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Accepted Manuscript
The effective elastic properties analysis of periodic microstructure
with hybrid uncertain parameters
Wenqing Zhu , Ning Chen , Jian Liu , Siyuan Xia
PII:
DOI:
Reference:
S0020-7403(18)31036-1
https://doi.org/10.1016/j.ijmecsci.2018.08.018
MS 4476
To appear in:
International Journal of Mechanical Sciences
Received date:
Revised date:
Accepted date:
31 March 2018
16 August 2018
18 August 2018
Please cite this article as: Wenqing Zhu , Ning Chen , Jian Liu , Siyuan Xia , The effective elastic
properties analysis of periodic microstructure with hybrid uncertain parameters, International Journal
of Mechanical Sciences (2018), doi: https://doi.org/10.1016/j.ijmecsci.2018.08.018
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Highlights

Hybrid uncertain-but-bounded model is used to treat the uncertainty of
microscopic materials in homogenization analysis.

Homogenization-based hybrid uncertain analysis method (HHUAM) is
with bounded hybrid uncertainty.

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presented to estimate the effective elastic tensor of microscopic materials
The influence of microscopic bounded hybrid material uncertainties on the
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homogenized macroscopic elastic property is investigated.
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The effective elastic properties analysis of periodic
microstructure with hybrid uncertain parameters
Wenqing Zhu, Ning Chen*, Jian Liu, Siyuan Xia
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(State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body,
Hunan University, Changsha, Hunan, 410082, People’s Republic of China)
Abstract: This paper presents a homogenization-based hybrid uncertain analysis
material
properties
with
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method (HHUAM) for the prediction of the effective elastic tensor for microscopic
uncertain-but-bounded
parameters.
For
those
uncertain-but-bounded parameters related to the microscopic material properties, the
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ones with sufficient statistic information are modelled as bounded random variables,
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and those without enough statistics to build the probability density functions are
defined as interval variables. Based on the finite element framework for
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homogenization method, the effective elastic tensor with bounded hybrid uncertain
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parameters can be expanded by using Gegenbauer series expansion. The variation
ranges of the expectation and variance of the effective elastic tensor can be obtained
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due to the orthogonality relationship of Gegenbauer polynomials. Two numerical
cases are carried out to verify the effectiveness and the efficiency of the HHUAM. The
influence of the bounded hybrid uncertainties in microstructures on homogenized
macroscopic elastic properties of heterogeneous materials is also investigated.
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1. Introduction
Composite materials are extensively used in various fields due to its more tunable
and versatile material properties. The macroscopic performance of a composite
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material is generally determined by the complex microstructures [1,2]. By using
homogenization method, the homogenized parameters of a periodic composite material
can be obtained, and the heterogeneous media can be transformed into an energetically
equivalent material model [3-5]. So far, the most commonly used method for
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homogenization problems is the finite element method [6-9].
In the manufacturing process of composite materials, the physical parameters of
its constituent materials and the morphological parameters of its microstructure are
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inevitably affected by uncertain factors. Therefore, these uncertainties existed in the
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microstructure or the microscopic material properties must be taken into consideration
during the process of homogenization. For this reason, many researchers focused on
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the uncertain analysis of homogenization problems. Koishi et al. presented a new
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finite element approach based on the homogenization method for composite materials
with stochastic uncertainties [10]. Kaminski et al. reported a stochastic analysis
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method to solve the homogenization problems by using perturbation technique [11].
Sakata et al. discussed how the homogenized elastic property influenced by some
microscopic uncertainty through the Monte-Carlo method [12]. Besides, Sakata et al.
used a perturbation-based method to predict the equivalent elastic properties of
composite materials [13]. Ostoja-Starzewski investigated the microstructural
randomness in thermomechanics using homogenization method [14]. Xu and Brady
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used Fourier Galerkin method to solve the stochastic homogenization of random
media elliptic problems [15]. Ma et al. investigated the homogenization analysis of
heterogeneous materials based on random factor method [16]. Kaminski reported the
homogenization for polymers with rubber particles considering probabilistic
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uncertainty in Poisson ratio [17]. Recently, Chen et al. proposed an interval
homogenization-based method for solving the homogenization problems when the
probabilistic information of the microscopic uncertain parameter cannot be obtained
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unambiguously [18].
From the overall perspective, research of homogenization problems for
composite materials with microscopic uncertainties are focused on either random
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parameters or interval parameters. But in some cases, random parameters and interval
parameters may exist at the meantime. Under this circumstance, a hybrid uncertain
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model was presented by Elishakoff and Colombi [19]. Up to now, the hybrid uncertain
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model was extensively used in structural reliability analysis and structural-acoustic
response analysis [20-23]. Hybrid uncertain methods were generally on the basis of
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the perturbation method (PM) or Monte-Carlo method (MCM) [24, 25]. It is well
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known that the MCM is the most straightforward approach for uncertainty analysis
[26]. However, the excessive computational burden of MCM limits its applications in
engineering problems. The perturbation method has a much higher efficiency than the
MCM, but it can only be applied to uncertain problems with small uncertainty level
because of its inherent drawback. To overcome the defects of the PM and MCM, the
polynomial approximation method has been introduced to handle the hybrid uncertain
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problems. Lately, Wu et al. proposed a Polynomial-Chaos-Chebyshev-Interval (PCCI)
method for vehicle dynamics with hybrid uncertain parameters [27]. In practical
engineering practice, uncertain parameters are always bounded because of the
tolerance design. Naturally, a bounded hybrid uncertain model is emerged, in which
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the range of both the random variables and interval variables are strictly defined.
Relatively speaking, the bounded hybrid uncertain model has more practical
application value. Yin et al. proposed an orthogonal polynomial approximation
for
structural-acoustic
system
and
acoustic
field
with
hybrid
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method
uncertain-but-bounded parameters, in which a derivative λ-PDF model based on
Gegenbauer polynomial expansion is employed[28, 29]. Compared with the
traditional orthogonal polynomial approximation methods in probabilistic analysis,
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the advantage of the derivative λ-PDF model is that it can approximate arbitrary
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mono-valley and mono-peak PDF. Furthermore, the accuracy of Gegenbauer
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polynomial expansion method is better compared with the perturbation method in
interval analysis, and the efficiency of the Gegenbauer polynomial expansion method
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is greater than the MCM.
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From these backgrounds mentioned above, research on the homogenization
problems with bounded hybrid uncertain parameters is promising and interesting,
which is still unreported. Thus, a homogenization-based hybrid uncertain analysis
method (HHUAM) for estimating the effective elastic properties of periodic
microstructure with bounded hybrid uncertain parameters is presented in this paper.
Based on the homogenization method and the Gegenbauer series expansion method,
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the formulation of HHUAM is deduced. In HHUAM, the effective elasticity tensor
with bounded hybrid uncertain parameters is approximated through a N-order
Gegenbauer series expansion. The expectation and variance of the effective elasticity
tensor with bounded hybrid uncertain parameters can be calculated due to the
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orthogonality of the Gegenbauer polynomials. The influence of bounded hybrid
uncertainties in microstructures on homogenized macroscopic elastic properties of
heterogeneous materials is also investigated.
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This paper is organized as follows. The homogenization method is introduced in
Section 2. The theory of Gegenbauer series expansion method is introduced in Section
3. The HHUAM is proposed for the homogenization problems with bounded hybrid
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uncertain parameters in Section 4. Two numerical cases are investigated in order to
drawn in Section 6.
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verify the accuracy and the efficiency of HHUAM in Section 5. The conclusions are
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2. Homogenization method
The homogenization method can be used to calculate the effective properties of
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periodic composite materials from their unit cell topology [30-32]. In this section, the
homogenization method will be introduced briefly. Two levels of coordinate system in
the homogenization method includes the macro-level x and the micro-level y = x / ε,
where ε denotes the small positive scaling parameter. Here, the representative volume
element is defined as Ω, and a body subject to the body force fi satisfies the following
equation
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 ij
xj
where
 fi  0 in 
(1)
 ij represents the stress, xj denotes the macro generalized coordinate system.
ui
The prescribed displacement
boundary
F
on the boundary
u
and the traction Fi on the
are
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ui  0 on u
(2)
 ij n j  Fi on F
(3)
where nj denotes the normal vector.
expressed as
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The relationship between the stress-strain and the strain-displacement can be

