Nonlinear stiffness optimization with prescribed deformed geometry and loads

Optimization based on traditional forward motion analysis to ensure a prescribed load distribution on a deformed geometry is challenging, since the load distribution is highly coupled to the deformed geometry, boundary conditions, and the optimized material layout. In contrast to traditional forward motion analysis, the deformed configuration is prescribed in the inverse motion analysis, and the undeformed configuration is the outcome of the analysis. Consequently, the inverse motion analysis is able to define an exact deformed geometry. In the present study, we make use of this key advantage to design structures with both an exact deformed geometry and a prescribed load distribution. The objective in the optimization is to minimize a general function of the nodal displacement vector. To formulate a well-posed optimization problem, the design is regularized using the partial differential equation filter and the sensitivity analysis is based on the adjoint method. The computational model is developed for neo-Hookean hyper-elasticity and the balance equations are discretized using the finite element method. The resulting nonlinear equations are solved using a conventional Newton–Raphson scheme. In the numerical examples, a cantilever beam with an embedded perfect circular shape is first considered. Next, a 2D gasket-like structure is designed, and finally, we consider a 3D structure with contact-like boundary conditions. In these examples, the prescribed deformed geometry is subject to a distributed external force. The examples show that the deformed geometry and load distribution can be exactly prescribed through stiffness optimization based on the inverse motion analysis.