Polynomial Chaos in Stochastic Finite Elements

Abstract

A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials. The proposed method involves a mathematical formulation which is a natural extension of the deterministic finite element concept to the space of random functions. A beam problem and a plate problem are investigated using the new method. The corresponding results are found in good agreement with those obtained through a Monte-Carlo simulation solution of the problems.

Application of Reciprocal Intervening Variables for Stochastic Finite Element Analysis

“…This example has been considered a number of times in the literature, see e.g. [4]. The model is represented schematically in fig.…”

Statistical Inverse Problems for Non-Gaussian Non-Stationary Stochastic Processes Defined by a Set of Realizations

“…Among these methods, the polynomial chaos expansion (PCE) method [3] has given very promising results, even in cases when M ν ( [5,4,7]). This technique is based on a direct projection of C on a polynomial hilbertian basis, { ψ j (ξ), 1 ≤ j }, of all the second-order random vectors with values in R M , such that:…”

“…In that context, it will be shown to what extent the polynomial chaos expansion (PCE) method (see [3,4,5]) can be used to identify such a distribution in very high dimension, even if the number of available realizations is very small. Section 2 introduces the method we propose to identify the statistical distribution of nonGaussian and non-stationary stochastic processes from a set of independent realizations.…”

Computation of connection coefficients and measure modifications for orthogonal polynomials

“…Assuming that the integral f 2 ω X dx is finite, a natural approximation space to consider is Π N , using elements p n [ω X ] as basis elements: this simplifies computations and allows straightforward computation of probabilistic moments. Several extensions of this idea have been considered for vector-valued parameters [25], Karhunen-Loeve expansions [12], random fields [3], etc., and arise in applications to micro-channel fluid flow [28], electrochemical processes [4], and electromagnetic systems [20], to name a few.…”