Academic literature on the topic 'Proper Generalized Decomposition (PGD)'
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Journal articles on the topic "Proper Generalized Decomposition (PGD)"
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Le-Quoc, C., Linh A. Le, V. Ho-Huu, P. D. Huynh, and T. Nguyen-Thoi. "An Immersed Boundary Proper Generalized Decomposition (IB-PGD) for Fluid–Structure Interaction Problems." International Journal of Computational Methods 15, no. 06 (September 2018): 1850045. http://dx.doi.org/10.1142/s0219876218500457.
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Proper generalized decomposition (PGD), a method looking for solutions in separated forms, was proposed recently for solving highly multidimensional problems. In the PGD, the unknown fields are constructed using separated representations, so that the computational complexity scales linearly with the dimension of the model space instead of exponential scaling as in standard grid-based methods. The PGD was proven to be effective, reliable and robust for some simple benchmark fluid–structure interaction (FSI) problems. However, it is very hard or even impossible for the PGD to find the solution of problems having complex boundary shapes (i.e., problems of fluid flow with arbitrary complex geometry obstacles). The paper hence further extends the PGD to solve FSI problems with arbitrary boundaries by combining the PGD with the immersed boundary method (IBM) to give a so-called immersed boundary proper generalized decomposition (IB-PGD). In the IB-PGD, a forcing term constructed by the IBM is introduced to Navier–Stokes equations to handle the influence of the boundaries and the fluid flow. The IB-PGD is then applied to solve Poisson’s equation to find the fluid pressure distribution for each time step. The numerical results for three problems are presented and compared to those of previous publications to illustrate the robustness and effectiveness of the IB-PGD in solving complex FSI problems.
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Passieux, J. C., and J. N. Périé. "High resolution digital image correlation using proper generalized decomposition: PGD-DIC." International Journal for Numerical Methods in Engineering 92, no. 6 (June 5, 2012): 531–50. http://dx.doi.org/10.1002/nme.4349.
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Sibileau, Alberto, Alberto García-González, Ferdinando Auricchio, Simone Morganti, and Pedro Díez. "Explicit parametric solutions of lattice structures with proper generalized decomposition (PGD)." Computational Mechanics 62, no. 4 (January 10, 2018): 871–91. http://dx.doi.org/10.1007/s00466-017-1534-9.
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Pineda-Sanchez, Manuel, Angel Sapena-Baño, Juan Perez-Cruz, Javier Martinez-Roman, Ruben Puche-Panadero, and Martin Riera-Guasp. "Internal inductance of a conductor of rectangular cross-section using the proper generalized decomposition." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 35, no. 6 (November 7, 2016): 2007–21. http://dx.doi.org/10.1108/compel-03-2016-0124.
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Purpose Rectangular conductors play an important role in planar transmission line structures, multiconductor transmission lines, in power transmission and distribution systems, LCL filters, transformers, industrial busbars, MEMs devices, among many others. The precise determination of the inductance of such conductors is necessary for their design and optimization, but no explicit solution for the AC resistance and internal inductances per-unit length of a linear conductor with a rectangular cross-section has been found, so numerical methods must be used. The purpose of this paper is to introduce the use of a novel numerical technique, the proper generalized decomposition (PGD), for the calculation of DC and AC internal inductances of rectangular conductors. Design/methodology/approach The PGD approach is used to obtain numerically the internal inductance of a conductor with circular cross-section and with rectangular cross-section, both under DC and AC conditions, using a separated representation of the magnetic vector potential in a 2D domain. The results are compared with the analytical and approximate expressions available in the technical literature, with an excellent concordance. Findings The PGD uses simple one-dimensional meshes, one per dimension, so the use of computational resources is very low, and the simulation speed is very high. Besides, the application of the PGD to conductors with rectangular cross-section is particularly advantageous, because rectangular shapes can be represented with a very few number of independent terms, which makes the code very simple and compact. Finally, a key advantage of the PGD is that some parameters of the numerical model can be considered as additional dimensions. In this paper, the frequency has been considered as an additional dimension, and the internal inductance of a rectangular conductor has been computed for the whole range of frequencies desired using a single numerical simulation. Research limitations/implications The proposed approach may be applied to the optimization of electrical conductors used in power systems, to solve EMC problems, to the evaluation of partial inductances of wires, etc. Nevertheless, it cannot be applied, as presented in this work, to 3D complex shapes, as, for example, an arrangement of layers of helically stranded wires. Originality/value The PGD is a promising new numerical procedure that has been applied successfully in different fields. In this paper, this novel technique is applied to find the DC and AC internal inductance of a conductor with rectangular cross-section, using very dense and large one-dimensional meshes. The proposed method requires very limited memory resources, is very fast, can be programmed using a very simple code, and gives the value of the AC inductance for a complete range of frequencies in a single simulation. The proposed approach can be extended to arbitrary conductor shapes and complex multiconductor lines to further exploit the advantages of the PGD.
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Falcó, Antonio, Lucía Hilario, Nicolás Montés, Marta C. Mora, and Enrique Nadal. "Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)." Mathematics 9, no. 1 (December 25, 2020): 34. http://dx.doi.org/10.3390/math9010034.
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A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.
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Alameddin, Shadi, Amélie Fau, David Néron, Pierre Ladevèze, and Udo Nackenhorst. "Toward Optimality of Proper Generalised Decomposition Bases." Mathematical and Computational Applications 24, no. 1 (March 5, 2019): 30. http://dx.doi.org/10.3390/mca24010030.
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The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.
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Poulhaon, Fabien, Francisco Chinesta, and Adrien Leygue. "A First Approach Toward a Proper Generalized Decomposition Based Time Parallelization." Key Engineering Materials 504-506 (February 2012): 461–66. http://dx.doi.org/10.4028/www.scientific.net/kem.504-506.461.
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Many models encountered in computer science remain intractable because of their tremendouscomplexity. Among them, the numerical modeling of manufacturing processes involving severalcharacteristic times is a challenging issue. Classical incremental methods often fail for solving efficientlysuch transient models. In that sense model reduction based simulation appears to be a verypromising alternative. Multidimensional parametric models can be solved within the context of theProper Generalized Decomposition (PGD). It opens new horizons regarding time parallelization. Indeed,by no more considering the initial condition of a transient problem as a static input data butas an extra- coordinate similarly to space and time, we demonstrate that it is possible to parallelizeefficiently the computation and even reach real-time in some cases.
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Le, Cuong Q., H. Phan-Duc, and Son H. Nguyen. "Immersed boundary method combined with proper generalized decomposition for simulation of a flexible filament in a viscous incompressible flow." Vietnam Journal of Mechanics 39, no. 2 (June 21, 2017): 109–19. http://dx.doi.org/10.15625/0866-7136/8120.
