We present an extensible Julia-based solver for the Euler equations that uses a summationby-parts (SBP) discretization on unstructured triangular grids. While SBP operators have been used for tensor-product discretizations for some time, they have only recently been extended to simplices. Here we investigate the accuracy and stability properties of simplexbased SBP discretizations of the Euler equations. Non-linear stabilization is a particular concern in this context, because SBP operators are nearly skew-symmetric. We consider an edge-based stabilization method, which has previously been used for advection-diffusionreaction problems and the Oseen equations, and apply it to the Euler equations. Additionally, we discuss how the development of our software has been facilitated by the use of Julia, a new, fast, dynamic programming language designed for technical computing. By taking advantage of Julia's unique capabilities, code that is both efficient and generic can be written, enhancing the extensibility of the solver.

We describe an algorithm for optimizing time-averaged objective functions that depend on a chaotic state variable. Such problems are ubiquitous in engineering design. They are challenging, because of the sensitive dependence of the state to perturbations in the design. One consequence of this sensitive dependence is that increasing the averaging period, which improves the accuracy of the objective, causes the gradient to diverge. To overcome this issue, the proposed algorithm uses an ensemble objective in a Newton-Krylov trust-region framework. The ensemble objective averages a set of objective functions, each of which uses a reduced time-averaging period and independent state variable; this independence permits each simulation and adjoint computation to be carried out in parallel. The novel aspect of the proposed method is the use of the ensemble objective within a Newton-Krylov algorithm; the latter helps avoid some of the issues presented by objectives governed by chaotic state variables. We demonstrate the proposed ensemble-Newton-Krylov algorithm on an optimization problem governed by the Lorenz dynamical system.

We present a sampling strategy suitable for optimization problems characterized by high-dimensional design spaces and noisy outputs. Such outputs can arise, for example, in time-averaged objectives that depend on chaotic states. The proposed sampling method is based on a generalization of Arnoldi's method used in Krylov iterative methods. We show that Arnoldi-based sampling can effectively estimate the dominant eigenvalues of the underlying Hessian, even in the presence of inaccurate gradients. This spectral information can be used to build a low-rank approximation of the Hessian in a quadratic model of the objective. We also investigate two variants of the linear term in the quadratic model: one based on step averaging and one based on directional derivatives. The resulting quadratic models are used in a trust-region optimization framework called the Stochastic Arnoldi's Method (SAM). Numerical experiments highlight the potential of SAM relative to conventional derivative-based and derivative-free methods when the design space is high-dimensional and noisy.

Optimization is an important phase of modern engineering design, and gradient-based optimization methods are especially effective for design problems involving a large number of design variables, such as aerodynamic shape optimization. However, if the system of interest exhibits chaotic dynamics and the quantity of interest is a long-time average, for example, drag in a turbulent flow, conventional sensitivity analysis methods will fail to provide useful gradients and gradient-based optimization becomes infeasible. Progress has been made in the sensitivity analysis of chaotic systems, but most proposed methods are computationally expensive. We propose a new method of stabilizing the tangent-sensitivity equations by introducing a minimal perturbation to the linearized equations. The goal is to eliminate linear instabilities in the sensitivity equations, while ensuring useful gradients are obtained. We implement the tangent sensitivity method on a model chaotic problem, demonstrate the effectiveness of our stabilizations, and validate the gradient information obtained from the stabilized tangent sensitivity. Finally, we describe how the method of stabilization can easily be adapted to the sensitivity analysis of large-scale computational fluid dynamics problems.

No abstract