An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Abstract

A vertex centred Jameson-Schmidt-Turkel (JST) finite volume algorithm was recently introduced by the authors (Aguirre et al., 2014 [1]) in the context of fast solid isothermal dynamics. The spatial discretisation scheme was constructed upon a Lagrangian two-field mixed (linear momentum and the deformation gradient) formulation presented as a system of conservation laws [2][3][4]. In this paper, the formulation is further enhanced by introducing a novel upwind vertex centred finite volume algorithm with three key novelties. First, a conservation law for the volume map is incorporated into the existing two-field system to extend the range of applications towards the incompressibility limit (Gil et al., 2014 [5]). Second, the use of a linearised Riemann solver and reconstruction limiters is derived for the stabilisation of the scheme together with an efficient edge-based implementation. Third, the treatment of thermo-mechanical processes through a Mie-Grüneisen equation of state is incorporated in the proposed formulation. For completeness, the study of the eigenvalue structure of the resulting system of conservation laws is carried out to demonstrate hyperbolicity and obtain the correct time step bounds for non-isothermal processes. A series of numerical examples are presented in order to assess the robustness of the proposed methodology. The overall scheme shows excellent behaviour in shock and bending dominated nearly incompressible scenarios without spurious pressure oscillations, yielding second order of convergence for both velocities and stresses.

A new Jameson–Schmidt–Turkel Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics

“…with E I is the I th unit vector of the Cartesian basis defined as: Notice that in the presence of non-smooth solutions, the above system (1) of local conservation laws must be accompanied by suitable jump conditions as described in References [1,35,36,42,43]. Moreover, it is clear from expressions (1b) and (1c) that two sets of involutions [49] need to be satisfied by the conservation variables F and H, that is:…”

“…References [35,36,[40][41][42][43][44] have considered simpler mixed-based systems {p, F } and {p, F , J}. These reduced mixed-based systems have proved robust.…”

“…For instance, Finite Volume based techniques, combined with either an upwinding Riemann solver [35,36] or a Jameson-Schmidt-Turkel (JST) stabilisation procedure [3,37]. In order to adapt these approaches to the field of solid dynamics, it is first necessary to reformulate the governing equations in the form of a system of first order conservation laws, as presented in References [1,2,31,36,[38][39][40][41][42][43][44][44][45][46].…”

“…Over the last few years, some of the authors of this paper have pursued the same {p, F } mixed conservationbased methodology spatially discretised using a wide variety of second order Finite Volume and Finite Element techniques. For instance, an Upwind Cell Centred Finite Volume Method (Upwind-CCFVM) [35,36], a JamesonSchmidt-Turkel Vertex Centred Finite Volume Method (JST-VCFVM) [42], an Upwind Vertex Centred Finite Volume Method (Upwind-VCFVM) [43], a two step Taylor-Galerkin Finite Element Method [44] and a stabilised PetrovGalerkin Finite Element Method (PG-FEM) [40] have been introduced.…”

Thermomechanical Numerical Simulation of Impacts on Elastic-Plastic Solids With the Finite Volume Method

“…The characteristic structure of the set of hyperbolic equations can be accounted for by the solution of a Riemann problem at interfaces between cells, and the same order of convergence is achieved for both the displacement and stress fields [2]. Several authors [3,4,5,6,7] have proposed many ways to simulate impacts on dissipative solid media such that elastic-plastic solids.…”