A semi-implicit finite element method for viscous lipid membranes. (English) Zbl 1349.76260
Summary: A finite element formulation to approximate the behavior of lipid membranes is proposed. The mathematical model incorporates tangential viscous stresses and bending elastic forces, together with the inextensibility constraint and the enclosed volume constraint. The membrane is discretized by a surface mesh made up of planar triangles, over which a mixed formulation (velocity-curvature) is built based on the viscous bilinear form (Boussinesq-Scriven operator) and the Laplace-Beltrami identity relating position and curvature. A semi-implicit approach is then used to discretize in time, with piecewise linear interpolants for all variables. Two stabilization terms are needed: The first one stabilizes the inextensibility constraint by a pressure-gradient-projection scheme [R. Codina and J. Blasco, Comput. Methods Appl. Mech. Eng. 143, No. 3–4, 373–391 (1997; Zbl 0893.76040)], the second couples curvature and velocity to improve temporal stability, as proposed by E. Bänsch [Numer. Math. 88, No. 2, 203–235 (2001; Zbl 0985.35060)]. The volume constraint is handled by a Lagrange multiplier (which turns out to be the internal pressure), and an analogous strategy is used to filter out rigid-body motions. The nodal positions are updated in a Lagrangian manner according to the velocity solution at each time step. An automatic remeshing strategy maintains suitable refinement and mesh quality throughout the simulation. Numerical experiments show the convergent and robust behavior of the proposed method. Stability limits are obtained from numerous relaxation tests, and convergence with mesh refinement is confirmed both in the relaxation transient and in the final equilibrium shape. Virtual tweezing experiments are also reported, computing the dependence of the deformed membrane shape with the tweezing velocity (a purely dynamical effect). For sufficiently high velocities, a tether develops which shows good agreement, both in its final radius and in its transient behavior, with available analytical solutions. Finally, simulation results of a membrane subject to the simultaneous action of six tweezers illustrate the robustness of the method.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76Z05 | Physiological flows |
| 82-08 | Computational methods (statistical mechanics) (MSC2010) |
| 74L15 | Biomechanical solid mechanics |
| 92C37 | Cell biology |
Keywords:
biological membranes; lipid bilayer; bending energy; Canham-Helfrich energy; surface viscosity; Boussinesq-scriven operator; inextensibility; tangential calculus; surface finite elementsReferences:
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