On consistency and energy conservation in SPH

Energy conservation and consistency are not easy to achieve simultaneously in smoothed particle hydrodynamics (SPH). In this study, an efficient strategy is proposed to achieve energy conservation in an SPH framework with a consistent basis function. Herein, at every particle pair interaction, a correction term is introduced such that energy conservation is restored locally and, at the same time, the total variation of different variables due to the correction term is minimum. The final form of the proposed formulation is such that no additional computational effort is required and the simplicity of SPH is preserved. The theoretical error estimate of the proposed formulation is performed. The proposed scheme is also compared with benchmark SPH formulations in terms of the L2 error norm for representative functional derivatives both in regular and irregular particle distributions. Finally, the efficacy of the proposed formulation in conserving energy and maintaining accuracy is demonstrated via few elastic, elastic-plastic impact and fracture problems.

Reference :

  • Islam, M.R.I., Chakraborty, S. and Shaw, A. “On Consistency and Energy Conservation in Smoothed Particle Hydrodynamics" International Journal of Numerical Methods in Engineering, 116 (9), 601-632, 2018 (doi.org/10.1002/nme.5937).

    Computational model for material damage and fracture

    An SPH-based computational framework is developed for studying the deformation and failure behaviour of brittle, ductile and composite armours under impact loading. Crack propagation is modelled through a pseudo-spring analogy wherein the interacting particles are assumed to be connected through pseudo springs, and the interaction is continuously modified through an order-parameter based on the accumulated damage in the spring. At the onset of crack formation, i.e., when the accrued damage reaches the critical value, the spring breaks which results in termination of interaction between particles on either side of the spring. A key feature of the computational model is that it can capture arbitrary propagating cracks without introducing any special treatment such as discontinuous enrichment, particle-splitting, etc. This computational framework is used herein to study adiabatic shear plugging in metal plates when modelling penetration under impact loading by a flat-ended, cylindrical projectile; different failure modes in metal plates under impact by sharp-nosed projectiles; arbitrarily oriented cracks, fragmentation and conoid formation of ceramics, etc. Computed results are compared with the experimental observation given in the literature, and the efficacy of the framework is demonstrated.

    References :

  • Islam, M.R.I., Chakraborty, S., Shaw, A. and Reid, S. “A Computational Model for Failure of Ductile Material Under Impact" International Journal of Impact Engineering, 108, 334-347, 2017 (doi: 10.1016/j.ijimpeng.2017.04.005).
  • Chakraborty, S., Islam, M.R.I., Shaw, A. Ramachandra L.S. and Reid, S.R. “A Computational Framework for Modelling Impact Induced Damage in Ceramic and Ceramic-Metal Composite Structures" Composite Structures, 164, 263-276, 2017 (doi: 10.1016/j.compstruct.2016.12.064).

    Tracing snap through and snap back load displacement paths

    A systematic finite element analysis procedure has been developed for the nonlinear analysis of truss structures undergoing finite deformation. The Total Lagrangian formulation has been used and the complete Green-Lagrange strain tensor has been considered in this work. Only geometric nonlinearity has been considered. In order to identify the post-buckling path, the displacement control and the arc-length continuation scheme has been adopted. The procedures outlined by Wagner et al. (1988) and Wriggers et al. (1990) for the computation of stability points, has been embedded inside the arc-length continuation in the present work. Therefore, it has been possible to compute the critical points. The effectiveness of the above mentioned procedures for computation of critical points have also been reported. The effectiveness of the present analysis procedure has been illustrated through several numerical examples consisting of two and three dimensional trusses.


    Fluid simulation in SPH