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Journal articles on the topic 'Proper Generalized Decomposition (PGD)'

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Published: 4 June 2021

Last updated: 9 February 2022

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1

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.

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Garikapati, Hasini, Sergio Zlotnik, Pedro Díez, Clemens V. Verhoosel, and E. Harald van Brummelen. "A Proper Generalized Decomposition (PGD) approach to crack propagation in brittle materials: with application to random field material properties." Computational Mechanics 65, no. 2 (October 26, 2019): 451–73. http://dx.doi.org/10.1007/s00466-019-01778-0.

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

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Montés, Nicolas, Francisco Chinesta, Marta C. Mora, Antonio Falcó, Lucia Hilario, Nuria Rosillo, and Enrique Nadal. "Real-Time Path Planning Based on Harmonic Functions under a Proper Generalized Decomposition-Based Framework." Sensors 21, no. 12 (June 8, 2021): 3943. http://dx.doi.org/10.3390/s21123943.

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This paper presents a real-time global path planning method for mobile robots using harmonic functions, such as the Poisson equation, based on the Proper Generalized Decomposition (PGD) of these functions. The main property of the proposed technique is that the computational cost is negligible in real-time, even if the robot is disturbed or the goal is changed. The main idea of the method is the off-line generation, for a given environment, of the whole set of paths from any start and goal configurations of a mobile robot, namely the computational vademecum, derived from a harmonic potential field in order to use it on-line for decision-making purposes. Up until now, the resolution of the Laplace or Poisson equations has been based on traditional numerical techniques unfeasible for real-time calculation. This drawback has prevented the extensive use of harmonic functions in autonomous navigation, despite their powerful properties. The numerical technique that reverses this situation is the Proper Generalized Decomposition. To demonstrate and validate the properties of the PGD-vademecum in a potential-guided path planning framework, both real and simulated implementations have been developed. Simulated scenarios, such as an L-Shaped corridor and a benchmark bug trap, are used, and a real navigation of a LEGO®MINDSTORMS robot running in static environments with variable start and goal configurations is shown. This device has been selected due to its computational and memory-restricted capabilities, and it is a good example of how its properties could help the development of social robots.

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Müller, Fabian, Lucas Crampen, Thomas Henneron, Stephane Clénet, and Kay Hameyer. "Model order reduction techniques applied to magnetodynamic T-Ω-formulation." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 39, no. 5 (May 11, 2020): 1057–69. http://dx.doi.org/10.1108/compel-01-2020-0025.

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Purpose The purpose of this paper is to use different model order reduction techniques to cope with the computational effort of solving large systems of equations. By appropriate decomposition of the electromagnetic field problem, the number of degrees of freedom (DOF) can be efficiently reduced. In this contribution, the Proper Generalized Decomposition (PGD) and the Proper Orthogonal Decomposition (POD) are used in the frame of the T-Ω-formulation, and the feasibility is elaborated. Design/methodology/approach The POD and the PGD are two methods to reduce the model order. Particularly in the context of eddy current problems, conventional time-stepping algorithms can lead to many numerical simulations of the studied problem. To simulate the transient field, the T-Ω-formulation is used which couples the magnetic scalar potential and the electric vector potential. In this paper, both methods are studied on an academic example of an induction furnace in terms of accuracy and computational effort. Findings Using the proposed reduction techniques significantly reduces the DOF and subsequently the computational effort. Further, the feasibility of the combination of both methods with the T-Ω-formulation is given, and a fundamental step toward fast simulation of eddy current problems is shown. Originality/value In this paper, the PGD is combined for the first time with the T-Ω-formulation. The application of the PGD and POD and the following comparison illustrate the great potential of these techniques in combination with the T-Ω-formulation in context of eddy current problems.

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Dumon, A., C. Allery, and A. Ammar. "Simulation of Heat and Mass Transport in a Square Lid-Driven Cavity with Proper Generalized Decomposition (PGD)." Numerical Heat Transfer, Part B: Fundamentals 63, no. 1 (January 2013): 18–43. http://dx.doi.org/10.1080/10407790.2012.724991.

