Phase field topology optimisation for 4D printing

Harald Garcke, Kei Fong Lam*, Robert Nürnberg, Andrea Signori

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

2 Citations (Scopus)


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.

Original languageEnglish
Article number24
Number of pages46
JournalESAIM - Control, Optimisation and Calculus of Variations
Publication statusPublished - 30 Mar 2023

Scopus Subject Areas

  • Control and Systems Engineering
  • Control and Optimization
  • Computational Mathematics

User-Defined Keywords

  • 4D printing
  • printed active composites
  • topology optimisation
  • phase field
  • linear elasticity
  • optimal control


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