CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models

We propose a method for finding optimal quadrature/cubature rules with positive weights for parameterized functions in 1D, 2D or 3D spatial domains. The method takes as starting point the values of the functions at the Gauss points of a finite element (FE) mesh of the spatial domain for a representa...

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Autores: Hernández Ortega, Joaquín Alberto|||0000-0001-9334-4002, Bravo Martínez, José Raúl|||0000-0002-4465-7536, Ares de Parga Regalado, Sebastian|||0000-0001-5709-4683
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/396913
Acceso en línea:https://hdl.handle.net/2117/396913
https://dx.doi.org/10.1016/j.cma.2023.116552
Access Level:acceso abierto
Palabra clave:Finite element method
Decomposition method
Gaussian distribution
Empirical Cubature Method
Hyperreduction
Reduced-order modeling
Singular Value Decomposition
Quadrature
Elements finits, Mètode dels
Descomposició, Mètode de
Distribució de Gauss
Àrees temàtiques de la UPC::Enginyeria dels materials
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repository_id_str
dc.title.none.fl_str_mv CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
title CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
spellingShingle CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
Hernández Ortega, Joaquín Alberto|||0000-0001-9334-4002
Finite element method
Decomposition method
Gaussian distribution
Empirical Cubature Method
Hyperreduction
Reduced-order modeling
Singular Value Decomposition
Quadrature
Elements finits, Mètode dels
Descomposició, Mètode de
Distribució de Gauss
Àrees temàtiques de la UPC::Enginyeria dels materials
title_short CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
title_full CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
title_fullStr CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
title_full_unstemmed CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
title_sort CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
dc.creator.none.fl_str_mv Hernández Ortega, Joaquín Alberto|||0000-0001-9334-4002
Bravo Martínez, José Raúl|||0000-0002-4465-7536
Ares de Parga Regalado, Sebastian|||0000-0001-5709-4683
author Hernández Ortega, Joaquín Alberto|||0000-0001-9334-4002
author_facet Hernández Ortega, Joaquín Alberto|||0000-0001-9334-4002
Bravo Martínez, José Raúl|||0000-0002-4465-7536
Ares de Parga Regalado, Sebastian|||0000-0001-5709-4683
author_role author
author2 Bravo Martínez, José Raúl|||0000-0002-4465-7536
Ares de Parga Regalado, Sebastian|||0000-0001-5709-4683
author2_role author
author
dc.subject.none.fl_str_mv Finite element method
Decomposition method
Gaussian distribution
Empirical Cubature Method
Hyperreduction
Reduced-order modeling
Singular Value Decomposition
Quadrature
Elements finits, Mètode dels
Descomposició, Mètode de
Distribució de Gauss
Àrees temàtiques de la UPC::Enginyeria dels materials
topic Finite element method
Decomposition method
Gaussian distribution
Empirical Cubature Method
Hyperreduction
Reduced-order modeling
Singular Value Decomposition
Quadrature
Elements finits, Mètode dels
Descomposició, Mètode de
Distribució de Gauss
Àrees temàtiques de la UPC::Enginyeria dels materials
description We propose a method for finding optimal quadrature/cubature rules with positive weights for parameterized functions in 1D, 2D or 3D spatial domains. The method takes as starting point the values of the functions at the Gauss points of a finite element (FE) mesh of the spatial domain for a representative sample of input parameters, and then construct an elementwise continuous orthogonal basis for such functions using the truncated Singular Value Decomposition (SVD) along with element polynomial fitting. To avoid possible memory bottlenecks in computing the SVD, we propose a Sequential Randomized SVD (SRSVD) in which the matrix is provided in a column-partitioned format, and which uses randomization to accelerate the processing of each individual block. After computing the basis functions, the method determines an exact integration rule for such functions, featuring as many points as functions, and in which the points are selected among the Gauss points of the FE mesh. Finally, the desired optimal rule is obtained by an sparsification process in which the algorithm zeroes one weight at a time while readjusting the positions and weights of the remaining points so that the constraints of the problem are satisfied. We apply this methodology to multivariate polynomials in cartesian domains to demonstrate that the method is indeed able to produce optimal rules – i.e., Gauss product rules –, and to a 2-parameters, 3D sinusoidal–exponential function to illustrate the use of the SRSVD in scenarios in which the standard SVD cannot handle the operation because of memory limitations. Lastly, the fact that the method does not require the analytical expression of the integrand functions – just their values at the FE Gauss points – makes it suitable for dealing with the so-called hyperreduction of parameterized finite element models. We exemplify this by showing its performance in the derivation of low-dimensional surrogate models in the context of the multiscale FE method. The Matlab source codes of both the CECM and the SRSVD, along with the scripts for launching the numerical tests, are openly accessible in the public repository https://github.com/Rbravo555/CECM-continuous-empirical-cubature-method.
