Parametric FEM for geometric biomembranes. (English) Zbl 1307.76049
Summary: 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.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76Z05 | Physiological flows |
| 92C15 | Developmental biology, pattern formation |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
moving finite elements; isoparametric elements; geometric flow; shape differential calculus; gradient flow; Helfrich; Willmore; bending energy; biomembrane; red blood cellSoftware:
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