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Published in:
Mechanics and Physics of Lipid Bilayers Read chapter Read first chapter

2018 | OriginalPaper | Chapter

On the Computational Modeling of Lipid Bilayers Using Thin-Shell Theory

Author : Roger A. Sauer

Published in: The Role of Mechanics in the Study of Lipid Bilayers

Publisher: Springer International Publishing

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Abstract

This chapter discusses the computational modeling of lipid bilayers based on the nonlinear theory of thin shells. Several computational challenges are identified and various theoretical and computational ingredients are proposed in order to counter them. In particular, \(C^1\)-continuous, NURBS-based, LBB-conforming surface finite element discretizations are discussed. The constitutive behavior of the bilayer is based on in-plane viscosity and (near) area-incompressibility combined with the Helfrich bending model. Various shear stabilization techniques are proposed for quasi-static computations. All ingredients are formulated in the curvilinear coordinate system characterizing general surface parameterizations. The consistent linearization of the formulation is presented, and several numerical examples are shown.

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previous chapter The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws
next chapter Onsager’s Variational Principle in Soft Matter: Introduction and Application to the Dynamics of Adsorption of Proteins onto Fluid Membranes
Footnotes
1
Note that \(\det [c^{\alpha }_\beta ]=\det [c^\alpha _{~\beta }]=\det [c^{~\alpha }_\beta ]\) even if \(c^\alpha _{~\beta }\ne c^{~\alpha }_\beta \).
 
2
For an extension to changing mass, e.g., due to protein binding, see Sahu et al. (2017).
 
3
Per current length of the cut face.
 
4
Since \(\sigma ^{\alpha \beta }_\mathrm {visc}\,\dot{a}_{\alpha \beta }=4\eta \,\varvec{d}:\varvec{d}=4\eta \Vert \varvec{d}\Vert ^2>0\) due to (65) and (66).
 
5
Strictly, \(G^1\)-continuity (i.e., continuity in \({\varvec{n}}\) but not necessary in \({\varvec{a}}_\alpha \)) is sufficient.
 
6
Named after Ladyzhenskaya, Babuška, and Brezzi.
 
7
Assuming that the tube is sufficiently long and can be idealized by a perfect cylinder.
 
8
The shear stresses are now physical and need to be applied both in-plane and out-of-plane.
 
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Metadata
Title
On the Computational Modeling of Lipid Bilayers Using Thin-Shell Theory
Author
Roger A. Sauer
Copyright Year
2018
Book
The Role of Mechanics in the Study of Lipid Bilayers

Print ISBN: 978-3-319-56347-3

Electronic ISBN: 978-3-319-56348-0

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-56348-0_5

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