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This paper presents a bidirectional evolutionary structural optimization (BESO) method for designing periodic 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 electromagnetic properties separately or simultaneously. The sensitivity analysis of the objective function is conducted using the adjoint method. Based on the established sensitivity 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 resulting topologies are clear black-and-white solutions (with no 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.
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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
2, the Hashin-
Shtrikman (HS) bounds in 2D and 3D are
μ1+c
2μ1+1c
μ1+μ21
μ≤−μ2+c
μ1+μ2+1c
2μ21
(1)
μ1+3(1c)μ1(μ2μ1)
3μ1+c(μ2μ1)μμ2
+3cμ2(μ1μ2)
3μ2+(1c)(
μ1μ2)(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=∇×HJ
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
iεNe
i
x
Ne
j
x+Ne
i
y
Ne
j
y+Ne
i
z
Ne
j
z
=−
e
n
j=1
εNe
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
iεNe
i
x
Ne
j
x+Ne
i
y
Ne
j
y+Ne
i
z
Ne
j
z
=−
e
M
j=1
εNe
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
iεNe
i
x
Ne
j
x+Ne
i
y
Ne
j
y+Ne
i
z
Ne
j
z
=−
e
M
j=1
εNe
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
1N
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
2and μ1
2, the relationships
between the physical properties and the element density are
interpolated as
ε(xi)=ε1xp
i+ε21xp
i
μ(xi)=μ1xp
i+μ21xp
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
2and μ1
2, the rela-
tionships between the electromagnetic properties and the
element density are reformulated as
ε(xi)=ε1xp
i+ε21xp
i
1
μ(xi)=1
μ1xp1
i+1
μ21xp1
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)εyχ1(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 (8a8c). 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,kαi,k1(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(1ER)(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: +(1w)ε
Subject to: μ=kε
V
1N
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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IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion Appendix 4.1References
Book
Contents: Preface.- Introduction.- Basic Evolutionary Structural Optimization.- ESO for Multiple Load Cases and Multiple Support Environments.- Structures with Stiffness or Desplacement Contraints.- Frequency Optimization.- Optimization Against Buckling.- ESO for Pin- and Rigid-Jointed Frames.- ESO for Shape Optimization and the Reduction of Stress Concentrations.- ESO Computer Program Evolve97.- Author Index.- Subject Index.
Chapter
The evolutionary structural optimization (ESO) and basic ESO (BESO) processes and given various illustrative examples are described. Such processes are based on the concept of slowly removing inefficient materials from a structures so that the residual structure evolves towards the optimum. It is shown that the simple ESO and BESO algorithms are capable of solving a wide range of shape and topology optimization problems.
Article
We present a method for the generation of periodic embedded surfaces of nonpositive Gaussian curvature and multiply continuous phases. The structures are related to the local minima of the scalar order parameter Landau-Ginzburg Hamiltonian for microemulsions. In the bicontinuous structure the single surface separates the volume into two disjoint subvolumes. In some of our phases (multiply continuous) there is more than one periodic surface that disconnects the volume into three or more disjoint subvolumes. We show that some of these surfaces are triply periodic minimal surfaces. We have generated known minimal surfaces (e.g., Schwarz primitive P, diamond D, and Schoen-Luzatti gyroid G and many surfaces of high genus. We speculate that the structure of microemulsion can be related to the high-genus gyroid structures, since the high-genus surfaces were most easily generated in the phase diagram close to the microemulsion stability region. We study in detail the geometrical characteristics of these phases, such as genus per unit cell, surface area per unit volume, and volume fraction occupied by oil or water in such a structure. Our discovery calls for new experimental techniques, which could be used to discern between bicontinuous and multiply continuous structures. We observe that multiply continuous structures are most easily generated close to the water-oil coexistence region. © 1996 The American Physical Society.
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This paper presents a new approach to designing periodic microstructures of cellular materials. The method is based on the bidirectional evolutionary structural optimization (BESO) technique. The optimization problem is formulated as finding a micro-structural topology with the maximum bulk or shear modulus under a prescribed volume constraint. Using the homogenization theory and finite element analysis within a periodic base cell (PBC), elemental sensitivity numbers are established for gradually removing and adding elements in PBC. Numerical examples in 2D and 3D demonstrate the effectiveness of the proposed method for achieving convergent microstructures of cellular materials with maximum bulk or shear modulus. Some interesting topological patterns have been found for guiding the cellular material design.
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This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.
Article
This paper presents two computational models to design the periodic microstructure of cellular materials for optimal elastic properties. The material equivalent mechanical properties are obtained through a homogenization model. The two formulations address the problem of finding the optimal representative microstructural element for periodic media that maximizes either the weighted sum of the equivalent strain energy density for specified multiple macroscopic strain fields, or a linear combination of the equivalent mechanical properties. Constraints on material volume fraction and material symmetries are considered. The computational models are established using finite elements and mathematical programming techniques and tested in several numerical examples.