Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

Ma, Lin; Klug, William S.

doi:10.1016/j.jcp.2008.02.019

Cited by 78 publications

(108 citation statements)

References 55 publications

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“…For a brief review of these methods, in comparison with integral equation formulations, see [9]. Several groups have considered stationary shapes of three-dimensional vesicles using semi-analytic [10,15,56], or numerical methods like the phase-field [8,22,23] and finite element methods [25,43]. These approaches are based on a constrained variational approach (i.e., minimizing the bending energy subject to area and volume constraints) and have not been used for resolving fluid-structure interactions.…”

Section: Related Workmentioning

confidence: 99%

“…Ma and Klug [43] mention that local inextensibility combined with large deformations still leads to instability, and describe how viscous stabilization of the mesh can be achieved by minimizing an energy measuring the deviation of edge length from previous values. This approach is suitable for computing equilibrium shapes in the absence of external forces, but not for dynamic simulations.…”

Section: Related Workmentioning

confidence: 99%

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A fast algorithm for simulating vesicle flows in three dimensions

Veerapaneni

Rahimian

Biros

et al. 2011

Journal of Computational Physics

137

143

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Vesicles are locally-inextensible fluid membranes that can sustain bending. In this paper, we extend "A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows", Veerapaneni et al. Journal of Computational Physics, 228(19), 2009 to general non-axisymmetric vesicle flows in three dimensions.Although the main components of the algorithm are similar in spirit to the axisymmetric case (spectral approximation in space, semi-implicit time-stepping scheme), important new elements need to be introduced for a full 3D method. In particular, spatial quantities are discretized using spherical harmonics, and quadrature rules for singular surface integrals need to be adapted to this case; an algorithm for surface reparameterization is neeed to ensure sufficient of the timestepping scheme, and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs. To characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi-implicit time-stepping scheme, we perform a spectral analysis of the evolution operator on the unit sphere.By introducing these algorithmic components, we obtain a time-stepping scheme that, in our numerical experiments, is unconditionally stable. We present results to analyze the cost and convergence rates of the overall scheme. To illustrate the applicability of the new method, we consider a few vesicle-flow interaction problems: a single vesicle in relaxation, sedimentation, shear flows, and many-vesicle flows.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

A fast algorithm for simulating vesicle flows in three dimensions

Veerapaneni

Rahimian

Biros

et al. 2011

Journal of Computational Physics

137

143

View full text Add to dashboard Cite

show abstract

“…Several groups have focused on determining stationary shapes of three-dimensional vesicles using semianalytic [19,4,6], or numerical methods like the phase-field [9,8] and membrane finite element methods [10,13]. These approaches are based on a constrained variational approach (i.e., minimizing the bending energy subject to area and volume constraints) and cannot be used for interactions of multiple vesicles in shear flows.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows

Veerapaneni

Gueyffier

Biros

et al. 2009

Journal of Computational Physics

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We extend "A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D", Veerapaneni et al. Journal of Computational Physics, 228(7), 2009 to the case of three dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows. Although the main components of the algorithm are similar in spirit to the 2D case-spectral approximation in space, semi-implicit time-stepping scheme-the main differences are that the bending and viscous force require new analysis, the linearization for the semi-implicit schemes must be rederived, a fully implicit scheme must be used for the toroidal topology to eliminate a CFL-type restriction, and a novel numerical scheme for the evaluation of the 3D Stokes single-layer potential on an axisymmetric surface is necessary to speed up the calculations. By introducing these novel components, we obtain a time-scheme that experimentally is unconditionally stable, has low cost per time step, and is third-order accurate in time. We present numerical results to analyze the cost and convergence rates of the scheme.To verify the solver, we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid. To illustrate the applicability of method, we consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and three vesicles with a background Poiseuille flow.

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“…We obtain equilibrium configurations of both intact and whiffleball shells by relaxation of the energy, computed numerically by finite element approximation on triangular meshes, using C 1 -conforming subdivision-surface shape functions for bending, and Lagrange interpolation for stretching [12,13]. The triangular finite-element meshes were generated by recursive subdivision of the Caspar-Klug T-number triangulations, to obtain convergence to the continuum limit, which, for an intact shell, is insensitive to the base T-number.…”

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confidence: 99%