In the present work, a new approach for designing graded lattice structures is developed under the moving morphable components/voids (MMC/MMV) topology optimization framework. The essential idea is to make a coordinate perturbation to the topology description functions (TDF) that are employed for the description of component/void geometries in the design domain. Then, the optimal graded structure design can be obtained by optimizing the coefficients in the perturbed basis functions. Our numerical examples show that the proposed approach enables a concurrent optimization of both the primitive cell and the graded material distribution in a straightforward and computationally effective way. Moreover, the proposed approach also shows its potential in finding the optimal configuration of complex graded lattice structures with a very small number of design variables employed under various loading conditions and coordinate systems.

References

1.
Lakes
,
R.
,
1993
, “
Materials With Structural Hierarchy
,”
Nature
,
361
(
6412
), pp.
511
515
.
2.
Sigmund
,
O.
, and
Torquato
,
S.
,
1996
, “
Composites With Extremal Thermal Expansion Coefficients
,”
Appl. Phys. Lett.
,
69
(
21
), pp.
3203
3205
.
3.
Knight
,
J. C.
,
2003
, “
Photonic Crystal Fibres
,”
Nature
,
424
(
6950
), pp.
847
851
.
4.
Kushwaha
,
M. S.
,
Halevi
,
P.
,
Dobrzynski
,
L.
, and
Djafari-Rouhani
,
B.
,
1993
, “
Acoustic Band Structure of Periodic Elastic Composites
,”
Phys. Rev. Lett.
,
71
(
13
), pp.
2022
2025
.
5.
Liu
,
C.
,
Du
,
Z. L.
,
Sun
,
Z.
,
Gao
,
H. J.
, and
Guo
,
X.
,
2015
, “
Frequency-Preserved Acoustic Diode Model With High Forward-Power-Transmission Rate
,”
Phys. Rev. Appl.
,
3
(
6
), p.
064014
.
6.
Zheng
,
X.
,
Lee
,
H.
,
Weisgraber
,
T. H.
,
Shusteff
,
M.
,
DeOtte
,
J.
,
Duoss
,
E. B.
,
Kuntz
,
J. D.
,
Biener
,
M. M.
,
Ge
,
Q.
,
Jackson
,
J. A.
,
Kucheyev
,
S. O.
,
Fang
,
N. X.
, and
Spadaccini
,
C. M.
,
2014
, “
Ultralight, Ultrastiff Mechanical Metamaterials
,”
Science
,
344
(
6190
), pp.
1373
1377
.
7.
Kim
,
T.
,
Hodson
,
H. P.
, and
Lu
,
T. J.
,
2004
, “
Fluid-Flow and Endwall Heat-Transfer Characteristics of an Ultralight Lattice-Frame Material
,”
Int. J. Heat Mass Transfer
,
47
(
6–7
), pp.
1129
1140
.
8.
Zheng
,
J.
,
Zhao
,
L.
, and
Fan
,
H.
,
2012
, “
Energy Absorption Mechanisms of Hierarchical Woven Lattice Composites
,”
Compos. Part B: Eng.
,
43
(
3
), pp.
1516
1522
.
9.
Vasiliev
,
V. V.
, and
Razin
,
A. F.
,
2006
, “
Anisogrid Composite Lattice Structures for Spacecraft and Aircraft Applications
,”
Compos. Struct.
,
76
(
1–2
), pp.
182
189
.
10.
Zheludev
,
N. I.
, and
Kivshar
,
Y. S.
,
2012
, “
From Metamaterials to Metadevices
,”
Nat. Mater.
,
11
(
11
), pp.
917
924
.
11.
Bendsøe
,
M. P.
, and
Kikuchi
,
N.
,
1988
, “
Generating Optimal Topologies in Structural Design Using a Homogenization Method
,”
Comput. Methods Appl. Mech. Eng.
,
71
(
2
), pp.
197
224
.
12.
Rodrigues
,
H.
,
Guedes
,
J. M.
, and
Bendsøe
,
M. P.
,
2002
, “
Hierarchical Optimization of Material and Structure
,”
Struct. Multidisciplinary Optim.
,
24
(
1
), pp.
1
10
.
13.
Coelho
,
P. G.
,
Fernandes
,
P. R.
,
Guedes
,
J. M.
, and
Rodrigues
,
H. C.
,
2007
, “
A Hierarchical Model for Concurrent Material and Topology Optimisation of Three-Dimensional Structures
,”
Struct. Multidiscip. Optim.
,
35
(
2
), pp.
107
115
.
14.
Coelho
,
P. G.
,
Guedes
,
J. M.
, and
Rodrigues
,
H. C.
,
2015
, “
Multiscale Topology Optimization of Bi-Material Laminated Composite Structures
,”
Compos. Struct.
,
132
, pp.
