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Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516
www.elsevier.com/locate/cma
Topology optimization for microstructures of viscoelastic composite
materials
Xiaodong Huanga,b,∗, Shiwei Zhoub, Guangyong Suna, Guangyao Lia, Yi Min Xieb
aState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan, 410082, PR China
bCentre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476,
Melbourne 3001, Australia
Received 1 July 2014; received in revised form 2 October 2014; accepted 8 October 2014
Available online 16 October 2014
Highlights
•An extended BESO method for designing microstructures of viscoelastic composites.
•Unambiguous microstructures of viscoelastic composites are obtained.
•Composites with desirable viscoelastic properties are presented.
•Comparison with theoretical bounds of storage and loss moduli.
Abstract
The viscoelastic response of materials is often utilized for wide applications such as vibration reduction devices. This paper
extends the bi-directional evolutionary structural optimization (BESO) method to the design of composite microstructure with
optimal viscoelastic characteristics. Both storage and loss moduli of composite materials are calculated through the homogenization
theory using complex variables. Then, the BESO method is established based on the sensitivity analysis. Through iteratively
redistributing the base material phases within the unit cell, optimized microstructures of composites with the desirable viscoelastic
properties will be achieved. Numerical examples demonstrate the effectiveness of the proposed optimization method for the design
of viscoelastic composite materials. Various microstructures of optimized composites are presented and discussed. Meanwhile, the
storage and loss moduli of the optimized viscoelastic composites are compared with available theoretical bounds.
c
⃝2014 Elsevier B.V. All rights reserved.
Keywords: Topology optimization; Viscoelastic composite; Microstructure; Bi-directional evolutionary structural optimization (BESO)
∗Corresponding author at: Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT
University, GPO Box 2476, Melbourne 3001, Australia. Tel.: +61 3 99253320; fax: +61 3 96390138.
E-mail address: huang.xiaodong@rmit.edu.au (X. Huang).
http://dx.doi.org/10.1016/j.cma.2014.10.007
0045-7825/ c
⃝2014 Elsevier B.V. All rights reserved.
504 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516
1. Introduction
Vibration is often undesirable for structures due to the demands for structural stability, durability and noise re-
duction. Viscoelastic materials such as rubbers are often applied for reducing the vibration level through damping
mechanisms [1,2]. Those viscoelastic materials have favorable damping characteristics but often lack stiffness for
constructing engineering products. Composites may produce the high damping and high stiffness by mixing two or
more constituent materials with different physical properties [3]. The resulting viscoelastic composites will be of great
interest to various industries such as automobile, aerospace, etc. The viscoelastic response of these artificial compos-
ites mainly depends on their microstructures apart from the proportion and physical properties of their constituents
[1,3]. Designing viscoelastic composites with high damping and stiffness could be achieved by formulating a topology
optimization problem for micro-structural topology and material properties at the macro scale.
Topology optimization methods, e.g. homogenization method [4], Solid Isotropic Material with Penalization
(SIMP) [5–8], level set method [9–11], Evolutionary Structural Optimization (ESO) [12,13] and its later version Bi-
directional ESO (BESO) [14,15], were originally developed to find a stiffest structural layout under given constraints.
Topology optimization for the material design was initially proposed by Sigmund [16,17]. It was assumed that the
material was microscopically composed of periodical unit cells (PUCs) and its effective macroscopic properties
could be calculated through the homogenization theory. The inverse homogenization problem for seeking the best
microstructure of the unit cell with the prescribed constitutive properties was then solved by topology optimization
technique. Since then, extensive research has been carried out to investigate the material design with prescribed or
extreme effective mechanical properties [18,19], thermal conductivity [20], permeability [21] and electromagnetic
properties [22,23], the combination of properties [24–26], and so on.
