Multiplicative reduced bases for hyperelasticity

Patrice Hauret

doi:10.5802/crmath.584

Article de recherche - Analyse numérique, Équations aux dérivées partielles

Multiplicative reduced bases for hyperelasticity
[Bases réduites multiplicatives en hyperélasticité]

Patrice Hauret ^1,²

¹ SiMatLab, Campus Universitaire des Cézeaux, 24 Avenue Blaise Pascal, TSA 60026 / CS 60026, 63178 Aubière Cedex, France
² Centre de Technologies Michelin, Place des Carmes, 63000 Clermont-Ferrand, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 593-605.

Résumé
Abstract

Cet article introduit un principe de bases réduites multiplicatives en hyperélasticité s’appuyant sur le développement de Baker–Campbell–Hausdorff. Nous montrons que cette construction produit des interpolations identiques pour les gradients de déformation, et les mesures de déformation surfaciques et volumiques en grandes déformations. Cette méthode est établie dans un cadre variationel consistant et nous montrons une borne supérieure pour l’erreur en norme de l’énergie. Numériquement, l’approche se distingue par une décroissance efficace de la n-épaisseur de Kolmogorov, particulièrement en présence de grandes rotations pour des comportements incompressibles.

The present paper introduces multiplicative reduced bases for hyperelasticity relying on a truncated version of the Baker–Campbell–Hausdorff’s expansion. We show such a construction is equally interpolatory (in a multiplicative way) for the fields of deformation gradients, surfacic and volumetric deformation measures involved in large deformation mechanics. The method is naturally derived from a fully consistent variational setting and we establish an upper bound of the error in the energy norm. From a computational standpoint, the approach achieves efficient Kolmogorov $n$ -width decay when very large rotations and incompressibility are involved.

Reçu le : 2023-08-03
Révisé le : 2023-10-14
Accepté le : 2023-10-20
Publié le : 2024-07-09

DOI : 10.5802/crmath.584

Affiliations des auteurs :

Patrice Hauret ^{1, 2}

¹ SiMatLab, Campus Universitaire des Cézeaux, 24 Avenue Blaise Pascal, TSA 60026 / CS 60026, 63178 Aubière Cedex, France
² Centre de Technologies Michelin, Place des Carmes, 63000 Clermont-Ferrand, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Patrice Hauret},
     title = {Multiplicative reduced bases for hyperelasticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {593--605},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.584},
     language = {en},
}

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Patrice Hauret. Multiplicative reduced bases for hyperelasticity. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 593-605. doi : 10.5802/crmath.584. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.584/

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