Phase field topology optimisation for 4D printing

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