Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods

The paper discusses the problem of tension instability of particle-based methods such as smooth particle hydrodynamics (SPH)or corrected SPH (CSPH). It is shown that tension instability is a property of a continuum where the stress tensor is isotropic and the value of the pressure is a function of the density or volume ratio. The paper will show that, for this material model, the non-linear continuum equations fail to satisfy the stability condition in the presence of tension. Consequently, any discretization of this continuum will result in negative eigenvalues in the tangent stixed reference con>guration. The resulting formulation maintains the momentum preservation properties of its Eulerian equivalent. Finally a simple 1-D wave propagation example will be used to demonstrate that a stable solution can be obtained using Lagrangian CSPH without the need for any arti>cial viscosity