1  ui u j
eij (u )     
2  x j xi




(4)
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
 ij  Dijkl
ekl (u )
(5)
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where eij represents the strain, Dijkl denotes the constitutive matrix.
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By the use of asymptotic expansion, the displacement
ui
can be approximated
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with respect to ε and expressed as follow
 0
0
1
(6)
1
is the macroscopic displacement, ui denotes the first order variation
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in which, ui
ui (x)  ui  (x, y)   ui  (x, y) 
of average displacement and is defined as follow [33].
where
 ik l
 
ui    ik l ekl ui
1
0
(7)
denotes the characteristic displacement function, and can be obtained by
the auxiliary equation as follow
 kmn


D
 y j ijkl yl d    y j Dijmn d 
(8)
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After solving the
 ik l
through Eq. (8), the effective elasticity tensor can be
calculated by
H
ijmn
D

 pkl
  Dijkl  Dijpq yq

1



d 

(9)
expressed as
Kχ  F
1

 D  I-bχ  d 

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DH =
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By the use of finite element method, Eqs. (8) and (9) can be rewritten and
(10)
(11)
in which, b denotes the strain matrix at the microscale, and the stiffness matrix K and
the force vector F can be represented as
K   bT Dbd 
(12)
F   bT Dd 
(13)
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
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
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3. Gegenbauer series expansion method (GSEM)
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3.1. Gegenbauer polynomials
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The Gegenbauer polynomials of n degree denoted by Gn (ξ) can be defined as
below [34]:
G0 ( )  1
 
G1 ( )  2




(n  1)Gn 1 ( )  2(n   ) Gn ( )  (n  2  1)Gn 1 ( ), n  2
(14)
where λ is a polynomial parameter and λ > 0.
The orthogonality relationship of Gegenbauer polynomials on ξ  [-1, 1] related to
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the weight function ρλ (ξ) can be expressed as

1
1
hi , i  j
  ( )Gi ( )G j ( )dx  
0,
i j
(15)
where
(  1)
(1/ 2)(  (1/ 2))
21 2  (  1)
hi 
i !(i   ) 2 ( )

in which,  () represents the Gamma function.
(16)
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  ( )  k (1   2 ) (1/2) , k 
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3.2. Gegenbauer series expansion (GSE) for the approximation of a function
On the basis of the orthogonal relationships of the Gegenbauer polynomials, a
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continuous function f (ξ) defined on ξ  [-1, 1] can be expressed as
N
f ( )   N ( )   fiGi ( )
(17)
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i 0
where fi represents the i th (i = 0, 1,…, N) expansion coefficient, N is the retained order
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of GSE.
N1
NL
i1  0
iL  0
f ( )    fi1 ,...,iL Gi1 1 (1 )  GiLL ( L ).
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As for the L-dimensional problem, f (ξ) can be expanded as
where Nl (l = 1, 2,…, L) denotes the retained order of GSE in relation to ξl,
(18)
fi1 ,...,iL
denotes the expansion coefficient.
3.3. Computation of the coefficients of the Gegenbauer series expansion
The coefficient of the GSE can be obtained by the use of the Gauss-Gegenbauer
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integration method and the weighted least squares method [35]. The weighted least
squares error E can be expressed as
1
E     ( )(rN )2 d
(19)
1
where rN is the residual error of the Gegenbauer series expansion, which can be
N
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expressed as
rN  f ( )   N ( )  f ( )   f j G j ( ) .
j 0
(20)
To minimize E, the necessary condition of the coefficient fi (i = 0, 1,…, N) is
i  0,1,..., N
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E
 0,
fi
(21)
Substituting Eqs. (19) and (20) into Eq. (21), we can obtain

N

j 0






1  ( ) f ( )Gi d  1   f jG j ( ) Gi ( )  ( )d ,
Gi ( )

simplified as
i  0,1,..., N .
(22)

is orthogonal to each G j ( ) for j = 0, 1,…, N, Eq. (22) can be
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Since
1
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1
1
hi
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fi 

1
1
  ( ) f ( )Gi d ,
i  0,1,..., N
(23)
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Through the Gauss-Gegenbauer integration formula [35], Eq(23) can be rewritten
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as follow
fi 
1
hi

1
1
  ( ) f ( )Gi d 



1  m
f ( j )Gi ( j ) Aj 
 
hi  j 1

(24)

In the above equation,  j ( j  1, 2,..., m) represents the interpolation point;
Aj ( j  1,2,..., m) denotes the weight which is defined by
Aj  222   ( ) 
2
2
(2  m)
(1  x 2j )1  Gm  ( x j )  ,
(1  m)
j  0,1,..., m
(25)
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where
Gm ( x j )  2Gm 11
(26)
Likewise, through the least-squares approximation and the Gauss-Gegenbauer
integration formula, the expansion coefficient
fi1 ,...,iL
can be calculated and
fi1 ,...,iL 
1
1
h1  hLL
1
 1
h1  hLL

