Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

In this paper, we use the first-order virtual element method (VEM) to investigate the effect of shape quality of polyhedra in the estimation of the critical time step for explicit three-dimensional elastodynamic finite element (FE) simulations. Low-quality finite elements are common when meshing realistic complex components, and while tetrahedral meshing technology is generally robust, meshing algorithms cannot guarantee high-quality meshes for arbitrary geometries or for non-water-tight computer-aided design models. For reliable simulations on such meshes, we consider FE meshes with tetrahedral and prismatic elements that have badly-shaped elements$-$tetrahedra with dihedral angles close to $0^\circ$ and $180^\circ$, and slender prisms with triangular faces that have short edges$-$and agglomerate such `bad' elements with neighboring elements to form a larger polyhedral virtual element. On each element, the element-eigenvalue inequality is used to estimate the critical time step. For a suite of illustrative finite element meshes with $\epsilon$ being a mesh-coordinate parameter that leads to poor mesh quality, we show that adopting VEM on the agglomerated polyhedra yield critical time steps that are insensitive as $\epsilon \rightarrow 0$. The significant reduction in solution time on meshes with agglomerated virtual elements vis-$\`a$-vis tetrahedral meshes is demonstrated through explicit dynamics simulations on a tapered beam.