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• Imposing essential boundary conditions in mesh-free methods

  S. Fernández-Méndez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (12-14), pp. 1257-1275

  
  

  Abstract

  Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations.

  Abstract
  Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition [...]

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• Pseudo-divergence-free element free Galerkin method for incompressible fluid flow

  A. Huerta, Y. Vidal, P. Villon

  Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (12-14), pp. 1119-1136

  
  

  Abstract

  Incompressible modeling in finite elements has been a major concern since its early developments and has been extensively studied. However, incompressibility in mesh-free methods is still an open topic. Thus, instabilities or locking can preclude the use of mesh-free approximations in such problems. Here, a novel mesh-free formulation is proposed for incompressible flow. It is based on defining a pseudo-divergence-free interpolation space. That is, the finite dimensional interpolation space approaches a divergence-free space when the discretization is refined. Note that such an interpolation does not include any overhead in the computations. The numerical evaluations are performed using the inf–sup numerical test and two well-known benchmark examples for Stokes flow.

  Abstract
  Incompressible modeling in finite elements has been a major concern since its early developments and has been extensively studied. However, incompressibility in mesh-free [...]

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• Computing bounds for linear functionals of exact weak solutions to Poisson's equation

  A. Sauer-Budge, J. Bonet, A. Huerta, J. Peraire

  SIAM J. Numer. Anal. (2004). Vol. 42 (4), pp. 1610-1630

  
  

  Abstract

  We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of piecewise-polynomial linear functional outputs of the exact weak solution of the infinite-dimensional continuum problem with piecewise-polynomial forcing. The method results from exploiting the Lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem. Localization is achieved by Lagrangian relaxation and the bounds are computed by appeal to a local dual problem. The proposed method computes approximate Lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers. The computed bounds hold uniformly for any level of refinement, and in the asymptotic convergence regime of the finite element method, the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization. The elemental contributions to the bound gap are always positive and hence lend themselves to be used as adaptive indicators, as we demonstrate with a numerical example.

  Abstract
  We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of piecewise-polynomial linear functional outputs of the exact [...]

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• Numerical modelling of void inclusions in porous media

  A. Rodríguez-Ferran, J. Sarrate, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2004). Vol. 59 (4), pp. 577-596

  
  

  Abstract

  Numerical modelling of porous flow in a low‐permeability matrix with high‐permeability inclusions is a challenging task because the large ratio of permeabilities ill‐conditions the finite element system of equations. We propose a coupled model where Darcy flow is used for the porous matrix and potential flow is used for the inclusions. We discuss appropriate interface conditions in detail and show that the head drop in the inclusions can be prescribed in a very simple way. Algorithmic aspects are treated in full detail. Numerical examples show that this coupled approach precludes ill‐conditioning and is more efficient than heterogeneous Darcy flow.

  Abstract
  Numerical modelling of porous flow in a low‐permeability matrix with high‐permeability inclusions is a challenging task because the large ratio of permeabilities ill‐conditions [...]

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• Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

  Y. Vidal, P. Villon, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2003). Vol. 19 (9), pp. 725-735

  
  

  Abstract

  Locking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This paper proposes a new formulation that asymptotically suppresses locking for the element free Galerkin (EFG) method in incompressible limit, i.e. the so-called volumetric locking. Originally it was claimed that EFG did not present volumetric locking. However, recently, performing a modal analysis, the senior author has shown that EFG presents volumetric locking. In fact, it is concluded that an increase of the dilation parameter attenuates, but never suppresses, the volumetric locking and that, as in standard finite elements, an increase in the order of reproducibility (interpolation degree) reduces the relative number of locking modes. Here an improved formulation of the EFG method is proposed in order to alleviate volumetric locking.Diffuse derivatives are defined in the thesis of the second author and their convergence to the derivatives of the exact solution, when the radius of the support goes to zero (for a fixed dilation parameter), it is proved. Therefore, diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids corroborate this issue.

  Abstract
  Locking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This [...]

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• Numerical representation of the quality measures of triangles and triangular meshes

  J. Sarrate, J. Palau, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2003). Vol. 19 (7), pp. 551-561

  
  

  Abstract

  In this note a new procedure to represent the quality measure for triangles is proposed. The triangles are identified by their three angles and are represented in a bounded domain, called angle representation region, according to the area co-ordinates, which are common and well known by finite element users. The developed representation can also be used in order to visualize the characteristics of any quality measure. This new procedure is extended to graphically represent triangular meshes in the angle representation region.

