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Struct Multidisc Optim

DOI 10.1007/s00158-012-0766-8

RESEARCH PAPER

Evolutionary topology optimization of periodic composites

for extremal magnetic permeability and electrical permittivity

X. Huang ·Y. M. Xie ·B. Jia ·Q. Li ·S. W. Zhou

Received: 21 June 2011 / Revised: 14 December 2011 / Accepted: 6 January 2012

c

Springer-Verlag 2012

Abstract This paper presents a bidirectional evolutionary

structural optimization (BESO) method for designing peri-

odic microstructures of two-phase composites with extremal

electromagnetic permeability and permittivity. The effective

permeability and effective permittivity of the composite are

obtained by applying the homogenization technique to the

representative periodic base cell (PBC). Single or multiple

objectives are defined to maximize or minimize the elec-

tromagnetic properties separately or simultaneously. The

sensitivity analysis of the objective function is conducted

using the adjoint method. Based on the established sen-

sitivity number, BESO gradually evolves the topology of

the PBC to an optimum. Numerical examples demonstrate

that the electromagnetic properties of the resulting 2D and

3D microstructures are very close to the theoretical Hashin-

Shtrikman (HS) bounds. The proposed BESO algorithm is

computationally efficient as the solution usually converges

in less than 50 iterations. The proposed BESO method

can be implemented easily as a post-processor to standard

commercial finite element analysis software packages, e.g.

ANSYS which has been used in this study. The result-

ing topologies are clear black-and-white solutions (with no

X. Huang (B)·Y. M. X i e

School of Civil, Environmental and Chemical Engineering,

RMIT University, GPO Box 2476, Melbourne 3001, Australia

e-mail: huang.xiaodong@rmit.edu.au

B. Jia

Centre for Micro-Photonics, Faculty of Engineering & Industrial

Science, Swinburne University of Technology, PO Box 218,

Hawthorn, Victoria, 3122 Australia

Q. Li ·S. W. Zhou

School of Aerospace, Mechanical and Mechatronic Engineering,

The University of Sydney, Sydney NSW 2006, Australia

grey areas). Some interesting topological patterns such as

Vigdergauz-type structure and Schwarz primitive structure

have been found which will be useful for the design of

electromagnetic materials.

Keywords Topology optimization ·Bidirectional

evolutionary structural optimization (BESO) ·

Homogenization ·Effective permeability ·

Effective permittivity

1 Introduction

Composite materials have attracted growing attention in

recent years due to their extraordinary physical proper-

ties desired for various applications. Composites could be

artificially made by a large number of metallic or dielectric

particles periodically placed in a homogenous host medium

or free space for a desirable electromagnetic permeability or

permittivity (Caloz and Itoh 2006). For given compositional

materials, the effective permeability and effective permit-

tivity of the resulting composites mainly depends on the

distribution of the constituent phases within the design cell.

Various numerical methods such as the homogenization

method (Ouchetto et al. 2006a,b), the moment-method-

based technique (Whites and Wu 2002), the effective-

medium-based method (Brosseau and Beroual 2001)etc.

have been developed to calculate the effective permittivity

or permeability of the composites. However, in the design

setting, one may wish to seek optimal distribution of the

constituent phases of the composites. This objective could

be achieved by formulating a topology optimization prob-

lem for microstructural topologies and their corresponding

material properties.

X. Huang et al.

The commonly used topology optimization meth-

ods, e.g. homogenization method (Bendsøe and Kikuchi

1988), Solid Isotropic Material with Penalization (SIMP)

(Bendsøe 1989; Rozvany et al. 1992; Zhou and Rozvany

1991; Bendsøe and Sigmund 2003), level set method (Wang

et al. 2003,2004; Sethian and Wiegmann 2000) and Evo-

lutionary Structural Optimization (ESO) (Xie and Steven

1993,1997) and its later version bidirectional ESO (BESO)

(Querin et al. 1998; Huang and Xie 2007,2009,2010a),

were traditionally developed to find stiffest structures under

given constraints. ESO was originally developed based on

removing inefficient materials from a structure so that the

resulting topology evolves toward an optimum (Xie and

Steven 1993,1997). BESO allows for not only removing

material from the least efficient regions, but also adding

material to the most needed areas simultaneously (Huang

and Xie 2007,2009,2010a;Querinetal.1998). It has

been demonstrated that the current BESO method (Huang

and Xie 2007,2010a) is capable of generating reliable and

practical topologies for various applications with high com-

putational efficiency. Also, the BESO algorithm is very

simple and can be easily understood by practitioners (such

as engineers and architects) who may not have strong back-

ground in mathematics. The whole BESO procedure can

be readily implemented as a “post-processor” to commer-

cial finite element analysis (FEA) software packages such

as ABAQUS, ANSYS and NASTRAN.

