Curvilinear Virtual Elements for 2D solid mechanics applications

TL;DR

This paper extends the curvilinear Virtual Element Method to 2D solid mechanics, enabling precise geometry approximation and flexible constitutive modeling, with promising numerical results for elastic and inelastic problems.

Contribution

It introduces a novel Virtual Element space for displacements on curvilinear elements, generalizing previous scalar problem work to complex solid mechanics applications.

Findings

Accurate approximation of curved geometries using curved VEM.

Effective handling of elastic and inelastic material behaviors.

Enhanced scheme performance over standard VEM in curved geometries.

Abstract

In the present work we generalize the curvilinear Virtual Element technology, introduced for a simple linear scalar problem in a previous work, to generic 2D solid mechanic problems in small deformations. Such generalization also includes the development of a novel Virtual Element space for displacements that contains rigid body motions. Our approach can accept a generic black-box (elastic or inelastic) constitutive algorithm and, in addition, can make use of curved edges thus leading to an exact approximation of the geometry. Rigorous theoretical interpolation properties for the new space on curvilinear elements are derived. We undergo an extensive numerical test campaign, both on elastic and inelastic problems, to assess the behavior of the scheme. The results are very promising and underline the advantages of the curved VEM approach over the standard one in the presence of curved geometries.