Improved non-intrusive uncertainty propagation in complex fluid flow problems

The problem of non-intrusive uncertainty quantification is studied, with a focus on two computational fluid dynamics cases. A collocation method using quadrature or cubature rules is applied, where the simulations are selected deterministically. A one-dimensional quadrature rule is proposed which is nested, symmetric, and has positive weights. The rule is based on the removal of nodes from an existing symmetric quadrature rule with positive weights. The set of rules can be used to generate high-dimensional sparse grids using a Smolyak procedure, but such a procedure introduces negative weights. Therefore a new cubature rule is generated, also based on the removal of nodes. Again the rules are symmetric, positive, and nested. In low-dimensional cases, the number of nodes is approximately equal to the number of nodes of a sparse grid. If weight-positivity is dropped, it also has less nodes in high-dimensional cases. Moreover a method is proposed to determine the convergence criterion for each individual node. Because the weights for each node differ, varying the convergence criterion for each node results in less computational time without changing the quadrature or cubature rule. Two CFD cases are studied that show the properties of the proposed methods.