Over the last decade, topology optimization methods

have been successfully extended to the design of peri-

odic microstructures of cellular materials or composites

(Sigmund 1994,1995; Neves et al. 2000; Zhou and Li

2008a,b; Torquato et al. 2002; Huang et al. 2011). The

effective physical properties at a specific point in a mate-

rial can be obtained by applying the homogenization theory

(Hassani and Hinton 1998a,b) to its microstructure. In con-

trast, the inverse homogenization procedure is to seek a

periodic microstructure of a material with extremal phys-

ical properties where the microstructure of the material

is represented by a periodic base cell (PBC) (Sigmund

1995). Tailoring materials with prescribed constitutive prop-

erties could be achieved by formulating an optimization

problem where the PBC was modelled with frame ele-

ments (Sigmund 1995) or solid elements (Sigmund 1994;

Neves et al. 2000). Using the SIMP method, Zhou and

Li (2008a,b) investigated the computational design of

microstructural composites for extremal conductivity or

graded mechanical property. Torquato et al. (2002) opti-

mized microstructures for simultaneous transport of heat

and electricity in two-phase composites. The BESO proce-

dure for maximizing the bulk modulus or shear modulus of

cellular materials was recently developed by Huang et al.

(2011).

Topology optimization has also been extended to the

microstructural design for periodic electromagnetic mate-

rials. Jensen and Sigmund (2004) successfully designed

photonic crystal waveguides for the wave propagation prob-

lems in optical waveguide. Diaz and Sigmund (2010)and

Zhou et al. (2010a) presented the design of electromagnetic

metamaterials with negative permeability by considering the

resonance resulted from internal capacitance and inductance

(Pendry 2004). The effective permeability and effective

permittivity of periodic composites can also be predicted

by the homogenization theory using the gradient opera-

tor when the period of microstructure is small compared

to the wavelength (Ouchetto et al. 2006a,b). Zhou et al.

(2010b) proposed a new homogenization method, in which

the effective permittivity and permeability were expressed

by the curl operator rather than the gradient operator, and

the new homogenization technique was further developed as

an inverse procedure for optimizing 3D periodic metamate-

rials for desirable permittivity and permeability. Compared

with the size optimization for the ratio of the width the cubic

inclusion to the corner radius in Whites and Wu (Whites and

Wu 2002), this inverse homogenization method was more

versatile for optimizing the size and shape of the microstruc-

tures simultaneously. However, the homogenization method

using the curl operator was only applicable for 3D models

and the computational cost of the curl operator was much

more expensive compared with that of the gradient operator

(Zhou et al. 2010b).

The bounds of the effective magnetic permeability were

derived by Hashin and Shtrikman (Hashin and Shtrikman

1962) for isotropic multiphase composites. For two-phase

composites with permeabilities μ1<μ

2, the Hashin-

Shtrikman (HS) bounds in 2D and 3D are

−μ1+c

2μ1+1−c

μ1+μ2−1

≤μ≤−μ2+c

μ1+μ2+1−c

2μ2−1

(1)

μ1+3(1−c)μ1(μ2−μ1)

3μ1+c(μ2−μ1)≤μ≤μ2

+3cμ2(μ1−μ2)

3μ2+(1−c)(

μ1−μ2)(2)

respectively, where cdenotes the volume fraction of

phase 1. In fact, the HS bounds are also applicable to other

transport properties such as thermal expansion (Sigmund

and Torquato 1997) and conductivity (Zhou and Li 2008a).

However, the HS bounds can only provide a range of pos-

sible effective physical properties rather than a specific

Evolutionary topology optimization of periodic composites

periodic microstructure of composites attaining the bound

values.