1
1
1
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represented as
 i1 ,...,iL ( ) f ( )Gi1 ,...,iL ( )d1  d 2
1
m1
mL
j1 1
jL 1
  f (
(27)
j1
,...,  jL )Gi1 ,...,iL ( j1 ,...,  jL ) Aj1 ,..., jL
Aj1 ,..., jL
can be expressed as
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in which,  j1 ,...,  jL are the interpolation points,
Aj1 ,..., jL  Aj11  Aj22  AjLL .
(28)
Generally, when the number of the integration points related to ξi increases to Ni +
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1, the accuracy of the integration method can be acceptable. Therefore, the integration
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points are set as mi = Ni + 1 in this paper. Through Eq. (27), the total number of
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integration points is Ntotol = (N1 + 1) (NL + 1).
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4.HHUAM for homogenization problems with hybrid
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uncertain-but-bounded parameters
In this part, by introducing the GSEM to the FEM of homogenization, a hybrid
uncertain homogenization-based method (HHUAM) is proposed for homogenization
problems with hybrid uncertain-but-bounded parameters.
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4.1. Definition of bounded hybrid uncertain model.
In this section, the uncertainties in constituent material properties are treated as
bounded hybrid variables. On the one hand, when the PDFs of the bounded uncertain
parameters cannot be constructed because of the insufficient information, the bounded
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uncertain parameters are defined by an interval vector a = [a1, a2, …, ai]. The interval
variable ai satisfies
ai  aiI  ai , ai  , i  1, 2,..., L1
(29)
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where L1 denotes the total number of the interval variables.
On the other hand, when the PDFs of the bounded uncertain parameters can be
structured unambiguously, the bounded uncertain parameters are defined by a bounded
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random vector b = [b1, b2, …, bi]. The bounded random variable bi (i = 1, 2, …, L2)
satisfies
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
 Pb (bi )  0, bi  bi , bi 


 Pb (bi )  0, else
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(30)
where Pb (bi) denotes the PDF of bi.
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Thus, the hybrid uncertain vector x = [x1, x2, …, xL] can be expressed as
L1  L2  L
(31)
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x  [a, b]  a1 , a2 ,..., aL1 , b1 , b2 ,..., bL2  ,
where L denotes the total number of the hybrid uncertain-but-bounded parameters; L1
and L2 denote the interval and the bounded random variables, respectively.
4.2 Transformation of bounded hybrid uncertain variables
The key point of random polynomial expansion method is that the PDFs of
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random variables can be represented by the weight function of orthogonal polynomials.
The Gegenbauer polynomial can be applied to stand for a family of bounded PDFs,
termed as λ-PDF, because it is a parametric polynomial [34]. Thus, uncertain problems
with bounded random variables following λ-PDFs can be solved through the
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Gegenbauer series expansion.
An arbitrary bounded random variable β with λ-PDF is defined as
k (1  x 2 ) (1/2) x   1,1
P ( x)    ( x)   
else
0
(32)
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where ρλ () denotes the weight function as shown in Eq. (16).
Gegenbauer series expansion can be directly employed for random problems with
bounded random parameters following λ-PDFs. However, when the bounded random
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variables are following arbitrary probability distribution, we should transform the
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bounded random variables into a function of β [36]. By using the transformation
process, an arbitrary random variable b is represented as
   1 ,  2 ,...,  L 
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b  b(  ) ,
bi  bi ,k ( i )  bi ,0  bi ,1i  bi ,2 i 2    bi , B i B , i  1, 2,..., L2
(33)
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where βi is the i th bounded random variable with λ-PDF; B denotes the polynomial
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order; bi,k (k = 0, 1,…, B) is the transformation coefficient of the function bi,k (βi), which
can be obtained by
To find  , bi ,k (k  0,1,..., B)
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Min
  Pori ( xi )  P ( xi )
i 1

b
2
(34)
In the abovementioned equation, xi are discrete points in the domain of b, Pori (xi)
is the original PDF of b, Pb ( xi ) denotes the PDF of the polynomial function bi,k (βi) .
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Eq. (34) can be solved by numerous optimization methods. When the parameter λ and
the transformation coefficients bi,k are chosen properly, the PDF of the function bi,k (βi)
can approximate the mono-valley and mono-peak PDFs in any interval [36].
Similarly, the interval variable bi in the arbitrary interval vector a can be
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transformed as a function of interval variable i  [-1, 1]. The arbitrary interval vector a
is represented as:
  1 ,2 ,...,i 
a  a( ),
where
ai ,0 
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ai  ai (i )  ai ,0  ai ,1i , i  1, 2,..., L1
ai  ai
2
,
ai ,1 
ai  ai
2
(35)
(36)
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As with GSEM for the random problem, the polynomial parameter λ in GSEM for
interval analysis is also determined by Eq. (34). Ref. [28] shows that the GSE can
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achieve high precision for interval analysis with a very small value of λ. Therefore, the
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parameter λ of the Gegenbauer series related to interval variables is taken as 0.001 and
named as λ0 in this paper.
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Thus, the hybrid uncertain vector x = [x1, x2 ,…, xL] can be transformed as
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x  [a( ), b(  )]  a1 (1 ), a2 (2 ),..., aL1 (L1 ), b1,k ( 1 ), b2,k ( 2 ),..., bL2 ,k (  L )  , L1  L2  L

2 
(37)
4.3 HHUAM for hybrid uncertain analysis of homogenization problem
Based on the theory of GSEM and the transformation formula of bounded hybrid
variables in Eq. (37), the effective elasticity matrix related to bounded hybrid uncertain
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variable x can be expressed as
DH  DH  x   DH  a( ), b(  ) 
N1
N L1 L2
i1  0
iL1 L2  0
=  


fi1 ,...,iL L Gi1 ,...,iL ( )GiL 1 ,...,iL L (  ),
0
1
2
1
1
1
k  1, 2,..., Ntotal
2
(38)
where the constant coefficients of the expansion fi1 ,...,iL L can be obtained through Eq.
1
2
fi1 ,...,iL L 
1
2
m1
1
 
L1 L2
L1  L2 j1 1
h11  h
mL1 L2

jL1 L2 1

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(27) and expressed as

0

0

DH a(ˆ ), b( ˆ ) Gi1 ,...,iL ( )GiL 1 ,...,iL L ( ) A A
1
1
1
2
(39)


denotes the value of the
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In the abovementioned equation, DH a(ˆ ), b( ˆ )
effective elasticity matrix at the interpolation point. L1 and L2 are the number of interval
variables and bounded random variables, respectively.
ˆ
and
A
ˆ
and
represent the
A

represent
M
integration points and weights with respect to interval variable.
0
ED
the integration points and weights related to the bounded random variable.
d kH of the DH is written as follow
The k th element
PT
d kH  d kH  x   d kH  a( ), b(  ) 
N1
N L1 L2
i1  0
iL1 L2  0
CE
= 


fi1k,...,iL L Gi1 ,...,iL ( )GiL 1 ,...,iL L (  ),
0
1
2
1
1
1
k  1, 2,..., Ntotal
2
(40)
AC
where
Gi1 ,...,iL ( )
0
1


iL11 ,...,iL1 L2
G
0
i0
0
 Gi1 i1 Gi2i2  GiL L1
1
i
i
i
1
1
1
(  )  GiL 1 GiL 2  GiL L
L11
L1 2
(41)
L1 L2
2
It needs two steps to obtain the bounds of the expectation and variance of the
effective elasticity matrix related to bounded hybrid uncertain variables [28]. First of all,
the interval parameters are treated as constant parameters, then Eq. (40) can be
rewritten as
ACCEPTED MANUSCRIPT
 N1 N L1 k
 
0
d         fi1 ,...,iL L Gi1 ,...,iL ( ) GiL 1 ,...,iL L (  )
1 2
1

 1 1 2
iL11  0
iL1 L2  0  i1  0
iL1 =0

N L11
N L1 L2
H
k
N L11
=


iL11  0
N L1 L2


iL1 L2  0
cikL 1 ,...,iL L GiL 1 ,...,iL L (  )
1
1
2
1
1
(42)
2
where
c
Thus, the expectation of
H
k
N L1
i1  0
iL1 =0
    fi1k,...,iL L Gi1 ,...,iL ( )
d kH
0
1
2
1
can be expressed as
 N L11 N L1L2 k


 E     ciL 1 ,...,iL L GiL 1 ,...,iL L (  ) 
1
1 2
1
1 2
iL11 0 iL1L2 0

(43)
AN
US
d
N1
CR
IP
T
k
iL11 ,...,iL1 L2
N
N L1 L2


  L11



       cikL 1 ,...,iL L GiL 1 ,...,iL L (  ) P (  L1 1 )  P (  L1  L2 ) d  L1 1  d  L1  L2
1
1 2
1
1 2
L11
L1 L2

 

 iL11 0 iL1L2 0

M
(44)
Since the orthogonality of the Gegenbauer polynomial basis, the expectation of
PT
ED
d kH can be calculated analytically [36] and expressed as follow
For calculating the variance of
k
d  c0,...,0
(45)
H
k
d kH , the expectation of the mean square of d kH
AC
CE
need to be calculated and can be expressed as [36]
2
 N L11 N L1L2
 

k

E (d )   E     ciL 1 ,...,iL L GiL 1 ,...,iL L (  )  
 
 iL11 0 iL1L2 0 1 1 2 1 1 2
 

H 2
k

N L11
N L1 L2
  
iL11  0
Accordingly, the variance of
iL1 L2  0
d kH
c
k
iL11 ,...,iL1 L2
2
h
L11
iL11

 hiLL1LL2 .
1
2
can be calculated and expressed as
(46)
ACCEPTED MANUSCRIPT
 d2  E (d kH )2   ( d ) 2
H
k
H
k
N L11
N L1 L2
  

iL11  0
iL1 L2  0
2
c
h
k
iL11 ,...,iL1 L2
L11
iL11
 hiL L   c
L1 L2
1
(47)

2
k
0,...,0
2
To substitute Eq. (43) into Eq. (44) and Eq. (47), we can obtain
N1
N L1
i1  0
iL1 =0
d H  d H ( )     fi1k,...,iL1 ,0,...,0Gi1 ,...,iL1 ( )

2
d kH
k
(48)
CR
IP
T
k
0
2
 N1 N L1 k
 
0

        fi1 ,...,iL L Gi1 ,...,iL ( )  hiLL111  hiLL1LL2
1
2
1
1
1 2


iL11  0
iL1 L2  0  i1  0
iL1 =0

N L11
N L1 L2
2
(49)
AN
US
 N1 N L1

0
     fi1k,...,iL ,0,...,0Gi1 ,...,iL ( ) 
1
1
 i 0 i =0

L1
1

Secondly, the bounds of the expectation and variance of
d kH
can be obtained
through Monte-Carlo simulation and expressed as
I
H
k
I
ED
2
d kH






 ,

  minI d H ( η) , maxI d H ( η)
 ηη
k
k
ηη
  minI  d2H ( η) , maxI  d2H ( η)
 ηη
k
k
ηη
M
 
