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Exploring Flapping Flight For Micro Aerial Vehicles

Potential flow codes represent an extremely efficient and accurate means to compute un-separated flow around complex geometries. The aim of this code project was to develop a code capable of rapidly solving potential problems around complex geometries in a rapid, hands-off manner.


FastAero2D is a two-dimensional potential flow solver coupled with a simple spring-mass representation of the structural dynamics. The 2-Dimensional potential flow code exploits and iterative GMRES [6] solver for rapidly solving the linear system. The potential flow problem is discretized using linear basis functions on flat panels. Both a thin surface (velocity) formulation and a thick body (Greens Theorem Thick Body Formulation) have been developed. The unsteady 2-D wake vorticity is represented using a linear element freely convecting, self-influencing wake. As a result the wake will convect and roll-up under the local velocity influence (as seen in figure 1).

Figure 1: An example of the 2-Dimensional FastAero code applied to a simple heaving wing motion. In this example the fluid structure interaction is simplified using a leading edge spring to allow compliance. Although this model is a realistic model of natural flight, the example lends insight into the effect of leading edge torsion rod or bone compliance.


FastAero3D[1] is an advanced, 3-Dimensional unsteady panel method solver developed at MIT. The code incorporates an accelerated iterative solution approach through the use of the precorrected-FFT[3] and Fast Multipole Algorithms[4] (Matrix Vector Product (MVP) acceleration).  Through the use of matrix-multiply accelerations, intermediate time-step solutions can be performed very rapidly. As such, the solution of highly resolved unsteady potential flows is possible.

Figure: A plot of the FastAero3D solution around a business jet configuration. In this image the features of the FastAero3D solution are illustrated. The body of the business jet is represented using a boundary element surface triangular discretization. The localized vorticity in the wake is represented using the vortex particle method (VPM).


[1] Willis, D.J., Peraire, J. White, J.K., A combined pFFT-multipole tree code, unsteady panel method with vortex particle wakes, 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2005-0854, Reno, NV, Jan. 2005.

[2] Willis, D.J., Peraire, J. White, J.K., 'A quadratic basis function, quadratic geometry, high order panel method' presented at 44th AIAA Aerospace Sciences Meeting, AIAA-2006-1253, Reno , Nevada , 2006.

[3] Philips, J.R., White, J.K., A Precorrected-FFT Method for Capacitance Extraction of Complicated 3-D Structures, Proc. of the IEEE/ACM International Conference on Computer-Aided Design, pp. 268-271, November 1994.

[4] Greengard, L., Rokhlin, V. A Fast Algorithm for Particle Simulations,  J. Comput. Phys. 73, 325 (1987).

[5]  Rehbach, C. Calcul numerique d’ecoulement tridimensionels instationaires avec nappes tourbillonnaires, La Recherche Aerospatiale, pp. 289-298, 1977.

[6] Saad, Y. and Schultz, M. H. 1986. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linea systems. SIAM J. Sci. Stat. Comput. 7, 3 (Jul. 1986), 856-869.



The current project has been funded by several different sources at different times. We are very thankful to the following funding sources:

Singapore-MIT Alliance (SMA), National Science Foundation (NSF), Natural Science and Engineering Councile of Canada (NSERC), and the Air Force Office of Scientific Research.


 Copyright 2007, MIT. All rights reserved.

Aerospace Computational Design Laboratory, Department of Aeronautics and Astronuatics. Massachusetts Institute of Technology.

Sample Simulations