Documents published in Scipedia

• Explicit parametric solutions of lattice structures with proper generalized decomposition (PGD)

  A. Sibileau, A. García-González, F. Auricchio, S. Morganti, P. Diez

  Computational Mechanics (2018). (Preprint) Vol. 62 (4), pp. 871-891

  
  

  Abstract

  Architectured materials (or metamaterials) are constituted by a unit-cell with a complex structural design repeated periodically forming a bulk material with emergent mechanical properties. One may obtain specific macro-scale (or bulk) properties in the resulting architectured material by properly designing the unit-cell. Typically, this is stated as an optimal design problem in which the parameters describing the shape and mechanical properties of the unit-cell are selected in order to produce the desired bulk characteristics. This is especially pertinent due to the ease manufacturing of these complex structures with 3D printers. The proper generalized decomposition provides explicit parametic solutions of parametric PDEs. Here, the same ideas are used to obtain parametric solutions of the algebraic equations arising from lattice structural models. Once the explicit parametric solution is available, the optimal design problem is a simple post-process. The same strategy is applied in the numerical illustrations, first to a unit-cell (and then homogenized with periodicity conditions), and in a second phase to the complete structure of a lattice material specimen.

  Abstract
  Architectured materials (or metamaterials) are constituted by a unit-cell with a complex structural design repeated periodically forming a bulk material with emergent mechanical [...]

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• Numerical Modelling of Multi-Phase Multi-Component. Reactive Transport in the Earth’s interior

  B. Oliveira, J. Afonso, S. Zlotnik, P. Diez

  Geophysical Journal International (2018). (Preprint) Vol. 212 (1), pp. 345-388

  
  

  Abstract

  We present a conceptual and numerical approach to model processes in the Earth’s interior that involve multiple phases that simultaneously interact thermally, mechanically and chemically. The approach is truly multiphase in the sense that each dynamic phase is explicitly modelled with an individual set of mass, momentum, energy and chemical mass balance equations coupled via interfacial interaction terms. It is also truly multicomponent in the sense that the compositions of the system and its constituent phases are expressed by a full set of fundamental chemical components (e.g. SiO2, Al2O3, MgO, etc.) rather than proxies. These chemical components evolve, react with and partition into different phases according to an internally consistent thermodynamic model. We combine concepts from Ensemble Averaging and Classical Irreversible Thermodynamics to obtain sets of macroscopic balance equations that describe the evolution of systems governed by multiphase multicomponent reactive transport (MPMCRT). Equilibrium mineral assemblages, their compositions and physical properties, and closure relations for the balance equations are obtained via a ‘dynamic’ Gibbs free-energy minimization procedure (i.e. minimizations are performed on-the-fly as needed by the simulation). Surface tension and surface energy contributions to the dynamics and energetics of the system are taken into account. We show how complex rheologies, that is, visco-elasto-plastic, and/or different interfacial models can be incorporated into our MPMCRT ensemble-averaged formulation. The resulting model provides a reliable platform to study the dynamics and nonlinear feedbacks of MPMCRT systems of different nature and scales, as well as to make realistic comparisons with both geophysical and geochemical data sets. Several numerical examples are presented to illustrate the benefits and limitations of the model.

  Abstract
  We present a conceptual and numerical approach to model processes in the Earth’s interior that involve multiple phases that simultaneously interact thermally, mechanically [...]

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• Generalized parametric solutions in Stokes flow

  P. Diez, S. Zlotnik, A. Huerta

  Comput. Methods Appl. Mech. Engrg. (2017). (Preprint) Vol. 326, pp. 223-240

  
  

  Abstract

  Design optimization and uncertainty quantification, among other applications of industrial interest, require fast or multiple queries of some parametric model. The Proper Generalized Decomposition (PGD) provides a separable solution, a computational vademecum explicitly dependent on the parameters, efficiently computed with a greedy algorithm combined with an alternated directions scheme and compactly stored. This strategy has been successfully employed in many problems in computational mechanics. The application to problems with saddle point structure raises some difficulties requiring further attention. This article proposes a PGD formulation of the Stokes problem. Various possibilities of the separated forms of the PGD solutions are discussed and analyzed, selecting the more viable option. The efficacy of the proposed methodology is demonstrated in numerical examples for both Stokes and Brinkman models.

  Abstract
  Design optimization and uncertainty quantification, among other applications of industrial interest, require fast or multiple queries of some parametric model. The Proper [...]

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• A semi‐analytical scheme for highly oscillatory integrals over tetrahedra

  R. Hospital-Bravo, J. Sarrate, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2017). (Preprint) Vol. 111 (8), pp. 703-723

  
  

  Abstract

  This paper details a semi‐analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the partition of unity method enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non‐oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared with standard Gauss–Legendre quadratures.

  Abstract
  This paper details a semi‐analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly [...]

