Adaptive Uncertainty Quantification for Computational Fluid Dynamics

Abstract

Two different approaches to propagating uncertainty are considered, both with application to CFD models, both adaptive in the stochastic space. The first is “Adaptive Stochastic Finite Elements”, which approximates the model response in the stochastic space on a triangular grid with quadratic reconstruction on the elements. Limiting reduces the reconstruction to linear in the presence of discontinuities, and the mesh is refined using the Hessian of the response as an indicator. This construction allows for UQ in the presence of strong shocks in the response, for example in case of a transonic aerofoil with uncertain Mach number, and where variability in surface pressure is of interest. The second approach is “Adaptive Gradient-Enhanced Kriging for UQ”, which uses a Gaussian process model as a surrogate for the CFD model. To deal with the high cost of interpolating in high-dimensional spaces we make use of the adjoint of the CFD code, which provides derivatives in all stochastic directions at a cost independent of dimension. The Gaussian process framework allows this information to be incorporated into the surrogate, as well as providing regression of both CFD output values and derivatives according to error estimates. It also provides an interpolation error estimate on which an adaptivity indicator is based, weighted with the input uncertainty. A transonic aerofoil with four uncertain shape parameters is given as an example case.