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In this paper, a combination of the Proper Generalized Decomposition (PGD) with the Immersed Boundary method (IBM) for solving fluid-filament interaction problem is proposed. In this combination, a forcing term constructed by the IBM is introduced to Navier-Stokes equations to handle the influence of the filament on the fluid flow. The PGD is applied to solve the Poission's equation to find the fluid pressure distribution for each time step. The numerical results are compared with those by previous publications to illustrate the robustness and effectiveness of the proposed method.
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Falcó, Antonio, Lucía Hilario, Nicolás Montés, Marta C. Mora, and Enrique Nadal. "A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition." Mathematics 8, no. 12 (December 19, 2020): 2245. http://dx.doi.org/10.3390/math8122245.
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A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem has been studied by different researchers suggesting different techniques to solve the problem of how to design a trajectory of a mobile robot avoiding collisions with dynamic obstacles. One of these algorithms is the artificial potential field (APF), proposed by O. Khatib in 1986, where a set of an artificial potential field is generated to attract the mobile robot to the goal and to repel the obstacles. This is one of the best options to obtain the trajectory of a mobile robot in real-time (RT). However, the main disadvantage is the presence of deadlocks. The mobile robot can be trapped in one of the local minima. In 1988, J.F. Canny suggested an alternative solution using harmonic functions satisfying the Laplace partial differential equation. When this article appeared, it was nearly impossible to apply this algorithm to RT applications. Years later a novel technique called proper generalized decomposition (PGD) appeared to solve partial differential equations, including parameters, the main appeal being that the solution is obtained once in life, including all the possible parameters. Our previous work, published in 2018, was the first approach to study the possibility of applying the PGD to designing a path planning alternative to the algorithms that nowadays exist. The target of this work is to improve our first approach while including dynamic obstacles as extra parameters.
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Nasri, Mohamed Aziz, Camille Robert, Amine Ammar, Saber El Arem, and Franck Morel. "Proper Generalized Decomposition (PGD) for the numerical simulation of polycrystalline aggregates under cyclic loading." Comptes Rendus Mécanique 346, no. 2 (February 2018): 132–51. http://dx.doi.org/10.1016/j.crme.2017.11.009.
Full textDissertations / Theses on the topic "Proper Generalized Decomposition (PGD)"
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Retat, Françoise. "Proper generalized decomposition based dynamic data driven application systems." Ecole centrale de Nantes, 2013. http://www.theses.fr/2013ECDN0025.
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De nos jours, en sciences des simulations, le besoin de réponses en temps réel se fait de plus en plus ressentir. Les applications basées sur des données dynamiques - Dynamic Data-Driven Application Systems ou DDDAS- nous permettent d'obtenir des résultats en temps réel, grâce aux liens existant entre les outils de simulation et les instruments de mesure. Mais pour cela, les DDDAS ont besoin d'outils de simulations précis et rapides. La solution proposée ici consiste à développer une fois pour toute, en différé (ou 'off-line'), la solution la plus générale possible du modèle, intégrant tous les paramètres comme extra-coordonnées. Cet abaque est alors utilisé pour les calculs 'en-ligne'. Mais ceci soulève à son tour le problème des espaces de grande dimension. La technique de la PGD -Proper Generalized Decomposition- permet d'éviter ce fléau, grâce à la représentation séparée des solutions. Le but de ce travail est d'approfondir les possibilités d'estimation des paramètres, de vérification et de contrôle en temps réel. Le champ d'application est le développement d'une nouvelle méthode de contrôle des couches limites, c'est-à-dire le contrôle des écoulements laminaires au-dessus d'une aile d'avion. Cette nouvelle approche est liée au chauffage en discontinu de certain partie de l'aile en utilisant des bandes de résistance électrique. Contrôler le point de transition et de séparation de la couche limite permettrait de réduire la trainée de frottement, ce qui entrainerait une réduction dans la consommation du carburant, ce qui répondrait aux recommandations de ACARE 2020Nowadays, in simulation-based engineering science, the need of real-time responses is felt more than ever. Dynamic Data-Driven Application systems -DDDAS, thanks to the linkage of the simulation tools with the measurement devices, enable us to achieve real-time computation. But to do so, DDDAS need accurate and fast simulation tools. The solution presented here consists in first computing once and for all, off-line the model's most general solution, introducing all the parameters as extra-coordinates. This abacus is then considered for the on-line purposes. But this, in turn, raises the issue of highly multidimensional spaces. The Proper Generalized Decomposition technique, thanks to its separated representation, allows circumventing this redoubtable curse. The focus of this work is to explore some possibilities in the context of parameter estimation, verification and control in real time. The application of this research is the development of a new boundary control method, i. E. Laminar-flow control over an airfoil. This new approach is associated with the unsteady surface heating regime using electrically resistant strips embedded in the wing skin. The control of the boundary layer separation and transition will provide a reduction in friction drag, and hence a reduction in the fuel consumption, which would comply with the ACARE 2020 requirements
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Lázaro, García Juan. "Contribución al cálculo de elementos en instalaciones eléctricas mediante PGD (Proper Generalized Decomposition)." Doctoral thesis, Universitat Politècnica de València, 2016. http://hdl.handle.net/10251/61966.
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[EN] Thesis exposition and summary.
This thesis focuses on giving light to the current state of traditional numerical methods, the constraints we face, and the different solutions that are being proposed for the simulation of the electromagnetic behaviour of different materials as electrical conductors in transmission lines and grounding facilities, based on the formulation that defines the Electromagnetic Field Theory (Maxwell Laws), and the different conditions of each particular problem to solve.
The main aim of the thesis is to investigate the application of numerical techniques very recently applied, known as the Proper Generalized Decomposition (PGD). Based on a novel technique of decomposition of multidimensional variables (such as in electromagnetic field) in a sum of products (modes) of one-dimensional variables, and using iterative algorithms, PGD can address with a reduced need for computational media, complex problems whose solution requires extraordinary means using traditional techniques. These new techniques have been successfully applied in other domains, such as the simulation of mechanical components and materials science. The aim of this thesis is the application of these new techniques to the simulation of electromagnetic phenomena in the different elements designed for the use of electricity.
The thesis focuses on the development of modelling power transmission conductor energy and grounding networks, basic structures in electrical technology but serve to analyze and observe in detail, as well as to validate with traditional methods of proven reliability, the great potential of PGD, leaving open the application of the technique to technically complex as transformers and rotating machines in future works of the Electrical Equipment, Systems and Facilities Research Group (ISEE) of the Polytechnic University of Valencia (UPV).
