A sharp interface direct-forcing immersed boundary method using the moving least square approximation

Based on the Moving Least Square (MLS) approximation, we propose a sharp interface direct-forcing immersed boundary method for incompressible fluid flows with fixed and moving boundaries. Since the domain of definition for the interpolation is highly flexible and the MLS approximation provides an accurate reconstructed approximation of the solution, the proposed method serves the precision and versatility required for a numerical framework to study the fluid-structure interaction problems. To alleviate the inherent spurious numerical oscillation that occurs in the calculated forces on moving boundary embedded objects, we use a two step predictor-corrector method in which the direct forcing terms are calculated after the predictor step and imposed on the whole solid domain as well as at the immediate vicinity of the solid boundary inside the fluid domain. To represent the arbitrary geometries, we adopt a signed distance function representation of the rigid body and an interpolation strategy to considerably reduce the computational cost of the re-initialization of the distance function at every time step. The potential capability of the method is demonstrated for both fixed and moving boundary problems. We also solve a sedimentation of a single cylinder to demonstrate the ability of the present method in solving fluid-structure interaction problems. These numerical experiments show that the proposed moving least square immersed boundary method can handle relatively complex moving problems while enjoying a versatile interpolation strategy and keeping the boundary conditions sharp with remarkable accuracy.