This paper attempts to develop a BESO method for the

design of periodic microstructure of two-phase composites

with extremal effective permeability and permittivity. The

microstructure of the composite is represented by a PBC

discretized with finite elements. Starting from the homog-

enization theory using the gradient operator, the sensitivity

of the physical properties of composites is derived using the

adjoint method. With the relative rankings of the sensitiv-

ity numbers, elements within the PBC are swapped between

two phases and the resulting topology of the PBC gradually

evolves to an optimal solution. Towards the end of the paper,

a series of 2D and 3D examples are presented to demon-

strate the effectiveness of the proposed BESO method. It

is noted that the proposed BESO procedure is implemented

as a “post-processor” to the commercial software package

ANSYS.

2 Homogenization and finite element formulation

When a composite is composed of homogeneous and

isotropic materials, the electromagnetic fields (E,H)are

governed by Maxwell’s equations (Jin 2002). The consti-

tutive relations are the electric flux density D=εEand

the magnetic flux density B=μH. By assuming the time

dependence of eiωtwe have

iωεE=∇×H−J

iωμH=−∇×E(3)

where the constitutive parameters εand μare the permit-

tivity and permeability of the material, and Jis the source

of electric current. When the period of the microstructure is

quite small compared with the wavelength, the electromag-

netic fields (u=[E,H]) can be approximated by the formal

asymptotic expansion as

u(x,y)=u(x)+∇yv(x,y)+γw(x,y)+··· (4)

where yis the microscopic variable and x=γyis the macro-

scopic variable. According to the homogenization theory,

the first two terms provides an approximation to the electro-

magnetic fields within an acceptable error when γtends to

be zero. By assuming v(x,y)=u(x)χ1(y),χ2(y)and

inserting (4)into(3), the effective permittivity εHand the

effective permeability μHof PBC can be found according

to the convergence in the homogenization theory (Ouchetto

et al. 2006a)as

εH=1

|Y|YεI+∇yχ1(y)dy

μH=1

|Y|YμI+∇yχ2(y)dy(5)

where Iis the identity matrix and |Y|is the volume of PBC.

χ1and χ2are the solutions of the following equations.

YεI+∇yχ1(y)∇yϕ1(y)dy=0

YμI+∇yχ2(y)∇yϕ2(y)dy=0(6)

ϕ1and ϕ2are the arbitrary test periodic functions. Due to

the interchangeable relationship between permittivity and

permeability, the following formulations on permittivity are

also applicable to permeability. To solve (6), the whole PBC

is discretized into finite elements. The components such as

χ1and ϕ1are expressed as (Jin 2002)

χ1(y)=

e

n

i=1

Ne

i(y)χe

iand ϕ1(y)=

e

n

j=1

Ne

j(y)ϕe

j

(7)

where edenotes element and nis the total number of nodes

for each element. Thus, we can rewrite (6) in a matrix form

by considering the following three separate cases for 3D

models:

Case 1:

e

n

i,j=1

χe

iε∂Ne

i

∂x

∂Ne

j

∂x+∂Ne

i

∂y

∂Ne

j

∂y+∂Ne

i

∂z

∂Ne

j

∂z

=−

e

n

j=1

ε∂Ne

j

∂x(8a)

with the periodic boundary conditionχy0

1,y2,y3=

χy0

1+Y1,y2,y3=0 which is derived from

the direction method (Hassani 1996). Equa-

tion (8a) is equivalent to an electrostatic problem

imposed with a unit initial electric field in the x

direction.

Case 2:

e

n

i,j=1

χe

iε∂Ne

i

∂x

∂Ne

j

∂x+∂Ne

i

∂y

∂Ne

j

∂y+∂Ne

i

∂z

∂Ne

j

∂z

=−

e

M

j=1

ε∂Ne

j

∂y(8b)

with the periodic boundary conditionχy1,y0

2,y3=

χy1,y0

2+Y2,y3=0. The electric charge to

X. Huang et al.

be imposed in this case is a unit initial electric

field in the ydirection.

Case 3:

e

M

i,j=1

χe

iε∂Ne

i

∂x

∂Ne

j

∂x+∂Ne

i

∂y

∂Ne

j

∂y+∂Ne

i

∂z

∂Ne

j

∂z

=−

e

M

j=1

ε∂Ne

j

∂z(8c)

with the periodic boundary conditionχy1,y2,y0

3=

χy1,y2,y0

3+Y3=0. The electric charge to

be imposed in this case is a unit initial electric

field in the zdirection.