The main novelties of the thesis on previous work are part of the objectives, and are as follows:
-Optimization on PGD technique. In this thesis has been chosen by an application of PGD with maximum decomposition in elementary functions, i.e., modes will be considered consisting of products of functions exclusively one-dimensional (x, y, z, t, frequency, etc.), then discretized with uniform dimensional meshes. This will lead us to obtain simple codes, which require easy deployment and reduced computational resources.
-Applications of PGD to electromagnetism field, since the vast majority of references that can be found in the application of PGD concern the field of mechanics and materials. This work aims to use advances made in these fields, and apply to the field of electromagnetism, where only very few works have been published in recent years, with the aim of contributing to further open a new front in the development and application of technology that allows to overcome the limitations and problems that far presented with traditional techniques resolution.[ES] Planteamiento y resumen de la tesis doctoral. La presente tesis se centra en dar luz al estado actual de los métodos numéricos tradicionales, las limitaciones a las que nos enfrentamos, y las diferentes soluciones que se están planteando para la simulación del comportamiento electromagnético de diferentes materiales como conductores eléctricos en líneas de transmisión e instalaciones de puesta a tierra, basándose en la formulación que define la Teoría de Campos Electromagnéticos (Leyes de Maxwell), y las diferentes condiciones de cada problema particular a resolver. El objetivo principal de la tesis es el investigar la aplicación de técnicas numéricas de muy reciente aplicación, conocidas como la Descomposición Propia Generalizada (Proper Generalized Decomposition PGD). Basándose en una técnica novedosa de descomposición de las variables multidimensionales (como en el campo electromagnético) en una suma de productos (modos) de variables unidimensionales, y mediante algoritmos iterativos, la PGD permite abordar, con una reducida necesidad de medios computacionales, problemas complejos cuya solución requiere medios extraordinarios empleando las técnicas tradicionales. Estas nuevas técnicas han sido aplicadas con éxito en otros dominios, como el de la simulación de elementos mecánicos y en ciencia de los materiales. El objetivo de la presente tesis es precisamente el de la aplicación de estas novedosas técnicas a la simulación de fenómenos electromagnéticos en los diferentes elementos diseñados para la utilización de la energía eléctrica. La tesis se centra en el desarrollo de la modelización de conductores de transmisión de energía eléctricas y redes de puesta a tierra, estructuras básicas en la tecnología eléctrica pero que sirven para analizar y observar con detalle además de validar con métodos tradicionales, de demostrada fiabilidad, el gran potencial de la PGD, dejando abierta la aplicación de la técnica a elementos técnicamente más complejos como transformadores y máquinas rotativas en futuros trabajos del Grupo de Investigación de Instalaciones, Sistemas y Equipos Eléctricos (ISEE) de la Universidad Politécnica de Valencia (UPV). Las principales novedades que aporta la tesis sobre trabajos realizados anteriormente son parte de los objetivos que persigue, y son las siguientes: - Optimización de la técnica de la PGD. En la presente tesis se ha optado por una aplicación de la PGD con la máxima descomposición posible en funciones elementales, es decir, los modos se considerarán formados por productos de funciones exclusivamente unidimensionales (x, y, z, t, frecuencia, etc.), discretizadas posteriormente con mallas unidimensionales uniformes. Esto nos llevará a obtener códigos simples, de sencilla implementación y que necesitarán de reducidos recursos computacionales. - Aplicación de la PGD al campo del Electromagnetismo, ya que la gran mayoría de las referencias que se pueden encontrar en la aplicación de la PGD se refieren al campo de la mecánica y los materiales. Este trabajo pretende utilizar avances logrados en esos campos, y aplicarlos al campo del electromagnetismo, donde sólo muy pocos trabajos han sido publicados en los últimos años, con el objetivo de contribuir a seguir abriendo un nuevo frente en el desarrollo y aplicación de la técnica, que permita vencer las limitaciones y problemas que hasta el momento se presentan con las técnicas de resolución tradicionales.
[CAT] Plantejament i resum de la tesi doctoral. La present tesi se centra a donar llum a l'estat actual dels mètodes numèrics tradicionals, les limitacions a què ens enfrontem, i les diferents sol¿lucions que s'estan plantejant per a la simulació del comportament electromagnètic de diversos materials com a conductors elèctrics en linies de transmissió i instal¿lacions d'enclavament a terra, basant-se en la formulació que defineix la Teoria de Camps Electromagnètics (Lleis de Maxwell) , i les diferents condicions de cada problema particular a resoldre. L'objectiu principal de la tesi és investigar l'aplicació de tècniques numèriques de molt recent aplicació, conegudes com la Descomposició Pròpia Generalitzada (Proper Generalized Decomposition PGD). Basant-se en una tècnica nova de descomposició de les variables multidimensionales (com en el camp electromagnètic) en una suma de productes (modes) de variables unidimensionals, i per mitjà d'algoritmes iteratius. La PGD permet abordar, amb una reduïda necessitat de mitjans computacionals, problemes complexos la sol¿lució de la qual requereix mitjans extraordinaris emprant les tècniques tradicionals. Tals tècniques han sigut aplicades amb èxit en altres dominis, com el de la simulació d'elements mecànics i en ciència dels materials. L'objectiu de la present tesi és precisament el de l'aplicació d'estes noves tècniques a la simulació de fenòmens electromagnètics en els diversos elements dissenyats per a l'utilització de l'energia elèctrica. La tesi es centra en el desenrotllament de la modelització de conductors de transmissió d'energia eléctrica i xarxes d'enclavament a terra, estructures bàsiques en la tecnologia elèctrica però que serveixen per a analitzar i observar amb detall a més de validar amb mètodes tradicionals, de demostrada fiabilitat, el gran potencial de la PGD, deixant oberta l'aplicació de la tècnica a elements tècnicament més complexos com a transformadors i màquines rotatives en futures treballs del Grup d'Investigació d'Instal¿lacions, Sistemes i Equips Elèctrics (ISEE) de la Universitat Politècnica de València (UPV). Les principals novetats que aporta la tesi sobre treballs realitzats anteriorment són part dels objectius que persegueix, i són les següents: -Optimització de la tècnica de la PGD. En la present tesi s'ha optat per una aplicació de la PGD amb la màxima descomposició possible en funcions elementals, és a dir, els modes es consideraran formats per productes de funcions exclusivament unidimensionals (x, y, z, t, freqüència, etc.), discretizadas amb malles unidimensionals uniformes. Açò ens portarà a obtindre còdics simples, de senzilla implementació i que necessitaran de reduïts recursos computacionals. -Aplicació de la PGD al camp de l'Electromagnetisme, ja que la gran majoria de les referències que es poden trobar en l'aplicació de la PGD es refereixen al camp de la mecànica i els materials. Este treball pretén utilitzar avanços aconseguits en esses camps, i aplicar-los al camp de l'electromagnetisme, on només molt pocs treballs han sigut publicats en els últims anys, amb l'objectiu de contribuir a continuar obrint un nou front en el desentrollament i aplicació de la tècnica, que permeta véncer les limitacions i problemes que fins al moment es presenten amb les tècniques de resol¿lució tradicionals.
