Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes

Journal of Computational Physics

Bonet

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: Conservation Law Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Journal of Computational Physics

Bonet

et al. 2015

152

“…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. Third, the presence of numerical instabilities in the form of volumetric locking, shear locking and spurious hydrostatic pressure fluctuations [12,13] when large Poisson's ratios are used.…”

Section: Introductionmentioning

confidence: 99%

“…However, the resulting approach was reported to behave poorly in bending dominated scenarios. To overcome this difficulty, reference [11] proposed the nodal based uniform strain tetrahedral element by applying a nodal averaging process to the whole small strain tensor. Reference [31] extended this application to the large strain regime with the idea of employing both a nodal average Jacobian and a nodal average deformation gradient in the calculation of the stress tensor.…”

Section: Introductionmentioning

confidence: 99%

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

Computer Methods in Applied Mechanics and Engineering

Lee

Bonet

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.

“…Additionally, following Gil and Ortigosa [1,2], we propose an extended Hu-Washizu [44,52,54,54,[69][70][71][71][72][73][74][75][76][77][78][79][80] mixed variational principle for nearly and truly incompressible scenarios. Considering now interpolation spaces which satisfy the LBB condition, a comparison of this enhanced methodology is carried out in this paper against the three-field formulation, where both LBB compliant and non-compliant interpolation spaces are considered.…”

Section: Introductionmentioning

confidence: 99%

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

Ortigosa

Computer Methods in Applied Mechanics and Engineering

Lee

2016

Please cite this article as: R. Ortigosa, A.J. Gil, C.H. Lee, A computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies, Comput. Methods Appl. Mech. Engrg. (2016), http://dx.doi.org/10. 1016/j.cma.2016.06.025 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 computational framework for large strain nearly and truly incompressible electromechanics based on convex multi-variable strain energies. AbstractThe series of papers published by Gil and Ortigosa [1-3] introduced a new convex multi-variable variational and computational framework for the numerical simulation of Electro Active Polymers (EAPs) in scenarios characterised by extreme deformations and/or extreme electric fields. Building upon this body of work, five key novelties are incorporated in this paper. First, a generalisation of the concept of multi-variable convexity to energy functionals additively decomposed into isochoric and volumetric components. This decomposition is typical of nearly and truly incompressible materials, group which represents the majority of the most relevant EAPs. Second, convexification or regularisation strategies are applied to a priori non-convex multi-variable isochoric functionals to yield physically meaningful convex multi-variable functionals. Third, based on the mixed variational principles introduced in Reference [1] in the context of compressible electro-elasticity, a novel extended Hu-Washizu mixed variational principle for nearly and truly incompressible scenarios is presented. From the computational standpoint, a static condensation procedure is applied in order to condense out the element-wise extra fields, the resulting formulation having a comparable cost to the more standard three-field displacement-potential-pressure mixed formulation. Fourth, the computational framework for the three-field mixed variational principle in nearly and truly incompressible scenarios is also presented. In this case, the novelty resides in the consideration of convex multi-