Proper Generalized Decomposition computational methods on a benchmark problem: introducing a new strategy based on Constitutive Relation Error minimization

Allier, Pierre-Eric; Chamoin, Ludovic

doi:10.1186/s40323-015-0038-4

Cited by 25 publications

(25 citation statements)

References 23 publications

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“…In what follows, we consider also the possibility of a parametric problem, where the conductivity k acts itself as a parameter of the problem. This problem has also been analyzed, among others, by Allier et al…”

Section: Examplesmentioning

confidence: 89%

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Local proper generalized decomposition

Badías

González

Alfaro

et al. 2017

Numerical Meth Engineering

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One of the main difficulties a reduced order method could face is the poor separability of the solution. This problem is common to both a posteriori model order reduction (Proper Orthogonal Decomposition, Reduced Basis) and a priori (Proper Generalized Decomposition) model order reduction. Early approaches to solve it include the construction of local reduced order models in the framework of POD. We present here an extension of local models in a PGD -and thus, a priori-context. Three different strategies are introduced to estimate the size of the different patches or regions in the solution manifold where PGD is applied. As will be noticed, no gluing or special technique is needed to deal with the resulting set of local reduced order models, in contrast to most POD local approximations. The resulting method can be seen as a sort of a priori manifold learning or non-linear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.KEY WORDS: local model order reduction, proper generalized decomposition, kernel principal component analysis, non linear dimensionality reduction.

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Section: Examplesmentioning

confidence: 89%

“…In general, this basis, even if it is globally optimal, can lead to very poor results if the manifold structure of the solution is very intricate. Previous works in the field are included . To avoid these problems, the idea of local basis arises naturally.…”

Section: Introductionmentioning

confidence: 99%

Local proper generalized decomposition

Badías

González

Alfaro

et al. 2017

Numerical Meth Engineering

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show abstract

“…In the context of model verification, the CRE concept was already used to control PGD approximations (see a posteriori error estimates developed in other studies()) or to directly drive the PGD process with CRE minimization . Nevertheless, the use of PGD in CRE implementation has never been investigated and we wish to show here that there are major advantages to do so, in particular for the construction of equilibrated fields.…”

Section: Introductionmentioning

confidence: 99%

Towards simplified and optimized a posteriori error estimation using PGD reduced models

Allier

Chamoin

2017

Numerical Meth Engineering

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Summary The paper deals with the use of model order reduction within a posteriori error estimation procedures in the context of the finite element method. More specifically, it focuses on the constitutive relation error concept, which has been widely used over the last 40 years for FEM verification of computational mechanics models. A technical key‐point when using constitutive relation error is the construction of admissible fields, and we propose here to use the proper generalized decomposition to facilitate this task. In addition to making the implementation into commercial FE software easier, it is shown that the use of proper generalized decomposition enables to optimize the verification procedure and to get both accurate and reasonably expensive upper bounds on the discretization error. Numerical illustrations are presented to assess the performance of the proposed approach.

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“…On the one hand, in the context of model verification, the CRE concept was already used to control PGD approximations (see a posteriori error estimates developed in [36,37]) or to directly drive the PGD process with CRE minimization [38]. Nevertheless, the use of PGD in CRE implementation has never been investigated and we wish to show here that there are major advantages to do so, in particular for the construction of equilibrated fields.…”

Section: Introductionmentioning

confidence: 99%

Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures

Chamoin

Allier

Marchand

2016

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

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The paper deals with the constitutive relation error (CRE) concept which has been widely used over the last 40 years for verification and validation of computational mechanics models. It more specifically focuses on the beneficial use of model reduction based on proper generalized decomposition (PGD) into this CRE concept. Indeed, it is shown that a PGD formulation can facilitate the construction of so-called admissible fields which is a technical key-point of CRE. Numerical illustrations, addressing both model verification and model updating, are presented to assess the performances of the proposed approach.

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Proper Generalized Decomposition computational methods on a benchmark problem: introducing a new strategy based on Constitutive Relation Error minimization

Cited by 25 publications

References 23 publications

Local proper generalized decomposition

Local proper generalized decomposition

Towards simplified and optimized a posteriori error estimation using PGD reduced models

Synergies between the constitutive relation error concept and PGD model reduction for simplified V&V procedures

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