Computation of Two-Phase Biomembranes with Phase Dependent Material Parameters Using Surface Finite Elements
Year: 2013
Communications in Computational Physics, Vol. 13 (2013), Iss. 2 : pp. 325–360
Abstract
The shapes of vesicles formed by lipid bilayers with phase separation are governed by a bending energy with phase dependent material parameters together with a line energy associated with the phase interfaces. We present a numerical method to approximate solutions to the Euler-Lagrange equations featuring triangulated surfaces, isoparametric quadratic surface finite elements and the phase field approach for the phase separation. Furthermore, the method involves an iterative solution scheme that is based on a relaxation dynamics coupling a geometric evolution equation for the membrane surface with a surface Allen-Cahn equation. Remeshing and grid adaptivity are discussed, and in various simulations the influence of several physical parameters is investigated.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.170611.130112a
Communications in Computational Physics, Vol. 13 (2013), Iss. 2 : pp. 325–360
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 36
-
On the condition number of the finite element method for the Laplace–Beltrami operator
Nguemfouo, Marcial | Ndjinga, MichaëlJournal of Elliptic and Parabolic Equations, Vol. 10 (2024), Iss. 1 P.59
https://doi.org/10.1007/s41808-023-00251-7 [Citations: 1] -
Curve shortening flow coupled to lateral diffusion
Pozzi, Paola | Stinner, BjörnNumerische Mathematik, Vol. 135 (2017), Iss. 4 P.1171
https://doi.org/10.1007/s00211-016-0828-8 [Citations: 14] -
Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme
Poochinapan, Kanyuta | Wongsaijai, BenApplied Mathematics and Computation, Vol. 434 (2022), Iss. P.127374
https://doi.org/10.1016/j.amc.2022.127374 [Citations: 4] -
Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation
Deng, Yangfang | Weng, ZhifengJournal of Applied Mathematics and Computing, Vol. 68 (2022), Iss. 5 P.3347
https://doi.org/10.1007/s12190-021-01666-y [Citations: 6] -
A Sharp Interface Limit of a Nonlocal Variational Model for Pattern Formation in Biomembranes
Ginster, Janusz | Hayrapetyan, Gurgen | Pešić, Anastasija | Zwicknagl, BarbaraSIAM Journal on Mathematical Analysis, Vol. 56 (2024), Iss. 3 P.2818
https://doi.org/10.1137/23M1559099 [Citations: 1] -
Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case
Pozzi, Paola | Stinner, BjörnIMA Journal of Numerical Analysis, Vol. (2018), Iss.
https://doi.org/10.1093/imanum/dry004 [Citations: 0] -
Domain formation via phase separation for spherical biomembranes with small deformations
ELLIOTT, C. M. | HATCHER, L.European Journal of Applied Mathematics, Vol. 32 (2021), Iss. 6 P.1127
https://doi.org/10.1017/S0956792520000297 [Citations: 4] -
SS-DNN: A hybrid strang splitting deep neural network approach for solving the Allen–Cahn equation
Singh, Anjali | Sinha, Rajen KumarEngineering Analysis with Boundary Elements, Vol. 169 (2024), Iss. P.105944
https://doi.org/10.1016/j.enganabound.2024.105944 [Citations: 1] -
A Lagrangian Thin-Shell Finite Element Method for Interacting Particles on Fluid Membranes
Dharmavaram, Sanjay | Wan, Xinran | Perotti, Luigi E.Membranes, Vol. 12 (2022), Iss. 10 P.960
https://doi.org/10.3390/membranes12100960 [Citations: 2] -
Multioutput FOSLS Deep Neural Network for Solving Allen–Cahn Equation
