Mechanical and mathematical basis for using negative thickness in finite element analysis

Topology optimization is a powerful and widely used tool in lightweight design. It can be applied to optimize the surface of an existing part by modeling a shell onto an existing finite element (FE) model and optimizing the thickness of the shell. However, if the goal is to reduce the mass, it is currently necessary to remodel the existing model with smaller dimensions before performing this type of optimization. This additional step is necessary because most available optimization tools only allow for the increase of the thickness of the shell and thus cannot be used to remove material. To address this limitation, we propose elements with negative thickness values to eliminate this time-consuming remodeling step. As a first step towards this goal, we prove that the fundamental laws of mechanics remain valid when areas are subtracted from an existing solid body and that there are no mechanical and mathematical inconsistencies. Our analysis is based on established principles of engineering mechanics. In conclusion of this paper, examples are given for the necessary next steps in preparation of the transfer of the approach to FE simulations. We also emphasize the necessity of investigating the subsequent numerical challenges resulting from our approach.