Phase field topology optimisation for 4D printing

TL;DR

This paper develops a phase field topology optimization method for 4D printing, enabling the design of multi-material structures that morph into targeted shapes upon environmental stimuli, validated through numerical simulations.

Contribution

It introduces a PDE-constrained optimization framework for 4D printing, linking phase field models to sharp interface descriptions, and proves existence and optimality conditions for designs.

Findings

Existence of optimal material distributions proven.

First order necessary conditions derived.

Numerical simulations validate the approach in 2D and 3D.

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 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.