**Abstract** : The design of optimal and reliable systems is an objective which is pursued in several fields of engineering. Optimality is expressed in terms of a system cost which needs to be minimized, including the partial costs involved at each instant of the system's life, e.g. construction costs, maintenance costs and costs resulting from potential failures. In addition, the system is expected to satisfy performance requirements considering all the criteria which could cause it to fail. In a probabilistic framework this performance is assessed through a failure probability w.r.t. such criteria, accounting for all sources of uncertainty in the problem inputs. The tradeoff to make between optimality and safety constitutes nowadays one the greatest challenges to solve in the engineering field. The set of works presented in this document brings contributions to the efficient solving of the above-defined problem, by addressing either the calculation of the system failure probability (problem referred to as reliability assessment) or the finding of its optimal design under constraints expressed in terms of this failure probability (problem referred to as reliability-based design optimization).
The document is organized in three chapters, with highlights on my contributions.
Chapter I concerns methods which are well established in reliability. My objective has been to make a clear and almost self-contained presentation of the main available methods with unified notations.
My contributions to these methods concern sensitivities of failure probability w.r.t. distribution parameters, exact sensitivities in FORM with the Nataf model, including those regarding correlation, and sensitivities in the standard normal space with subset simulation. I am also providing the reader with a comparison between subset simulation and the cross-entropy method applied to variates of the standard normal space on a set of selected examples, which illustrates the performances of each method.
Chapter II gives a presentation of kernel-based techniques used as surrogates of costly-to-evaluate models involved in failure criteria. These surrogate models are used for the purpose of reliability assessment and reliability-based design optimization in adaptive approaches. This type of approach starts from an initial set of evaluated points in the input space and then sequentially selects a few new points to evaluate, with updates to the surrogate model until a sufficient accuracy is reached. Two surrogate techniques are presented in this document: support vector machines (SVMs) and kriging, whose juxtaposition will hopefully help the reader in understanding the connections between them, in addition to the well-known and shared concept of kernel.
Chapter III compiles some results obtained in two specific fields of structural mechanics, namely buckling and crack propagation, which are known for their uncertain character as observed in experimental works. Several types of challenge were addressed in the selected problems, e.g. (i) identification of a probabilistic model from statistical data, (ii) reliability assessment not so easy to carry out due to the reliability problem formulation or intricacy of the limit-state surface, (iii) high computational cost of the numerical model involved in the analysis.