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The completeness of Smooth Particle Hydrodynamics (SPH) and its modiÿcations is investigated. Completeness, or the reproducing conditions, in Galerkin approximations play the same role as consistency in ÿnite-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 PetrovGalerkin 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. ? 1998 John Wiley & Sons, Ltd.

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A technique for incorporating discontinuities in derivatives into meshless methods is presented. The technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives. The special shape functions have compact support which results in banded matrix equations. The technique is described in element-free Galerkin context, but is easily applicable to other meshless methods and projections. Numerical results for problems in one and two dimensions are reported.1998 John Wiley & Sons, Ltd.