As additive manufacturing (AM) expands into multimaterial, there is a demand for efficient multimaterial topology optimization (MMTO), where one must simultaneously optimize the topology and the distribution of various materials within the topology. The classic approach to multimaterial optimization is to minimize compliance or stress while imposing two sets of constraints: (1) a total volume constraint and (2) individual volume-fraction constraint on each of the material constituents. The latter can artificially restrict the design space. Instead, the total mass and compliance are treated as conflicting objectives, and the corresponding Pareto curve is traced; no additional constraint is imposed on the material composition. To trace the Pareto curve, first-order element sensitivity fields are computed, and a two-step algorithm is proposed. The effectiveness of the algorithm is demonstrated through illustrative examples in 3D.

References

References
1.
Eschenauer
,
H. A.
, and
Olhoff
,
N.
,
2001
, “
Topology Optimization of Continuum Structures: A Review
,”
ASME Appl. Mech. Rev.
,
54
(
4
), pp.
331
389
.
2.
Rozvany
,
G. I. N.
,
2009
, “
A Critical Review of Established Methods of Structural Topology Optimization
,”
Struct. Multidiscip. Optim.
,
37
(
3
), pp.
217
237
.
3.
Bendsøe
,
M.
, and
Sigmund
,
O.
,
2003
,
Topology Optimization: Theory, Methods and Application
,
2nd ed.
,
Springer
,
Berlin
.
4.
Kesseler
,
E.
, and
Vankan
,
W. J.
,
2006
, “
Multidisciplinary Design Analysis and Multi-Objective Optimisation Applied to Aircraft Wing
,”
WSEAS Trans. Syst. Control
,
1
(
2
), pp.
221
227
.
5.
Alonso
,
J. J.
,
2009
, “
Aircraft Design Optimization
,”
Math. Comput. Simul.
,
79
(
6
), pp.
1948
1958
.
6.
Coverstone-Carroll
,
V. H.
,
Hartmann
,
J. W.
, and
Mason
,
W. J.
,
2000
, “
Optimal Multi-Objective Low-Thrust Spacecraft Trajectories
,”
Comput. Methods Appl. Mech. Eng.
,
186
(
2–4
), pp.
387
402
.
7.
Wang
,
L.
,
2004
, “
Automobile Body Reinforcement by Finite Element Optimization
,”
Finite Elem. Anal. Des.
,
40
(
8
), pp.
879
893
.
8.
Ananthasuresh
,
G. K.
,
Kota
,
S.
, and
Gianchandani
,
Y.
,
1994
, “
A Methodical Approach to the Design of Compliant Micromechanisms
,” Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, June 13–16, pp.
189
192
.
9.
Nishiwaki
,
S.
,
1998
, “
Topology Optimization of Compliant Mechanisms Using the Homogenization Method
,”
Int. J. Numer. Methods Eng.
,
42
(3), pp.
535
559
.
10.
Bruns
,
T. E.
, and
Tortorelli
,
D. A.
,
2001
, “
Topology Optimization of Non-Linear Elastic Structures and Compliant Mechanisms
,”
Comput. Methods Appl. Mech. Eng.
,
190
(
26–27
), pp.
3443
3459
.
11.
Luo
,
Z.
,
2005
, “
Compliant Mechanism Design Using Multi-Objective Topology Optimization Scheme of Continuum Structures
,”
Struct. Multidiscip. Optim.
,
30
(
2
), pp.
142
154
.
12.
Gibson
,
I.
,
Rosen
,
D. W.
, and
Stucker
,
B.
,
2010
,
Additive Manufacturing Technologies
,
Springer
,
New York
.
13.
Vidimce
,
K.
,
Wang
,
S.-P.
,
Ragan-Kelley
,
J.
, and
Matusik
,
W.
,
2013
, “
openfab: A Programmable Pipeline for Multi-Material Fabrication
,”
SIGGRAPH/ACM Trans. Graphics
,
32
(4), pp.
136:1
136:12
.
14.
Thomsen
,
J.
,
1992
, “
Topology Optimization of Structures Composed of One or Two Materials
,”
J. Struct. Optim.
,
5
(
1–2
), pp.
108
115
.
15.
Suresh
,
K.
