Second order three-dimensional serendipity virtual elements for hyperelasticity: Static and dynamic analysis

In this work, a three-dimensional (3D) second-order serendipity virtual element method (S-VEM) is developed for the static and dynamic analysis of hyperelastic materials. The VEM framework is based on the projection of unknown basis functions onto polynomial spaces, allowing for flexible discretization with arbitrary polyhedral meshes. While most existing VEM formulations for 3D mechanical problems are discretized using first-order formulations, higher-order schemes offer improved precision, especially for nonlinear problems. However, conventional second-order VEM formulations introduce additional degrees of freedom (DOFs), such as body and surface moments, which complicate the implementation and reduce computation efficiency. To address this challenge, we propose a novel 3D second-order serendipity VEM that avoids any extra moment-related DOFs. This is the first application of a serendipity VEM to 3D static and dynamic problems in hyperelasticity. Furthermore, by integrating advanced mesh generation techniques, the proposed method enables hybrid simulations that combine second-order serendipity VEM and FEM to efficiently handle complex geometries.