Simultaneous determination of shape and material distribution for shell structures under large elastic deformations via topology optimisation and inverse finite element analysis

Designing thin-walled and shell-like structures undergoing geometric non-linearities is fundamentally challenging, particularly when simultaneously determining manufacturing geometries for prescribed configurations and optimal material distributions. This work addresses this challenge by integrating Topology Optimisation (TO) and the Inverse Finite Element Method (IFEM) for geometrically non-linear degenerated-solid shell formulations. By employing the inverse counterpart of the MITC4 element, the framework enables the concurrent computation of the undeformed manufacturing shape and optimal material distribution. IFEM ensures the structure recovers a target geometry after large elastic deformations, while the material layout is determined via the Solid Isotropic Material with Penalisation (SIMP) technique. The sensitivity of the resulting large scale, non-linear, continuous optimisation problem is formulated via the adjoint method. Both mechanical (primal) and adjoint (dual) problems are solved using IFEM quadrilateral elements, with the Method of Moving Asymptotes (MMA) updating the design variables. This methodology delivers a unified, robust structural design strategy for determining manufacturing configurations and material layouts in large-deformation scenarios. Its practical significance and originality are demonstrated through benchmark problems and demanding applications, including compliant mechanisms and large elastic structures, encompassing both isotropic and multi-layered orthotropic materials—highlighted by the design of an intraocular lens (IOL) folder and a wind turbine blade.