Documents published in Scipedia

• A stable XFEM formulation for multi-phase problems enforcing the accuracy of the fluxes through Lagrange multipliers

  P. Diez, R. Cottereau, S. Zlotnik

  Int. J. Numer. Meth. Engng. (2013). (Preprint) Vol. 96 (5), pp. 303-322

  
  

  Abstract

  When applied to diffusion problems in a multi‐phase setup, the popular extended finite element method (XFEM) strategy suffers from an inaccurate representation of the local fluxes in the vicinity of the interface. The XFEM enrichment improves the global quality of the solution but it is not enforcing any local feature to the fluxes. Thus, the resulting numerical fluxes in the vicinity of the interface are not realistic, in particular when conductivity ratios between the different phases are very high. This paper introduces an additional restriction to the XFEM formulation aiming at properly reproducing the features of the local fluxes in the transition zone. This restriction is implemented through Lagrange multipliers, and the stability of the resulting mixed formulation is tested satisfactorily through the Chapelle–Bathe numerical procedure. Several examples are presented, and the solutions obtained show a spectacular improvement with respect to the standard XFEM.

  Abstract
  When applied to diffusion problems in a multi‐phase setup, the popular extended finite element method (XFEM) strategy suffers from an inaccurate representation of the local [...]

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• An algorithm for mesh refinement and un-refinement in fast transient dynamics

  L. Casadei, P. Diez, F. Verdugo

  Int. J. Computational Meths. (2013). (preprint) Vol. 10 (4), pp. 1-31

  
  

  Abstract

  A procedure to locally refine and un-refine an unstructured computational grid of four-node quadrilaterals (in 2D) or of eight-node hexahedra (in 3D) is presented. The chosen refinement strategy generates only elements of the same type as their parents, but also produces so-called hanging nodes along non-conforming element-to-element interfaces. Continuity of the solution across such interfaces is enforced strongly by Lagrange multipliers. The element split and un-split algorithm is entirely integer-based. It relies only upon element connectivity and makes no use of nodal coordinates or other real-number quantities. The chosen data structure and the continuous tracking of the nature of each node facilitate the treatment of natural and essential boundary conditions in adaptivity. A generalization of the concept of neighbor elements allows transport calculations in adaptive fluid calculations. The proposed procedure is tested in structure and fluid wave propagation problems in explicit transient dynamics.

  Abstract
  A procedure to locally refine and un-refine an unstructured computational grid of four-node quadrilaterals (in 2D) or of eight-node hexahedra (in 3D) is presented. The chosen [...]

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• SUPG-based stabilization of Proper Generalized Decompositions for the high-dimensional Advection-Diffusion Equations

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

  Int. J. Numer. Meth. Engng. (2013). Vol. 94 (13), pp. 1216-1232

  
  

  Abstract

  This work is a first attempt to address efficient stabilizations of high dimensional advection–diffusion models encountered in computational physics. When addressing multidimensional models, the use of mesh-based discretization fails because the exponential increase of the number of degrees of freedom related to a multidimensional mesh or grid, and alternative discretization strategies are needed. Separated representations involved in the so-called proper generalized decomposition method are an efficient alternative as proven in our former works; however, the issue related to efficient stabilizations of multidimensional advection–diffusion equations has never been addressed to our knowledge. Thus, this work is aimed at extending some well-experienced stabilization strategies widely used in the solution of 1D, 2D, or 3D advection–diffusion models to models defined in high-dimensional spaces, sometimes involving tens of coordinates.

  Abstract
  This work is a first attempt to address efficient stabilizations of high dimensional advection–diffusion models encountered in computational physics. When addressing [...]

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• Computable exact bounds for linear outputs from stabilized solutions of the advection-diffusion-reaction equation

  N. Parés, P. Diez, A. Huerta

  Int. J. Numer. Meth. Engng. (2013). Vol. 93 (5), pp. 483-509 (Preprint)

  
  

  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 bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.

  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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• Enforcing interface flux continuity in enhanced XFEM: stability analysis

  P. Diez, S. Zlotnik, R. Cottereau

  Oberwolfach Reports (2012). Vol. 9 (1), pp. 523-526

  
  

  Abstract

  The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications.

  Abstract
  The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical [...]

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• Computable bounds of functional outputs in linear visco-elastodynamics

  F. Verdugo, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2012). (preprint) Vol. 245-246, pp. 313-330

  
  

  Abstract

  This work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco-elastodynamics. A novel expression for the error representation is introduced, alternative to the previous ones using the Cauchy–Schwarz inequality. The proposed formulation utilizes symmetrized forms of the error equations to derive error bounds in terms of energy error measures. The practical implementation of the method is based on constructing admissible fields for both the original problem and the adjoint problem associated with the quantity of interest. Here, the flux-free technique is considered to compute the admissible stress fields. The proposed methodology yields estimates with better quality than the ones based on the Cauchy–Schwarz inequality. In the studied examples the bound gaps obtained are approximately halved, that is the estimated intervals of confidence are reduced

  Abstract
  This work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco-elastodynamics. A novel expression for the error representation [...]

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• Adaptive reduced basis strategy based on goal oriented error assessment for stochastic problems

  E. Florentin, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2012). (preprint) Vol. 225, pp. 116-127

  
  

  Abstract

  In the framework of stochastic non-intrusive finite element modeling, a common practice is using Monte Carlo simulation. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. The different Monte Carlo samplings correspond to realizations of the random variables characterizing the stochastic behavior of the model. Thus, this requires solving a set deterministic problems with the same structure, that is with variations concerning the material parameters and the loading data. Consequently, the different problems to be solved are in practice similar to each other. The reduced basis strategy is therefore a sensible option to reduce computational cost, provided that the quality of the numerical solution is guaranteed. The paper introduces a goal-oriented strategy allowing to successively enrich the reduced basis along the Monte Carlo process. The method is based on assessing the error of the reduced basis solution with a residual estimate for the prescribed quantity of interest. The efficiency of the proposed approach, which is particularly important if the number of independent random variables is large, is illustrated in 1D and 2D mechanical examples.

  Abstract
  In the framework of stochastic non-intrusive finite element modeling, a common practice is using Monte Carlo simulation. The main drawback of this approach is the computational [...]

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• 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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