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This is the peer reviewed version of the following article: Nasika, C. [et al.]. Discrete empirical interpolation for hyper-reduction of hydro-mechanical problems in groundwater flow through soil. "International journal for numerical and analytical methods in geomechanics", 10 Abril 2023, vol. 47, núm. 5, p. 667-693, which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/nag.3487. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited. The recent surge in the availability of sensor data and computational resources has fostered the development of technologies for optimization, control, and monitoring of large infrastructures, integrating data and numerical modeling. The major bottleneck in this type of technologies is the model response time, since repetitive solutions are typically required. To reduce the computational time, reduced order models (ROMs) are used as surrogates for expensive finite element (FE) simulations enabling the use of complex models in this type of applications. In this work, ROMs are explored for the solution of the fully coupled hydro-mechanical system of equations that governs the water flow through partially saturated soil. The POD-based Reduced Basis Method and the Discrete Empirical Interpolation Method (DEIM), as well as its localized version (LDEIM), are examined in solving a parametrized problem simulating the mechanical loading of an embankment dam. Hydraulic and mechanical soil properties are considered as parameters. It is shown that the combination of these methods results in simulations that require 1/10 to 1/100 of the FE response time. Moreover, the method is shown to yield scaling efficiency gains with increasing problem size.

One of the goals of geophysicists is mapping and understanding the current structure of the Earth including its variations in composition, temperature and dynamical state. This structure is only accessible via indirect observations and, therefore, the mathematical problem to be solved is of an inverse kind. Within the inverse solver, many forward problems will be tested until finding a configuration compatible with the observations. This work deals with the problem statement and numerical solution of the forward thermal problem that arises from an inverse solver. In this case, we will use a simple parameterization of the Lithosphere-Asthenosphere Boundary (LAB), but the results are useful for other parametric description (e.g. one parameter per each cell). A simplified model is used to show the ill-posedness of the mathematical problem arising when the LAB --an isotherm whose location is determined by the input parameters-- is imposed within the domain, over-constraining the forward problem. This is well-known in the community and several authors have proposed different approaches to circumvent it. Nevertheless, the strategies used in practice usually involve some non-physical procedures such as transitional regions where two different temperature fields are made compatible by smearing out differences. Generally, the solution in these regions does not comply with the governing equation and exhibits a non-physical behaviour. In this work, we propose a specific problem statement for the temperature with interior essential conditions. The resulting problem is mathematically sound and results in a two-step numerical solver. This guarantees a self-consistent temperature field, in the sense that it respects the thermal governing equations everywhere. The numerical domain is divided into two subdomains (lithosphere and asthenosphere) that are solved separately in the same mesh, using an unfitted mesh methodology. First, the temperature of the lithosphere is computed using the essential condition on the LAB. Second, the temperature in the mantle is obtained by minimizing a residual that measures the compatibility between the two subdomains in terms of LAB temperatures and across-LAB fluxes. This is done by adjusting the proper fluxes at the bottom of the numerical domain. Several examples are presented showing that the obtained temperature fields are stable and oscillation-free. Moreover, the resulting fluxes at the bottom of the domain are reasonable and compatible with the expected values.

This is the peer reviewed version of the following article: Diez, P. [et al.]. Nonlinear dimensionality reduction for parametric problems: a kernel proper orthogonal decomposition. "International journal for numerical methods in engineering", 30 Desembre 2021, vol. 122, núm. 24, p. 7306-7327, which has been published in final form at DOI: 10.1002/nme.6831. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy. Generalitat de Catalunya, 2017-SGR-1278; Ministerio de Ciencia e Innovación, CEX2018-000797-S; PID2020-113463RB-C32; PID2020-113463RB-C33

