Time-dependent failure credibility analysis and its optimization based computational methods

To measure the safety degree of the time-dependent structure under the fuzzy uncertainty, a time-dependent failure credibility (TDFC) is defined. The credibility measure has the excellent property of self-duality, thus the sum of the TDFC and the credibility of the time-dependent safety event, i.e., the complementary of the time-dependent failure event, equals to one. To solve the TDFC, two methods, including direct double-loop optimization (DLOB) method and transformed single-loop optimization (SLOB) method, are proposed. In two methods, the TDFC is firstly expressed as a bi-level problem by its definition, where the outer is the one-dimensional rooting and the inner is the min-min or max-min optimization for different cases. The outer one-dimensional rooting is solved by the dichotomy in DLOB method and SLOB method. The inner min-min or max-min problem is solved by different strategies in DLOB method and SLOB method. In the DLOB method, the min-min or max-min problem is directly solved by double-loop nested optimization strategy. In the SLOB method for improving the computational efficiency of the DLOB method, the min-min and max-min problems are respectively transformed to the single-loop optimizations by corresponding simple transition and trajectory-following algorithm, and in the trajectory-following algorithm, the complex-step method is employed for estimating the first order derivative for high precision. Several examples are used to demonstrate the rationality of the TDFC and the significance of two solutions. The results of the examples show that the proposed TDFC can reasonably describe the safety degree, and the SLOB method can drastically reduce the computational cost compared with the DLOB method under the acceptable precision.