In this paper we present and discuss the results of some numerical simulations in order to investigate the mean curvature flow problem in presence of a nonconvex anisotropy. Mathematically, nonconvexity of the anisotropy leads to the ill-posedness of the evolution problem, which becomes forward-backward parabolic. Simulations presented here refer to two different settings: curvature driven vertical motion of graphs (nonparametric setting) and motion in the normal direction by anisotropic mean curvature of surfaces (parametric setting). In the latter case we first relax the problem via an Allen-Cahn type reaction-diffusion equation, in the context of Finsler geometry (diffused interface approximation). Our results suggest three main points. A nonconvex anisotropy and its convexification give rise, for both settings and the discretizations considered, to different evolutions. Wrinkled regions seem to appear only in correspondence to locally concave parts of the anisotropy. Moreover, locally convex regions (interior to the convexification of the so-called Frank diagram) seem to play an important role.

Goglione, R., Paolini, M., Fierro, F., Numerical simulations of mean curvature flow in the presence of a nonconvex anisotropy, <<MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES>>, 1998; (Maggio): 573-601 [http://hdl.handle.net/10807/20943]

Numerical simulations of mean curvature flow in the presence of a nonconvex anisotropy

Paolini, Maurizio;
1998

Abstract

In this paper we present and discuss the results of some numerical simulations in order to investigate the mean curvature flow problem in presence of a nonconvex anisotropy. Mathematically, nonconvexity of the anisotropy leads to the ill-posedness of the evolution problem, which becomes forward-backward parabolic. Simulations presented here refer to two different settings: curvature driven vertical motion of graphs (nonparametric setting) and motion in the normal direction by anisotropic mean curvature of surfaces (parametric setting). In the latter case we first relax the problem via an Allen-Cahn type reaction-diffusion equation, in the context of Finsler geometry (diffused interface approximation). Our results suggest three main points. A nonconvex anisotropy and its convexification give rise, for both settings and the discretizations considered, to different evolutions. Wrinkled regions seem to appear only in correspondence to locally concave parts of the anisotropy. Moreover, locally convex regions (interior to the convexification of the so-called Frank diagram) seem to play an important role.
Inglese
Goglione, R., Paolini, M., Fierro, F., Numerical simulations of mean curvature flow in the presence of a nonconvex anisotropy, <<MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES>>, 1998; (Maggio): 573-601 [http://hdl.handle.net/10807/20943]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/20943
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 21
  • ???jsp.display-item.citation.isi??? 18
social impact