The version of record is available online at: http://dx.doi.org/10.1007/s00603-022-02925-5 In this paper, a reduced-order model built from the reduced basis (RB) method is used as a surrogate for an inverse identification problem related to coupled thermo-hydro-mechanical (THM) processes. The RB space—spanned by solutions of the governing THM equations—is constructed using a greedy adaptive procedure guided by an a posteriori error estimator that selects the optimal snapshot points in a given parametric space. The RB model is assessed in terms of accuracy and computational cost reduction for the three-dimensional transient coupled problem described by the ATLAS III small-scale in situ heating test. The substantial system size reduction and the associated significant computational gain result in a surrogate model suitable for parameter identification procedures in the Boom Clay material. The effectiveness of the proposed strategy is demonstrated by performing inverse analysis based on direct-search and genetic algorithm (GA) optimization supported by real sensor measurement data where 800 times faster computational speed-up was achieved. We acknowledge the funding support from the European Commission Education, Audiovisual and Culture Executive Agency (EACEA) under Erasmus Mundus Joint Doctorate Simulation in Engineering and Entrepreneurship Development (SEED), FPA 2013-0043 and SCK CEN for funding the 4th year PhD study of the first author. S. Zlotnik and P. Díez would like to thank the funding from the Generalitat de Catalunya 2017-SGR-1278, the project DPI2017-85139-C2-2-R funded by the Spanish Ministry and the project H2020-RISE MATHROCKS GA no. 777778.

This is the peer reviewed version of the following article: Cavaliere, F. [et al.]. Nonintrusive reduced order model for parametric solutions of inertia relief problems. "International journal for numerical methods in engineering", 30 Agost 2021, vol. 122, núm. 16, p. 4270-4291., which has been published in final form at DOI:10.1002/nme.6702. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. The Inertia Relief (IR) technique is widely used by industry and produces equilibrated loads allowing to analyze unconstrained systems without resorting to the more expensive full dynamic analysis. The main goal of this work is to develop a computational framework for the solution of unconstrained parametric structural problems with IR and the Proper Generalized Decomposition (PGD) method. First, the IR method is formulated in a parametric setting for both material and geometric parameters. A reduced order model using the encapsulated PGD suite is then developed to solve the parametric IR problem, circumventing the so-called curse of dimensionality. With just one offline computation, the proposed PGD-IR scheme provides a computational vademecum that contains all the possible solutions for a predefined range of the parameters. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. The applicability and potential of the developed technique is shown using a three-dimensional test case and a more complex industrial test case. The first example is used to highlight the numerical properties of the scheme, whereas the second example demonstrates the potential in a more complex setting and it shows the possibility to integrate the proposed framework within a commercial FE package. In addition, the last example shows the possibility to use the generalized solution in a multi-objective optimization setting. This project is part of the Marie Sk lodowska-Curie ITN-EJD ProTechTion funded by the European Union Horizon 2020 research and innovation program with grant number 764636. The work of Fabiola Cavaliere, Sergio Zlotnik and Pedro D´ıez is partially supported by the Spanish Ministry of Economy and Competitiveness, Spain (Grant number: DPI2017-85139- C2-2-R) and by the Generalitat de Catalunya, Spain (Grant number: 2017-SGR-1278). Ruben Sevilla also acknowledges the support of the Engineering and Physical Sciences Research Council (Grant number: EP/P033997/1).

The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-019-01778-0 Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in materialproperties are frequently modeled through random fields, which typically induces the need to solve finite element problemsfor a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at anaffordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittlematerials 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 forthe physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a post-processing operation on the explicit solution provided by PGD. We first validate the framework using a global energy approachon a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced ordermodeling approach to a stochastic fracture propagation problem.

A model order reduction technique is proposed in order to optimise the design process of a car body structure with respect to the noise and vibration characteristics of the vehicle. The final goal is to support designers in the decision-making process, such that they can evaluate in real-time the impact of certain parameters on the global response of the structure.

We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced allowing to derive alternative guaranteed bounds from nearly-arbitrary H ( div ; Ω ) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.

