A finite volume cell‐centered Lagrangian hydrodynamics approach for solids in general unstructured grids

Abstract

SUMMARYA finite volume cell‐centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto‐plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie‐Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo‐elastic stress‐strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi‐dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J‐2 von Mises yield condition. The scheme presented in this work is second‐order accurate both in space and time. The suitability of the scheme is evinced by solving one‐ and two‐dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

“…It can be seen that the numerical dissipation is clearly reduced as the mesh is refined. The frequency and amplitude of the oscillations are in good agreement with the results reported in [1,48]. Comparison of the numerical total energy obtained using two CFD-like methodologies, namely JST and upwind FVM, is shown in Figure 16.…”

“…The main aim of this example is to evaluate the dissipation properties of the method in the linear elastic regime [1,48,92]. A plate of 6 cm length and 1 cm width is made of Beryllium [1,48] using a compressible Neo-Hookean material with properties ρ 0 = 1845 kg/m 3 , E = 3.1827 × 10 11 Pa and ν = 0.05390.…”

“…A plate of 6 cm length and 1 cm width is made of Beryllium [1,48] using a compressible Neo-Hookean material with properties ρ 0 = 1845 kg/m 3 , E = 3.1827 × 10 11 Pa and ν = 0.05390. The plate is free of constraints and has an initial velocity that excites the first flexural mode (see Figure 13) described by The plate is left oscillating at time t = 0.…”

“…The use of a mixed approach proved to be very efficient in large strain solid dynamics, circumventing the above-mentioned drawbacks for the traditional displacement based techniques. Early attempts at applying CFD-like numerical techniques in the context of displacement based computational solid dynamics are reported in references [2,[43][44][45][46][47][48]. Eulerian Finite Volume Godunov methods, typically used for modelling compressible gas dynamics, were employed to model plastic flows in solid dynamics [49][50][51].…”

A vertex centred Finite Volume Jameson–Schmidt–Turkel (JST) algorithm for a mixed conservation formulation in solid dynamics

“…Early attempts at applying the Finite Volume Method (FVM) in the context of solid dynamics date to references [5][6][7][8][9], using displacement based formulations for linear elasticity. Eulerian Finite Volume Godunov methods, classically used for modelling compressible gas dynamics, have been also adapted to model plastic flows in solid dynamics [10][11][12][13].…”