The objective of this work is to establish a cluster-based optimization method for the optimal design of cellular materials and structures for crashworthiness, which involves the use of nonlinear, dynamic finite element models. The proposed method uses a cluster-based structural optimization approach consisting of four steps: conceptual design generation, clustering, metamodel-based global optimization, and cellular material design. The conceptual design is generated using structural optimization methods. K-means clustering is applied to the conceptual design to reduce the dimensional of the design space as well as define the internal architectures of the multimaterial structure. With reduced dimension space, global optimization aims to improve the crashworthiness of the structure can be performed efficiently. The cellular material design incorporates two homogenization methods, namely, energy-based homogenization for linear and nonlinear elastic material models and mean-field homogenization for (fully) nonlinear material models. The proposed methodology is demonstrated using three designs for crashworthiness that include linear, geometrically nonlinear, and nonlinear models.

References

References
1.
Gibson
,
L. J.
, and
Ashby
,
M. F.
,
1997
,
Cellular Solids: Structure and Properties
(Cambridge Solid State Science Series),
2nd ed.
,
Cambridge University Press
, Cambridge, UK.
2.
Gibson
,
L. J.
,
Ashby
,
M. F.
, and
Harley
,
B. A.
,
2010
,
Cellular Materials in Nature and Medicine
,
Cambridge University Press
,
Cambridge, UK
.
3.
Wu
,
T.
,
Liu
,
K.
, and
Tovar
,
A.
,
2017
, “
Multiphase Topology Optimization of Lattice Injection Molds
,”
Comput. Struct.
,
192
, pp.
71
82
.
4.
Mehta
,
P. S.
,
S.
,
Ocampo
,
J.
,
Tovar
,
A.
, and
Chaudhari
,
P.
,
2016
, “
Bio-Inspired Design of Lightweight and Protective Structures
,”
SAE
Paper No. 2016-01-0396
.
5.
Orringer
,
O.
,
Knadle
,
K. T.
, and
Mandell
,
J. F.
,
1984
, “
Crash Padding Mechanical Properties and Impact Response Analysis
,”
SAE
Paper No. 840866
.
6.
Cui
,
L.
,
Kiernan
,
S.
, and
Gilchrist
,
M. D.
,
2009
, “
Designing the Energy Absorption Capacity of Functionally Graded Foam Materials
,”
Mater. Sci. Eng.: A
,
507
(
1–2
), pp.
215
225
.
7.
Yin
,
H.
,
Wen
,
G.
,
Liu
,
Z.
, and
Qing
,
Q.
,
2014
, “
Crashworthiness Optimization Design for Foam-Filled Multi-Cell Thin-Walled Structures
,”
Thin-Walled Struct.
,
75
, pp.
8
17
.
8.
Bi
,
J.
,
Fang
,
H.
,
Wang
,
Q.
, and
Ren
,
X.
,
2010
, “
Modeling and Optimization of Foam-Filled Thin-Walled Columns for Crashworthiness Designs
,”
Finite Elem. Anal. Des.
,
46
(
9
), pp.
698
709
.
9.
Zhang
,
Z.
,
Liu
,
S.
, and
Tang
,
Z.
,
2011
, “
Comparisons of Honeycomb Sandwich and Foam-Filled Cylindrical Columns Under Axial Crushing Loads
,”
Thin-Walled Struct.
,
49
(
9
), pp.
1071
1079
.
10.
Paz
,
J.
,
Díaz
,
J.
,
Romera
,
L.
, and
Costas
,
M.
,
2015
, “
Size and Shape Optimization of Aluminum Tubes With Gfrp Honeycomb Reinforcements for Crashworthy Aircraft Structures
,”
Compos. Struct.
,
133
, pp.
499
507
.
11.
Ahmad
,
Z.
, and
Thambiratnam
,
D.
,
2009
, “
Crushing Response of Foam-Filled Conical Tubes Under Quasi-Static Axial Loading
,”
Mater. Des.
,
30
(
7
), pp.
2393
2403
.
12.
Yin
,
H.
,
Fang
,
H.
,
Xiao
,
Y.
,
Wen
,
G.
, and
Qing
,
Q.
