Advancement of additive manufacturing is driving a need for design tools that exploit the increasing fabrication freedom. Multimaterial, three-dimensional (3D) printing allows for the fabrication of components from multiple materials with different thermal, mechanical, and “active” behavior that can be spatially arranged in 3D with a resolution on the order of tens of microns. This can be exploited to incorporate shape changing features into additively manufactured structures. 3D printing with a downstream shape change in response to an external stimulus such as temperature, humidity, or light is referred to as four-dimensional (4D) printing. In this paper, a design methodology to determine the material layout of 4D printed materials with internal, programmable strains is introduced to create active structures that undergo large deformation and assume a desired target displacement upon heat activation. A level set (LS) approach together with the extended finite element method (XFEM) is combined with density-based topology optimization to describe the evolving multimaterial design problem in the optimization process. A finite deformation hyperelastic thermomechanical model is used together with an higher-order XFEM scheme to accurately predict the behavior of nonlinear slender structures during the design evolution. Examples are presented to demonstrate the unique capabilities of the proposed framework. Numerical predictions of optimized shape-changing structures are compared to 4D printed physical specimen and good agreement is achieved. Overall, a systematic design approach for creating 4D printed active structures with geometrically nonlinear behavior is presented which yields nonintuitive material layouts and geometries to achieve target deformations of various complexities.

References

References
1.
Tibbits
,
S.
,
2014
, “
4D Printing: Multi-Material Shape Change
,”
Archit. Des.
,
84
(
1
), pp.
116
121
.
2.
Ge
,
Q.
,
Qi
,
H. J.
, and
Dunn
,
M. L.
,
2013
, “
Active Materials by Four-Dimension Printing
,”
Appl. Phys. Lett.
,
103
(
13
), p.
131901
.
3.
Ge, Q.
,
Dunn, C. K.
,
Qi, H. J.
, and
Dunn, M. L.
, 2014, “
Active Origami by 4D Printing
,”
Smart Mater. Struct.
,
23
, p. 094007.
4.
Maute
,
K.
,
Tkachuk
,
A.
,
Wu
,
J.
,
Jerry Qi
,
H.
,
Ding
,
Z.
, and
Dunn
,
M. L.
,
2015
, “
Level Set Topology Optimization of Printed Active Composites
,”
ASME J. Mech. Des.
,
137
(
11
), p.
111402
.
5.
Ding
,
Z.
,
Yuan
,
C.
,
Peng
,
X.
,
Wang
,
T.
,
Qi
,
H. J.
, and
Dunn
,
M. L.
,
2017
, “
Direct 4D Printing Via Active Composite Materials
,”
Sci. Adv.
,
3
(
4
), p. e1602890.
6.
Zhao
,
Z.
,
Wu
,
J.
,
Mu
,
X.
,
Chen
,
H.
,
Qi
,
H. J.
, and
Fang
,
D.
,
2017
, “
Desolvation Induced Origami of Photocurable Polymers by Digit Light Processing
,”
Macromol. Rapid Commun.
,
38
(
13
), p. 1600625.
7.
Zhao
,
Z.
,
Wu
,
J.
,
Mu
,
X.
,
Chen
,
H.
,
Qi
,
H. J.
, and
Fang
,
D.
,
2017
, “
Origami by Frontal Photopolymerization
,”
Sci. Adv.
,
3
(
4
), p. e1602326.
8.
Weeger
,
O.
,
Kang
,
Y. S. B.
,
Yeung
,
S.-K.
, and
Dunn
,
M. L.
,
2016
, “
Optimal Design and Manufacture of Active Rod Structures With Spatially Variable Materials
,”
3D Print. Addit. Manuf.
,
3
(
4
), pp.
204
215
.
9.
Ding
,
Z.
,
Weeger
,
O.
,
Qi
,
H. J.
, and
Dunn
,
M. L.
,
2018
, “
4D Rods: 3D Structures Via Programmable 1D Composite Rods
,”
Mater. Des.
,
137
, pp.
256
265
.
10.
Tolley
,
M. T.
,
Felton
,
S. M.
,
Miyashita
,
S.
,
Aukes
,
D.
,
Rus
,
D.
, and
Wood
,
R. J.
,
2014
, “
Self-Folding Origami: Shape Memory Composites Activated by Uniform Heating
,”
Smart Mater. Struct.
,
23
(
9
), p.
94006
.
11.
Van Dijk
,
N. P.
,
Maute
,
K.
,
Langelaar
,
M.
, and
Van Keulen
,
F.
,
2013
, “
Level-Set Methods for Structural Topology Optimization: A Review
,”
Struct. Multidiscip. Optim.
,
48
(
3
), pp.
437
472
.
12.
Sigmund
,
O.
, and
Maute
,
K.
,
2013
, “
Topology Optimization Approaches: A Comparative Review
,”
Struct. Multidiscip. Optim.
,
48
(
6
), pp.
1031
1055
.
13.
