TY - JOUR
T1 - Numerical simulations of mean curvature flow in the presence of a nonconvex anisotropy
AU - Fierro, Francesca
AU - Goglione, Roberta
AU - Paolini, Maurizio
PY - 1998
Y1 - 1998
N2 - 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.
AB - 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.
KW - nonconvex anisotropy
KW - nonconvex anisotropy
UR - http://hdl.handle.net/10807/20943
U2 - 10.1142/S0218202598000263
DO - 10.1142/S0218202598000263
M3 - Article
SN - 0218-2025
SP - 573
EP - 601
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
ER -