Journal
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Volume 347, Issue -, Pages 103-119Publisher
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2018.12.025
Keywords
Lipid biomembrane; Canham-Helfrich energy; Geometric partial differential equation; Isogeometric analysis; Backward differentiation formulas; Lagrange multiplier
Funding
- Swiss National Science Foundation [147033]
- INdAM-GNCS Project 2017 (Italy) Modellistica numerica di fenomeni idro/geomeccanici per la simulazione di eventi sismici
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We consider the numerical approximation of lipid biomembranes at equilibrium described by the Canham-Helfrich model, according to which the bending energy is minimized under area and volume constraints. Energy minimization is performed via L-2-gradient flow of the Canham-Helfrich energy using two Lagrange multipliers to weakly enforce the constraints. This yields a highly nonlinear, high order, time dependent geometric Partial Differential Equation (PDE). We represent the biomembranes as single-patch NURBS closed surfaces. We discretize the geometric PDEs in space with NURBS-based Isogeometric Analysis and in time with Backward Differentiation Formulas. We tackle the nonlinearity in our formulation through a semi-implicit approach by extrapolating, at each time level, the geometric quantities of interest from previous time steps. We report the numerical results of the approximation of the Canham-Helfrich problem on ellipsoids of different aspect ratio, which leads to the classical biconcave shape of lipid vesicles at equilibrium. We show that this framework permits an accurate approximation of the Canham-Helfrich problem, while being computationally efficient. (C) 2019 Elsevier B.Y. All rights reserved.
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