Documents published in Scipedia

• 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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• Error estimation for adaptive computations on shell structures

  P. Diez, A. Huerta

  European Journal of Finite Elements (2000). Vol. 9 (1-3), pp. 49-66

  
  

  Abstract

  The finite element discretization of a shell structure introduces two kinds of errors: the error in the functional approximation and the error in the geometry approximation. The first is associated with the finite dimensional interpolation space and it is present in any finite element computation. The latter is associated with the piecewise polynomial approximation of a curved surface and is much more relevant in shell problems than in any other standard 2D or 3D computation. In the shells framework, formerly the quality control of the finite element solution has been carried out using flux projection a posteriori error estimators. This technique exhibits two main drawbacks: 1) the flux smoothing averages stress components over different elements that may have different physical meaning if the tangent planes are different and 2) the error estimation process uses only the approximate solution and hence, the discretized forces and the computational mesh: the data describing the real geometry and load is therefore not accounted for. In this work, a residual type error estimator introduced for standard 2D finite element analysis is generalized to shell problems. This allows to easily account for the real original geometry of the problem in the error estimation procedure and precludes the necessity of comparing generalized stress components between non coplanar elements. This estimator is based on approximating a reference error associated with a refined reference mesh. In order to build up the residual error equation the computed solution must be represented (projected) on the reference mesh. The use of thin shell finite elements requires a proper formulation in order to preclude shear locking. Following an idea of Donea and Lamain, the interpolation of the rotations is not unique and requires a particular technique to transfer the information from the computational mesh to the reference mesh. This technique is also developed in this work and may be used in any adaptive evolution problem where the solution must be transferred from one mesh to another.

  Abstract
  The finite element discretization of a shell structure introduces two kinds of errors: the error in the functional approximation and the error in the geometry approximation. [...]

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• Adaptive finite element strategies based on error assessment

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

  Int. J. Numer. Meth. Engng. (1999). Vol. 46 (10), pp. 1803-1818

  
  

  Abstract

  Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics.

  Abstract
  Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error [...]

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• A unified approach to remeshing strategies for finite element h-adaptivity

  P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (1999). Vol. 176 (1-4), pp. 215-229

  
  

  Abstract

  In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced different remeshing strategies yielding very different results. In this work these methods are included in a unified framework, emphasizing the role of the underlying hypotheses. Moreover, an objective tool to evaluate the accuracy of the resulting finite element solution is presented. Thus, a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions. The different remeshing strategies are compared with well-known numerical examples.

  Abstract
  In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced [...]

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• Vulnerabilidad sísmica y toma de decisiones

  J. Egozcue, G. Grande, P. Diez

  Revista Internacional de Cálculo de Estructuras (1997). Vol. 2 (1), pp. 35-58

  
  

  Abstract

  Vulnerability of civil structures and systems under seismic actions have been extensively used and studied in the last decade. However, the goals of those studies are very diverse and it have caused some confusion on the meaning and definition of vulnerability. Naturally, different scopes of vulnerability lead to different methodologies in its study which may be incompatible. The present aim is to analyse the definition of seismic vulnerability in the framework of Bayesian decision making. Three different examples of decision making related to seismic vulnerability, which are qualitatively different, guide the discussion. They have an increasing complexity: decisions on normative seismic design; decisions on insurances against seismic risk and, finally, those decisions that should be taken to lead alarm, mitigation and rescue systems for seismic disasters. The three analised examples correspond to three different decision making schemes, respectively named: a priori, a posteriori and preposterior. In each case the definition of seismic vulnerability depends on the concept of considered utility (cost-benefit), on the involved random states and the scale in which the phenomena are to be considered. Most of the concepts associated with seismic vulnerability are reviewed through these three examples. A conclusion is that seismic vulnerability studies should consider the associated decision making problem. Simultaneously, the study of methodological and estimation aspects on vulnerability are remarked in order to improve both guidelines for seismic design and disaster mitigation.

  Abstract
  Vulnerability of civil structures and systems under seismic actions have been extensively used and studied in the last decade. However, the goals of those studies are very [...]

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• Adaptive finite element strategies based on error assessment

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

  (1998). Research Report, Nº PI136

  
  

  Abstract

  Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or errors indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are elearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics.  

  Abstract
  Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or errors [...]

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