An adaptive Stochastic Finite Elements approach based on Newton–Cotes quadrature in simplex elements

In this paper an adaptive Stochastic Finite Elements approach with Newton–Cotes quadrature and simplex elements is developed for resolving the effect of random parameters in flow problems. The stochastic response is represented by a piecewise polynomial approximation by subdividing probability space into simplex elements. The quadrature approximation in the elements leads to solving uncoupled deterministic problems for varying parameter values. The elements are refined adaptively using a refinement measure based on the curvature of the approximation of the response weighted by the probability represented by the elements. Due to the Newton–Cotes quadrature the required number of deterministic solves is relatively low, since (i) the deterministic samples are reused in successive refinement steps due to the location of the quadrature points, and (ii) the samples are used in approximating the response in multiple elements, because most quadrature points are located on the boundaries of the elements. Monotonicity and extrema of the samples are preserved in the piecewise polynomial approximation of the response by subdividing elements where necessary in subelements with a linear approximation of the response. Applications to flows in a piston problem, a stall flutter model and transonic flow over a NACA0012 airfoil with uniformly and lognormally distributed random parameters demonstrate that the method is capable of resolving complex problems with singularities in probability space effectively. Resolving singularities is important since they can result in high sensitivities, and oscillatory or unphysical predictions.