Evolution by mean curvature is recently attracting large attention especially when the underlying anisotropic structure degenerates to become crystalline. In such situation new phenomena must be taken into account: the evolution law becomes nonlocal and "hyperbolic" across facets; moreover events like face breaking or bending have to be considered especially in three dimensions. For this reason the ODE approach suggested by J. Taylor long ago cannot be used directly and the required modifications to the algorithm are not clear at the moment. The well known diffused interface approximation for the classical mean curvature flow, which leads to the Allen-Cahn equation, can be applied in this contex, resulting in a bistable reaction-diffusion equation with good convergence properties to the sharp interface evolution. This equation can then be discretized using finite elements in space and forward differences in time. Numerical simulations with the resulting scheme seem to recover the face breaking and face bending phenomena.

Paolini, M., Pasquarelli, F., Numerical simulation of crystalline curvature flow in 3D by interface diffusion, Comunicazione, in Free Boundary Problems: theory and applications II, GAKUTO Internat. Ser. Math. Sci. Appl. 14, (Chiba, 07-13 November 1999), Gakkotosho, Tokyo 2000: 376-389 [http://hdl.handle.net/10807/21117]

Numerical simulation of crystalline curvature flow in 3D by interface diffusion

Paolini, Maurizio;Pasquarelli, Franco
2000

Abstract

Evolution by mean curvature is recently attracting large attention especially when the underlying anisotropic structure degenerates to become crystalline. In such situation new phenomena must be taken into account: the evolution law becomes nonlocal and "hyperbolic" across facets; moreover events like face breaking or bending have to be considered especially in three dimensions. For this reason the ODE approach suggested by J. Taylor long ago cannot be used directly and the required modifications to the algorithm are not clear at the moment. The well known diffused interface approximation for the classical mean curvature flow, which leads to the Allen-Cahn equation, can be applied in this contex, resulting in a bistable reaction-diffusion equation with good convergence properties to the sharp interface evolution. This equation can then be discretized using finite elements in space and forward differences in time. Numerical simulations with the resulting scheme seem to recover the face breaking and face bending phenomena.
Inglese
Free Boundary Problems: theory and applications II, GAKUTO Internat. Ser. Math. Sci. Appl. 14
Free Boundary Problems, Theory and Applications II
Chiba
Comunicazione
7-nov-1999
13-nov-1999
Paolini, M., Pasquarelli, F., Numerical simulation of crystalline curvature flow in 3D by interface diffusion, Comunicazione, in Free Boundary Problems: theory and applications II, GAKUTO Internat. Ser. Math. Sci. Appl. 14, (Chiba, 07-13 November 1999), Gakkotosho, Tokyo 2000: 376-389 [http://hdl.handle.net/10807/21117]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/21117
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