  Abstract
  In this note a new procedure to represent the quality measure for triangles is proposed. The triangles are identified by their three angles and are represented in a bounded [...]

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• Efficient and accurate approach for powder compaction problems

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

  Computational Mechanics (2003). Vol. 30 (3), pp. 220-234

  
  

  Abstract

  In this paper, a new approach for powder cold compaction simulations is presented. A density-dependent plastic model within the framework of finite strain multiplicative hyperelastoplasticity is used to describe the highly nonlinear material behaviour; the Coulomb dry friction model is used to capture friction effects at die-powder contact; and an Arbitrary Lagrangian–Eulerian (ALE) formulation is used to avoid the (usual) excessive distortion of Lagrangian meshes caused by large mass fluxes. Several representative examples, involving structured and unstructured meshes are simulated. The results obtained agree with the experimental data and other numerical results reported in the literature. It is shown that, contrary to other Lagrangian and adaptive h-remeshing approaches recently reported for this type of problems, the present approach verifies the mass conservation principle with very low relative errors (less than 1% in all ALE examples and exactly in the pure Lagrangian examples). Moreover, thanks to the use of an ALE formulation and in contrast with other simulations, the presented density distributions do not present spurious oscillations.

  Abstract
  In this paper, a new approach for powder cold compaction simulations is presented. A density-dependent plastic model within the framework of finite strain multiplicative hyperelastoplasticity [...]

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• Time accurate consistently stabilized mesh-free methods for convection dominated problems

  A. Huerta, S. Fernández-Méndez

  Int. J. Numer. Meth. Engng. (2003). Vol. 56 (9), pp. 1225-1242

  
  

  Abstract

  The behaviour of high‐order time stepping methods combined with mesh‐free methods is studied for the transient convection–diffusion equation. Particle methods, such as the element‐free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high‐order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates.

  Abstract
  The behaviour of high‐order time stepping methods combined with mesh‐free methods is studied for the transient convection–diffusion equation. Particle methods, such [...]

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• Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

  Y. Vidal, P. Villon, A. Huerta

  European Journal of Computational Mechanics (2002). Vol. 11 (7-8), pp. 869-892

  
  

  Abstract

  Locking in finite elements has been a major concern since its early developments and has been extensively studied. However, locking in mesh-free methods is still an open topic. Until now the remedies proposed in the literature are extensions of already developed methods for finite elements. Here a new approach is explored and an improved formulation that asymptotically suppresses volumetric locking for the EFG method is proposed. The diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids and fluids corroborate this issue.

  Abstract
  Locking in finite elements has been a major concern since its early developments and has been extensively studied. However, locking in mesh-free methods is still an open topic. [...]

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• The efficient computation of bounds for functionals of finite element solutions in large strain elasticity

  J. Bonet, A. Huerta, J. Peraire

  Comput. Methods Appl. Mech. Engrg., (2002). Vol. 191 (43), pp. 4807-4826

  
  

  Abstract

  We present an implicit a posteriori finite element procedure to compute bounds for functional outputs of finite element solutions in large strain elasticity. The method proposed relies on the existence of a potential energy functional whose local minima, over a space of suitably chosen continuous functions, corresponds to the problem solution. The output of interest is cast as a constrained minimization problem over an enlarged discontinuous finite element space. A Lagrangian is formed were the multipliers are an adjoint solution, which enforces equilibrium, and hybrid fluxes, which constrain the solution to be continuous. By computing approximate values for the multipliers on a coarse mesh, strict upper and lower bounds for the output of interest on a suitably refined mesh, are obtained. This requires a minimization over a discontinuous space, which can be carried out locally at low cost. The computed bounds are uniformly valid regardless of the size of the underlying coarse discretization. The method is demonstrated with two applications involving large strain plane stress incompressible neo-Hookean hyperelasticity.

  Abstract
  We present an implicit a posteriori finite element procedure to compute bounds for functional outputs of finite element solutions in large strain elasticity. The method proposed [...]

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