Probabilistic finite element methods for problems in solid mechanics

A general strategy for the development of probabilistic finite element methods in solid mechanics is presented. Discretization techniques based on finite element shape functions and the spectral decomposition theorem are used to describe continuous fields of correlated random variables. Existing methods for the solution of stochastic problems are surveyed, with particular emphasis on the sampling patterns employed by these methods. An efficient family of iterative perturbation algorithms is proposed in order to reduce the cost of computing the finite element solutions at these sampling points. The iterative perturbation algorithms are residual-driven, do not involve explicit derivatives of the stiffness equations, and are easily extended to problems where the unperturbed solution is also obtained by an iterative strategy. The combination of an efficient sampling strategy with a robust perturbation algorithm will permit the use of detailed finite element models, which more accurately represent the underlying mechanics of the problem. Applications to both linear and non-linear probabilistic finite element problems in solid mechanics are discussed with the aid of numerical examples.