Improved stresses for the 4-node tetrahedral element

Journal of Computational Physics

et al. 2015

152

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.

Section: Mie-grüneisen Equation Of Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Journal of Computational Physics

et al. 2015

152

“…In this paper, a simple Dirichlet-based Lagrange multiplier projection procedure is employed where the value of the conjugate stresses (Σ F and Σ J ) on the boundary of the continuum is projected via the minimisation of a functional Π defined as follows 7 ,…”

Section: Two Dimensional Symmetric Equations For Elastodynamicsmentioning

confidence: 99%

“…Linear tetrahedral elements tend to be preferred due to the maturity of the existing unstructured mesh generators [5]. However, this approach presents a number of well-known shortcomings, namely, a reduced order of convergence for strains and stresses [6,7], poor performance in nearly incompressible bending dominated scenarios [8][9][10] and numerical instabilities in the form of shear locking, volumetric locking and spurious hydrostatic pressure fluctuations [11][12][13]. In addition, displacement-based methods used in conjunction with Newmark-type time integrators [14] have a tendency to introduce high frequency noise and accuracy is degraded once numerical artificial damping is employed [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity

Computer Methods in Applied Mechanics and Engineering

Lee

et al. 2015

257

This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors [1], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the use of a new tensor cross product greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov-Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney-1 Corresponding author: j.bonet@swansea.ac.uk 2 Corresponding author: a.j.gil@swansea.ac.uk 3 Corresponding author: c.h.lee@swansea.ac.uk Preprint submitted to Computer Methods in Applied Mechanics and EngineeringSeptember 16, 2014Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke-Hu-Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes.

“…second order for displacements but first order for stresses), requiring some form of stress recovery procedure if these are of interest [8,9]. Second, the performance of these formulations in bending dominated scenarios can be very poor [10,11] yielding unacceptable results.…”

Section: Introductionmentioning

confidence: 99%

A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Lee

Computer Methods in Applied Mechanics and Engineering

et al. 2014

198

A mixed second order stabilised Petrov-Galerkin finite element framework was recently introduced by the authors (C.H.Lee, A.J.Gil and J.Bonet. "Development of a stabilised Petrov-Galerkin formulation for conservation laws in Lagrangian fast solid dynamics", CMAME, 268:40-64, 2014). The new mixed formulation, written as a system of conservation laws for the linear momentum and the deformation gradient, performs extremely well in bending dominated scenarios (even when linear tetrahedral elements are used) yielding equal order of convergence for displacements and stresses. In this paper, this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties. First, a new conservation law for the Jacobian of the deformation is added into the system providing extra flexibility to the scheme. Second, a variationally consistent PetrovGalerkin stabilisation methodology is derived. Third, an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics. For completeness and ease of understanding, these three improvements are presented both in small and large strain regimes, studying the eigenstructure of the resulting systems. A series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors.