TY - JOUR

T1 - On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials

AU - Wriggers, P.

AU - Hudobivnik, B.

AU - Allix, O.

N1 - Funding Information: O. Allix would like to thank the Alexander von Humbolt Foundation for its support through the the Gay-Lussac-Humboldt prize which made it possible to closely interact with the colleagues from the Institute of Continuum Mechanics at Leibniz University Hannover in this joint work. B. Hudobivnic and P. Wriggers were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD, EXC 2122 (Project No.: 390833453).

PY - 2022/2

Y1 - 2022/2

N2 - The virtual element method allows to revisit the construction of Kirchhoff-Love elements because the C1-continuity condition is much easier to handle in the VEM framework than in the traditional Finite Elements methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in Brezzi and Marini (Comput Methods Appl Mech Eng 253:455–462, 2013). The formulation contains new ideas and different approaches for the stabilisation needed in a virtual element, including classic and energy stabilisations. An efficient stabilisation is crucial in the case of C1-continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for C-continuous elements. This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. It will be shown that these C1-continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.

AB - The virtual element method allows to revisit the construction of Kirchhoff-Love elements because the C1-continuity condition is much easier to handle in the VEM framework than in the traditional Finite Elements methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in Brezzi and Marini (Comput Methods Appl Mech Eng 253:455–462, 2013). The formulation contains new ideas and different approaches for the stabilisation needed in a virtual element, including classic and energy stabilisations. An efficient stabilisation is crucial in the case of C1-continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for C-continuous elements. This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. It will be shown that these C1-continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.

KW - Finite plate element

KW - Isotropic/anisotropic material

KW - Kirchhoff-Love theory

KW - Stabilisation

KW - Virtual element method (VEM)

KW - Virtual plate element

UR - http://www.scopus.com/inward/record.url?scp=85120856444&partnerID=8YFLogxK

U2 - 10.1007/s00466-021-02106-1

DO - 10.1007/s00466-021-02106-1

M3 - Article

AN - SCOPUS:85120856444

VL - 69

SP - 615

EP - 637

JO - Computational mechanics

JF - Computational mechanics

SN - 0178-7675

IS - 2

ER -