Microstructures are stochastic by their nature. These aleatoric uncertainties can alter the expected material performance substantially and thus they must be considered when designing materials. One safe approach would be assuming the worst case scenario of uncertainties in design. However, design under the worst case conditions can lead to over-conservative solutions that provide less effective material properties. Here, a more powerful design approach can be developed by implementing reliability constraints into the optimization problem to achieve superior material properties while satisfying the prescribed design criteria. This is known as reliability-based design optimization (RBDO), and it has not been studied for microstructure design before. In this work, an analytical formulation that models the propagation of microstructural uncertainties to the material properties is utilized to compute the probability of failure. Next, the analytical uncertainty solution is integrated into the optimization problem to define the reliability constraints. The presented optimization under uncertainty scheme is exercised to maximize the yield stress of α-Titanium and magnetostriction of Galfenol, respectively.

References

References
1.
Xu
,
H.
,
Liu
,
R.
,
Choudhary
,
A.
, and
Chen
,
W.
,
2015
, “
A Machine Learning-Based Design Representation Method for Designing Heterogeneous Microstructures
,”
ASME J. Mech. Des.
,
137
(
5
), p.
051403
.
2.
Cang
,
R.
,
Xu
,
Y.
,
Chen
,
S.
,
Liu
,
Y.
,
Jiao
,
Y.
, and
Ren
,
M. Y.
,
2017
, “
Microstructure Representation and Reconstruction of Heterogeneous Materials Via Deep Belief Network for Computational Material Design
,”
ASME J. Mech. Des.
,
139
(
7
), p.
071404
.
3.
Huyse
,
L.
, and
Maes
,
M. A.
,
2001
, “
Random Field Modeling of Elastic Properties Using Homogenization
,”
J. Eng. Mech.
,
127
(
1
), pp.
27
36
.
4.
Sakata
,
S.
,
Ashida
,
F.
,
Kojima
,
T.
, and
Zako
,
M.
,
2008
, “
Three-Dimensional Stochastic Analysis Using a Perturbation-Based Homogenization Method for Elastic Properties of Composite Material Considering Microscopic Uncertainty
,”
Int. J. Solids Struct.
,
45
(
3–4
), pp.
894
907
.
5.
Creuziger
,
A.
,
Syed
,
K.
, and
Gnaupel-Herold
,
T.
,
2014
, “
Measurement of Uncertainty in Orientation Distribution Function Calculations
,”
Scr. Mater.
,
72–73
, pp.
55
58
.
6.
Juan
,
L.
,
Liu
,
G.
,
Wang
,
H.
, and
Ullah
,
A.
,
2011
, “
On the Sampling of Three-Dimensional Polycrystalline Microstructures for Distribution Determination
,”
J. Microsc.
,
44
(
2
), pp.
214
222
.
7.
Hiriyur
,
B.
,
Waisman
,
H.
, and
Deodatis
,
G.
,
2011
, “
Uncertainty Quantification in Homogenization of Heterogeneous Microstructures Modeled by XFEM
,”
Int. J. Numer. Methods Eng.
,
88
(
3
), pp.
257
278
.
8.
Kouchmeshky
,
B.
, and
Zabaras
,
N.
,
2009
, “
The Effect of Multiple Sources of Uncertainty on the Convex Hull of Material Properties of Polycrystals
,”
Comput. Mater. Sci.
,
47
(
2
), pp.
342
352
.
9.
Madrid
,
P. J.
,
Sulsky
,
D.
, and
Lebensohn
,
R. A.
,
2014
, “
Uncertainty Quantification in Prediction of the in-Plane Young's Modulus of Thin Films With Fiber Texture
,”
J. Microelectromech. Syst.
,
23
(
2
), pp.
380
390
.
10.
Niezgoda
,
S. R.
,
Yabansu
,
Y.
, and
Kalidindi
,
S. R.
,
2011
, “
Understanding and Visualizing Microstructure and Microstructure Variance as a Stochastic Process
,”
Acta Mater.
,
59
(
16
), pp.
6387
6400
.
11.
Sakata
,
S.
,
Ashida
,
F.
, and
Zako
,
M.
,
2008
, “
Kriging-Based Approximate Stochastic Homogenization Analysis for Composite Materials
,”
Comput. Methods Appl. Mech. Eng.
