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

Abstract

This paper presents an approach to enhance volume conservation in the immersed boundary (IB) method by employing regularized delta functions derived from composite B-splines. These delta functions are constructed using tensor product kernels, similar to the conventional IB method. However, the kernels are B-splines whose polynomial degree varies according to the normal and tangential directions of each velocity component. The conventional IB method, while effective for fluid-structure interaction applications, has long been challenged by poor volume conservation which is 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. Through a series of numerical experiments, we show that sufficiently regular composite B-spline kernels can 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 C1 regularity produce results comparable to the DFIB method in dynamic simulations, with errors in volume conservation primarily dominated by truncation error of the employed time-stepping scheme. This work offers a computationally efficient alternative for improving volume conservation in IB methods which is particularly beneficial for large-scale, three-dimensional simulations. The proposed approach requires only changing the functional form of the regularized delta function in an existing IB code, making it an accessible improvement for a wide range of applications in computational fluid dynamics and fluid-structure interaction.