A novel Arbitrary Lagrangian Eulerian Smooth Particle Hydrodynamics algorithm for nonlinear solid dynamics

This paper introduces a novel Smooth Particle Hydrodynamics (SPH) computational framework that incorporates an Arbitrary Lagrangian Eulerian (ALE) formalism, expressed through a system of first-order conservation laws. In addition to the standard material and spatial configurations, an additional (fixed) referential configuration is introduced. The ALE conservative framework is established based on the fundamental conservation principles, including mass, linear momentum and the first law of thermodynamics represented through entropy density. A key contribution of this work lies in the evaluation of the physical deformation gradient tensor, which measures deformation from material to spatial configuration through a multiplicative decomposition into two auxiliary deformation gradient tensors. Both of the deformation tensors are obtained via additional first-order conservation equations. Interestingly, the new ALE conservative formulation will be shown to degenerate into alternative mixed systems of conservation laws for solid dynamics: particle-shifting, velocity-shifting and Eulerian formulations. The framework also considers path- and/or strain rate-dependent constitutive models, such as isothermal plasticity and thermo-visco-plasticity, by integrating evolution equations for internal state variables. Another contribution of this paper is the evaluation of ALE motion (known as smoothing procedure) by solving a conservation-type momentum equation. This procedure is indeed useful for maintaining a regular particle distribution and enhancing solution accuracy in regions characterised by large plastic flows. The hyperbolicity of the underlying system is ensured and accurate wave speed bounds in the context of ALE description are presented, crucial for ensuring the stability of explicit time integrators. For spatial discretisation, a Godunov-type SPH method is employed and adapted. To guarantee stability from the semi-discretisation standpoint, a carefully designed numerical stabilisation is introduced. The Lyapunov stability analysis is carried out by assessing the time rate of the Ballistic energy of the system, aiming to ensure non-negative entropy production. In order to ensure the global conservation of angular momentum, we employ a three-stage Runge–Kutta time integrator together with a discrete angular momentum projection algorithm. Finally, a range of three dimensional benchmark problems are examined to illustrate the robustness and applicability of the framework. The developed ALE SPH scheme outperforms the Total Lagrangian SPH framework, particularly excelling in capturing plasticity regimes with optimal computational efficiency.