A multi-fidelity boundary element method for structural reliability analysis with higher-order sensitivities

A novel multi-fidelity modelling methodology for structural reliability analysis using the Boundary Element Method (BEM) with an Implicit Differentiation Method (IDM) is presented. Reliability analyses are conducted with methods such as Monte Carlo Simulation (MCS) and the First-Order Reliability Method (FORM). The higher-order sensitivities of the elastostatic Boundary Element Method equations with respect to changes in several geometric variables have been derived for the first time for use with the IDM for the purpose of conducting reliability analyses with the Second-Order Reliability Method (SORM), a more accurate alternative to FORM for problems with non-linear limit state functions. Multi-fidelity formulations involving the IDM have also been derived for the first time, making use of the metamodelling technique Kriging. The use of multi-fidelity modelling enables the creation of a model that has similar accuracy to a high-fidelity model, but with a computational cost similar to that of a low-fidelity model. By combining the accuracy of the IDM with the efficiency of multi-fidelity modelling the proposed methodology has the capability to be very effective when used for structural reliability analysis. The IDM is validated through a numerical example for which the analytical solution is known. A further two numerical examples featuring an I-beam section and a triangular support bracket with a large number of variables are also investigated. Results show that the employed multi-fidelity models were up to 6000 times faster in terms of CPU-time than the high-fidelity model, while also providing probabilities of failure that were up to 2225 times more accurate than the low-fidelity model. Overall, it has been shown that the use of the proposed IDM/multi-fidelity modelling methodology significantly improved the efficiency and accuracy of the above reliability analysis techniques when applied to complex problems involving a large number of random variables under high levels of uncertainty.