Lázaro García, J. (2016). Contribución al cálculo de elementos en instalaciones eléctricas mediante PGD (Proper Generalized Decomposition) [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/61966
TESIS
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Sandino, de Benito Carlos. "Global-local separated representations based on the Proper Generalized Decomposition." Thesis, Ecole centrale de Nantes, 2019. http://www.theses.fr/2019ECDN0064.
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L'un des principaux avantages de la méthode «Proper Generalized Decomposition», par rapport à d'autres méthodes de réduction de modèles, réside dans son adéquation pour calculer des représentations séparées dans l’espace pour des domaines dégénérés de type cartésien, tels que des plaques ou des coques. L'objectif principal de cette thèse est de généraliser les représentations séparées dans l’espace aux domaines non cartésiens, en introduisant la notion de représentations séparées. Les représentations séparées de type global-local peuvent être comprises comme une décomposition multiplicative dans laquelle les modes locaux capturent la solution à une échelle fine, tandis que les modes globaux résolvent une échelle grossière. Pour ce faire, deux stratégies sont proposées. La première proposition est basée sur la partition de l'unité et combine les niveaux de discrétisation globale et locale, basés sur une partition du domaine. Cette approche construit une représentation séparée qui fournit l'enrichissement local, sans qu'il soit nécessaire de connaître a priori la solution, ni de mettre en oeuvre des problèmes locaux auxiliaires pour déterminer l'enrichissement. La deuxième stratégie est consacrée à la construction de représentations séparées de type global-local de manière moins intrusive, compatible avec le standard des éléments finis. Par conséquent, nous nous basons sur l’assemblage FEM standard des opérateurs et utilisons la PGD comme résolveur algébrique itératif. La continuité sur les limites de la partition du domaine n'a pas besoin d'être imposée explicitement, car elle constitue une propriété intégrée dans les opérateurs FEMOne of the main advantages of the Proper Generalized Decomposition method, when compared to other model reduction methods, lies in its adequacy to compute space separated representations in Cartesian-like degenerated domains, such as plates or shells. The main objective of this thesis is to generalize space separated representations to non-Cartesian domains, by introducing the notion of Global-Local separated representations. Global-Local separated representations can be understood as a multiplicative decomposition in which the local modes capture the solution at the finer scale, while the global modes solve the coarser scale. To this aim, two strategies are proposed. The first proposal is based on the partition of unity, and combines the global and local discretization levels, based on a partition of the domain. It builds a separated representation that provides the local enrichment, without the need for a priori knowledge of the solution, nor the implementation of auxiliary local problems to determine the enrichment. The second strategy is devoted to the construction of Global-Local separated representations in a less intrusive manner, compatible with the finite element standard. Therefore, we rely on standard FEM assembly of the operators and use the PGD as an algebraic iterative solver. Continuity on the boundaries of the domain’s partition does not need to be imposed explicitly, as it comes as a built-in property of the FEM operators
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Barbarulo, Andrea. "On a PGD model order reduction technique for mid-frequency acoustic." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00822643.
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In many industrial contexts, such as aerospace applications or cars design, numerical prediction techniquesbecome more and more useful. They restrict the use of real prototypes to a minimum and make easier thedesign phase. In such industries and in the specific for acoustic, engineers are interested in computing theresponses of systems on frequency bands. In order to predict the vibration behavior of systems overfrequency bands, standard numerical techniques usually involve many frequency-fixed computations, atmany different frequencies. Although it is a straightforward and natural mean to answer to the posed problem,such a strategy can easily lead to huge computations, and the amount of data to store often increasessignificantly. This is particularly true in the context of medium frequency bands, where these responses havea strong sensitivity to the frequency. In this work PGD (Proper Generalized Decomposition), in a first time, isapplied to found a separate functional representation over frequency and space of the unknown amplitude ofVTCR (Variational Theory of Complex Rays) formulation on a reduced frequency space. This allows tocalculate an high quality mid-frequency response over a wide band without a fine frequency discretization,saving computational resources. Moreover the PGD representation of the solution allows to save a hugeamount of space in term of stored data. In a second time, PGD technique as been applied to extend itspeculiarity to mid-frequency wide band with uncertainty.
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Allier, Pierre-Eric. "Contrôle d’erreur pour et par les modèles réduits PGD." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLN063/document.
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De nombreux problèmes de mécanique des structures nécessitent la résolution de plusieurs problèmes numériques semblables. Une approche itérative de type réduction de modèle, la Proper Generalized Decomposition (PGD), permet de déterminer l’ensemble des solutions en une fois, par l’introduction de paramètres supplémentaires. Cependant, un frein majeur à son utilisation dans le monde industriel est l’absence d’estimateur d’erreur robuste permettant de mesurer la qualité des solutions obtenues. L’approche retenue s’appuie sur le concept d’erreur en relation de comportement. Cette méthode consiste à construire des champs admissibles, assurant ainsi l’aspect conservatif et garanti de l’estimation de l’erreur en réutilisant le maximum d’outils employés dans le cadre éléments finis. La possibilité de quantifier l’importance des différentes sources d’erreur (réduction et discrétisation) permet de plus de piloter les principales stratégies de résolution PGD. Deux stratégies ont été proposées dans ces travaux. La première s’est principalement limitée à post-traiter une solution PGD pour construire une estimation de l’erreur commise, de façon non intrusive pour les codes PGD existants. La seconde consiste en une nouvelle stratégie PGD fournissant une approximation améliorée couplée à une estimation de l’erreur commise. Les diverses études comparatives sont menées dans le cadre des problèmes linéaires thermiques et en élasticité. Ces travaux ont également permis d’optimiser les méthodes de construction de champs admissibles en substituant la résolution de nombreux problèmes semblables par une solution PGD, exploitée comme un abaqueMany structural mechanics problems require the resolution of several similar numerical problems. An iterative model reduction approach, the Proper Generalized Decomposition (PGD), enables the control of the main solutions at once, by the introduction of additional parameters. However, a major drawback to its use in the industrial world is the absence of a robust error estimator to measure the quality of the solutions obtained.The approach used is based on the concept of constitutive relation error. This method consists in constructing admissible fields, thus ensuring the conservative and guaranteed aspect of the estimation of the error by reusing the maximum number of tools used in the finite elements framework. The ability to quantify the importance of the different sources of error (reduction and discretization) allows to control the main strategies of PGD resolution.Two strategies have been proposed in this work. The first was limited to post-processing a PGD solution to construct an estimate of the error committed, in a non-intrusively way for existing PGD codes. The second consists of a new PGD strategy providing an improved approximation associated with an estimate of the error committed. The various comparative studies are carried out in the context of linear thermal and elasticity problems.This work also allowed us to optimize the admissible fields construction methods by substituting the resolution of many similar problems by a PGD solution, exploited as a virtual chart
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Barasinski, Anaïs. "Modélisations du procédé de placement de fibres composites à matrice thermoplastique." Ecole centrale de Nantes, 2012. http://www.theses.fr/2012ECDN0032.