,
2015
, “
Multi-Objective Robust Optimization of Foam-Filled Tapered Multi-Cell Thin-Walled Structures
,”
Struct. Multidiscip. Optim.
,
52
(
6
), pp.
1051
1067
.
13.
Ahmad
,
Z.
, and
Thambiratnam
,
D. P.
,
2009
, “
Dynamic Computer Simulation and Energy Absorption of Foam-Filled Conical Tubes Under Axial Impact Loading
,”
Comput. Struct.
,
87
(
3–4
), pp.
186
197
.
14.
Correa
,
D. M.
,
Klatt
,
T.
,
Cortes
,
S.
,
Haberman
,
M.
,
Kovar
,
D.
, and
Seepersad
,
C.
,
2015
, “
Negative Stiffness Honeycombs for Recoverable Shock Isolation
,”
Rapid Prototyping J.
,
21
(
2
), pp.
193
200
.
15.
Correa
,
D. M.
,
Seepersad
,
C. C.
, and
Haberman
,
M. R.
,
2015
, “
Mechanical Design of Negative Stiffness Honeycomb Materials
,”
Integr. Mater. Manuf. Innovation
,
4
(
1
), p.
10
.
16.
Cortes
,
S.
,
Allison
,
J.
,
Morris
,
C.
,
Haberman
,
M.
,
Seepersad
,
C.
, and
Kovar
,
D.
,
2017
, “
Design, Manufacture, and Quasi-Static Testing of Metallic Negative Stiffness Structures Within a Polymer Matrix
,”
Exp. Mech.
,
57
(
8
), pp.
1183
1191
.
17.
Matthews
,
J.
,
Klatt
,
T.
,
Morris
,
C.
,
Seepersad
,
C. C.
,
Haberman
,
M.
, and
Shahan
,
D.
,
2016
, “
Hierarchical Design of Negative Stiffness Metamaterials Using a Bayesian Network Classifier
,”
ASME J. Mech. Des.
,
138
(
4
), p.
041404
.
18.
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
.
19.
Rodrigues
,
H. C.
,
Guedes
,
J. M.
, and
Bendsøe
,
M. P.
,
2002
, “
Hierarchical Optimization of Material and Structure
,”
Struct. Multidiscip. Optim.
,
24
(
1
), pp.
1
10
.
20.
Coelho
,
P. G.
,
Fernandes
,
P. R.
,
Guedes
,
J. M.
, and
Rodrigues
,
H. C.
,
2008
, “
A Hierarchical Model for Concurrent Material and Topology Optimisation of Three-Dimensional Structures
,”
Struct. Multidiscip. Optim.
,
35
(
2
), pp.
107
115
.
21.
Miehe
,
C.
,
2002
, “
Strain-Driven Homogenization of Inelastic Microstructures and Composites Based on an Incremental Variational Formulation
,”
Int. J. Numer. Methods Eng.
,
55
(
11
), pp.
1285
1322
.
22.
Feyel
,
F.
, and
Chaboche
,
J.-L.
,
2000
, “
Fe2 Multiscale Approach for Modelling the Elastoviscoplastic Behaviour of Long Fibre Sic/Ti Composite Materials
,”
Comput. Methods Appl. Mech. Eng.
,
183
(
3–4
), pp.
309
330
.
23.
Geers
,
M. G.
,
Kouznetsova
,
V. G.
, and
Brekelmans
,
W.
,
2010
, “
Multi-Scale Computational Homogenization: Trends and Challenges
,”
J. Comput. Appl. Math.
,
234
(
7
), pp.
2175
2182
.
24.
Xia
,
L.
, and
Breitkopf
,
P.
,
2017
, “
Recent Advances on Topology Optimization of Multiscale Nonlinear Structures
,”
Arch. Comput. Methods Eng.
,
24
(
2
), pp.
227
249
.
25.
Fang
,
J.
,
Sun
,
G.
,
Qiu
,
N.
,
Kim
,
N. H.
, and
Li
,
Q.
,
2017
, “
On Design Optimization for Structural Crashworthiness and Its State of the Art
,”
Struct. Multidiscip. Optim.
,
55
(
3
), pp.
1091
1119
.
26.
