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

TL;DR

This work evaluates composite B-spline (CBS) kernels within the immersed finite element/finite difference (IFED) framework for fluid–structure interaction and demonstrates that CBS kernels preserve a discrete divergence-free velocity field, yielding superior volume conservation without the need for volumetric stabilization. By contrasting CBS with traditional isotropic kernels (Peskin IB and B-spline) across diverse benchmarks—pressurized membranes, compression and Cook's membrane tests, slanted-channel flow, modified Turek–Hron, and bioprosthetic heart valve dynamics—the study shows CBS achieve accurate results on coarser fluid grids and are less sensitive to mesh ratios. The paper details the numerical machinery (diagonal Galerkin projection, nodal coupling, and tensor-product CBS delta functions that decouple velocity components) and stabilization strategies (volumetric penalization and modified invariants) to contextualize performance depends on kernel choice. Overall, CBS kernels offer a favorable balance between accuracy and efficiency for FSI problems, with practical recommendations on kernel order and solid–fluid mesh ratios for two-dimensional, lower-order simulations and clear paths for extension to three dimensions and higher-order elements.

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.

Figures (35)

• Figure 1: Pressurized membrane setup. The circular membrane (radius 0.25) at equilibrium experiences a pressure difference across its interface, with interior pressure $p_i$ and exterior pressure $p_o$.
• Figure 2: Vorticity comparison between $\text{IB}_{4}$ (left) and $\text{CBS}_{43}$ (right) of for the pressurized thin elastic membrane. The vorticity should ideally be zero everywhere. The CBS kernel exhibits a noticeably smaller error induced by non-zero velocity due to errors in the discrete force spreading operator.
• Figure 3: Volume conservation comparison of pressurized thin elastic membrane for different kernels.
• Figure 4: The effects of MFAC on the volume conservation error of pressurized thin elastic membrane for different kernels.
• Figure 5: Schematic of a two-dimensional pressure-loaded elastic band. The elastic band is fixed at the top and bottom and experiences pressure differences across the band.