Modern infrastructure monitoring technologies are often based on the fast response of a digital model of the asset for real-time predictions and/or control. In the case of earthfill dams, particularly tailings dams, the dynamic design and functioning of the structure can give rise to a need for realtime model calibration and data assimilation for accurate monitoring. However, these techniques suppose many queries on large models and are often too costly to be performed in real-time. To accelerate the model response and make feasible the incorporation of numerical modeling within the monitoring system, we propose using Model Order Reduction techniques in the transient thermohydro-mechanical system.The scope of this work is to create a model to support a sensor monitoring system of a tailings dam, by receiving sensor data of local pressure and/or displacement measurements in real time, and evaluating the full stress and deformation fields in the body of the structure, as well as predicting the location of the phreatic surface. Such model can be used for the detection hazardous pressure-stress states, for gaining insights in the most critical potential failure mechanisms, but also for the optimal design of the sensor network and the overall optimization of the monitoring system.POD-based model reduction, combined with Discrete Empirical Interpolation (DEIM) is used to tackle the coupled hydro-mechanical, non-linear, transient problem. The model is implemented in the FEniCS computing platform, an open-source platform that automates the solution of Partial Differential Equations with FEM. It is used for data assimilation applications, parameter identification (soil mechanical and hydraulic properties) and optimal sensor placement. The efficiency gains in inverse problem solving and the accuracy of the resulting ROM are examined and discussed., info:eu-repo/semantics/inPress

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Uncertainty Quantification (UQ) is a booming discipline for complex computational models based on the analysis of robustness, reliability and credibility. UQ analysis for nonlinear crash models with high dimensional outputs presents important challenges. In crashworthiness, nonlinear structural behaviours with multiple hidden modes require expensive models (18 h for a single run). Surrogate models (metamodels) allow substituting the full order model, introducing a response surface for a reduced training set of numerical experiments. Moreover, uncertain input and large number of degrees of freedom result in high dimensional problems, which derives to a bottle neck that blocks the computational efficiency of the metamodels. Kernel Principal Component Analysis (kPCA) is a multidimensionality reduction technique for non-linear problems, with the advantage of capturing the most relevant information from the response and improving the efficiency of the metamodel. Aiming to compute the minimum number of samples with the full order model. The proposed methodology is tested with a practical industrial problem that arises from the automotive industry. This work is partially funded by Generalitat de Catalunya (Grant Number 1278 SGR 2017-2019 and Pla de Doctorats Industrials 2017 DI 058) and Ministerio de Economía y Empresa and Ministerio de Ciencia, Innovación y Universidades (Grant Number DPI2017-85139-C2-2-R).

© 2021 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Uncertainty Quantification (UQ) is a key discipline for computational modeling of complex systems, enhancing reliability of engineering simulations. In crashworthiness, having an accurate assessment of the behavior of the model uncertainty allows reducing the number of prototypes and associated costs. Carrying out UQ in this framework is especially challenging because it requires highly expensive simulations. In this context, surrogate models (metamodels) allow drastically reducing the computational cost of Monte Carlo process. Different techniques to describe the metamodel are considered, Ordinary Kriging, Polynomial Response Surfaces and a novel strategy (based on Proper Generalized Decomposition) denoted by Separated Response Surface (SRS). A large number of uncertain input parameters may jeopardize the efficiency of the metamodels. Thus, previous to define a metamodel, kernel Principal Component Analysis (kPCA) is found to be effective to simplify the model outcome description. A benchmark crash test is used to show the efficiency of combining metamodels with kPCA. This work is partially funded by Generalitat de Catalunya (Grant Number 1278 SGR 2017-2019 and Pla de Doctorats Industrials 2017 DI 058) and Ministerio de Econom´ıa y Empresa and Ministerio de Ciencia, Innovaci´on y Universidades (Grant Number DPI2017-85139-C2-2-R).

