The divergence-conforming immersed boundary method: Application to vesicle and capsule dynamics

Abstract

We extend the recently introduced divergence-conforming immersed boundary (DCIB) method [1] to fluid-structure interaction (FSI) problems involving closed co-dimension one solids. We focus on capsules and vesicles, whose discretization is particularly challenging due to the higher-order derivatives that appear in their formulations. In two-dimensional settings, we employ cubic B-splines with periodic knot vectors to obtain discretizations of closed curves with C 2 inter-element continuity. In three-dimensional settings, we use analysis-suitable bi-cubic T-splines to obtain discretizations of closed surfaces with at least C 1 inter-element continuity. Large spurious changes of the fluid volume inside closed codimension one solids is a well-known issue for IB methods. The DCIB method results in volume changes orders of magnitude lower than conventional IB methods. This is a byproduct of discretizing the velocity-pressure pair with divergence-conforming B-splines, which lead to negligible incompressibility errors at the Eulerian level. The higher interelement continuity of divergence-conforming B-splines is also crucial to avoid the quadrature/interpolation errors of IB methods becoming the dominant discretization error. Benchmark and application problems of vesicle and capsule dynamics are solved, including meshindependence studies and comparisons with other numerical methods.

A note on the accuracy of the generalized-$α$ scheme for the incompressible Navier-Stokes equations

“…This choice of temporal discretization has in fact become quite popular in both CFD and FSI simulations (see e.g. [1,2,3,4,6,5,10,11,22,19,26,30,32,34,35]). In geometrically multiscale coupled problems, this dichotomy of time discretization treatment further leads to a competing question: whether one should collocate the boundary traction at the time step t n+1 [25] or at the intermediate time step [23].…”