Reliability-based analysis and design via failure domain bounding

This paper proposes a methodology for bounding and approximating failure probabilities for systems with parametric uncertainty that are subject to multiple design requirements. The two fundamental developments in this work are (1) a method to explicitly compute upper bounds on failure probability based on solving an optimization problem, and (2) a hybrid method, utilizing the upper bounds together with conditional sampling, to achieve highly accurate estimates of failure probabilities. The computation of failure probability bounds is accomplished by the deformation of hyper-spherical or hyper-rectangular sets in the standard normal space and utilizes a combination of numerical optimization and analytic tools. The methods for calculating these bounds are applicable to systems having multiple limit state functions and are not subject to difficulties when there are multiple critical parameter points. Numerical experiments are presented to demonstrate that the gains in accuracy and efficiency of the method are considerable when compared to alternative methods. The tools proposed are especially suited for design optimization due to their efficiency and the resulting continuity of the upper bounds. Since only standard optimization algorithms are required for implementation, these strategies are easily applicable to a variety of complex engineering problems.