A univariate decomposition method for higher-order reliability analysis and design optimization

Abstract

The objective of this research is to develop new stochastic \t\t\t\tmethods based on most probable points (MPPs) for general \t\t\t\treliability analysis and reliability-based design optimization \t\t\t\tof complex engineering systems. The current efforts involves: \t\t\t\t(1) univariate method with simulation for reliability analysis; \t\t\t\t(2) univariate method with numerical integration for reliability \t\t\t\tanalysis; (3) multi-point univariate for reliability analysis \t\t\t\tinvolving multiple MPPs; and (4) univariate method for design \t\t\t\tsensitivity analysis and reliability-based design optimization.\nTwo MPP-based univariate decomposition methods were developed \t\t\t\tfor component reliability analysis with highly nonlinear \t\t\t\tperformance functions. Both methods involve novel function \t\t\t\tdecomposition at MPP that facilitates higher-order univariate \t\t\t\tapproximations of a performance function in the rotated Gaussian \t\t\t\tspace. The first method entails Lagrange interpolation of \t\t\t\tunivariate component functions that leads to an explicit \t\t\t\tperformance function and subsequent Monte Carlo simulation. \t\t\t\tBased on linear or quadratic approximations of the univariate \t\t\t\tcomponent function in the direction of the MPP, the second \t\t\t\tmethod formulates the performance function in a form amenable to \t\t\t\tan efficient reliability analysis by multiple one-dimensional \t\t\t\tintegrations. Although both methods have comparable \t\t\t\tcomputational efficiency, the second method can be extended to \t\t\t\tderive analytical sensitivity of failure probability for design \t\t\t\toptimization. For reliability problems entailing multiple MPPs, \t\t\t\ta multi-point univariate decomposition method was also \t\t\t\tdeveloped. In addition to the effort of identifying the MPP, the \t\t\t\tunivariate methods require a small number of exact or numerical \t\t\t\tfunction evaluations at selected input. Numerical results \t\t\t\tindicate that the MPP-based univariate methods provide accurate \t\t\t\tand/or computationally efficient estimates of failure \t\t\t\tprobability than existing methods.\nFinally, a new univariate decomposition method was developed for \t\t\t\tdesign sensitivity analysis and reliability-based design \t\t\t\toptimization subject to uncertain performance functions in \t\t\t\tconstraints. The method involves a novel univariate \t\t\t\tapproximation of a general multivariate function in the rotated \t\t\t\tGaussian space; analytical sensitivity of failure probability \t\t\t\twith respect to design variables; and standard gradient-based \t\t\t\toptimization algorithms. In both reliability and sensitivity \t\t\t\tanalyses, the proposed effort has been reduced to performing \t\t\t\tmultiple one-dimensional integrations. Numerical results \t\t\t\tindicate that the proposed method provides accurate and \t\t\t\tcomputationally efficient estimates of the sensitivity of \t\t\t\tfailure probability and leads to accurate design optimization of \t\t\t\tuncertain mechanical systems.

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