Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods

The numerical simulation of the deformation of vesicle membranes under simple shear external fluid flow is considered in this paper. A saddle-point approach is proposed for the imposition of the fluid incompressibility and the membrane inextensibility constraints, through Lagrange multipliers defined in the fluid and on the membrane respectively. Using a level set formulation, the problem is approximated by mixed finite elements combined with an automatic adaptive mesh procedure at the vicinity of the membrane boundary. Numerical experiments show that this combination of the saddle-point and adaptive mesh method enhances the robustness of the method. The effect of inertia on the stability of the vesicle in a shear flow is also investigated.

[1]  Q. Du,et al.  Energetic variational approaches in modeling vesicle and fluid interactions , 2009 .

[2]  Simona Perotto,et al.  New anisotropic a priori error estimates , 2001, Numerische Mathematik.

[3]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[4]  Gwennou Coupier,et al.  Tumbling of viscous vesicles in a linear shear field near a wall , 2009 .

[5]  Dierk Raabe,et al.  Author's Personal Copy Computers and Mathematics with Applications , 2022 .

[6]  T. Biben,et al.  Tumbling of vesicles under shear flow within an advected-field approach. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Vanessa Lleras,et al.  DYNAMICS OF RED BLOOD CELLS IN 2D , 2009 .

[8]  Gene H. Golub,et al.  Matrix computations , 1983 .

[9]  G. Cottet,et al.  EULERIAN FORMULATION AND LEVEL SET MODELS FOR INCOMPRESSIBLE FLUID-STRUCTURE INTERACTION , 2008 .

[10]  Duc-Vinh Le,et al.  A front-tracking method with Catmull-Clark subdivision surfaces for studying liquid capsules enclosed by thin shells in shear flow , 2011, J. Comput. Phys..

[11]  S. Hysing,et al.  A new implicit surface tension implementation for interfacial flows , 2006 .

[12]  C Misbah,et al.  Toward a thermodynamically consistent picture of the phase-field model of vesicles: curvature energy. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Ming-Chih Lai,et al.  Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method , 2010, J. Comput. Phys..

[14]  Hong Zhao,et al.  The dynamics of a vesicle in simple shear flow , 2011, Journal of Fluid Mechanics.

[15]  W. Helfrich,et al.  Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. , 1989, Physical review. A, General physics.

[16]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[17]  G. Aubert,et al.  Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences) , 2006 .

[18]  Jian Zhang,et al.  Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations , 2008, SIAM J. Sci. Comput..

[19]  W. Helfrich.  Elastic Properties of Lipid Bilayers: Theory and Possible Experiments , 1973, Zeitschrift fur Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie.

[20]  Michael J. Miksis,et al.  Reynolds number effects on lipid vesicles , 2012, Journal of Fluid Mechanics.

[21]  Christophe Prud'homme,et al.  Simulation of vesicle using level set method solved by high order finite element , 2012 .

[22]  Emmanuel Maitre,et al.  Applications of level set methods in computational biophysics , 2009, Math. Comput. Model..

[23]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[24]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[25]  Harald Garcke,et al.  Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations , 2008, SIAM J. Sci. Comput..

[26]  Rudolf Podgornik,et al.  Statistical thermodynamics of surfaces, interfaces, and membranes , 1995 .

[27]  Samuel A. Safran,et al.  Statistical Thermodynamics of Surfaces , 1994 .

[28]  Raluca E. Rusu.  An algorithm for the elastic flow of surfaces , 2005 .

[29]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[30]  Q. Du,et al.  A phase field approach in the numerical study of the elastic bending energy for vesicle membranes , 2004 .

[31]  Gjlles Aubert,et al.  Mathematical problems in image processing , 2001 .

[32]  P. Canham.  The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. , 1970, Journal of theoretical biology.

[33]  Ricardo H. Nochetto,et al.  Parametric FEM for geometric biomembranes , 2010, J. Comput. Phys..

[34]  Aymen Laadhari,et al.  On the equilibrium equation for a generalized biological membrane energy by using a shape optimization approach , 2010 .

[35]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[36]  David Salac,et al.  A level set projection model of lipid vesicles in general flows , 2011, J. Comput. Phys..

[37]  Marc Jaeger,et al.  3D vesicle dynamics simulations with a linearly triangulated surface , 2011, J. Comput. Phys..

[38]  R. Skalak,et al.  Motion of a tank-treading ellipsoidal particle in a shear flow , 1982, Journal of Fluid Mechanics.

[39]  G. Burton.  Sobolev Spaces , 2013 .

[40]  Pierre Saramito,et al.  Improving the mass conservation of the level set method in a finite element context , 2010 .

[41]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[42]  Lin Ma,et al.  Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics , 2007, J. Comput. Phys..

[43]  Weizhang Huang,et al.  Metric tensors for anisotropic mesh generation , 2005 .

[44]  P. Saramito.  Efficient C++ finite element computing with Rheolef , 2012 .

[45]  George Biros,et al.  Author ' s personal copy Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain , a boundary integral method , 2010 .

[46]  Klaus Kassner,et al.  Phase-field approach to three-dimensional vesicle dynamics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Weizhang Huang,et al.  An anisotropic mesh adaptation method for the finite element solution of variational problems , 2010 .

[48]  Udo Seifert,et al.  Configurations of fluid membranes and vesicles , 1997 .

[49]  C. Pozrikidis,et al.  Numerical Simulation of the Flow-Induced Deformation of Red Blood Cells , 2003, Annals of Biomedical Engineering.

[50]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..

[51]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[52]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[53]  Gene H. Golub,et al.  Matrix Computations, Third Edition , 1996 .

[54]  Martin Rumpf,et al.  A finite element method for surface restoration with smooth boundary conditions , 2004, Comput. Aided Geom. Des..

[55]  Petia M. Vlahovska,et al.  Vesicles and red blood cells in flow: From individual dynamics to rheology , 2009 .

[56]  T. Biben,et al.  Steady to unsteady dynamics of a vesicle in a flow. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  Seifert,et al.  Fluid Vesicles in Shear Flow. , 1996, Physical review letters.

[58]  Pierre Saramito,et al.  Vesicle tumbling inhibited by inertia , 2012 .

[59]  Samuel A. Safran,et al.  Statistical Thermodynamics Of Surfaces, Interfaces, And Membranes , 1994 .

[60]  Victor Steinberg,et al.  Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow. , 2006, Physical review letters.