An entropy-stable updated reference Lagrangian smoothed particle hydrodynamics algorithm for thermo-elasticity and thermo-visco-plasticity

Abstract

This paper introduces a novel upwind Updated Reference Lagrangian Smoothed Particle Hydrodynamics (SPH) algorithm for the numerical simulation of large strain thermo-elasticity and thermo-visco-plasticity. The deformation process is described via a system of first-order hyperbolic conservation laws expressed in referential description, chosen to be an intermediate configuration of the deformation. The linear momentum, the three incremental geometric strains measures (between referential and spatial domains), and the entropy density of the system are treated as conservation variables of this mixed coupled approach, thus extending the previous work of the authors in the context of isothermal elasticity and elasto-plasticity. To guarantee stability from the SPH discretisation standpoint, appropriate entropy-stable upwinding stabilisation is suitably designed and presented. This is demonstrated via the use of the Ballistic free energy of the coupled system (also known as Lyapunov function), to ensure the satisfaction of numerical entropy production. An extensive set of numerical examples is examined in order to assess the applicability and performance of the algorithm. It is shown that the overall algorithm eliminates the appearance of spurious modes (such as hour-glassing and non-physical pressure fluctuations) in the solution, typical limitations observed in the classical Updated Lagrangian SPH framework.