Comparison of second-order serendipity and Lagrange tetrahedral elements for nonlinear explicit methods

Abstract

This paper evaluates the performances of second-order finite elements for nodal lumped-mass explicit methods in nonlinear solid dynamics, with a particular emphasis on 10-node “serendipity” and 15-node “Lagrange” tetrahedral elements. Historically, many nonlinear explicit finite element codes have exclusively used first-order elements, until a fairly recent flurry of activity that has resulted in higher-order elements becoming available in explicit codes including the authors' in-house one, ParaAble, and the production software EPIC, IMPETUS, LS-DYNA, and Abaqus. A major attractiveness of tetrahedrons is their ease in meshing and higher-order elements can facilitate the avoidance of severe volumetric locking with unstructured C0 meshes, which are generally used with these codes for the discontinuities of inelasticity, contact, etc. They also can improve modeling of flexure and curved shapes as well as eliminate spurious modes without artificial stabilization. The inclusion of face and body centroid nodes with Lagrange interpolants, including the 15-node tetrahedron, has proven to provide robust overall performance with lumped-mass explicit methods and with contact. Nevertheless, versions of the 10-node tetrahedron have also emerged in lumped-mass explicit software. In contrast to hexahedrons, an important observation about tetrahedrons is that the 10-node serendipity version uses about four times fewer quadrature points and a larger time increment than their 15-node Lagrange counterpart, which could result in tremendous computational differences. Serendipity elements, however, notoriously do not nodal mass lump well and tetrahedron versions have not been rigorously evaluated/documented for their effectiveness, as will be done herein with comparisons of those using 15-node tetrahedrons. Using row-summation lumping for Lagrange elements and the ad hoc HRZ scheme for serendipity ones, performances are assessed in common benchmark problems and practical applications using various elastic and inelastic material models and involving large strains/deformations/rotations and severe distortions. Whereas the 10-node tetrahedrons were found to perform much better than their 20-node serendipity hexahedral counterparts, specifically with substantial computational reductions and reasonable predictions, they are not generally quite as accurate or robust as the 15-node tetrahedral elements. The results thus indicate benefits of including both 10- and 15-node tetrahedrons in an explicit code's element library.