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

• DOI: 10.1007/s00466-017-1404-5
• Publication Date: 2017-04-05T11:23:02Z
• Authors: E. Artioli; L. Beirão da Veiga; C. Lovadina; E. Sacco
• Keywords: Plasticity; 0211 other engineering and technologies; Ocean Engineering; 02 engineering and technology; 01 natural sciences; Shape memory alloy; Settore ICAR/08 - SCIENZA DELLE COSTRUZIONI; Material nonlinearity; Plasticity; Shape memory alloy; Virtual element method; Viscoelasticity;; Computational Theory and Mathematic; FOS: Mathematics; Virtual element method; Mathematics - Numerical Analysis; 0101 mathematics; Virtual element method; Elasticity ; Static analysis; Mechanical Engineering; Applied Mathematics; Polygonal meshes; Material nonlinearity; Viscoelasticity; Numerical Analysis (math.NA); Static analysis; Virtual element method; Plasticity; Viscoelasticity; Elasticity; Static analysi; Material nonlinearity; Plasticity; Shape memory alloy; Virtual element method; Viscoelasticity; Ocean Engineering; Mechanical Engineering; Computational Theory and Mathematics; Computational Mathematics; Applied Mathematics; Computational Mathematic; Elasticity; Polygonal meshes; Static analysis; Virtual element method; Ocean Engineering; Mechanical Engineering; Computational Theory and Mathematics; Computational Mathematics; Applied Mathematics; Elasticity; Polygonal meshes; Static analysis; Virtual element method;; Polygonal meshe

The present work deals with the formulation of a Virtual Element Method (VEM) for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II [3] 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 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.