Structural Reliability Analysis Based On Homotopy Algorithm

The uncertainty caused by manufacturing and measurement is ubiquitous inengineering practice. The accurate measurement, analysis and control for parameteruncertainty and whereby conducting corresponding reliability design are effectivemeans of ensuring the normal functions of products or structures under complexworking conditions. The first order reliability method (FORM) is the classic and mostcommonly used method in the field of reliability, and it has been widely used instructural analysis and design in various industrial fields. In FORM, the problem ofreliability analysis is usually converted into an optimization problem to solve for thereliability index in the standard normal space, based on a probability transformation.The key problem of FORM-based reliability analysis is thus reduced to solving theoptimization problem. A series of algorithms aimed at solving this optimizationproblem have been developed. In particular, the Hasofer-Lind-Rackwitz-Fiessler(HL-RF) iterative algorithm has been widely used due to its advantages of clearconcept, simplicity, and high efficiency, as well as a requirement for less memoryduring operation. However, when the nonlinearity of limit-state functions is relativelyhigh or non-convex, it is difficult to guarantee the convergence of the HL-RF method.In this paper, we introduce a homotopy analysis into the FORM and propose a newalgorithm different from the traditional HL-RF type of methods to solve for thereliability index. The main contents of this paper are as follows:(1) In this paper, we introduce a homotopy analysis method into the probabilisticstructural reliability analysis and propose a different algorithm from the traditionalHasofer-Lind-Rackwitz-Fiessler (HL-RF) type of iterations to solve the reliabilityindex. Due to the powerful function of homotopy analysis in solving nonlinearequations, the method presented in this paper yields high efficiency and strongconvergence of the reliability analysis. We first establish the Karush–Kuhn–Tucker(KKT) condition of the optimization problem for reliability index in the first orderreliability method (FORM). Next, we construct the corresponding combinedhomotopy equations for the KKT condition and apply the path-tracking algorithm toefficiently solve the equations. Finally, several numerical examples and anengineering application are provided to validate the effectiveness of the presentmethod. (2) A new hybrid reliability analysis method based on the homotopy analysismethod is introduced to solve the reliability problem of the outer loop, the powerfulfunction method makes full use of the homotopy algorithm for solving nonlinearequations, the convergence of the method is stable and excellent. Finally, numericalexamples are provided to verify the effectiveness of this method.