The Proper Generalized Decomposition is a well established reduced order method, used to efficiently obtain approximate solutions of multi-dimensional problems in a procedure that controls the effects of the "curse of dimensionality". The question of assessing the quality of the solutions obtained and adapting the approximations assumed, for example the finite element meshes used, so that the best result is obtained a minimal cost, remains a relevant challenge. This paper deals with finite element solutions for solid mechanics problems, using the error obtained from a dual analysis, the difference between complementary solutions, to bound the error of the solutions and to drive an optimal adaptivity process, which obtains meshes with errors significantly lower than those obtained using a uniform refinement. Education, Audiovisual and Culture Executive Agency, EMJD ‐ FPA 2013‐0043 (SEED); Generalitat de Catalunya, 2017‐SGR‐1278; Horizon 2020 Framework Programme, 777778; Ministerio de Economía y Competitividad, DPI2017‐ 85139‐C2‐2‐R

Methodologies for multidimensionality reduction aim at discovering low-dimensional manifolds where data ranges. Principal Component Analysis (PCA) is very effective if data have linear structure. But fails in identifying a possible dimensionality reduction if data belong to a nonlinear low-dimensional manifold. For nonlinear dimensionality reduction, kernel Principal Component Analysis (kPCA) is appreciated because of its simplicity and ease implementation. The paper provides a concise review of PCA and kPCA main ideas, trying to collect in a single document aspects that are often dispersed. Moreover, a strategy to map back the reduced dimension into the original high dimensional space is also devised, based on the minimization of a discrepancy functional.

A goal-oriented a-posteriori error estimator is developed for transient coupled Thermo-Hydro-Mechanical (THM) parametric problems solved with a reduced basis approximation. The estimator assesses the error in some specific Quantity of Interest (QoI). The goal-oriented error estimate is derived based on explicitly-computed weak residual of the primal problem and implicitly-computed adjoint solution associated with the QoI. The time-dependence of the coupled THM system poses an additional complexity as the auxiliary adjoint problem evolves backwards in time. The error estimator guides a greedy adaptive procedure that constructs progressively an optimal reduced basis by smartly selecting snapshot points over a given parametric training sample. The reduced basis obtained is used to drastically reduce the coupled system spatial degrees of freedom by several orders of magnitude. The computational gain obtained from the developed methodology is demonstrated through applications in 2D and 3D parametrized problems simulating the evolution of coupled THM processes in rock masses., info:eu-repo/semantics/published

Reduced order methods are powerful tools for the design and analysis of sophisticated systems, reducing computational costs and speeding up the development process. Among these reduced order methods, the Proper Generalized Decomposition is a well-established one, commonly used to deal with multi-dimensional problems that often suffer from the curse of dimensionality. Although the PGD method has been around for some time now, it still lacks mechanisms to assess the quality of the solutions obtained. This paper explores the dual error analysis in the scope of the PGD, using complementary solutions to compute error bounds and drive an adaptivity process. The energy of the error obtained from the dual analysis is used to determine the quality of the PGD approximations. We define a new adaptivity indicator based on the energy of the error and use it to drive parametric h- and p- adaptivity processes. The results are positive, with the indicator accurately capturing the parameter that will lead to lowest errors.

Markov chain Monte Carlo (MCMC) methods have become standard in Bayesian inference and multi-observable inversions in almost every discipline of the Earth sciences. In the case of geodynamic and/or coupled geophysical-geodynamic inverse problems, however, the computational cost associated with the solution of large-scale 3-D Stokes forward problems has rendered probabilistic formulations impractical. Here we present a novel and extremely efficient method to produce ultrafast solutions of the 3-D Stokes problem for MCMC simulations. Our approach combines the individual benefits of Reduced Basis techniques, goal-oriented error formulations, and MCMC algorithms to produce an accurate and computationally efficient surrogate for the forward problem. Importantly, the surrogate adapts itself during the MCMC simulation according to the history of the chain and the goals of the inversion. This maximizes the efficiency of the forward problem and removes the need for preinversion off-line computations to build a surrogate. We demonstrate the benefits and limitations of the method with several numerical examples and show that in all cases the computational cost is of the order of

