Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading

TL;DR

This paper introduces a new regularized delta function based on composite B-splines for the immersed boundary method, significantly improving volume conservation by ensuring divergence-free velocity interpolation and gradient-preserving force spreading.

Contribution

It develops a composite B-spline regularized delta function that enhances volume conservation and maintains computational efficiency without requiring non-local Poisson solves.

Findings

Composite B-spline kernels of at least C^1 regularity achieve volume conservation comparable to DFIB.

The approach maintains local computations, avoiding additional Poisson solves.

Results show near-machine precision volume conservation in simulations.

Abstract

This paper presents an approach to enhance volume conservation in the immersed boundary (IB) method by using regularized delta functions derived from composite B-splines. The conventional IB method, while effective for fluid-structure interaction applications, has long been challenged by poor volume conservation, particularly evident in simulations of pressurized, closed membranes. We demonstrate that composite B-spline regularized delta functions significantly enhance volume conservation through two complementary properties: they provide continuously divergence-free velocity interpolants and maintain the gradient character of forces corresponding to mean pressure jumps across interfaces. By correctly representing these forces as discrete gradients, they eliminate a key source of spurious flows that typically plague immersed boundary computations. Our approach maintains the local nature of the classical IB method, avoiding the computational overhead associated with the non-local Divergence-Free Immersed Boundary (DFIB) method's construction of an explicit velocity potential which requires additional Poisson solves for interpolation and force spreading operations. We show that sufficiently regular composite B-spline kernels maintain initial volumes to within machine precision. We provide a detailed analysis of the relationship between kernel regularity and the accuracy of force spreading and velocity interpolation operations. Our findings indicate that composite B-splines of at least $C^1$ regularity produce results comparable to the DFIB method in dynamic simulations, with errors in volume conservation dominated by truncation error of the time-stepping scheme. The proposed approach requires minimal modifications to an existing IB code, making it an accessible improvement for a wide range of applications in computational fluid dynamics and fluid-structure interaction.