,
2010
, “
A 199-Line matlab Code for Pareto-Optimal Tracing in Topology Optimization
,”
Struct. Multidiscip. Optim.
,
42
(
5
), pp.
665
679
.
16.
Suresh
,
K.
,
2013
, “
Efficient Generation of Large-Scale Pareto-Optimal Topologies
,”
Struct. Multidiscip. Optim.
,
47
(
1
), pp.
49
61
.
17.
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
.
18.
Sigmund
,
O.
,
2001
, “
A 99 Line Topology Optimization Code Written in matlab
,”
Struct. Multidiscip. Optim.
,
21
(
2
), pp.
120
127
.
19.
Allaire
,
G.
,
Jouve
,
F.
, and
Toader
,
A. M.
,
2002
, “
A Level-Set Method for Shape Optimization
,”
C. R. Math.
,
334
(
12
), pp.
1125
1130
.
20.
Allaire
,
G.
, and
Jouve
,
F.
,
2005
, “
A Level-Set Method for Vibration and Multiple Loads Structural Optimization
,”
Struct. Des. Optim.
,
194
(
30–33
), pp.
3269
3290
.
21.
He
,
L.
,
Kao
,
C.-Y.
, and
Osher
,
S.
,
2007
, “
Incorporating Topological Derivatives Into Shape Derivatives Based Level Set Methods
,”
J. Comput. Phys.
,
225
(
1
), pp.
891
909
.
22.
Wang
,
M. Y.
,
Wang
,
X.
, and
Guo
,
D.
,
2003
, “
A Level Set Method for Structural Topology Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
192
(
1
), pp.
227
246
.
23.
Hamda
,
H.
,
2002
, “
Application of a Multi-Objective Evolutionary Algorithm to Topology Optimum Design
,”
Fifth International Conference on Adaptive Computing in Design and Manufacture
(
ACDM'02
), Exeter, UK, Apr. 16–18.
24.
Xie
,
Y. M.
,
1993
, “
A Simple Evolutionary Procedure for Structural Optimization
,”
Comput. Struct.
,
49
(
5
), pp.
885
896
.
25.
Xie
,
Y. M.
, and
Steven
,
G. P.
,
1997
,
Evolutionary Structural Optimization
,
1st ed.
,
Springer-Verlag
,
Berlin
.
26.
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
.
27.
Sigmund
,
O.
, and
Maute
,
K.
,
2013
, “
Topology Optimization Approaches
,”
Struct. Multidiscip. Optim.
,
48
(
6
), pp.
1031
1055
.
28.
Guest
,
J. K.
,
2004
, “
Achieving Minimum Length Scale in Topology Optimization Using Nodal Design Variables and Projection Functions
,”
Int. J. Numer. Methods Eng.
,
61
(
2
), pp.
238
254
.
29.
Nguyen
,
T.
,
Paulino
,
G.
,
Song
,
J.
, and
Le
,
C.
,
2010
, “
A Computational Paradigm for Multiresolution Topology Optimization (MTOP)
,”
Struct. Multidiscip. Optim.
,
41
(
4
), pp.
525
539
.
30.
Stump
,
F. V.
,
Silva
,
E. C. N.
, and
Paulino
,
G.
,
2007
, “
Optimization of Material Distribution in Functionally Graded Structures With Stress Constraints
,”
Commun. Numer. Methods Eng.
,
23
(
6
), pp.
535
551
.
31.
Sigmund
,
O.
,
2001
, “
Design of Multiphysics Actuators Using Topology Optimization—Part II: Two-Material Structures
,”
Comput. Methods Appl. Mech. Eng.
,
190
(
49–50
), pp.
6605
6627
.
32.
Yin
,
L.
, and
Ananthasuresh
,
G. K.
,
2001
, “
Topology Optimization of Compliant Mechanisms With Multiple Materials Using a Peak Function Material Interpolation Scheme
,”
Struct. Multidiscip. Optim.
,
23
(
1
), pp.
49
62
.
33.
Jeong
,
S. H.
,
Choi
,
D.-H.
, and
Yoon
,
G. H.
,
2014
, “
Separable Stress Interpolation Scheme for Stress-Based Topology Optimization With Multiple Homogenous Materials
,”
Finite Elem. Anal. Des.
,
82
, pp.
16
31
.
34.
Chaves
,
L. P.
, and
Cunha
,
J.