3 The optimization problem and sensitivity analysis

The effective permeability (μ=μI) and the effective per-

mittivity (ε=εI) of an isotropic composite depends on

its constituent phases such as two materials with different

permeabilities (μ1=μ1Iand μ2=μ2I) and different

permittivities (ε1=ε1Iand ε2=ε2I). To obtain a com-

posite with extremal effective permeability or permittivity,

an inverse homogenization procedure (e.g. Sigmund 1994)

can be applied for the following optimization problem

Maximize or Minimize: μor ε

Subject to: V∗

1−N

i=1

Vixi=0

xi=0or1

(9)

where Viis the volume of element iand V∗

1is the prescribed

volume for material 1. Nis the total number of elements in

PBC. The binary design variable xidenotes the (artificial)

relative density of element i. If the element is made of

material 1, xi=1. If the element is made of material 2,

xi=0.

The local material of an element within PBC can be

treated to be isotropic and its physical properties are

assumed to be a function of the element density. Zhou

and Li (2008a) and Pendry (2004) adopted the lower HS

bound as the interpolation scheme when optimizing the

material towards the upper HS bound and vice versa. For

simplicity, a SIMP model (Bendsøe and Sigmund 2003)

is adopted here. When the two-phases have well-ordered

properties such as ε1>ε

2and μ1>μ

2, the relationships

between the physical properties and the element density are

interpolated as

ε(xi)=ε1xp

i+ε21−xp

i

μ(xi)=μ1xp

i+μ21−xp

i(10)

where pis the penalty exponent. The penalty exponent is

artificially applied to make sure that the solution indeed

converges to a two-phase design without any intermediate

density. Instead of using different interpolation schemes

as adopted in Zhou and Li (2008a) and Pendry (2004),

we select the penalty exponent pto be 3 for maximiz-

ing the permeability or permittivity and 1 for minimizing

the permeability or permittivity. When the two-phases have

ill-ordered properties such as ε1>ε

2and μ1<μ

2, the rela-

tionships between the electromagnetic properties and the

element density are reformulated as

ε(xi)=ε1xp

i+ε21−xp

i

1

μ(xi)=1

μ1xp1

i+1

μ21−xp1

i(11)

For maximizing permeability or permittivity p=3and

p1=1 are selected; while for minimizing permeability or

permittivity p=1and p1=3 are selected.

To derive the sensitivity of the physical properties, the

adjoint method (Haug et al. 1986) can be used. Taking the

effective permittivity as an example, (5) can be rewritten by

adding a zero term as

εH=1

YY

εI+∇yχ1(y)dy

−Y

εI+∇yχ1(y)∇y˜

χ1(y)dy(12)

where the adjoint variable ˜

χ1(y)can be an arbitrary test

function. Thus, the derivative of εHcan be obtained by

rearrangement of terms as

∂εH

∂xi=1

YYI−∇y˜

χ1(y)∂ε

∂xiI+∇yχ1(y)dy

+YI−∇y˜

χ1(y)ε∇y∂χ1(y)

∂xidy(13)

To eliminating the term ∂χ1

∂xi, the solution of ˜

χ1can be

selected from the solution of the following adjoint equation

by letting ϕ1(y)=∂χ1(y)

∂xi

Y

εI−∇y˜

χ1(y)∇yϕ1(y)dy=0 (14)

This adjoint equation has the same form as (6) except for

the negative sign before the term ∇y˜

χ1(y). Therefore, the

adjoint variable can be found as ˜

χ1=−χ1where χ1

is solved by (8a–8c). Substituting it back into (13), the

sensitivity of the effective permittivity is simplified as

∂εH

∂xi=1

|Y|YI+∇yχ1(y)∂ε

∂xiI+∇yχ1(y)dy

(15a)

Evolutionary topology optimization of periodic composites

Similarly, the sensitivity of the effective permeability μH

can be expressed as

∂μH

∂xi=1

|Y|YI+∇yχ2(y)∂μ

∂xiI+∇yχ2(y)dy

(15b)

The sensitivity number used in BESO denotes the rela-

tive ranking of the elemental sensitivity. For example, the

sensitivity number for the ith element for maximizing the

effective permeability of 3D composites can be defined as

αi=∂μ11

∂xi+∂μ22

∂xi+∂μ33

∂xi(16)

And the sensitivity number for the ith element for mini-

mizing the effective permeability of 3D composites can be

defined as

αi=−∂μ11

∂xi+∂μ22

∂xi+∂μ33

∂xi(17)

In the BESO method, the design variable xiis restricted

to be either 0 or 1. Thus, the optimality criterion can be

stated as that the sensitivity numbers of elements with xi=1

should be higher than those of elements with xi=0(Huang

and Xie 2010a). Accordingly, we devise a simple scheme

for updating the design variable xi=0 for elements with

the lowest sensitivity numbers and xi=1 for elements with

the highest sensitivity numbers.