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Le procédé de placement de fibres thermoplastiques permet de fabriquer des pièces composites de grandes dimensions ayant toutes sortes de géométries (double courbure par exemple). Ce procédé est basé sur le soudage continu par fusion d’un pli composite à matrice thermoplastique sur un substrat. Il reçoit un intérêt croissant au cours des dernières années en raison de sa capacité à réaliser des pièces hors autoclave. Cette étude s’inscrit dans le cadre d’un projet aéronautique qui vise à comprendre et développer ce procédé de dépose automatisé en étudiant l’histoire thermo-mécanique de la matière au cours de la dépose, le champ de contraintes résiduelles dans la pièce après fabrication et la capabilité du procédé pour un soudage in-situ. Cette étude permet d’autre part de contribuer au développement de la tête de dépose. Afin de contrôler et d'optimiser la qualité de la pièce fabriquée, la prédiction de l’histoire thermo-mécanique appliquée durant la fabrication du stratifié est nécessaire. Dans ce travail, une modélisation thermique originale de ce procédé est proposée. L’adhésion interplis évoluant au cours de la mise en forme et ayant une influence significative sur les transferts thermiques, ce paramètre est intégré dans le modèle au moyen de résistances thermiques de contact évolutives. Les résultats numériques sont validés par des mesures expérimentales. En raison de la géométrie particulière du problème (fines bandes de pré-imprégnés de grande longueur), une méthode de résolution numérique PGD ou (Proper Generalized Decomposition) est utilisée. Cette méthode permet de contourner les problèmes liés au grand ratio existant entre la longueur de matière et son épaisseur. Elle permet par ailleurs de prendre en compte de nombreux paramètres (matériaux, ou procédés) comme extra coordonnées du modèle. Finalement, différentes fenêtres procédé, adaptées au niveau de spécifications requis sur la pièce, sont définies. Les questions de maximisation de la cristallinité et minimisation des contraintes résiduelles sont abordées afin de pouvoir s’approcher du procédé in-situ recherchéThermoplastic automated tape placement process allows the manufacture of large composite parts with all kinds of geometries (eg double curvature). This method is based on the continuous welding by fusion bonding of a thermoplastic matrix composite ply on a substrate. It received an increasing interest in recent years due to its ability to produce parts out of autoclave. This study is part of an aeronautic project that aims to understand and develop this automated deposition process through the study of the thermo-mechanical history of the material during its deposition, the residual stresses in the workpiece after manufacturing and the process capability for in-situ welding, while contributing to the development of the deposit head. In order to control and optimize the quality of the part, the prediction of the thermo-mechanical history applied during the manufacturing of the laminate is required. In this work, an original thermal modeling of this process is proposed. The bonding degree evolves during the deposition and has a significant impact on heat transfer, this parameter is included in the model using evolving thermal contact resistance. The numerical results are validated by experimental measurements. Due to the particular geometry of the problem (thin pre-impregnated tape of great length), a numerical PGD method (Proper Generalized Decomposition) is used. This method overcomes the problems associated with large ratio between the length of material and its thickness. It also allows to take into account many parameters (material, or process) as extra coordinates of the model. Finally, various process windows, appropriate to the level required specifications on the part, are defined. Questions related to the maximization of the crystallinity and the minimization of residual stresses are discussed in order to approach the one-step process sought
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Heyberger, Christophe. "PGD espace-temps adaptée pour le traitement de problèmes paramétrés." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2014. http://tel.archives-ouvertes.fr/tel-01048636.
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Cette thèse s'intéresse à la question récurrente qu'est la résolution d'un problème pour un grand nombre de configurations différentes. Malgré l'augmentation constante de la puissance de calcul que l'on connait aujourd'hui, le traitement direct d'un tel problème reste souvent hors de portée. La technique qui est développée ici est basée sur l'utilisation de la Proper Generalized Decomposition (PGD) dans le cadre de la méthode LATIN. On étudie tout d'abord la capacité de cette technique de réduction de modèle à résoudre un problème paramétré pour un espace de conception donné. Lors du traitement d'un tel problème, on génère une base réduite que l'on peut réutiliser et éventuellement enrichir en traitant un par un les problèmes correspondants aux jeux de paramètres étudiés. Le but devient alors de développer une stratégie, inspirée par la méthode " Reduced Basis ", afin d'explorer de façon rationnelle l'espace des paramètres. L'objectif étant de construire, avec le minimum de résolutions, une base réduite " complète " qui permet de résoudre tous les autres problèmes de l'espace de conception sans enrichir cette base. On commence dès lors par montrer l'existence d'une telle base complète en extrayant les informations les plus pertinentes des solutions PGD d'un problème pour tous les jeux de paramètres de l'espace de conception. On propose ensuite une stratégie rationnelle pour construire cette base complète sans la nécessité préalable de la résolution du problème pour tous les jeux de paramètres. Enfin, les performances de la méthode proposée sont illustrées sur plusieurs exemples, montrant des gains conséquents lorsque des études récurrentes doivent être menées.
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Dumon, Antoine. "Réduction dimensionnelle de type PGD pour la résolution des écoulements incompressibles." Phd thesis, Université de La Rochelle, 2011. http://tel.archives-ouvertes.fr/tel-00644565.
