This paper introduces an approach to improve volume conservation in the immersed boundary (IB) method using regularized delta functions derived from composite B-splines. These delta functions employ tensor product kernels using B-splines, whose polynomial degrees vary in normal and tangential directions based on the corresponding velocity component. Our method addresses the long-standing volume conservation issues in the conventional IB method, particularly evident in simulations of pressurized, closed membranes. We demonstrate that our approach significantly enhances volume conservation, rivaling the performance of the non-local Divergence-Free Immersed Boundary (DFIB) method introduced by Bao et al. while maintaining the local nature of the classical IB method. This avoids the computational overhead associated with the DFIB method's construction of an explicit velocity potential which requires additional Poisson solves. Numerical experiments show that sufficiently regular composite B-spline kernels can maintain initial volumes to within machine precision. We analyze the relationship between kernel regularity and the accuracy of force spreading and velocity interpolation operations. Our findings indicate that composite B-splines of at least $C^1$ regularity produce results comparable to the DFIB method in dynamic simulations, with volume conservation errors primarily dominated by the time-stepping scheme's truncation error. This work offers a computationally efficient alternative for improving volume conservation in IB methods, particularly beneficial for large-scale, three-dimensional simulations. The proposed approach requires minimal modifications to an existing IB code, making it an accessible improvement for a wide range of applications in computational fluid dynamics and fluid-structure interaction.

翻译：本文提出一种利用复合B样条导出的正则化δ函数来改进浸没边界法中体积守恒性的方法。这些δ函数采用基于B样条的张量积核函数，其多项式次数根据对应速度分量的法向与切向分别变化。本方法解决了传统浸没边界法中长期存在的体积守恒问题，在受压封闭膜模拟中尤为显著。我们证明该方法在保持经典浸没边界法局部特性的同时，显著提升了体积守恒性能，其效果可与Bao等人提出的非局部无散浸没边界法相媲美。这避免了DFIB方法构建显式速度势所需额外泊松求解的计算开销。数值实验表明，充分正则的复合B样条核函数能将初始体积保持至机器精度水平。我们分析了核函数正则性与力铺展及速度插值运算精度之间的关系。研究结果表明，具有至少$C^1$正则性的复合B样条在动态模拟中可获得与DFIB方法相当的结果，其体积守恒误差主要受时间推进格式的截断误差主导。这项工作为改进浸没边界法的体积守恒性提供了一种计算高效的替代方案，特别有利于大规模三维模拟。所提方法仅需对现有浸没边界代码进行最小修改，使其成为计算流体力学和流固耦合领域广泛应用中易于实现的改进方案。