Singh, Anjali | Sinha, Rajen KumarMathematical Models and Computer Simulations, Vol. 15 (2023), Iss. 6 P.1132
https://doi.org/10.1134/S2070048223060066 [Citations: 0] -
A C1 finite element method for axisymmetric lipid membranes in the presence of the Gaussian energy
Ebrahimi, Faezeh
Computer Methods in Applied Mechanics and Engineering, Vol. 391 (2022), Iss. P.114472
https://doi.org/10.1016/j.cma.2021.114472 [Citations: 1] -
Modeling of multicomponent three-dimensional vesicles
Gera, Prerna | Salac, DavidComputers & Fluids, Vol. 172 (2018), Iss. P.362
https://doi.org/10.1016/j.compfluid.2018.04.003 [Citations: 10] -
Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation
Feng, Xinlong | Song, Huailing | Tang, Tao | Yang, JiangInverse Problems & Imaging, Vol. 7 (2013), Iss. 3 P.679
https://doi.org/10.3934/ipi.2013.7.679 [Citations: 62] -
Finite element methods for surface PDEs
Dziuk, Gerhard | Elliott, Charles M.Acta Numerica, Vol. 22 (2013), Iss. P.289
https://doi.org/10.1017/S0962492913000056 [Citations: 398] -
A two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential
Wang, Danxia | Li, Yanan | Jia, HongenNumerical Methods for Partial Differential Equations, Vol. 39 (2023), Iss. 2 P.1251
https://doi.org/10.1002/num.22932 [Citations: 6] -
Error analysis for an ALE evolving surface finite element method
Elliott, Charles M. | Venkataraman, ChandrasekharNumerical Methods for Partial Differential Equations, Vol. 31 (2015), Iss. 2 P.459
https://doi.org/10.1002/num.21930 [Citations: 14] -
Discrete Maximum Principle and Energy Stability Analysis of Du Fort-Frankel Scheme for 1D Allen-Cahn Equation
林, 树华
Pure Mathematics, Vol. 12 (2022), Iss. 09 P.1501
https://doi.org/10.12677/PM.2022.129164 [Citations: 0] -
Unconditional Energy Stability Analysis of a Second Order Implicit–Explicit Local Discontinuous Galerkin Method for the Cahn–Hilliard Equation
Song, Huailing | Shu, Chi-WangJournal of Scientific Computing, Vol. 73 (2017), Iss. 2-3 P.1178
https://doi.org/10.1007/s10915-017-0497-5 [Citations: 27] -
Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles
Healey, Timothy J. | Dharmavaram, SanjaySIAM Journal on Mathematical Analysis, Vol. 49 (2017), Iss. 2 P.1027
https://doi.org/10.1137/15M1043716 [Citations: 8] -
A new family of A-stable Runge-Kutta methods with equation-dependent coefficients for stiff problems
Fang, Yonglei | Yang, Yanping | You, Xiong | Wang, BinNumerical Algorithms, Vol. 81 (2019), Iss. 4 P.1235
https://doi.org/10.1007/s11075-018-0619-7 [Citations: 4] -
Dynamics of a multicomponent vesicle in shear flow
Liu, Kai | Marple, Gary R. | Allard, Jun | Li, Shuwang | Veerapaneni, Shravan | Lowengrub, JohnSoft Matter, Vol. 13 (2017), Iss. 19 P.3521
https://doi.org/10.1039/C6SM02452A [Citations: 18] -
A class of monotone and structure-preserving Du Fort-Frankel schemes for nonlinear Allen-Cahn equation
Deng, Dingwen | Lin, Shuhua | Wang, QihongComputers & Mathematics with Applications, Vol. 170 (2024), Iss. P.1
https://doi.org/10.1016/j.camwa.2024.06.023 [Citations: 0] -
Well posedness for the Poisson problem on closed Lipschitz manifolds
Ndjinga, Michaël | Nguemfouo, MarcialPartial Differential Equations and Applications, Vol. 4 (2023), Iss. 5
https://doi.org/10.1007/s42985-023-00263-x [Citations: 0] -
Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature
Barrett, John W. | Garcke, Harald | Nürnberg, RobertIMA Journal of Numerical Analysis, Vol. (2017), Iss.