Uncertainty Quantification (UQ) is a key discipline for computational modeling of complex systems, enhancing reliability of engineering simulations. In crashworthiness, having an accurate assessment of the behavior of the model uncertainty allows reducing the number of prototypes and associated costs. Carrying out UQ in this framework is especially challenging because it requires highly expensive simulations. In this context, surrogate models (metamodels) allow drastically reducing the computational cost of Monte Carlo process. Different techniques to describe the metamodel are considered, Ordinary Kriging, Polynomial Response Surfaces and a novel strategy (based on Proper Generalized Decomposition) denoted by Separated Response Surface (SRS). A large number of uncertain input parameters may jeopardize the efficiency of the metamodels. Thus, previous to define a metamodel, kernel Principal Component Analysis (kPCA) is found to be effective to simplify the model outcome description. A benchmark crash test is used to show the efficiency of combining metamodels with kPCA.

This is the peer reviewed version of the following article: Reis, J. [et al.]. Error estimation for proper generalized decomposition solutions: dual analysis and adaptivity for quantities of interest. "International journal for numerical methods in engineering", 15 Febrer 2021, vol. 122, núm. 3, p. 752-776, which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/nme.6559. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. When designing a structure or engineering a component, it is crucial to use methods that provide fast and reliable solutions, so that a large number of design options can be assessed. In this context, the proper generalized decomposition (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected by the “curse of dimensionality.” Assessing the accuracy of the solutions obtained with the PGD is still a relevant challenge, particularly when seeking quantities of interest with guaranteed bounds. In this work, we compute compatible and equilibrated PGD solutions and use them in a dual analysis to obtain quantities of interest and their bounds, which are guaranteed. We also use these complementary solutions to compute an error indicator, which is used to drive a mesh adaptivity process, oriented for a quantity of interest. The corresponding solutions have errors that are much lower than those obtained using a uniform refinement or an indicator based on the global error, as the proposed approach focuses on minimizing the error in the quantity of interest. Education, Audiovisual and Culture Executive Agency, FPA 2013‐0043; Generalitat de Catalunya, No. 2017‐SGR‐1278; H2020 Marie Skłodowska‐Curie Actions, MATH‐ROCKS GA no 777778.; Ministerio de Economía y Competitividad, No. DPI2017‐85139‐C2‐2‐R

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Efficient and optimal design of radar-based Advanced Driver Assistant Systems (ADAS) needs the evaluation of many different electromagnetic solutions for evaluating the impact of the radome on the electromagnetic wave propagation. Because of the very high frequency at which these devices operate, with the associated extremely small wavelength, very fine meshes are needed to accurately discretize the electromagnetic equations. Thus, the computational cost of each numerical solution for a given choice of the design or operation parameters, is high (CPU time consuming and needing significant computational resources) compromising the efficiency of standard optimization algorithms. In order to alleviate the just referred difficulties the present paper proposes an approach based on the use of reduced order modeling, in particular the construction of a parametric solution by employing a non-intrusive formulation of the Proper Generalized Decomposition, combined with a powerful phase-angle unwrapping strategy for accurately addressing the electric and magnetic fields interpolation, contributing to improve the design, the calibration and the operational use of those systems.

