Alexandre Noll Marques

Umberto Villa, Alexandre Noll Marques

Numerical solution of the system of partial differential equations arising from the Reynolds-Averaged Navier-Stokes (RANS) equations with $k-ε$ turbulence model presents several challenges due to the advection dominated nature of the problem and the presence of highly nonlinear reaction terms. State-of-the-art software for the numerical solution of the RANS equations address these challenges by introducing non-differentiable perturbations in the model to ensure numerical stability. However, this approach leads to difficulties in the formulation of the higher-order forward/adjoint problems, which are needed for scalable Hessian-based uncertainty quantification (UQ) methods. In this note, we present the construction of a UQ-ready flow solver, i.e., one that is smoothly differentiable and provides not only forward solver capabilities but also adjoint and higher derivatives capabilities.

Alexandre N. Marques, Remi R. Lam, Karen E. Willcox

We introduce an algorithm to locate contours of functions that are expensive to evaluate. The problem of locating contours arises in many applications, including classification, constrained optimization, and performance analysis of mechanical and dynamical systems (reliability, probability of failure, stability, etc.). Our algorithm locates contours using information from multiple sources, which are available in the form of relatively inexpensive, biased, and possibly noisy approximations to the original function. Considering multiple information sources can lead to significant cost savings. We also introduce the concept of contour entropy, a formal measure of uncertainty about the location of the zero contour of a function approximated by a statistical surrogate model. Our algorithm locates contours efficiently by maximizing the reduction of contour entropy per unit cost.