Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme

Kluth, Gilles; Després, Bruno

doi:10.1016/j.jcp.2010.08.024

Cited by 69 publications

(133 citation statements)

References 40 publications

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“…Therefore, the study of the eigenvalue structure of the system of conservation laws becomes of paramount importance. The study of the eigenstructure of the hyperbolic system has been presented previously by other authors [2,3,5]. Extension of the analysis to include the conservation law of Jacobian and the consideration of thermo-mechanical processes is presented in this section.…”

Section: Eigenvalue Structurementioning

confidence: 99%

“…This ensures that F corresponds to the material gradient of a real mapping [2,3,42]. Notice that the material CURL of a second order two-point tensor is defined in the usual fashion by:…”

Section: Conservation Law Formulationmentioning

confidence: 99%

“…Several authors have presented an alternative p-F mixed formulations in both Lagrangian solid and gas dynamics [1][2][3][4][40][41][42]. Specifically, in references [1,3,40,41], the authors presented a mixed conservation law for applications in Lagrangian fast solid dynamics, which are spatially discretised using tailor-made CFD technology.…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Gil

Bonet

et al. 2015

Journal of Computational Physics

152

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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.

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Section: Eigenvalue Structurementioning

confidence: 99%

“…This ensures that F corresponds to the material gradient of a real mapping [2,3,42]. Notice that the material CURL of a second order two-point tensor is defined in the usual fashion by:…”

Section: Conservation Law Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Gil

Bonet

et al. 2015

Journal of Computational Physics

152

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

Sambasivan

Shashkov

Burton

2013

Numerical Methods in Fluids

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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.

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A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

Georges

Breil

Maire

2016

Numerical Methods in Fluids

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Summary The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.

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Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme

Cited by 69 publications

References 40 publications

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

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

A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

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