Documents published in Scipedia

• Modeling steel fiber reinforced concrete: numerical immersed boundary approach and a phenomenological mesomodel for concrete-fiber interaction

  A. Pros, P. Diez, C. Molins

  Comput. Methods Appl. Mech. Engrg., (2012). (preprint) Vol. 90 (1), pp. 65-86

  
  

  Abstract

  Steel fiber reinforced concrete (SFRC) allows overcoming brittleness and weakness under tension, the main drawbacks of plain concrete. The influence of the fibers on the behavior of SFRC depends on their shape, length, slenderness, and also on their orientation and distribution into the plain concrete. The goal of this paper is to develop an ad hoc numerical strategy to account for the contribution of the fibers in the simulation of the mechanical response of SFRC. In the model presented, the individual fibers immersed in the concrete bulk are accounted for in their actual location and orientation. The selected approach is based on the ideas introduced in the immersed boundary (IB) methods. These methods were developed to account for 1D (or 2D) solids immersed in 2D (or 3D) fluids. Here, the concrete bulk is playing the role of the fluid and the cloud of steel fibers is acting as the immerse boundary (that is, a 1D structure in a 2D or 3D continuous). Thus, the philosophy of the IB methodology is used to couple the behavior of the two systems, the concrete bulk and fiber cloud, precluding the need of matching finite element meshes. Note that, considering the different size scales and the intricate geometry of the fiber cloud, the conformal matching of the meshes would be a restriction resulting in a practically unaffordable mesh. In the proposed approach, the meshes of the concrete bulk and fiber cloud are independent, and the models are coupled imposing displacement compatibility and equilibrium of the two systems. In the applications presented here, the concrete bulk is modeled using a standard nonlinear damage model. The constitutive model for the fibers is designed to account for the complex interaction between fibers and concrete. The fiber models are based on the previous investigations describing the concrete-fiber interaction and its dependence on the factors identified to be relevant: shape of the fiber (straight or hooked) and angle between the fiber and crack plane.

  Abstract
  Steel fiber reinforced concrete (SFRC) allows overcoming brittleness and weakness under tension, the main drawbacks of plain concrete. The influence of the fibers on the behavior [...]

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• Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation

  L. Steffens, N. Parés, P. Diez

  Int. J. Numer. Meth. Engng. (2011). Vol. 86 (10), pp. 1197-1224

  
  

  Abstract

  An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in [28]. In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post-processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in [28] is based in a polynomial least squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem.

  Abstract
  An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension [...]

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• Numerical modeling of the Double Punch Test for plain concrete

  A. Pros, P. Diez, C. Molins

  Int. J. Solids and Structures (2011). Vol. 48 (7-8), pp. 1229-1238

  
  

  Abstract

  Double punch test is used to indirectly assess the tensile strength of plain concrete, ft. For this normalized test, the tensile strength is obtained as a function of the failure load, P, which is expressed as ft = F(P). Different authors have proposed different expressions for the relation F(.), yielding scattered values of ft. None of these alternatives is universally recognized as being more suitable than the others. In fact, these expressions are mainly based on elastic models considering the maximum tensile stress under the load P and ft is obtained as an output of the linear model. A numerical simulation allows using models in which ft is an input of the material model and the corresponding failure load P is obtained associated with each value of ft. In the present work, double punch test is simulated numerically considering two alternatives for modeling plain concrete accounting for damage and cracking: (a) the nonlocal Mazars damage model and (b) an heuristic crack model including joint elements in an a priori defined crack pattern. Numerical results are validated with experimental data and compared with the analytical expressions available in the literature.

  Abstract
  Double punch test is used to indirectly assess the tensile strength of plain concrete, ft. For this normalized test, the tensile strength is obtained as a function of the [...]

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• 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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