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• Time‐accurate solution of stabilized convection–diffusion–reaction equations: II. Accuracy analysis and examples

  A. Huerta, B. Roig, J. Donea

  Comput. Methods Appl. Mech. Engrg., (2002). Vol. 18 (8), pp. 575-584

  
  

  Abstract

  The paper addresses the development of time‐accurate methods for solving transient convection–diffusion –reaction problems using finite elements. The accuracy characteristics of the spatially stabilized implicit multi‐stage time‐stepping schemes developed in a companion paper (Part I of this work) are analysed and compared here. This is done by means of a Fourier analysis. An important improvement is observed when the order of the method is increased. Moreover, the stabilization techniques proposed (streamline‐upwind Petrov–Galerkin (SUPG), Galerkin least‐square (GLS), sub‐grid scale (SGS) and least squares) do not degrade the phase accuracy. Finally, some examples are presented to show the applicability of these schemes. 

  Abstract
  The paper addresses the development of time‐accurate methods for solving transient convection–diffusion –reaction problems using finite elements. The accuracy [...]

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• Time-accurate solution of stabilized convection-difusion-reaction equations: I. Time and space discretization

  A. Huerta, J. Donea

  Comput. Methods Appl. Mech. Engrg., (2002). Vol. 18 (8), pp. 565-573

  
  

  Abstract

  The paper addresses the development of time-accurate methods for solving transient convection-diffusion-reaction problems using finite elements. Multi-stage time-stepping schemes of high accuracy are used. They are first combined with a Galerkin formulation to briefly recall the time-space discretization. Then spatial stabilization techniques are combined with high-order time-stepping schemes. Moreover, a least-squares formulation is also developed for these high-order time schemes combined with C0 finite elements (in spite of the diffusion operator and without reducing the strong form into a system of first-order differential equations). The weak forms induced by the SUPG, GLS, SGS and least-squares formulations are presented and compared. In a companion paper (Part II of this work), the phase and damping properties of the developed schemes are analysed and numerical examples are included to confirm the effectiveness of the proposed methodology for solving time-dependent convection-diffusion-reaction problems.

  Abstract
  The paper addresses the development of time-accurate methods for solving transient convection-diffusion-reaction problems using finite elements. Multi-stage time-stepping [...]

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• Error estimation and adaptivity for nonlinear FE analysis

  A. Huerta, A. Rodríguez-Ferran, P. Diez

  Int. J. Applied Mathematics and Computer Science (2002). Vol. 12 (1), pp. 59-70

  
  

  Abstract

  An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type error estimator and an unstructured quadrilateral mesh generator. The error estimator is based on simple local computations over the elements and the so-called patches. In contrast to other residual estimators, no flux splitting is required. The adaptive strategy is illustrated by means of a complex nonlinear problem: the failure analysis of a single-edge notched beam. The quasi-brittle response of concrete is modelled by means of a nonlocal damage model.

  Abstract
  An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type [...]

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• Arbitrary Lagrangian-Eulerian (ALE) formulation for hyperelastoplasticity

  A. Rodríguez-Ferran, A. Pérez-Foguet, A. Huerta

  Int. J. Numer. Meth. Engng. (2001). Vol. 53 (8), pp. 1831-1851

  
  

  Abstract

  The arbitrary Lagrangian–Eulerian (ALE) description in non‐linear solid mechanics is nowadays standard for hypoelastic–plastic models. An extension to hyperelastic–plastic models is presented here. A fractional‐step method—a common choice in ALE analysis—is employed for time‐marching: every time‐step is split into a Lagrangian phase, which accounts for material effects, and a convection phase, where the relative motion between the material and the finite element mesh is considered. In contrast to previous ALE formulations of hyperelasticity or hyperelastoplasticity, the deformed configuration at the beginning of the time‐step, not the initial undeformed configuration, is chosen as the reference configuration. As a consequence, convecting variables are required in the description of the elastic response. This is not the case in previous formulations, where only the plastic response contains convection terms. In exchange for the extra convective terms, however, the proposed ALE approach has a major advantage: only the quality of the mesh in the spatial domain must be ensured by the ALE remeshing strategy; in previous formulations, it is also necessary to keep the distortion of the mesh in the material domain under control. Thus, the full potential of the ALE description as an adaptive technique can be exploited here. These aspects are illustrated in detail by means of three numerical examples: a necking test, a coining test and a powder compaction test.

  Abstract
  The arbitrary Lagrangian–Eulerian (ALE) description in non‐linear solid mechanics is nowadays standard for hypoelastic–plastic models. An extension to hyperelastic–plastic [...]