publishDate 2023
dc.date.none.fl_str_mv 2023
2023-11-22
2024
2024-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/396913
https://dx.doi.org/10.1016/j.cma.2023.116552
url https://hdl.handle.net/2117/396913
https://dx.doi.org/10.1016/j.cma.2023.116552
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PCI2021-121944 ENABLING DYNAMIC AND INTELLIGENT WORKFLOWS IN THE FUTURE EUROHPCECOSYSTEM
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial 4.0 International
http://creativecommons.org/licenses/by-nc/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial 4.0 International
http://creativecommons.org/licenses/by-nc/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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spelling CECM: a continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element modelsHernández Ortega, Joaquín Alberto|||0000-0001-9334-4002Bravo Martínez, José Raúl|||0000-0002-4465-7536Ares de Parga Regalado, Sebastian|||0000-0001-5709-4683Finite element methodDecomposition methodGaussian distributionEmpirical Cubature MethodHyperreductionReduced-order modelingSingular Value DecompositionQuadratureElements finits, Mètode delsDescomposició, Mètode deDistribució de GaussÀrees temàtiques de la UPC::Enginyeria dels materialsWe propose a method for finding optimal quadrature/cubature rules with positive weights for parameterized functions in 1D, 2D or 3D spatial domains. The method takes as starting point the values of the functions at the Gauss points of a finite element (FE) mesh of the spatial domain for a representative sample of input parameters, and then construct an elementwise continuous orthogonal basis for such functions using the truncated Singular Value Decomposition (SVD) along with element polynomial fitting. To avoid possible memory bottlenecks in computing the SVD, we propose a Sequential Randomized SVD (SRSVD) in which the matrix is provided in a column-partitioned format, and which uses randomization to accelerate the processing of each individual block. After computing the basis functions, the method determines an exact integration rule for such functions, featuring as many points as functions, and in which the points are selected among the Gauss points of the FE mesh. Finally, the desired optimal rule is obtained by an sparsification process in which the algorithm zeroes one weight at a time while readjusting the positions and weights of the remaining points so that the constraints of the problem are satisfied. We apply this methodology to multivariate polynomials in cartesian domains to demonstrate that the method is indeed able to produce optimal rules – i.e., Gauss product rules –, and to a 2-parameters, 3D sinusoidal–exponential function to illustrate the use of the SRSVD in scenarios in which the standard SVD cannot handle the operation because of memory limitations. Lastly, the fact that the method does not require the analytical expression of the integrand functions – just their values at the FE Gauss points – makes it suitable for dealing with the so-called hyperreduction of parameterized finite element models. We exemplify this by showing its performance in the derivation of low-dimensional surrogate models in the context of the multiscale FE method. The Matlab source codes of both the CECM and the SRSVD, along with the scripts for launching the numerical tests, are openly accessible in the public repository https://github.com/Rbravo555/CECM-continuous-empirical-cubature-method.This work is sponsored in part by the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2018-000797-S)”. The authors acknowledge the support of the European High-Performance Computing Joint Undertaking (JU) under grant agreement No. 955558 (the JU receives, in turn, support from the European Union’s Horizon 2020 research and innovation programme and Spain, Germany, France, Italy, Poland, Switzerland, Norway), as well as the R&D project PCI2021-121944, financed by MCIN/AEI/10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. J.A. Hernández expresseses gratitude by the support of, on the one hand, the “MCIN/AEI/10.13039/501100011033/y por FEDER una manera de hacer Europa” (PID2021-122518OBI00), and, on the other hand, the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 952966 (project FIBREGY). Lastly, both J.R. Bravo and S. Ares de Parga acknowledge the Departament de Recerca i Universitats de la Generalitat de Catalunya for the financial support through doctoral grants FI-SDUR 2020 and FI SDUR-2021, respectively.Peer ReviewedElsevier20242024-01-0120232023-11-22journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/396913https://dx.doi.org/10.1016/j.cma.2023.116552reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PCI2021-121944 ENABLING DYNAMIC AND INTELLIGENT WORKFLOWS IN THE FUTURE EUROHPCECOSYSTEMopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial 4.0 Internationalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3969132026-05-27T15:37:01Z
score 15.301603

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