A new generalized displacement control for nonlinear spectral stochastic finite element problems

Abstract The influence of uncertain material parameters on the mechanical behavior of structures is investigated using stochastic structural analysis. A widely used method for estimating the stochastic characteristics of the structural response is the Monte Carlo simulation (MCS). However, this approach exhibits a slow convergence rate. In this paper, we investigate geometrically nonlinear finite element problems using the spectral stochastic finite element method (SSFEM). This application of the SSFEM remains comparatively unexplored to date, see [1, 2, 14]. The SSFEM combines the polynomial chaos expansion (PCE) with the finite element method (FEM). The basis of the SSFEM is an extended variational formulation, which is discretized in the stochastic domain using the PCE and in the spatial domain using the FEM. This paper focuses on the development and implementation of solution algorithms for tracing nonlinear equilibrium paths within the framework of the SSFEM. In addition to existing standard solution methods, a generalized displacement control is developed for the application within the SSFEM. This algorithm allows the investigation of mechanical problems that cannot be solved using standard solution methods. The applicability of the developed solution algorithm is demonstrated through three numerical examples involving different structural elements. Besides a shell formulation similar to [13], a mixed-hybrid two-dimensional solid element and a geometrically exact beam element are used, see [12]. Furthermore, the implementation of the SSFEM into a general finite element program is shown. All algorithms required for implementing a geometrically nonlinear spectral stochastic FE formulation for a two-dimensional solid element are provided in the appendices.