Discrete Gradient Flows for Shape Optimization and Applications

We present a variational framework for shape optimization problems that hinges
on devising energy decreasing flows, based on shape differential calculus,
followed by suitable space and time discretizations (discrete gradient flows).
A key ingredient is the flexibility in choosing appropriate descent directions
by varying the scalar products, used for computation of normal velocity, on the
deformable domain boundary. We discuss applications to optimal shape design for
PDE, surface diffusion, and image segmentation, along with several simulations
exhibiting large deformations as well as pinching and topological changes in
finite time. This work is joint with E. Baensch, G. Dogan, P. Morin, and M.
Verani. An important resolution issue, critical for large domain deformations
and new paradigm in adaptivity, is geometrically consistent mesh refinement of
polyhedral surfaces. We discuss recent results joint with A. Bonito and M.S.
Pauletti.