Volume 29, 2023
|Number of page(s)||46|
|Published online||30 March 2023|
Phase field topology optimisation for 4D printing*
Fakultät für Mathematik, Universität Regensburg,
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: email@example.com
Accepted: 16 February 2023
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.
Mathematics Subject Classification: 49J20 / 49K40 / 49J50
Key words: 4D printing / printed active composites / topology optimisation / phase field / linear elasticity / optimal control
The authors HG and AS gratefully acknowledge the support by the Graduiertenkolleg 2339 IntComSin of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 321821685. The work of KFL is supported by the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No.: HKBU 14302218 and HKBU 12300321].
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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