The Monte Carlo method. Application to the stochastic analysis of sheet stamping processes

Abstract

Stochastic Mechanics is a rapidly growing area of research, whose importance is being recognized not only in academic circles but also in industrial practice. This is no doubt due to the fact that most structural properties and loads are either random or uncertain. The first term refers to a natural chaotic variation of the parameter, while the second is associated to the human lack of knowledge about it. Both kinds of unpredictability work together in rendering doubtful the results of a (usually single) deterministic mechanical analysis. When thinking about the randomness and uncertainty linked to all physical parameters and phenomena a big question mark closes the large list of numbers produced by a finite element calculation.

In Stochastic Mechanics there are several techniques to analyse the natural scatter of strains and stresses caused by the dispersion in the given loads and/or the structural parameters. The most general one is the Monte Carlo method. However, it must be recognized that is as well the most costly in computational terms. Nevertheless, this cost has becoming feasible with the advance in Computer Science, specially with the advent of parallel computing, due to the fact that a Monte Carlo calculation is intrinsically a task that can be performed in parallel.

The present report is intended to provide the reader an introduction to the Monte Carlo method in the context of Computational Mechanics. The technique is applied to the analysis of the uncertainty spread in a stamping process. The first chapter summarises the Monte Carlo method and its theoretical backgrounds. The second chapter is devoted to the case study, namely, the stochastic analysis of a square cup deep drawing problem. Finally, the basic equations governing the mechanical modelling of the stamping process are summarized in the appendix.