A fast algorithm for simulating vesicle flows in three dimensions

S. Veerapaneni, A. Rahimian, G. Biros and D. Zorin

*Journal of Computational Physics, Vol. 230(14), 2011. *

Vesicles are locally-inextensible fluid membranes that can sustain
bending. In this paper, we extend *``A numerical method
for simulating the dynamics of 3{D} 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
time-stepping 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 evolutions operator on the unit sphere.

By introducing these algorithmic components, we obtain a
time-stepping scheme that experimentally is unconditionally
stable and has a low cost per time step. We present numerical results
to analyze the cost and convergence rates of the scheme. To
illustrate the applicability of method, we consider a few vesicle-flow
interaction problems: a single vesicle in relaxation, sedimentation,
and shear flows, and many-vesicle flows.