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L'objectif de ce travail consiste à développer la méthode de résolution PGD (Proper Generalized Decomposition), qui est une méthode de réduction de modèle où la solution est recherchée sous forme séparée, à la résolution des équations de Navier-Stokes. Dans un premier temps, cette méthode est appliquée à la résolution d'équations modèles disposant d'une solution analytique. L' équation de diffusion stationnaire 2D et 3D, l'équation de diffusion instationnaire 2D et les équations de Burgers et Stokes sont traitées. Nous montrons que dans tous ces cas la méthode PGD permet de retrouver les solutions analytiques avec une précision équivalente au modèle standard. Nous mettons également en évidence la supériorité de la PGD par rapport au modèle standard en terme de temps de calcul. En effet, dans tous ces cas, laPGD se montre beaucoup plus rapide que le solveur standard (plusieurs dizaine de fois). La résolution des équations de Navier-Stokes isothermes et anisothermes est ensuite effectuée par une discrétisation volumes finis sur un maillage décalé où le couplage vitesse-pression a été géré à l'aide d'un schéma de prédiction-correction. Dans ce cas une décomposition PGD sur les variables d'espaces uniquement a été choisie. Pour les écoulements incompressibles 2D stationnaire ou instationnaire, de type cavité entrainée et/ou différentiellement chauffé, les résultats obtenus par résolution PGD sont similaires à ceux du solveur standard avec un gain de temps significatif (la PGD est une dizaine de fois plus rapide que le solveur standard). Enfin ce travail introduit une première approche de la résolution des équations de transferts par méthode PGD en formulation spectrale. Sur les différents problèmes traités, à savoir l'équation de diffusion stationnaire, l'équation de Darcy et les équations de Navier-Sokes, la PGD a montré une précision aussi bonne que le solveur standard. Un gain de temps a été observé pour le cas de l'équation de Poisson, par contre, concernant le problème de Darcy ou les équations de Navier-Stokes les performances de la PGD en terme de temps de calcul peuvent encore être améliorées.
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Saleh, Marwan. "Étude mathématique et numérique des méthodes de réduction dimensionnelle de type POD et PGD." Thesis, La Rochelle, 2015. http://www.theses.fr/2015LAROS004/document.
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Ce mémoire de thèse est formé de quatre chapitres. Un premier chapitre présente les différentes notions et outils mathématiques utilisés dans le corps de la thèse ainsi qu’une description des résultats principaux que nous avons obtenus. Le second chapitre présente une généralisation d’un résultat obtenu par Rousselet-Chénais en 1990 qui décrit la sensibilité des sous-espaces propres d’opérateurs compacts auto-adjoints. Rousselet-Chénais se sont limités aux sous-espaces propres de dimension 1 et nous avons étendu leur résultat aux dimensions supérieures. Nous avons appliqué nos résultats à la Décomposition par Projection Orthogonale (POD) dans le cas de variation paramétrique, temporelle ou spatiale (Gappy-POD). Le troisième chapitre traite de l’estimation du flot optique avec des énergies quadratiques ou linéaires à l’infini. On montre des résultats mathématiques de convergence de la méthode de Décomposition Progressive Généralisée (PGD) dans le cas des énergies quadratiques. Notre démonstration est basée sur la décomposition de Brézis-Lieb via la convergence presque-partout de la suite gradient PGD. Une étude numérique détaillée est faite sur différents type d’images : sur les équations de transport de scalaire passif, dont le champ de déplacement est solution des équations de Navier-Stokes. Ces équations présentent un défi pour l’estimation du flot optique à cause du faible gradient dans plusieurs régions de l’image. Nous avons appliqué notre méthode aux séquences d’images IRM pour l’estimation du mouvement des organes abdominaux. La PGD a présenté une supériorité à la fois au niveau du temps de calcul (même en 2D) et au niveau de la représentation correcte des mouvements estimés. La diffusion locale des méthodes classiques (Horn & Schunck, par exemple) ralentit leur convergence contrairement à la PGD qui est une méthode plus globale par nature. Le dernier chapitre traite de l’application de la méthode PGD dans le cas d’équations elliptiques variationnelles dont l’énergie présente tous les défis aux méthodes variationnelles classiques : manque de convexité, manque de coercivité et manque du caractère borné de l’énergie. Nous démontrons des résultats de convergence, pour la topologie faible, des suites PGD (lorsqu’elles sont bien définies) vers deux solutions extrémales sur la variété de Nehari. Plusieurs questions mathématiques concernant la PGD restent ouvertes dans ce chapitre. Ces questions font partie de nos perspectives de rechercheThis thesis is formed of four chapters. The first one presents the mathematical notions and tools used in this thesis and gives a description of the main results obtained within. The second chapter presents our generalization of a result obtained by Rousselet-Chenais in 1990 which describes the sensitivity of eigensubspaces for self-adjoint compact operators. Rousselet-Chenais were limited to sensitivity for specific subspaces of dimension 1, we have extended their result to higher dimensions. We applied our results to the Proper Orthogonal Decomposition (POD) in the case of parametric, temporal and spatial variations (Gappy- POD). The third chapter discusses the optical flow estimate with quadratic or linear energies at infinity. Mathematical results of convergence are shown for the method Progressive Generalized Decomposition (PGD) in the case of quadratic energies. Our proof is based on the decomposition of Brézis-lieb via the convergence almost everywhere of the PGD sequence gradients. A detailed numerical study is made on different types of images : on the passive scalar transport equations, whose displacement fields are solutions of the Navier-Stokes equations. These equations present a challenge for optical flow estimates because of the presence of low gradient regions in the image. We applied our method to the MRI image sequences to estimate the movement of the abdominal organs. PGD presented a superiority in both computing time level (even in 2D) and accuracy representation of the estimated motion. The local diffusion of standard methods (Horn Schunck, for example) limits the convergence rate, in contrast to the PGD which is a more global approach by construction. The last chapter deals with the application of PGD method in the case of variational elliptic equations whose energy present all challenges to classical variational methods : lack of convexity, lack of coercivity and lack of boundedness. We prove convergence results for the weak topology, the PGD sequences converge (when they are well defined) to two extremal solutions on the Nehari manifold. Several mathematical questions about PGD remain open in this chapter. These questions are part of our research perspectives
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Oulghelou, Mourad. "Développement de modèles réduits adaptatifs pour le contrôle optimal des écoulements." Thesis, La Rochelle, 2018. http://www.theses.fr/2018LAROS014/document.
