This paper presents an implementation of the approximate statistical moment method for uncertainty propagation and robust optimization for a quasi 1-D Euler CFD code. Given uncertainties in statistically independent, random, normally distributed input variables, a first-and second-order statistical moment matching procedure is performed to approximate the uncertainty in the CFD output. Efficient calculation of both first-and second-order sensitivity derivatives is required. In order to assess the validity of the approximations, the moments are compared with statistical moments generated through Monte Carlo simulations. The uncertainties in the CFD input variables are also incorporated into a robust optimization procedure. For this optimization, statistical moments involving firstorder sensitivity derivatives appear in the objective function and system constraints. Second-order sensitivity derivatives are used in a gradient-based search to successfully execute a robust optimization. The approximate methods used throughout the analyses are found to be valid when considering robustness about input parameter mean values.

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This paper presents a brief overview of some of the more recent advances in steady aerodynamic shape-design sensitivity analysis and optimization, based on advanced computational fluid dynamics.

We describe algorithms to compute self-similar measures associated to iterated function systems (i.f.s.) on an interval, and more general self-replicating measures that include Hausdorff measure on the attractor of a nonlinear i.f.s. We discuss a variety of error measurements for these algorithms. We then use the algorithms to study density properties of these measures experimentally. By density we mean the behavior of the ratio (B r (x))=(2r) as r ! 0, were is an appropriate dimension. It is well-known that a limit usually does not exist. We have found an intriguing structure associated to these ratios that we call density diagrams. We also use density computations to approximate the exact Hausdorff measure of the attractor of an i.f.s.