,
2014
, “
Design of Carbon Fiber Reinforcement of Concrete Slabs Using Topology Optimization
,”
Constr. Build. Mater.
,
73
, pp.
688
698
.
35.
de Kruijf
,
N.
,
Zhou
,
S.
,
Li
,
Q.
, and
Mai
,
Y.-W.
,
2007
, “
Topological Design of Structures and Composite Materials With Multiobjectives
,”
Int. J. Solids Struct.
,
44
(22–23), pp.
7092
7109
.
36.
Blasques
,
J. P.
, and
Stolpe
,
M.
,
2012
, “
Multi-Material Topology Optimization of Laminated Composite Beam Cross Sections
,”
Compos. Struct.
,
94
(
11
), pp.
3278
3289
.
37.
Park
,
J.
, and
Sutradhar
,
A.
,
2014
, “
A Multi-Resolution Method for 3D Multi-Material Topology Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
285
, pp.
571
586
.
38.
Wang
,
M. Y.
, and
Wang
,
X.
,
2004
, “
‘Color’ Level Sets: A Multi-Phase Method for Structural Topology Optimization With Multiple Materials
,”
Comput. Methods Appl. Mech. Eng.
,
193
(
6–8
), pp.
469
496
.
39.
Wang
,
M. Y.
,
Chen
,
S.
,
Wang
,
X.
, and
Mei
,
Y.
,
2005
, “
Design of Multimaterial Compliant Mechanisms Using Level-Set Methods
,”
ASME J. Mech. Des.
,
127
(
5
), pp.
941
956
.
40.
Wang
,
X.
,
Mei
,
Y.
, and
Wang
,
M. Y.
,
2004
, “
Level-Set Method for Design of Multi-Phase Elastic and Thermoelastic Materials
,”
Int. J. Mech. Mater. Des.
,
1
(
3
), pp.
213
239
.
41.
Allaire
,
G.
,
Dapogny
,
C.
,
Delgado
,
G.
, and
Michailidis
,
G.
,
2014
, “
Multi-Phase Structural Optimization Via a Level Set Method
,”
ESAIM: Contr. Optim. Calcul. Variat.
,
20
(
2
), pp.
576
611
.
42.
Vermaak
,
N.
,
Michailidis
,
G.
,
Parry
,
G.
,
Estevez
,
R.
,
Allaire
,
G.
, and
Bréchet
,
Y.
,
2014
, “
Material Interface Effects on the Topology Optimization of Multi-Phase Structures Using a Level Set Method
,”
Struct. Multidiscip. Optim.
,
50
(
4
), pp.
623
644
.
43.
Wang
,
M. Y.
, and
Zhou
,
S.
,
2005
, “
Synthesis of Shape and Topology of Multi-Material Structures With a Phase-Field Method
,”
J. Comput. Aided Mater. Des.
,
11
(
2–3
), pp.
117
138
.
44.
Wang
,
Y.
,
Luo
,
Z.
,
Kang
,
Z.
, and
Zhang
,
N.
,
2015
, “
A Multi-Material Level Set-Based Topology and Shape Optimization Method
,”
Comput. Methods Appl. Mech. Eng.
,
283
(
1
), pp.
1570
1586
.
45.
Xie
,
Y. M.
, and
Steven
,
G. P.
,
1993
, “
A Simple Evolutionary Procedure for Structural Optimization
,”
Comput. Struct.
,
49
(
5
), pp.
885
896
.
46.
Chu
,
D. N.
,
Xie
,
Y. M.
,
Hira
,
A.
, and
Steven
,
G. P.
,
1996
, “
Evolutionary Structural Optimization for Problems With Stiffness Constraints
,”
Finite Elem. Anal. Des.
,
21
(
4
), pp.
239
251
.
47.
Querin
,
O. M.
,
Steven
,
G. P.
, and
Xie
,
Y. M.
,
1998
, “
Evolutionary Structural Optimisation (ESO) Using a Bidirectional Algorithm
,”
Eng. Comput.
,
15
(
8
), pp.
1031
1048
.
48.
Querin
,
O. M.
,
Steven
,
G. P.
, and
Xie
,
Y. M.
,
2000
, “
Evolutionary Structural Optimisation Using an Additive Algorithm
,”
Finite Elem. Anal. Des.