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Ghnatios, Chady, Christian H. Mathis, Rok Simic, Nicholas D. Spencer, and Francisco Chinesta. "Modeling soft, permeable matter with the proper generalized decomposition (PGD) approach, and verification by means of nanoindentation." Soft Matter 13, no. 25 (2017): 4482–93. http://dx.doi.org/10.1039/c7sm00246g.

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Giner, Eugenio, Brice Bognet, Juan J. Ródenas, Adrien Leygue, F. Javier Fuenmayor, and Francisco Chinesta. "The Proper Generalized Decomposition (PGD) as a numerical procedure to solve 3D cracked plates in linear elastic fracture mechanics." International Journal of Solids and Structures 50, no. 10 (May 2013): 1710–20. http://dx.doi.org/10.1016/j.ijsolstr.2013.01.039.

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Niroomandi, Siamak, David Gonzalez, Iciar Alfaro, Felipe Bordeu, Adrien Leygue, Elias Cueto, and Francisco Chinesta. "Real Time Simulation of Non-Linear Solids by PGD Techniques." Key Engineering Materials 504-506 (February 2012): 467–72. http://dx.doi.org/10.4028/www.scientific.net/kem.504-506.467.

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We analyse here how Dynamic Data Driven Application Systems (DDDAS) can constitute a valuable tool in the field of forming processes. Simulation tools in the field of DDDAS are required to provide a response in real-time, a requisite that is often too severe for complex problems. Here, we consider that of hyperelasticity, commonly used in different fields such as rubber manufacturing or surgery, for instance. We analyse here how model reduction techniques, and particularly Proper Generalized Decompositions (PGD) methods can provide a suitable response to the strong requirements posed by DDDAS. We will consider two different approaches to the problem. The first one is an explicit approach that nevertheless provides with good results. The second one is based on the use of Asymptotic Numerical Method.

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Canales, Diego, Elias Cueto, Eric Feulvarch, and Francisco Chinesta. "First Steps towards Parametric Modeling of FSW Processes by Using Advanced Separated Representations: Numerical Techniques." Key Engineering Materials 611-612 (May 2014): 513–20. http://dx.doi.org/10.4028/www.scientific.net/kem.611-612.513.

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Friction Stir Welding (FSW) is a welding technique the more and more demanded in industry by its multiple advantages. Despite its wide use, its physical foundations and the effect of the process parameters have not been fully elucidated. Numerical simulations are a powerful tool to achieve a greater understanding in the physics of the problem. Although several approaches can be found in the literature for simulating FSW, all of them present different limitations that restrict their applicability in industrial applications. This paper presents a new solution strategy that combines a robust approximation method, based on natural neighborhood interpolation, with a solution separated representation making use of the Proper Generalized Decomposition (PGD), for creating a new 3D updated-Lagrangian strategy for addressing the 3D model while keeping a 2D computational complexity

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Kuznetsov, A. V., D. G. Blinov, A. A. Avramenko, I. V. Shevchuk, A. I. Tyrinov, and I. A. Kuznetsov. "Approximate modelling of the leftward flow and morphogen transport in the embryonic node by specifying vorticity at the ciliated surface." Journal of Fluid Mechanics 738 (December 13, 2013): 492–521. http://dx.doi.org/10.1017/jfm.2013.588.

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AbstractIn this paper, we have developed an approximate method for modelling the flow of embryonic fluid in a ventral node. We simplified the problem as flow in a two-dimensional cavity; the effect of rotating cilia was modelled by specifying a constant vorticity at the edge of the ciliated layer. We also developed an approximate solution for morphogen transport in the nodal pit. The solutions were obtained utilizing the proper generalized decomposition (PGD) method. We compared our approximate solutions with the results of numerical simulation of flow caused by the rotation of 81 cilia, and obtained reasonable agreement in most of the flow domain. We discuss locations where agreement is less accurate. The obtained semi-analytical solutions simplify the analysis of flow and morphogen distribution in a nodal pit.