495
505
.
15.
Liu
,
L.
,
Yan
,
J.
, and
Cheng
,
G.
,
2008
, “
Optimum Structure With Homogeneous Optimum Truss-Like Material
,”
Comput. Struct.
,
86
(
13–14
), pp.
1417
1425
.
16.
Niu
,
B.
,
Yan
,
J.
, and
Cheng
,
G.
,
2009
, “
Optimum Structure With Homogeneous Optimum Cellular Material for Maximum Fundamental Frequency
,”
Struct. Multidiscip. Optim.
,
39
(
2
), pp.
115
132
.
17.
Deng
,
J.
,
Yan
,
J.
, and
Cheng
,
G.
,
2012
, “
Multi-Objective Concurrent Topology Optimization of Thermoelastic Structures Composed of Homogeneous Porous Material
,”
Struct. Multidiscip. Optim.
,
47
(
4
), pp.
583
597
.
18.
Yan
,
J.
,
Guo
,
X.
, and
Cheng
,
G.
,
2016
, “
Multi-Scale Concurrent Material and Structural Design Under Mechanical and Thermal Loads
,”
Comput. Mech.
,
57
(
3
), pp.
437
446
.
19.
Guo
,
X.
,
Zhao
,
X.
,
Zhang
,
W.
,
Yan
,
J.
, and
Sun
,
G.
,
2015
, “
Multi-Scale Robust Design and Optimization Considering Load Uncertainties
,”
Comput. Methods Appl Mech. Eng.
,
283
, pp.
994
1009
.
20.
Sivapuram
,
R.
,
Dunning
,
P. D.
, and
Kim
,
H. A.
,
2016
, “
Simultaneous Material and Structural Optimization by Multiscale Topology Optimization
,”
Struct. Multidiscip. Optim.
,
54
(
5
), pp.
1267
1281
.
21.
Tan
,
T.
,
Rahbar
,
N.
,
Allameh
,
S. M.
,
Kwofie
,
S.
,
Dissmore
,
D.
,
Ghavami
,
K.
, and
Soboyejo
,
W. O.
,
2011
, “
Mechanical Properties of Functionally Graded Hierarchical Bamboo Structures
,”
Acta Biomater.
,
7
(
10
), pp.
3796
3803
.
22.
Rho
,
J.-Y.
,
Kuhn-Spearing
,
L.
, and
Zioupos
,
P.
,
1998
, “
Mechanical Properties and the Hierarchical Structure of Bone
,”
Med. Eng. Phys.
,
20
(
2
), pp.
92
102
.
23.
Gibson
,
L. J.
,
2012
, “
The Hierarchical Structure and Mechanics of Plant Materials
,”
J. R. Soc. Interface
,
9
(
76
), pp.
2749
2766
.
24.
Norris
,
A. N.
,
2008
, “
Acoustic Cloaking Theory
,”
Proc. R. Soc. A: Math. Phys. Eng. Sci.
,
464
(
2097
), pp.
2411
2434
.
25.
Maldovan
,
M.
,
2013
, “
Sound and Heat Revolutions in Phononics
,”
Nature
,
503
(
7475
), pp.
209
217
.
26.
Conway, J. H., Burgiel, H., and Goodman-Strauss, C., 2016,
The Symmetries of Things
, CRC Press, Boca Raton, FL, p. 218.
27.
M. C. Escher, “M. C. Escher,” The M. C. Escher Company B.V., Baarn, The Netherlands, accessed June 13, 2017, http://www.mcescher.com/
28.
Zhou
,
S.
, and
Li
,
Q.
,
2008
, “
Design of Graded Two-Phase Microstructures for Tailored Elasticity Gradients
,”
J. Mater. Sci.
,
43
(
15
), pp.
5157
5167
.
29.
Lin
,
D.
,
Li
,
Q.
,
Li
,
W.
,
Zhou
,
S.
, and
Swain
,
M. V.
,
2009
, “
Design Optimization of Functionally Graded Dental Implant for Bone Remodeling
,”
Compos. Part B: Eng.
,
40
(
7
), pp.
668
675
.
30.
Radman
,
A.
,
Huang
,
X.
, and
Xie
,
Y. M.
,
2013
, “
Topology Optimization of Functionally Graded Cellular Materials
,”
J. Mater. Sci.
,
48
(
4
), pp.
1503
1510
.
31.
Radman
,
A.
,
Huang
,
X.
, and
Xie
,
Y. M.
,
2014
, “
Maximizing Stiffness of Functionally Graded Materials With Prescribed Variation of Thermal Conductivity
,”
Comput. Mater. Sci.
,
82
, pp.
457
463
.
32.
Wang
,
Y.
,
Chen
,
F.
, and
Wang
,
M. Y.
,
2017
, “
Concurrent Design With Connectable Graded Microstructures
,”
Comput. Methods Appl. Mech. Eng.