Different from the pure elastic materials, viscoelastic materials have complex moduli, namely storage modulus and
loss modulus. Early studies on viscoelastic composites focused on the bounds of effective complex moduli and found
that multi-scale microstructures such as the Hashin–Shtrikman coated spheres assemblage or rank-N laminates could
achieve high stiffness and high damping [27–30]. With the development of the modern manufacture technologies
such as 3D printers, it is worthwhile to optimally design one-length scale microstructures with clear boundaries for
viscoelastic composites. Topology optimization was firstly applied to the design of microstructures of viscoelastic
composites for optimal damping characteristics by Yi et al. [31] and obtained the microstructures of viscoelastic
composites. Prasad and Diaz [32] conducted the topology optimization of viscoelastic materials utilizing negative
stiffness components. Recently, Chen and Liu [33] investigated topology optimization for the design of viscoelastic
cellular materials with prescribed properties. Most recently, Andreasen et al. [34] investigated microstructures of
viscoelastic composites which achieve the theoretical upper bound by topology optimization and Andreassen and
Jensen [35] further studied viscoelastic composites for maximizing the loss/attenuation of propagating waves.
It has been revealed that optimized material microstructures highly depended on the used optimization parameters
and algorithm because a number of different microstructures could possess the same physical property [16–19].
Because of the simplicity and computational efficiency of the BESO method [15,36,37], this paper will investigate
the topology optimization of viscoelastic composites by using the BESO method with discrete design variables.
Composite materials are assumed to be composed of two base materials (at least one is viscoelastic material) and
their microstructures are uniformly represented by corresponding periodic unit cells. The optimization objective is to
find the optimal distribution of two base materials within the unit cell so that the resulting composite has the maximum
damping and/or stiffness. The homogenization theory will be used to calculate the effective properties of viscoelastic
composites and then the BESO method will be applied for finding their optimal microstructures. Several numerical
examples will be presented and compared with the theoretical bounds to demonstrate the effectiveness of the proposed
BESO method.
2. Homogenization for viscoelastic composites
2.1. Properties of viscoelastic materials in the frequency domain
When a uniform viscoelastic material is subjected to a sinusoidally varying stress with the operation frequency, ω,
the resulting strain also varies sinusoidally with the same frequency when a steady state is eventually reached. The
X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516 505
Fig. 1. (a) Macroscopic structure; (b) composite microstructure; (c) periodic unit cell.
stress and strain varying with time at the steady state are expressed by
σi j (ω, t)= ¯σi j (ω) exp(iωt)(1)
εkl (ω, t)= ¯εkl (ω) exp(iωt)(2)
where ¯σij and ¯εk l are the spatial part of the stress and strain and their relationship is given by [2]
¯σij (ω) =Ei jk l (ω)¯εkl (ω) (3)
where Ei j kl (ω) is the complex modulus which also depends on the operation frequency. The complex modulus
Ei j kl (ω) in the frequency domain can be measured by the relaxation modulus in the time domain through the Fourier
transform as
Ei jk l (ω)=iω∞
0
Ei j kl (t)exp(−iωt)dt .(4)
The complex modulus can be explicitly divided into the real and imaginary parts as
Ei j kl (ω) =E′
i jk l (ω)+i E′′
i j kl (ω) (5)
where E′
i jk l (ω)is the storage modulus and E′′
i j kl (ω) is the loss modulus. The loss tangent tan δi jk l as a measure of
damping is the ratio of the loss modulus to the storage modulus, which is proportional to the energy loss per cycle
within the framework of linear viscoelasticity.
2.2. Effective complex modulus of viscoelastic composites
It is assumed that the macroscopic structure as shown in Fig. 1(a) is made by a composite which is composed of
two-phase base materials where at least one material phase is viscoelastic. The microstructure of the composite as
shown in Fig. 1(b) is spatially repeated with the periodic unit cell in Fig. 1(c). When the size of the periodic unit cell is
quite small compared with the wavelengths of all relevant elastic waves, the macroscopic properties of the viscoelastic
composite can be homogenized over the unit cell. At any given frequency, the homogenized relationship between the
spatial parts of the stress and strain in the heterogeneous composite is then expressed by
¯σij (ω) =EH
i jk l (ω)¯εk l (ω) (6)
where EH
i j kl (ω) is the effective complex modulus which depends on the properties of base materials, the volume
fractions and spatial distribution of material phases in the unit cell. With the asymptotic approximations, the effective
complex modulus can be obtained by the homogenization theory [38–40] as
EH
i j kl (ω) =1
|Y|Ω
Ei j pq (ω)(¯εkl
pq − ˜εkl
pq )dΩ(7)
506 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516
where |Y|denotes the area (or volume in 3D) of the unit cell domain Ω.¯εkl
pq defines the four linearly independent unit
strain fields as ¯ε11
pq =1000T,¯ε22
pq =0100T,¯ε12
pq =0010Tand ¯ε21
pq =0001T
for 2D cases.
