Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method
ACM Transactions on Mathematical Software (TOMS)
A phase field approach in the numerical study of the elastic bending energy for vesicle membranes
Journal of Computational Physics
Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions
Journal of Computational Physics
Finite element modeling of lipid bilayer membranes
Journal of Computational Physics
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Journal of Computational Physics
Journal of Computational Physics
Computational parametric Willmore flow
Numerische Mathematik
Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations
SIAM Journal on Scientific Computing
Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces
SIAM Journal on Numerical Analysis
Parametric FEM for geometric biomembranes
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.45 |
Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L^2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen-Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H^1 spaces, and we employ triangulated surfaces and H^1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.