Effects of finite element formulation on optimal plate and shell structural topologies

The effects of selected membrane, plate and flat shell finite element formulations on optimal topologies are numerically investigated. Two different membrane components are considered. The first is a standard 4-node bilinear quadrilateral, and the other is a 4-node element accounting for in-plane (drilling) rotations. Plate elements selected for evaluation include the discrete Kirchhoff quadrilateral (DKQ) element and two Mindlin–Reissner-based elements, one employing selective reduced integration (SRI), and the other an assumed natural strain (ANS) formulation. The effect of hourglass control on SRI elements is also evaluated. The flat shell elements consist of an assemblage of these membrane and plate components. The Mindlin–Reissner elements are shown to recover the thin plate result computed using DKQ elements. However, optimal topology is shown to be dependent on plate element formulation as thickness increases. Furthermore, a new benchmark problem is introduced illustrating the deficiencies of Mindlin–Reissner elements employing SRI without hourglass control. For shell problems, elements which properly account for in-plane rotations are shown to be insensitive to the penalty parameter which enforces the relationship between in-plane rotations and displacements, in contrast to the situation when an ad hoc treatment of drilling degrees of freedom is used.