Documents published in Scipedia

• Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

  S. Zlotnik, P. Diez, D. Gonzalez, E. Cueto, A. Huerta

  Adv. Model. and Simul. in Eng. Sci. (2015). Vol. 2, art. 28

  
  

  Abstract

  The Proper Generalized Decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases the input data is not expressed in a separated form and it has to be replaced by some separable approximation. These approximations constitute a new error source that, in some cases, may dominate the standard ones (discretization, truncation…) and control the nal accuracy of the PGD solution. In this work the relation between errors in the separated input data and the errors induced in the PGD solution is discussed. Error estimators proposed for homogenized problems and oscillation terms are adapted to asses the behaviour of the PGD errors resulting from approximated input data. The PGD is stable with respect to error in the separated data, with no critical amplification of the perturbations. Interestingly, we identified a high sensitiveness of the resulting accuracy on the selection of the sampling grid used to compute the separated data. The separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy. For the case of a Poisson problem separated in the spatial coordinates (a complex diffusivity function requires a separable approximation), the final PGD error is linear with the truncation error of the separated data. This relation is used to estimate the number of terms required in the separated data, that has to be in good agreement with the truncation error accepted in the PGD truncation (tolerance for the stoping criteria in the enrichment procedure). A sensible choice or the prescribed accuracy of the PGD solution has to be kept within the limits set by the errors in the separated input data.

  Abstract
  The Proper Generalized Decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases [...]

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• Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition

  E. Nadal Soriano, F. Chinesta, P. Diez, F. Fuenmayor, F. Denia

  Comput. Methods Appl. Mech. Engrg., (2015). (preprint) Vol. 296 (1), pp. 113-128

  
  

  Abstract

  Some industrial processes are modelled by parametric partial differential equations. Integrating computational modelling and data assimilation into the control process requires obtaining a solution of the numerical model at the characteristic frequency of the process (real-time). This paper introduces a computational strategy allowing to efficiently exploit measurements of those industrial processes, providing the solution of the model at the required frequency. This is particularly interesting in the framework of control algorithms that rely on a model involving a set of parameters. For instance, the curing process of a composite material is modelled as a thermo-mechanical problem whose corresponding parameters describe the thermal and mechanical behaviours. In this context, the information available (measurements) is used to update the parameters of the model and to produce new values of the control variables (data assimilation). The methodology presented here is devised to ensure the possibility of having a response in real-time of the problem and therefore the capability of integrating it in the control scheme. The Proper Generalized Decomposition is used to describe the solution in the multi-parametric space. The real-time data assimilation requires a further simplification of the solution representation that better fits the data (reconstructed solution) and it provides an implicit parameter identification. Moreover, the analysis of the assimilated data sensibility with respect to the points where the measurements are taken suggests a criterion to locate the sensors.

  Abstract
  Some industrial processes are modelled by parametric partial differential equations. Integrating computational modelling and data assimilation into the control process [...]

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• An efficient and general approach for implementing thermodynamic phase equilibria information in geophysical and geodynamic studies

  J. Alfonso, S. Zlotnik, P. Diez

  Geochmistry, geophysics, geosystems (2015). (preprint) Vol. 16 (10), pp. 3767-3777

  
  

  Abstract

  We present a flexible, general, and efficient approach for implementing thermodynamic phase equilibria information (in the form of sets of physical parameters) into geophysical and geodynamic studies. The approach is based on Tensor Rank Decomposition methods, which transform the original multidimensional discrete information into a separated representation that contains significantly fewer terms, thus drastically reducing the amount of information to be stored in memory during a numerical simulation or geophysical inversion. Accordingly, the amount and resolution of the thermodynamic information that can be used in a simulation or inversion increases substantially. In addition, the method is independent of the actual software used to obtain the primary thermodynamic information, and therefore, it can be used in conjunction with any thermodynamic modeling program and/or database. Also, the errors associated with the decomposition procedure are readily controlled by the user, depending on her/his actual needs (e.g., preliminary runs versus full resolution runs). We illustrate the benefits, generality, and applicability of our approach with several examples of practical interest for both geodynamic modeling and geophysical inversion/modeling. Our results demonstrate that the proposed method is a competitive and attractive candidate for implementing thermodynamic constraints into a broad range of geophysical and geodynamic studies. MATLAB implementations of the method and examples are provided as supporting information and can be downloaded from the journal's website.

  Abstract
  We present a flexible, general, and efficient approach for implementing thermodynamic phase equilibria information (in the form of sets of physical parameters) into geophysical [...]

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• Fast r-adaptivity for multiple queries of heterogeneous stochastic material fields

  R. Cottereau, P. Diez

  Computational Mechanics (2015). (preprint) Vol. 56 (4), pp. 601-612

  
  

  Abstract

  We present an r-adaptivity approach for boundary value problems with randomly fluctuating material parameters solved through the Monte Carlo or stochastic collocation methods. This approach tailors a specific mesh for each sample of the problem. It only requires the computation of the solution of a single deterministic problem with the same geometry and the average parameter, whose numerical cost becomes marginal for large number of samples. Starting from the mesh used to solve that deterministic problem, the nodes are moved depending on the particular sample of mechanical parameter field. The reduction in the error is small for each sample but sums up to reduce the overall bias on the statistics estimated through the Monte Carlo scheme. Several numerical examples in 2D are presented.

  Abstract
  We present an r-adaptivity approach for boundary value problems with randomly fluctuating material parameters solved through the Monte Carlo or stochastic collocation [...]

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• Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications

  S. Zlotnik, P. Diez, D. Modesto, A. Huerta

  Int. J. Numer. Meth. Engng. (2015). (preprint) Vol. 103 (10), pp. 737-758

  
  

  Abstract

  The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the material properties. These parameters are considered as new independent variables. The problem is therefore stated in a multidimensional setup. The proper generalized decomposition (PGD) provides an approximation scheme especially well suited to preclude dramatically increasing the computational complexity with the number of dimensions. The PGD strategy is reviewed for the standard case dealing only with material parameters. Then, the ideas presented in [Ammar et al., “Parametric solutions involving geometry: A step towards efficient shape optimization.” Comput. Methods Appl. Mech. Eng., 2014; 268:178–193] to deal with parameters describing the domain geometry are adapted to a more general case including parametrization of the location of internal interfaces. Finally, the formulation is extended to combine the two types of parameters. The proposed strategy is used to solve a problem in applied geophysics studying the temperature field in a cross section of the Earth crust subsurface. The resulting problem is in a 10‐dimensional space, but the PGD solution provides a fairly accurate approximation (error ≤1%) using less that 150 terms in the PGD expansion.

  Abstract
  The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the [...]

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• A recovery-explicit error estimator in energy norm for linear elasticity

  E. Nadal Soriano, P. Diez, J. Ródenas, M. Tur, F. Fuenmayor

  Comput. Methods Appl. Mech. Engrg., (2015). (preprint) Vol. 287, pp. 172-190

  
  

  Abstract

  Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recovery-based estimates is proposed. The idea is (1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, (2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and (3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications.

  Abstract
  Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the [...]

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• An error estimator for separated representations of highly multidimensional models

  A. Ammar, F. Chinesta, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (25-28), pp. 1872-1880

  
  

  Abstract

  Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most of these models consist of a partial differential equation defined in a highly multidimensional domain (e.g. Schrodinger equation, Fokker-Planck equations among many others). The main challenge related to these models is their associated curse of dimensionality. We proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations (also known as finite sums decomposition). This technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates. The issue related to error estimation has never been addressed. This paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques.

  Abstract
  Fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics. Most [...]

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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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