The complex-step derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to algorithmic differentiation theory. A general procedure for the implementation of the complex-step method is described in detail and a script is developed that automates its implementation. Automatic implementations of the complex-step method for Fortran and C/C++ are presented and compared to existing algorithmic differentiation tools. The complex-step method is tested in two large multidisciplinary solvers and the resulting sensitivities are compared to results given by finite differences. The resulting sensitivities are shown to be as accurate as the analyses. Accuracy, robustness, ease of implementation and maintainability make these complex-step derivative approximation tools very attractive options for sensitivity analysis.

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This paper describes the history, objectives, structure, and current capabilities of the Stanford University Unstructured (SU 2) tool suite. This computational analysis and design software collection is being developed to solve complex, multi-physics analysis and optimization tasks using arbitrary unstructured meshes, and it has been designed so that it is easily extensible for the solution of Partial Differential Equation-based (PDE) problems not directly envisioned by the authors. At its core, SU 2 is an open-source collection of C++ software tools to discretize and solve problems described by PDEs and is able to solve PDE-constrained optimization problems, including optimal shape design. Although the toolset has been designed with Computational Fluid Dynamics (CFD) and aerodynamic shape optimization in mind, it has also been extended to treat other sets of governing equations including potential flow, electrodynamics, chemically reacting flows, and several others. In our experience, capabilities for computational analysis and optimization have improved considerably over the past two decades. However, the ability to integrate the resulting software packages into coupled multi-physics analysis and design optimization solvers has remained a challenge: the variety of approaches chosen for the independent components of the overall problem (flow solvers, adjoint solvers, optimizers, shape parameterization, shape deformation, mesh adaption, mesh deformation, etc) make it difficult to (a) expand the range of applicability to situations not originally envisioned, and (b) to reduce the overall burden of creating integrated applications. By leveraging well-established object-oriented software architectures (using C++) and by enabling a common interface for all the necessary components, SU 2 is able to remove these barriers for both the beginner and the seasoned analyst. In this paper we attempt to describe our efforts to develop SU 2 as an integrated platform. In some senses, the paper can also be used as a software reference manual for those who might be interested in modifying it to suit their own needs. We carefully describe the C++ framework and object hierarchy, the sets of equations that can be currently modeled by SU 2 , the available choices for numerical discretization, and conclude with a set of relevant validation and verification test cases that are included with the SU 2 distribution. We intend for SU 2 to remain open source and to serve as a starting point for new capabilities not included in SU 2 today, that will hopefully be contributed by users in both academic and industrial environments.

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Turbulence modeling in a Reynolds Averaged Navier-Stokes (RANS) setting has traditionally evolved through a combination of theory, mathematics, and empiricism. The problem of closure, resulting from the averaging process, requires an infusion of information into the various models that is often managed in an ad-hoc way or that is focused on particular classes of problems, thus diminishing the predictive capabilities of a model in other flow contexts. In this work, a proof-of-concept of a new data-driven approach of turbulence model development is presented. The key idea in the proposed framework is to use supervised learning algorithms to build a representation of turbulence modeling closure terms. The learned terms are then inserted into a Computational Fluid Dynamics (CFD) numerical simulation with the aim of offering a better representation of turbulence physics. But while the basic idea is attractive, modeling unknown terms by increasingly large amounts of data from higher-fidelity simulations (LES, DNS, etc) or even experiment, the details of how to make the approach viable are not at all obvious. In this work, we investigate the feasibility of such an approach by attempting to reproduce, through a machine learning methodology, the results obtained with the well-established Spalart-Allmaras model. In other words, the key question that we seek to answer is the following: Given a number of observations of CFD solutions using the Spalart-Allmaras model (our truth model), can we reproduce those solutions using machine-learning techniques without knowledge of the structure, functional form, and coefficients of the actual model? We discuss the challenges of applying machine learning techniques in a fluid dynamic setting and possible successful approaches. We also explore the potential for machine learning as an enhancement to or replacement for traditional turbulence models. Our results highlight the potential and viability of machine learning approaches to aid turbulence model development.