This paper presents an approach to enhance volume conservation in the immersed boundary (IB) method by using regularized delta functions derived from composite B-splines. The conventional IB method, while effective for fluid-structure interaction applications, has long been challenged by poor volume conservation, particularly evident in simulations of pressurized, closed membranes. We demonstrate that composite B-spline regularized delta functions significantly enhance volume conservation through two complementary properties: they provide continuously divergence-free velocity interpolants and maintain the gradient character of forces corresponding to mean pressure jumps across interfaces. By correctly representing these forces as discrete gradients, they eliminate a key source of spurious flows that typically plague immersed boundary computations. Our approach maintains the local nature of the classical IB method, avoiding the computational overhead associated with the non-local Divergence-Free Immersed Boundary (DFIB) method's construction of an explicit velocity potential which requires additional Poisson solves for interpolation and force spreading operations. We show that sufficiently regular composite B-spline kernels maintain initial volumes to within machine precision. We provide a detailed analysis of 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 errors in volume conservation dominated by truncation error of the time-stepping scheme. 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：本文提出了一种利用复合B样条导出的正则化δ函数来增强浸没边界法体积守恒性的方法。传统浸没边界法虽然在流固耦合应用中表现有效，但其较差的体积守恒性长期存在挑战，在受压封闭膜的模拟中尤为明显。我们证明复合B样条正则化δ函数通过两个互补特性显著提升了体积守恒性：它们提供连续无散的速度插值函数，并保持界面平均压力差对应力的梯度特性。通过将这些力正确表示为离散梯度，该方法消除了浸没边界计算中通常存在的虚假流动关键来源。我们的方法保持了经典浸没边界法的局部特性，避免了无散浸没边界法中因构建显式速度势而引入的非局部计算开销——该构造需要额外的泊松求解来完成插值和力延拓操作。研究表明，充分正则的复合B样条核函数可将初始体积误差控制在机器精度范围内。我们详细分析了核函数正则性与力延拓及速度插值操作精度之间的关系。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无散浸没边界法相当的结果，其体积守恒误差主要受时间推进格式的截断误差主导。所提方法仅需对现有浸没边界代码进行最小修改，为计算流体力学和流固耦合领域的广泛应用提供了易于实现的改进方案。