 
d
(50)
PT
Various methods can be used to calculate the extremum in Eq. (50). Interval
arithmetic is efficient to obtain the bounds of interval functions. Nonetheless, the
CE
wrapping effect of interval arithmetic may lead overestimation [37]. Up to now, the
AC
Monte-Carlo simulation (MCS) is regarded as the most robust method for the interval
problems [38]. Therefore, in this paper, the MCS is adopted to calculate the extreme
value of the functions in Eq. (50). It can be noted that the interval functions in Eq. (50)
are simple functions. Thus, the computational cost of the MCS for the interval
functions in Eq. (50) is acceptable.
ACCEPTED MANUSCRIPT
4.4 The procedure of HHUAM for homogenization problem with hybrid
uncertain-but-bounded parameters
Step 1. Calculating the polynomials parameter λ and the transformation coefficient
related to each bounded uncertain parameter. For the bounded random variable
λ0 = 0.001 [28], ai,k is determined by Eq. (36).
CR
IP
T
bi, λ and bi,k can be obtained by Eq. (34). For the interval variable ai, λ is set as
integration through Eq. (25).
AN
US
Step 2. Calculating the weights and the interpolation points of Gauss-Gegenbauer
Step 3. Calculating the effective elasticity matrix at the interpolation points based on Eq.
(11).
M
Step 4. Calculating the expansion coefficients of the GSE by Eq. (39).
ED
Step 5. Calculating the bounds of the expectation and variance of the effective elasticity
matrix with bounded hybrid uncertain parameters according to Eq. (50).
PT
It is worth mentioning that the accuracy of GSE can be improved via employing
CE
higher retained order [36]. Thus, the proposed HHUAM can be conveniently applied to
the bounded hybrid uncertain problem with large uncertain level. Considering the
AC
computational accuracy and the computational cost, the relative improvement criterion
is used to estimate the retained order of HHUAM [28].
5. Numerical examples
In this section, two cases are used to investigate the accuracy and efficiency of the
ACCEPTED MANUSCRIPT
HHUAM. On account of the convergence of the Monte-Carlo simulation (MCS) [39],
the results calculated by the MCS with enough sampling points for homogenization
with bounded hybrid uncertain parameters are used as reference solution. In MCS, the
interval variables are given random sampling values within the intervals, and the
CR
IP
T
expectation and variance can be obtained by sampling the bounded random variables.
With time repeating, we can obtain the bounds of the expectation and variance.
Simulations in numerical cases are carried out by using MATLAB R2015b on a
5.1. Rectangular microscale voids
AN
US
3.30GHz Intel(R) Xeon(R) CPU E3-1230 V2.
M
In this case, a unit square cell contains a 0.4×0.6 rectangular hole as shown in
Fig. 1 is used to demonstrate the HHUAM. The finite element model consists of 304
ED
elements and 364 nodes. The Young’s modulus E and the Poisson’s ratio ν of the solid
PT
phase material are 4.5 GPa and 0.39, respectively. Considering the microscopic
uncertainty, the Young’s modulus E and the Poisson’s ratio ν are treated as bounded
CE
uncertain parameters. The uncertain level of E and ν are defined as E and ν,
AC
respectively.
ACCEPTED MANUSCRIPT
0.6
CR
IP
T
0.4
Fig. 1 Unit cell with rectangular hole
For simplicity, the PDF of the bounded random Young’s modulus E is assumed to
AN
US
be a linear function of E [36], Em represents the mean value, ∆E represents the
deviation amplitude, E is a random variable which is defined on [-1, 1] and follows
λE-PDF. The polynomial parameter λ is assumed to be 3. The information of
M
uncertain-but-bounded parameters is listed in Table 1.
ED
Table 1
The information of uncertain-but-bounded parameters.
Uncertain parameters
PT
E (GPa)
PDF
EI = 4.5 [1- E , 1+ E ]
E=Em + E ∆E, λ=3
νI = 0.39 [1- ν , 1+ ν ]
Unknown
CE
ν
Range
AC
In this numerical case, the uncertain level of input is assumed as E =ν =0. By
using the relative improvement criterion shown in the appendix, the retained order
vector N
req
estimated
 [ N Ereq , Nreq ] of HHUAM with respect to the uncertain parameter can be
[28].
When
Nreq  [ N Ereq , Nreq ]
0 =0.2, respectively.
setting
the
tolerance
to
10-2,
the
estimation
of
are [2, 2], [2, 3] and [2, 5] at uncertain level 0 =0.05, 0 =0.1 and
ACCEPTED MANUSCRIPT
The effective elastic tensor of the unit cell is calculated by the MCS and the
HHUAM. In the MCS, when the sample points are up to 5106, the results calculated
by MCS almost no longer change. It worth mentioning that the interval and random
sample points are respectively set as 1000 and 5000. Therefore, the total number of
CR
IP
T
sample points is 5106. The bounds of the expectation and variance of the effective
elastic tensor are listed in Tables 2a, 2b and 2c when 0 = 0.05, 0 = 0.1 and 0 = 0.2,
respectively.
AN
US
Tables 2a-2c show that results obtained by HHUAM are very close to the
referenced results obtained by MCS. It can be found out that the errors of the
expectation of the effective tensor are much less than that of the variance of the
M
effective tensor. The reason for this phenomenon is that the variance is related to the
square of the expectation. On the basis of the error propagation rule, the relative error of
ED
variance is larger than that of expectation. Besides, note that the statistics information
PT
of the effective elastic tensor is not deterministic. It is owing to the PDF information of
the interval parameter is missing. Therefore, the statistics information of the effective
CE
elastic tensor is an interval. Furthermore, it can be found that the error may be larger
AC
than the tolerance tol =10-2. Nevertheless, the largest error is 1.92E-02, which is
acceptable. It is indicated that the accuracy of HHUAM for predicting the effective
elastic tensor is good.
ACCEPTED MANUSCRIPT
Table 2a
Bounds of the expectation and variance of the effective elastic tensor (0 = 0.05).
Bounds
H
D12
H
D22
H
D66
Variance
MCS
HHUAM
Error
MCS
HHUAM
Error
Lower
2.7315
2.7316
3.66E-05
2.30E-03
2.33E-03
1.30E-02
Upper
2.9945
2.9940
1.67E-04
2.76E-03
2.80E-03
1.45E-02
Lower
1.1517
1.1519
1.74E-04
4.08E-04
4.15E-04
1.72E-02
Upper
1.4571
1.4567
2.75E-04
6.54E-04
6.63E-04
1.38E-02
Lower
3.6968
3.6968
1.62E-05
4.21E-03
4.27E-03
1.43E-02
Upper
4.0461
4.0454
1.73E-04
5.04E-03
5.11E-03
1.39E-02
Lower
0.5211
0.5211
5.76E-05
8.36E-05
8.49E-05
1.56E-02
Upper
0.5334
0.5334
9.37E-05
8.76E-05
8.89E-05
1.48E-02
CR
IP
T
H
D11
Expectation
Bounds
H
D22
H
MCS
HHUAM
Error
MCS
HHUAM
Error
Lower
2.6307
2.6301
2.28E-04
8.52E-03
8.65E-03
1.53E-02
Upper
3.1705
3.1698
2.21E-04
1.24E-02
1.26E-04
1.61E-02
Lower
1.0318
1.0315
2.91E-04
1.31E-03
1.33E-03
1.53E-02
Upper
1.6574
1.6568
3.62E-04
3.38E-03
3.43E-03
1.48E-02
Lower
3.5621
3.5614
1.97E-04
1.56E-02
1.59E-02
1.92E-02
Upper
4.2780
4.2770
2.34E-04
2.25E-02
2.29E-02
1.78E-02
Lower
0.5164
0.5163