4 Numerical implementation and BESO procedure

To circumvent checkerboard pattern and mesh-dependency

problems, a numerical filter is applied to the elemental

sensitivity number. The mesh-independent filter, originally

developed for structural topology optimization (Bendsøe

and Sigmund 2003), is adopted in the current procedure.

The elemental sensitivity number is modified by the follow-

ing equation

ˆαi=

N

j=1

wrijαe

M

j=1

wrij(18)

where rij denotes the distance between the centre of ele-

ments iand j.w(rij)is the weight factor given as

wrij=rmin −rij for rij <rmin

0forrij ≥rmin (19)

where rmin is the filter radius.

Due to the discrete design variables used in the BESO

algorithm, Huang and Xie (2007,2010a) proposed that the

elemental sensitivity number should be further modified by

averaging with its historical information to improve the con-

vergence of the solution. Thus, the sensitivity number after

the first iteration is calculated by

˜αi=1

2ˆαi,k+ˆαi,k−1(20)

where kis the current iteration number. Then let ˆαi,k=

˜αiwhich will be used for the next iteration. Therefore the

modified sensitivity number takes into consideration of the

sensitivity information from the previous iterations.

The BESO procedure for the design of two-phase elec-

tromagnetic composites is outlined as follows:

Step 1: Define BESO parameters: the objective volume

of material 1, V∗

1, the evolutionary ratio ER (nor-

mally ER =2%) and the filter radius rmin .

Step 2: Discretize the PBC domain into a finite element

mesh and construct an initial design.

Step 3: Carry out finite element analysis for all Cases 1, 2

and 3 as described in Section 2.

Step 4: Calculate the elemental sensitivity numbers αi

according to (16)or(17).

Step 5: Filter sensitivity numbers using (18)andthen

average with its historical information using (20).

Step 6: Determine the target volume of material 1 for the

next design. When the current volume of mate-

rial 1, Vk, is larger than the objective volume V∗

1,

reduce the volume of material 1 so that

Vk+1=Vk(1−ER)(21a)

If the resulting Vk+1is less than V∗

1,thenVk+1

is set to be equal to V∗

1. Similarly, the volume of

material 1 should be increased when Vkis less than

the objective volume V∗

1so that

Vk+1=Vk(1+ER)(21b)

If the resulting Vk+1is larger than V∗

1,thenVk+1

issettobeequaltoV∗

1.

Step 7: Based the relative rankings of all elements accord-

ing to the sensitivity numbers, reset the design

variable xito 1 for elements with highest sensitiv-

ity numbers and to 0 (material 2) for elements with

lowest sensitivity numbers so that the resulting

volume of material 1 equals to Vk+1.

Step 8: Repeat Steps 3–7 until both the volume con-

straint is satisfied and the objective function is

convergent.

X. Huang et al.

5 Results and discussion

5.1 2D examples for maximizing or minimizing

the effective permeability

For the examples considered in this sub-section, it is

assumed that the microstructure is composed of two-phase

well-ordered materials with the permeabilities of μ1=10

and μ2=1 (air) and permittivities of ε1=10 and ε2=1

(air). Therefore, the following conclusions drawn on the

effective permeability are also applicable to the effective

permittivity. The square base cell of dimensions 100 ×100

is discretized into 100 ×100 4-node quadrilateral elements

(PLANE121 in ANSYS). To illustrate microstructures in

figures, red and blue colours denote the materials with per-

meability μ1and μ2, respectively. To start the optimization

procedure, three different initial designs shown in Fig. 1are

considered. The BESO parameters are: the evolution rate

ER =0.02, filter radius rmin =10 and the objective vol-

ume of material 1 equal to 50% of the total volume (i.e. the

volume fraction of material 1, V1f, is equal to 50%).

Figure 2shows the microstructures of the PBC and their

4×4 unit cells with the highest effective permeabilities

obtained from three different initial designs. It is seen that

the low permeability phase (material 2, blue) is separated

by the surrounding high permeability phase (material 1,

red) in order to maximize the effective permeability of

the composites. The resulting effective permeabilities for

these three microstructures are 4.20, 4.20 and 4.19. They

are all very close to the upper bound μupper =4.19 even

though the three microstructural topologies seem different.

