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Symmetry considerations for topology

design in the elastic inverse

homogenization problem

J.M. Podest´a1, C. M´endez1, S. Toro1, A.E. Huespe1,2,3∗

1CIMEC-UNL-CONICET, Predio Conicet “Dr Alberto Cassano”, CP 3000 Santa Fe, Argentina

2Centre Internacional de Metodes Numerics en Enyinyeria (CIMNE),Campus Nord UPC.

3E.T.S d’Enginyers de Camins, Canals i Ports, Technical University of Catalonia (Barcelona Tech)

Campus Nord UPC, M`odul C-1, c/ Jordi Girona 1-3, 08034, Barcelona, Spain

Keywords: metamaterial design; elastic symmetry; topology optimization; inverse

homogenization; tailored elastic properties.

Abstract

An analysis of the symmetries characterizing the micro-architecture topolo-

gies and the elastic material properties is performed. The goal is to elucidate a

systematic procedure that facilitates the design of elastic metamaterial with a

prescribed target elasticity tensor via inverse homogenization methodologies.

This systematic procedure, which is deﬁned through a set of rules, is based

on the relationship established between the elasticity tensor symmetries and

the symmetry displayed by the micro-architecture topology.

Following this procedure, it can be guaranteed that the designed com-

posites, with the attained micro-structures, have eﬀective elasticity tensors

that possess the same or higher symmetries than those shown by the target

elasticity tensors. Furthermore, the micro-architectures designed through this

technique display simple topologies.

Both properties that are supplied by the procedure, i.e., the accomplish-

ment of the required symmetry of the composite homogenized elasticity tensor

combined with the topology simplicity, are assessed through numerical simu-

lations of several micro-architecture design problems. They are designed by

formulating the inverse homogenization problem as a topology optimization

problem which is solved with two diﬀerent standard algorithms. The proposed

procedure and the conclusions here obtained do not depend on the algorithm

adopted for solving this problem.

∗Corresponding author. E-mail address: ahuespe@cimec.unl.edu.ar (A.E. Huespe).

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

1 Introduction

The goal of the inverse homogenization problem addressed in this work is the micro-

architecture design of a two-phase periodic elastic composite whose eﬀective elastic-

ity tensor Chis identical to a target elasticity tensor ˆ

C. In this context, a systematic

procedure for restricting the search space of variables that are meaningful in this

problem is described.

An approach to solve the mentioned inverse homogenization problem has been

reported in the past by formulating it as a topology optimization problem, see for

instance the landmark works by Sigmund ([1] and [2]) who has coined the name

inverse homogenization to this kind of problem. See also the reference book by

Bendsoe and Sigmund [3] and the bibliography cited therein related to this topic.

According to this approach, the topology optimization problem is formulated by

choosing a design domain, Ωµ, assumed as a base cell of the composite, which is also

used to compute its overall elastic properties. The algorithm solving the topology

optimization problem tests diﬀerent distributions of given hard and soft materials to

satisfy a proposed optimality condition, see for example the methodologies proposed

by Huang et al. [4], Amstutz et al. [5], Ferrer et al. [6], and [7] where the design of

graded micro-architectures has been addressed.

The selection of the cell Ωµis a decision that has to be taken by the designer

and is a particular aspect of the inverse problem that we want to highlight in this

work. There are several issues related to this choice. For example, Coelho et al. [8]

investigate the cell size inﬂuence on the designed topology.

Also, Diaz and Benard [9] mention that the shape of the design domain plays an

important role in widening the range of attainable micro-architectures with unusual

properties. This has been the case for developing new classes of extreme material

micro-structures by using rectangular unit cells and certain material distribution

symmetries, see the cases studied in [10]. The relationship between tensor sym-

metry and unit cell shape has already been discussed by Barbarosie et al. [11] and

Lukkassen et al. [12]. These authors exploit the symmetry of the material conﬁgura-

tion within the unit cell to get a less expensive computation of the eﬀective material

properties. But, in these cases, the morphological symmetries have not been used

to facilitate the search of a topology satisfying the inverse homogenization problem

requirement.

In this work, we make an exhaustive analysis of the information associated with

the target elasticity tensor. This information could be used to make easier the inverse

design procedure. In particular, we aim to study the elasticity tensor symmetry and

its connection with the base cell geometry and the material conﬁguration within

this cell. Then, based on this analysis, we introduce a set of guidelines, which

can be taken as a protocol to guarantee the accomplishment of the required elastic

symmetry.

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

To reach this goal, arguments taken from crystallography are employed. Typi-

cally, each crystal is associated with a point group through its Bravais lattice and its

motif. This association helps to determine the crystal system, as well as, the sym-

metry of any eﬀective material property, according to Neumann’s principle ([13]).

See the discussion about the connection between material and physical symmetries

reported by Zheng and Boehler [14].

In a complete parallelism, a Bravais lattice and a motif can be assigned to each

periodic material conﬁguration. Therefore, by following the same arguments to

that given in crystallography, the material micro-architecture can be classiﬁed by

possessing one of the seventeen plane groups with a given point group. Also, in

this case, the point group of the material conﬁguration geometry is connected with

the eﬀective elasticity tensor symmetry class. Hence, when this notion is applied

to the inverse homogenization problem, the plane group characterization of the

micro-structure could give a hint for distributing the material within the design

domain such that the eﬀective properties of the composite display a similar or higher

symmetry than that shown by the target elasticity tensor. This is the principal line

of argument taken by this work.

Another aspect related to the choice of the design domain has to be remarked.

The symmetry properties of the target elasticity tensor give useful information to

select the base cell shape between a family of cells. However, the slenderness of

the adopted cell, taken from one particular family, should be deﬁned by introducing

more information. We propose to build a database storing eﬀective elasticity tensors

of composites with a set of representative and simple topologies. Then, the stored

information in this database is used to choose the cell slenderness, as well as, to

provide an initial conﬁguration to solve the topology optimization algorithm.

