3D numerical simulations of vesicle and inextensible capsule dynamics

Vesicles are locally-inextensible fluid membranes, capsules are endowed with in-plane shear elasticity mimicking the cytoskeleton of red blood cells (RBCs), but are extensible, while RBCs are inextensible. We use boundary integral (BI) methods based on the Green function techniques to model and solve numerically their dynamics. We regularize the single layer integral by subtraction of exact identities for the terms involving the normal and the tangential components of the force. The stability and precision of BI calculation is enhanced by taking advantage of additional quadrature nodes located in vertices of an auxiliary mesh, constructed by a standard refinement procedure from the main mesh. We extend the partition of unity technique to boundary integral calculation on triangular meshes. The proposed algorithm offers the same treatment of near-singular integration regardless whether the source and the target points belong to the same surface or not. Bending forces are calculated by using expressions derived from differential geometry. Membrane incompressibility is handled by using two penalization parameters per suspended entity: one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex. Extensible or inextensible capsules, a model of RBC, are studied by storing the position in the reference configuration for each vertex. The elastic force is then calculated by direct variation of the elastic energy. Various nonequilibrium physical examples on vesicles and capsules will be presented and the convergence and precision tests highlighted. Overall, a good convergence is observed with numerical error inversely proportional to the number of vertices used for surface discretization, the highest order of convergence allowed by piece-wise linear interpolation of the surface.