The nonlocal, local and mixed forms of the SPH method

From its early days the SPH method has been criticised for its shortcomings namely tensile instability and consistency. Without thorough understanding of the method attempts were made to make the classical SPH method consistent and stable which resulted in the local and Total Lagrangian forms of SPH similar to the finite element method. In this paper we derived and analysed a consistent nonlocal SPH which has similarity with Bazant's imbricate continuum. In addition, the paper provides comparison and discussion of different SPH forms including: Classical SPH, Nonlocal, Local and Mixed SPH. The partition of unity approach was used to define the following two mixed forms: Local–Nonlocal and Local–Classical SPH. These mixed forms were intended for modelling of physical processes characterised with local and nonlocal effects (local and nonlocal constitutive equations), e.g. progressive damage and failure. The stabilising effect of the Local form on the Classical SPH, which is inherently unstable (tensile instability), are also illustrated. The stability analysis, presented in appendices A and B, demonstrate stability of the continuous and discrete form of the nonlocal SPH based on Eulerian kernels for elastic continuum.