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• Coupling finite elements and particles for adaptivity: an application to consistently stabilized convection–diffusion

  S. Fernández-Méndez, A. Huerta

  Lecture Notes in Computational Sciences and Engineering (2002). Vol. 26, pp. 117-129

  
  

  Abstract

  A mixed approximation coupling finite elements and mesh-less methods is presented. It allows selective refinement of the finite element solution without remeshing cost. The distribution of particles can be arbitrary. Continuity and consistency is preserved. The behaviour of the mixed interpolation in the resolution of the convection-diffusion equation is analyzed.

  Abstract
  A mixed approximation coupling finite elements and mesh-less methods is presented. It allows selective refinement of the finite element solution without remeshing cost. The [...]

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• Arbitrary Lagrangian-Eulerian simulation of powder compaction processes

  A. Pérez-Foguet, A. Rodríguez-Ferran, A. Huerta

  Int. J. Forming Processes (2001). Vol. 4 (3-4), pp. 535-549

  
  

  Abstract

  In this paper, a new strategy for the simulation of quasi-static cold compaction processes of powders is presented. A material model formulated within the framework of isotropic finite strain multiplicative hyperelastoplasticity is used. An elliptic plastic model expressed in terms of the Kirchhoff stresses and the relative density models the transition between the loose powder and the compacted sample. The Coulomb dry friction model is used to capture friction effects at powder-die contact. Excessive distortion of Lagrangian meshes due to large mass fluxes is usual in powder compaction problems. Moreover, Lagrangian approaches cannot deal properly with mass fluxes around sharp corners. For these reasons, an Arbitrary Lagrangian-Eulerian (ALE) formulation is used here. The present results illustrate that this approach allows simulating highly demanding powder compaction processes without mesh distortion and spurious oscillations in the results. Moreover, it is shown that the mass conservation principle is verified with a low relative error.

  Abstract
  In this paper, a new strategy for the simulation of quasi-static cold compaction processes of powders is presented. A material model formulated within the framework of isotropic [...]

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• Consistent tangent matrices for density-dependent finite plasticity models

  A. Pérez-Foguet, A. Rodríguez-Ferran, A. Huerta

  Int. J. Numer. Anal. Meth. Geomech. (2001). Vol. 25 (11), pp. 1045-1075

  
  

  Abstract

  The consistent tangent matrix for density-dependent plastic models within the theory of isotropic multiplicative hyperelastoplasticity is presented here. Plastic equations expressed as general functions of the Kirchhoff stresses and density are considered. They include the Cauchy-based plastic models as a particular case. The standard exponential return-mapping algorithm is applied, with the density playing the role of a fixed parameter during the nonlinear plastic corrector problem. The consistent tangent matrix has the same structure as in the usual density-independent plastic models. A simple additional term takes into account the influence of the density on the plastic corrector problem. Quadratic convergence results are shown for several representative examples involving geomaterial and powder constitutive models.

  Abstract
  The consistent tangent matrix for density-dependent plastic models within the theory of isotropic multiplicative hyperelastoplasticity is presented here. Plastic equations [...]

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• Consistent tangent matrices for substepping schemes

  A. Pérez-Foguet, A. Rodríguez-Ferran, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2001). Vol. 190 (35-36), pp. 4627-4647

  
  

  Abstract

  A very simple and general expression of the consistent tangent matrix for substepping time-integration schemes is presented. If needed, the derivatives required for the computation of the consistent tangent moduli can be obtained via numerical differentiation. These two strategies (substepping and numerical differentiation) lead to quadratic convergence in complex nonlinear inelasticity problems.

  Abstract
  A very simple and general expression of the consistent tangent matrix for substepping time-integration schemes is presented. If needed, the derivatives required for the computation [...]

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• Adaptive analysis based on error estimation for nonlocal damage models

  A. Rodríguez-Ferran, I. Arbós, A. Huerta

  Revue Européenne des Éléments Finis (2001). Vol. 10 (2,3,4), pp. 193-207

  
  

  Abstract

  An adaptive finite element strategy for nonlocal damage computations is presented. The proposed approach is based on the combination of a residual-type error estimator and quadrilateral h-remeshing. A distinctive feature of the error estimator is that it consists in solving simple local problems over elements and so-called patches. The paper focuses on how the nonlocality of the constitutive model should be accounted for when solving these local problems. It is shown that the nonlocal damage models must be slightly modified. The resulting adaptive strategy is illustrated by means of some numerical examples involving the single-edge notched beam test.

  Abstract
  An adaptive finite element strategy for nonlocal damage computations is presented. The proposed approach is based on the combination of a residual-type error estimator and [...]

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• Locking in the incompressible limit for the element-free Galerkin method

  A. Huerta, S. Fernández-Méndez

  Int. J. Numer. Meth. Engng. (2001). Vol. 51 (11), pp. 1361-1383

  
  

  Abstract

  Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis developed here shows that the number of non-physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non-physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element-free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example.

  Abstract
  Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis [...]

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