Documents published in Scipedia

• Reduced-Order Models in Bayesian solvers for inverse problems

  P. Diez, S. Zlotnik, A. Muixí, A. García-González

  coupled2023.

  
  

  Abstract

  Challenging inverse problems aim at identifying large sets of parameters using data from different sources and diverse accuracy. This is the case of data assimilation for geophysical crust dynamics, were the number of parameters to identify amounts to thousands. In this context, Bayesian inverse solvers combined with Markov-Chain Monte Carlo (MCMC) strategies are an affordable strategy, accounting for the uncertainty of the input data and quantifying also uncertainty of the output. Despite the efficiency of the MCMC approach, the direct problem has to be evaluated an extremely large number of times, many (after the burn-in phase) with the input parameters lying in a narrow range. This is the ideal situation for Reduced-Order Models (ROM): many repeated queries to the model corresponding to parameters lying in a limited manifold. Thus, we aim at applying ROM to large-dimensional parametric forward problems. In this case, it is important optimising the dimensionality reduction technique inherent to the ROM strategy. For instance, Proper Orthogonal Decomposition (POD) is associated with a linear Principal Component Analysis (PCA). PCA is linear in the sense that assumes the reduceddimension manifold to be Euclidean. We explore using kernel PCA (kPCA) to further reduce the dimension, thus devising a kPOD approach. Different options to select physically inspired kernels, based on the knowledge of the problem under consideration, are discussed. Moreover, the computational strategy to explore the feature space (the reduced-dimensional space) is also discussed. 

  Abstract
  Challenging inverse problems aim at identifying large sets of parameters using data from different sources and diverse accuracy. This is the case of data assimilation for [...]

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• Encapsulated PGD algebraic toolbox operating with high-dimensional data

  P. Diez, S. Zlotnik, A. Garcia, A. Huerta

  Archives of computational methods in engineering (2019).

  
  

  Abstract

  In its original conception, proper generalized decomposition (PGD) provides explicit parametric solutions, denoted as computational vademecums or digital abacuses, to parametric boundary value problems. The PGD approach is extended here to devise a set of algebraic tools enabling to operate with multidimensional tensor data. These tools are designed to store, compress and perform basic operations (in particular divisions) with tensors in separable format. These tools are directly producing the computational vademecums for the resulting high-dimensional tensor data. Thus, the general methodology enables performing nontrivial operations (storage, compression, division, solving linear systems of equations...) for multidimensional tensor data. The idea is based on the principle of the PGD separation, that produces a separable least squares approximation of any multidimensional function. The PGD compression is a particular case, extensively used in practice to compact the separable solution without loss of accuracy. Here, this concept is applied to algebraic tensor structures that are also seen as functions in multidimensional Cartesian domains. Moreover, a straightforward extension of this concept is devised to operate with multidimensional objects stored in the separable format. That allows creating a toolbox of PGD arithmetic operators that is publicly released at https://git.lacan.upc.edu/zlotnik/algebraicPGDtools. Numerical tests demonstrate the performance and efficiency of the toolbox, both for tensor data handling and operation and also in applications pertaining to the discretized version of boundary value problems.

  Abstract
  In its original conception, proper generalized decomposition (PGD) provides explicit parametric solutions, denoted as computational vademecums or digital abacuses, to parametric [...]

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• Error estimation and adaptivity for nonlinear FE analysis

  A. Huerta, A. Rodríguez-Ferran, P. Diez

  Int. J. Applied Mathematics and Computer Science (2002). Vol. 12 (1), pp. 59-70

  
  

  Abstract

  An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type error estimator and an unstructured quadrilateral mesh generator. The error estimator is based on simple local computations over the elements and the so-called patches. In contrast to other residual estimators, no flux splitting is required. The adaptive strategy is illustrated by means of a complex nonlinear problem: the failure analysis of a single-edge notched beam. The quasi-brittle response of concrete is modelled by means of a nonlocal damage model.

  Abstract
  An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type [...]

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• A Proper Generalized Decomposition (PGD) approach to crack propagation in brittle materials: with application to random field material properties

  H. Garikapati, S. Zlotnik, C. Verhoosel, E. Brummelen, P. Diez

  Computational Mechanics (2019). (Preprint) pp. 1-23

  
  

  Abstract

  Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a postprocessing operation on the explicit solution provided by PGD.We first validate the framework using a global energy approach on a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.

  Abstract
  Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random [...]

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• A new 3D equilibrated residual method improving accuracy and efficiency of flux‐free error estimates

  N. Parés, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2019). (Preprint) Vol. 120, pp. 391-432

  
  

  Abstract

  The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three‐dimensional linear finite element approximations of the reaction‐diffusion equation. The upper bounds are guaranteed regardless the size of the finite element mesh and the given data, and all the constants involved are fully computable. The upper bound property holds if the shape of the domain is polyhedral and the Dirichlet boundary conditions are piecewise‐linear. The new approach is an extension of the flux‐free methodology introduced by Parés and Díez in the paper “A new equilibrated residual method improving accuracy and efficiency of flux‐free error estimates”, which introduces a guaranteed, low‐cost, and efficient flux‐free method substantially reducing the computational cost of obtaining guaranteed bounds using flux‐free methods while retaining the good quality of the bounds. Besides extending the 2D methodology, specific new modifications are introduced to further reduce the computational cost in the three‐dimensional setting. The presented methodology also provides a new strategy to obtain equilibrated boundary tractions, which improves the quality of standard techniques while having a similar computational cost.

