The divergence-conforming immersed boundary method: Application to vesicles, capsules, and red blood cells under flow

The divergence-conforming immersed boundary (DCIB) method is presented to tackle a long-standing issue of immersed boundary (IB) numerical methods for fluid-structure interaction (FSI), namely, the challenge of accurately imposing the incompressibility constraint at the discrete level. In the DCIB method, the Eulerian velocity-pressure pair is discretized using divergence-conforming B-splines, which leads to inf-sup stable, H¹-conforming, and pointwise divergence-free Eulerian solutions. In order to discretize the higher-oder derivatives that appear in vesicle and capsule formulations, we use C²-continuous cubic B-splines with periodic knot vectors and C¹-continuous bi-cubic analysis-suitable T-splines in 2D and 3D settings, respectively. Non-negligible spurious changes of the fluid volume inside of closed co-dimension one solids is a well-known issue of IB methods. The DCIB method results in volume changes various orders of magnitude lower than conventional IB methods. Benchmark and application problems of vesicle, capsules and red blood cells are solved, including mesh-independence studies and comparisons with other numerical methods.