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

## TL;DR

This paper develops a Virtual Element Method (VEM) for 2D elastic problems, enabling flexible polygonal elements with higher-order approximations, and compares its performance with classical finite element methods.

## Contribution

It introduces a standardized procedure for constructing VEM terms for polygonal elements with higher-order interpolation, expanding the applicability of VEM in structural analysis.

## Key findings

- VEM performs comparably to classical FEM in numerical tests.
- Higher-order polygonal elements improve approximation accuracy.
- The method effectively handles various polygonal shapes and edge counts.

## Abstract

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.