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La résolution des problèmes de contrôle optimal nécessite des temps de calcul et des capacités de stockage très élevés. Pour s’affranchir de ces contraintes, il est possible d’utiliser les méthodes de réduction de modèles comme la POD (Proper Orthogonal Decomposition). L’inconvénient de cette approche est que la base POD n’est valable que pour des paramètres situés dans un voisinage proche des paramètres pour lesquels elle a été construite. Par conséquent, en contrôle optimal, cette base peut ne pas être représentative de tous les paramètres qui seront proposés par l’algorithme de contrôle. Pour s’affranchir de cet handicap, une méthodologie de contrôle optimal utilisant des modèles réduits adaptatifs a été proposée dans ce manuscrit. Les bases réduites adaptées sont obtenues à l’aide de la méthode d’interpolation ITSGM (Interpolation on Tangent Subspace of Grassman Manifold) ou de la méthode d’enrichissement PGD (Proper Generalized Decomposition). La robustesse de cette approche en termes de précision et de temps de calcul a été démontrée pour le contrôle optimal (basé sur les équations adjointes) des équations 2D de réaction-diffusion et de Burgers. L’approche basée sur l’interpolation ITSGM a également été appliquée avec succès pour contrôler l’écoulement autour d’un cylindre 2D. Deux méthodes de réduction non intrusives, ne nécessitant pas la connaissance des équations du modèle étudié, ont également été proposées. Ces méthodes appelées NIMR (Non Intrusive Model Reduction) et HNIMR (Hyper Non Intrusive Model Reduction) ont été couplées à un algorithme génétique pour résoudre rapidement un problème de contrôle optimal. Le problème du contrôle optimal de l’écoulement autour d’un cylindre 2D a été étudié et les résultats ont montré l’efficacité de cette approche. En effet, l’algorithme génétique couplé avec la méthode HNIMR a permis d’obtenir les solutions avec une bonne précision en moins de 40 secondesThe numerical resolution of adjoint based optimal control problems requires high computational time and storage capacities. In order to get over these high requirement, it is possible to use model reduction techniques such as POD (Proper Orthogonal Decomposition). The disadvantage of this approach is that the POD basis is valid only for parameters located in a small neighborhood to the parameters for which it was built. Therefore, this basis may not be representative for all parameters in the optimizer’s path eventually suggested by the optimal control loop. To overcome this issue, a reduced optimal control methodology using adaptive reduced order models obtained by the ITSGM (Interpolation on a Tangent Subspace of the Grassman Manifold) method or by the PGD (Proper Generalized Decomposition) method, has been proposed in this work. The robustness of this approach in terms of precision and computation time has been demonstrated for the optimal control (based on adjoint equations) of the 2D reaction-diffusion and Burgers equations. The interpolation method ITSGM has also been validated in the control of flow around a 2D cylinder. In the context of non intrusive model reduction, two non intrusive reduction methods, which do not require knowledge of the equations of the studied model, have also been proposed. These methods called NIMR (Non-Intrusive Model Reduction) and HNIMR (Hyper Non-Intrusive Model Reduction) were developed and then coupled to a genetic algorithm in order to solve an optimal control problem in quasi-real time. The problem of optimal control of the flow around a 2D cylinder has been studied and the results have shown the effectiveness of this approach. Indeed, the genetic algorithm coupled with the HNIMR method allowed to obtain the solutions with a good accuracy in less than 40 seconds
Books on the topic "Proper Generalized Decomposition (PGD)"
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. The Proper Generalized Decomposition for Advanced Numerical Simulations. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-02865-1.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer. Springer, 2013.
Find full textBook chapters on the topic "Proper Generalized Decomposition (PGD)"
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Cueto, Elías, David González, and Icíar Alfaro. "PGD for Dynamical Problems." In Proper Generalized Decompositions, 65–89. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29994-5_5.
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Cueto, Elías, David González, and Icíar Alfaro. "PGD for Non-linear Problems." In Proper Generalized Decompositions, 39–63. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29994-5_4.
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Cueto, Elías, David González, and Icíar Alfaro. "To Begin With: PGD for Poisson Problems." In Proper Generalized Decompositions, 7–19. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29994-5_2.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. "PGD Versus SVD." In The Proper Generalized Decomposition for Advanced Numerical Simulations, 47–56. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02865-1_3.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. "PGD Solution of the Poisson Equation." In The Proper Generalized Decomposition for Advanced Numerical Simulations, 25–46. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02865-1_2.
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Cueto, Elías, Francisco Chinesta, and Antonio Huerta. "Model Order Reduction based on Proper Orthogonal Decomposition." In Separated Representations and PGD-Based Model Reduction, 1–26. Vienna: Springer Vienna, 2014. http://dx.doi.org/10.1007/978-3-7091-1794-1_1.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. "Introduction." In The Proper Generalized Decomposition for Advanced Numerical Simulations, 1–24. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02865-1_1.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. "The Transient Diffusion Equation." In The Proper Generalized Decomposition for Advanced Numerical Simulations, 57–69. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02865-1_4.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. "Parametric Models." In The Proper Generalized Decomposition for Advanced Numerical Simulations, 71–88. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02865-1_5.
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Chinesta, Francisco, Roland Keunings, and Adrien Leygue. "Advanced Topics." In The Proper Generalized Decomposition for Advanced Numerical Simulations, 89–110. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02865-1_6.
Full textConference papers on the topic "Proper Generalized Decomposition (PGD)"
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Grolet, Aurelien, and Fabrice Thouverez. "On the Use of the Proper Generalised Decomposition for Solving Nonlinear Vibration Problems." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-87538.
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This paper presents the use of the so called Proper Generalized Decomposition method (PGD) for solving nonlinear vibration problems. PGD is often presented as an a priori reduction technique meaning that the reduction basis for expressing the solution is computed during the computation of the solution itself. In this paper, the PGD is applied in addition with the Harmonic Balance Method (HBM) in order to find periodic solutions of nonlinear dynamic systems. Several algorithms are presented in order to compute nonlinear normal modes and forced solutions. Application is carried out on systems containing geometrical nonlinearity and/or friction damping. We show that the PGD is able to compute a good approximation of the solutions event with a projection basis of small size. Results are compared with a Proper Orthogonal Decomposition (POD) method showing that the PGD can sometimes provide an optimal reduction basis relative to the number of basis components.
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Nazeer, S. Mohamed, Francisco Chinesta, and Adrien Leygue. "A Novel Proper Generalized Decomposition (PGD) Based Approach for Non-Matching Grids." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-82506.
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Proper Generalized Decomposition (PGD) is a recent model reduction technique, successfully employed to solve many multidimensional problems. This method is able to circumvent, or at least alleviate, the curse of dimensionality. This method is based on the use of separated representations. By avoiding the exponential complexity of standard grid based discretization techniques, the PGD circumvents the curse of dimensionality in a variety of problems. With the PGD, the problem’s usual coordinates (e.g. space, time), but also model parameters, boundary conditions, and other sources of variability can be viewed globally as coordinates of a high-dimensional space wherein an approximate solution can efficiently be computed at once. Non-matching grids are very common in advanced scientific computing (e.g. contact problems, sub-domains coupling,).In this framework, approximate solutions from one grid to a non-matching second grid needs to be projected. This approach poses substantial numerical complexity which increases when going from one to higher dimensional interfaces. In this paper, we try to simulate a domain, which has a coarse mesh on one side and a fine mesh on other side by PGD. We show that PGD can handle these non -matching grids by using a smooth transition of the separated representation description.
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Néron, David, and Pierre Ladevèze. "A Data Compression Approach for PGD Reduced-Order Modeling." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-83008.
