Adaptive Total Lagrangian Eulerian SPH for high-velocity impacts

This paper presents a novel coupling between the Total Lagrangian and Eulerian formulations of the Smooth Particle Hydrodynamics (SPH) method. It is implemented such that a Total Lagrangian formulation can adaptively convert to an Eulerian formulation (referred to as AdapTLE in this work) and applied to high-velocity impact problems. The novel coupling makes use of the mixed kernel-and-gradient correction scheme, which is applied in order to improve consistency and conserve momentum. Additionally, the symmetrical terms, which are often found in SPH, are included in the derivation of the conservation equations to reduce the number of calculations required. A novel implementation of artificial viscosity is suggested for the Total Lagrangian formulation, which makes use of the kernel gradient in the undeformed state and therefore does not require neighbour lists to be updated. This form of artificial viscosity can be applied to a singularly Total Lagrangian formulation or a coupled one. Numerical examples including a patch test, and two-dimensional and three-dimensional high-velocity impact problems were simulated to evaluate this novel coupling method and demonstrate the benefits of AdapTLE, which was found to produce superior results to either a Total Lagrangian or Eulerian formulation independently. This was due to the retention of a Total Lagrangian formulation until severe distortions required the conversion to an Eulerian formulation, which significantly reduced the effect of the tensile instability in the latter.