SUMMARY The consistency condition for the nodal derivatives in traditional meshfree Galerkin methods is only the differentiation of the approximation consistency (DAC). One missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms. In this paper, a consistency framework for the meshfree nodal derivatives including the DAC and the discrete divergence consistency (DDC) is proposed. The summation of the linear DDC over the whole computational domain leads to the so‐called integration constraint in the literature. A three‐point integration scheme using background triangle elements is developed, in which the corrected derivatives are computed by the satisfaction of the quadratic DDC. We prove that such smoothed derivatives also meet the quadratic DAC, and therefore, the proposed scheme possesses the quadratic consistency that leads to its name QC3. Numerical results show that QC3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability, convergence, accuracy, and efficiency among all the tested methods. Particularly, it shows a huge improvement for the existing linearly consistent one‐point integration method in some examples. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARYA time-discontinuous Galerkin ÿnite element method (DGFEM) for dynamics and wave propagation in non-linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the speciÿc P3-P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non-linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non-linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin ÿnite element method using the Newmark algorithm in the time domain.

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Purpose -To present a discrete particle model for granular materials. Design/methodology/approach -Starting with kinematical analysis of relative movements of two typical circular grains with different radii in contact, both the relative rolling and the relative sliding motion measurements at contact, including translational and angular velocities (displacements) are defined. Both the rolling and sliding friction tangential forces, and the rolling friction resistance moment, which are constitutively related to corresponding relative motion measurements defined, are formulated and integrated into the framework of dynamic model of the discrete element method. Findings -Numerical results demonstrate that the importance of rolling friction resistance, including both rolling friction tangential force and rolling friction resistance moment, in correct simulations of physical behavior in particulate systems; and the capability of the proposed model in simulating the different types of failure modes, such as the landslide (shear bands), the compression cracking and the mud avalanching, in granular materials. Research limitations/implications -Each grain in the particulate system under consideration is assumed to be rigid and circular. Do not account for the effects of plastic deformation at the contact points. Practical implications -To model the failure phenomena of granular materials in geo-mechanics and geo-technical engineering problems; and to be a component model in a combined discrete-continuum macroscopic approach or a two-scale discrete-continuum micro-macro-scopic approach to granular media. Originality/value -This paper develops a new discrete particle model to describe granular media in several branches of engineering such as soil mechanics, power technologies or sintering processes.

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