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• Monitoring a PGD solver for parametric power flow problems with goal-oriented error assessment

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

  Int. J. Numer. Meth. Engng. (2017). Vol. 111 (6), pp. 529-552

  
  

  Abstract

  The parametric analysis of electric grids requires carrying out a large number of Power Flow computations. The different parameters describe loading conditions and grid properties. In this framework, the Proper Generalized Decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to 1) iterating algebraic solver, 2) number of terms in the separable greedy expansion and 3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal-oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end- user. The paper discusses how to compute the goal-oriented error estimates. This requires linearizing the error equation and the Quantity of Interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems.

  Abstract
  The parametric analysis of electric grids requires carrying out a large number of Power Flow computations. The different parameters describe loading conditions and grid properties. [...]

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• Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

  K. Serafin, B. Magnain, E. Florentin, N. Parés, P. Diez

  Int. J. Numer. Meth. Engng. (2017). (Preprint) Vol. 110 (5), pp. 440-466

  
  

  Abstract

  This work focuses on providing accurate low-cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In [E. Florentin, P. Diez, Adaptive reduced basis strategy based on goal oriented error assessment for stochastic problems, Comput. Methods Appl. Mech. Engrg. 225-228 (2012) 116-127], an adaptive strategy has been introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU cost gain. Technical insight of these two topics are presented in details and numerical examples show the interest of these new developments.

  Abstract
  This work focuses on providing accurate low-cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In [E. Florentin, P. Diez, [...]

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• Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems

  M. Signorini, S. Zlotnik, P. Diez

  Int. J. Numer. Meth. Engng. (2017). (Preprint) Vol. 109, (8), pp. 1085-1102

  
  

  Abstract

  The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are described by material and geometric parameters, which are taken as input data for a predictive model. Here, the model is based on the Helmholtz equation, describing the acoustic response of the system for a given wave length. Thus, the inverse problem consists in identifying the values of these parameters such that the output of the model agrees the best with observations. This optimization algorithm requires multiple queries to the model with different values of the parameters. Reduced order models are especially well suited to significantly reduce the computational overhead of the multiple evaluations of the model. In particular, the proper generalized decomposition produces a solution explicitly stating the parametric dependence, where the parameters play the same role as the physical coordinates. A proper generalized decomposition solver is devised to inexpensively explore the parametric space along the iterative process. This exploration of the parametric space is in fact seen as a post‐process of the generalized solution. The approach adopted demonstrates its viability when tested in two illustrative examples.

  Abstract
  The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are [...]

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• A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates

  N. Parés, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2017). (Preprint) Vol. 313 (1), pp. 785-816

  
  

  Abstract

  This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction-diffusion equation. The new approach revisits the ideas in [20, 18] with the goal of substantially reducing the com- putational cost of the flux-free method while retaining the good quality of the bounds. The new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies. The zeroth-order equilibration conditions are imposed using an alternative less restrictive form of the first-order equilibration conditions, along with a new efficient minimization criterion. This new equilibration strategy provides much more accurate up- per bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved.

  Abstract
  This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction-diffusion [...]

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• A reduced order modeling approach for optimal allocation of Distributed Generation in power distribution systems

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

  International Energy Conference (ENERGYCON) (2016). (preprint) pp, 1-6

  
  

  Abstract

  This paper presents an “offline-online” strategy for optimal allocation and sizing of Distributed Generation. In traditional optimization approaches, each function evaluation requires the solution of a power flow problem, which makes global optimality a computationally challenging goal. In the proposed strategy the power flow solver is invoked only once and a parametric solution is constructed with a monolithic solver. Despite the fact that the parametrized power flow equations result in a high-dimensional problem, the proposed algorithm is specifically designed to circumvent the curse of dimensionality. This is achieved through the application of Model Reduction, in particular the Proper Generalized Decomposition combined with a nonlinear solver. Numerical examples are carried out for showing the validity of the proposed method.

  Abstract
  This paper presents an “offline-online” strategy for optimal allocation and sizing of Distributed Generation. In traditional optimization approaches, each function [...]

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• Unified formulation of a family of iterative solvers for power system analysis

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

  Electric Power Systems Research (2016). (Preprint) Vol. 140, pp. 201-208

  
  

  Abstract

  This paper illustrates the construction of a new class of iterative solvers for power flow calculations based on the method of Alternating Search Directions. This method is fit to the particular algebraic structure of the power flow problem resulting from the combination of a globally linear set of equations and nonlinear local relations imposed by power conversion devices, such as loads and generators. The choice of the search directions is shown to be crucial for improving the overall robustness of the solver. A noteworthy advantage is that constant search directions yield stationary methods that, in contrast with Newton or Quasi-Newton methods, do not require the evaluation of the Jacobian matrix. Such directions can be elected to enforce the convergence to the high voltage operative solution. The method is explained through an intuitive example illustrating how the proposed generalized formulation is able to include other nonlinear solvers that are classically used for power flow analysis, thus offering a unified view on the topic. Numerical experiments are performed on publicly available benchmarks for large distribution and transmission systems.

  Abstract
  This paper illustrates the construction of a new class of iterative solvers for power flow calculations based on the method of Alternating Search Directions. This method is [...]

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