An adaptive finite element method for the modeling of the equilibrium of red blood cells

Abstract

International audienceThis contribution is concerned with a the numerical modeling of an isolated red blood cell (RBC), and more generally of phospholipid membranes. We propose an adaptive Eulerian finite element approximation, based on the level set method, of a shape optimization problem arising in the study of RBC's equilibrium. We simulate the equilibrium shapes that minimize the elastic bending energy under prescribed constraints of fixed volume and surface area. An anisotropic mesh adaptation technique is used in the vicinity of the cell's membrane to enhance the robustness of the method. Efficient time and spatial discretizations are considered and implemented. We address in detail the main features of the proposed method and finally we report several numerical experiments in the two-dimensional and the three-dimensional axisymmetric cases. The effectiveness of the numerical method is further demonstrated through numerical comparisons with semi-analytical solutions provided by a reduced order model

Modeling Membrane-Protein Interactions

“…Simplifying these mass and momentum governing equations in continuum framework results in partial differential equations (PDEs) [103]. To solve the PDEs, first step we need to define the constitutive relationship for membrane deformation such element methods [106][107][108], Monte Carlo methods [109][110][111], finite difference methods [112,113], and the phase field representation of the surface [114][115][116]. Each of these methods has its own advantage and disadvantage and depending on the complexity of the problem, one or more of them can be implemented.…”

Eulerian finite element method for the numerical modeling of fluid dynamics of natural and pathological aortic valves

“…To avoid tackling remeshing issues resulting from the large deformations and quick movement of the leaflets, all surface integrals are replaced by their regularized counterparts, see e.g. [30]. Let ε be a regularization parameter proportional to the local mesh size.…”

“…Based on the computation of the metric tensor of a given criterion ς, the mesh is adapted in such a way that the interpolation error becomes equidistributed, and the maximal and minimal directions of stretching become adjusted to the directions of maximal and minimal errors. We refer to [30] for a detailed description of the meshing procedure. In the present work, we introduce a different meshing criterion aggregating the kinetic and viscous energies at every time step.…”

Fully implicit finite element method for the modeling of free surface flows with surface tension effect

“…We use the bidimensional anisotropic mesh generator BAMG based a Delaunay‐type triangulation . We use a metric‐based approach, and we follow the mesh adaptation procedure described in . However, we consider a different meshing criterion more appropriate to the present capillary problem.…”

“…In , adapted unstructured meshes are employed together with the immersed boundary method and the level set method, resulting in a more accurate flow description especially near the wall. In , an anisotropic mesh adaptation technique is presented and allows to adapt a general unstructured mesh in the vicinity of the interface. Numerical results in the cases of LeVeque's test and the mean curvature motion depict that the rate of convergence of the method is improved by the mesh adaptation.…”