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• Subdomain-based flux-free a posteriori error estimators

  N. Parés, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (4-6), pp. 297-323

  
  

  Abstract

  A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks of standard residual type estimators, circumvents the need of flux-equilibration and results in a simple implementation that uses standard resources available in .nite element codes. This is specially interesting for 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. The bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. The presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest.

  Abstract
  A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks [...]

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• Accurate upper and lower error bounds by solving flux-free local problems in “stars”

  P. Diez, N. Parés, A. Huerta

  European Journal of Finite Elements (2004). Vol. 13 (5-7), pp. 497-507

  
  

  Abstract

  Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced in [MAC 00, CAR 99, MOR 03, PRU 02]. The new approach introduced here is based on using the estimated error function rather than the estimated error norms. Once the upper bounds are computed, also lower bounds for the error are obtained with little supplementary effort.

  Abstract
  Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced [...]

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• Convergence of finite elements enriched with meshless methods

  S. Fernández-Méndez, P. Diez, A. Huerta

  Numerische Mathematik (2003). Vol. 96 (1), pp. 43-59

  
  

  Abstract

  A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions based on a particle distribution. Convergence results are presented and proved.

  Abstract
  A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions [...]

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• A note on the convergence of the secant method for simple and multiple roots

  P. Diez

  Applied Numerical Letters (2003). Vol. 16 (8), pp. 1211-1215

  
  

  Abstract

  The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is super linear: the rate of convergence is set by the gold number. Nevertheless, this property holds only for simple roots. If the multiplicity of the root is larger than one, the convergence of the secant method becomes linear. This communication includes a detailed analysis of the secant method when it is used to approximate multiple roots. Thus, a proof of the linear convergence is shown. Moreover, the values of the corresponding asymptotic convergence factors are determined and are found to be also related with the golden ratio.

  Abstract
  The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is super linear: the rate of [...]

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• Goal-oriented adaptivity for shell structures: Error assessment and remeshig criteria

  P. Diez, I. Morata, A. Huerta

  European Journal of Finite Elements (2003). Vol. 12 (6), pp. 691-715

  
  

  Abstract

  The reliable computation of shell structures requires a tool to assess and control the quality of the finite element solution. For practical purposes, the quality of the numerical solution must be measured using a quantity of engineering interest rather than in the standard energy norm. However, the assessment of the error in an output of interest is based on a standard energy norm error estimator. The standard error estimator has to be applied to both the original problem (primal) and a dual problem related with the selected engineering quantity. In shells with assumed-strain models, the combination of primal and dual error estimation is performed differently than in the continuum mechanics case. Moreover, a part from the goal-oriented error estimator, the adaptive process requires a remeshing criterion. This work introduces an specific remeshing criterion for goal-oriented adaptivity and its particularization to the context of shell elements.

  Abstract
  The reliable computation of shell structures requires a tool to assess and control the quality of the finite element solution. For practical purposes, the quality of the numerical [...]

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• Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates

  P. Diez, N. Parés, A. Huerta

  Int. J. Numer. Meth. Engng. (2003). Vol. 56 (10), pp. 1465-1488

  
  

  Abstract

  Classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or reference) error. Lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs. This work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates. This lower bound may also be used to assess the effectivity of the former estimate and to improve it.

  Abstract
  Classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or reference) error. Lower bounds of the error [...]

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• Enrichment of the finite element interpolation with mesh-free methods

  A. Huerta, S. Fernández-Méndez, P. Diez

  Mathematical Modelling and Numerical Analysis (2002). Vol. 36 (6), pp. 1027-1042

  
  

  Abstract

  In the framework of meshless methods, the interpolation is based on a distribution of particles: it is not necessary to define connectivities. In these methods the interpolation can be easily enriched, increasing the number of particles (as in h-refinement of finite elements) or increasing the order of consistency (as in p-refinement of finite elements). However, comparing with finite elements, particle methods suffer from an increase in the computational cost, mainly due to the computation of the shape functions. In this paper, a mixed interpolation that combines finite elements and particles is presented. The objective is to take advantage of both methods. In order to define h- p refinement strategies an a priori error estimate is needed, and thus, some convergence results are presented and proved for this mixed interpolation

  Abstract
  In the framework of meshless methods, the interpolation is based on a distribution of particles: it is not necessary to define connectivities. In these methods the interpolation [...]

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• Probabilistic analysis of an a posteriori error estimator for Finite Elements

  P. Diez, J. Egozcue

  Mathematical Models and Methods in Applied Sciences (2001). Vol. 11 (5), pp. 841-854

  
  

  Abstract

  A residual type a posteriori error estimator for finite elements is analyzed using a new technique. In this case, the error estimate is the result of two consecutive projections of the exact error on two finite-dimensional subspaces. The analysis introduced in this paper is based on a probabilistic approach, that is, the idea is to assess the average value of the effectivity index (the ratio estimated error over exact error) by assuming the randomness of the exact error. The average value characterizes the mean behavior of the estimator and it is found to be related with some geometric properties of the subspaces. These geometric properties are obtained from the standard matrices of the linear systems arising in the formulation of the finite element method.

  Abstract
  A residual type a posteriori error estimator for finite elements is analyzed using a new technique. In this case, the error estimate is the result of two consecutive [...]

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• Adaptivity based on error estimation for viscoplastic softening materials

  P. Diez, M. Arroyo, A. Huerta

  Mechanics of Cohesive-Frictional Materials (2000). Vol. 5 (2), pp. 87-112

  
  

  Abstract

  This paper focuses on the numerical simulation of strain softening mechanical problems. Two problems arise: (1) the constitutive model has to be regular and (2) the numerical technique must be able to capture the two scales of the problem (the macroscopic geometrical representation and the microscopic behavior in the localization bands). The Perzyna viscoplastic model is used in order to obtain a regularized softening model allowing to simulate strain localization phenomena. This model is applied to quasistatic examples. The viscous regularization of quasistatic processes is also discussed: in quasistatics, the internal length associated with the obtained band width is no longer only a function of the material parameters but also depends on the boundary value problem (geometry and loads, specially loading velocity). An adaptive computation is applied to softening viscoplastic materials showing strain localization. As the key ingredient of the adaptive strategy, a residual type error estimator is generalized to deal with such highly nonlinear material model. In several numerical examples the adaptive process is able to detect complex collapse modes that are not captured by a first, even if fine, mesh. Consequently, adaptive strategies are found to be essential to detect the collapse mechanism and to assess the optimal location of the elements in the mesh.

  Abstract
  This paper focuses on the numerical simulation of strain softening mechanical problems. Two problems arise: (1) the constitutive model has to be regular and (2) the numerical [...]

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• Error estimation including pollution assessment for nonlinear finite element analysis

  A. Huerta, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2000). Vol. 181 (1-3), pp. 21-41

  
  

  Abstract

  A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure. Pollution errors are taken into account by a feedback strategy, that is, an error estimate based on local computations is used as the input of the pollution analysis. This estimator is used in the frame of an adaptive procedure. Numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions. Moreover, one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms.

  Abstract
  A residual type error estimator for nonlinear finite element analysis is introduced. This error estimator solves local problems avoiding both the computation of the flux jumps [...]

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