Explicit boundary thickening direct forcing immersed boundary method

For simulating the fluid-structure interaction (FSI) problems with moving boundaries and large deformations, the original implicit direct forcing (IDF) immersed boundary method (IBM) has to construct and inverse a large correlation matrix which contains the local relationships between the Lagrangian points and their surrounding Eulerian points. Consequently, as the number of Lagrangian points increases, the IBM procedure experiences exponential growth in both memory consumption and computational time. In this work, we first proposed an implicit boundary thickening direct forcing (BTDF) scheme that combines the correlation between the thicknesses of the solid and fluid forcing shells with the IDF scheme. By employing Taylor series expansion, a second-order approximation is derived through error analysis, which allows the force density to be explicitly calculated with second-order accuracy without the construction and inversion of the correlation matrix. With this second-order approximation, we proposed an explicit BTDF scheme to achieve high computational efficiency and maintain similar accuracy as the implicit BTDF scheme. The comparisons of the memory consumption and computational cost between the explicit and implicit BTDF-IBs demonstrate that the explicit BTDF scheme is not only computational efficient, but also has memory saving performance. The proposed explicit and implicit BDTF-IBs integrated with the reconstructed lattice Boltzmann flux solver (RLBFS) (Lu et al., 2022) are validated by the numerical simulations of flow past an in-line oscillating cylinder, particle sedimentation, and flow past a 3D flexible plate. The results show that both the explicit and implicit BTDF-IBs provide almost identical solutions, and the instantaneous boundary velocity errors in the explicit BTDF scheme maintain the second-order accuracy.