Computational stochastic statics of an uncertain curved structure with geometrical nonlinearity in three-dimensional elasticity

Abstract

A methodology for analyzing the large static deformations of geometrically nonlinear structural systems in the presence of both system parameters uncertainties and model uncertainties is presented. It is carried out in the context of the identification of stochastic nonlinear reducedorder computational models using simulated experiments. This methodology requires the knowledge of a reference calculation issued from the mean nonlinear computational model in order to determine the POD basis (Proper Orthogonal Decomposition) used for the mean nonlinear reducedorder computational model. The construction of such mean reduced-order nonlinear computational model is explicitly carried out in the context of three-dimensional solid finite elements. It allows the stochastic nonlinear reduced-order computational model to be constructed in any general case with the nonparametric probabilistic approach. A numerical example is then presented for a curved beam in which the various steps are presented in details.

Prediction of displacement and stress fields of a notched panel with geometric nonlinearity by reduced order modeling

“…Equation (5) (4) and imposing the condition that the error be orthogonal to the basis (Galerkin approach), a set of nonlinear ordinary differential equations for the generalized coordinates ( ) t q n can be obtained [8], they are the reduced order model equations [8]. They can be used for their estimation (see [26]) but an indirect evaluation of many of these coefficients from the finite element model is also possible (see [1,8,22,29]) and is adopted here.…”

“…In addition, validation studies with experiments have been carried out for different types of panels [3,[23][24]. The introduction of uncertainty in the reduced order model has finally been formulated and implemented [25,26].…”

Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation

“…[28], see [29] for a recent review. The parameters of the nonlinear reduced-order model of the mean structure can then be either deduced using the STEP procedure (which is based on the smart non-intrusive use of standard commercial finite element codes) [30,22,29] or from explicit construction as shown in [31] in the context of three-dimensional solid finite elements.…”

“…The resulting stochastic nonlinear computational model is characterized by a single scalar dispersion parameter, quantifying the level of uncertainty in the stiffness properties which can easily be identified with experiments. Experimental validations based on this theory can be found in [31,32] for slender elastic bodies, e.g. beams.…”

“…Section 2 summarizes the main steps leading to the mean non-linear reduced-order computational model following the approach of [31]. Section 3 is devoted to the construction of the stochastic nonlinear computational model using the nonparametric probabilistic approach for modeling the random uncertainties.…”