Mathematical Physics - Volume II - Numerical Methods

5.12 Summary

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5.12 Summary This chapter provides an overview of the development of the SPH method. Especial attention is given to the main shortcomings of the original form of the method namely consistency, tensile instability and zero energy modes. These are important to understand as the SPH method is increasingly used in engineering analysis, and without an understanding of the method inappropriate conclusions may be drawn from numerical results. The topics covered include: • A discussion of the kernel interpolation that forms the basis of the SPH spatial interpolation method. In kernel interpolation the variable at a particle is calcu lated from summing the contribution from all neighbouring particles. The use of a differentiable kernel function allows the spatial gradient of the variable to be approximated. • An overview of the conventional SPH forms of the conservation equations of Lagrangian continuum mechanics. It should be noted that in these forms the boundary is diffuse and not clearly defined, which is important to understand when modelling solids or liquids. • A summary of the kernel function, varying smoothing length and neighbour search algorithm that are commonly used in SPH implementations. • An example of the derivation of a correction necessary to assure first order consis tency is given. The conventional SPH method is not first order consistent and is not even zero order consistent except in the special case when particles are evenly distributed. • The introduction of corrected SPH required boundary conditions to rigorously treated. In the conventional SPH method the deficiency in neighbour particles at the boundary of the domain leads to an error in the interpolation that allows free surfaces to be approximately treated. • A summary of a stability analysis of SPH is presented and used to explain the so called tensile instability problem. This problem is relevant for solid mechanics simulations where the instability can lead to fracture occurring and preventing accurate analyses of problems involving material fracture. A few proposed solutions to this problem are described. Similar consideration is given with respect to the zero energy modes typical for the collocational SPH method. • In the SPH simulations of the softening materials, damage distribution is controlled by the smoothing domain used in the kernel interpolation. Consequently, the smoothing domain size represents a material characteristic length. The user can control the damage localisation process by varying the smoothing length h. Stress waves propagated through the softening zone. • The numerical results in Section 8 demonstrate that, in the problem considered, SPH performed as nonlocal method and did not suffer from the same instabilities as FE. The sensitivity of results to the spatial discretisation can be removed in SPH by adjusting the smoothing length appropriately, as the smoothing length represents a characteristic length that controls damage/softening localisation. Consequently, physically representative values for a material should be used when modelling damage.