Higher-order shell elements based on a Cosserat model, and their use in the topology design of structures

Abstract

For the purpose of carrying out topology optimization of shell structures, we show that it is advantageous to use shell elements based on a classical shell theory (such as the Cosserat theory) rather than elements based on the degenerated solid approach, since the shell thickness appears explicitly in the formulation, thereby greatly simplifying the sensitivity analysis. One of the well-known shell elements based on the Cosserat shell theory is the four-node element presented by Simo et al. [Comput. Methods Appl. Mech. Engrg. 72 (1989) 267; 73 (1989) 53]. Although one could use this element, the use of lower-order elements often results in instabilities (such as the “checkerboard” instability) in the resulting topologies; this provides the motivation for developing higher-order shell elements based on a classical shell theory. In this work, we present the formulation and implementation details for six-node and seven-node triangular, and nine-node quadrilateral shell elements, which are based on the variational formulation of Simo et al., and show that good accuracy is obtained even with coarse meshes in fairly demanding problems. We also present the topology optimization formulation, and some examples of optimal topologies that are obtained using this formulation.