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Giunta, Gaetano, Salim Belouettar, Olivier Polit, Laurent Gallimard, Philippe Vidal, and Michele D’ottavio. "Hierarchical Beam Finite Elements Based Upon a Variables Separation Method." International Journal of Applied Mechanics 08, no. 02 (March 2016): 1650026. http://dx.doi.org/10.1142/s1758825116500265.

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A family of hierarchical one-dimensional beam finite elements developed within a variables separation framework is presented. A Proper Generalized Decomposition (PGD) is used to divide the global three-dimensional problem into two coupled ones: one defined on the cross-section space (beam modeling kinematic approximation) and one belonging to the axis space (finite element solution). The displacements over the cross-section are approximated via a Unified Formulation (UF). A Lagrangian approximation is used along the beam axis. The resulting problems size is smaller than that of the classical equivalent finite element solution. The approach is, then, particularly attractive for higher-order beam models and refined axial meshes. The numerical investigations show that the proposed method yields accurate yet computationally affordable three-dimensional displacement and stress fields solutions.

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Al Takash, Ahmad, Marianne Beringhier, Mohammad Hammoud, and Jean-Claude Grandidier. "A mixed PGD-a priori time basis strategy for the simulation of cyclic transient thermal behavior." Mechanics & Industry 21, no. 6 (2020): 606. http://dx.doi.org/10.1051/meca/2020082.

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The knowledge of the service life of polymers under cyclic loading, widely used in industrial applications, is required and usually based on the use of methods necessitating an accurate prediction of the stabilized cycle. This implies a large computation time using the Finite Element Method (FEM) since it requires a large number of cycles for polymers. To alleviate this difficulty, a model order reduction method can be used. In this paper, a mixed strategy is investigated. Through the Proper Generalized Decomposition Method (PGD) framework, this strategy combines the Fast Fourier Transform (FFT) to create a priori time basis and the FEM to compute the related spatial modes. The method is applied to 3D thermal problems under cyclic loadings. The robustness of the proposed strategy is discussed for various boundary conditions, multi-times, and different cyclic loadings. A large time saving is obtained proving the interest of this alternative strategy to deal with fatigue simulations.

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Germoso, Claudia, Jean Louis Duval, and Francisco Chinesta. "Harmonic-Modal Hybrid Reduced Order Model for the Efficient Integration of Non-Linear Soil Dynamics." Applied Sciences 10, no. 19 (September 27, 2020): 6778. http://dx.doi.org/10.3390/app10196778.

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Nonlinear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis, and more concretely laboratory data, indicate that the stress-strain relationship of soils is nonlinear and exhibits hysteresis. An equivalent linearization method, in which non-linear characteristics of shear modulus and damping factor of soils are modeled as equivalent linear relations of the shear strain is usually applied, but this assumption, however, may lead to a conservative approach of the seismic design. In this paper, we propose an alternative analysis formulation, able to address forced response simulation of soils exhibiting their characteristic nonlinear behavior. The proposed approach combines ingredients of modal and harmonic analyses enabling efficient time-integration of nonlinear soil behaviors based on the offline construction of a dynamic response parametric solution by using Proper Generalized Decomposition (PGD)-based model order reduction technique.

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Sancarlos-González, Abel, Manuel Pineda-Sanchez, Ruben Puche-Panadero, Angel Sapena-Bano, Martin Riera-Guasp, Javier Martinez-Roman, Juan Perez-Cruz, and Jose Roger-Folch. "Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors." Open Physics 15, no. 1 (December 29, 2017): 929–35. http://dx.doi.org/10.1515/phys-2017-0113.

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AbstractAC lines of industrial busbar systems are usually built using conductors with rectangular cross sections, where each phase can have several parallel conductors to carry high currents. The current density in a rectangular conductor, under sinusoidal conditions, is not uniform. It depends on the frequency, on the conductor shape, and on the distance between conductors, due to the skin effect and to proximity effects. Contrary to circular conductors, there are not closed analytical formulas for obtaining the frequency-dependent impedance of conductors with rectangular cross-section. It is necessary to resort to numerical simulations to obtain the resistance and the inductance of the phases, one for each desired frequency and also for each distance between the phases’ conductors. On the contrary, the use of the parametric proper generalized decomposition (PGD) allows to obtain the frequency-dependent impedance of an AC line for a wide range of frequencies and distances between the phases’ conductors by solving a single simulation in a 4D domain (spatial coordinatesxandy, the frequency and the separation between conductors). In this way, a general “virtual chart” solution is obtained, which contains the solution for any frequency and for any separation of the conductors, and stores it in a compact separated representations form, which can be easily embedded on a more general software for the design of electrical installations. The approach presented in this work for rectangular conductors can be easily extended to conductors with an arbitrary shape.