The use of Internet of Things (IoT) technologies is becoming a preferred solution for the assessment of tailings dams safety. Real-time sensor monitoring proves to be a key tool for reducing the risk related to these ever-evolving earth-fill structures, that exhibit a high rate of sudden and hazardous failures. In order to optimally exploit real-time embankment monitoring, one major hindrance has to be overcome: the creation of a supporting numerical model for stability analysis, with rapid-enough response to perform data assimilation in real time. A model should be built, such that its response can be obtained faster than the physical evolution of the analyzed phenomenon. In this work, Reduced Order Modelling (ROM) is used to boost computational efficiency in solving the coupled hydro-mechanical system of equations governing the problem. The Reduced Basis method is applied to the coupled hydro-mechanical equations that govern the groundwater flow, that are made non-linear as a result of considering an unsaturated soil. The resulting model's performance is assessed by solving a 2D and a 3D problem relevant to tailings dams safety. The ROM technique achieves a speedup of 3 to 15 times with respect to the full-order model (FOM) while maintaining high levels of accuracy, info:eu-repo/semantics/published

The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-017-1534-9 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.

Summary 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. This article is protected by copyright. All rights reserved.

The final publication is available at Springer via http://dx.doi.org/10.1007/s11831-017-9223-6 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.

This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and Diez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

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 Parés et al. (2009) [6, 4], with the goal of substantially reducing the computational 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 upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved.

International audience; 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.

This is an Accepted Manuscript of an article published by Taylor & Francis Group in "European journal of environmental and civil engineering" on 2011, available online at: http://www.tandfonline.com/doi/abs/10.1080/19648189.2011.9714848 We present in this paper a numerical model of the erosion of a soil that accounts for both the flow in the open fluid and the flow of fluid through the porous soil. The interface between the open fluid and the soil is represented using a level-set function, and the erosion is controlled by the shear stress vector. The evaluation of the approximate value of this gradient is particularly focused on, and an improved method, called XFE+ method, is presented. Numerical results in 2D and 3D illustrate the accuracy and the potentiality of this method.

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.

International audience; This paper builds on the flexibility of the level-set representation to model in a unified manner the expansion of a hollow in the ground under different physical phenomena. In particular, the dissolving action of a flow of water in a saturated soil, and that of a jet of particles of water in a non-saturated one, are represented in a common framework. In that manner, the complex geometrical evolutions of the hollow can be followed without the need for remeshing and this approach allows for a smooth transition between saturated and non-saturated models of the soil. Implementation and numerical difficulties are discussed and two applications of industrial interest are considered. The first one describes the modeling of the piping phenomenon, and the second one the evolution of an excavation created by a leaking duct.

The assembly of sparse matrices is a key operation in finite element methods. In this study we analyze several factors that may have an influence on the efficiency of the assembly procedure. Different insertion strategies are compared using two metrics: a Cost function (the number of memory movements) and actual computing time. An improved algorithm implemented in MATLAB is proposed. It reduces both memory operations and computing time for all tested cases. The efficiency of the assembly process is found to be highly dependent on node and element numbering. The effect of the classic reverse Cuthill–McKee algorithm is, in most cases, positive and reduces computation costs. Finally, the case where a sparse matrix has to be re-assembled at each time step is studied. The efficiency of the assembly is improved if the matrix pattern is entirely or partially inherited from previous steps.

Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques.

A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are described: A level set technique to track the position of the materials and an enrichment of the solution to allow the strain rate to be discontinuous across the interface. These techniques have low computational cost and can be used in standard desktop PCs. Examples of phase tracking with level set are presented in two and three dimensions to study slab detachment in subduction processes and Rayleigh–Taylor instabilities, respectively. The modelling of slab detachment processes includes realistic rheology with viscosity depending on temperature, pressure and strain rate; shear and adiabatic heating mechanisms; density including mineral phase changes and varying thermal conductivity. Detachment models show a first prolonged period of thermal diffusion until a fast necking of the subducting slab results in the break–off. The influence of several numerical and physical parameters on the detachment process is analyzed: The shear heating exerts a major influence accelerating the detachment process, reducing the onset time to one half and lubricating the sinking of the detached slab. The adiabatic heating term acts as a thermal stabilizer. If the mantle temperature follows an adiabatic gradient, neglecting this heating term must be included, otherwise all temperature contrasts are overestimated. As expected, the phase change at 410 km depth (olivine–spinel transition) facilitates the detachment process due to the increase in negative buoyancy. Finally, simple plume simulations are used to show how the presented numerical methodologies can be extended to three dimensions., This research has been supported by Ministerio de Educacio´n y Ciencia, Grants DPI2007-62395, SAGAS CTM2005-08071-C03-03/MAR and the Spanish Team η1/η2 = 0.01, h = 1/3 η1/η2 = 100, h = 1/3 η1/η2 = 1, h = 0. 1 η1/η2 = 1, h = 0.05 (a) (b) (c) (d) Figure 8 Two snapshots of the evolution of four different Rayleigh–Taylor instabilities. The viscosity ratios and thickness h of the lower layer determines the shape of the plume. 1508 Sergio Zlotnik et al. Pure appl. geophys., Consolider-Ingenio 2010 nrCSD2006-00041. We thank Vlad Manea and an anonymous reviewer for discussion and comments

