Parametric FEM for geometric biomembranes

• Published in: Journal of computational physics Vol. 229; no. 9; pp. 3171 - 3188
• Main Authors: Bonito, Andrea, Nochetto, Ricardo H., Sebastian Pauletti, M.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.05.2010; Elsevier
• Subjects: Bending; Bending energy; Biomembrane; Computational techniques; Computer simulation; Derivation; Discretization; Exact sciences and technology; Finite element method; Geometric flow; Gradient flow; Helfrich; Isoparametric elements; Mathematical analysis; Mathematical methods in physics; Mathematical models; Moving finite elements; Physics; Red blood cell; Shape differential calculus; Vectors (mathematics); Willmore; Willmore; Geometric flow; Shape differential calculus; Moving finite elements; Bending energy; Red blood cell; Biomembrane; Isoparametric elements; Gradient flow; Helfrich; Differential calculus; Deformation; Equilibrium shape; Euler scheme; Calculation methods; Discretization; Toroidal shape; Models; Calculation; Performance
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.12.036

We consider geometric biomembranes governed by an L 2 -gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using quadratic isoparametric elements and a semi-implicit Euler method. We document the performance of the new parametric FEM with a number of simulations leading to dumbbell, red blood cell and toroidal equilibrium shapes while exhibiting large deformations.