The strain fields ˜εkl
pq induced by the test strains can be found from the following equation
Ω
Ei j pq (ω)εi j (v)˜εkl
pq dΩ=Ω
Ei j pq (ω)εi j (v)¯εkl
pq dΩ(8)
where v∈H1
per(Ω)which is the Y-periodic admissible displacement field. The above equation is the weak form of
the standard elasticity equation applied to the unit cell with periodic boundary conditions subject to the independent
cases of pre-strain given by ¯εkl
pq .
The effective complex modulus, EH
i jk l can be found by substituting the solution of Eq. (8),˜εkl
pq into Eq. (7). It can
be seen that the above homogenization expressions are the same to those for pure elastic materials except that the
properties of the viscoelastic phase depends on the operation frequency. Furthermore, it should also note that both
Ei j pq and ˜εk l
pq in the equations are complex and thus Eq. (8) will be solved by finite element analysis with complex
variables in this paper.
3. Topology optimization
3.1. Design variables and material interpolation scheme
The effective stiffness and damping of a two-phase composite highly depend on the spatial distribution of material
phases, therefore, how to optimally distribute material phases within the unit cell would be critical in the design of
viscoelastic composites. In this paper, the unit cell is discretized into finite elements and each element is assigned with
either material 1 or material 2. An artificial design variable, xe, is introduced by assuming that xe=1 if an element is
made of material 1 and xe=0 if an element is made of material 2. With such an assumption, Yi et al. [31] employed
a linear artificial two-phase material model as
Ei j kl (xi)=xeE(1)
i jk l +(1−xe)E(2)
i jk l (9)
where E(1)
i jk l and E(2)
i jk l are the moduli of material 1 and material 2 respectively. Due to lack of proper penalization
scheme, the resulting solutions contained a large volume of “gray area” with intermediate design variables [31]. In the
BESO method, we will use the binary design variable xe=0 or 1 only and the solutions will give clearly boundaries
between material 1 and material 2. However, it should be noted that BESO can only find a convergent 0/1 solution
when the solution exists [41,42]. In the solid isotropic material penalization (SIMP) method [5–8], the well-known
SIMP model [5–8] makes elements with intermediate design variables density uneconomical in the optimization
process and thus the solution naturally tends to be 0/1. However, our numerical tests indicated that the solutions are
hardly convergent to 0/1 designs by directly employing the SIMP model for viscoelastic material design. Andreasen
et al. [34] have investigated the influence from the penalization parameter for the complex modulus and suggested
to use a power-law exponent less than 1 for the imaginary part. As a compromise no penalization is used for both
real and imaginary moduli [34]. Here, we establish artificial two-phase material models for storage modulus and loss
modulus separately. The storage modulus uses the SIMP model as
E′
i j kl (xi)=xp
eE′(1)
i jk l +(1−xp
e)E′(2)
i jk l (10)
where E′(1)
i jk l >E′(2)
i j kl .pis the exponent of penalization and it has been verified that the optimization solution tends to
0/1 as p>1 [8]. Therefore, p=3 is used throughout this paper. Meanwhile, the linear relationship is defined for the
loss modulus by
E′′
i j kl (xe)=xeE′′(1)
i jk l +(1−xe)E′′(2)
i jk l (11)
where E′′(1)
i jk l and E′′(2)
i jk l are the loss moduli of material 1 and material 2 respectively. Our numerical examples will
demonstrate that the above material interpolation schemes work well for maximizing damping and/or stiffness of
viscoelastic composites reported in this paper.
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3.2. Statement of the optimization problem
The high stiffness and/or high damping at the operation frequency are desirable for the design of viscoelastic ma-
terials. For example, we want to obtain a composite with maximum damping or maximum stiffness along direction 1
for a 2D composite. The optimization problem can be defined as
Maximize: f(xe)=tan δ1111(ω) or E′
1111(ω)
Subject to: V1∗
f=
N
e=1
VexeN
e=1
Ve(12)
where f(xe)is the objective function. Veis the volume of the eth element and V1∗
fis the prescribed volume fraction
of material 1 which can be specified by the user. Nis the total number of elements in the unit cell. It should be noted
that the numerical examples in this paper will consider maximizing damping and/or stiffness in both directions (1 and
2). Certainly, other optimization problems with damping and stiffness properties of composites can also be formulated
and equally solved by the proposed BESO algorithm in this paper.
