Improved finite element method for stochastic problems

The conventional finite element method dealing with stochastic problems is based on series expansion of stochastic quantities with respect to basic stochastic deviations, by means of either Taylor expansion, perturbation technique or Neumann expansion. The first-order approximation of the mean response, which is utilized to calculate the required probabilistic characteristics of the response, is just the deterministic solution obtained by fixing stochastic parameters at their mean value. However, such a mean response differs from the exact mean value and its use may cause significant error when the coefficients of variation of stochastic parameters are relatively large. In this paper, we propose an improved finite element method for stochastic problems. The method takes into account the first-order and second-order probabilistic information of stochastic parameters for computing the mean solution. The variance and covariance of the solution are calculated by utilizing the improved mean solution instead of ‘deterministic’ mean solution. The present improved method requires only means, variances and covariances of stochastic parameters. However, it has been found that the proposed improved method is much more accurate than the conventional first-order approximate method. Moreover, the present method is more accurate than the conventional second-order approximation method, which requires third and fourth-order probabilistic characteristics of stochastic parameters