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Sancarlos, Abel, Chady Ghnatios, Jean-Louis Duval, Nicolas Zerbib, Elias Cueto, and Francisco Chinesta. "Fast Computation of Multi-Parametric Electromagnetic Fields in Synchronous Machines by Using PGD-Based Fully Separated Representations." Energies 14, no. 5 (March 7, 2021): 1454. http://dx.doi.org/10.3390/en14051454.

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A novel Model Order Reduction (MOR) technique is developed to compute high-dimensional parametric solutions for electromagnetic fields in synchronous machines. Specifically, the intrusive version of the Proper Generalized Decomposition (PGD) is employed to simulate a Permanent-Magnet Synchronous Motor (PMSM). The result is a virtual chart allowing real-time evaluation of the magnetic vector potential as a function of the operation point of the motor, or even as a function of constructive parameters, such as the remanent flux in permanent magnets. Currently, these solutions are highly demanded by the industry, especially with the recent developments in the Electric Vehicle (EV). In this framework, standard discretization techniques require highly time-consuming simulations when analyzing, for instance, the noise and vibration in electric motors. The proposed approach is able to construct a virtual chart within a few minutes of off-line simulation, thanks to the use of a fully separated representation in which the solution is written from a series of functions of the space and parameters coordinates, with full space separation made possible by the use of an adapted geometrical mapping. Finally, excellent performances are reported when comparing the reduced-order model with the more standard and computationally costly Finite Element solutions.

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Ghnatios, Chady, Khalil El Rai, Nicolas Hascoet, Pierre-Adrien Pires, Jean-Louis Duval, Jon Lambarri, Jean-Yves Hascoet, and Francisco Chinesta. "Reduced order modeling of selective laser melting: from calibration to parametric part distortion." International Journal of Material Forming 14, no. 5 (March 31, 2021): 973–86. http://dx.doi.org/10.1007/s12289-021-01613-z.

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AbstractAdditive manufacturing is an appealing solution to produce geometrically complex parts, difficult to manufacture using traditional technologies. The extreme process conditions, in particular the high temperature, complex interactions and couplings, and rich metallurgical transformations that this process entails, are at the origin of numerous process defects. Therefore, the numerical simulation of the process is gaining the interest of both the scientific and the industrial communities. However, simulating that process demands impressive computational resources, limiting high resolution simulations to the microscopic and mesoscopic scales. This paper proposes a thermo-mechanical modeling framework at the process scale as well as its associated reduced order simulation counterpart, enabling the parametric evaluation of the part distortion. It deeply addresses the process calibration using a high-resolution computational procedure based on the use of an in-plane-out-of-plane separated representation at the heart of the so-called Proper Generalized Decomposition (PGD), as well as the analysis of the transient thermal effects, defining the conditions in which the thermal and mechanical analyses can be decoupled.

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Montero-Chacón, Francisco, José Sanz-Herrera, and Manuel Doblaré. "Computational Multiscale Solvers for Continuum Approaches." Materials 12, no. 5 (February 26, 2019): 691. http://dx.doi.org/10.3390/ma12050691.

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Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest—e.g., mechanical, fluid, thermal, or electromagnetic—involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper.

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Badías, Alberto, David González, Iciar Alfaro, Francisco Chinesta, and Elias Cueto. "Local proper generalized decomposition." International Journal for Numerical Methods in Engineering 112, no. 12 (June 20, 2017): 1715–32. http://dx.doi.org/10.1002/nme.5578.

