A first‐order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

Abstract

Citation for this version held on GALA:Haider, Jibran, Lee, Chun Hean, Gil, Antonio J. and Bonet, Javier (2016) A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme. London: Greenwich Academic Literature Archive. , a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Després [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications.

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.…”

A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies

“…From the numerical standpoint, in the context of incompressible elasticity, some authors resort to interpolation spaces for both geometry and pressure fields which a priori do not satisfy the Ladyzenskaja-Babuska-Brezzi (LBB) [54][55][56] condition but then recover the ellipticity of the problem via appropriate stabilisation techniques [41,47,53,[57][58][59][60][61][62][63]. In this regard, the Stream-Upwind-Petrov-Galerkin (SUPG) method can be utilised as a robust technique for the circumvention of the LBB condition [64].…”