https://doi.org/10.1093/imanum/drx006 [Citations: 0] -
Mathematical modelling in cell migration: tackling biochemistry in changing geometries
Stinner, Björn | Bretschneider, TillBiochemical Society Transactions, Vol. 48 (2020), Iss. 2 P.419
https://doi.org/10.1042/BST20190311 [Citations: 3] -
A gauge-fixing procedure for spherical fluid membranes and application to computations
Dharmavaram, Sanjay
Computer Methods in Applied Mechanics and Engineering, Vol. 381 (2021), Iss. P.113849
https://doi.org/10.1016/j.cma.2021.113849 [Citations: 5] -
Structure-preserving discretizations of gradient flows for axisymmetric two-phase biomembranes
Garcke, Harald | Nürnberg, RobertIMA Journal of Numerical Analysis, Vol. 41 (2021), Iss. 3 P.1899
https://doi.org/10.1093/imanum/draa027 [Citations: 2] -
The Role of Mechanics in the Study of Lipid Bilayers
Onsager’s Variational Principle in Soft Matter: Introduction and Application to the Dynamics of Adsorption of Proteins onto Fluid Membranes
Arroyo, Marino | Walani, Nikhil | Torres-Sánchez, Alejandro | Kaurin, Dimitri2018
https://doi.org/10.1007/978-3-319-56348-0_6 [Citations: 12] -
An efficient time adaptivity based on chemical potential for surface Cahn–Hilliard equation using finite element approximation
Zhao, Shubo | Xiao, Xufeng | Feng, XinlongApplied Mathematics and Computation, Vol. 369 (2020), Iss. P.124901
https://doi.org/10.1016/j.amc.2019.124901 [Citations: 8] -
An unconditionally energy stable second order finite element method for solving the Allen–Cahn equation
Li, Congying | Huang, Yunqing | Yi, NianyuJournal of Computational and Applied Mathematics, Vol. 353 (2019), Iss. P.38
https://doi.org/10.1016/j.cam.2018.12.024 [Citations: 30] -
Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles
Zhao, Siming | Healey, Timothy | Li, QingduComputer Methods in Applied Mechanics and Engineering, Vol. 314 (2017), Iss. P.164
https://doi.org/10.1016/j.cma.2016.07.011 [Citations: 5] -
A reduced order method for Allen–Cahn equations
Song, Huailing | Jiang, Lijian | Li, QiuqiJournal of Computational and Applied Mathematics, Vol. 292 (2016), Iss. P.213
https://doi.org/10.1016/j.cam.2015.07.009 [Citations: 20] -
A Highly Anisotropic Nonlinear Elasticity Model for Vesicles I. Eulerian Formulation, Rigidity Estimates and Vanishing Energy Limit
Merlet, Benoît
Archive for Rational Mechanics and Analysis, Vol. 217 (2015), Iss. 2 P.651
https://doi.org/10.1007/s00205-014-0839-5 [Citations: 3] -
Computational modeling of coupled interactions of fluid membranes with embedded filaments
Sharma, Basant Lal | Perotti, Luigi E. | Dharmavaram, SanjayComputer Methods in Applied Mechanics and Engineering, Vol. 417 (2023), Iss. P.116441
https://doi.org/10.1016/j.cma.2023.116441 [Citations: 0] -
On the equivalence of local and global area-constraint formulations for lipid bilayer vesicles
Dharmavaram, Sanjay | Healey, Timothy J.Zeitschrift für angewandte Mathematik und Physik, Vol. 66 (2015), Iss. 5 P.2843
https://doi.org/10.1007/s00033-015-0523-0 [Citations: 4] -
A weak Galerkin finite element method for Allen–Cahn equation with a nonuniform two‐step backward differentiation formula scheme
Wang, Xiuping | Gao, Fuzheng | Cui, Jintao | Sun, ZhengjiaNumerical Methods for Partial Differential Equations, Vol. 39 (2023), Iss. 3 P.2227
https://doi.org/10.1002/num.22964 [Citations: 0] -
Finite element approximation for the dynamics of fluidic two-phase biomembranes
Barrett, John W. | Garcke, Harald | Nürnberg, RobertESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51 (2017), Iss. 6 P.2319
https://doi.org/10.1051/m2an/2017037 [Citations: 20] -
A Variational Approach to Particles in Lipid Membranes
Elliott, Charles M. | Gräser, Carsten | Hobbs, Graham | Kornhuber, Ralf | Wolf, Maren-WandaArchive for Rational Mechanics and Analysis, Vol. 222 (2016), Iss. 2 P.1011
https://doi.org/10.1007/s00205-016-1016-9 [Citations: 12] -
Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods
Feng, Xinlong | Tang, Tao | Yang, JiangSIAM Journal on Scientific Computing, Vol. 37 (2015), Iss. 1 P.A271
https://doi.org/10.1137/130928662 [Citations: 73] -
Evolving surface finite element method for the Cahn–Hilliard equation
Elliott, Charles M. | Ranner, ThomasNumerische Mathematik, Vol. 129 (2015), Iss. 3 P.483
https://doi.org/10.1007/s00211-014-0644-y [Citations: 47] -
Transport phenomena in fluid films with curvature elasticity
Mahapatra, Arijit | Saintillan, David | Rangamani, PadminiJournal of Fluid Mechanics, Vol. 905 (2020), Iss.
https://doi.org/10.1017/jfm.2020.711 [Citations: 11] -
Geometric Partial Differential Equations - Part I
Parametric finite element approximations of curvature-driven interface evolutions
Barrett, John W. | Garcke, Harald | Nürnberg, Robert2020
https://doi.org/10.1016/bs.hna.2019.05.002 [Citations: 22] -
Three-dimensional multicomponent vesicles: dynamics and influence of material properties
Gera, Prerna | Salac, DavidSoft Matter, Vol. 14 (2018), Iss. 37 P.7690
https://doi.org/10.1039/C8SM01087K [Citations: 6]