In the class of immersed boundary (IB) methods, the choice of the delta function plays a crucial role in transferring information between fluid and solid domains. Most prior work has used isotropic kernels that do not preserve the divergence-free condition of the velocity field, leading to loss of incompressibility of the solid when interpolating velocity to Lagrangian markers. To address this issue, in simulations involving large deformations of incompressible hyperelastic structures immersed in fluid, researchers often use stabilization approaches such as adding a volumetric energy term. Composite B-spline (CBS) kernels offer an alternative by maintaining the discrete divergence-free property. This work evaluates CBS kernels in terms of volume conservation and accuracy, comparing them with isotropic kernel functions using a construction introduced by Peskin (IB kernels) and B-spline (BS) kernels. Benchmark tests include pressure-loaded and shear-dominated flows, such as an elastic band under pressure loads, a pressurized membrane, a compressed block, Cook's membrane, and a slanted channel flow. Additionally, we validate our methodology using a complex fluid-structure interaction model of bioprosthetic heart valve dynamics. Results demonstrate that CBS kernels achieve superior volume conservation compared to isotropic kernels, eliminating the need for stabilization techniques. Further, CBS kernels converge on coarser fluid grids, while IB and BS kernels need finer grids for comparable accuracy. Unlike IB and BS kernels, which perform better with larger mesh ratios, CBS kernels improve with smaller mesh ratios. Wider kernels provide more accurate results across all methods, but CBS kernels are less sensitive to grid spacing variations than isotropic kernels.

翻译：在浸入边界法（IB方法）类中，δ函数的选择对于流体域与固体域之间的信息传递起着至关重要的作用。大多数先前工作采用各向同性核函数，这类函数无法保持速度场的无散条件，导致在将速度插值到拉格朗日标记点时固体不可压缩性的丧失。为解决此问题，在涉及浸入流体中的不可压缩超弹性结构大变形模拟时，研究者常采用稳定化方法，例如添加体积能项。复合B样条（CBS）核函数通过保持离散无散特性提供了一种替代方案。本研究从体积守恒和精度方面评估CBS核函数，并将其与使用Peskin引入的构造方法（IB核函数）和B样条（BS）核函数构造的各向同性核函数进行比较。基准测试包括压力载荷主导和剪切主导的流动，例如压力载荷下的弹性带、受压薄膜、压缩块体、Cook膜以及倾斜槽道流。此外，我们使用一个生物假体心脏瓣膜动力学的复杂流固耦合模型验证了我们的方法。结果表明，与各向同性核函数相比，CBS核函数实现了更优的体积守恒，从而无需稳定化技术。此外，CBS核函数在较粗的流体网格上即可收敛，而IB和BS核函数需要更细的网格才能达到相当的精度。与IB和BS核函数在较大网格比下表现更好不同，CBS核函数在较小网格比下性能更优。更宽的核函数在所有方法中都能提供更准确的结果，但CBS核函数对网格间距变化的敏感性低于各向同性核函数。