Dynamic stochastic finite element method using time‐dependent generalized polynomial chaos

Abstract

Summary We present an intrusive formulation for the dynamic stochastic finite‐element method to propagate the epistemic uncertainty in material properties into a finite‐element system over time. The stochastic finite‐element method, originally developed for the static case, uses generalized polynomial chaos (gPC) expansions to represent the uncertainty in both material/load fields and displacement fields and solves for the unknown PC coefficients of displacement at each degree of freedom of the finite‐element system. In this case, the gPC basis used to represent the solution is optimal and can be kept the same throughout the static simulation; however, when integrating a stochastic system over time, it is proven that using gPC tends to break down for integrations over long times. The reason is that the solution's complexity increases in time, and the set of polynomials used in the gPC expansion to represent the solution, therefore, does not stay optimal. In this formulation, new stochastic variables and orthogonal polynomials are constructed as time progresses. These variables are obtained as the Kahrunen‐Loeve expansion of the finite‐element solution at the times of update, thus, optimally representing the distribution of the solution using a minimal set of orthogonal polynomials. The result from the method is a time‐domain polynomial chaos representation of the entire finite‐element solution. A fast post‐processing phase can be used to (a) obtain the probability distribution of the solution at each degree of freedom and (b) generate any number of time series realizations of the solution, which correspond to the same time series that would be obtained from a set of simulations based on direct Monte‐Carlo simulations of the uncertain material properties.

On the dynamic simulation of viscoelastic structures with random material properties using time‐separated stochastic mechanics

“…where the first moment ⟨F i ⟩ is given by Equation (12). The second moment ⟨F 2 i ⟩ can be calculated as…”

“…This approach was used for example to analyze the frequency response of a rod in [10] and a simplified rotor system in [11]. Some approaches to deal with an optimal basis for long time integration are presented in [12,13]. A non-intrusive approach to calculate the polynomial chaos expansion is the stochastic collocation method.…”

A stochastic Galerkin method with adaptive time-stepping for the Navier–Stokes equations

“…In particular, long-term integration was addressed by Gerritsma et al [10], Heuveline, Schick and Song [19,34,36], Wilkins [38], Özen nad Bal [29,30], and most recently by Esquivel et al [9], among others. Methods for flows exhibiting uncertain periodic dynamics were proposed, e.g., by Bonnaire et al [2], Lacour et al [22] and Schick et al [33]. These methods typically entail time-dependent or other variants of gPC expansions that are tailored to the changing character of the solution.…”