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• Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

  F. Verdugo, N. Parés, P. Diez

  Computational Mechanics (2014). Vol. 54 (2), pp. 331-352

  
  

  Abstract

  This article presents a space-time adaptive strategy for transient elastodynamics. The method aims at computing an optimal space-time discretization such that the computed solution has an error in the quantity of interest below a user-defined tolerance. The methodology is based on a goal-oriented error estimate that requires accounting for an auxiliary adjoint problem. The major novelty of this paper is using modal analysis to obtain a proper approximation of the adjoint solution. The idea of using a modal-based description was introduced in a previous work for error estimation purposes. Here this approach is used for the first time in the context of adaptivity. With respect to the standard direct time-integration methods, the modal solution of the adjoint problem is highly competitive in terms of computational effort and memory requirements. The performance of the proposed strategy is tested in two numerical examples. The two examples are selected to be representative of different wave propagation phenomena, one being a 2D bulky continuum and the second a 2D domain representing a structural frame.

  Abstract
  This article presents a space-time adaptive strategy for transient elastodynamics. The method aims at computing an optimal space-time discretization such that the computed [...]

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• Error assessment in structural transient dynamics

  F. Verdugo, N. Parés, P. Diez

  Archives of Comp. Meths. Engng. (2014). Vol. 21 (1), pp. 59-90

  
  

  Abstract

  This paper presents in a unified framework the most representative state-of-the-art techniques on a posteriori error assessment for second order hyperbolic problems, i.e., structural transient dynamics. For the sake of presentation, the error estimates are grouped in four types: recovery-based estimates, the dual weighted residual method, the constitutive relation error method and error estimates for timeline-dependent quantities of interest. All these methodologies give a comprehensive overview on the available error assessment techniques in structural dynamics, both for energy-like and goal-oriented estimates.

  Abstract
  This paper presents in a unified framework the most representative state-of-the-art techniques on a posteriori error assessment for second order hyperbolic problems, i.e., [...]

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• Modal based goal-oriented error assessment for timeline-dependent quantities in transient dynamics

  F. Verdugo, N. Parés, P. Diez

  Int. J. Numer. Meth. Engng. (2013). (preprint) Vol. 95 (8), pp. 685-720

  
  

  Abstract

  This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline-dependent quantities) is proposed. These quantities are scalar time-dependent outputs of the transient solution which are better suited to time-dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline-dependent quantities of interest. The key ingredient is the modal-based approximation of the associated adjoint problems which allows efficiently computing and storing the adjoint solution. The approximated adjoint solution is readily post-processed to produce an enhanced solution, requiring only one spatial post-process for each vibration mode and using the time-harmonic hypothesis to recover the time dependence. Thus the proposed goal-oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low.

  Abstract
  This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities [...]

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• 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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