用于浸入边界法的复合B样条正则化Delta函数：无散度插值与保梯度力扩散
Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading
AI总结：
本文提出利用复合B样条导出的正则化delta函数改进浸入边界法的体积守恒，通过提供无散度插值和保梯度的力扩散消除虚假流，在保持方法局部性的同时达到与DFIB方法相当的精度。
摘要：
本文提出了一种通过使用由复合B样条导出的正则化delta函数来增强浸入边界（IB）方法体积守恒的方法。传统的IB方法虽然有效适用于流固耦合应用，但长期以来一直受到体积守恒较差的困扰，这在加压闭合膜的模拟中尤为明显。我们证明，复合B样条正则化delta函数通过两个互补特性显著增强了体积守恒：它们提供连续无散度的速度插值，并保持与界面平均压力跳跃相对应的力的梯度特性。通过将这些力正确表示为离散梯度，它们消除了通常困扰浸入边界计算的一个虚假流的关键来源。我们的方法保持了经典IB方法的局部特性，避免了与非局部Divergence-Free Immersed Boundary (DFIB)方法相关的计算开销，该方法构建显式速度势需要在插值和力扩散操作中进行额外的Poisson求解。我们表明，足够平滑的复合B样条核能将初始体积保持在机器精度范围内。我们详细分析了核平滑性与力扩散及速度插值操作精度之间的关系。我们的研究结果表明，至少具有$C^1$平滑性的复合B样条在动态模拟中产生与DFIB方法相当的结果，体积守恒误差由时间步进方案的截断误差主导。所提出的方法对现有IB代码仅需极小修改，使其成为计算流体力学和流固耦合广泛应用中的一项易于实现的改进。
英文摘要：
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.
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