Stability analysis of particle methods with corrected derivatives

The stability of discretizations by particle methods with corrected derivatives is analyzed. It is shown that the standard particle method (which is equivalent to the element-free Galerkin method with an Eulerian kernel and nodal quadrature) has two sources of instability: 1. rank deficiency of the discrete equations, and 2. distortion of the material instability. The latter leads to the so-called tensile instability. It is shown that a Lagrangian kernel with the addition of stress points eliminates both instabilities. Examples that verify the stability of the new formulation are given.