Gaussian-Like Immersed Boundary Kernels with Three Continuous Derivatives and Improved Translational Invariance

TL;DR

This paper introduces new 5-point and 6-point immersed-boundary kernels with three continuous derivatives that significantly enhance translational invariance in fluid-structure interaction simulations.

Contribution

The paper develops novel IB kernels with higher smoothness and improved translational invariance, advancing numerical accuracy and efficiency in immersed boundary methods.

Findings

New 5-point and 6-point kernels with $ ext{C}^3$ continuity.

Significant improvement in translational invariance.

Enhanced numerical accuracy in fluid-structure simulations.

Abstract

The immersed boundary (IB) method is a general mathematical framework for studying problems involving fluid-structure interactions in which an elastic structure is immersed in a viscous incompressible fluid. In the IB formulation, the fluid described by Eulerian variables is coupled with the immersed structure described by Lagrangian variables via the use of the Dirac delta function. From a numerical standpoint, the Lagrangian force spreading and the Eulerian velocity interpolation are carried out by a regularized, compactly supported discrete delta function, which is assumed to be a tensor product of a single-variable immersed-boundary kernel. IB kernels are derived from a set of postulates designed to achieve approximate grid translational invariance, interpolation accuracy and computational efficiency. In this note, we present new 5-point and 6-point immersed-boundary kernels that are $\mathscr{C}^3$ and yield a substantially improved translational invariance compared to other common IB kernels.