Stochastic finite element methods in dynamic

When studying mechanical systems using the finite element method, one of the main assumptions is that the model is deterministic. As the performance of computing machines increases considerably over time, it becomes possible to take into account complex phenomena that were previously neglected or simplified. In particular, it is becoming possible to work on problems with uncertain data, enabling the uncertainties intrinsic to the properties of materials and structures to be taken into account. The need to take these uncertainties into account has led to the development of stochastic finite element methods (SFEM) and mechano-fiabilistic couplings. Two main families of numerical methods currently exist:

• first and second moment methods for sensitivity analysis of the response of a structure whose parameters are uncertain, using a combination of finite element analysis and probability theory;

• mechano-fiabilistic couplings for reliability analysis of complex structures, combining finite element and reliability methods.

Irrespective of the analysis method used, uncertainties in the mechanical properties of materials, in the geometric characteristics of the structure and in the loading applied, vary according to their spatial position. These uncertainties are then modeled by random fields. However, reliability and sensitivity analysis require the discretization of random fields.

The first section of this article recalls the principles of discretizing random fields in finite element models, as well as the computational methods used. Spectral stochastic finite element methods are presented in the second section, focusing on the Muscolino perturbation method and the spectral modal synthesis method, and illustrating them on a simple mechanical example. The third section deals with an industrial gear system, comparing the different methods presented, and discussing the interest and limitations of SFFM for sensitivity and reliability analyses in mechanical design.