Karen E. Willcox Associate Professor Department of Aeronautics & Astronautics Massachusetts Institute of Technology 77 Massachusetts Ave Room 37-447 Cambridge, MA 02139
Phone: (617) 253-3503
FAX: (617) 258-5143
Email: kwillcox@mit.edu
PhD, 2000, MIT
S.M., 1996, MIT
B.E. in Engineering Science, 1994, University of Auckland, New Zealand
Research Interests:
Reduced-Order Modeling: Reduced-order modeling is a powerful technique in which small yet accurate models of complicated systems are derived. The basic idea is to start with a high-order model (e.g. a CFD model of a fluid flow), extract a reduced-space basis which captures the desired dynamics accurately, and project the high-order system onto the reduced space to obtain a low-order model. By considering the inputs and outputs that are relevant for the particular application, a huge reduction in the number of states can be obtained, e.g. for a two-dimensional Euler aeroelastic model a reduction from 300,000 to 200 flow states was achieved while retaining the required accuracy. The basis can be calculated via a variety of methods, including eigenmodes, proper orthogonal decomposition (POD), Arnoldi methods, balanced truncation and Fourier expansion.
Projects and Publications in Reduced-Order Modeling:
Reduced-order models for unsteady aerodynamic applications
Gappy POD for flow reconstruction and flow sensing
Reduced-order models for active flow control
Reduced-order aerodynamic models for mistuned compressor blades
Multidisciplinary Design Optimization: MDO is a tool that has been used successfully throughout the design process to enable improvements in aircraft performance. By simultaneously considering the effects of aerodynamics, structural dynamics and controls, and the complicated interaction between them, substantially improved performance can be achieved. The concept of MDO can be extended beyond just including the engineering disciplines. In current design practices, optimal is almost exclusively synonymous with minimum weight. If accurate cost models can be developed and integrated to an optimization framework, then we can begin to consider the trade-offs between performance and cost, and arrive at solutions that are more favorable. In addition, the effects of environmental impact are becoming increasingly important. A number of the projects listed below use MDO extensively.
Environmental Issues in Aircraft Design: Environmental issues are becoming increasingly important, both for current and future aircraft. It is essential that the early design process take account of effects on noise, local air quality, and global climate change. An interesting example that demonstrates the importance of environmental factors is that the A380 is rumored to have sacrificed 2% in fuel burn in order to meet noise restrictions at London Heathrow. While noise is a primary concern at Heathrow, the implication of this tradeoff for local air quality is not clear. In the context of new technology development, it is particularly important to understand the tradeoffs between the various environmental factors and traditional design metrics such as fuel burn.
Projects and Publications in Environmental Issues in Aircraft Design:
Environmental Design Space
Aircraft System Design for Value: As the aerospace industry moves from the era of "Higher, Faster, Farther" to the challenge of "Better, Faster, Cheaper" the definition of best design has evolved considerably. The best strategy is that which optimizes value to the enterprise including not only the aircraft manufacturer, but also the suppliers, the airlines, the consumers and the community. How do we make the best decisions, not only at a detailed design level, but also at a program level (e.g. on fleet mix or speed)? Moreover, how do we quantify the value of intangibles such as system commonality, quietness or environmental impact?
Projects and Publications in System Design for Value:
Economic models in MDO for commercial aircraft design
Design Space Visualization: The most effective use of MDO is not as a “push-button” tool where one can specify the problem and get the “best” answer. Instead, MDO is a valuable complement to other design tools, and as such should be constructed to provide relevant information for informed design decisions rather than just a solution to a problem. Rather than being used to eke out a 5% improvement in the design solution (where model fidelity is often an issue anyway), MDO ought to be used as a way of gaining insight to the design space, quantitatively identifying trades and finding innovative design options. Often, in practical applications, it is the solution concept suggested by the optimizer but not the actual details of the design that are most interesting. In this research, we are developing methodology to improve the way in which designers can use MDO. By coupling the optimizer with a visualization framework, we aim to automatically convey important information about the design space, such as:
- What path did the optimizer take from the initial to the final solution – what design decisions and tradeoffs were made and why?
- Which constraints are active, and which constraints are particularly constraining to the design?
- What is the sensitivity of the solution to various problem parameters?
Projects and Publications in Design Space Visualization:
Deremaux, Y., Willcox, K., and Haimes, R., “Real-Time Visualization and Constraint Analysis in Multidisciplinary Design Optimization,” AIAA Paper 2003-3876, presented at 15th Computational Fluid Dynamics Conference, Orlando, FL, June 2003.
Blended-Wing-Body: The Boeing BWB is a revolutionary concept for transport that integrates wing, fuselage, engines, and tail to achieve a substantial improvement in performance over a conventional transport. This aircraft provides not only many exciting technical challenges, but also many thought-provoking opportunities to revolutionize the way aircraft are designed and built.
Projects and Publications related to the BWB:
Willcox, K. and Wakayama, S., “Simultaneous Optimization of a Multiple-Aircraft Family”, Journal of Aircraft, Vol. 40, No. 4, July 2003, pp. 616-622.
Teaching Interests:
16.060 Principles of Automatic Control
A junior-level core Aero/Astro class taught with Prof. John Deyst.
16.888 Multidisciplinary System Design Optimization
A graduate-level class taught with Prof. Olivier de Weck.
Mathematics for Aerospace Engineers
An initiative to diagnose problems with the math skills of Aero/Astro undergraduates and to develop resources to address these issues. Funded by the MIT Alumni Fund.
Publications and Resources:
A paper describing the mapping of the implicit mathematics curriculum throughout the core Aero/Astro engineering classes:
Willcox, K. and Bounova, G., “Mathematics in Engineering: Identifying, Enhancing and Linking the Implicit Mathematics Curriculum,” presented at ASEE Annual Conference, Salt Lake City, UT, June 2004.
A report describing research to create explicit links between engineering courses and upstream mathematics courses, and to incorporate “flashbacks”. The approach was implemented in Principles of Automatic Control.
Allaire, D. and Willcox, K., “Explicit Linking of Mathematics in the Undergraduate Engineering Curriculum,”
Supplementary math notes developed to help create explicit links between Principles of Automatic Control and upstream mathematics courses.
Allaire, D. and Willcox, K., “16.06 Supplementary Math Notes”
An OpenCourseWare resource to use the mathematics mapping and to help students navigate through mathematics material relevant to Principles of Automatic Control:
Keynote address at 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, VA, September 2006.
de Weck, O. and Willcox, K., “Trends in Multidisciplinary Engineering Education: 2006 and Beyond”.
Other Activities:
Member of the AIAA MDO Technical Committee.
Associate faculty fellow of the Singapore-MIT Alliance.
Board member MIT Muddy Charles Pub.
Research : People : Publications: Seminars 