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This work concerns the Proper Generalized Decomposition (PGD) which is used to solve problems defined over the time-space domain and which are possibly nonlinear. The PGD is an a priori model reduction technique which allows to decrease CPU costs drastically by seeking the solution of a problem in a reduced-order basis generated automatically by a dedicated algorithm. The algorithm which is used herein is the LATIN method, a non incremental iterative strategy which enables to generate the approximations of the solution over the entire time-space domain by successive enrichments. The problematics which is addressed in this paper is the construction of the resulting reduced-order models along the iterations, which can represent a large part of the remaining global CPU cost. To make easier the construction of these models, we propose an algebraic framework adapted to PGD which allows to define a “compressed” version of the data. The space of the compressed fields exhibits some very interesting properties that lead to an important increase of the performances of the global strategy.
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Niroomandi, Siamak, Felipe Bordeu, Iciar Alfaro, David Gonzalez, Adrien Leygue, Elias Cueto, and Francisco Chinesta. "Real-Time Simulation for Virtual Surgery in a PGD Framework." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-82541.
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We analyze here the use of proper generalized decompositions (PGD) for real-time simulation of living soft tissues in virtual surgery environments. These tissues are usually modeled as hyperelastic solids, and therefore present important difficulties for their simulation under real-time constraints (i.e., feedback rates on the order of1 kHz). PGD techniques provide with physics-based meta-models without any prior computer experiment, that can be used on-line for the simulation under such severe constraints. These metemodels are constructed on the assumption of the problem to be multi-dimensional, with parameters as additional space dimensions. These parameters, in this case, are taken as the position of contact of surgical tool and organ, modulus of the contact force and orientation (a 9D problem). PGD techniques allow to solve efficiently these high-dimensional problems without the burden associated to the application of mesh-based techniques to these problems.
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Behzad, Fariduddin, Brian T. Helenbrook, and Goodarz Ahmadi. "On Reduced Order Modeling of Transient Flows Using the Proper Orthogonal Decomposition." In ASME 2012 Fluids Engineering Division Summer Meeting collocated with the ASME 2012 Heat Transfer Summer Conference and the ASME 2012 10th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/fedsm2012-72141.
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Reduced-order modeling (ROM) of transient fluid flows using the proper orthogonal decomposition (POD) was studied. Particular attention was given to incompressible, unsteady flow over a two-dimensional NACA0015 airfoil in the laminar regime. When the airfoil sheds vortices, a transient blowing through a jet placed at the 10% chord location was imposed. POD modes were derived from the numerical solution of the flow obtained using an hp-finite element method. The ROM was obtained by a streamwise-upwind-Petrov-Galerkin (SUPG) projection of the incompressible Navier–Stokes equations onto the space spanned by the POD modes. The extraction of accurate POD-based reduced order model of this flow was explored using three different POD mode generation methods. The first approach was the split method, which superposes modes derived from simulations of the blowing jet with no flow and simulations of the baseline flow with no jet. The second method combined POD modes derived from simulations having both the jet and flow with modes obtained from simulation of only the flow. These modes were generated after the simulations reached the periodic state. The third and newly proposed approach was to generate a set of modes called “Generalized POD basis functions.” These modes were derived from simulations where the jet’s flow amplitude is varied slowly. For each method, the results were compared with detailed Finite Element solutions and the accuracy and efficiency of different methods were evaluated. The newly proposed “Generalized POD basis functions” approach predicted the transient response of the system most accurately.
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Canales, D., Ch Ghnatios, A. Leygue, F. Chinesta, I. Alfaro, E. Cueto, E. Feulvarch, and J. M. Bergheau. "Numerical Simulation of Friction Stir Welding by an Efficient 3D Updated Lagrangian Technique." In ASME 2014 12th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/esda2014-20285.
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Friction Stir Welding (FSW) is a welding technique which since its invention in 1991 is of great interest to the industry for its many advantages. Despite being widely used, its physical foundations and its relation to the technological parameters of the process are not known in detail. Numerical simulations are a powerful tool to achieve a greater understanding of the physics of the problem. Although several approaches can be found in the literature for FSW, all of them present different limitations that restrict their applicability to the industry. This paper presents a new solution strategy that combines a meshless method, the Natural Element Method (NEM), with a solution separated representation making use of the Proper Generalized Decomposition (PGD), for creating a new powerful updated-Lagrangian method for addressing the 3D model while maintaining a 2D computational complexity.
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Sabetghadam, Fereydoun. "Generalization of the RANS Equations Using Mean Modal Decomposition of the Navier Stokes Equations." In ASME 7th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2004. http://dx.doi.org/10.1115/esda2004-58395.
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A generalization in the Reynolds decomposition and averaging are proposed in this paper. The method is directly applied to the Navier Stokes (N-S) equations to construction of a generalized Reynolds Averaged Navier Stokes (RANS) equations. The formulation which is presented for the fields realized in a suitable ensemble, is based on a two part decomposition. One part is an approximate unique representation of the field and when reconstruction of the field, will repeat in all ensemble elements. The other part represents deviation of the real field from the approximate part and therefore is different in any mode and each ensemble element. The decomposition is applied in both spatial and temporal fashions. In the temporal decomposition, a system of Partial Differential Equations (PDEs) is obtained that is nonclosed, coupled and second order in space and its zeroth mode is the classical Reynolds averaged values of the field. In the spatial decomposition whereas, a first order system of nonclosed PDEs is obtained which could be seen as an alternative version of the Proper Orthogonal Decomposition (POD) or the Coherent Vortex Simulation (CVS) methods. In both fashions however, there are some terms that must be modeled just like as the classical closure problem in the RANS method. The method is applied on a one dimensional mixed random-nonrandom field and successfully extracted the coherent part of the field.
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Badías, A., D. González, I. Alfaro, F. Chinesta, and E. Cueto. "Local proper generalized decomposition." In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE OF GLOBAL NETWORK FOR INNOVATIVE TECHNOLOGY AND AWAM INTERNATIONAL CONFERENCE IN CIVIL ENGINEERING (IGNITE-AICCE’17): Sustainable Technology And Practice For Infrastructure and Community Resilience. Author(s), 2017. http://dx.doi.org/10.1063/1.5008205.
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Henneron, Thomas, and Stephane Clenet. "Parametric analysis of magnetoharmonic problem based on proper generalized decomposition." In 2016 IEEE Conference on Electromagnetic Field Computation (CEFC). IEEE, 2016. http://dx.doi.org/10.1109/cefc.2016.7816328.
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Ghnatios, C., F. Chinesta, E. Cueto, A. Leygue, A. Poitou, Francisco Chinesta, Yvan Chastel, and Mohamed El Mansori. "Optimizing Composites Forming Processes by Applying the Proper Generalized Decomposition." In INTERNATIONAL CONFERENCE ON ADVANCES IN MATERIALS AND PROCESSING TECHNOLOGIES (AMPT2010). AIP, 2011. http://dx.doi.org/10.1063/1.3552440.
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