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Arbitrary order 2D virtual elements for polygonal meshes: part I, elastic problem

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Abstract

The present work deals with the formulation of a virtual element method for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II (Artioli et al. in Comput Mech, 2017) the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a computer code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical results are also compared with the one recovered using the classical finite element method.

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Authors and Affiliations

  1. Department of Civil Engineering and Computer Science, University of Rome Tor Vergata, Via del Politecnico 1, 00133, Rome, Italy

    E. Artioli

  2. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 53, 20125, Milan, Italy

    L. Beirão da Veiga

  3. Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133, Milan, Italy

    C. Lovadina

  4. Dipartimento di Ingegneria Civile e Meccanica, Università di Cassino e del Lazio Meridionale, Via G. Di Biasio 43, 03043, Cassino, Italy

    E. Sacco

Authors
  1. E. Artioli
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  2. L. Beirão da Veiga
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  3. C. Lovadina
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  4. E. Sacco
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Corresponding author

Correspondence to E. Artioli.

Additional information

E. Artioli gratefully acknowledges the partial financial support of the Italian Minister of University and Research, MIUR (Program: Consolidate the Foundations 2015; Project: BIOART; Grant number (CUP): E82F16000850005).

L. Beirão da Veiga and C. Lovadina where partially supported by IMATI-CNR of Pavia, Italy. This support is gratefully acknowledged.

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Artioli, E., Beirão da Veiga, L., Lovadina, C. et al. Arbitrary order 2D virtual elements for polygonal meshes: part I, elastic problem. Comput Mech 60, 355–377 (2017). https://doi.org/10.1007/s00466-017-1404-5

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  • DOI: https://doi.org/10.1007/s00466-017-1404-5

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