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The objective of this research is to develop new stochastic methods based on most probable points (MPPs) for general reliability analysis and reliability-based design optimization of complex engineering systems. The current efforts involves: (1) univariate method with simulation for reliability analysis; (2) univariate method with numerical integration for reliability analysis; (3) multi-point univariate for reliability analysis involving multiple MPPs; and (4) univariate method for design sensitivity analysis and reliability-based design optimization.
Two MPP-based univariate decomposition methods were developed for component reliability analysis with highly nonlinear performance functions. Both methods involve novel function decomposition at MPP that facilitates higher-order univariate approximations of a performance function in the rotated Gaussian space. The first method entails Lagrange interpolation of univariate component functions that leads to an explicit performance function and subsequent Monte Carlo simulation. Based on linear or quadratic approximations of the univariate component function in the direction of the MPP, the second method formulates the performance function in a form amenable to an efficient reliability analysis by multiple one-dimensional integrations. Although both methods have comparable computational efficiency, the second method can be extended to derive analytical sensitivity of failure probability for design optimization. For reliability problems entailing multiple MPPs, a multi-point univariate decomposition method was also developed. In addition to the effort of identifying the MPP, the univariate methods require a small number of exact or numerical function evaluations at selected input. Numerical results indicate that the MPP-based univariate methods provide accurate and/or computationally efficient estimates of failure probability than existing methods.
Finally, a new univariate decomposition method was developed for design sensitivity analysis and reliability-based design optimization subject to uncertain performance functions in constraints. The method involves a novel univariate approximation of a general multivariate function in the rotated Gaussian space; analytical sensitivity of failure probability with respect to design variables; and standard gradient-based optimization algorithms. In both reliability and sensitivity analyses, the proposed effort has been reduced to performing multiple one-dimensional integrations. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of the sensitivity of failure probability and leads to accurate design optimization of uncertain mechanical systems.