  Abstract
  The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three‐dimensional linear finite element approximations [...]

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• Sampling-based stochastic analysis of the PKN model for hydraulic fracturing

  H. Garikapati, C. Verhoosel, E. Brummelen, S. Zlotnik, P. Diez

  Computational Geosciences (2019). (Preprint) Vol. 23 (1), pp. 81-105

  
  

  Abstract

  Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture.

  Abstract
  Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic [...]

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• Algebraic and parametric solvers for the power flow problem: towards real-time and accuracy-guaranteed simulation of electric systems

  R. García-Blanco, P. Diez, D. Borzacchiello, F. Chinesta

  Archives of Comp. Meths. Engng. (2018). (Preprint) Vol. 25 (4), pp. 1003-1026

  
  

  Abstract

  The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers for the power flow equations, in both algebraic and parametric version. The application of the Alternating Search Direction method to the power flow problem is also detailed. This results in a family of iterative solvers that combined with Proper Generalized Decomposition technique allows to solve the parametric version of the equations. Once the solution is computed using this strategy, analyzing the network state or solving optimization problems, with inclusion of generation in real-time, becomes a straightforward procedure since the parametric solution is available. Complementing this approach, an error strategy is implemented at each step of the iterative solver. Thus, error indicators are used as an stopping criteria controlling the accuracy of the approximation during the construction process. The application of these methods to the model IEEE 57-bus network is taken as a numerical illustration.

  Abstract
  The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers [...]

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• Algebraic PGD for tensor separation and compression: An algorithmic approach

  P. Diez, S. Zlotnik, A. García-González, A. Huerta

  Comptes Rendus Mécanique (2018). (Preprint) Vol. 346 (7), pp. 501-514

  
  

  Abstract

  Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated with the space of parameters defining the problem). The PGD philosophy consists in providing a separated representation of the multidimensional solution using a greedy approach combined with an alternated directions scheme to obtain the successive rank-one terms. This paper presents an algorithmic approach to high-dimensional tensor separation based on solving the Least Squares approximation in a separable format of multidimensional tensor using PGD. This strategy is usually embedded in a standard PGD code in order to compress the solution (reduce the number of terms and optimize the available storage capacity), but it stands also as an alternative and highly competitive method for tensor separation.

  Abstract
  Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensional boundary value problems (where many dimensions are associated [...]

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• Numerical estimation of the bearing capacity of resistance spot welds in martensitic boron steels using a J-integral fracture criterion

  D. Dorribo, L. Greve, P. Diez, I. Arias, X. Larráyoz-Izcara

  Theoretical and Applied Fracture Mechanics (2018). (Preprint) Vol. 96, pp. 497-508

  
  

  Abstract

  Predicting the bearing capacity of resistance spot welds (RSW) during vehicle crash tests has become a crucial task for the automotive industry, since the recent introduction of advanced high strength steels (AHSS) such as martensitic boron steels (e.g. 22MnB5). The spot weld joints of these steels exhibit relatively low bearing strengths, compared to those of more ductile high strength steels. Currently, the bearing capacity of spot weld joints is characterized through extensive experimental campaigns. In this article, a model for quantification of the bearing capacity of RSW using a finite-element J-integral fracture criterion is presented. The model takes into account geometric and mechanical features of the spot weld, namely the weld diameter and the mechanical properties distribution resulting from the welding process. An experimental loading test campaign is carried out for calibration and validation purposes, considering multiple sheet thickness combinations, loading angles and weld sizes. Experimental observations of the failed spot welds and preliminary simulations show that failure is caused mostly by stress concentration around the sharp weld notch. Consequently, the J-integral obtained from detailed finite element simulations is used to asses the stress/strain concentration along the first crack advance direction predicted by the acoustic tensor. The computed J-integral values are compared to the material toughness to obtain the joint’s maximum force. The resulting simulated and experimental bearing capacities show a good agreement for all tested configurations.

  Abstract
  Predicting the bearing capacity of resistance spot welds (RSW) during vehicle crash tests has become a crucial task for the automotive industry, since the recent introduction [...]

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• A non-intrusive proper generalized decomposition scheme with application in biomechanics

  X. Zou, M. Conti, P. Diez, F. Auricchio

  Int. J. Numer. Meth. Engng. (2018). (Preprint) Vol. 113 (2), pp. 230-251

  
  

  Abstract

  Proper generalized decomposition (PGD) is often used for multi-query and fast-response simulations. It is a powerful tool alleviating the curse of dimensionality affecting multi-parametric partial differential equations. Most implementations of PGD are intrusive extensions based on in-house developed finite element (FE) solvers. In this work, we propose a non-intrusive PGD scheme using off-the-shelf FE codes (such as certified commercial software) as an external solver. The scheme is implemented and monitored by in-house flow-control codes. A typical implementation is provided with downloadable codes. Moreover, a novel parametric separation strategy for the PGD resolution is presented. The parametric space is split into two- or three-dimensional subspaces, to allow PGD technique solving problems with constrained parametric spaces, achieving higher convergence ratio. Numerical examples are provided. In particular, a practical example in biomechanics is included, with potential application to patient-specific simulation.

  Abstract
  Proper generalized decomposition (PGD) is often used for multi-query and fast-response simulations. It is a powerful tool alleviating the curse of dimensionality affecting [...]

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