,
197
(
21–24
), pp.
1953
1964
.
12.
Chen
,
W.
,
Yin
,
X.
,
Lee
,
S.
, and
Liu
,
W. K.
,
2010
, “
A Multiscale Design Methodology for Hierarchical Systems With Random Field Uncertainty
,”
ASME J. Mech. Des.
,
132
(
4
), p.
041006
.
13.
Clement
,
A.
,
Soize
,
C.
, and
Yvonnet
,
J.
,
2012
, “
Computational Nonlinear Stochastic Homogenization Using a Nonconcurrent Multiscale Approach for Hyperelastic Heterogenous Microstructure Analysis
,”
Int. J. Numer. Methods Eng.
,
91
(
8
), pp.
799
824
.
14.
Clement
,
A.
,
Soize
,
C.
, and
Yvonnet
,
J.
,
2013
, “
Uncertainty Quantification in Computational Stochastic Multi-Scale Analysis of Nonlinear Elastic Materials
,”
Comput. Methods Appl. Mech. Eng.
,
254
, pp.
61
82
.
15.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2017
, “
Uncertainty Quantification of Microstructural Properties Due to Variability in Measured Pole Figures
,”
Acta Mater.
,
124
, pp.
100
108
.
16.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2017
, “
Uncertainty Quantification of Microstructural Properties Due to Experimental Variations
,”
AIAA J.
,
55
(
8
), pp.
2824
2832
.
17.
Acar
,
P.
, and
Sundararaghavan
,
V.
,
2017
, “
Uncertainty Quantification of Microstructural Properties Due to Experimental Variations
,”
AIAA
Paper No. AIAA 2017-0815
.
18.
Acar
,
P.
,
Srivastava
,
S.
, and
Sundararaghavan
,
V.
,
2017
, “
Stochastic Design Optimization of Microstructures With Utilization of a Linear Solver
,”
AIAA
Paper No. AIAA 2017-1939.
19.
Acar
,
P.
,
Srivastava
,
S.
, and
Sundararaghavan
,
V.
,
2017
, “
Stochastic Design Optimization of Microstructures With Utilization of a Linear Solver
,”
AIAA J.
,
55
(
9
), pp.
3161
3168
.
20.
Scarth
,
C.
,
Sartor
,
P. N.
,
Cooper
,
J. E.
,
Weaver
,
P. M.
, and
Silva
,
G. H. C.
,
2017
, “
Robust and Reliability-Based Aeroelastic Design of Composite Plate Wings
,”
AIAA J.
,
55
(
10
), pp.
3539
3552
.
21.
Luo
,
Y.
,
Kang
,
Z.
, and
Yue
,
Z.
,
2012
, “
Maximal Stiffness Design of Two-Material Structures by Topology Optimization With Nonprobabilistic Reliability
,”
AIAA J.
,
50
(
9
), pp.
1993
2003
.
22.
Jianqiao
,
C.
,
Yuanfu
,
T.
,
Rui
,
G.
,
Qunli
,
A.
, and
Xiwei
,
G.
,
2013
, “
Reliability Design Optimization of Composite Structures Based on PSO Together With FEA
,”
Chin. J. Aeronaut.
,
26
(
2
), pp.
343
349
.
23.
Lopez
,
R. H.
,
Lemosse
,
D.
,
de Cursi
,
J. E. S.
,
Rojas
,
J.
, and
El-Hami
,
A.
,
2011
, “
An Approach for the Reliability Based Design Optimization of Laminated Composites
,”
Eng. Optim.
,
43
(
10
), pp.
1079
1094
.
24.
Abumeri
,
G.
,
Munir
,
N.
, and
Rognin
,
F.
,
2009
, “
Reliability Based Design Optimization of Composite Joint Structures
,” AIAA Paper No. AIAA 2009-2240.
25.
Honarmandi
,
P.
,
Zu
,
J. W.
, and
Behdinan
,
K.
,
2007
, “
Reliability-Based Design Optimization of Cantilever Beams Under Fatigue Constraint
,”
AIAA J.
,
45
(
11
), pp.
2737
2746
.
26.
Hao
,
P.
,
Wang
,
B.
,
Li
,
G.
