Brendan Tracey

Stanford University Unstructured (SU2): Analysis and Design Technology for Turbulent Flows

This paper presents a comprehensive set of test cases for the verification and validation (V & V) of the Stanford University Unstructured (SU) software suite within the context of compressible, turbulent flows described by the Reynolds-averaged Navier-Stokes (RANS) equations. SU is an open-source (Lesser General Public License, version 2.1), integrated analysis and design tool for solving multi-disciplinary problems governed by partial differential equations (PDEs) on general, unstructured meshes. As such, SU is able to handle arbitrarily complex geometries, mesh adaptation, and a variety of physical problems. At its core, the software suite is a collection of C++ modules embedded within a Python framework that are built specifically for both PDE analysis and PDE-constrained optimization, including surface gradient computations using the continuous adjoint technique. V & V studies of twoand three-dimensional problems are presented for turbulent flows across a wide range of Mach numbers (from subsonic flat plate studies to a complex, transonic aircraft configuration). The presentation of this comprehensive V & V of SU is intended to be the main contribution of this paper: the results generated with SU in a variety of standard test cases compare well with experimental data and established flow solvers that have undergone similar V & V efforts. For completeness, the adjoint-based shape design capability within SU is also illustrated.

A machine learning strategy to assist turbulence model development

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 SpalartAllmaras 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.

Cyber-Physical Security: A Game Theory Model of Humans Interacting Over Control Systems

Recent years have seen increased interest in the design and deployment of smart grid devices and control algorithms. Each of these smart communicating devices represents a potential access point for an intruder spurring research into intruder prevention and detection. However, no security measures are complete, and intruding attackers will compromise smart grid devices leading to the attacker and the system operator interacting via the grid and its control systems. The outcome of these machine-mediated human-human interactions will depend on the design of the physical and control systems mediating the interactions. If these outcomes can be predicted via simulation, they can be used as a tool for designing attack-resilient grids and control systems. However, accurate predictions require good models of not just the physical and control systems, but also of the human decision making. In this manuscript, we present an approach to develop such tools, i.e., models of the decisions of the cyber-physical intruder who is attacking the systems and the system operator who is defending it, and demonstrate its usefulness for design.

Application of Supervised Learning to Quantify Uncertainties in Turbulence and Combustion Modeling

The accuracy of low-fidelity models of turbulent flow such as those based on the Reynolds Averaged Navier–Stokes (RANS) equations can be questionable, especially when these models are applied in situations different from those in which the models were calibrated. At present, there is no general method to quantify structural uncertainties in such models. Greater accuracy and a reliable quantification of modeling errors is much needed to expand the use of affordable simulation models in engineering design and analysis. In this paper, we introduce a methodology aimed at improving low-fidelity models of turbulence and combustion and obtaining error bounds. Towards this end, we first develop a new machine learning algorithm to construct a stochastic model of the error of low-fidelity models using information from higher-fidelity data sets. Then, by applying this error model to the lowfidelity result, we obtain better approximations of uncertain model outputs and generate confidence intervals on the prediction of simulation outputs. We apply this technique to two representative flow problems. The first application is in flamelet-based simulations to model combustion in a turbulent mixing layer; and the second application is in the prediction of the anisotropy of turbulence in a non-equilibrium boundary layer flow. We demonstrate that our methodology can be used to improve aspects of predictive modeling while offering a route towards obtaining error bounds.

Air Vehicle Design and Technology Considerations for an Electric VTOL Metro-Regional Public Transportation System

This paper examines the potential of VTOL aircraft to supplement commuter rail services in a metropolitan or regional transportation system. An interdisciplinary study was conducted to examine the feasibility of integrating an aerial mass transit system into the existing airspace using a fleet of electrically powered rotorcraft. A notional network of stations and overall operating schedule were constructed based on the existing regional rail networks serving the San Francisco Bay Area. To define the VTOL vehicles, the rotorcraft sizing code NDARC has been modified to accommodate electric propulsion sizing. Initial sizing results indicate that battery technologies available by 2030, coupled with the “shorthop” ranges of the proposed aerial network, result in feasible aircraft designs. These vehicle designs, while significantly heavier than their Jet A powered turboshaft equivalents, may become economically viable in a business environment dominated by fuel costs. Finally, these initial study results are informing follow-on study efforts.

Estimating Mixture Entropy with Pairwise Distances

Mixture distributions arise in many parametric and non-parametric settings—for example, in Gaussian mixture models and in non-parametric estimation. It is often necessary to compute the entropy of a mixture, but, in most cases, this quantity has no closed-form expression, making some form of approximation necessary. We propose a family of estimators based on a pairwise distance function between mixture components, and show that this estimator class has many attractive properties. For many distributions of interest, the proposed estimators are efficient to compute, differentiable in the mixture parameters, and become exact when the mixture components are clustered. We prove this family includes lower and upper bounds on the mixture entropy. The Chernoff α -divergence gives a lower bound when chosen as the distance function, with the Bhattacharyaa distance providing the tightest lower bound for components that are symmetric and members of a location family. The Kullback–Leibler divergence gives an upper bound when used as the distance function. We provide closed-form expressions of these bounds for mixtures of Gaussians, and discuss their applications to the estimation of mutual information. We then demonstrate that our bounds are significantly tighter than well-known existing bounds using numeric simulations. This estimator class is very useful in optimization problems involving maximization/minimization of entropy and mutual information, such as MaxEnt and rate distortion problems.

Counter-Factual Reinforcement Learning: How to Model Decision-Makers That Anticipate the Future

This chapter introduces a novel framework for modeling interacting humans in a multi-stage game. This ”iterated semi network-form game” framework has the following desirable characteristics: (1) Bounded rational players, (2) strategic players (i.e., players account for one another’s reward functions when predicting one another’s behavior), and (3) computational tractability even on real-world systems. We achieve these benefits by combining concepts from game theory and reinforcement learning. To be precise, we extend the bounded rational ”level-K reasoning” model to apply to games over multiple stages. Our extension allows the decomposition of the overall modeling problem into a series of smaller ones, each of which can be solved by standard reinforcement learning algorithms. We call this hybrid approach ”level-K reinforcement learning”. We investigate these ideas in a cyber battle scenario over a smart power grid and discuss the relationship between the behavior predicted by our model and what one might expect of real human defenders and attackers.

Probabilistic simulation of multi-stage decisions for operation of a fractionated satellite mission

We present two complementary analysis models to study the effect of programmatic management decisions on the distribution of net present value for a fractionated satellite constellation. The goal is to begin development of an approach to quantify when system attributes associated with design flexibility have realizable benefits for space systems. The first approach is a heuristics-based decision model, which utilizes a Monte Carlo simulation to produce value distributions for satellite operator decision sets; the second approach is a multi-stage decision process model, which utilizes a dynamic programming algorithm to find value-optimal decisions. We use a generic Department of Defense (DoD) terrestrial weather satellite program as a case study for analysis. We find evidence that technological evolution of a fractionated satellite system within the scope of a single program may not be desirable due to cost and schedule risks.