A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks of standard residual type estimators, circumvents the need of flux-equilibration and results in a simple implementation that uses standard resources available in finite element codes. This is specially interesting for 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. The bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. The presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest.

In h-adaptivity, a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution. Several authors have introduced different remeshing strategies yielding very different results. In this work these methods are included in a unified framework, emphasizing the role of the underlying hypotheses. Moreover, an objective tool to evaluate the accuracy of the resulting finite element solution is presented. Thus, a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions. The different remeshing strategies are compared with well-known numerical examples.

This is a post-peer-review, pre-copyedit version of an article published in Archives of computational methods in engineering. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11831-019-09378-0 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.

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.

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-elementJ-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, theJ-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 computedJ-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.

Summary 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. This article is protected by copyright. All rights reserved.

This is the peer reviewed version of the following article: [Signorini, M., Zlotnik, S., and Diez, P. (2017) Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems. Int. J. Numer. Meth. Engng, 109: 1085–1102. doi: 10.1002/nme.5313], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5313/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-015-1257-8 A new 2D numerical model to predict the underwater acoustic propagation is obtained by exploring the potential of the Partition of Unity Method (PUM) enriched with plane waves. The aim of the work is to obtain sound pressure level distributions when multiple operational noise sources are present, in order to assess the acoustic impact over the marine fauna. The model takes advantage of the suitability of the PUM for solving the Helmholtz equation, especially for the practical case of large domains and medium frequencies. The seawater acoustic absorption and the acoustic reflectance of the sea surface and sea bottom are explicitly considered, and perfectly matched layers (PML) are placed at the lateral artificial boundaries to avoid spurious reflexions. The model includes semi-analytical integration rules which are adapted to highly oscillatory integrands with the aim of reducing the computational cost of the integration step. In addition, we develop a novel strategy to mitigate the ill-conditioning of the elemental and global system matrices. Specifically, we compute a low-rank approximation of the local space of solutions, which in turn reduces the number of degrees of freedom, the CPU time and the memory footprint. Numerical examples are presented to illustrate the capabilities of the model and to assess its accuracy.

This study presents the application of proper generalized decomposition on transmission line models involving frequency-dependent parameters. Frequency dependence of parameters can be due to a number of reasons including phenomena such as “ground return effects,” “proximity effects,” and “skin effects.” In the current study, simplified methods of skin effects based on Bessel functions are used to showcase the method. Although we implement our study using a specific model for skin effects to demonstrate the effectiveness of the proposed method to accommodate frequency-dependent effects in proper generalized decomposition, the present work is not meant to discuss the merits of any particular skin effects model. The method can easily accommodate other effects, which induces frequency dependence in the transmission line parameters. In time-domain modeling, the parameters are assumed constant, and these models prove inefficient when incorporating these parameters as function of frequency. Therefore, a frequency-domain simulation is implemented using harmonic analysis. Proper generalized decomposition (PGD) presents a separated representation and provides a quick and accurate solution for such problems.