Proper Generalized Decomposition for Multiscale and Multiphysics Problems

Néron, David; Ladevèze, Pierre

doi:10.1007/s11831-010-9053-2

Cited by 78 publications

(49 citation statements)

References 71 publications

(112 reference statements)

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“…We conclude this section by mentioning also some additional techniques quite close to POD for generating efficiently reduced spaces, the Centroidal Voronoi Tessellation (CVT) [25,26,46] and the Proper Generalized Decomposition (PGD) method [92], which has been recently applied to the solution of Navier-Stokes equations [47,117]. Recent contributions are also contained in MS&A Vol.…”

Section: Reduced Basis Construction By Greedy Algorithmsmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

162

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Section: Reduced Basis Construction By Greedy Algorithmsmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

162

179

View full text Add to dashboard Cite

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“…PGD was originally introduced into the LATIN method as part of the solver under the name ''radial approximation'' [28]. It has been shown in previous works that PGD leads to a significant reduction in computation time [36,37,44].…”

Section: Proper Generalized Decompositionmentioning

confidence: 99%

“…If the quality of the solution is insufficient, the ROB is enriched by adding new PGD functions using a greedy power algorithm based on the minimization of a functional built from the verification of the search direction (A.8) and (A.10). Further details on the introduction of PGD into the microproblems and examples showing the ability of the method to reduce the computation cost of the microproblems can be found in [37,44,48]. The novelty of the present work lies in the reduction of the computation and storage cost of the homogenized operator by extending the use of the PGD technique to its construction.…”

Section: Proper Generalized Decompositionmentioning

confidence: 99%

A PGD-based homogenization technique for the resolution of nonlinear multiscale problems

Cremonesi

Néron

Guidault

et al. 2013

Computer Methods in Applied Mechanics and Engineering

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This paper deals with the offline resolution of nonlinear multiscale problems leading to a PGD-reduced model from which one can derive microinformation which is suitable for design. Here, we focus on an improvement to the Proper Generalized Decomposition (PGD) technique which is used for the calculation of the homogenized operator and which plays a central role in our multiscale computational strategy. This homogenized behavior is calculated offline using the LATIN multiscale strategy, and PGD leads to a drastic reduction in CPU cost. Indeed, the multiscale aspect of the LATIN method is based on splitting the unknowns between a macroscale and a microcomplement. This macroscale is used to capture the homogenized behavior of the structure and then to accelerate the convergence of the iterative algorithm. The novelty of this work lies in the reduction of the computation cost of building the homogenized operator. In order to do that, the problems defined within each subdomain are solved using a new PGD representation of the unknowns in which the separation of the unknowns is performed not only in time and in space, but also in terms of the interface macrodisplacements. The numerical example presented shows that the convergence rate of the approach is not affected by this new representation and that a significant reduction in computation cost can be achieved, particularly in the case of nonlinear behavior. The technique used herein is then particularly important to enhance the performances of the LATIN strategy but is not limited to this method. The more general path which is followed in this paper is to solve the microproblems arising in the homogenization for all the configurations that can be encountered in the calculation. An additional and novel benefit of this approach is that it includes the calculation of the homogenized tangent operator which gives, over the whole time interval and for any cell, the relation between the perturbation in terms of macroforces and the perturbation in terms of macrodisplacements. In linear viscoelasticity, this operator represents the homogenized behavior of the structure exactly

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“…In this context, the need of optimising non-linear multiphysics problems makes necessary to develop numerical techniques which can efficiently deal with the high computational cost characterising such applications. A widespread strategy is to consider the formulation of Reduced Order Models, which can be implemented by adopting either the Proper Orthogonal Decomposition (POD) method [2,3], or the proper generalised decomposition (PGD) technique [4,5]. The discussion in this paper only considers POD-based ROMs, from now on referred to as ROMs.…”

Section: Introductionmentioning

confidence: 99%

General treatment of essential boundary conditions in reduced order models for non-linear problems

Cosimo

Cardona

Idelsohn

2016

Adv. Model. and Simul. in Eng. Sci.

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Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.

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Proper Generalized Decomposition for Multiscale and Multiphysics Problems

Cited by 78 publications

References 71 publications

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

A PGD-based homogenization technique for the resolution of nonlinear multiscale problems

General treatment of essential boundary conditions in reduced order models for non-linear problems

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