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

Abstract

In this article, 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 FEs 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∘$$ {0}^{\circ } $$ and 180∘$$ 18{0}^{\circ } $$, 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 FE meshes with ϵ$$ \epsilon $$ 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 ϵ→0$$ \epsilon \to 0 $$. The significant reduction in solution time on meshes with agglomerated virtual elements vis‐à‐vis tetrahedral meshes is demonstrated through explicit dynamics simulations on a tapered beam.

A co‐rotational virtual element method for 2D elasticity and plasticity

“…The notation given closely follows references 23 and, 22 with some exceptions. Sukumar and Tupek 24 describe aspects of VEM as follows. In VEM, "the basis functions are defined as the solution of a local elliptic partial differential equation", yet they are not ever actually calculated in construction of the method.…”

“…As with finite element analysis, in VEM an element stiffness matrix is constructed for each polygonal element. The element stiffness in VEM is composed of two parts: a consistency part (8) and a stability part (24) or (26).…”

“…Another form of the stability part of the element stiffness, suggested by Sukumar and Tupek, 24 is expressed as…”

“…The identity matrix $$$ \mathbf{I} $$$ is $$$ 2{n}_v\times 2{n}_v $$$. The projection matrix, $$$ \boldsymbol{\Pi} $$$, is determined as follows: $$$$ \boldsymbol{\Pi} =\mathbf{D}\tilde{\boldsymbol{\Pi}}.\kern2.00em \left[2{n}_v\times 2{n}_v\right] $$$$ Another form of the stability part of the element stiffness, suggested by Sukumar and Tupek, 24 is expressed as $$$$ {\mathbf{k}}_E^s={\left(\mathbf{I}-\boldsymbol{\Pi} \right)}^T{\mathbf{S}}_E^d\left(\mathbf{I}-\boldsymbol{\Pi} \right). $$$$ where a $$$ 2{n}_v\times 2{n}_v $$$ diagonal matrix, $$$ {\mathbf{S}}_E^d $$$, is scaled as needed.…”

Stabilization-free virtual element method for 3D hyperelastic problems

“…As a generalization of the Finite Element Method (FEM), the virtual element method allows very general arbitrarily shaped elements, which makes the method suitable for calculating some special problems such as crack propagation, hanging nodes, and adaptive meshes. In the framework of structural mechanics, the method has been applied to different fields including linear elastic problems [3][4][5][6][7], hyperelastic materials at finite deformations [8][9][10][11], contact problems [12][13][14][15], phase field fracture [16,17], elastodynamics problems [18][19][20][21], and finite elastoplastic deformations [22][23][24]. Specific work of VEM in engineering can be found in the latest published VEM book [25].…”