,
Meng
,
Z.
, and
Wang
,
L.
,
2015
, “
Hybrid Framework for Reliability-Based Design Optimization of Imperfect Stiffened Shells
,”
AIAA J.
,
53
(
10
), pp.
2878
2889
.
27.
Gomes
,
H. M.
,
Awruch
,
A. M.
, and
Lopes
,
P. A. M.
,
2011
, “
Reliability Based Optimization of Laminated Composite Structures Using Genetic Algorithms and Artificial Neural Networks
,”
Struct. Saf.
,
33
(
3
), pp.
186
195
.
28.
Hao
,
W.
,
Ying
,
Y.
, and
Yujia
,
L.
,
2008
, “
Reliability Based Optimization of Composite Laminates for Frequency Constraint
,”
Chin. J. Aeronaut.
,
21
(
4
), pp.
320
327
.
29.
Qu
,
X.
,
Haftka
,
R. T.
,
Venkataraman
,
S.
, and
Johnson
,
T. F.
,
2003
, “
Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments
,”
AIAA J.
,
41
(
10
), pp.
2029
2036
.
30.
Dehmous
,
H.
,
Karama
,
M.
, and
Welemane
,
H.
,
2014
, “
Contribution of a Micromechanics-Based Approach for Reliability Assessment
,” International Conference on Accelerated Life Testing and Degradation Models-ALT 2014
, Pau, France, June 11–13, pp.
20
27
.
31.
Du
,
J.
, and
Sun
,
C.
,
2017
, “
Reliability-Based Vibro-Acoustic Microstructural Topology Optimization
,”
Struct. Multidiscip. Optim.
,
55
(
4
), pp.
1195
1215
.
32.
Raulli
,
M.
, and
Maute
,
K.
,
2009
, “
Reliability Based Design Optimization of MEMS considering Pull-In
,”
ASME J. Mech. Des.
,
131
(
6
), p.
061014
.
33.
Kumar
,
A.
, and
Dawson
,
P. R.
,
2000
, “
Computational Modeling of F.C.C. Deformation Textures Over Rodrigues' Space
,”
Acta Mater.
,
48
(
10
), pp.
2719
2736
.
34.
Bunge
,
H. J.
,
1982
,
Texture Analysis in Materials Science
,
Butterworths
,
London, UK
.
35.
Wenk
,
H. R.
,
1985
,
Preferred Orientation in Deformed Metals and Rocks
,
Academic Press
,
London, UK
.
36.
Adams
,
B. L.
,
Henrie
,
A.
,
Henrie
,
B.
,
Lyon
,
M.
,
Kalidindi
,
S. R.
, and
Garmestani
,
H.
,
2001
, “
Microstructure-Sensitive Design of a Compliant Beam
,”
J. Mech. Phys. Solids
,
49
(
8
), pp.
1639
1663
.
37.
Kalidindi
,
S. R.
,
Houskamp
,
J.
,
Lyons
,
M.
, and
Adams
,
B. L.
,
2004
, “
Microstructure Sensitive Design of an Orthotropic Plate Subjected to Tensile Load
,”
Int. J. Plast.
,
20
(
8–9
), pp.
1561
1575
.https://pdfs.semanticscholar.org/6da7/dab4e34e997969b99444f1edb2811f3f7751.pdf
38.
Taylor
,
G. I.
,
1938
, “
Plastic Strain in Metals
,”
J. Inst. Met.
,
62
, pp.
307
324
.
39.
Liu
,
R.
,
Kumar
,
A.
,
Chen
,
Z.
,
Agrawal
,
A.
,
Sundararaghavan
,
V.
, and
Choudhary
,
A.
,
2015
, “
A Predictive Machine Learning Approach for Microstructure Optimization and Materials Design
,”
Nat. Sci. Rep.
,
5
(
1
), p.
11551
.
40.
Ross
,
S. M.
,
2010
,
Introduction to Probability Models
,
10th ed.
,
Elsevier
, New York.
41.
Kumar
,
A.
, and
Sundararaghavan
,
V.
,
2017
, “
Simulation of Magnetostrictive Properties of Galfenol Under Thermomechanical Deformation
,”
Finite Elem. Anal. Des.
,
127
, pp.
1
5
.
You do not currently have access to this content.