Efficient Adaptive Importance Sampling Estimation of Time Dependent Probability of Failure with Inspections for Damage Tolerant Aircraft Structures

Probabilistic Damage Tolerance Analysis (PDTA) addresses the need for a quantitative basis on which to set periodic inspections including the type of inspection and the interval between inspections and to determine when the aircraft structure is no longer economically viable. Ideally, PDTA includes all sources of random variation such as the initial flaw size, the crack growth rate variation, da/dN, the geometry, the material properties, and the loading, ensuring that the estimated probability of failure (POF) incorporates all known uncertainties, however, this requires evaluating the crack growth for each sample. In practice, due to the computational expense of performing a crack growth analysis for every realization, PDTA applications use a single crack growth analysis which limits the random variables to initial crack size, fracture toughness and maximum stress per flight. The objective of this research was to develop an adaptive sampling method which reduces the number of samples required for accurate PDTA by several orders of magnitude. This would enable comprehensive PDTA analyses that consider all sources of random variations. Research objectives were met through the following accomplishments; 1) a review of the current state of the art in PDTA, 2) the research and implementation of advanced sampling methods, 3) the development of the most promising sampling method and 4) application and verification of the chosen Adaptive Multiple Importance Sampling (AMIS) algorithm to multiple PDTA problems. The PDTA AMIS algorithm that has been developed in this research is 6 orders of magnitude more efficient than standard Monte Carlo (SMC) for estimating probabilities on the order of 10-7 with a coefficient of variation of 0.1. In addition, because of the efficiency of the PDTA AMIS algorithm, analysis outputs can be stored and reused for additional estimates such as a modified inspection program, and sensitivity analysis.