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 discrete Kirchhoff quadrilateral (DKQ) element as well as two Mindlin-Reissner based elements, one employing selective reduced integration (SRI), and the other an assumed natural strain (ANS) formulation. The flat shell elements consist of an assemblage of these membrane and plate components. Both Mindlin-Reissner elements are shown to recover the thin plate result computed using DKQ elements for popular benchmark topology optimization plate problems. However, a new benchmark problem is introduced illustrating the deficiencies of Mindlin-Reissner elements employing SRI on transverse shear terms. 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.