Biomathematics Seminar: Phase field models with topological constraints
1 November 2011 14:00 in CM105 (Mathematical Sciences)
Elastic structures confined to a certain volume or area appear in many situations. For example inner membranes in biological cells separate an inner region from the rest of the cell and consist of an elastic bilayer. The inner structures are confined by the outer cell membrane. Since the inner membrane contributes to the biological function it is often advantageous to include a large membrane area in the cell. In two dimensions elastic structures confined to a plane ball have been experimentally produced by Boue et al. They show that with increasing length the structures become more and more complex.
We are considering here the problem of computationally modeling a one-dimensional closed elastic wire constrained in a two-dimensional container of circular shape. In order to treat this problem, we adopt a phase field approach: a closed elastic wire separates an inner region from an outer region. We approximate these sets by a smooth field with value +1 on the inside and -1 on the outside and derive an approximate energy for the system. The approximation of the elastica energy by an interfacial energy functional is well known, the length of the curve can be evaluated using a Modica-Mortola functional. Implementing the topological constraint thus becomes the main difficulty here. We propose a first solution based on a diffuse approximation of the winding number and present a proof that one can approximate a given sharp interface using a sequence of phase fields. We then show some numerical results using finite elements based on subdivision surfaces in order to demonstrate the feasibility of our approach. Finally, we consider a refined approach to dealing with the topological constraint and present a proof of convergence of energies and energy minimizers.
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