Abstract:
Using shape differential calculus, which expresses variations of
bulk and surface energy with respect to domain changes, and Euler
implicit time discretization, we formulate gradient flows for
these energies which yield geometric laws for the motion of
domain boundaries (curves or surfaces). We next present a
semi-implicit variational formulation which requires no explicit
parametrization of the surface, and is sufficiently flexible to
accommodate several scalar products for the computation of normal
velocity, depending on the application. This leads to linear
systems of lower order elliptic PDE to solve at each time step,
in both the surface and bulk. We develop adaptive finite element
methods (AFEM), and propose a Schur complement approach to solve
the resulting linear SPD systems.
We first apply this idea to surface diffusion, namely to the geometric motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature, and couple it with elasticity in the bulk; this is a simple model for epitaxially stressed films. We present several numerical experiments for surface diffusion including pinch-off in finite time and topological changes. We also present computations for the coupled system exhibiting formation of inclusions. We next discuss applications to shape optimization and image processing, that is to the minimization of functionals subject to differential constraints, and present preliminary simulations. We briefly discuss time and space adaptivity to handle the multiscale nature of these problems, as well as mesh generation, mesh distortion and mesh smoothing.
This work is joint with G. Dogan, P. Morin, and M. Verani.
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