A multiple linearization method based on the quasi-Newton algorithm for hybrid reliability analysis with evidence and random variables

This paper aims to develop a novel, efficient, and accurate method for structural reliability assessment that involves both aleatory and epistemic uncertainties, represented by random variables and evidence variables, respectively. A significant challenge in structural reliability assessment involving evidence variables is the high computational cost associated with multidimensional problems and an excessive number of joint focal elements. To address this, first- and second-order approximate reliability methods are often employed to enhance computational efficiency, albeit at the expense of some accuracy, particularly for highly nonlinear problems. In this research, a multiple linearization method is proposed to strike a balance between accuracy and efficiency in hybrid reliability assessment. This method involves identifying a set of hyperplane approximations to the limit state function at the most probable points (MPPs) and additional linearization points (ALPs). The belief and plausibility measures of the structure are subsequently evaluated through the intersection or union of the failure domains defined by these hyperplanes. To optimally determine the ALPs based on the curvature of the original limit state at the MPPs and to avoid calculating second derivatives at the MPPs, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm and Symmetric Rank 1 (SR1) algorithm are respectively incorporated into the first-order reliability method to locate the MPPs and the suitable ALPs around the MPPs. A screening criterion is presented to avoid the computation burden on too many joint focal elements in hybrid reliability analysis. The effectiveness of the proposed methodology is demonstrated through one numerical example and four engineering applications.