Local divergence-free velocity interpolation for the immersed boundary method using composite B-splines

This paper introduces an approach to improve volume conservation in the immersed boundary (IB) method using regularized delta functions derived from composite B-splines. These delta functions employ tensor product kernels using B-splines, whose polynomial degrees vary in normal and tangential directions based on the corresponding velocity component. Our method addresses the long-standing volume conservation issues in the conventional IB method, particularly evident in simulations of pressurized, closed membranes. We demonstrate that our approach significantly enhances volume conservation, rivaling the performance of the non-local Divergence-Free Immersed Boundary (DFIB) method introduced by Bao et al. while maintaining the local nature of the classical IB method. This avoids the computational overhead associated with the DFIB method's construction of an explicit velocity potential which requires additional Poisson solves. Numerical experiments show that sufficiently regular composite B-spline kernels can maintain initial volumes to within machine precision. We analyze 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 volume conservation errors primarily dominated by the time-stepping scheme's truncation error. This work offers a computationally efficient alternative for improving volume conservation in IB methods, particularly beneficial for large-scale, three-dimensional simulations. 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.