3.3. Sensitivity analysis
To implement the BESO optimization technique, sensitivity analysis is necessary for guiding the search direction
during the iteration process. As the given objective functions in Eq. (12) can be calculated from the modulus matrix,
it is necessary to compute the sensitivity of the complex modulus with regard to design variables. With the help of
the material interpolation scheme in Eqs. (10) and (11), the derivation of the complex modulus EH
i jk l with respect to
design variables xecan be easily obtained by using the adjoint method [31,33,43].
∂EH
i jk l
∂xe
=1
|Y|Ωepx p−1
e(E′(1)
i j pq −E′(2)
i j pq )+i(E′′(1)
i j pq −E′′(2)
i j pq )(¯εkl
pq − ˜εkl
pq )dΩe(13)
where Ωeis the domain of the eth element. It can be seen that the resulting sensitivity is also complex where the real
part is the sensitivity of storage modulus and the imaginary part is the sensitivity of loss modulus. Here, the detailed
derivations for sensitivity analysis are overlooked and the reader may refer to Refs. [31,33]. In the proposed BESO
method, the design variables xeare restricted to be binary values either 0 or 1, thus the elemental sensitivity can be
expressed explicitly as
∂EH
i jk l
∂xe
=
1
|Y|Ωe(E′(1)
i j pq −E′(2)
i j pq )+i(E′′(1)
i j pq −E′′(2)
i j pq )(¯εkl
pq − ˜εkl
pq )dΩewhen xe=1
1
|Y|Ωei(E′′(1)
i j pq −E′′(2)
i j pq )(¯εkl
pq − ˜εkl
pq )dΩewhen xe=0.
(14)
3.4. Numerical implementation
The BESO method [41] normally used sensitivity numbers to update the design variable xewhere sensitivity num-
bers denote the relative ranking of elemental sensitivities. For the maximization optimization problem, elemental
sensitivity number for the eth element can be simply expressed by
αe=∂f(xe)
∂xe
.(15)
As the objective function is composed of the combination of the components of EH
i j kl , the elemental sensitivity
numbers can be easily obtained by substituting Eq. (14) into Eq. (15). According to the relative ranking of elemental
sensitivity numbers, BESO will update the design variables xe=0 for elements with the lowest sensitivity numbers
and xe=1 for elements with highest sensitivity numbers.
Numerical instabilities such as checkerboard pattern and mesh-dependency problem are common phenomenon in
the topology optimization techniques based on the finite element analysis [44]. Here, a mesh-independent filter for
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Table 1
The properties of materials 1 and 2 at the operation frequencies.
Operation frequency
ω(rad/s)
Storage modulus
E′
1111 (GPa)
Loss modulus
E′′
1111 (GPa)
Loss tangent Bulk modulus
κ(GPa)
Material 1 All 73.56 0.00 0.00 44.87
Material 2 0 1 0.00 0.00 0.77
0.5 1.71 1.14 0.67 1.32 +0.88i
discrete design variables [14] is applied for the sensitivity numbers. The modified elemental sensitivity number can
be expressed by
ˆαi=
N
j=1
w(ri j )α j
N
j=1
w(ri j )
(16)
where ri j denotes the distance between the centers of elements iand j.w(ri j )is the weight factor of the jth sensitivity
number as
w(ri j )=rmin −ri j for ri j <rmin
0 for ri j ≥rmin (17)
where rmin is the filter radius which can be specified by the user.
Due to the discrete design variables used in the BESO algorithm, Huang and Xie [14] proposed that the sensitivity
number can be further modified by averaging with its historical information to improve the convergence of the solution.
Thus, the sensitivity number after the first iteration can be further modified by
˜αi,k=1
2ˆαi,k+ ˜αi,k−1(18)
where kis the current iteration number.
BESO starts from an initial design and update the topology of the unit cell according to the calculated sensitivity
numbers step by step. The whole iteration process is stopped until both the volume fraction constraint is satisfied and
the objective function is convergent. For the detailed BESO procedure one can also refer to Refs. [15,19,23].
