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• A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation

  F. Larsson, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (37-40), pp. 2383-2402

  
  

  Abstract

  In this contribution, we present an a posteriori error estimator for the incom- pressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point property of the problem, conventional error estimators de- veloped for pure minimization problems cannot be utilized straight-forwardly. The new estimator is built up by two key ingredients. At first, a computed error approximation, exactly fulfilling the continuity equation for the error, is obtained via local Dirichlet problems. Secondly, we adopt the approach of solving local equilibrated flux-free problems in order to bound the remaining, incompressible, error. In this manner, guaranteed upper and lower bounds, of the velocity ”energy norm” of the error as well as goal-oriented (linear) output functionals, with respect to a reference (overkill) mesh are obtained. In particular, it should be noted that this approach requires no computa- tion of hybrid fluxes. Furthermore, the estimator is applicable to mixed FE formulations using continuous pressure approximations, such as the Mini and Taylor-Hood class of elements. In conclusion, a few simple numerical examples are presented, illustrating the accuracy of the error bounds.

  Abstract
  In this contribution, we present an a posteriori error estimator for the incom- pressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point [...]

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• Modeling, with a unified level-set representation, of the expansion of a hollow in the ground under different physical phenomena

  R. Cottereau, P. Diez, A. Huerta

  Computational Mechanics (2010). Vol. 46 (2), pp. 315-327

  
  

  Abstract

  This paper builds on the flexibility of the level-set representation to model in a unified manner the expansion of a hollow in the ground under different physical phenomena. In particular, the dissolving action of a flow of water in a saturated soil, and that of a jet of particles of water in a non-saturated one, are represented in a common framework. In that manner, the complex geometrical evolutions of the hollow can be followed without the need for remeshing and this approach allows for a smooth transition between saturated and non-saturated models of the soil. Implementation and numerical difficulties are discussed and two applications of industrial interest are considered. The first one describes the modeling of the piping phenomenon, and the second one the evolution of an excavation created by a leaking duct.

  Abstract
  This paper builds on the flexibility of the level-set representation to model in a unified manner the expansion of a hollow in the ground under different physical phenomena. [...]

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• Accurate recovery-based upper error bounds for the extended finite element framework

  J. Ródenas, A. Estrada, P. Diez, F. Fuenmayor

  Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199, pp. 2607-2621

  
  

  Abstract

  This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests.

  Abstract
  This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended [...]

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• SUPG-based stabilization using a separated representations approach

  D. Gonzalez, L. Debeugny, E. Cueto, F. Chinesta, P. Diez, A. Huerta

  Int. J. of Material Forming (2010). Vol. 3 (1), pp. 883-886

  
  

  Abstract

  We have developed a new method for the construction of Streamline Upwind Petrov Galerkin (SUPG) stabilization techniques for the resolution of convection-diffusion equations based on the use of separated representations inside the Proper Generalized Decompositions (PGD) framework. The use of SUPG schemes produces a consistent stabilization adding a parameter to all the terms of the equation (not only the convective one). SUPG obtains an exact solution for problems in 1D, nevertheless, a generalization does not exist for elements of high order or for any system of convection-diffusion equations. We introduce in this paper a method that achieves stabilization in the context of Proper Generalzied Decomposition (PGD). This class of approximations use a representation of the solution by means of the sum of a finite number of terms of separable functions. Thus it is possible to use the technique of separation of variables in the context of problems of convection-diffusion that will lead to a sequence of problems in 1D where the parameter of stabilization is well known.

  Abstract
  We have developed a new method for the construction of Streamline Upwind Petrov Galerkin (SUPG) stabilization techniques for the resolution of convection-diffusion equations [...]

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• Error assessment and mesh adaptivity for regularized continuous failure models

  T. Pannachet, P. Diez, H. Askes, L. Sluys

  Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (17-20), pp. 961-978

  
  

  Abstract

  This paper deals with the adaptive finite element analysis of structural failure. A gradient-enhanced damage model has been chosen to simulate material degradation. Since this model is regularized in the post-peak regime, the finite element solution does not suffer from pathological mesh dependence and thus converges to an objective solution upon mesh refinement. However, the error analyses have shown that the error in the nonlocal equivalent strain field becomes dominant during the postpeak loading stages. The accuracy of the nonlocal equivalent strain field (and the corresponding damage quantity) also greatly influences the accuracy of the quantity of interest. Two error measures have been proposed. The goal-oriented error estimates have provided similar error distributions, although some small differences have been found in the softening regime. Objective error estimates, together with adaptive criteria, have been used to perform automated h-adaptivity during computation.

  Abstract
  This paper deals with the adaptive finite element analysis of structural failure. A gradient-enhanced damage model has been chosen to simulate material degradation. Since [...]

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• Assembling sparse matrices in MATLAB

  S. Zlotnik, P. Diez

  Commun. Numer. Meth. Engng (2010). Vol. 26 (6), pp. 760-769

  
  

  Abstract

  The assembly of sparse matrices is a key operation in finite element methods. In this study we analyze several factors that may have an influence on the efficiency of the assembly procedure. Different insertion strategies are compared using two metrics: a Cost function (the number of memory movements) and actual computing time. An improved algorithm implemented in MATLAB is proposed. It reduces both memory operations and computing time for all tested cases. The efficiency of the assembly process is found to be highly dependent on node and element numbering. The effect of the classic reverse Cuthill–McKee algorithm is, in most cases, positive and reduces computation costs. Finally, the case where a sparse matrix has to be re‐assembled at each time step is studied. The efficiency of the assembly is improved if the matrix pattern is entirely or partially inherited from previous steps.

  Abstract
  The assembly of sparse matrices is a key operation in finite element methods. In this study we analyze several factors that may have an influence on the efficiency of the [...]

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• Guaranteed energy error bounds for the Poisson equation using a flux-free approach: solving the local problems in subdomains

  N. Parés, H. Santos, P. Diez

  Int. J. Numer. Meth. Engng. (2009). Vol. 79 (10), pp. 1203-1244

  
  

  Abstract

  A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain based a posteriori error estimate which yields self-equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux-free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds.

  Abstract
  A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed [...]

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• Exact bounds for linear outputs of the advection-diffusion-reaction equation using flux-free error estimates

  N. Parés, P. Diez, A. Huerta

  Journal on Scientific Computing (SIAM) (2009). Vol. 31 (4), pp. 3064-3089

  
  

  Abstract

  The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The proposed approach is an alternative to the standard residual type estimators (hybrid-flux), circumventing the need of flux-equilibration following a fluxfree error estimation strategy. The presented estimator provides sharper estimates than the ones provided by both the standard hybrid-flux techniques and other flux-free techniques.

  Abstract
  The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. [...]

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• Hierarchical X-FEM for n-phase flow (n>2)

  S. Zlotnik, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (30-32), pp. 2329-2338

  
  

  Abstract

  The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more than two phases, the X-FEM has to be further eXtended. The proposed approach is presented in the case of a quasi-static Stokes n-phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases. A level set hierarchy allows describing triple junctions avoiding overlapping or “voids” between materials. Moreover, an enriched solution accounting for several simultaneous phases inside one element is proposed. The interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets. The computational implementation of this scheme involves calculating integrals in elements having several materials inside. An adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals. Examples of the hierarchical X-FEM approach are given for a n-phase Stokes problem in 2 and 3 dimensions.

  Abstract
  The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more [...]

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• Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

  R. Cottereau, P. Diez, A. Huerta

  Computational Mechanics (2009). Vol. 44 (4), pp. 533-547

  
  

  Abstract

  We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented

  Abstract
  We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are [...]

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