**Abstract** : These lectures are devoted to the presentation of a new computational procedure for fatigue analysis of structures. This method, which is based on the theory of hys-teresis operators, consists to reduce computation of the damage D caused by a time varying stress t ∈ [0, T ] → Σ e (t) to the energy dissipated in the hysteresis loops of the image H µ (Σ e) of Σ e by an appropriately calibrated Preisach operator H µ. We then see that this formalism allows to reduce the structure optimization problem, which consists to seek design parameters u minimizing the damage in some given parts of a structure, to the minimization of the mapping u → D(u) = T 0 H µ (Σ e , t) d t where Σ e (x u) is a numerical mapping governed by a system of second order differential equations M uẍ + W uẋ + K u x = F (t) describing the dynamical behavior of the considered structure. Furthermore, we provide and validate a series of algorithms allowing to solve the optimization problem by a steepest descent method tailored to manage large dynamical problems derived from finite element models. At last, the theoretical results obtained in this course are illustrated with the help of numerous examples, intended for supporting the relevancy of the approach and providing implementation templates in design engineering software.