Variational Approach to Probabilistic Finite Elements

Abstract

A probabilistic Hu‐Washizu variational principle (PHWVP) formulation for the probabilistic finite element method (PFEM) is presented. The formulation is developed for nonlinear elasticity using the Saint Venant Kirchhoff model. The PHWVP allows incorporation of probabilistic distributions for the compatibility condition, constitutive law, equilibrium, domain, and boundary conditions into the PFEM. Solution of the three stationary conditions for the compatibility relation, constitutive law, and equilibrium yield the variations in displacement, strain, and stress. Finally, the statistics such as expectation, autocovariance, and correlation of displacement, strain, and stress are determined. Thus, a probabilistic analysis can be performed in which all aspects of the problem are treated as random variables and/or fields. The Hu‐Washizu variational formulation is amenable to many conventional finite element codes, thereby enabling the extension of present codes to probabilistic problems.