Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications

Zlotnik, Sergio; Dı́ez, Pedro; Modesto, David; Huerta, Antonio

doi:10.1002/nme.4909

Cited by 55 publications

(92 citation statements)

References 25 publications

(52 reference statements)

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“…The absorption coe cient ↵, see (6b), is a natural candidate to evaluate possible remediation actions in physical boundaries. More challenging implementations can include geometrical parameters, see [42,43].…”

Section: The Parameterized Wave Propagation Weak Formmentioning

confidence: 99%

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Modesto

Zlotnik

Huerta

2015

Computer Methods in Applied Mechanics and Engineering

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116

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Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies.The Proper Generalized Decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the contributions of the paper include the PGD implementation into a Perfectly Matched Layer framework to model the unbounded domain, and the separability of the operator which is addressed here using an e cient higher-order projection scheme.Then, the performance of the PGD in this framework is discussed and improved using the higher-order projection and a Petrov-Galerkin approach to construct the separated basis. Moreover, the e ciency of the higherorder projection scheme is demonstrated and compared with the higher-order singular value decomposition.

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Section: The Parameterized Wave Propagation Weak Formmentioning

confidence: 99%

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Modesto

Zlotnik

Huerta

2015

Computer Methods in Applied Mechanics and Engineering

Self Cite

116

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“…Following the approach described in [51][52][53], we reformulate the weak problem (27) which can be used for any element geometry.…”

Section: Use Of the Pgd To Solve Problems At The Element Levelmentioning

confidence: 99%

“…The progressive Galerkin approach described in "Background" section is used with bilinear form B and linear form F constructed from the parameterized separated variable Jacobian transformation (all technical details can be found in [51,52]). Introducing the interval I α (resp.…”

Section: Implementation Of the Pgdmentioning

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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“…Figure 1 shows a parameter dependant geometry for an airfoil and the objective is to find the air flow around it. The method applied was proposed in [3] and later extended in [4]. It is based on the idea of having a reference domain T and a mapping function that relates all possible geometries to the reference domain.…”

Section: Motivation Examplesmentioning

confidence: 99%

“…As an example the function k(x, y) = sin 1 2 (x + y) 2 + 2 introduced in (3) is separated using SVD to obtain k sep (x, y) as defined in (4). This function is chosen because it does not admits an exact separated representation (Fig.…”

Section: Separation Of the Input Datamentioning

confidence: 99%

Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

Zlotnik

Dı́ez

González

et al. 2015

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

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The proper generalized decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases the input data is not expressed in a separated form and it has to be replaced by some separable approximation. These approximations constitute a new error source that, in some cases, may dominate the standard ones (discretization, truncation…) and control the final accuracy of the PGD solution. In this work the relation between errors in the separated input data and the errors induced in the PGD solution is discussed. Error estimators proposed for homogenized problems and oscillation terms are adapted to asses the behaviour of the PGD errors resulting from approximated input data. The PGD is stable with respect to error in the separated data, with no critical amplification of the perturbations. Interestingly, we identified a high sensitiveness of the resulting accuracy on the selection of the sampling grid used to compute the separated data. The separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy. For the case of a Poisson problem separated in the spatial coordinates (a complex diffusivity function requires a separable approximation), the final PGD error is linear with the truncation error of the separated data. This relation is used to estimate the number of terms required in the separated data, that has to be in good agreement with the truncation error accepted in the PGD truncation (tolerance for the stoping criteria in the enrichment procedure). A sensible choice for the prescribed accuracy of the PGD solution has to be kept within the limits set by the errors in the separated input data.

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Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications

Cited by 55 publications

References 25 publications

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

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

Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

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