Composite B-spline regularized delta functions for the immersed boundary method: Divergence-free interpolation and...

Area of Science:

• Computational Fluid Dynamics
• Fluid-Structure Interaction
• Numerical Methods

Background:

• The immersed boundary (IB) method is widely used for fluid-structure interaction (FSI) problems.
• Conventional IB methods suffer from poor volume conservation, especially in simulations of pressurized, closed membranes.
• This limitation hinders accuracy in applications like biological flows and flexible structures.

Purpose of the Study:

• To enhance volume conservation in the immersed boundary method.
• To introduce a computationally efficient approach using regularized delta functions derived from composite B-splines.
• To provide an alternative to non-local methods like the Divergence-Free Immersed Boundary (DFIB) method.

Main Methods:

• Employed regularized delta functions constructed from composite B-splines with direction-dependent polynomial degrees.
• Utilized tensor product kernels, similar to conventional IB methods, but with enhanced B-spline properties.
• Maintained the local nature of the IB method, avoiding complex Poisson solves required by DFIB.

Main Results:

• Composite B-spline regularized delta functions significantly improve volume conservation.
• The method provides continuously divergence-free velocity interpolants and correctly represents pressure jump forces as discrete gradients.
• Simulations with $C^1$ regular composite B-splines achieve volume conservation comparable to DFIB, with errors dominated by time-stepping truncation.

Conclusions:

• The proposed approach offers a computationally efficient and accessible enhancement for IB methods.
• It effectively eliminates spurious flows by accurately modeling forces and interpolating velocities.
• This method is particularly beneficial for large-scale, 3D FSI simulations requiring high volume conservation.