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CONSTRUCTION OF geometric partial differential equations FOR LEVEL SETS

Journal of Computational Mathematics 2010年第1期28卷 105-121页

作者： Chong Chen Guoliang Xu LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Graduate University of Chinese Academy of Sciences Beijing 100190 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.

关键词： geometric partial differential equations Level set differential geometry operators.

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A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution equations

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Communications in Computational Physics 2010年第1期7卷 195-211页

作者： Karol Mikula Daniel Sevcovic Martin Balazovjech Department of Mathematics Slovak University of TechnologyRadlinskeho 1181368 BratislavaSlovak Republic Department of Applied Mathematics and Statistics Comenius University84248 BratislavaSlovak Republic

A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.

关键词： geometric partial differential equations evolving plane curves mean curvature flow anisotropy Willmore flow surface diffusion finite volume method semi-implicit scheme tangen-tial redistribution

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