On the completeness of meshfree particle methods

The completeness of Smooth Particle Hydrodynamics (SPH) and its modi cations is investigated. Com- pleteness, or the reproducing conditions, in Galerkin approximations play the same role as consistency in nite-di erence approximations. Several techniques which restore various levels of completeness by satisfying reproducing conditions on the approximation or the derivatives of the approximation are examined. A Petrov{ Galerkin formulation for a particle method is developed using approximations with corrected derivatives. It is compared to a normalized SPH formulation based on kernel approximations and a Galerkin method based on moving least-square approximations. It is shown that the major di erence is that in the SPH discretization, the function which plays the role of the test function is not integrable. Numerical results show that approximations which do not satisfy the completeness and integrability conditions fail to converge for linear elastostatics, so convergence is not expected in non-linear continuum mechanics.