In fact the topologies shown in Fig. 2aandcarealmost

identical if one looks at the images of the 4 ×4base

cells. The microstructures shown in Fig. 2are known as

Vigdergauz-type structures (Vigdergauz 1999), which have

been identified as having the maximum fluid permeabil-

ity (Guest and Prévost 2007) and the maximum thermal

conductivity (Zhou and Li 2008a). Figure 3shows the evo-

lution histories of the effective permeability and volume

fraction for the BESO process starting from initial design

1. It is noted that the solution converges after 44 iterations,

indicating that the proposed BESO algorithm is of high

computational efficiency. Numerical experiments indicated

that the used filter eliminates the possible small members

and checkerboard pattern of the final topologies so that the

resulting topologies are mesh-independent.

To minimize the effective permeability, the penalty expo-

nent p=1 is used. BESO starting from initial designs 1 and

2 have resulted in the same PBC microstructure and 4 ×4

unit cells as shown in Fig. 4a. Figure 4bshowsthePBC

microstructure and 4×4 unit cells resulting from BESO

starting from initial design 3. The corresponding effective

permeabilities of the microstructures are 2.48 for Fig. 4a

and 2.50 for Fig. 4b, respectively, both of which are close

to the HS lower bound μlower =2.38. The results in Figs. 3

and 4also show that the microstructures corresponding to

the maximum and minimum permeabilities simply swap the

high and low permeability materials topologically.

To further illustrate the effectiveness and robustness of

the proposed BESO method, we now try to maximize the

effective permeability by adopting the microstructure shown

in Fig. 4a as the initial design which is the worst guess

design as it has the minimum permeability. Figure 5shows

evolution histories of the effective permeability and the

topology. The high permeability phase (red) evolves grad-

ually from the isolated core to members connected with

boundaries and the low permeability phase (blue) becomes

gradually isolated. As a result, the effective permeability

gradually increases to the maximum value 4.22 which is

close to the HS upper bound, μupper =4.19 (the effective

permeability is a little higher than HS upper bound because

the final volume fraction of material 1 is about 50.3%).

(a) (b) (c)

Fig. 1 Initial designs: ainitial design 1 with four corner elements for material 2; binitial design 2 with eight elements at mid-sides of PBC for

material 2; cinitial design 3 with four elements at the centre of PBC for material 2

Evolutionary topology optimization of periodic composites

(a) (b) (c)

Fig. 2 Microstructures of the composites with maximum permeabilities: athe optimized base cell resulted from initial design 1 (above)and4×4

base cells (below); bthe optimized base cell resulted from initial design 2 (above)and4×4 base cells (below); cthe optimized base cell resulted

from initial design 3 (above)and4×4 base cells (below)

Similarly, we can minimize the effective permeability

starting from the initial design shown Fig. 2a which has

the maximum permeability (the worst guess design). As

showninFig.6, the high permeability phase (red) gradually

becomes isolated and surrounded by the low permeability

phase (blue) and the effective permeability finally converges

to the minimum value of 2.58 which is about 8% above the

HS lower bound μlower =2.38.

Fig. 3 Evolution histories of

the effective permeability and

the volume fraction of material

1 for maximizing permeability

starting from initial design 1

X. Huang et al.

Fig. 4 Microstructures of the

composites with minimum

permeabilities: athe optimized

base cell resulted from initial

designs 1 and 2 (above)and4×

4 base cells (below); bthe

optimized base cell resulted

from initial design 3 (above)

and 4 ×4 base cells (below)

(a) (b)

Fig. 5 Evolution histories of

the effective permeability and

the base cell for maximizing

permeability starting from the

worst guess design with the

lowest permeability

Evolutionary topology optimization of periodic composites

Fig. 6 Evolution histories of

the effective permeability and

the base cell for minimizing

permeability starting from the

worst guess design with the

largest permeability

(a) (b) (c)

Fig. 7 Microstructures of the composites for maximizing permeability

and permittivity simultaneously: athe optimized base cell (above)and

4×4 base cells (below)fork=0.7 (resulting in μ=2.70 and ε=

3.86); bthe optimized base cell (above)and4×4 base cells (below)

for k=1 (resulting in μ=ε=3.25); cthe optimized base cell (above)

and 4 ×4 base cells (below)fork=1.4 (resulting in μ=3.84 and ε=

2.74)

X. Huang et al.

It should be noted that the resulting topologies and prop-

erties depend on the physical properties and the volume frac-

tions of the constituent phases. As the material properties are

assumed to be with the permeability of μ1=10 and μ2=1and

permittivity of ε1=10 and ε2=1 in the above examples,

different topology and effective permeability and permittiv-

ity may be obtained when the actual properties of material

phases are used in the BESO program and FE analysis.

