A general purpose library for stochastic finite element computations

Abstract

This chapter presents a paper that introduces the elements of a software library for the integration of stochastic prediction capabilities with general-purpose finite element analysis software. The library is written in C++ and can be linked to standard finite element method (FEM) software. The stochastic finite element permits the characterization of predictions from numerical models as random variables and processes. This allows the quantification of scatters in the predictions that are consistent with observed scatters in the data used for calibrating the model. While this task of uncertainty propagation can be accomplished through Monte Carlo sampling (MCS), the format of the output from standard MCS cannot be readily interpreted for decision purposes. The spectral stochastic finite element method (SSFEM) provides an approximation of the predictions from mechanics-based simulations in a format that lends itself to these postprocessing tasks. The methodology permits the extension of concepts developed for adaptive mesh refinement to the realm of data and information refinement. The theoretical foundation of the method consists of a blend of measure theory and functional analysis, permitting the definition of orthogonal projections and meaningful approximants for random variables and vectors. This paper presents a review of the SSFEM, highlighting its connectivity requirements to legacy deterministic FEM software.