Variable resolution for SPH in three dimensions: Towards optimal splitting and coalescing for dynamic adaptivity

As smoothed particle hydrodynamics (SPH) becomes increasingly popular for complex flow analysis the need to improve efficiency particularly for 3-D problems is becoming greater. Automatic adaptivity with variable particle size is therefore desirable. In this paper, a novel 3-D splitting and coalescing algorithm is developed which minimises density error while conserving both mass and momentum using a variational principle. Accuracy is increased in refined areas unaffected by coarser particle distributions elsewhere. For particle splitting, the key criteria are the number of split (daughter) particles, their distribution, spacing and kernel size. Four different splitting arrangements are investigated including a cubic stencil with 8 particles, a cubic stencil with an additional 6 located at the face centres, an icosoahedron-shaped arrangement with 14 particles, and a dodecahedron-shaped arrangement with 20 particles where particles are located at the vertices. The error analysis also examines whether retaining a particle at the centre of the arrangement is necessary revealing that regardless of the stencil adopted, to minimize the density error a daughter particle should be placed at the same position of the original particle. The optimum configuration is found to be the iconsahedron-shaped arrangement while commonly used smoothing kernels such as the cubic and quintic splines and Wendland produce similar density errors, so that the optimal refinement stencil is effectively independent of the kernel choice. A new 3-D coalescing scheme completes the algorithm such that the particle resolution can be either increased or reduced locally. The SPH splitting and coalescing scheme, is tested with Poiseuille flow showing negligible loss of convergence accuracy in the refined area and the lid driven cavity for a wide range of Reynolds number showing good agreement with reference solutions again with local accuracy defined by particle distribution