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 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 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.

中文翻译：

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符合发散的浸入边界法：在囊泡和胶囊动力学中的应用

我们将最近引入的符合发散性的浸入边界 (DCIB) 方法 [1] 扩展到涉及封闭共维一维固体的流固耦合 (FSI) 问题。我们专注于胶囊和囊泡，由于它们的配方中出现高阶导数，因此它们的离散化特别具有挑战性。在二维设置中，我们使用具有周期性节点向量的三次 B 样条来获得具有 C^2 元素间连续性的闭合曲线的离散化。在三维设置中，我们使用适合分析的双三次 T 样条来获得具有至少 C^1 元素间连续性的闭合曲面的离散化。封闭共维一固体内流体体积的大量虚假变化是 IB 方法的一个众所周知的问题。DCIB 方法导致体积变化的数量级低于传统 IB 方法。这是使用符合发散度的 B 样条对速度-压力对进行离散化的副产品，这导致在欧拉水平上的不可压缩性误差可以忽略不计。符合发散的 B 样条的更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。囊泡和胶囊动力学的基准和应用问题得到解决，包括网格独立性研究和与其他数值方法的比较。符合发散的 B 样条的更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。囊泡和胶囊动力学的基准和应用问题得到解决，包括网格独立性研究和与其他数值方法的比较。符合发散的 B 样条的更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。囊泡和胶囊动力学的基准和应用问题得到解决，包括网格独立性研究和与其他数值方法的比较。