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Dumon, A., C. Allery, and A. Ammar. "Proper general decomposition (PGD) for the resolution of Navier–Stokes equations." Journal of Computational Physics 230, no. 4 (February 2011): 1387–407. http://dx.doi.org/10.1016/j.jcp.2010.11.010.

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Chinesta, F., A. Ammar, and E. Cueto. "Proper generalized decomposition of multiscale models." International Journal for Numerical Methods in Engineering 83, no. 8-9 (December 1, 2009): 1114–32. http://dx.doi.org/10.1002/nme.2794.

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Leon, Angel, Anais Barasinski, Emmanuelle Abisset-Chavanne, Elias Cueto, and Francisco Chinesta. "Wavelet-based multiscale proper generalized decomposition." Comptes Rendus Mécanique 346, no. 7 (July 2018): 485–500. http://dx.doi.org/10.1016/j.crme.2018.04.013.

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Massarotti, N., A. Mauro, and V. Trombetta. "Proper generalized decomposition for geothermal applications." Thermal Science and Engineering Progress 23 (June 2021): 100882. http://dx.doi.org/10.1016/j.tsep.2021.100882.

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Huerta, Antonio, Enrique Nadal, and Francisco Chinesta. "Proper generalized decomposition solutions within a domain decomposition strategy." International Journal for Numerical Methods in Engineering 113, no. 13 (January 19, 2018): 1972–94. http://dx.doi.org/10.1002/nme.5729.

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Ammar, Amine, Francisco Chinesta, Elías Cueto, and Manuel Doblaré. "Proper generalized decomposition of time-multiscale models." International Journal for Numerical Methods in Engineering 90, no. 5 (December 12, 2011): 569–96. http://dx.doi.org/10.1002/nme.3331.

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Aquino, W., J. C. Brigham, C. J. Earls, and N. Sukumar. "Generalized finite element method using proper orthogonal decomposition." International Journal for Numerical Methods in Engineering 79, no. 7 (August 13, 2009): 887–906. http://dx.doi.org/10.1002/nme.2604.

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Néron, David, and Pierre Ladevèze. "Proper Generalized Decomposition for Multiscale and Multiphysics Problems." Archives of Computational Methods in Engineering 17, no. 4 (October 27, 2010): 351–72. http://dx.doi.org/10.1007/s11831-010-9053-2.

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Heyberger, Christophe, Pierre-Alain Boucard, and David Néron. "Multiparametric analysis within the proper generalized decomposition framework." Computational Mechanics 49, no. 3 (September 17, 2011): 277–89. http://dx.doi.org/10.1007/s00466-011-0646-x.

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González, D., F. Masson, F. Poulhaon, A. Leygue, E. Cueto, and F. Chinesta. "Proper Generalized Decomposition based dynamic data driven inverse identification." Mathematics and Computers in Simulation 82, no. 9 (May 2012): 1677–95. http://dx.doi.org/10.1016/j.matcom.2012.04.001.

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Falcó, A., L. Hilario, N. Montés, and M. C. Mora. "Numerical strategies for the Galerkin–proper generalized decomposition method." Mathematical and Computer Modelling 57, no. 7-8 (April 2013): 1694–702. http://dx.doi.org/10.1016/j.mcm.2011.11.012.

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El Hamidi, Abdallah, Marwan Saleh, Nicolas Papadakis, and B. Denis Senneville. "A proper generalized decomposition approach for optical flow estimation." Mathematical Methods in the Applied Sciences 43, no. 8 (February 8, 2020): 5339–56. http://dx.doi.org/10.1002/mma.6275.

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Montier, L., T. Henneron, S. Clenet, and B. Goursaud. "Proper Generalized Decomposition Applied on a Rotating Electrical Machine." IEEE Transactions on Magnetics 54, no. 3 (March 2018): 1–4. http://dx.doi.org/10.1109/tmag.2017.2761359.

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Qin, Zhi, Hakeim Talleb, and Zhuoxiang Ren. "A Proper Generalized Decomposition-Based Solver for Nonlinear Magnetothermal Problems." IEEE Transactions on Magnetics 52, no. 2 (February 2016): 1–9. http://dx.doi.org/10.1109/tmag.2015.2492462.