A brief description of this paper is given as follows. Initially, we analyze the

point and plane groups properties of diﬀerent crystal systems in two-dimensions, as

well as the point groups of the Bravais lattices associated with these crystals. We

perform an exhaustive analysis of the full set of possible symmetries that can display

a periodic crystal.

Next, we deﬁne a criterion to build the database. Based on this database and

the symmetry notions previously discussed, in Section 4 we present the procedure

through a series of rules that facilitates the inverse homogenization process.

The inﬂuence of these rules on the topology design process are assessed in Section

5 by means of a set of numerical simulations. We solve some representative inverse

homogenization problems with techniques that follow the proposed rules and the

solutions are compared with those obtained using conventional approaches that do

not follow these rules.

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

1.1 Inverse material design as a topology optimization prob-

lem

Material design via inverse homogenization refers to the problem of ﬁnding the

micro-architecture conﬁguration of a composite whose eﬀective elasticity tensor is

identical to a target elasticity tensor. This problem involves two characteristic scale

lengths; the macro-scale length, `, which is of the same order of magnitude as that

of the structure size, and the micro-scale length, `µ, which is of the same order

of magnitude as that of the material micro-architecture characteristic length. We

assume that `µ`. The eﬀective elasticity tensor is deﬁned at the macro-scale

level, such as sketched in Figure 1, and the material design is performed at the scale

`µ.

Material design via inverse homogenization has been formulated as a topology

optimization problem in a given design domain and we follow a similar formulation

in this paper.

Let us consider a structure whose material is a periodic composite constituted by

two isotropic elastic phases M1and M2. We take a basic micro-cell of this material

identiﬁed by Ωµ. In this micro-cell, phases M1and M2occupy the domains Ω1

µand

Ω2

µ, respectively, see Figure 1.

The characteristic function χ(y) in Ωµidentiﬁes the positions where the phase

M1is placed and is deﬁned by:

χ(y) = 0∀y∈Ω2

µ

1∀y∈Ω1

µ

.(1)

Evidently, the homogenized elasticity tensor of the composite, Ch, depends on the

geometrical conﬁguration of the phases M1and M2in Ωµ. This dependence is made

explicit by introducing the notation Ch(χ). This tensor can be evaluated in Ωµby

enforcing periodic boundary conditions in displacements ﬂuctuations. Then, stan-

dard computational techniques based on ﬁnite elements ([15], [16]) or Fast Fourier

Transform ([17]) can be used to get this goal.

Next, we formulate the micro-architecture inverse design problem as a topology

optimization problem expressed as follows: given the design domain Ωµand the

target eﬀective elasticity tensor ˆ

C, ﬁnd the characteristic function χsatisfying:

min

χ

1

|Ωµ|ZΩµ

χ dΩ

such that: kCh(χ)−ˆ

Ck= 0 .

(2)

The cost function represents the stiﬀ phase volume fraction. In particular, con-

sidering that the soft phase is void, the problem (2) identiﬁes a minimum weight

problem.

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

ℓ

ℓ >> ℓ

Scale of the material

structure

Macro-structure scale

x

y

s=C:e

h

W

ℓ

Wm

2

Phase M (c=0)

2 2

(domain )

Phase M (c=1)

11

(domain )

W1

m

e;s

m

m

Micro-cell W

m

Figure 1: The material design is carried out by solving an inverse homogenization technique

formulated as a topology optimization problem which involves two characteristic scale lengths.

The micro-cell Ωµis used as the design domain for the topology optimization problem. The

symbols σand εrepresent the macro-stress and macro-strain tensors, respectively.

There are several implicit variables in the problem (2). One of particular interest

here is the design domain, Ωµ, where this problem is posed. The shape of this domain

is a variable that should be ﬁxed in advance; it results from a decision taken by the

designer. Also, the enforcing of periodic boundary conditions along pre-established

directions to ﬁnd the eﬀective properties of the composite, Ch, is a decision taken

in advance by arguing that the micro-architecture is periodic along these directions.

Due to this arguments, the full material architecture must result from a spatial

replica, by tessellation, of the cell Ωµ.

We remark that both decisions play an important role to govern the complexity

of the attained micro-architecture topology.

2 Eﬀective elastic symmetry inherited from the

micro-architecture conﬁguration

Eﬀective elastic properties of composites constituted by two isotropic phases show

diﬀerent classes of symmetries. These symmetries are a result of the micro-structure

geometry, due to the spatial distribution of phases.

In this Section, by resorting to a complete parallelism with crystallography, we

analyze the connection between the overall elastic properties and the micro-structure

geometry, from the common perspective involving the symmetry properties.

First, we categorize the periodic material micro-architectures according to their

point groups. To reach this goal, we identify the underlying Bravais lattice, the motif

and the crystal system of the material conﬁguration. This classiﬁcation involves only

the geometrical features of the composite. See the book of S´olyom [18] for additional

information about this topic. Then, the elastic symmetry classes of these composites

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

are brieﬂy discussed and presented. Finally, we close this Section by discussing the

connection between physical and material conﬁguration symmetries stated in terms

of Neumann’s principle.

Point group symmetry

An isometric transformation imposed on the material conﬁguration, which leaves

invariant its spatial distribution, is a symmetry operation. The composition of suc-

cessive symmetry operations is also a symmetry operation for that material conﬁg-

uration. So, from an algebraic point of view, this set of transformations constitutes

a group. One simple operation of this group is a symmetry element.

First, we consider a restricted set of symmetry transformations which leaves a

point of the material ﬁxed. In this case, the symmetry elements are the rotations

around a ﬁxed axis (orthogonal to the plane of analysis), reﬂections across straight

lines intersecting the ﬁxed point and inversion in the ﬁxed point. The last one can

be ignored as it corresponds to a rotation through an angle of π[rad]. The element

of the group denoted nis a rotation through an angle 2π/n [rad] and the element

denoted mis a reﬂection through a plane.