4. Numerical examples and discussions
Some numerical examples are presented in this section to illustrate the microstructural design of viscoelastic com-
posites and demonstrate the effectiveness of the proposed optimization approach. It is assumed that the viscoelastic
composite is composed of two materials: one is pure elastic and another is viscoelastic. Both materials are assumed
to be isotropic and their material properties are Young’s modulus E(1)=70 GPa, Poisson’s ratio v(1)=0.22 for
material 1 (glass) and E(2)=1+2.5e−tGPa, v (2)=0.35 for material 2 (epoxy). The storage and loss moduli, loss
tangent and bulk modulus for both materials at the frequencies ω=0 and 0.5 rad/s are listed in Table 1. However, it
should be noted that the proposed optimization algorithm can be equally applied for two viscoelastic materials.
The unit cell which represents the microstructure of a composite is discretized with 80 ×80 square bilinear finite
elements. To initialize the optimization process, the initial guess is assumed to be that the unit cell is full of material 1
except for four elements at the center of the unit cell with material 2. The evolution rate ER =2% and rmin is selected
to be 5 times of the typical size of elements. In the following figures of the unit cells, black elements represent material
1 (stiff and elastic) and white elements represent material 2 (soft and viscoelastic).
4.1. Examples for maximizing damping of composites
The high damping at the operation frequency is mostly desirable for the design of viscoelastic materials. The ob-
jective of numerical examples in this section is to find microstructures so that the resulting composites yield damping
X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516 509
Table 2
Optimized results for maximizing damping of composites.
along directions 1 and 2 at ω=0.5 rad/s as large as possible. The optimization objective is defined as
maximize: f=tan δ1111(ω) +tan δ2222 (ω) at ω=0.5 rad/s.(19)
The proposed BESO method was applied for the above optimization problem under various given volume fractions
of material 1. Table 2 lists the resulting microstructures of optimized composites and their properties. The optimized
microstructures show that the stiff and elastic material 1 is surrounded by soft and viscoelastic material 2 so as to
maximize the damping of the resulting composites. It is interesting to note that the material damping is significantly
enhanced by adding a small amount of the viscoelastic material into the elastic material without any damping. For
instance, the loss tangent δ1111 =δ2222 =0.53 when V1
f=0.8. Further increasing the volume fraction of the
viscoelastic material only causes the insignificant increase of the loss tangent. When V1
f=0.4, the loss tangent
δ1111 =δ2222 =0.64 which is very close to that of the pure viscoelastic material 2, δ1111 =δ2222 =0.67.
Nevertheless, the microstructures with disconnected stiff material 1 as shown in Table 2 inevitably lead to composites
with low stiffness (storage modulus) because maximizing material damping is equivalent to maximizing loss modulus
and minimizing storage modulus simultaneously (the loss tangent is the ratio of the loss modulus to the storage
modulus). Fig. 2 gives the changes of damping, storage modulus and loss modulus with the variation of the volume
fraction of material 1. It can be seen that the material damping decreases as the volume fraction of material 1 increases
and it becomes zero when the microstructure is full of material 1. Both the storage modulus and loss modulus increase
with the increase of volume fraction of material 1, but the loss modulus finally returns to zero when V1
f=100%.
The maximum value of loss modulus in Fig. 2 corresponds to that for V1
f=90% because we only conducted limited
cases for the given volume fraction constraints. However, it should note that the actual maximum loss modulus should
occur at some volume fraction 90% <V1
f<100%. This maximum loss modulus is less important with regard to the
material design because the corresponding composite must be with low damping as shown in Fig. 2.
510 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516
Fig. 2. Variations of loss tangent, storage and loss moduli for maximizing damping under various volume fraction constraints.
Fig. 3. Evolution histories of loss tangent, volume fraction and topology for maximizing damping with the volume fraction Vf
1=50%.
Fig. 3 plots the evolution histories of the objective function and volume fraction of material 1 when the objective
volume fraction of material 1 is set to be 50%. BESO starts from the initial design being almost full of material
1, gradually decreases the volume fraction of material 1 to its prescribed value 50% and then keeps constant. The
composite damping is generally improved as more and more viscoelastic material 2 is added to the unit cell. At the
latest stage of the optimization, both the composite damping and microstructure are stably convergent to the solutions
while the volume fraction of material 1 keeps its constraint value 50%.