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Niroomandi, Siamak, Icíar Alfaro, David González, Elías Cueto, and Francisco Chinesta. "Model order reduction in hyperelasticity: a proper generalized decomposition approach." International Journal for Numerical Methods in Engineering 96, no. 3 (July 15, 2013): 129–49. http://dx.doi.org/10.1002/nme.4531.

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Zou, X., M. Conti, P. Díez, and F. Auricchio. "A nonintrusive proper generalized decomposition scheme with application in biomechanics." International Journal for Numerical Methods in Engineering 113, no. 2 (September 7, 2017): 230–51. http://dx.doi.org/10.1002/nme.5610.

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Ibáñez, Rubén, Amine Ammar, Elías Cueto, Antonio Huerta, Jean‐Louis Duval, and Francisco Chinesta. "Multiscale proper generalized decomposition based on the partition of unity." International Journal for Numerical Methods in Engineering 120, no. 6 (July 23, 2019): 727–47. http://dx.doi.org/10.1002/nme.6154.

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Reis, Jonatha, J. P. Moitinho de Almeida, Pedro Díez, and Sergio Zlotnik. "Error estimation for proper generalized decomposition solutions: A dual approach." International Journal for Numerical Methods in Engineering 121, no. 23 (August 4, 2020): 5275–94. http://dx.doi.org/10.1002/nme.6452.

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Ammar, Amine. "The proper generalized decomposition: a powerful tool for model reduction." International Journal of Material Forming 3, no. 2 (September 24, 2009): 89–102. http://dx.doi.org/10.1007/s12289-009-0647-x.

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Berger, Julien, Marx Chhay, Sihem Guernouti, and Monika Woloszyn. "Proper generalized decomposition for solving coupled heat and moisture transfer." Journal of Building Performance Simulation 8, no. 5 (August 15, 2014): 295–311. http://dx.doi.org/10.1080/19401493.2014.932012.

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Henneron, T., and S. Clenet. "Application of the Proper Generalized Decomposition to Solve Magnetoelectric Problem." IEEE Transactions on Magnetics 54, no. 3 (March 2018): 1–4. http://dx.doi.org/10.1109/tmag.2017.2762702.

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Ghnatios, Chady, Ilige Hage, and Najib Metni. "Modeling the human knee joint using the Proper Generalized Decomposition." MATEC Web of Conferences 261 (2019): 01006. http://dx.doi.org/10.1051/matecconf/201926101006.

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Abstract:

Nowadays, human joints specifically movable are active research topics. The lack of effective replacements and the inefficient natural healing of these joints hinders any athlete from pursuing his career if injured in his joints. Therefore, researchers are testing innovative soft materials and biphasic materi- als as replacements of human joints. However, the lack of effective mechanical modeling is slowing the development of new replacements. In this work, we tackle the mechanical modeling of the synovial joint in a human knee. The tibiofemoral joint is modelled during impact. This joint is basically made of a cartilage, a meniscus (both a biphasic material) and the synovial fluid. The modeling is performed using Brinkman equation. However, the rich physics in- volved in the thickness direction requires a large number of degrees of freedom in the mesh to represent the physical phenomenon taking place in a knee joint. Thus, the use of model order reduction techniques appears to be an appealing approach in this situation. In fact, the proper generalized decomposition re- duced the number of degrees of freedom by using domain decomposition. The result of this work shows the pressure and fluid flow in the synovial joint under impact. A post treatment of the solution estimates the force held by each of the fluid and solid components of the cartilage joint. This model could be used to the human knee to estimate its components’ velocities and pressure fields while performing an activity.

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Prince, Zachary M., and Jean C. Ragusa. "PARAMETRIC NEUTRON DIFFUSION USING PROPER GENERALIZED DECOMPOSITION FOR UNCERTAINTY QUANTIFICATION." Proceedings of the International Conference on Nuclear Engineering (ICONE) 2019.27 (2019): 2114. http://dx.doi.org/10.1299/jsmeicone.2019.27.2114.

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