Now, consider the groups that contain mirror lines and rotations. Any line

obtained from a mirror line via a rotation with the angle 2π/n [rad], around the

n−fold axis, is also a mirror line. Then:

- for nodd, the angular separation of the nmirror lines obtained in this way

is π/n [rad]. Thus, the group has 2nsymmetry elements, nrotations and n

reﬂection planes, and is denoted by nm;

- for neven, rotations of a mirror line will yield only n/2 diﬀerent mirror lines

whose angular separation is 2π/n [rad]. Nevertheless, in this case, there must

exist another set of mirror lines, i.e., the angle bisectors of the previously

obtained lines. This is so because the composition of a reﬂection, in a mirror

line, and a rotation through 2π/n [rad] is equivalent to a reﬂection in a mirror

line that makes an angle π/n with the original mirror line. Therefore, there

are two independent sets of mirror lines. This is expressed by the notation

nmm of such groups.

Considering the material distribution in the space, we can identify all the iso-

metric transformation with ﬁxed point leaving invariant this distribution. The set

of all the symmetry elements of a given material is called its point group.

2.1 Materials with periodic micro-architecture

Let us consider materials having periodic micro-architectures with a given trans-

lational symmetry. The micro-architecture is invariant under discrete translations

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

along two directions deﬁned by the non-proportional primitive vectors a1and a2.

The translation of the material along directions that result from integer multiples

of a1and a2takes the material into itself. Therefore, the characteristic function χi,

for a given i-th phase, satisﬁes

χi(x+t) = χi(x) ; where t=ω1a1+ω2a2(3)

and the scaling factors ω1and ω2are two arbitrary integers.

2.1.1 Underlying Bravais lattices of a periodic pattern

Every periodic conﬁguration of material possesses an underlying Bravais lattice and

a motif. The procedure to identify the underlying lattice of a periodic structure and

its motif is indicated in the artistic wallpapers reproduced in Figure 2. The lattice

and the primitive vectors are found by identifying the equivalent set of points of

the material periodic array. For example, equivalent sets of points are the eyes of

the ﬁshes in Figure 2-a and the red points in Figure 2-b. Once the Bravais lattice

is characterized, the unit cells can be easily recognized. The motif is the material

conﬁguration pattern within a unit cell.

a1

a2

Bravais lattice: Oblique

(a) (b)

Wigner-Seitz cell

Wigner-Seitz cell

Motif

p2mm

Symmetry

lines

Motif

p2

a1

a2

Bravais lattice: Rectangular centered

Figure 2: Identiﬁcation of the underlying Bravais lattices, primitive vectors, Weigner-Seitz

(Voronoi) cells and plane groups of artistic wallpapers by: (a) Escher; b) Geometrical draw.

According to the angle that the primitive vectors a1and a2form and the ratio

between their magnitudes, only ﬁve diﬀerent types of Bravais lattices can be identi-

ﬁed. They are depicted in Figure 3 and are called: Hexagonal, Square, Rectangular

primitive, Rectangular centered and Oblique.

Lattice point group

Due to the characteristic translational symmetry of Bravais lattices, and consid-

ering the crystallographic restriction theorem, there can only be a ﬁnite number of

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

point groups for the ﬁve types of lattices. They are n-fold angles of symmetries,

where n∈ {2,3,4,6}, with one or two systems of symmetry lines. Then, the only

possible point groups that Bravais lattices can have are the following four types: 2,

2mm, 4mm, 6mm. In this notation1, the number identiﬁes the n-fold angle of ro-

tational symmetry and mand mm means one or two mirror line systems. A mirror

line system is the set of reﬂection lines which are obtained by the n-fold rotation of

one mirror line.

a

a

1

2

a = a

<p/2

12a = a

=p/2

12

a

a

a

a

a

a

1

1

1

2

2

a

1

aa1

a

a = a

=p/2

12

a = a

=3p/2

12

2

2

Voronoi

cell

Voronoi

cell

Voronoi

cell

Voronoi

cell

Voronoi

cell

2(a a )= a

11

2

2

2

Unit-cell

Unit-cell

Unit-cell

Unit-cell

<p/2

D

Oblique

Rectangular

Centered

Rectangular

Primitive

Hexagonal

Square

Figure 3: The ﬁve Bravais lattice types in the plane.

All the symmetry elements with a ﬁxed point of the ﬁve types of plane lattices are

shown in Figure 4. The ﬁxed point can be any one of the lattice atoms. Therefore,

the respective point group of each lattice can also be identiﬁed. They are shown in

the same Figure.

The lattice system is formed by collecting the lattice types sharing similar point

groups. So, due to the fact that the Rectangular primitive and the Rectangular

centered lattices share the same point group, 2mm, they are grouped into one lattice

system identiﬁed as the Rectangular lattice systems. Therefore, the four lattice

systems are Oblique, Rectangular, Square and Hexagonal.

A Bravais lattice is the tessellation of a repetitive pattern, i.e., the juxtaposition

of a given domain by the translation along the primitive directions. The domains

with the repetitive patterns which have the smallest areas are the unit cells of the

lattice, such as shown in Figure 4. In particular, we focus on the Weigner-Seitz2(or

Voronoi) unit-cell of the Bravais lattice.

1We use the International or Hermann-Mauguin notation, see [19].

2The Weigner-Seitz cell of a lattice is the domain centered in a lattice atom and comprising all

the spatial points that are closer to the central atom than to any other atom of the lattice.

