Some critical observations and attendant new results in the finite element method for stochastic problems

Extensive motivation to do additional work in the finite element method in stochastic problems (FEMSP) is discussed. The qualitative comparison of FEMSP with the state of the art of the deterministic FEM is given. These critical observations and thoughts are then realized in several manners. We first present the exact inverse FEMSP, which however cannot serve as a general tool for stochastic analysis of complex structures. Therefore, an improved FEMSP is presented. Finally, the variational principles for stochastic beams, for the mean response function, as well as the responseʹs auto-correlation function are formulated for the first time in the literature, it is concluded that much work needs to be done in order for FEMSP to be at the level compared to that of the deterministic FEM. The advantage of the main methods presented here over the conventional ones lies in their non-perturbational nature. Numerical examples are presented to illustrate the superiority (if not yet universality) of the presented methods over the conventional ones.