Liu
,
K.
,
Detwiler
,
D.
, and
Tovar
,
A.
,
2017
, “
Optimal Design of Nonlinear Multimaterial Structures for Crashworthiness Using Cluster Analysis
,”
ASME J. Mech. Des.
,
139
(
10
), p.
101401
.
27.
Bendsøe
,
M. P.
, and
Sigmund
,
O.
,
2003
,
Topology Optimization: Theory, Method and Applications
,
Springer
,
New York
.
28.
Xia
,
L.
, and
Breitkopf
,
P.
,
2015
, “
Multiscale Structural Topology Optimization With an Approximate Constitutive Model for Local Material Microstructure
,”
Comput. Methods Appl. Mech. Eng.
,
286
, pp.
147
167
.
29.
Perdahc ioğlu
,
E. S.
, and
Geijselaers
,
H. J.
,
2011
, “
Constitutive Modeling of Two Phase Materials Using the Mean Field Method for Homogenization
,”
Int. J. Mater. Form.
,
4
(
2
), pp.
93
102
.
30.
Digimat
,
2016
, “
User Manual, version 2017.0 ed
,” e-Xstream Engineering, Luxembourg.
31.
MacQueen
,
J. B.
,
1967
, “
Some Methods for Classification and Analysis of Multivariate Observations
,”
5th Berkeley Symposium on Mathematical Statistics and Probability
, Vol.
1
,
University of California Press
,
Berkeley, CA
, pp.
281
297
.
32.
MacKay
,
D.
,
2003
,
Information Theory, Inference, and Learning Algorithms
,
Cambridge University Press
,
Cambridge, UK
.
33.
Liu
,
K.
,
Tovar
,
A.
,
Nutwell
,
E.
, and
Detwiler
,
D.
,
2015
, “
Thin-Walled Compliant Mechanism Component Design Assisted by Machine Learning and Multiple Surrogates
,”
SAE
Paper No. 2015-01-1369
.
34.
Forrester
,
A. I. J.
,
Sóbester
,
A.
, and
Keane
,
A. J.
,
2008
,
Engineering Design Via Surrogate Models
,
Wiley
,
Chichester, UK
.
35.
Lophaven
,
S. N.
,
Nielsen
,
H. B.
, and
Sondergaard
,
J.
,
2002
,
“Dace”—A Matlab Kriging Toolbox. Tech. Rep., Informatics and Mathematical Modelling
,
Technical University of Denmark
,
Kongens Lyngby, Denmark
.
36.
Jones
,
D. R.
,
Schonlau
,
M.
, and
Welch
,
W. J.
,
1998
, “
Efficient Global Optimization of Expensive Black-Box Functions
,”
J. Global Optim.
,
13
(
4
), pp.
455
492
.
37.
Mori
,
T.
, and
Tanaka
,
K.
,
1973
, “
Average Stress in Matrix and Average Elastic Energy of Materials With Misfitting Inclusions
,”
Acta Metall.
,
21
(
5
), pp.
571
574
.
38.
Eshelby
,
J.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. London A: Math., Phys. Eng. Sci.
,
241
(
1226
), pp.
376
396
.
39.
Hassani
,
B.
, and
Hinton
,
E.
,
1998
,
Homogenization and Structural Topology Optimization: Theory, Practice and Software
,
Springer
,
New York
.
40.
Sigmund
,
O.
,
1994
, “
Materials With Prescribed Constitutive Parameters—An Inverse Homogenization Problem
,”
Int. J. Solids Struct.
,
31
(
17
), pp.
2313
2329
.
41.
Andreassen
,
E.
,
Clausen
,
A.
,
Schevenels
,
M.
,
Lazarov
,
B. S.
, and
Sigmund
,
O.
,
2011
, “
Efficient Topology Optimization in MATLAB Using 88 Lines of Code
,”
Struct. Multidiscip. Optim.
,
43
(
1
), pp.
1
16
.
42.
Liu
,
K.
, and
Tovar
,
A.
,
2014
, “
An Efficient 3D Topology Optimization Code Written in Matlab
,”
Struct. Multidiscip. Optim.
,
50
(
6
), pp.
1175
1196
.
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