On kernel functions for the Immersed Boundary Method in the Lattice Boltzmann framework

The Immersed Boundary Method (IBM) is a renowned approach for numerical fluid-structure interactions, whose methodology and numerical techniques have been refined and commonly integrated into classical Navier-Stokes solvers such as FDM, FEM, and FVM. However, the effects of kernel functions and their implementation are seldom discussed in the framework of the Lattice Boltzmann Method (LBM). The theoretical foundation of LBM has distinctions from classical Navier-Stokes solvers, which require a conscientious use of IBM in the LBM regime. Streaming and collision processes prevent fluid particles from traveling more than one node per time step. For direct-forcing IBM-LBM, commonly used kernel functions cover more than a single fluid node, which leads to unstable force fluctuations. In this work, we investigate force fluctuation issues caused by existing Dirac delta functions. We propose more compact kernel functions, such as linear shape functions, Gaussian kernels, and error functions, to overcome observed fluctuations. We validate the proposed kernels through 2D benchmark tests. In addition, the proposed kernels show better convergence regarding the number of boundary points, compared to existing delta-kernel functions.