ECCOMAS 2024

A mixed hybrid-stress virtual element for plane elasticity

  • Liguori, Francesco S (University of Calabria)
  • Madeo, Antonio (University of Calabria)
  • Marfia, Sonia (Roma Tre University)
  • Sacco, Elio (University of Naples Federico II)

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The enhanced Virtual Element Method (VEM) formulation, recently proposed by D’Altri et al.[1] for plane elasticity, provides numerous advantages over the classic VEM. The two main results are the absence of stabilisation terms and a considerable accuracy improvement. Upon this premise, this work shows how the enhanced VEM formulation can be obtained within a Hellinger-Reissner (HR) variational principle. The primary unknowns are the displacements along the element contour, as usual in classic VEM, and the stress field. The latter is assumed to within a polynomial basis that a-priori satisfies the equilibrium equations expressed on a Cartesian frame and corresponds to the divergence-free projector operator in enhanced VEM formulations. Then, all the discrete operators can be evaluated through line integrals, facilitating the formulation for polygonal elements with a generic number of edges [2]. The HR reformulation of enhanced VEM not only provides a wider theoretical backgrounds, but also reveals a close relationship with mixed hybrid stress Finite Elements (FEs) [3], traditionally limited to triangular or quadrilateral geometries. Then, leveraging the broad literature on the hybrid-stress FEs behaviour, we discuss the critical role of selecting appropriate stress interpolation functions, not only for the element stability, but also for the accuracy improvement. In particular, a so-called isostatic interpolation, that adopts the minimum number of stress parameters to ensure the element stability turns out to be the optimal choice for achieving accurate solutions also for coarse meshes. REFERENCES [1] A.M. D’Altri, S. de Miranda, L. Patruno, E. Sacco, An enhanced VEM formulation for plane elasticity. Comput. Methods Appl. Mech. Engrg., Vol. 376, pp. 113663, 2021. [2] S. Marfia, E. Monaldo, E. Sacco, Cohesive fracture evolution within virtual element method Engineering Fracture Mechanics, Vol. 269, pp. 108464, 2022. [3] A. Madeo, G. Zagari, R. Casciaro, An isostatic quadrilateral membrane finite element with drilling rotations and no spurious modes FinEl, Vol. 50, pp. 21–32, 2012.