On the bending algorithms for soft objects in flows

Abstract

International audienceOne of the most challenging aspects in the accurate simulation of three-dimensional soft objects such as vesicles or biological cells is the computation of membrane bending forces. The origin of this difficulty stems from the need to numerically evaluate a fourth order derivative on the discretized surface geometry. Here we investigate six different algorithms to compute membrane bending forces, including regularly used methods as well as novel ones. All are based on the same physical model (due to Canham and Helfrich) and start from a surface discretization with flat triangles. At the same time, they differ substantially in their numerical approach. We start by comparing the numerically obtained mean curvature, the Laplace-Beltrami operator of the mean curvature and finally the surface force density to analytical results for the discocyte resting shape of a red blood cell. We find that none of the considered algorithms converges to zero error at all nodes and that for some algorithms the error even diverges. There is furthermore a pronounced influence of the mesh structure: Discretizations with more irregular triangles and node connectivity present serious difficulties for most investigated methods. To assess the behavior of the algorithms in a realistic physical application, we investigate the deformation of an initially spherical capsule in a linear shear flow at small Reynolds numbers. To exclude any influence of the flow solver, two conceptually very different solvers are employed: the Lattice-Boltzmann and the Boundary Integral Method. Despite the largely different quality of the bending algorithms when applied to the static red blood cell, we find that in the actual flow situation most algorithms give consistent results for both hydrodynamic solvers. Even so, a short review of earlier works reveals a wide scattering of reported results for, e.g., the Taylor deformation parameter. Besides the presented application to biofluidic systems, the investigated algorithms are also of high relevance to the computer graphics and numerical mathematics communities

Bending models of lipid bilayer membranes: Spontaneous curvature and area-difference elasticity

“…However, a non-zero H 0 introduces a length scale and we define h 0 = H 0 /R. It is important to remark that most simulations of vesicles or RBCs [39,38,40] use H 0 = 0, thus assuming no chemical differences between the two sides of the lipid bilayer. However it has been noted [6] that such an assumption lacks a physical realization.…”

“…However, for the general energy/force in the SC and BC/ADE models considered in this work, we need an explicit discrete definition of n i . We consider a discrete n i [56,57,40]. where the unit normal vector for vertex i is the sum of neighboring normal vectors weighted by incident angles [56,57].…”

“…In this work, we represent the surface as a triangulated mesh, which has been used extensively to model vesicle and RBC membranes and they have been coupled to fluid solvers [35,36,37,38,39,40]. However, the majority of vesicle and RBC simulations have employed only the minimal model.…”

Computational modeling of active deformable membranes embedded in three-dimensional flows

“…Our implementation of the force calculation and the LBM/IBM have extensively been tested for passive, elastic membranes in previous publications [52,56,78,[80][81][82].…”

“…The algorithm for the passive, elastic force calculation has been validated in references [52,78] by comparison with exact results in static situations as well as for a capsule in shear flow. For red blood cells flowing through a rectangular channel very good agreement with in vitro experiments has been found [56,80].…”