Improvement on stress instability in smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) is a Lagrangian meshfree particle method. It has special advantages in modeling large deformation and free surface flow, and has been widely applied to different problems in engineering and science. However, the classical SPH suffers from stress instability which resticts its further development and applications. The fundamental reason of stress instability is that the stress state and the kernel do not match each other. For frequently used bell-shaped kernel function, in tensile state the attraction between particles increases as particle spacing decreases, thereby leading to tensile instability. In a compressible state, the repulsive force between particles increases, and then decreases as particle spacing decreases, thereby leading to compressible instability. In this paper is presented an approach to removing stress instability in SPH by proposing a new kernel function and a modified SPH discrete form. In the modified SPH, the force between particles is always repulsive and it increases as particle spacing decreases. Two numerical examples are given to test the proposed approachs, and the obtained numerical results clearly demonstrate that the new approach can eliminate stress instability effectively.