4.2. Examples for maximizing stiffness of composites
Apart from the viscoelastic damping, the material stiffness (storage modulus) also has the significant effect on the
reduction of vibration and noise. The optimization objective of numerical examples in this section is to maximizing
the storage modulus of composites along both directions 1 and 2 at the operation frequency ω=0.5 rad/s. Thus, the
optimization objective is expressed by
maximize: f=E1111(ω) +E2222 (ω) at ω=0.5 rad/s.(20)
Table 3 gives the optimized microstructures of composites and their material properties under various volume
fractions of material 1. Totally different from the optimized microstructures for maximizing damping in Section 4.1,
maximizing stiffness of composites always leads to the optimized microstructures with connected stiff and elastic
material 1. Fig. 4 shows the variations of the resulting loss tangent, storage modulus and loss modulus against the
volume fraction of material 1. It can be seen that the loss modulus always keeps at a very low level (less than 1.16)
X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516 511
Table 3
Optimized results for maximizing stiffness of composites.
Fig. 4. Variations of loss tangent, storage and loss moduli for maximizing stiffness under various volume fraction constraints.
for all cases. As the result, the loss tangent quickly decreases to a low level even if a small amount of elastic material
1 adds to the viscoelastic material 2, e.g. δ1111 =δ2222 =0.125 for Vf
1=20%.
4.3. Comparison with bounds of effective bulk moduli
To further check the optimized solutions, the obtained results can be compared with the theoretical bounds of
effective bulk moduli. Unfortunately, the theoretical bounds for viscoelastic composites cannot be easily expressed by
512 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516
Fig. 5. Bounds of effective bulk modulus in the complex plane.
explicit formulae. Gibiansky and Lakes [29,30] investigated the set of possible values of the complex effective bulk
modulus κshould fill a region in the complex plane because κis described by two numbers, i.e. its real and imaginary
parts. At the operation frequency ω=0.5 rad/s, bounds of the effective bulk modulus in the complex plane can be
obtained numerically as shown in Fig. 5 where the solid and dash lines denote the upper and lower bounds of loss bulk
modulus respectively. In Fig. 5, the hollow squares denote the effective bulk moduli for the composites with maximum
damping given in Table 2, and solid circles denote the effective bulk moduli of the composites with maximum stiffness
given in Table 3. It clearly shows that the effective bulk moduli for maximizing damping are coincident with the upper
bound of loss bulk modulus, however the effective bulk moduli for maximizing stiffness is located at the lower bound
of loss bulk modulus. As discussed in Section 4.1, the loss bulk modulus can be further increased by increasing the
volume fraction of material 1. According to the theoretical upper bound of the loss bulk modulus, the maximum loss
bulk modulus is about 6.9 GPa and the corresponding storage bulk modulus and loss tangent are about 22.6 GPa and
0.3 respectively.
The bounds for the storage modulus against volume fraction can be described by Hashin–Shtrikman (H–S) bounds
through the replacement of elastic moduli with corresponding viscoelastic complex moduli [27,28]. The numerical
H–S upper and lower bounds of the storage bulk modulus are plotted in Fig. 6. The storage bulk moduli of the
optimized composites in Tables 2 and 3are given in Fig. 6 with hollow squares and solid circles. It can be seen that
the storage bulk moduli of the optimized composites for maximizing stiffness approach the H–S upper bound, and the
storage bulk moduli of the optimized composites for maximizing damping are very close to the H–S lower bound.
4.4. Examples for maximizing damping with stiffness constraint
Examples in the above sections indicate that maximizing damping and stiffness are somewhat conflicting opti-
mization objectives for the design of viscoelastic composites and the design of viscoelastic composites is therefore
an inherent multi-objective optimization problem. The possible ranges of damping and stiffness of composites under
a given volume fraction can be estimated by previous examples as they approach the theoretical bounds of storage
and loss moduli, for instance, 0.047 ≤δ1111 (=δ2222)≤0.63 and 4.02 GPa 6E′
1111 =E′
2222623.98 GPa when
V1
f=50%. With the above limits in mind, we will maximize composite damping by setting a series of stiffness
constraints in this section. To this end, the optimization problem can be stated as
maximize: f=tan δ1111(ω) +tan δ2222 (ω) at ω=0.5
subject to: E′
1111(ω) =E′
2222(ω)=E∗at ω=0.5 (21)
V1∗
f=
N
e=1
VexeN
e=1
Ve=50%
here, E∗is the constraint value of the storage modulus in directions 1 and 2. Such an additional constraint on the
storage modulus can be easily satisfied by introducing a Lagrange multiplier as given in Ref. [45].
