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Simulations of 3D vesicles and capsules with boundary integral method using singularity subtraction technique

Alexander Farutin (LIPhy, CNRS & UJF Grenoble), Chaouqi Misbah (LIPhy, CNRS & UJF Grenoble)

Computational Mechanics of Biomembranes

Mon 2:40 - 4:00

Barus-Holley 160

Vesicles are locally-inextensible fluid membranes while inextensible capsules are in addition endowed with in-plane shear elasticity mimicking the cytoskeleton of red blood cells. Boundary integral methods based on the Green's function techniques are used to describe their dynamics, which is usually highly nonlinear and nonlocal due to the free-boundary nature of the problem. Numerical solutions raise several obstacles and challenges that strongly impact the results. Of particular complexity is (i) the membrane inextensibility, (ii) the mesh stability and (iii) numerical precision of evaluation of the boundary integral equations. Despite the intense research, these questions are still a matter of intense debate. We simulate vesicles and inextensible capsules using boundary integral method on a triangular mesh. We calculate the membrane force using local coordinates in vicinity of each vertex and approximating the surface and the curvature by a second-degree polynomial of the local coordinates. We use exact identities to regularize the boundary integrals. An auxiliary mesh with additional quadrature nodes is used to increase the stability and precision of integration over almost-singular triangles. We solve for the Lagrange multiplier to ensure local and global conservation of the membrane area by an optimized penalization method, taking at each time step the optimal values of penalization constants for local and global strains of the membrane. For the simulation of capsules, the reference coordinates are stored for each vertex of the mesh, which allows us to calculate the elastic forces. This method has allowed us to solve many problems of vesicle and capsule dynamics under flow. In particular, we shall discuss the dynamics of very deflated vesicles under shear flow, the shapes and migration velocity of vesicles in Poiseuille flow, as well as the behavior of inextensible capsules in shear flow and during stretching by optical tweezers.