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A line element algorithm for curve flow problems in the plane

Published online by Cambridge University Press:  17 February 2009

Stephen Roberts
Stephen Roberts
Affiliation:
Mathematics Department, The Faculties, Australian National University.
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Abstract

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In this paper we shall describe a numerical method for the solution of curve flow problems in which the normal velocity of the curve depends locally on the position, normal and curvature of the curve. The method involves approximating the curve by a number of line elements (segments) which are only allowed to move in a direction normal to the element. Hence the normal of each line element remains constant throughout the evolution. In regions of high curvature elements naturally tend to accumulate. The method easily deals with the formation of cusps as found in flame propagation problems and is computationally comparable to a naive marker particle method. As a test of the method we present a number of numerical experiments related to mean curvature flow and flows associated with flame propagation and bushfires.

Type
Research Article
Information
The ANZIAM Journal , Volume 35 , Issue 2 , October 1993 , pp. 244 - 261
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Anderson, D. H., Catchpole, E. A., Mestre, N. J. de and Parkes, T., “Modelling the spread of grass fires”, J. Austral. Math. Soc. (Series B) 23 (1982) 451–466.CrossRefGoogle Scholar
[2]Catchpole, E. A., Mestre, N. J. de and Gill, A. M., “Intensity of fire at its perimeter”, Aust. For. Res. 12 (1982) 47–54.Google Scholar
[3]Chorin, A. J., “Flame advection and propagation algorithms”, J. Comp. Phys. 35 (1980) 1–11.CrossRefGoogle Scholar
[4]Ghoniem, A. F., Chorin, A. J. and Oppenheim, A. K., “Numerical modelling of turbulent flow in a combustion tunnel”, Philos. Trans. Roy. Soc. London Ser. A 304 (1982) 303–325.Google Scholar
[5]Grayson, M., “The heat equation shrinks embedded plane curves to points”, J. Diff. Geom. 26 (1987) 285–314.Google Scholar
[6]Markstein, G. H., “Experimental and theoretical studies of flame front stability”, J. Aero Sci. 18 (1951) 199–209.CrossRefGoogle Scholar
[7]Osher, S. and Sethian, J. A., “Fronts propagating with curvature dependent speed: Algorithms based on hamiltonian-jacobi formulations”, J. Comput. Phys. 79 (1988) 12–49.CrossRefGoogle Scholar
[8]Sethian, J. A., “Curvature and the evolution of fronts”, Commun. Math. Phys. 101 (1985) 487–499.CrossRefGoogle Scholar
[9]Smith, C. S., “Grain shape and other metallurgical applications of topology”, in Metal Interfaces, (American Society for Metal, Cleveland, 1952).Google Scholar
[10]Taylor, J. E.. “Crystals, in equilibrium and otherwise”, 1990, Videotape of 1989 AMS-MMA lecture, published by AMS.Google Scholar