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

Oblique Rectangular Hexagonal

Square

n

n=2

Point group: 2

n=2 ;

Point group: 2mm

n=6

Point group 6mm

n=2 ;

Point group: 2mm

n=4 ;

Point group: 4mm

Lattice systems

Rectangular

Centered

Rectangular

Primitive

n

n

n

n

Figure 4: Rotational and reﬂection symmetries of Bravais lattices. Mirror planes (reﬂection

symmetries) are depicted with two parallel lines; rotational symmetries are identiﬁed with the

rotation angle 2π/n around an axis perpendicular to the plane. Parameters nand the lattice point

groups are depicted. Voronoi cells have the same point groups as that shown by the associated

lattices, i.e., they preserve the symmetries of the lattices. According to the lattice point group, there

are four Bravais lattice systems: Oblique (2), Rectangular (2mm), Square (4mm) and Hexagonal

(6mm). The Rectangular system has two subsystems: centered and primitive.

Remark: the Weigner-Seitz cell and the associated lattice have the same point

group. This statement is graphically observed in Figure 4.

Parametrization of Bravais lattices

All Bravais lattices can be characterized through the following two parameters:

ω=ka2k/ka1k; (4)

ς= arccos [(a2·a1)/(ka2kka1k)] .(5)

Each pair of values ω, ς deﬁnes a Bravais lattice. It can be easily proven that the

range of parameters ω, ς identifying the full set of Bravais lattices is limited to

the points displayed in gray in Figure 5-a. We refer to this region as the reduced

domain of parameters. In fact, due to the symmetry properties of these lattices, the

points that are outside this reduced domain characterize lattices which can always

be parametrized with points in the reduced domain.

Figure 5-b depicts the Voronoi cells of the Bravais lattices in several points on

the reduced domain (ω, ς ). Note that Oblique Bravais lattices are only represented

by points in the interior of the reduced domain, while Rectangular, Square and

Hexagonal lattices are only represented by points on its boundary.

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

0

60

70

90

0.250.1 0.5 0.75

80

S

H

[deg]

Rp

O

0.

60

70

90

0.250.1 0.5 0.75

80

[deg]

1.

W’

W

0.5

0.517

W=(0.5,30) W'=(0.967,75)

(b)(a)

Rp Rp Rp

Rc

Rc

Rc

Rc

O

O

O

O

O

O

Rc

Rc

W’

W

Figure 5: Space of parameters (ω, ς) characterizing the Bravais lattices. a) The gray

region is the reduced domain. Points outside of the reduced domain (W) identify

lattices that can be parameterized with equivalent points in the reduced domain

(W’). b) Voronoi cells of lattices characterized by diﬀerent parameters (ω, ς): O

(Oblique), Rc (Rectangular centered), Rp (Rectangular primitive), S (Square) and

H (Hexagonal).

2.1.2 Plane groups

The material conﬁguration, or crystal motif, can be deﬁned by identifying a unit cell.

When the motif is taken into account, an additional symmetry element, the glide

reﬂection, has to be contemplated. It consists of a geometrical reﬂection, through

a mirror line, followed by a translation, parallel to the same line. The translation

distance is half of the periodicity distance, or unit cell size, parallel to the mirror

line.

A plane group is the set of symmetry elements, including glide reﬂection, which

identiﬁes a wallpaper3. Here, we use the word wallpaper to denote a speciﬁc con-

ﬁguration of the material distribution of a periodic composite. Therefore, every

wallpaper has an underlying Bravais Lattice and a motif that deﬁnes a plane group.

Similar to Bravais Lattice point groups, there are only a ﬁnite number of plane

groups identifying all possible wallpapers. After introducing the motif and the glide

reﬂection symmetries, the number of plane groups is seventeen. All of them are

shown in Figure 6, where we depict the Voronoi cells of the underlying lattices

with diﬀerent motifs and the symmetry elements characterizing each plane group:

reﬂection symmetry lines, the n-fold angle of rotational symmetry and the glide

lines.

These plane groups are denoted by the letters por cwhich indicates that the

underlying lattice is either primitive or centered. The existence of one or two glide

reﬂection lines are identiﬁed with the letters gand gg, respectively. And, similar

3The words wallpaper and plane crystal have an identical meaning in this work.

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

to the notation of point groups, the number identiﬁes the n-fold angle of rotational

symmetry, and the letters mand mm indicates if there are one or two mirror line

systems, respectively.

p1

p2

pm

pg

cm

p2mm

p2mg

p2gg

c2mm

p4

p4mm

p4gm

p3

p3m1

p31m

p6

p6mm

3

3

3

4

4

4

Oblique Rectangular Square

Hexagonal

Six-fold

symmetry angle

4Four-fold

symmetry angle

Two-fold

symmetry angle

Reﬂection

symmetry line

glide

reﬂection line

3Three-fold

symmetry angle

6

6

6

2

2

2

2

2

2

Figure 6: Voroni cells of crystals with the seventeen plane groups in 2D. The symmetry elements

of each crystal, i.e. the Voronoi cell and the motif, are shown. Gray and white colors in the

wallpapers represent the distribution of the composite phases.

The Bravais lattice symmetry of the wallpaper may be broken when the motif

has a lower symmetry than that of the underlying lattice. Therefore, the wallpaper

symmetry group is a subgroup of that characterizing the underlying lattice. Ac-

cording to this property, we next establish the relationship that assigns one point

group to each one of the seventeen plane groups that represent arbitrary wallpapers

and motifs. This relationship is shown in Table 2. Each plane group in the fourth

column is associated (in the same line) with one point group in the third column.

Note that wallpapers with plane group pg has a point group m. This relationship

comes from the fact that performing a mirror symmetry of a crystal with symmetry

pg results in the same crystal with a translation of half of the unit cell size parallel

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Accepted in Journal of the Mechanics and Physics of Solids, March, 2019

to the glide line. Therefore, by considering an inﬁnite crystal, both crystals, the

original one and the reﬂected and translated one, are indistinguishable when the

eﬀective material properties are evaluated. A similar conclusion can be given to

wallpapers with plane group p2mg,p2gg and p4gm by changing the gsymmetry

operation by a mirror symmetry operation m.

