An averaged nodal deformation gradient linear tetrahedral element for large strain explicit dynamic applications

Bonet, Javier; Marriott, Hannah R.; Hassan, O.

doi:10.1002/cnm.429

Cited by 124 publications

(117 citation statements)

References 7 publications

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“…Four different types of boundary conditions are considered: Fix (condition 1), Free (condition 2), Skew symmetric (condition 3) and Symmetric (condition 4). 6 field, contrary to the classical displacement based formulation [33,35,[37][38][39]. Crucially, the evolution of the fibre map (1b) must be advanced in time satisfying a set of compatibility conditions (known as involutions [62]), namely CURL(F ) = 0.…”

Section: Conservation Law Formulationmentioning

confidence: 99%

“…Extensive efforts have since been made to further develop this class of averaged nodal strain technologies with the use of various types of stabilisation [34][35][36][37][38][39]. Despite exhibiting geometric locking-free behaviours, the resulting formulation still suffers from spurious hydrostatic pressure fluctuations when simulating nearly incompressible materials.…”

Section: Introductionmentioning

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: Conservation Law Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Gil

Bonet

et al. 2015

Journal of Computational Physics

152

View full text Add to dashboard Cite

show abstract

“…Low-order (4-node) tetrahedral meshes are well-known to behave poorly for the analysis of incompressible solids, which has spurred research on special finite element formulations to overcome this deficiency [43,44,51,52,53,45,54,46]. In three dimensions it is possible to obtain poor tessellations (i.e, near slivers), which can lead to inaccurate results using finite elements.…”

Section: Numerical Examplesmentioning

confidence: 99%

Maximum-entropy meshfree method for incompressible media problems

Ortiz

Puso

Sukumar

2011

Finite Elements in Analysis and Design

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A novel maximum-entropy meshfree method that we recently introduced in Ref.[1] is extended to Stokes flow in two dimensions and to three-dimensional incompressible linear elasticity. The numerical procedure is aimed to remedy two outstanding issues in meshfree methods: the development of an optimal and stable formulation for incompressible media, and an accurate cell-based numerical integration scheme to compute the weak form integrals.On using the incompressibility constraint of the standard u-p formulation, a u-based formulation is devised by nodally averaging the hydrostatic pressure around the nodes. A modified Gauss quadrature scheme is employed, which results in a correction to the stiffness matrix that alleviates integration errors in meshfree methods, and satisfies the patch test to machine accuracy.The robustness and versatility of the maximum-entropy meshfree method is demonstrated in three-dimensional computations using tetrahedral background meshes for integration. The meshfree formulation delivers optimal * Corresponding author Email address: alortiz@ucdavis.edu (A. Ortiz) December 25, 2010 rates of convergence in the energy and L 2 -norms. Inf-sup tests are presented to demonstrate the stability of the maximum-entropy meshfree formulation for incompressible media problems. Preprint submitted to Elsevier

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“…An alternative procedure, which avoids the abovementioned difficulties, is the positional version of the FEM (see, for instance, [15], [16] and [9]). In this case, the seven nodal degrees of freedom of the shell are: three components of the current shell reference surface; three components of the final deformed unconstrained director; and the final linear strain rate across the thickness.…”

Section: Introductionmentioning

confidence: 99%

A shell finite element formulation to analyze highly deformable rubber-like materials

Pascon

Coda

2013

Lat. Am. j. solids struct.

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In this paper, a shell finite element formulation to analyze highly deformable shell structures composed of homogeneous rubberlike materials is presented. The element is a triangular shell of anyorder with seven nodal parameters. The shell kinematics is based on geometrically exact Lagrangian description and on the ReissnerMindlin hypothesis. The finite element can represent thickness stretch and, due to the seventh nodal parameter, linear strain through the thickness direction, which avoids Poisson locking. Other types of locking are eliminated via high-order approximations and mesh refinement. To deal with high-order approximations, a numerical strategy is developed to automatically calculate the shape functions. In the present study, the positional version of the Finite Element Method (FEM) is employed. In this case, nodal positions and unconstrained vectors are the current kinematic variables, instead of displacements and rotations. To model near-incompressible materials under finite elastic strains, which is the case of rubber-like materials, three nonlinear and isotropic hyperelastic laws are adopted. In order to validate the proposed finite element formulation, some benchmark problems with materials under large deformations have been numerically analyzed, as the Cook's membrane, the spherical shell and the pinched cylinder. The results show that the mesh refinement increases the accuracy of solutions, high-order Lagrangian interpolation functions mitigate general locking problems, and the seventh nodal parameter must be used in bending-dominated problems in order to avoid Poisson locking.Keywo rds large deformation analysis; homogeneous rubber-like materials; shell finite elements.

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An averaged nodal deformation gradient linear tetrahedral element for large strain explicit dynamic applications

Cited by 124 publications

References 7 publications

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Maximum-entropy meshfree method for incompressible media problems

A shell finite element formulation to analyze highly deformable rubber-like materials

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