(a)

(b)

(c)

Fig. 8 Microstructures of composites with the maximum isotropic per-

meability: athe optimized base cell (left), base cell with material 1

only (centre)and2×2×2 base cells with material 1 only (right)for

V1f=80%; bthe optimized base cell (left), base cell with material 1

only (centre)and2×2×2 base cells with material 1 only (right)for

V1f=50%; cthe optimized base cell (left), base cell with material 1

only (centre)and2×2×2 base cells with material 1 only (right)for

V1f=20%

Evolutionary topology optimization of periodic composites

5.2 2D examples for maximizing permeability

and permittivity of ill-ordered composites

The composite is composed of two-phase ill-ordered mate-

rials with properties such as permeability μ1=10, μ2=1

and permittivity ε1=1, ε2=10. The optimization objec-

tive is to maximize both the effective permeability μand the

effective permittivity εsimultaneously. Thus, the optimiza-

tion problem is expressed by

Maximize: wμ +(1−w)ε

Subject to: μ=kε

V∗

1−N

i=1

Vixi=0

xi=0or1

(22)

where wis the weight factor to indicate the relative impor-

tance of permeability and permittivity. For example, when

w=1 the effective permeability attains its upper HS bound,

but the effective permittivity attains its lower HS bound due

to the competing properties of the two phases. kis a given

value which describes the relationship between the objective

effective permeability and the effective permittivity. Instead

of a random selection of the weight factor for multiobjec-

tives as adopted in literature (e.g. Torquato et al. 2002; Zhou

et al. 2010b), here we determine the weight factor wnumer-

ically solely from the additional constraint μ=kεas in

(Huang and Xie 2010b).

The microstructure shown in Fig. 1a is used as the initial

design for the BESO process for this multiobjective opti-

mization problem. Figure 7shows the final microstructures

of composites obtained for maximizing the effective perme-

ability and effective permittivity for k=0.7, 1.0 and 1.4

respectively. It can be seen that the microstructure for k=

0.7 has the isolated phase 1 (red) and continuous phase 2

(blue) so that the effective permeability, μ=2.70 is smaller

than the effective permittivity, ε=3.86. On the contrary, the

microstructure for k=1.4 has the continuous phase 1 (red)

and isolated phase 2 (blue) so that the effective permeabil-

ity, μ=3.84 is larger than the effective permittivity, ε=

2.74. In the microstructure for k=1.0, the two phases can

be exchanged with each other and therefore the microstruc-

ture has equal permeability and permittivity, μ=ε=3.25.

It should be noted that, due to the competing properties of

permeability and permittivity of the two phases, it is impos-

sible to obtain a 2D microstructure of which the effective

permeability and permittivity attain their HS upper bounds

simultaneously.

5.3 3D examples for maximizing the effective permeability

In the following 3D examples, the composite is composed

of two-phase well-order materials with permeability μ1=

10, μ2=1 and permittivity ε1=10, ε2=1. The cubic

base cell with dimensions of 50 ×50 ×50 is discretized

into 50 ×50 ×50 brick elements (SOLID122 in ANSYS).

BESO starts from an initial design which is fully occupied

by material 1 except for the eight corner elements which are

composed of material 2. Other BESO parameters are ER =

0.02 and rmin =5.

The objective in the examples considered in this sub-

section is to maximize the effective magnetic permeability

of the two phase composite. Figure 8shows the result-

ing microstructures for the volume constraint V1f=80%,

50% and 20% respectively where the red colour denotes

Fig. 9 Evolution histories of

the effective permeability and

the volume fraction of material

1 for maximizing permeability

with V1f=50%

X. Huang et al.

material 1 with high permeability and blue colour denotes

material 2 with low permeability. These microstructures are

typical 3D Vigdergauz-type structures (Vigdergauz 1999).

