Stochastic finite element response analysis using random eigenfunction expansion. (November 2017) - Pryse, S.E.; Adhikari, S.
Highlights: A solution to discretized PDEs in the form of u ( ω ) = ∑ j = 1 t a j ( ω ) g j ( ω ) is achieved. A random eigenfunction approach has been established to calculate a j ( ω ) and g j ( ω ) . The computational cost of the approach is reduced by approximating and truncating. The error arising due to the reduction is addressed through a Galerkin approach. In comparison to direct MCS, the approach produces accurate results in a faster time. Abstract: A mathematical form for the response of the stochastic finite element analysis of elliptical partial differential equations has been established through summing products of random scalars and random vectors. The method is based upon the eigendecomposition of a system's stiffness matrix. The computational reduction is achieved by only summing the dominant terms and by approximating the random eigenvalues and the random eigenvectors. An error analysis has been conducted to investigate the effect of the truncation and the approximations. Consequently, a novel error minimisation technique has been applied through the Galerkin error minimisation approach. This has been implemented by utilising the orthogonal nature of the random eigenvectors. The proposed method is used to solve three numerical examples: the bending of a stochastic beam, the flow through a porous media with stochastic permeability and the bending of a stochastic plate. The results obtained through the proposed random eigenfunction expansion approach are compared with those obtained by using direct Monte Carlo Simulations and by using polynomial chaos.