A fast algorithm for simulating vesicle flows in three dimensions

Abstract

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.

On the damped oscillations of an elastic quasi-circular membrane in a two-dimensional incompressible fluid

“…Taking advantage of the linearity of the Stokes equations, boundary integral methods (BIM) (Pozrikidis 1992(Pozrikidis , 2010 are efficient techniques to solve the dynamics of fluidic particles in an external flow. These methods have been successfully applied to red blood cells (Pozrikidis 1995;Zhao et al 2010;Zhao & Shaqfeh 2011), capsules (Lac et al 2007;Walter et al 2011;Hu et al 2012) and vesicles (Ghigliotti et al 2010;Veerapaneni et al 2011;Biben et al 2011;Boedec et al 2012). An appealing feature of BIM is their precision, as they do not need the discretisation of the fluid domain (Pozrikidis 1992).…”

Nearly Singular Integrals in 3D Stokes Flow

“…Low Reynolds number flows are fundamental in a large class of problems, for example, particle and drop motion, the swimming of microorganisms, vesicle flows [8,14,19,21]. These phenomena are modeled by the Stokes equations, and a wide variety of numerical techniques have been employed to find solutions, among which boundary integral equation and singularity based methods are most popular.…”

“…Boundary integral equation methods have several well-known advantages, such as reduction in the dimensionality of the problem and high achievable accuracy of the solution. They have been used effectively to simulate the behavior of drops or vesicles in Stokes flow; comprehensive work includes [18,21,24].…”

High-order computation of burning propellant surface and simulation of fluid flow in solid rocket chamber

“…The spectral method was inspired by prior work on Wilmore flows for surfaces which are topologically equivalent to a sphere or a torus [28][29][30]. Unlike eikonal flows, Wilmore flows tend to smooth out geometric singularities.…”

“…For small and medium #DoF, convergence is indeed spectral. The error saturates around 10 -10 for large #DoF (=640) because of the truncation error in the FFT-differentiation (see [29,30]). …”