Documents published in Scipedia

• Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies

  L. Steffens, N. Parés, P. Diez

  Computational Mechanics (2011). Vol. 47 (6), pp. 681-699

  
  

  Abstract

  This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: 1) different error representations, 2) assessment of the dispersion error, and 3) different remeshing criteria.

  Abstract
  This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character [...]

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• Numerical modeling of erosion using an improvement of the extended finite element method

  R. Cottereau, P. Diez

  European Journal of Environmental and Civil Engineering (2011). Vol. 15 (8), pp. 1187-1206

  
  

  Abstract

  We present in this paper a numerical model of the erosion of a soil that accounts for both the flow in the open fluid and the flow of fluid through the porous soil. The interface between the open fluid and the soil is represented using a level-set function, and the erosion is controlled by the shear stress vector. The evaluation of the approximate value of this gradient is particularly focused on, and an improved method, called XFE+ method, is presented. Numerical results in 2D and 3D illustrate the accuracy and the potentiality of this method.

  Abstract
  We present in this paper a numerical model of the erosion of a soil that accounts for both the flow in the open fluid and the flow of fluid through the porous soil. The interface [...]

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

  R. Cottereau, L. Chamoin, P. Diez

  Mécanique & Industries (2010). Vol. 11 (3-4), pp. 249-254

  
  

  Abstract

  We discuss, in this paper, a common flux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they refer to the exact solution of the continuous equations, rather than to some FE computation over a refined esh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements

  Abstract
  We discuss, in this paper, a common flux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the [...]

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• Error estimation and quality control

  P. Diez, N. Parés, A. Huerta

  Encyclopedia of Aerospace Engineering (2010). Vol. 3 (15), pp. 1725-1734

  
  

  Abstract

  This chapter provides a description of the error assessment tools available for general finite element analysis, in particular those for solid and structural mechanics. The text focuses on goal-oriented error estimation, in terms of general quantities of interest rather than on energy norm. However, the energy norm estimates are also discussed because they are seen as basic tools that must be used for assessing the error in arbitrary functional outputs of the solution. Attention is paid to the classification of the different methodologies and their main characteristics.

  Abstract
  This chapter provides a description of the error assessment tools available for general finite element analysis, in particular those for solid and structural mechanics. The [...]

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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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