From these comments, there are only ten point groups characterizing the full set

of wallpapers. They are depicted in Table 2, third column, and are : 1, 2, m, 2mm,

4, 4mm, 3, 3mm, 6 and 6mm.

The system of crystals: the crystals are next classiﬁed by their point group sym-

metry. We identify all crystals which have a given point group. This identiﬁcation is

shown in Table 2, see also Landwehr [20]. Note that each crystal is composed of an

underlying lattice, shown in column 5, and a motif having the symmetry given by

the plane group in column 4. For example, crystals which lattices are Rectangular

or Square and their plane groups are p2mm,p2mg,p2gg and c2mm have a point

group 2mm. Also, observe that a crystal with hexagonal lattice and plane group p1

has the point group 1.

Once this relationship has been established, the systems of crystals can be deﬁned

as follows. A crystal system is the collection of crystals sharing the same point groups

with an identical set of compatible lattices. Using this criterion, we can classify the

crystals into four systems: Oblique, Rectangular, Square and Hexagonal4.

Again, as an example, the crystals having point groups mand 2mm are compat-

ible with the same lattices, and therefore, they belong to the Rectangular crystal

system.

2.2 Elasticity tensor structures according with their sym-

metries

The symmetry of the overall elastic properties of heterogeneous materials are well

established and are classiﬁed according to their point group, such as described in

the work of Ting [21], see also [13].

The methodology followed by Ting to classify the elasticity tensor structures with

comparable symmetries, a very conventional procedure in solid mechanics, consists

of applying isometric transformations, compatible with a given point group, to the

elasticity tensor. Then, the elastic tensor coeﬃcients have to satisfy the necessary

invariance conditions derived from these transformations.

In plane elasticity, this methodology determines four diﬀerent elastic symmetry

classes. They are denoted O(2) for Isotropic, D4for Tetragonal, D2for Orthotropic

4In plane crystals, there is a one-to-one relationship between the lattice systems and the crystal

systems. However, when the same classiﬁcation is extended to three-dimensions, this one-to-one

relationship is not preserved.

12

Elasticity tensors CN

O(2) (2)

c1c20

c2c10

0 0 (c1−c2)

D4(4)

c1c20

c2c10

0 0 c3

D2(5)

c1c20

c2c40

0 0 c3

Z2(6)

c1c2c5

c2c4−c5

c5−c5c3

Table 1: a) Symmetry classes in plane elasticity. Structure of the elasticity tensors, CN,

expressed in the Normal coordinates (material axis): O(2) for Isotropic, D4for Tetragonal, D2

for Orthotropic and Z2for fully Anisotropic materials. The total numbers of elastic coeﬃcients

c1, c2, ... deﬁning the elasticity tensors are shown in parenthesis for each symmetry class. The

rotation angle transforming Cinto CNis also considered as an additional coeﬃcient of the elasticity

tensor.

and Z2for fully Anisotropic materials. From higher to lower symmetry classes, they

are: O(2) ⊂D4⊂D2⊂Z2. Furthermore, the point group with lower symmetry

that is compatible with each one of these classes is: 3 for O(2), 4 for D4,mfor D2,

and 1 for Z2, respectively. This association between the four elastic classes and the

point groups is shown in Table 2 ([21]).

The criterion followed to ﬁnd the four classes of elastic symmetries implicitly

introduces a coordinate system which is aligned with the symmetry planes of the

point group. This coordinate system is called here as the Normal coordinate system.

We use the notation CNto indicate that the elasticity tensor is described in this

coordinate system.

The Table 1 displays the elasticity tensor coeﬃcients of CNfor the four elastic

symmetry classes5. The numbers in parenthesis indicate the quantity of independent

elastic coeﬃcients that deﬁne the elasticity tensors.

Considering an arbitrary elasticity tensor Cdescribed in the Cartesian coor-

dinate system, Auﬀray et al. [22] describe an algorithm to determine its elastic

symmetry class as well as to computes the rotation angle transforming Cinto its

normal form CN. See also the Appendix in the paper by Podest´a et al. [23].

2.3 Neumann’s principle

The connection existing between the symmetry elements of the micro-architecture

geometry and the symmetry elements of the overall elasticity tensor can be stated

with a fundamental postulate of the crystal physics known as Neumann’s principle.

5Here, the elasticity tensor CNis described as a matrix in Kelvin’s notation. The stress and

strain vectors in Kelvin’s notations are given by: [σss ;σnn;√2σsn ]Tand [εss;εnn;√2εsn]T, respec-

tively, where sand nare the Natural coordinates. Therefore, the components of the matrix CN

are the corresponding elastic coeﬃcients that are deﬁned in accordance with the notation of these

vectors.

13

Elastic Crystal Point Plane Compatible Bravais lattice and Wigner-Seitz unit cell

symmetry system group group Oblique Rectangular Rectangular Square Hexa

class primitive centered

Z2 oblique 1 p1

2 p2

D2 rectangular

m

pm

pg

cm

2mm

p2mm

p2mg

p2gg

c2mm

D4 square

4 p4

4mm p4mm

p4gm

O(2) hexagonal

3 p3

3m p3m1

p31m

6 p6

6mm p6mm

Table 2: Elastic symmetry classes (column 1) are determined by the point group

(column 3) of the crystal system (column 2). Also, each one of the seventeen plane

groups (column 4), depicted in Tables 3 and 4, corresponds to a point group. Ad-

ditionally, each plane group is compatible with several Bravais lattices (column 5).

The symmetries (number of symmetry elements) increase from top to bottom. The

higher symmetry is in the bottom of the Table.

14

This principle states that “the symmetry elements of any eﬀective physical prop-

erty (optical, magnetic, thermal, mechanical properties) of a crystal must include the

symmetry elements of the point group of the crystal”, see Nye [13].

