A stabilized Smoothed Particle Hydrodynamics, Taylor–Galerkin algorithm for soil dynamics problems

SUMMARY Modelling of failure under dynamic conditions in geomaterials with finite elements presents a series of complex problems, among which we can mention those of (i) volumetric locking, which results on higher failure loads, (ii) influence of mesh alignment, resulting to unrealistic failure surfaces, (iii) diffusion of the shear band over some element widths, (iv) nonoptimal propagation properties (numerical diffusion and dispersion), (v) fulfilling Babuska–Brezzi conditions when using the same order of interpolation for displacement and pressures in coupled problems and (vi) large deformation analysis. This paper is based on previous work done by the authors, who developed a mixed approximation based on (i) casting the dynamic problem in the form of a system of first order PDEs and (ii) using stresses and velocities as nodal variables. The equations were discretized following a Taylor–Galerkin algorithm, first in time using a Taylor expansion and then in space using Galerkin method. The model was limited to small deformations. The purpose of this paper is to show how Taylor–Galerkin method can be extended to meshless formulations, such as the SPH method. The algorithm consists of (i) discretizing in time using a Taylor series expansion complemented with integration of source terms using a Runge–Kutta scheme and then (ii) discretizing in space using the SPH method. It is shown how the proposed method keeps the advantages of the Taylor–Galerkin method in Finite Elements (good propagation properties and capturing of shear bands) and avoids the tensile instability. A set of test problems ranging from elastic propagation of a wave in a bar to failure of a slope on a cohesive softening material are used to assess the performance of the method. Copyright © 2011 John Wiley & Sons, Ltd.