
AeroAstro Magazine HighlightThe following article appears in the 2007–2008 issue of AeroAstro, the annual report/magazine of the MIT Aeronautics and Astronautics Department. © 2008 Massachusetts Institute of Technology. Getting real(time) with simulation and optimizationBy Karen Willcox Computation for simulation and optimization is essential to the design and operation of aerospace systems. Improvements in computational methods, together with a substantial increase in computing power, have led to widespread use of highfidelity methods, such as computational fluid dynamics, for analysis and design.
The next generation of computational methods must address the challenges of probabilistic design and simulation/optimization in real time. My research program is developing new approaches to address these challenges, with a broad set of applications ranging from active flow control, to turbomachinery aeroelasticity, to astronaut motion control, to tools to support international policymaking for aviation environmental impact. Challenges in realtime simulation and uncertainty quantificationThe need for simulation and optimization in real time is critical for many applications, including control of dynamical processes, adaptive systems, and data assimilation — that is, situations where realtime data must be assimilated with computational tools for rapid decision making. For example, one challenge related to homeland security applications is to solve a contaminant transport inverse problem on a grid of millions of cells, with limited measurements to determine the probable upstream source of a contaminant release, and the potential downstream impact areas — all within a few minutes to allow for emergency response. Effective tools to support decisionmaking under uncertainty are also becoming essential in the design and operation of aerospace systems. This is particularly true as we move toward advanced technologies and unconventional aircraft configurations (i.e., unproven regions of the design space). To gain a sense of the immense challenge of carrying out probabilistic analysis with typical computational tools, consider two examples. The first example is quantification of the effects of manufacturing variations on the unsteady aerodynamic performance of a compressor in an aircraft engine — a problem critical to understanding and preventing engine high cycle fatigue. A simple linear twodimensional CFD model of just a handful of blade rows might take just a few minutes to simulate a single blade geometry. However, to quantify uncertainty, we want to run Monte Carlo simulations, requiring the analyses of many thousands of different blade geometries. A Monte Carlo simulation with our simple model using 10,000 samples would translate into around three weeks of computation time. Even with simple models, the computational burden is immense; to tackle this problem with more sophisticated models that include nonlinearities, threedimensional effects, or more blade rows is computationally intractable.
The second example draws on computational models to support aviation environmental policy decisionmaking. Together with Professor Ian Waitz and a large international team of collaborators, I am working under the sponsorship of the FAA within the Partnership for AiR Transportation Noise and Emissions Reduction, an FAA Center of Excellence headquartered in AeroAstro, to develop the Aviation Environmental Portfolio Management Tool, which will provide support to the international policy decisionmaking process through assessments of interdependencies among aviationrelated noise and emissions, impacts on health and welfare, and industry and consumer costs, under different policy, technology, operations, and market scenarios. The scale and complexity of this problem is immense; for example, simulation of one year involves more than two million flights with 350 aircraft types, analyzed with blackbox models spanning airline economics, environmental economics, aircraft operations, aircraft performance and emissions, noise, local air quality, and global climate. Furthermore, while just simulating the system is a daunting task, uncertainty must be characterized, computed and communicated in a way tangible to the domestic and international policy decisionmakers. Model reductionTo tackle these realtime and uncertainty quantification challenges, we need a way to come up with more efficient system models — models that are much cheaper to solve, but retain high levels of predictive accuracy for the system dynamics of interest. This is often referred to as “surrogate modeling.” My research focuses largely on model reduction, which uses mathematical techniques to exploit the structure of the system at hand in order to systematically generate surrogate models. With model reduction we can derive surrogate models for largescale complex systems, with the benefit that the underlying mathematics and physics of the problem are driving the choice of reducedorder model. For example, for active flow control of a supersonic inlet, we developed a new model reduction method to identify the important modes that describe the relationship between system inputs (control actuation, flow disturbances) and system outputs (describing the stability of the inlet). With this method, we can replace our CFD model of dimension 12,000 flow states (the unknown densities, velocities and pressures at each point in our computational domain) with a reducedorder model that has just 30 states, but retains very high accuracy for the specific input/output mappings of interest. This reducedorder model is a key enabler for achieving active control, which in turn enables the design of inlets that achieve efficiency gains by operating closer to stability margins. We have also developed methodology that allows us to apply model reduction in settings where the system has many input parameters. In these cases, sampling the space of possible parameter variations is a considerable challenge that is not addressed by existing methods. The realtime contaminant transport inverse problem described above is one such setting—here, the input comprises the space of possible initial conditions, of which there are many thousands (or even millions). In collaboration with researchers at Sandia National Laboratories, we have demonstrated our approach on a model problem and are working to implement the method at full scale in Sandia tools. For the case of uncertainty quantification of the effects of manufacturing variations on the unsteady aerodynamic performance of a compressor, our approach yields an accurate reducedorder model that performs a Monte Carlo simulation in fewer than 15 minutes, compared to three weeks for the CFD code. To ensure relevance and applicability to realworld problems, my research includes substantive engagement with government and industry. In 2006, I established the Research Consortium for Multidisciplinary System Design with Stanford University Professors Ilan Kroo and Juan Alonso. EducationPerhaps one of my biggest surprises in joining the MIT faculty was to discover the problems that our undergraduate students have with mathematics. For example, poor abilities to apply concepts from differential equations can greatly hinder a student in understanding the key physical principles and central material of a controls course. During my time on the MIT faculty, I have carried out educational research to address this issue, with a particular focus on creating better linkages between mathematics and engineering subjects. By working closely with faculty members in the mathematics department who teach freshman and sophomore required mathematics subjects, I established a process for explicit linking of mathematics courses and other mathematics resources in engineering courses. For Principles of Automatic Control (Course 16.06), I created a lecturebylecture mapping that details the specific mathematical skills required in each of my lectures and the associated upstream mathematics course where the concept was previously introduced or taught. I developed a set of supplementary mathematics notes and linkages to provide students with remedial resources for selfstudy and reference, and modified my lecture content to incorporate “flashbacks,” or specific references to materials used in upstream mathematics courses. We have found that class performance shows that the combination of these notes and an increased emphasis on linkages during lecture helps considerably with students’ grasp of underlying mathematical concepts. The supplementary notes appear to be an important resource—not for all students, but for those students who struggle with the underlying mathematics. In addition, I have worked with MIT OpenCourseWare to create an online version of the supplementary mathematics notes that provides direct links to online mathematics resources. This year, I moved teaching responsibilities from 16.06 to our Unified Engineering course; in the coming years I hope to have a chance to apply some of these ideas in Unified. Karen Willcox is an Associate Professor in the MIT Aeronautics and Astronautics Department. Originally from New Zealand, she has a Bachelor of Engineering (Hons.) from the University of Auckland, and S.M. and Ph.D. degrees from MIT. She has been on the faculty at MIT since 2001. Prior to that, she worked at Boeing with the BlendedWingBody design group. In her spare time she climbs mountains and trains for ultramarathons. She may be reached at kwillcox@mit.edu. 
