A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics

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: Discussionmentioning

confidence: 99%

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Journal of Computational Physics

et al. 2015

152

“…This formulation enables the use of low order elements without exhibiting volumetric locking in nearly incompressible situations [16] and, therefore, it is proposed as an alternative to nodal Finite Element formulations [17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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

Aguirre

Journal of Computational Physics

et al. 2014

223

Please cite this article in press as: M. Aguirre et al., A vertex centred Finite Volume Jameson-Schmidt-Turkel (JST) algorithm for a mixed conservation formulation in solid dynamics, Journal of Computational Physics (2013), http://dx.doi.org/10. 1016/j.jcp.2013.12.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A mixed formulation based upon the use of the linear momentum, the deformation gradient tensor and the total energy as conservation variables is discretised in space using linear triangles and tetrahedra in two-dimensional and three-dimensional computations, respectively. The scheme is implemented using central differences for the evaluation of the interface fluxes in conjunction with the Jameson-Schmidt-Turkel (JST) artificial dissipation term. The discretisation in time is performed by using a Total Variational Diminishing (TVD) two-stage Runge-Kutta time integrator. The JST algorithm is adapted in order to ensure the preservation of linear and angular momenta. The framework results in a low order computationally efficient solver for solid dynamics, which proves to be very competitive in nearly incompressible scenarios and bending dominated applications.

“…This mixed methodology is formulated in the form of a system of first order conservation laws [10,44,45], where the linear momentum p and the deformation gradient tensor F of the system are regarded as the main conservation variables of this mixed p-F approach. In [44], a robust and stable PG implementation is presented, derived with the help of the Variational Multi-Scale (VMS) method [46][47][48][49]. Unfortunately, in the case of extreme deformations in the incompressible limit (i.e.…”

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.