According to this principle, the relation between the point group and the elastic

symmetry class is recognized in Table 2 between the ﬁrst and third column. Note

that, in the elastic case we are considering, identical lines of the Table relates an

elastic class (column 1) with a crystal system (column 2).

This principle is conﬁrmed by the computed results presented in Tables 3 and

4. In these Tables, the eﬀective elasticity tensors of seventeen composites depicted

in column 1 are shown. The micro-architectures of these composites show diﬀerent

plane groups. The Voronoi cells of these micro-architectures are shown in Figure 6

and their plane groups are displayed in column 2 of the Tables 3 and 4. The related

point group to each plane group is presented in column 3 of the same Tables.

The computed eﬀective elasticity tensors in Normal coordinates6are denoted by

Ch

Nand are presented in Column 5 of Tables 3 and 4. In the same column, the

angle between Cartesian and Normal coordinates are only shown for those cases

where both coordinate systems are not coincident. According to the expressions

of Ch

Nin the Tables, and comparing with Table 1, we can identify their elastic

symmetry class which are depicted in column 4. Also, and according to the same

characterization of the elastic symmetry, the point groups of the eﬀective elasticity

tensors, Ch

N, are compatible with the symmetry anticipated by Neumann’s principle

and the information given in Table 2. Finally, it is noted that the Normal axes

coincide with the Cartesian ones when one of the symmetry axes of the plane group

is parallel to one Cartesian axis.

Remark: Neumann’s principle does not state that the symmetry elements of a

physical property must be the same as the corresponding ones to the crystal point

group. Very often, the physical properties may possess higher symmetry than the

crystal point group. For example, some periodic materials with square unit-cells

and motifs having a plane group p1, with a point group 1, may have an isotropic

eﬀective elasticity. Examples of this feature are the micro-structures displayed in

Figure 1 of reference [24] having a plane group p2mg, with a point group 2mm, and

an isotropic eﬀective elasticity tensor. See also Figure 11 below, case Spm, instance

2.

As a remarkable consequence of this principle, a procedure based on the following

premise is developed: given the number of parameters characterizing the elastic

response, or similarly, the symmetry class of the target elasticity tensor, we design

6The composites are constituted by a stiﬀ material, displayed in gray color, and void displayed

in white color. The elasticity tensor is normalized with an adequate Young’s modulus of the stiﬀ

material to attain a coeﬃcient C11 with value 1 in all cases. The Poisson’s ratio of the stiﬀ material

is 0.3.

15

the micro-structure by appealing to Voronoi cells and plane groups guaranteeing the

attainment of eﬀective properties having the same or higher symmetries to those of

the target tensor.

3 Database of elasticity tensors

An additional ingredient introduced in the present inverse design methodology is

the use of a database containing homogenized elasticity tensors. This database is

built oﬀ-line by sampling a spectrum of periodic composite materials.

The homogenized elasticity tensors stored in the database are computed from

composites with a stiﬀ phase and void. Their micro-structures are identiﬁed with

the Bravais lattices parameters (ω, ς) which are deﬁned by expressions (4) and (5).

The motifs are determined by a set of hard material bars. These bars are distributed

into the Voronoi cell of the lattice by following two patterns P, denoted PAand PB.

The pattern PAconsists of a set of bars placed on the boundaries of the cell

such as shown in Figure 7-a. Pattern PBis similar to PAbut with a re-entrant

architecture of bars in the vertices of the Voronoi cells, such as shown in Figure 7-b.

The geometry of pattern PBis deﬁned with the relative position of points V1, V2

and the size rrespect to the cell size. Patterns PAand PBare arbitrarily chosen.

However, pattern PAis a simple topology with a lattice-type micro-architecture.

This type of micro-architecture has attracted enormous interest in last years, see for

example [25] and [26]. Also, the pattern PBis a lattice-type micro-architecture that

is introduced to obtain elastic properties with negative Poisson’s ratios ([27], [28]).

The bar thickness ein both patterns PAand PBdeﬁnes the volume fraction fof

the composite.

In this way, the database contains homogenized elasticity tensors Ch

db of materials

whose micro-structures are deﬁned by four parameters ω,ς,Pand f. Therefore, each

element of the database is identiﬁed through its dependence with these parameters,

Ch

db(ω, ς, P, f ). The range of parameters (ω, ς, P, f ) that is utilized to compute the

database are the following:

•The ﬁrst two parameters, ωand ς, are varied such that the parametrized

reduced domain of Figure 5-a is swept by deﬁning a regular mesh of 6283

sampling points which cover the full reduced space.

•Several volume fractions fvarying from f= 0.005 to f= 0.8. The volume

fraction is given in term of the thickness eof the bars.

•The variable P takes the values, PAand PB.

The stiﬀ material properties utilized to build the database are deﬁned with a nor-

malized Young’s modulus E= 1.GP a and Poisson’s ratio ν= 0.3.

16

Assemblage Plane Point Elastic Sym- Ch

Nand angle

group group metry class

p1 1 Z2

1.0000 0.2604 −0.0027

0.2604 0.8797 0.0027

−0.0027 0.0027 0.6126

ang = 10.0835◦

p2 2 Z2

1.0000 0.2269 0.0187

0.2269 0.7257 −0.0187

0.0187 −0.0187 0.5638

ang = 9.0361◦

pm m D2

1.0000 0.2468 0.0000

0.2468 0.9530 0.0000

0.0000 0.0000 0.4584

pg m D2

1.0000 0.2349 0.0000

0.2349 0.7311 0.0000

0.0000 0.0000 0.5930

cm m D2

1.0000 0.2450 0.0000

0.2450 0.9495 0.0000

0.0000 0.0000 0.4545

p2mm 2mm D2

1.0000 0.1929 0.0000

0.1929 0.7839 0.0000

0.0000 0.0000 0.3692

p2mg 2mm D2

1.0000 0.2166 0.0000

0.2166 0.8250 0.0000

0.0000 0.0000 0.4226

p2gg 2mm D2

1.0000 0.2248 0.0000

0.2248 0.5645 0.0000

0.0000 0.0000 0.5736

c2mm 2mm D2

1.0000 0.1893 0.0000

0.1893 0.7667 0.0000

0.0000 0.0000 0.3653

Table 3: Column 1 depicts the microstructure of the composites resulting from the

