TY - JOUR
T1 - A Phase Field Based PDE Constrained Optimization Approach to Time Discrete Willmore Flow
JO - International Journal of Numerical Analysis and Modeling
VL - 1
SP - 116
EP - 138
PY - 2013
DA - 2013/10
SN - 10
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/561.html
KW - phase field approach, Willmore flow, image restoration, PDE constrained optimization.
AB - <p style="text-align: justify;">A novel phase field model for Willmore flow is proposed based
on a nested variational time discretization. Thereby, the mean curvature in
the Willmore functional is replaced by an approximate speed of mean curvature
motion, which is computed via a fully implicit variational model for time
discrete mean curvature motion. The time discretization of Willmore flow is
then performed in a nested fashion: in an outer variational approach a natural
time discretization is setup for the actual Willmore flow, whereas for the
involved mean curvature the above variational approximation is taken into account.
Hence, in each time step a PDE-constrained optimization problem has
to be solved in which the actual surface geometry as well as the geometry resulting
from the implicit curvature motion time step are represented by phase field
functions. The convergence behavior is experimentally validated and compared
with rigorously proved convergence estimates for a simple linear model problem.
Computational results in 2D and 3D underline the robustness of the new
discretization, in particular for large time steps and in comparison with a semi-implicit
convexity splitting scheme. Furthermore, the new model is applied as
a minimization method for elastic functionals in image restoration.</p>