An extension of elemental-level uncertainty modeling to nonlinear geometric structures

The present work focuses on uncertainty modeling strategies in finite element analyses of structures exhibiting geometrical nonlinearities, in continuity of recent developments concerning the introduction of uncertainties at the level of the elemental matrices in the linearized case, using a maximum-entropy–based stochastic modeling framework. Such an approach preserves essential physical constraints, such as the signature of the matrices, as well as invariance with respect to node and degree-of-freedom ordering. It further allows for controlling the global level of uncertainty but also its spatial repartition within the structure. The extension to the nonlinear case requires the explicit knowledge of the multi-dimensional arrays of components that represent the quadratic and cubic elemental stiffness tensors. Unlike their linear counterpart, these higher-order tensors exhibit a specific topological structure together with subsequent sparsity. A suitable reparameterization of this data is thus introduced in order to construct a new operator with reasonable size, from which the uncertainty modeling framework, including both preservation of node ordering and physical constraints, can be efficiently applied. With this strategy, the uncertainties are still controlled by two parameters that govern both the amount and the correlation length of the random fluctuations, but the nonlinear extension requires the generation of a supplemental independent Gaussian random field. A complete set of uncertain linear and nonlinear operators is thus generated at the elemental level, before assembling them into global stochastic operators within the nonlinear finite element formulation. Finally, the strategy is applied to a three-dimensional curved beam.