Practical approach of stochastic finite element method. Random variables

The finite element method has established itself as a major tool for the numerical representation of behavior in mechanics and physics in general. While these models are now more and more advanced, including ever broader hypotheses and requiring ever greater computing resources, they only calculate with fourteen significant digits an answer for which the designer has little information about the data. Probability theory offers a way of representing the uncertainty of the data by means of variables or random fields. The quantities of interest resulting from the calculation (stresses, displacements, etc.) are therefore also random, and the question is how to propagate the inputs into the behavioral model, which is supposed to represent physical reality as closely as possible. This led to the development of the Stochastic Finite Element Method (SFEM), a broad term encompassing different approaches, depending on whether they concern variables or fields, static or dynamic phenomena.

The objective is therefore simple in principle: knowing the stochastic model of the data, how to calculate the model of the output response; but it involves complex methodologies depending on the goal pursued. Sensitivity analysis looks for central properties (mean, median or variance of random variables), whereas reliability analysis looks at extreme quantiles and even goes so far as to search for the probability law. Because of its global nature, the stochastic finite element method is particularly well suited to sensitivity analysis.

The current legacy of finite element codes is considerable. That's why we're proposing a FEAM that draws directly on the extensive expertise of these codes, even though it limits the possibilities for representing uncertainties. It is called non-intrusive because it decouples the resolution of the mechanical model – the calculation code is used without modification – from that of the stochastic model. The intrusive method, on the other hand, integrates the stochastic model into the mechanics equations, requiring in-depth modifications to the calculation code. The non-intrusive FSEM is the only one discussed in this article, which is limited to random variables after discretization.

After a first paragraph recalling the principles and notations of the finite element model, the stochastic data model is introduced in a second paragraph. The third paragraph describes the proposed methods: perturbation and polynomial chaos; illustrating them on a simple mechanical example. A few fields of application to industrial issues are presented in the fourth paragraph, and an example of a real study is commented on. Finally, the last paragraph discusses the interest and limitations of FEM for sensitivity and reliability analyses in mechanical design....