Finite element methods for director fields on flexible surfaces

Abstract

We introduce a nonlinear model for the evolution of biomembranes driven by the L 2-gradient flow of a novel elasticity functional describing the interaction of a director field on a membrane with its curvature. In the linearized setting of a graph we present a practical finite element method (FEM), and prove a priori estimates. We derive the relaxation dynamics for the nonlinear model on closed surfaces and introduce a parametric FEM. We present numerical experiments for both linear and nonlinear models, which agree well with the expected behavior in simple situations and allow predictions beyond theory.

A Highly Anisotropic Nonlinear Elasticity Model for Vesicles I. Eulerian Formulation, Rigidity Estimates and Vanishing Energy Limit

“…For simplicity, let us assume that Γ ⊂ R 2 is a piece of a plane and that f (Id) = 0. 1 Since, the membrane is a two dimensional fluid, the stored energy function has to be anisotropic. For an isotropic material, the symmetries of the problem are given by the relations f (DΦ(x)R) = f (DΦ(x)) for any R ∈ SO(3).…”

“…One can find solid phases, gel phases and liquid phases. Let us mention the existence of models for 2D membranes with phase coexistence [14,5] and of a model for 2D membranes in gel phase which includes the local orientation of the lipids [1]. In contrast, in the present articles we consider thick membranes in liquid phase.…”

AFEM for Geometric PDE: The Laplace-Beltrami Operator

“…This is precisely what has been accomplished in [10], via an L 2 -gradient flow (or relaxation dynamics) for J (γ, n):…”

“…The expression of δ γ J (γ, n), the first variation of J with respect to γ (or shape derivative) is now much more involved than (4), whereas δ n J (γ, n) is rather simple; we refer to [10] for details. This dynamics involves again the Laplace-Beltrami operator Δ γ .…”

“…In order to describe this situation we consider the simple model introduced by S. Bartels, G. Dolzmann, R.H. Nochetto, and A. Raisch [10], which is in turn inspired on the model by M. Laradji and O.G. Mouritsen [45] for flat membranes.…”