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 L2-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.

Biomembrane modeling with isogeometric analysis

Dedè, Luca;Quarteroni, Alfio
2019-01-01

Abstract

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 L2-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.
2019
Backward differentiation formulas; Canham–Helfrich energy; Geometric partial differential equation; Isogeometric analysis; Lagrange multiplier; Lipid biomembrane; Computational Mechanics; Mechanics of Materials; Mechanical Engineering; Physics and Astronomy (all); Computer Science Applications1707 Computer Vision and Pattern Recognition
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1072267
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 8
  • ???jsp.display-item.citation.isi??? 7
social impact