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 co-dimension one solids are 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 inter-element 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 mesh-independence studies and comparisons with other numerical methods.

我们将最近引入的发散一致沉浸边界（DCIB）方法[1]扩展到涉及封闭共维一固体的流固耦合（FSI）问题。我们专注于胶囊和囊泡，由于其配方中出现了较高阶的衍生物，其离散化尤其具有挑战性。在二维设置中，我们使用具有周期性结向量的三次B样条来获得闭合曲线的离散化$C^2$元素间的连续性。在三维环境中，我们使用适合分析的双三次T样条获得至少具有至少$C^{\mathrm{1个}}$元素间的连续性。封闭的一维固体中流体体积的大量杂散变化是IB方法的一个众所周知的问题。DCIB方法导致体积变化比常规IB方法低几个数量级。这是用散度符合的B样条离散化速度-压力对的副产品，这导致欧拉水平的不可压缩性误差可忽略不计。散度符合B样条的较高元素间连续性对于避免IB方法的正交/插值误差成为主要的离散化误差也至关重要。解决了囊泡和囊动力学的基准和应用问题，包括独立于网格的研究以及与其他数值方法的比较。