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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.

X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516 507

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

508 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516

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.

References

[1] D.D.L. Chung, Review materials for vibration damping, J. Mater. Sci. 36 (2001) 5733–5737.

[2] R.M. Christensen, Theory of Viscoelasticity, Dover, New York, 2010.

[3] R.S. Lake, High damping composite materials: effect of structural hierarchy, J. Compos. Mater. 36 (3) (2002) 287–297.

[4] M.P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech.

Engrg. 71 (1988) 197–224.

[5] M.P. Bendsøe, Optimal shape design as a material distribution problem, Struct. Optim. 1 (1989) 193–202.

[6] G.I.N. Rozvany, M. Zhou, T. Birker, Generalized shape optimization without homogenization, Struct. Optim. 4 (1992) 250–254.

[7] M. Zhou, G.I.N. Rozvany, The COC algorithm, part II: topological, geometry and generalized shape optimization, Comput. Methods Appl.

Mech. Engrg. 89 (1991) 197–224.

[8] M.P. Bendsøe, O. Sigmund, Topology Optimization: Theory, Methods and Applications, Springer-Verlag, Berlin, 2003.

[9] M.Y. Wang, X. Wang, D. Guo, A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg. 192 (2003)

227–246.

[10] X. Wang, M.Y. Wang, D. Guo, Structural shape and topology optimization in a level-set-based framework of region representation, Struct.

Multidiscip. Optim. 27 (2004) 1–19.

[11] J.A. Sethian, A. Wiegmann, Structrual boundary design via level set and immersed interface methods, J. Comput. Phys. 163 (2) (2000)

489–528.

[12] Y.M. Xie, G.P. Steven, A simple evolutionary procedure for structural optimization, Comput. Struct. 49 (1993) 885–896.

[13] Y.M. Xie, G.P. Steven, Evolutionary Structural Optimization, Springer, London, 1997.

[14] X. Huang, Y.M. Xie, Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method, Finite

Elem. Anal. Des. 43 (14) (2007) 1039–1049.

[15] X. Huang, Y.M. Xie, Evolutionary Topology Optimization of Continuum Structures: Methods and Applications, John Wiley & Sons,

Chichester, 2010.

[16] O. Sigmund, Materials with prescribed constitutive parameters: an inverse homogenization problem, Internat. J. Solids Structures 31 (1994)

2313–2329.

[17] O. Sigmund, Tailoring materials with prescribed elastic properties, Mech. Mater. 20 (1995) 351–368.

[18] M.M. Neves, H. Rodrigues, J.M. Guedes, Optimal design of periodic linear elastic microstructures, Comput. Struct. 20 (2000) 421–429.

[19] X. Huang, A. Radman, Y.M. Xie, Topological design of microstructures of cellular materials for maximum bulk or shear modulus, Comput.

Mater. Sci. 50 (2011) 1861–1870.

516 X. Huang et al. / Comput. Methods Appl. Mech. Engrg. 283 (2015) 503–516

[20] S.W. Zhou, Q. Li, Computational design of multi-phase microstructural materials for extremal conductivity, Comput. Mater. Sci. 43 (2008)

549–564.

[21] J.K. Guest, J.H. Prevost, Design of maximum permeability material structures, Comput. Methods Appl. Mech. Engrg. 196 (4–6) (2007)

1006–1017.

[22] S.W. Zhou, W. Li, G. Sun, Q. Li, A level-set procedure for the design of electromagnetic metamaterials, Opt. Express 18 (7) (2010) 6693–6702.

[23] X. Huang, Y.M. Xie, B. Jia, Q. Li, S.W. Zhou, Evolutionary topology optimization of peridodic composites for extremal magnetic permeability

and electrical permittivity, Struct. Multidiscip. Optim. 46 (2010) 385–398.

[24] S. Torquato, S. Hyun, A. Donev, Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity,

Phys. Rev. Lett. 89 (2002) 266601.

[25] J.K. Guest, J.H. Prevost, Optimizing multifunctional materials: design of microstructures for maximized stiffness and ﬂuid permeability,

Internat. J. Solids Structures 43 (22–23) (2006) 7028–7047.

[26] V.J. Challis, A.P. Roberts, A.H. Wilkins, Design of three dimensional isotropic microstructures for maximized stiffness and conductivity,

Internat. J. Solids Structures 45 (2008) 4130–4146.

[27] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids 11 (2)

(1963) 127–140.

[28] Z. Hashin, Complex moduli of an viscoelastic composites: I General theory and application to particulate composites, Internat. J. Solids

Structures 6 (1965) 539–552.

[29] L.V. Gibiansky, G.W. Milton, On the effective viscoelastic moduli of two-phase media: I. Rigorous bounds on the complex bulk modulus,

Proc. R. Soc. Lond. A 440 (1993) 163–188.

[30] L.V. Gibiansky, R. Lakes, Bounds on the complex bulk modulus of a two-phase viscoelastic composite with arbitrary volume fractions of the

components, Mech. Mater. 16 (1993) 317–331.

[31] Y.-M. Yi, S.-H. Park, S.-K. Youn, Design of microstructrues of viscoelastic composites for optimal damping characteristics, Internat. J. Solids

Structures 37 (2000) 4791–4810.

[32] J. Prasad, A.R. Diaz, Viscoelastic material design with negative stiffness components using topology optimization, Struct. Multidiscip. Optim.

38 (2009) 583–597.

[33] W. Chen, S. Liu, Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus, Struct.

Multidiscip. Optim. (2014) http://dx.doi.org/10.1007/s00158-014-1049-3. online.

[34] C.S. Andreasen, E. Andreassen, J.S. Jensen, O. Sigmund, On the realization of the bulk modulus bounds for two-phase viscoelastic composites,

J. Mech. Phys. Solids 63 (2014) 228–241.

[35] E. Andreassen, J.S. Jesen, Topology optimization of periodic microstructures for enhanced dynamic properties of viscoelastic composite

materials, Struct. Multidiscip. Optim. 49 (2014) 695–705.

[36] X. Huang, S.W. Zhou, Y.M. Xie, Q. Li, Topology optimization of microstructures of cellular materials and composites for macrostructures,

Comput. Mater. Sci. 67 (2013) 397–407.

[37] L. Xia, P. Breitkopf, Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework,

Comput. Methods Appl. Mech. Engrg. 278 (2014) 524–542.

[38] Y.-M. Yi, S.-H. Park, S.-K. Youn, Asympotic homogenization of viscoelastic composites with periodic microstructures, Internat. J. Solids

Structures 35 (1998) 2039–2055.

[39] B. Hassani, E. Hinton, A review of homogenization and topology optimization I—homogenization theory for media with periodic structure,

Comput. Struct. 69 (1998) 707–717.

[40] B. Hassani, E. Hinton, A review of homogenization and topology optimization II—analytical and numerical solution of homogenization

equations, Comput. Struct. 69 (1998) 719–738.

[41] X. Huang, Y.M. Xie, Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials, Comput.

Mech. 43 (3) (2009) 393–401.

[42] X. Huang, Y.M. Xie, A further review of ESO type methods for topology optimization, Struct. Multidiscip. Optim. 41 (2010) 671–683.

[43] E.J. Haug, K.K. Choi, V. Komkov, Design Sensitivity Analysis of Structural Systems, Academic Press, Orlando, 1986.

[44] O. Sigmund, J. Petersson, Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-

dependencies and local minima, Struct. Optim. 16 (1) (1998) 68–75.

[45] X. Huang, Y.M. Xie, Evolutionary topology optimization of continuum structures with an additional displacement constraint, Struct.

Multidiscip. Optim. 40 (2012) 409–416.