The projection gradient algorithm with error control for structural reliability

Abstract Nowadays, probabilistic approaches are frequently used in the design of new civil engineering structures and the durability analysis of existing constructions. Hasofer–Lind’s reliability index is one of the most popular reliability measures due to its relevance and ease of use, and is now referred to in many structural design codes. This index can be determined by several algorithms dealing with minimization under constraint such as the well known Rackwitz–Fiessler algorithm based on the projection gradient method. The drawback of this algorithm lies in the estimation of the gradient vector of the limit-state function, which is often carried out by finite differences for non-explicit functions, resulting from the Finite Element Method for instance. If the perturbation chosen in the estimation of the gradient vector gives a variation of the output lower than the numerical accuracy of the limit-state function, the algorithm could lead to erroneous results or even not converge. In order to circumvent this problem, an original technique is suggested in this paper called “Projection gradient with error control”. The principle of the proposed technique is to attach to Rackwitz–Fiessler’s algorithm a procedure for judiciously determining the perturbation in the calculation of the gradient vector by finite differences, accounting for the numerical precision of the limit-state function. The efficiency and interest of the proposed procedure is emphasized through various examples.