1.94E-04
3.28E-04
3.33E-04
1.52E-02
Upper
0.5417
0.5416
1.85E-04
3.61E-04
3.67E-04
1.66E-02
PT
D66
Variance
M
D12H
Expectation
ED
H
D11
AN
US
Table 2b
Bounds of the expectation and variance of the effective elastic tensor (0 = 0.1).
CE
Table 2c
Bounds of the expectation and variance of the effective elastic tensor (0 = 0.2).
Bounds
H
D12
H
D22
H
D66
Variance
MCS
HHUAM
Error
MCS
HHUAM
Error
Lower
2.4705
2.4707
8.10E-05
3.01E-02
3.05E-02
1.33E-02
Upper
3.6954
3.6903
1.38E-03
6.72E-02
6.81E-02
1.34E-02
Lower
0.8362
0.8369
8.37E-04
3.44E-03
3.50E-03
1.74E-02
Upper
2.2387
2.2342
2.01E-03
2.47E-02
2.50E-02
1.21E-02
Lower
3.3474
3.3475
2.99E-05
5.52E-02
5.60E-02
1.45E-02
Upper
4.9549
4.9492
1.15E-03
1.21E-01
1.22E-01
8.26E-03
Lower
0.5089
0.5089
7.86E-05
1.28E-03
1.29E-03
7.81E-03
Upper
0.5697
0.5691
1.05E-03
1.60E-03
1.62E-03
1.25E-02
AC
H
D11
Expectation
ACCEPTED MANUSCRIPT
It can be seen from Fig. 2 that the uncertainty levels of the interval of the
expectation and the interval of variance of the effective elastic tensor are influenced
by different uncertain levels of input. When 0 = 0.15, the retained order vector
-2
Nreq  [ N Ereq , Nreq ] is Nreq
4  [2,4] with the prescribed tolerance tol = 10 . Fig. 2
CR
IP
T
shows that with the uncertain level of input increasing, the uncertainty levels of the
interval of the expectation and the interval of variance are monotonously increasing.
H
Besides, note that the D12 is the most influenced by the hybrid uncertain-but-bounded
H
H
H
AN
US
parameters in microscopic material properties, followed by D11 and D22 , and D66
is the least influenced. In the meantime, it can be found that the uncertainty levels of the
H
H
H
interval of the expectation and the interval of variance of the D11 , D12 and D22 are
H
M
much larger than the uncertain levels of input, whereas those of D66 are smaller than
the uncertain levels of input. This phenomenon is usually caused by the combined
ED
influences of Young’s modulus and Poison’s ratio. As an example, the D11 and D12
PT
terms in stiffness matrix for plain strain state are E(1-ν) / (1+ν)(1-2ν) and Eν /
(1+ν)(1-2ν), respectively. Comparing D11 with D12, the influence decreases for D11
CE
because of the negative correlation between E and -Eν in some degree. In detail, when
AC
the uncertain level of input increases to 20%, the uncertainty levels of the interval of the
H
expectation and the interval of variance of D12 are more than 45% and 75%,
respectively, whereas the uncertainty levels of the interval of the expectation and the
H
interval of variance of D66 are less than 6% and 12%, respectively. The uncertainty
H
H
levels of the expectation and the interval of variance of D11 and D22 are almost the
same as those of input.
ACCEPTED MANUSCRIPT
CR
IP
T
a
PT
ED
M
AN
US
b
AC
CE
Fig. 2 The uncertainty levels of the interval of expectation (a) and the interval of variance (b)
result from different uncertain levels of input
Except the accuracy, another important factor in the evaluation of a numerical
method is the computational efficiency, which is crucial for its application in practical
engineering. Hence, the computational time of MCS and HHUAM for predicting the
effective elastic tensor with respect to three uncertain levels of input is investigated
and listed in Table 3. We can observe form Table 3 that the execution time of MCS at
ACCEPTED MANUSCRIPT
different uncertain level of input are very close to each other. This is because the
sample points of MCS with respect to three uncertain levels of input are the same.
Besides, Table 3 shows that more execution time is spent for predicting the effective
elastic tensor through HHUAM with the uncertain level of input increasing. The
CR
IP
T
reason of this is that higher retained order of HUHAM is needed for reducing the
estimation error for homogenization problem with larger uncertain level. Nevertheless,
it can be observed from Table 3 that the computational cost of HHUAM for predicting
AN
US
the effective elastic tensor is much less than that of MCS under these three uncertain
levels of input, even if with a higher retained order. It can be indicated that the
computation efficiency of HHUAM for predicting the effective elastic tensor is
M
desirable.
Uncertainty of inputs
0 =0.1
Time of HHUAM (s)
Order of HHUAM
1.47410
6
6.727
[2, 2]
1.47410
6
7.899
[2, 3]
10.333
[2, 5]
1.474106
CE
0 =0.2
Time of MCS (s)
PT
0 =0.05
ED
Table 3
Execution time of MCS and HHUAM for predicting the effective elastic tensor.
AC
5.2 Particle filled polymers
In this case, a unit Representative Volume Element (RVE) of a unidirectional
fiber reinforced composite is investigated, as shown in Fig. 3. The similar finite
element mesh model has been studied in Refs. [40-43]. The radius of the fiber in the
center of the matrix is 0.2. The finite element model consists of 361 elements and 340
ACCEPTED MANUSCRIPT
AN
US
CR
IP
T
nodes.
Fig. 3 RVE of a unidirectional fiber reinforced composite
M
The Young’s modulus E1 and the Poisson’s ratio ν1 of the matrix are 4.5 GPa and
ED
0.39, respectively. The Young’s modulus E2 and the Poisson’s ratio ν2 of the fiber are 73
GPa and 0.21, respectively. Considering the microscopic uncertainty, the Young’s
PT
modulus E1, E2 and the Poisson’s ratio ν1, ν2 are treated as bounded uncertain
parameters. The uncertainty level of E1, E2 and ν1, ν2 are defined as
 E ,  E and  ,
1
2
1
2
CE
 , respectively. The information of these uncertain-but-bounded parameters is listed
AC
in Table 4.
For brevity but without loss of generality, the PDF of the bounded random
Young’s modulus E is assumed to be a linear function of E [36]. E1 and E2
m
m
denotes
I
I
the mean value of E1 and E2 , respectively. E1 and E2 denotes the deviation
I
I
amplitude of the E1 and E2 , respectively. 
E1
and 
E2
are random variables
which are defined on [-1, 1] and follows  E  PDF and  E  PDF , respectively.
1
2
ACCEPTED MANUSCRIPT
The values of polynomial parameter  E1 and  E2 are assumed to be 1 and 2,
respectively. The information of uncertain-but-bounded parameters is listed in Table
4.
Table 4
The information of the uncertain-but-bounded parameters.
Range
PDF
E1 (GPa)
E1  4.5[1   E , 1   E ]
I
1
1
1
 1  0.39 [1   , 1   ]
E2 (GPa)
E2  73[1   E , 1   E ]
I
1
1
I
2
2
 2  0.21[1   , 1   ]
I
2
2
2
E1  E1   1 E1 ,  1
m
E
E1
CR
IP
T
Uncertain parameters
Unknown
E2
 E2m   E E2 , E  2
2
2
Unknown
AN
US
The uncertain level of input is assumed as  E1   E2  1   2   0 , The
retained order of HHUAM can be estimated by the relative improvement criterion [28].
req
req
req
req
req
When setting the tolerance to 10-2, the estimation of N  [ N E1 , N1 , N E2 , N 2 ] are
M
[2, 2, 2, 2], [2, 2, 2, 3] and [3, 2, 2, 6] at uncertain level 0 = 0.05, 0 = 0.1 and 0 = 0.2,
respectively.
ED
In this case, the sample points of the MCS are 8106, the interval and random
PT