,
34
(
3–4
), pp.
291
308
.
49.
Liu
,
X.
,
Yi
,
W.-J.
,
Li
,
Q. S.
, and
Shen
,
P.-S.
,
2008
, “
Genetic Evolutionary Structural Optimization
,”
J. Constr. Steel Res.
,
64
(
3
), pp.
305
311
.
50.
Sigmund
,
O.
,
2011
, “
On the Usefulness of Non-Gradient Approaches in Topology Optimization
,”
Struct. Multidiscip. Optim.
,
43
(5), pp.
589
596
.
51.
Ramani
,
A.
,
2009
, “
A Pseudo-Sensitivity Based Discrete Variable Approach to Structural Topology Optimization With Multiple Materials
,”
Struct. Multidiscip. Optim.
,
41
(
6
), pp.
913
934
.
52.
Ramani
,
A.
,
2011
, “
Multi-Material Topology Optimization With Strength Constraints
,”
Struct. Multidiscip. Optim.
,
43
(5), pp.
597
615
.
53.
Novotny
,
A. A.
,
2006
, “
Topological-Shape Sensitivity Method: Theory and Applications
,”
Solid Mech. Appl.
,
137
, pp.
469
478
.
54.
Suresh
,
K.
, and
Takalloozadeh
,
M.
,
2013
, “
Stress-Constrained Topology Optimization: A Topological Level-Set Approach
,”
Struct. Multidiscip. Optim.
,
48
(
2
), pp.
295
309
.
55.
Deng
,
S.
, and
Suresh
,
K.
,
2014
, “
Multi-Constrained Topology Optimization Via the Topological Sensitivity
,”
Struct. Multidisc. Optim.
,
51
(
5
), pp.
987
1001
.
56.
Schneider
,
M.
, and
Andra
,
H.
,
2014
, “
The Topological Gradient in Anisotropic Elasticity With an Eye Towards Lightweight Design
,”
Math. Methods Appl. Sci.
,
37
(
11
), pp.
1624
1641
.
57.
Keeffe
,
G. D.
,
2014
, “
Optimization of Composite Structures: A Shape and Topology Sensitivity Analysis
,” Ph.D. thesis, Ecole Polytechnique, Universite Paris-Saclay, Saint-Aubin, France.
58.
Gao
,
T.
, and
Zhang
,
W.
,
2011
, “
A Mass Constraint Formulation for Structural Topology Optimization With Multiphase Materials
,”
Int. J. Numer. Methods Eng.
,
88
(
8
), pp.
774
796
.
59.
Amstutz
,
S.
,
2011
, “
Connections Between Topological Sensitivity Analysis and Material Interpolation Schemes in Topology Optimization
,”
Struct. Multidiscip. Optim.
,
43
(6), pp.
755
765
.
60.
Choi
,
K. K.
, and
Kim
,
N. H.
,
2005
,
Structural Sensitivity Analysis and Optimization I: Linear Systems
,
Springer
,
New York
.
61.
Hughes
,
T. J. R.
,
Levit
,
I.
, and
Winget
,
J.
,
1983
, “
An Element-by-Element Solution Algorithm for Problems of Structural and Solid Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
36
(
2
), pp.
241
254
.
62.
Yadav
,
P.
, and
Suresh
,
K.
,
2014
, “
Large Scale Finite Element Analysis Via Assembly-Free Deflated Conjugate Gradient
,”
ASME J. Comput. Inf. Sci. Eng.
,
14
(
4
), p.
041008
.
63.
Suresh
,
K.
, and
Yadav
,
P.
,
2012
, “
Large-Scale Modal Analysis on Multi-Core Architectures
,”
ASME
Paper No. DETC2012-70281.
64.
Saad
,
Y.
,
Yeung
,
M.
,
Erhel
,
J.
, and
Guyomarc'h
,
F.
,
2000
, “
A Deflated Version of the Conjugate Gradient Algorithm
,”
SIAM J. Sci. Comput.
,
21
(
5
), pp.
1909
1926
.
65.
Sigmund
,
O.
,
2007
, “
Morphology-Based Black and White Filters for Topology Optimization
,”
Struct. Multidiscip. Optim.
,
33
(4–5), pp.
401
424
.
You do not currently have access to this content.