X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516 513
Fig. 6. Bounds of storage bulk modulus under various volume fractions.
Table 4
Optimized results for maximizing damping with stiffness constraint.
The optimized microstructures and their material properties are given in Table 4 for E∗=5 GPa, 10 GPa, 15 GPa,
and 20 GPa respectively. It can be seen that the stiff and elastic material 1 in the optimized microstructures is
disconnected for E∗=5 GPa, weakly connected for E∗=10 GPa and strongly connected for E∗=15 GPa
and 20 GPa. Meanwhile, the loss tangent decreases from 0.6 for E∗=5 GPa to 0.08 for E∗=20 GPa. The elasticity
matrixes in Table 4 shows the resulting storage moduli are very close to the corresponding constraint values.
As mentioned above, the design of viscoelastic composite is in fact of a multi-objective optimization problem
which maximizes damping and stiffness simultaneously. To conveniently plot the Pareto front curve for this multi-
objective optimization problem, the damping and stiffness are inversely non-dimensionalized by their minimum values
δmin =0.047 and Emin =4.02 GPa so that 0 < δmin /δ ≤1 and 0 <Emin /E′≤1. The plotting of non-dimensional
damping against stiffness gives the Pareto front with a convex curve as shown in Fig. 7. The right above the Pareto
front curve gives the region for possible designs but the optimal solution should be on the Pareto front. The optimal
514 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516
Fig. 7. Relationship between loss tangent and storage modulus of optimized viscoelastic composites.
Table 5
Optimized results for maximizing damping with stiffness constraint under
different operation frequencies.
design of viscoelastic composite is therefore a matter of making a trade-off decision from a set of compromising
solutions on the Pareto front.
In some cases, viscoelastic material is required with high stiffness at a low frequency and high damping at a high
frequency. Thus, the optimization problem can be reformulated e.g. to maximize the damping at the high frequency
ω=0.5 rad/s subject to the static stiffness constraint.
maximize: f=tan δ1111(ω) +tan δ2222 (ω) at ω=0.5 rad/s
subject to: E′
1111(ω) =E′
2222(ω)=E∗at ω=0 (22)
V1∗
f=
N
e=1
VexeN
e=1
Ve=50%.
X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516 515
The proposed BESO method can also be applied for this optimization problem except for the homogenization
calculation for both frequencies which inevitably causes high computational cost. The resulting microstructures of
composites, elasticity matrixes and loss tangent are given in Table 5 when the static stiffness constraint is set to be
E∗=5 GPa, 10 GPa, 15 GPa, and 20 GPa respectively. The optimized microstructures are obviously different from
those in Table 4 and demonstrate that the design of composites depends on the application requirements.
5. Conclusions
In this paper, viscoelastic composites are supposed to be composed of a stiff and elastic material phase and a soft
and viscoelastic material. The BESO method is extended to designing microstructures of composites with desirable
viscoelastic properties. The given examples demonstrate the effectiveness of the proposed optimization algorithm to
obtain the clear microstructures of composites with maximum damping and/or stiffness. The numerical results indicate
that the damping property of composites can be greatly enhanced by properly mixing a small amount of a viscoelastic
material with an elastic material, but stiffness of composites has no significant improvement. When the optimization
objective changes to maximize the stiffness of composites, the damping property of viscoelastic material phase cannot
be fully utilized. Comparison with theoretical bounds reveal that maximizing damping results in designs at the upper
bound of loss modulus and the lower bound of storage modulus, but maximizing stiffness results in designs at the lower
bound of loss modulus and the upper bound of storage modulus. Therefore, the design of viscoelastic composites is
inherently a multi-objective optimization problem which is solved by maximizing damping subject to a stiffness
constraint in this paper. A set of Pareto optimal solutions is obtained in the presence of trade-offs between conflicting
damping and stiffness objectives for the design of viscoelastic composites.
Acknowledgment
The authors wish to acknowledge the financial support from the Australian Research Council (FT130101094) and
Key Program of National Natural Science Foundation of China (61232014) for carry out this work.
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