Inverse design framework for 4D printed structures using the finite element method

In this paper, we present an inverse design framework based on the finite element method (FEM) tailored for 4D printed structures capable of undergoing programmable, stimulus-responsive shape transformations over time. In contrast to conventional forward simulations, which predict deformation from a given initial configuration, our inverse approach seeks to determine the required printed geometry that will evolve into a desired final shape under specific external stimuli. The proposed framework incorporates viscoelasticity, geometric nonlinearity and time-dependent behavior into the FEM solver, formulating the inverse design task as an optimization problem. It enables users to specify custom target shapes and boundary conditions, thereby providing the flexibility needed to address a wide range of design scenarios. A comprehensive implementation workflow of the framework is also presented. To demonstrate the versatility and feasibility of our approach, we present two representative case studies: a classic bilayer actuator with 4D-printed normal hydrogel and a soft gripper with electrically responsive hydrogel. The results confirm the framework's accuracy and adaptability to different external stimulus, highlighting its potential as a practical computational design tool for the inverse design of 4D printed structures.