Local divergence-free immersed finite element-difference method using composite B-splines

Figure 1.  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

Figure 2.  Comparison of the influence of MFAC on the volume conservation of the pressure-loaded elastic band to isotropic kernels. (a) $ \text{IB}_5 $ and (b) $ \text{BS}_6 $. Generally, smaller MFAC yields better results regarding the volume conservation for all kernels. There are indistinct differences between $ \text{IB}_5 $ and $ \text{BS}_6 $ with the same MFAC

Figure 3.  Comparison of the influence of MFAC on the volume conservation of the pressure-loaded elastic band using $ \text{CBS}_{32} $ with MFAC = 0.2, 0.5, 0.75, and 1.0 (left to right). Compared with isotropic kernels (Figure 2), CBS kernels show a substantially smaller volume conservation error if MFAC is less than 1.0

Figure 4.  Deformation of the elastic band for $ \text{CBS}_{32} $ at $ t $ = 0.2 s for different MFAC values. The color map represents the elemental Jacobian values. At $ \text{MFAC} = 0.5 $, the elastic band deforms smoothly and the element Jacobians remain close to unity. As MFAC increases to 1.0 and 1.5, the element Jacobians deviate increasingly from unity. For $ \text{MFAC} \ge 1 $, as the simulation proceeds, eventually the band's deformation becomes nonphysical

Figure 5.  Comparison of stabilization treatments for $ \text{CBS}_{32} $ with MFAC = 0.5 ($ t $ = 10 s), visualized through element Jacobian distributions: (a) without a volumetric penalty term and using unmodified invariants, (b) including only a volumetric penalty term, (c) using modified invariants only, (d) using both a volumetric penalty and modified invariants

Figure 6.  Schematic of the compressed block benchmark

Figure 7.  Comparison of element Jacobians at t = 100 s for different kernels using two formulations: unmodified invariants without volumetric energy (left column) versus the fully stabilized approach with both treatments (right column). Note that while stabilization treatments substantially improve volume conservation for $ \text{IB}_3 $ and $ \text{BS}_3 $ kernels, they have minimal effect on $ \text{CBS}_{32} $ and $ \text{CBS}_{43} $ kernels, which maintain good volume conservation inherently

Figure 8.  Comparison of the vertical displacements at the center of the top surface with and without the volumetric stabilization treatments for different kernels with $ M = 32 $ and $ \text{MFAC} = 0.5 $ for the compressed block benchmark. The displacement of the top-mid point is similar for different kernels and treatments, though CBS kernels are much less sensitive to the treatments than IB and BS

Figure 9.  Grid convergence with different MFAC in terms of the displacement of the probed location. These are the results without modified invariants or volumetric energies. (1) Grid convergence is better for smaller MFACs for all kernels, but CBS kernels converge at a coarse mesh level. For large MFACs, CBS kernels do not converge to steady state values. (2) Wider kernels give a smaller volume error with the same MFAC

Figure 10.  Error norms of Jacobian against MFAC with $ N $ = 90. All kernel types show convergence under solid mesh refinement, with wider kernels yielding smaller errors. CBS kernels generally demonstrate superior volume conservation across different $ \text{MFAC} $ values, except for $ \text{IB}_6 $ which achieves the smallest overall error. CBS kernels show improved performance with decreasing $ \text{MFAC} $, while IB and BS kernels often perform better at larger $ \text{MFAC} $ values

Figure 11.  Setup of the Cook's membrane benchmark problem. The $ y $-displacement at the upper-right corner (indicated by the circle) is monitored for subsequent analysis. The initial configuration of the structure, denoted by $ \Omega_0^\text{s} $, is immersed within the fluid domain $ \Omega_0^\text{f} $. The entire computational domain is $ \Omega = \Omega_0^\text{s} \cup \Omega_0^\text{f} $, with zero fluid velocity enforced on $ \partial \Omega $

Figure 12.  Comparison of the Jacobians between the unmodified invariants with no volumetric energy (left column) and with both treatments (right column) for different kernels ($ \text{MFAC} $ = 0.5, $ t $ = 50 s). The special treatments improve $ \text{IB}_3 $ and $ \text{BS}_3 $ a lot for volume conservation but have little effect on $ \text{CBS}_{32} $ and $ \text{CBS}_{43} $

Figure 13.  Comparison of the displacement at the top-mid point of the compressed block with and without volumetric energy and stabilization treatments for different kernels. A clear difference is observed between cases with and without these treatments for IB and BS kernels, while CBS kernels show no distinguishable difference

Figure 14.  All kernels achieve grid convergence as the grid is refined across different MFAC values. CBS kernels demonstrate consistent performance and are less sensitive to MFAC, achieving convergence on coarser grids with better results at smaller MFAC values. In contrast, IB and BS kernels perform better with larger MFAC values and require finer meshes at smaller MFAC values to achieve results comparable to CBS. Among IB and BS kernels, those with wider supports converge more quickly than those with narrower supports

Figure 15.  Error norms of the Jacobian as a function of MFAC with a fixed background Cartesian grid spacing of $ h = \frac{13}{90} $ cm. IB and BS kernels show improved results with larger MFAC values, while CBS kernels display the opposite trend, with smaller MFAC values yielding lower errors. Additionally, different CBS kernels exhibit less discrepancy in error compared to IB and BS kernels

Figure 16.  Computational setup of the slanted channel flow problem (inclination angle $ \pi/6 $). The color map shows the velocity field computed using $ \text{CBS}_{32} $ kernel. The white vertical line at $ x = 0.5 $ indicates the location of velocity profile measurements, and the dots represent Lagrangian markers defining the top and bottom channel plates

Figure 17.  Comparison of velocity profiles at $ x = 0.5 $ for different kernel types. Kernels with narrower support regions show better accuracy in capturing peak velocities, with BS and CBS kernels of equal support performing similarly and outperforming their IB counterparts. The width of the numerical boundary layer decreases with decreasing kernel support size, with $ \text{CBS}_{21} $ demonstrating the thinnest numerical boundary layer among all tested kernels

Figure 18.  Schematic of the Turek-Hron benchmark

Figure 19.  Vorticity field field for the modified Turek-Hron benchmark generated using $ \text{CBS}_{32} $ with MFAC = 0.5. The gray colormap shows the displacement magnitude

Figure 20.  Vertical displacement of point $ A $ (as shown in Figure 18 under varying MFAC values for different kernels. Right panels show detailed oscillations during $ t = 6.5 $–7.0

Figure 21.  Representative cross section views of simulated axial velocity for the bioprosthetic heart valve model

Figure 22.  Comparison of (a) pressure waveforms and (b) flow rates for different kernels in the bioprosthetic heart valve model. All kernels show minimal discrepancy and good agreement with experimental data before valve flutter ($ t > 0.35 $ s). During early diastole (0.34 s to 0.39 s), CBS$ _{32} $ shows notable deviation from other kernels in both pressure and flow rate predictions. The wider CBS$ _{43} $ kernel demonstrates superior performance in capturing high-frequency flow characteristics associated with valve motion