assemblage of the Voronoi cells in Figure 6. These assemblages show the seven-

teen types of plane groups. Columns 2 and 3 depict the point and plane groups,

respectively, of the assemblages. The column 5 presents the homogenized elasticity

tensors in Natural coordinates, Ch

N, and the rotation angle from Cartesian to Nat-

ural coordinates, of the composites in column 1. These homogenized tensors have

the symmetry classes depicted in column 4.

17

Assemblage Plane Point Elastic Sym- Ch

N

group group metry class and angle

p4 4 D4

1.0000 0.2378 0.0000

0.2378 1.0000 0.0000

0.0000 0.0000 0.6594

ang =−80.0970◦

p4mm 4mm D4

1.0000 0.2803 0.0000

0.2803 1.0000 0.0000

0.0000 0.0000 0.6515

p4gm 4mm D4

1.0000 0.2444 0.0000

0.2444 1.0000 0.0000

0.0000 0.0000 0.6972

p3 3 O(2)

1.0000 0.2784 0.0000

0.2784 1.0000 0.0000

0.0000 0.0000 0.7216

p3m1 3m O(2)

1.0000 0.2991 0.0000

0.2991 1.0000 0.0000

0.0000 0.0000 0.7009

p31m 3m O(2)

1.0000 0.3016 0.0000

0.3016 1.0000 0.0000

0.0000 0.0000 0.6984

p6 6 O(2)

1.0000 0.2068 0.0000

0.2068 1.0000 0.0000

0.0000 0.0000 0.7932

p6mm 6mm O(2)

1.0000 0.3401 0.0000

0.3401 1.0000 0.0000

0.0000 0.0000 0.6599

Table 4: (Continuation of Table 3).

18

r

r = 0

V1

V

V2

V=

=V1 V2

=

=

r > 0

e

(b)(a)

Figure 7: Two-parametrized topologies used for the database construction. Pat-

terns: a) PA, and b) PB.

Partial results of this database have been presented in [23]. From that work, it

can be observed that auxetic materials are captured with the micro-architectures

characterized by the pattern PB.

Remark: The information stored in the database, of the order of 105elasticity

tensors, is utilized to provide a hint for selecting, from a predeﬁned Bravais lattice

system, the Voronoi cell used as the design domain of the topology optimization

problem, see item (3) of the next Section 4. The database information is managed

as follows:

•Once the target elasticity tensor ˆ

CNand its point group is known, we search

the parameters, ζ, of the database entry whose elasticity tensor Ch

db(ζ) satisﬁes

the solution of the problem:

ζ= arg min

ζkCh

db(ζ)−1

E1

ˆ

CNk; where ζ:= {ω, ς , P, f},(6)

with the parameters (ω, ς ) restricted to the region characterizing Bravais lat-

tices with identical point group to that of the target tensor. Regions in the

plane (ω, ς ) with the related point groups are shown in Figure 5-b. The factor

E1scaling the target elasticity tensor is the Young’s modulus of the composite

stiﬀ phase to be designed.

•Furthermore, the micro-architecture associated with the solution of problem

(6) could be taken as the initial conﬁguration for the iterative topology opti-

mization algorithm.

The database of homogenized elasticity tensors, the software to compute the entries

of the database for the parameter set ζand the software to attain the solution of

19

the problem (6) are freely available in the Mendeley dataset repository, see [29]. All

the codes are written in Matlab.

4 A systematic procedure for tailoring elastic com-

posites through inverse homogenization tech-

nique

Based on the symmetry properties of the target elasticity tensors, the discussion

given in Section 2 and taking into account the elasticity tensor database, four heuris-

tic rules are introduced to facilitate and enlarge the possible set of conﬁgurations

obtained with the inverse homogenization design problem. These rules deﬁne a

systematic procedure that can be adopted by the designer and are established as

follows.

1) The topology optimization problem (2) is solved in the Normal basis of the

target tensor. It means that the tensor ˆ

Cis replaced by ˆ

CNin the formulation

(2).

2) Voronoi cells of Bravais lattices, whose point groups are compatible with the

point groups of the target tensors, are chosen as the design domain Ωµ. The

relationship between the point groups of both entities, lattice and elasticity

tensor, are shown in Table 2. Entities sharing the same point group in the

Table are displayed in identical lines. For example, for a target tensor having

D2symmetry, the compatible cells are those associated with Rpor RcBravais

lattices, such as rectangles or irregular hexagons are shown in Figure 4. In

this case, note that square and regular hexagonal cells are also compatible.

3) The slenderness of the cell, i.e., the aspect ratio between the larger and shorter

size of the cell, can be determined through the database entries and the cri-

terion deﬁned by expression (6). Typically, this slenderness ratio should be

deﬁned in the Voronoi cells of Oblique, Rcand Rplattices.

4) The symmetry displaying the material distribution within the Voronoi cell

is decided in accordance with symmetry properties of the target elasticity

tensor. The material conﬁguration symmetry deﬁnes the micro-architecture

plane group. Then, this conﬁguration should be compatible with the target

elasticity tensor point group. The compatibility relationships between plane

groups and elasticity tensor point groups are presented in Table 2; the entries

displayed in the ﬁrst and the fourth columns of the same line are compatible.

Thus, recalling the previous example, i.e., a target tensor having D2symmetry,

the compatible plane groups are pm,pg,cm,p2mm,p2mg,p2gg or c2mm.