sample points are 1000 and 8000, respectively. The results obtained by MCS and
CE
HHUAM are listed in Table 5a-5c.
Tables 5a-5c show that the bounds of expectation and variance of the effective
AC
elastic tensor obtained by HHUAM match the referenced results obtained by MCS
perfectly. Note that even if the largest error of the bound of the effective elastic tensor
is less than 2%, which can certainly be accepted. It can be observed that when 0
increases from 0.05 to 0.1, the variations of the lower and upper bounds of the
expectation are small, whereas when 0 increases from 0.1 to 0.2, the variations of the
lower and upper bounds of the expectation are much larger and become
ACCEPTED MANUSCRIPT
disproportionate. This indicates that the bounds of the expectation are more sensitive to
large uncertainty. Besides, it can be noted that when 0 increases from 0.05 to 0.2, the
variations of the bounds of the variance are larger than those of the expectation
throughout. This phenomenon may be caused by that the variance is related to the
CR
IP
T
square of the expectation.
Fig. 4 shows the uncertainty levels of the interval of the expectation and the
interval of variance influenced by different uncertain levels of input. By using the
AN
US
req
req
req
req
req
relative improvement criterion, here the estimation of N  [ N E1 , N1 , N E2 , N 2 ]
is N4  [3, 2, 2,5] when the uncertain level of input 0 = 0.15 and the prescribed
req
tolerance is tol = 10-2. From Fig. 4, it can be observed that the uncertainty levels of the
M
interval of the expectation and the interval of variance increase monotonously with the
uncertain level of input increasing. It can be noted that the uncertainty levels of the
H
H
H
ED
interval of the expectation and the interval of variance of the D11 , D12 and D22
PT
obtained by MCS and HUHAM are much larger than the uncertain levels of input,
H
while those of D66 are smaller than the uncertain levels of input. In detail, when the
CE
uncertain level of input increases to 20%, the uncertainty levels of the interval of the
H
AC
expectation and the interval of variance of D12 are more than 70% and 90%,
respectively, whereas the uncertainty levels of the interval of the expectation and the
H
interval of variance of D66 are less than 5% and 10%, respectively. The uncertainty
H
H
levels of the interval of the expectation and the interval of variance of D11 and D22
are almost the same as those of input. Because of the combined influences of Young’s
H
modulus and Poison’s ratio, it can also be concluded that the D12 is the most
ACCEPTED MANUSCRIPT
influenced by the hybrid uncertain-but-bounded parameters in microscopic material
H
H
H
properties, followed by D11 and D22 , and D66 is the least influenced.
Table 5a
Bounds of the expectation and variance of the effective elastic tensor (0 = 0.05).
Bounds
H
D22
H
D66
MCS
HHUAM
Error
MCS
HHUAM
Error
Lower
8.8823
8.8915
1.04E-03
0.04740
0.04700
8.44E-03
Upper
11.2192
11.2119
6.51E-04
0.07510
0.07422
1.17E-02
Lower
4.8010
4.8091
1.69E-03
0.01387
0.01377
7.21E-03
Upper
7.1754
7.1672
1.14E-03
0.03075
0.03036
1.27E-02
Lower
8.8809
8.8901
1.04E-03
0.04739
0.04698
8.65E-03
Upper
11.2175
11.2102
6.51E-04
0.07508
0.07420
1.17E-02
Lower
1.8873
1.8877
2.12E-04
0.00218
0.00216
9.17E-03
Upper
1.9229
1.9233
2.08E-04
0.00227
0.00224
1.32E-02
CR
IP
T
H
D12
Variance
AN
US
H
D11
Expectation
Table 5b
Bounds of the expectation and variance of the effective elastic tensor (0 = 0.1).
Bounds
H
D66
MCS
HHUAM
Error
Lower
8.1605
8.1553
6.37E-04
0.1604
0.1585
1.18E-02
Upper
13.2819
13.2695
9.34E-04
0.4184
0.4132
1.24E-02
Lower
4.0530
4.0459
1.75E-03
0.0396
0.0391
1.26E-02
Upper
9.2449
9.2309
5.72E-02
0.2027
0.1999
1.38E-02
Lower
8.1592
8.1539
6.50E-04
0.1604
0.1585
1.18E-02
Upper
Lower
Upper
M
Error
ED
H
D22
HHUAM
PT
H
D12
Variance
MCS
13.2799
13.2676
9.26E-04
0.4183
0.4131
1.24E-02
1.8729
1.8737
4.27E-04
0.0086
0.0085
1.16E-02
1.9428
1.9437
4.63E-04
0.0093
0.0092
1.08E-02
CE
H
D11
Expectation
AC
Table 5c
Bounds of the expectation and variance of the effective elastic tensor (0 = 0.2).
Bounds
H
D11
H
D12
H
D22
Expectation
Variance
MCS
HHUAM
Error
MCS
HHUAM
Error
Lower
7.1687
7.1863
2.46E-03
0.4969
0.4932
7.45E-03
Upper
23.8403
23.8030
1.56E-03
5.2016
5.1276
1.42E-02
Lower
2.9967
3.0102
4.50E-03
0.0869
0.0865
4.60E-03
Upper
19.7493
19.7086
2.06E-03
3.5551
3.5014
1.51E-02
Lower
7.1675
7.1851
2.46E-03
0.4968
0.4931
7.45E-03
Upper
23.8377
23.8003
1.57E-03
5.2006
5.1266
1.42E-02
ACCEPTED MANUSCRIPT
H
D66
Lower
1.8638
1.8627
5.90E-04
0.0337
0.0333
1.19E-02
Upper
1.9867
1.9886
9.56E-04
0.0388
0.0385
7.73E-03
AN
US
CR
IP
T
a
AC
CE
PT
ED
M
b
Fig. 4 The uncertainty levels of the interval of expectation (a) and the interval of variance (b)
result from different uncertain levels of input
It is well known that the uncertain level of the material properties may be
different in practical engineering problems. Thus, a simple case is used to demonstrate
the HHUAM for bounded hybrid uncertain homogenization problems with
ACCEPTED MANUSCRIPT
non-uniform uncertain level of input. In this case, the uncertain level of input is
assumed as  E  0.05,   0.2,  E  0.15,   0.1 . The Young’s modulus E1 and E2 are
1
1
2
2
1
1
2
2
assumed to be the non-linear function of E ( E  E m  E   E E  E (  E ) 2 ), in which
Em represents the mean value of the interval of the Young’s modules, ∆E represents the
CR
IP
T
deviation amplitude, E is a random variable defined on [-1, 1] with λE-PDF. The
assumed information is listed in Table 6. When setting the tolerance to 10-2, the
req
req
req
req
req
estimation of N  [ N E1 , N1 , N E2 , N 2 ] is N5  [2,2,2,5] in the HHUAM. The
req
AN
US
MCS is also used to calculate the referenced results. The samples are 8106, the interval
and random sample points are 1000 and 8000, respectively. The bounds of the
expectation and variance of the effective elastic tensor of the RVE obtained by both
M
HHUAM and MCS is listed in Table 7. Table 7 shows that the bounds of expectation
ED
and variance of the effective elastic tensor obtained by HHUAM match the referenced
results obtained by MCS perfectly. Furthermore, the execution time of HHUAM and
PT
MCS for predicting the effective elastic tensor of the RVE in this case are 83 s and
CE
2.6106 s, respectively. From these analysis results, it can be concluded that both the
accuracy and efficiency of HUHAM are desirable for uncertain homogenization
AC
problems with uncertain-but-bounded parameters.
Table 6
The information of the uncertain-but-bounded parameters.
Uncertain parameters
Range
PDF
E1 (GPa)
[4.275,4.725]
E1 = 4.3875+0.225 +0.1125(
1
[0.312,0.468]
Unknown
E2 (GPa)
[62.05,83.95]
E2 = 67.525+10.95 +0.1125(
2
[0.189,0.231]
Unknown
E1
E2
E1
)2 , E  6
1
E2
)2 , E  4
2
ACCEPTED MANUSCRIPT
Table 7
Bounds of the expectation and variance of the effective elastic tensor of the RVE
