Gradient flows for general curvature-dependent surface energies
Tuesday, July 16, 2013 - 3:30pm - 4:30pm
Gunay Dogan (National Institute of Standards and Technology)
Our goal in this talk is to derive a method to compute surfaces that minimize general surface energies, in the form of weighted surface integrals with weights depending on the normal and the curvature of the surface. Energies of this form have applications in many areas, such as material science, biology and image processing. A well-known example of such energies is the Willmore functional, which is an integral of the squared mean curvature and is used as a regularization term in surface restoration and a model for the bending energy of vesicle membranes. Other examples are the weighted Willmore functional, the energy with spontaneous curvature and anisotropic surface energies. In this talk, I will derive the first variation of the general curvature-dependent surface energy using shape differential calculus. The first variation clearly delineates the influence of various components on the energy and can be used to write a gradient descent flow to compute surfaces that minimize the surface energy. However, it is difficult to discretize and implement numerically. Therefore, I will derive an alternate weak form. The weak form relies on computable quantities of the surface, and in particular, avoids tangential differentiation of the unit normal. I will use this new formulation to devise a finite element discretization of the gradient descent algorithm.