Neumann Expansion for Stochastic Finite Element Analysis

TLDR

The paper uses finite element analysis to study how spatial variability in material properties causes structural response variability under deterministic static loads, employing the Neumann expansion technique within a Monte Carlo framework. Material property variability is modeled as two‑dimensional stochastic fields, with a fine finite element mesh, and the Neumann expansion is used to derive the response variability, enabling detailed comparison of perturbation and Monte Carlo solutions for accuracy, convergence, and efficiency. The Monte Carlo results, compared with the perturbation method, reveal that the perturbation approach is only valid for small coefficient‑of‑variation cases, particularly when Young’s modulus is treated as a stochastic field.

Abstract

With the aid of the finite element method, the present paper deals with the problem of structural response variability resulting from the spatial variability of material properties of structures, when they are subjected to static loads of a deterministic nature. The spatial variabilities are modeled as two‐dimensional stochastic fields. The finite element discretization is performed in such a way that the size of each element is sufficiently small. Then, the present paper takes advantage of the Neumann expansion technique in deriving the finite element solution for the response variability within the framework of the Monte Carlo method. The Neumann expansion technique permits more detailed comparison between the perturbation and Monte Carlo solutions for accuracy, convergence, and computational efficiency. The result from such a Monte Carlo method is also compared with that based on the commonly used perturbation method. The comparison shows that the validity of the perturbation method is limited to the cases where the material property variation has a relatively small coefficient of variation, particularly when Young's modulus itself is assumed to form a stochastic field.