Phase field topology optimisation for 4D printing^*

¹ Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
² Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
³ Department of Mathematics, University of Trento, Trento, Italy
⁴ Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy

^** Corresponding author: andrewkflam@gmail.com

Received: 8 July 2022
Accepted: 16 February 2023

Abstract

This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage. Then, a change in an environmental stimulus and the removal of loads cause the object to deform in the programmed stage. The dynamic transition between the original and deformed shapes is achieved with appropriate applications of the stimulus. The mathematical interest is to find an optimal distribution for the materials such that the 3D printed object achieves a targeted configuration in the programmed stage as best as possible. Casting the problem as a PDE-constrained minimisation problem, we consider a vector-valued order parameter representing the volume fractions of the different materials in the composite as a control variable. We prove the existence of optimal designs and formulate first order necessary conditions for minimisers. Moreover, by suitable asymptotic techniques, we relate our approach to a sharp interface description. Finally, the theoretical results are validated by several numerical simulations both in two and three space dimensions.