An Hourglass Control Algorithm for Lagrangian Smooth Particle   Hydrodynamics
  This paper presents a stabilization scheme which addresses the rank-deficiency problem in meshless collocation methods for solid mechanics. Specifically, Smooth-Particle Hydrodynamics (SPH) in the Total Lagrangian formalism is considered. This method is rank-deficient in the sense that the SPH approximation of the deformation gradient is not unique with respect to the positions of the integration points. The non-uniqueness can result in the formation of zero-energy modes. If undetected, these modes can grow and completely dominate the solution. Here, an algorithm is introduced, which effectively suppresses these modes in a fashion similar to hour-glass control mechanisms in Finite-Element methods. Simulations utilizing this control algorithm result exhibit much improved stability, accuracy, and error convergence properties. In contrast to an alternative method which eliminates zero-energy modes, namely the use of additional integration points, the here presented algorithm is easy to implement and computationally very efficient. 
Georg C. Ganzenm\"uller
physics.comp-ph