Computing at the Edge
Capturing a flame's flicker, an ink jet's splatter, and other shifting shapes
Applied mathematicians use novel computer methods based on mathematical structures called level sets to model complex behavior at interfaces.
References:
Aleinov, I.D., E.G. Puckett, and M.M. Sussman. Preprint. Formation of droplets in microscale jetting devices.
Kimmel, R., and J.A. Sethian. 1998. Computing geodesic paths on manifolds. Proceedings of the National Academy of Sciences 95(July 21):8431.
Merriman, B., et al. Preprint. Island dynamics and level set methods for continuum modeling of epitaxial growth. Available at http://www.math.ucla.edu/applied/cam/index.html.
Sethian, J.A. 1999. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge, England: Cambridge University Press.
Sussman, M., and E.G. Puckett. Preprint. A coupled level set and volume of fluid method for computing 3D and axisymmetric incompressible two-phase flows.
Further Readings:
Caflisch, R.E., et al. 1998. Island dynamics at the level set method for epitaxial growth. Available at http://www.math.ucla.edu/applied/cam/index.html.
Malladi, R., and J.A. Sethian. 1995. Image processing via level set curvature flow. Proceedings of the National Academy of Sciences 92(July 18):7046.
Sethian, J.A. 1998. Tracking interfaces with level sets. American Scientist 85(May-June):254.
______. 1996. A fast marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Sciences 93(Feb. 20):1591.
Sussman, M., et al. 1999. An adaptive level set approach for incompressible two-phase flows. Journal of Computational Physics 148:81.
Sussman, M., and S. Uto. 1998. A computational study of the spreading of oil underneath a sheet of ice. Available at http://www.math.ucla.edu/applied/cam/index.html.
Sussman, M., P. Smereka, and S. Osher. 1994. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics 114(September):146.
Additional information about the level-set and fast-marching methods can be found at http://math.berkeley.edu/~sethian/level_set.html. In a series of interactive demos, you can build an obstacle course and watch a (virtual) robot find an optimal path, automatically extract shapes from medical scans, design and clean up noisy images, and observe curved interfaces evolve under various speed laws.
Sources:
Ron Kimmel
Computer Science Department
Technion Israel Institute of Technology
Haifa 32000
Israel
Web site: http://www.cs.technion.ac.il/~ron/
Ravi Malladi
1 Cyclotron Road
Mailstop 50A-2152
Lawrence Berkeley National Laboratory
Berkeley, CA 94720
Web site: http://www.lbl.gov/~malladi/
Stanley Osher
Department of Mathematics
University of California
Los Angeles, CA 90095-1555
Web site: http://www.math.ucla.edu/faculty/sjo.html
Leonid I. Rudin
Cognitech
Pasadena, CA 91101
Web site: http://www.cognitech.com/
James A. Sethian
Department of Mathematics
University of California
Berkeley, CA 94720
Web site: http://math.berkeley.edu/~sethian/level_set.html
Mark Sussman
Department of Mathematics
University of California
Davis, CA 95616
Web site: http://math.ucdavis.edu/~sussman/
Dave Wasson
Areté Image Software
5000 Van Nuys Boulevard
Suite 450
Sherman Oaks, CA 91403
Web site: http://www.areteis.com/
From Science News, Vol. 155, No. 15, April 10, 1999, p. 232. Copyright © 1999, Science Service.
