A unified high-order Eulerian method for continuum simulations of fluid flow and of elastic-plastic deformations in solids

We develop a new high-order method for Eulerian simulations of solids undergoing large, elastic-plastic deformations. Thermodynamically consistent constitutive relations of classical hyperelasticity are used to describe the behavior of solids, liquids and gases in a unified manner. Two kinematic formulations, one based on the inverse deformation gradient tensor, and a second based on the symmetric Finger tensor, are used for tracking large deformations in solids. Simulations based on the Finger tensor are shown to be equivalent to those using the full inverse deformation gradient tensor at much lower computational expense. The numerical algorithm employs a 10th-order compact finite-difference scheme for spatial discretization and a 4th-order Runge-Kutta time-stepping scheme. An improved form of the Localized Artificial Diffusivity (LAD) method is used for numerical regularization of shocks and contact discontinuities. We show that this high-order numerical framework, previously used for simulations of fluid flows, is suitable for problems involving large deformations in elastic-plastic solids as well. Particular emphasis is laid on the choice of the artificial diffusivity parameters in order to sufficiently capture shocks and discontinuities in all the aforementioned continuum media with minimal added dissipation. Test cases in one and two dimensions are shown to demonstrate the feasibility and accuracy of the proposed approach. In particular, this choice of algorithms is shown to lead to excellent numerical resolution properties, and to preserve mass-consistency and curl/compatibility constraints with high order of accuracy. Potential extensions of this numerical framework include application to multi-material problems, involving compressible flow of fluids coupled to elastic-plastic deformations in solids, that are of significant engineering interest.