Deaton
,
J. D.
, and
Grandhi
,
R. V.
,
2014
, “
A Survey of Structural and Multidisciplinary Continuum Topology Optimization: Post 2000
,”
Struct. Multidiscip. Optim.
,
49
(
1
), pp.
1
38
.
14.
Fuchi
,
K.
,
Ware
,
T. H.
,
Buskohl
,
P. R.
,
Reich
,
G. W.
,
Vaia
,
R. A.
,
White
,
T. J.
, and
Joo
,
J. J.
,
2015
, “
Topology Optimization for the Design of Folding Liquid Crystal Elastomer Actuators
,”
Soft Matter
,
11
(
37
), pp.
7288
7295
.
15.
Kwok
,
T.-H.
,
Wang
,
C. C. L.
,
Deng
,
D.
,
Zhang
,
Y.
, and
Chen
,
Y.
,
2015
, “
Four-Dimensional Printing for Freeform Surfaces: Design Optimization of Origami and Kirigami Structures
,”
ASME J. Mech. Des.
,
137
(
11
), p.
111413
.
16.
Xue
,
R.
,
Li
,
R.
,
Du
,
Z.
,
Zhang
,
W.
,
Zhu
,
Y.
,
Sun
,
Z.
, and
Guo
,
X.
,
2017
, “
Kirigami Pattern Design of Mechanically Driven Formation of Complex 3D Structures Through Topology Optimization
,”
Extreme Mech. Lett.
,
15
, pp.
139
144
.
17.
Guo
,
X.
,
Zhang
,
W.
, and
Zhong
,
W.
,
2014
, “
Doing Topology Optimization Explicitly and Geometrically a New Moving Morphable Components Based Framework
,”
ASME J. Appl. Mech.
,
81
(
8
), p.
081009
.
18.
Holzapfel
,
G. A.
,
2000
,
Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science
,
1st ed.
, Wiley, Hoboken, NJ.
19.
Villanueva
,
C. H.
, and
Maute
,
K.
,
2014
, “
Density and Level Set-XFEM Schemes for Topology Optimization of 3-D Structures
,”
Comput. Mech.
,
54
(
1
), pp.
133
150
.
20.
Lawry
,
M.
, and
Maute
,
K.
,
2018
, “
Level Set Shape and Topology Optimization of Finite Strain Bilateral Contact Problems
,”
Int. J. Numer. Methods Eng.
,
113
(
8
), pp.
1340
1369
.
21.
Villanueva
,
C. H.
, and
Maute
,
K.
,
2017
, “
CutFEM Topology Optimization of 3D Laminar Incompressible Flow Problems
,”
Comput. Methods Appl. Mech. Eng.
,
320
(
Suppl. C
), pp.
444
473
.
22.
Jenkins
,
N.
, and
Maute
,
K.
,
2015
, “
Level Set Topology Optimization of Stationary Fluid-Structure Interaction Problems
,”
Struct. Multidiscip. Optim.
,
52
(
1
), pp.
179
195
.
23.
Makhija
,
D.
, and
Maute
,
K.
,
2014
, “
Numerical Instabilities in Level Set Topology Optimization With the Extended Finite Element Method
,”
Struct. Multidiscip. Optim.
,
49
(
2
), pp.
185
197
.
24.
Kreissl
,
S.
, and
Maute
,
K.
,
2012
, “
Levelset Based Fluid Topology Optimization Using the Extended Finite Element Method
,”
Struct. Multidiscip. Optim.
,
46
(
3
), pp.
311
326
.
25.
Norato
,
J. A.
,
Bendsøe
,
M. P.
,
Haber
,
R. B.
, and
Tortorelli
,
D. A.
,
2007
, “
A Topological Derivative Method for Topology Optimization
,”
Struct. Multidiscip. Optim.
,
33
(
4–5
), pp.
375
386
.
26.
Bendsøe
,
M. P.
,
1989
, “
Optimal Shape Design as a Material Distribution Problem
,”
Struct. Optim.
,
1
(
4
), pp.
193
202
.
27.
Lazarov
,
B. S.
,
Wang
,
F.
, and
Sigmund
,
O.
,
2016
, “
Length Scale and Manufacturability in Density-Based Topology Optimization
,”
Arch. Appl. Mech.
,
86
(
1–2
), pp.
189
218
.
28.
Fries
,
T. P.
, and
Belytschko
,
T.
,
2010
, “
The Extended/Generalized Finite Element Method: An Overview of the Method and Its Applications
,”
Int. J. Numer. Methods Eng.
,
84
(
3
), pp.
253
304
.
29.
Hansbo
,
A.
, and
Hansbo
,
P.
,
2004
, “
A Finite Element Method for the Simulation of Strong and Weak Discontinuities in Solid Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
193
(
33–35
), pp.
3523
3540
.
30.
Terada
,
K.
,
Asai
,
M.
, and
Yamagishi
,
M.
,
2003
, “
Finite Cover Method for Linear and Non-Linear Analyses of Heterogeneous Solids
,”
Int. J. Numer. Methods Eng.
