Exact probability density function using multivariate transformation

The modeling of mechanical systems can be defined as the mathematical idealization of the physical phenomena controlling it. This implies to define the input variables (geometrical parameters, loading conditions...) and the output variables (displacements, stresses...) allow us to understand the evolution of the mechanical system. Indeed, we cannot admit to use the deterministic models where only the average parameters are considered, because it generally leads to wrong representation of the reality. Hence, it is interesting to introduce the uncertainties in parameter evaluation and to consider their variability. The fundamental issue of probabilistic studies is therefore to take into account the uncertain character and the spatial variability of parameters.The objective of this book is therefore to analyze and to study the probabilistic response of a mechanical system with uncertain parameters. Contrary to other numerical methods, the proposed technique couples the deterministic finite element method and the probabilistic transformation method, to evaluate analytically the probability density function of the response.