Local Divergence-Free Immersed Finite Element-Difference Method Using Composite B-Splines

Abstract: 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.