The author presents a hybrid continuum-particle framework (HyPAM) consisting of three components. The first is a second-order interface tracking method: the Polygonal Area Mapping (PAM) method [Zhang and Liu 2008. J. Comput. Phys. 227(8):4063–4088], which represents material areas explicitly as piecewise polygons and utilizes polygon-clipping algorithms from computational geometry. In addition to its mesh-topology independence, the PAM method is free of numerical diffusion and tracks interfaces with or without singularities naturally and accurately. Both numerical experiments and theoretical analyses show that the PAM method is superior to existing VOF methods. The second component is a hybrid formulation for free-surface flows [Zhang and Liu 2009. J. Comput. Phys. 228(4):1312–1342]. A graph-based algorithm decomposes the water phase into a continuum zone and a particle zone, where the Euler equations and the free fall of rigid bodies are used as the governing equations, respectively and separately. The third component is the Explicit Jump Approximation (EJA) method for handling irregular solid boundaries in viscous flows [Zhang and Liu 2010. J. Comput. Phys. in press]. An analysis shows that the widely-used linear and quadratic ghost-cell approaches (GCA) are inconsistent and first-order accurate, respectively. As a remedy, the jump corrections at the solid-fluid interface are explicitly approximated and incorporated into the discretization of the Laplacian operator to obtain second-order convergence. Further developments of HyPAM include a fourth-order Navier-Stokes solver with adaptive mesh refinement (AMR), incorporating a particle method, and multi-material three-dimensional interface tracking.
Viscous free-surface flows past cylinders