( E1  0.15, 1  0.2,  E2  0.05,  2  0.1).
Bounds
H
D12
H
D22
H
MCS
HHUAM
Error
MCS
HHUAM
Error
Lower
7.0080
7.0186
1.51E-03
0.0092
0.0091
1.09E-02
Upper
23.1949
23.1373
2.48E-03
0.0978
0.0961
1.74E-02
Lower
2.934
2.9445
3.58E-08
0.0016
0.0016
9.38E-03
Upper
19.1979
19.1409
2.97E-03
0.0669
0.0657
1.79E-02
Lower
7.0068
7.0174
1.51E-03
0.0092
0.0091
1.09E-02
Upper
23.1923
23.1348
2.48E-03
0.0978
0.0961
1.74E-02
Lower
1.8215
1.8189
1.43E-03
6.22E-04
6.14E-04
1.30E-02
Upper
1.9409
1.9410
5.15E-05
7.14E-02
7.06E-04
1.12E-02
AN
US
D66
Variance
CR
IP
T
H
D11
Expectation
It is worth mentioning that interval homogenization-based method (IHM) and
M
subinterval homogenization-based method (Sub-IHM) are only developed for
estimating the effective elastic tensor of periodic microstructure with interval
ED
uncertainty [19], while the proposed homogenization-based hybrid uncertain analysis
PT
method (HHUAM) can be used for predicting the effective elastic tensor for
microscopic material properties with bounded hybrid uncertainty. This is to say that the
CE
proposed HHUAM can deal with the uncertain homogenization problems which the
AC
interval and subinterval homogenization-based method cannot deal with. Furthermore,
with choosing the value of λ properly, the HHUAM can also be applied to determine the
effective elastic properties of periodic microstructure with interval parameters and have
a better performance than the IHM and Sub-IHM. Here, as a contrast, the bounds of the
effective elastic tensor of the RVE are calculated by IHM (Sub-IHM) and HHUAM. In
this case, all of the uncertain parameters are assumed to be interval variables and their
ACCEPTED MANUSCRIPT
uncertain level is assumed as  E     E     . The results calculated by IHM
1
1
2
2
and HHUAM when uncertain level α =5% are shown in Table 8. The results calculated
by Sub-IHM and HHUAM when uncertain level α =10% are shown in Table 9. The
reference results calculated by MCS are also listed in Table 8 and Table 9. Table 8 and
CR
IP
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Table 9 show that the bounds of the effective elastic tensor obtained by HHUAM match
the referenced results obtained by MCS better than IHM and Sub-IHM.
Table 8
H
H
D11
D12
Upper
Lower
MCS
8.4736
11.7364
4.5870
HHUAM
8.4404
11.7833
4.5621
Error
0.39%
0.40%
0.54%
IHM
8.3627
11.3451
4.4377
Error
1.31%
3.33%
3.25%
H
D22
H
D66
Upper
Lower
Upper
Lower
Upper
7.4985
8.4722
11.7346
1.7952
2.0180
7.5361
8.4390
11.7815
1.7934
2.0195
0.50%
0.39%
0.40%
0.09%
0.07%
7.1496
8.3613
11.3434
1.8406
1.9684
4.65%
1.31%
3.33%
2.53%
2.46%
ED
M
Lower
AN
US
Bounds of the effective elastic tensor ( = 0.05).
Table 9
PT
Bounds of the effective elastic tensor ( = 0.1).
H
D11
H
H
D22
D66
Upper
Lower
Upper
Lower
Upper
Lower
Upper
MCS
7.3814
14.4631
3.6703
10.0364
7.3802
14.4610
1.6911
2.1335
HHUAM
7.3404
14.5983
3.6416
10.1552
7.3392
14.5962
1.6864
2.1382
Error
0.56%
0.93%
0.78%
1.18%
0.56%
0.93%
0.28%
0.22%
AC
CE
Lower
H
D12
Sub-IHM
7.3172
13.8896
3.5766
9.5095
7.3159
13.8877
1.7287
2.0826
Error
0.87%
4.1%
1.77%
5.25%
0.87%
3.96%
2.22%
2.39%
6. Conclusions
A homogenization-based hybrid uncertain analysis method, termed as HHUAM,
is proposed for predicting the effective elastic properties of periodic microstructure
ACCEPTED MANUSCRIPT
with hybrid uncertain-but-bounded parameters in this paper. In the HHUAM, the
bounded hybrid random and interval variables are transformed into the unitary
uncertain-but-bounded variables related to the polynomial parameter λ. By choosing a
suitable λ, the effective elastic tensor can be expanded by GSE. The expansion
CR
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coefficients of the Gegenbauer polynomials of the HHUAM can be calculated through
the Gauss-Gegenbauer integration method. The influence of bounded hybrid
uncertainties in microstructures on homogenized macroscopic elastic properties of
AN
US
heterogeneous materials is also investigated through the MCS and HHUAM.
The HHUAM and MCS are used to calculate the effective elastic tensor of a unit
square cell with a rectangular hole and a unit RVE of a unidirectional fiber reinforced
M
composite with microscopic hybrid uncertain-but-bounded constituent material
properties. By comparing the results calculated by the HHUAM and the MCS, the
ED
accuracy and the efficiency of the HHAUM are verified. It should be pointed out that
PT
with the uncertain level of input increasing, more computational time is spent for
predicting the effective elastic tensor through HHUAM. This is because of that higher
CE
retained order of HUHAM is needed for reducing the estimation error for
AC
homogenization problem with larger uncertain level. But compared with MCS, the
computational efficiency of HHUAM is still significantly improved. Besides, it can be
observed from the numerical results that the uncertain levels of the interval of the
H
H
H
expectation and the interval of variance of the D11 , D12 and D22 are much larger than
H
the uncertain levels of the uncertain-but-bounded parameters, while those of D66 are
H
smaller than the uncertain levels of input. Moreover, the D12 is the most influenced by
ACCEPTED MANUSCRIPT
the hybrid uncertain-but-bounded parameters in microscopic material properties,
H
H
H
followed by D11 and D22 , and D66 is the least influenced. This phenomenon is due
CR
IP
T
to the combined influences of Young’s modulus and Poisson’s ratio.
Acknowledgement
The paper is supported by the Key Project of Science and Technology of
AN
US
Changsha (Grant No. KQ1703028) and the Fundamental Research Funds for the
Central Universities (Grant No. 531107051148). The author would also like to thank
AC
CE
PT
ED
M
reviewers for their valuable suggestions.
ACCEPTED MANUSCRIPT
Appendix
For an uncertain system with multiple uncertain-but-bounded parameters, the
relative improvement of response via increasing the retained order of HHUAM with
respect to the j th uncertain parameter is defined as [28]

Where
U (k , j )  U (k , e j )
;
U (k , j )  U (k , j )
Ir(k , j ) 
(51)
U (k , j )  U (k , e j )
.
U (k , j )  U (k , j )
(52)
AN
US
Ir(k , j ) 
k  1, 2,...
CR
IP
T

Ir(k , j )  max Ir(k , j ), Ir(k, j )  tol ,
In Eq(52), U (k, j ) and U (k, j ) denotes the lower and upper bounds of the
response obtained by GSEM, k=[k1, k2,…, kL] is the retained order vector, kj denotes
M
the retained order related to the jth uncertain parameter. ej denotes the L-dimension
vector, in which the j th element is equal to 1 while the other elements are zero.
ED
For each uncertain parameter, the least vector N
req
 [ N1req , N 2req ,..., N Lreq ] which
AC
CE
PT
satisfies Eq.(51) can be obtained by an iteration procedure shown as Fig. 5.
Fig. 5 The procedure of estimating the least order via the relative improvement criterion.
ACCEPTED MANUSCRIPT
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