,
58
(
9
), pp.
1321
1346
.
31.
Tran
,
A. B.
,
Yvonnet
,
J.
,
He
,
Q. C.
,
Toulemonde
,
C.
, and
Sanahuja
,
J.
,
2011
, “
A Multiple Level Set Approach to Prevent Numerical Artefacts in Complex Microstructures With Nearby Inclusions Within XFEM
,”
Int. J. Numer. Methods Eng.
,
85
(
11
), pp.
1436
1459
.
32.
Lang
,
C.
,
Makhija
,
D.
,
Doostan
,
A.
, and
Maute
,
K.
,
2014
, “
A Simple and Efficient Preconditioning Scheme for Heaviside Enriched XFEM
,”
Comput. Mech.
,
54
(
5
), pp.
1357
1374
.
33.
Nitsche
,
J. A.
,
1971
, “
Über Ein Variationsprinzip Zur Lösung Dirichlet-Problem Bei Verwendung Von Teilräumen, Die Keinen Randbedingungen Unteworfen Sind
,”
Abh. Math. Sem. Univ. Hamburg
,
36
(
1
), pp.
9
15
.
34.
Geiss
,
M. J.
, and
Maute
,
K.
,
2018
, “
Topology Optimization of Active Structures Using a Higher-Order Level-Set-XFEM-Density Approach
,”
AIAA
Paper No. AIAA-2018-4053.
35.
Arora
,
J. S.
, and
Wang
,
Q.
,
2005
, “
Review of Formulations for Structural and Mechanical System Optimization
,”
Struct. Multidiscip. Optim.
,
30
(
4
), pp.
251
272
.
36.
Kemmler
,
R.
,
2004
, “
Große Verschiebungen Und Stabilität in Der Topologie- Und Formoptimierung
,” Ph.D. thesis, Universität Stuttgart, Stuttgart, Germany, p.
187
.
37.
Gea
,
H. C.
, and
Luo
,
J.
,
2001
, “
Topology Optimization of Structures With Geometrical Nonlinearities
,”
Comput. Struct.
,
79
(
20–21
), pp.
1977
1985
.
38.
Kreissl
,
S.
,
Pingen
,
G.
, and
Maute
,
K.
,
2011
, “
An Explicit Level Set Approach for Generalized Shape Optimization of Fluids With the Lattice Boltzmann Method
,”
Int. J. Numer. Methods Fluids
,
65
(
5
), pp.
496
519
.
39.
Sharma
,
A.
,
Villanueva
,
H.
, and
Maute
,
K.
,
2017
, “
On Shape Sensitivities With Heaviside-Enriched XFEM
,”
Struct. Multidiscip. Optim.
,
55
(
2
), pp.
385
408
.
40.
Svanberg
,
K.
,
1987
, “
The Method of Moving Asymptotes—A New Method for Structural Optimization
,”
Int. J. Numer. Methods Eng.
,
24
(
2
), pp.
359
373
.
41.
Kuhn
,
H. W.
, and
Tucker
,
A. W.
,
1951
, “
Nonlinear Programming
,”
Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability
, University of California Press, Berkeley, CA, pp.
481
492
.
42.
Amestoy
,
P. R.
,
Duff
,
I. S.
,
L'Excellent
,
J.-Y.
, and
Koster
,
J.
,
2001
, “
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling
,”
SIAM J. Matrix Anal. Appl.
,
23
(
1
), pp.
15
41
.
43.
Amestoy
,
P. R.
,
Guermouche
,
A.
,
L'Excellent
,
J. Y.
, and
Pralet
,
S.
,
2006
, “
Hybrid Scheduling for the Parallel Solution of Linear Systems
,”
Parallel Comput.
,
32
(
2
), pp.
136
156
.
44.
Pajot
,
J. M.
,
Maute
,
K.
,
Zhang
,
Y.
, and
Dunn
,
M. L.
,
2006
, “
Design of Patterned Multilayer Films With Eigenstrains by Topology Optimization
,”
Int. J. Solids Struct.
,
43
(
6
), pp.
1832
1853
.
45.
Ahrens
,
J.
,
Geveci
,
B.
, and
Law
,
C.
,
2005
, “
ParaView: An End-User Tool for Large-Data Visualization
,” Los Alamos National Laboratory, Los Alamos, NM, Technical Report No. LA-UR-03-1560.
46.
Zhang
,
Y.
, and
Dunn
,
M. L.
,
2004
, “
Geometric and Material Nonlinearity During the Deformation of Micron-Scale Thin-Film Bilayers Subject to Thermal Loading
,”
J. Mech. Phys. Solids
,
52
(
9
), pp.
2101
2126
.
47.
Dunn
,
M. L.
,
Zhang
,
Y.
, and
Bright
,
V. M.
,
2002
, “
Deformation and Structural Stability of Layered Plate Microstructures Subjected to Thermal Loading
,”
J. Microelectromech. Syst.
,
11
(
4
), pp.
372
384
.
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