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Quadrature-free immersed isogeometric analysis

This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline...

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Detalles Bibliográficos
Autores principales: Antolin, P., Hirschler, T.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer London 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9653347/
https://www.ncbi.nlm.nih.gov/pubmed/36397879
http://dx.doi.org/10.1007/s00366-022-01644-3
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author Antolin, P.
Hirschler, T.
author_facet Antolin, P.
Hirschler, T.
author_sort Antolin, P.
collection PubMed
description This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.
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spelling pubmed-96533472022-11-15 Quadrature-free immersed isogeometric analysis Antolin, P. Hirschler, T. Eng Comput Original Article This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity. Springer London 2022-04-25 2022 /pmc/articles/PMC9653347/ /pubmed/36397879 http://dx.doi.org/10.1007/s00366-022-01644-3 Text en © The Author(s) 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Original Article
Antolin, P.
Hirschler, T.
Quadrature-free immersed isogeometric analysis
title Quadrature-free immersed isogeometric analysis
title_full Quadrature-free immersed isogeometric analysis
title_fullStr Quadrature-free immersed isogeometric analysis
title_full_unstemmed Quadrature-free immersed isogeometric analysis
title_short Quadrature-free immersed isogeometric analysis
title_sort quadrature-free immersed isogeometric analysis
topic Original Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9653347/
https://www.ncbi.nlm.nih.gov/pubmed/36397879
http://dx.doi.org/10.1007/s00366-022-01644-3
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