Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives

Abstract

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.

Finite element and sensitivity analysis of thermally induced flow instabilities

“…Therefore, Re is small compared with unity and we can drop the inertial term in the momentum equation (24). By dropping the ∼ over the non-dimensional variables, we finally obtain…”

“…In other words, it measures the importance of changes in the flow response to perturbations of the model parameters. There are several means of computing sensitivities [22][23][24].…”

A Continuous Second Order Sensitivity Equation Method for Time-Dependent Incompressible Laminar Flows

“…Automatic differentiation for first-order flow sensitivities is discussed by Sherman et al [10] and Putko et al [3]. Continuous SEMs may be found in Godfrey et al [11,12], Borggaard and Burns [4], Limache [13] and Turgeon [14] for aerodynamics problems.…”

“…There are several means of computing flow sensitivities: finite differences of flow solutions, the complex step method [2], automatic differentiation [3],a n dS E M s [4][5][6]. The first option is costly because the problem must be solved for two or more values of each parameter of interest.…”

Decision Support Methods and Tools

“…Uncertainty quantification [9][10][11][12][13][14] (UQ) can be used to both improve and clarify the evaluations of the probability of an occurrence, and of the consequence of an occurrence within a risk assessment. Uncertainty quantification begins in steps 1 and 2 of the 17 common technical processes above, with the examination of requirements for a given problem to identify "fuzzy" statements, those that cannot be definitively fulfilled.…”