Projection schemes in stochastic finite element analysis

In traditional computational mechanics, it is often assumed that the physical properties of the system under consideration are deterministic. This assumption of determinism forms the basis of most mathematical modeling procedures used to formulate partial differential equations (PDEs) governing the system response. In practice, however, some degree of uncertainty in characterizing virtually any engineering system is inevitable. In a structural system, deterministic characterization of the system properties and its environment may not be desirable due to several reasons, including uncertainty in the material properties due to statistically inhomogeneous microstructure, variations in nominal geometry due to manufacturing tolerances, and uncertainty in loading due to the nondeterministic nature of the operating environment. These uncertainties can be modeled within a probabilistic framework, which leads to PDEs with random coefficients and associated boundary and initial conditions governing the system dynamics. It is implicitly assumed here that uncertainty in the PDE coefficients can be described by random variables or random fields that are constructed using experimental data or stochastic micromechanical analysis