Note: Different Location
Abstract:
When lipid molecules are immersed in aqueous environment they
aggregate spontaneously into 2 mono-molecular layers or (bio)membranes that
form an encapsulating bag called vesicle. This happens because lipids consist
of a hydrophilic head group and a hydrophobic tail, which isolate itself in the
interior of the membrane.
As a first approach, we have studied a model based on geometry assuming
that the equilibrium shapes are the minimizers of the Willmore energy under
area and volume constraints. In this context, the biomembrane is the preponderant
factor influencing the shape of the vesicle. We propose a new model
derived from basic shape differential calculus. A 
-gradient flow is established
to reach these equilibrium shapes. Then, the effect of the inside (bulk) fluid is
taken into account leading to more physical dynamics. The boundary conditions
couple Stokes equations to the constrained Willmore force. Forth order,
highly nonlinear arising problems are solved using an adaptive finite element
method (AFEM). We analyse the method and present numerical simulations.
We finally insist on a new paradigm of adaptivity on manifolds. We discuss
how to execute mesh modifications with incomplete information about their
geometry and yet preserve position and curvature accuracy. We refer to this
as geometrically consistent mesh modification. We present the concept of
discrete geometric consistency, show the failure of naive approaches, propose
and analyze a simple geometrically consistent algorithm.
This is joint work with R.H. Nochetto and M.S. Pauletti.
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