,
317
, pp.
84
101
.
33.
Bendsøe
,
M. P.
,
1989
, “
Optimal Shape Design as a Material Distribution Problem
,”
Struct. Optim.
,
1
(
4
), pp.
193
202
.
34.
Bendsøe
,
M. P.
,
Guedes
,
J. M.
,
Haber
,
R. B.
,
Pedersen
,
P.
, and
Taylor
,
J. E.
,
1994
, “
An Analytical Model to Predict Optimal Material Properties in the Context of Optimal Structural Design
,”
ASME J. Appl. Mech.
,
61
(
4
), pp.
930
937
.
35.
Mlejnek
,
H. P.
,
1992
, “
Some Aspects of the Genesis of Structures
,”
Struct. Optim.
,
5
(
1–2
), pp.
64
69
.
36.
Zhou
,
M.
, and
Rozvany
,
G. I. N.
,
1991
, “
The COC Algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
89
(
1–3
), pp.
309
336
.
37.
Allaire
,
G.
,
Jouve
,
F.
, and
Toader
,
A.-M.
,
2004
, “
Structural Optimization Using Sensitivity Analysis and a Level-Set Method
,”
J. Comput. Phys.
,
194
(
1
), pp.
363
393
.
38.
Wang
,
M. Y.
,
Wang
,
X.
, and
Guo
,
D.
,
2003
, “
A Level Set Method for Structural Topology Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
192
(
1–2
), pp.
227
246
.
39.
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
.
40.
Zhang
,
W.
,
Yang
,
W.
,
Zhou
,
J.
,
Li
,
D.
, and
Guo
,
X.
,
2016
, “
Structural Topology Optimization Through Explicit Boundary Evolution
,”
ASME J. Appl. Mech.
, 84(1) p.
011011
.
41.
Guo
,
X.
,
Zhang
,
W.
,
Zhang
,
J.
, and
Yuan
,
J.
,
2016
, “
Explicit Structural Topology Optimization Based on Moving Morphable Components (MMC) With Curved Skeletons
,”
Comput. Methods Appl. Mech. Eng.
,
310
, pp.
711
748
.
42.
Zhang
,
W.
,
Li
,
D.
,
Yuan
,
J.
,
Song
,
J.
, and
Guo
,
X.
,
2017
, “
A New Three-Dimensional Topology Optimization Method Based on Moving Morphable Components (MMCs)
,”
Comput. Mech.
,
59
(
4
), pp.
647
665
.
43.
Zhang
,
W.
,
Yuan
,
J.
,
Zhang
,
J.
, and
Guo
,
X.
,
2015
, “
A New Topology Optimization Approach Based on Moving Morphable Components (MMC) and the Ersatz Material Model
,”
Struct. Multidiscip. Optim.
,
53
(
6
), pp.
1243
1260
.
44.
Zhang
,
W.
,
Zhang
,
J.
, and
Guo
,
X.
,
2016
, “
Lagrangian Description Based Topology Optimization—A Revival of Shape Optimization
,”
ASME J. Appl. Mech.
,
83
(
4
), p.
041010
.
45.
Zhang
,
W.
,
Li
,
D.
,
Zhang
,
J.
, and
Guo
,
X.
,
2016
, “
Minimum Length Scale Control in Structural Topology Optimization Based on the Moving Morphable Components (MMC) Approach
,”
Comput. Methods Appl. Mech. Eng.
,
311
, pp.
327
355
.
46.
Guo
,
X.
,
Zhou
,
J.
,
Zhang
,
W.
,
Du
,
Z.
,
Liu
,
C.
, and
Liu
,
Y.
,
2017
, “
Self-Supporting Structure Design in Additive Manufacturing Through Explicit Topology Optimization
,”
Comput. Methods Appl. Mech. Eng.
, epub.
47.
Zhang
,
W.
,
Chen
,
J.
,
Zhu
,
X.
,
Zhou
,
J.
,
Xue
,
D.
,
Lei
,
X.
, and
Guo
,
X.
,
2017
, “
Explicit Three Dimensional Topology Optimization Via Moving Morphable Void (MMV) Approach
,”
Comput. Methods Appl. Mech. Eng.
,
322
, pp.
590
614
.
48.
Svanberg
,
K.
,
1987
, “
The Method of Moving Asymptotes—A New Method for Structural Optimization
,”
Int. J. Numer. Methods Eng.
,
24
(
2
), pp.
359
373
.
49.
Wu
,
J.
,
Dick
,
C.
, and
Westermann
,
R.
,
2016
, “
A System for High-Resolution Topology Optimization
,”
IEEE Trans. Visualization Comput. Graphics
,
22
(
3
), pp.
1195
1208
.
You do not currently have access to this content.