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

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Book cover The Role of Mechanics in the Study of Lipid Bilayers

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 577))

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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Notes

  1. 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. 2.

    For an extension to changing mass, e.g., due to protein binding, see Sahu et al. (2017).

  3. 3.

    Per current length of the cut face.

  4. 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. 5.

    Strictly, \(G^1\)-continuity (i.e., continuity in \({\varvec{n}}\) but not necessary in \({\varvec{a}}_\alpha \)) is sufficient.

  6. 6.

    Named after Ladyzhenskaya, Babuška, and Brezzi.

  7. 7.

    Assuming that the tube is sufficiently long and can be idealized by a perfect cylinder.

  8. 8.

    The shear stresses are now physical and need to be applied both in-plane and out-of-plane.

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Acknowledgements

The author is grateful to the German Research Foundation (DFG) for supporting this research under grants GSC 111 and SA1822/5-1. The author also wishes to thank Kranthi Mandadapu, Thang Duong, and Amaresh Sahu for their valuable comments, and Yannick Omar for his help with the example in Sect. 13.3. Special thanks also go to David Steigmann for organizing the CISM summer school that led to this article.

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  1. Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Roger A. Sauer

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  1. Faculty of Mechanical Engineering, University of California, Berkeley, Berkeley, California, USA

    David J. Steigmann

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Sauer, R.A. (2018). On the Computational Modeling of Lipid Bilayers Using Thin-Shell Theory. In: Steigmann, D. (eds) The Role of Mechanics in the Study of Lipid Bilayers. CISM International Centre for Mechanical Sciences, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-56348-0_5

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