well as the statistical moments are then obtained for the above described system .

In order to investigate the applicability and the validation of the derived expressions, two dimensional stochastic problems were chosen and three case-studies were introduced in this context. The first and the second are plane strain problems for rectangular and nonrectangular plates respectively. The third is a plate bending problem for rectangular plates. In all of these problems, the plate is assumed to have stochastic operator and subjected to stochastic excitation. The first two statistical moments are evaluated using the two suggested expansions and the p.d.f. is plotted for these problems using Homogenous Chaos expansion. Also, parametric studies are introduced to investigate the effect of some stochastic parameters such as the covariance type, the process variance and the correlation lengths on the resultant random response.