Lie groups and continuum mechanics: where do we stand today?
The geometric methods have experienced a fast growth in the past few decades. In this survey, we discuss the use of Lie groups in continuum mechanics. We address both the theoretical and numerical aspects. We explore the classical symmetry groups of the mechanics, the covariant form of the equations...
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| Language: | English |
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mécanique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.242/ |
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| author | de Saxcé, Géry Razafindralandy, Dina |
| author_facet | de Saxcé, Géry Razafindralandy, Dina |
| author_sort | de Saxcé, Géry |
| collection | DOAJ |
| description | The geometric methods have experienced a fast growth in the past few decades. In this survey, we discuss the use of Lie groups in continuum mechanics. We address both the theoretical and numerical aspects. We explore the classical symmetry groups of the mechanics, the covariant form of the equations and the symmetry group of constitutive laws. We consider the Lie symmetry group of the equations of a mechanical problem and investigate how to take advantage of them in developping analytical models (self-similar solutions, conservation laws, turbulence, ...) of the physical phenomena encoded in these equations. Lastly, we present a method of constructing robust numerical integrators from the knowledge of the Lie symmetry group of the equations. |
| format | Article |
| id | doaj-art-db17706adf464f8a9a465628fbb377b1 |
| institution | Kabale University |
| issn | 1873-7234 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mécanique |
| spelling | doaj-art-db17706adf464f8a9a465628fbb377b12025-02-07T13:48:00ZengAcadémie des sciencesComptes Rendus. Mécanique1873-72342024-05-01351S313515910.5802/crmeca.24210.5802/crmeca.242Lie groups and continuum mechanics: where do we stand today?de Saxcé, Géry0https://orcid.org/0000-0002-0961-0513Razafindralandy, Dina1Univ. Lille, CNRS, Centrale Lille, UMR 9013 - LaMcube - Laboratoire de mécanique multiphysique multiéchelle, F59000, Lille, FranceLaboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE) - UMR CNRS 7356, Pôle Sciences et Technologie, Avenue Michel Crépeau, F17042 La Rochelle Cedex 1, FranceThe geometric methods have experienced a fast growth in the past few decades. In this survey, we discuss the use of Lie groups in continuum mechanics. We address both the theoretical and numerical aspects. We explore the classical symmetry groups of the mechanics, the covariant form of the equations and the symmetry group of constitutive laws. We consider the Lie symmetry group of the equations of a mechanical problem and investigate how to take advantage of them in developping analytical models (self-similar solutions, conservation laws, turbulence, ...) of the physical phenomena encoded in these equations. Lastly, we present a method of constructing robust numerical integrators from the knowledge of the Lie symmetry group of the equations.https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.242/Lie groupscontinuum mechanicssymmetry groupsgeometric integratorsturbulence modelling |
| spellingShingle | de Saxcé, Géry Razafindralandy, Dina Lie groups and continuum mechanics: where do we stand today? Comptes Rendus. Mécanique Lie groups continuum mechanics symmetry groups geometric integrators turbulence modelling |
| title | Lie groups and continuum mechanics: where do we stand today? |
| title_full | Lie groups and continuum mechanics: where do we stand today? |
| title_fullStr | Lie groups and continuum mechanics: where do we stand today? |
| title_full_unstemmed | Lie groups and continuum mechanics: where do we stand today? |
| title_short | Lie groups and continuum mechanics: where do we stand today? |
| title_sort | lie groups and continuum mechanics where do we stand today |
| topic | Lie groups continuum mechanics symmetry groups geometric integrators turbulence modelling |
| url | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.242/ |
| work_keys_str_mv | AT desaxcegery liegroupsandcontinuummechanicswheredowestandtoday AT razafindralandydina liegroupsandcontinuummechanicswheredowestandtoday |