The corresponding effective permeabilities are 7.65, 4.73

and 2.17 respectively, which are very close to their corre-

sponding upper HS bound 7.63, 4.71 and 2.34. Figure 9

shows evolution histories of the volume fraction and

effective permeability for the volume constraint of V1f=

(a)

(b)

(c)

Fig. 10 Microstructures of composites for maximizing the effective

permeability and effective permittivity simultaneously: athe optimized

base cell (left), base cell with material 1 only (centre)and2×2×2

base cells with material 1 only (right)forV1f=70% and μ=3ε;b

the optimized base cell (left), base cell with material 1 only (centre)

and 2 ×2×2 base cells with material 1 only (right)forV1f=50%

and μ=ε;cthe optimized base cell (left), base cell with material 1

only (centre)and2×2×2 base cells with material 1 only (right)for

V1f=30% and μ=1

3ε

Evolutionary topology optimization of periodic composites

50%. It is seen that the effective permeability converges

smoothly to its final value after 44 iterations.

It is worth pointing out that swapping the two phases

in these microstructures leads to the microstructures with

the minimum permeability. Due to the well-ordered mate-

rials used, the conclusions for the effective permeability

in the above examples are also applicable to the effective

permittivity.

5.4 3D examples for maximizing permeability

and permittivity of ill-ordered composites

Here we consider a more challenging problem of designing

composites of two-phase materials with competing proper-

ties such as permeability μ1=10, μ2=1 and permittivity

ε1=1, ε2=10. The objective is to maximize both

the effective permeability μand the effective permittiv-

ity εsimultaneously as defined in (22). The optimization

problem is solved for the following three cases:

(a) V1f=70% and μ=3ε

(b) V1f=50% and μ=ε

(c) V1f=30% and μ=1

3ε

BESO starts from an initial design which is fully occupied

by material 1 except for the eight corner elements which are

composed of material 2.

Figure 10 displays the resulting microstructures for the

three cases where red colour denotes material 1 with high

permeability but low permittivity and blue colour denotes

material 2 with low permeability but high permittivity.

When the unit cells are repeated periodically in space, both

material phases are continuous so that both permeability and

permittivity could be maximized. For example, in case (a)

the obtained effective permeability (6.53) is very close to its

HS upper bound (6.58) but the effective permittivity (2.18)

is much higher than its lower bound (1.87) even though the

two properties compete with each other. It is noted that

the microstructure for V1f=50% shown in Fig. 10bis

very similar to the Schwarz primitive structure (Gó´zd´zand

Holyst 1996), which is known to have the maximal ther-

mal conductivity and electrical conductivity (Torquato et al.

2002). The effective permeability and permittivity in this

case are the same: μ=ε=4.33. This result agrees well with

that obtained by the curl-operator-based homogenization

method (Zhou et al. 2010b).

The effective permeability and permittivity of the three

resulting 3D microstructures approach closely to, but do not

reach, their upper bounds due to the competing properties.

Compared with 2D cases, the 3D space provides more free-

dom for distributing two materials. For example, in 2D cases

it is impossible to obtain an optimal microstructure without

an isolated phase, unlike the above 3D microstructures.

6Conclusion

This paper has developed a BESO approach to design-

ing microstructures of two-phase composites with extremal

electromagnetic permeability and permittivity. The effective

permeability and permittivity are homogenized within the

PBC and the optimization problem is defined by max-

imizing or minimizing these electromagnetic properties

subject to a volume fraction constraint. Based on the

adjoint method, the sensitivity of the objective function

is determined and used in BESO process for evolving

the topology of PBC. The proposed BESO algorithm can

effectively and robustly derive 2D and 3D topologies with

the obtained effective permeability and effective permit-

tivity to be close to the HS upper or lower bound. The

computational efficiency of the algorithm is also high as the

solution usually converges in less than 50 iterations. For ill-

order composites, the effective permeability and effective

permittivity can be maximized simultaneously even though

the properties of the two-phase materials compete with

each other. Some interesting topological patterns such as

Vigdergauz-type structure and Schwarz primitive structure

have been found which will be useful for the design of elec-

tromagnetic materials. The proposed BESO method can be

implemented easily as a post-processor to standard commer-

cial finite element analysis software packages, e.g. ANSYS

which has been used in this study.

Acknowledgments This research is supported by the Australian

Research Council under its Discovery Projects funding scheme (project

number DP1094403).

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