The aim of this chapter is to provide an in-depth presentation and survey of meshfree particle methods. Several particle approximations are reviewed; the SPH method, corrected gradient methods and the moving least squares (MLS) approximation. The discrete equations are derived from a collocation scheme or a Galerkin method. Special attention is paid to the treatment of essential boundary conditions. A brief review of radial basis functions is given because they play a signi¯cant role in meshfree methods. Finally, di®erent approaches for modelling discontinuities in meshfree methods are described.

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