20

Our experience is that several conﬁgurations, with compatible plane groups,

have to be tested and solved. In some cases, improved designs are attained by

enforcing the higher compatible symmetry, though, this is not a general rule.

To apply the last rule, the symmetries of the material conﬁguration compatible

with the plane group can only be enforced if the topology optimization problem is

solved in the Normal basis of ˆ

C, such as mentioned in the ﬁrst rule. In this basis,

the symmetry planes of the plane groups are coincident with the symmetry axes of

the Voronoi cells, such as shown in Figure 6. Note also that the cost of the ﬁnite

element analysis for computing the homogenized elasticity tensor, in the topology

optimization problem, can signiﬁcantly be reduced after introducing the symmetries

compatible with the plane group. This issue has been addressed by Barbarosie et al.

[11] and Lukkassen et al. [12].

5 Numerical assessments

Some micro-structures designed via a topology optimization problem that follows the

systematic procedure previously described are analyzed and compared with micro-

structures obtained with alternative conventional procedures without enforcing those

rules. The eﬀects that several design variables have on the attained solutions are

assessed, such as:

i) Diﬀerent types of design domains. The tested cell shapes are square, rectangles

with diﬀerent aspect ratios and regular or irregular hexagons. Additionally,

the enforcement of diﬀerent plane groups is also tested.

ii) Voronoi cells with symmetry axes arbitrarily placed or aligned with the Normal

bases are tested.

iii) Diﬀerent material conﬁgurations are taken as the starting points of the topol-

ogy optimization algorithm.

The objective is to compare the solutions obtained in those diﬀerent situations.

5.1 Designing an extreme isotropic material with minimum

Poisson’s ratio

The target material, whose micro-structure is designed in this test, is a biphasic

isotropic composite with the minimum Poisson’s ratio that is estimated using the

Cherkaev and Gibiansky [30] bounds for this kind of composite. The properties of

the composite components and the volume fraction of the hard material are chosen

21

such that the Cherkaev et al.’s bounds estimate an elasticity tensor with negative

Poisson’s ratio, which makes this test more challenging.

Due to the interesting properties of auxetic materials, their micro-architecture

designs have been profusely studied in the literature, such as mentioned in the review

by Ren et al. [31]. Only to cite a few works closely related to the present contribution,

the paper of Larsen et al. [32] describes the micro-architecture design of auxetic

composites using topology optimization tools, particularly the SIMP technique. This

paper also shows the manufacture of such composites and their experimental testing.

More recently, and using a similar development, Andreassen et al. [24] have reported

the designs of 3D micro-architectures with negative Poisson’s ratios and describe

their manufacture attainability. Also, the work by Jiang and Li [28] gives a survey

of this topic, as well as a discussion about the fabrication of auxetic materials. An

interesting analysis of three typical micro-architecture topologies, re-entrant, chiral

and rotating, that can be used for designing auxetic materials has been reported in

the work by Kolken and Zadpoor [27], while a family of lattice-like metamaterials,

with macroscopic eﬀective Poisson’s ratio arbitrarily close to the stability limit −1,

has been studied by Sigmund [2] and more recently by Cabras and Brun [33]. Finally,

the initial contributions on auxetic materials of Lakes and Evans and co-workers in

the 80’s and 90’s must be here mentioned, see for example Evans and Alderson [34]

and reference cited therein.

5.1.1 Studied case

An isotropic composite constituted by two phases M1and M2is assumed. Material

M1has a bulk modulus7K1= 5/7[GP a] and shear modulus G1= 5/13[GP a].

Material M2has a bulk modulus K2=K1/200[GP a] and shear modulus G2=

G1/200[GP a]. The volume fraction of M1is f1= 0.5. Plane strain hypothesis is

considered.

According to the Cherkaev et al.’s analysis, the eﬀective moduli of an isotropic

composite constituted by M1and M2is bounded below and above with the curves

plotted in Figure 8-a, where ˆ

Kand ˆ

Gare the eﬀective bulk and shear moduli of the

composite, respectively. Taking into account these bounds, the minimum possible

Poisson’s ratio for such composites 8is attained at the point satisfying:

min ˆ

K−ˆ

G

ˆ

K+ˆ

G,(7)

with ˆ

Kand ˆ

Gbounded by the domain speciﬁed by the Cherkaev et al.’s anal-

ysis. This point should be on the curve ABC. The Poisson’s ratios of points

7The plane strain bulk modulus Kis given by K=κ+G/3, where κand Gare the conventional

bulk and shear moduli of the three-dimensional theory.

8The eﬀective Poisson’s ratio of plane elasticity theory if given by: ˆν∗= ( ˆ

K−ˆ

G)/(ˆ

K+ˆ

G).

22

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0

0.02

0.04

0.06

0.08

0.1

0 0.04 0.08 0.12 0.200.16 0 0.04 0.08 0.12 0.200.16

(a)

Eﬀective bulk modulus

(of planar elasticity) K

Eﬀective shear

modulus G

f = 0.5

C

MP

Eﬀective bulk modulus

(of planar elasticity) K

Eﬀective Poisson ratio

(of plane elasticity theory) v*

MP

0.0124

0.0703

(b)

AC

B

1

Walpole point

v*=0

v*=-0.25

A

B

-0.7002

v*=-0.5

Figure 8: a) Bounds of eﬀective properties for a two-phase isotropic composite

according to Cherkaev et al. Properties of phases M1and M2are K1= 5/7; K2=

K1/200; G1= 5/13; G2=G1/200 (units of moduli in GPa). Volume fraction of

phase M1is f1= 0.5. Iso-lines of ˆν∗with values 0, −0.25 and −0.5 are shown. b)

Poisson’s ratios of isotropic composites whose eﬀective properties are characterized

by the points on the curve ABC in a).

on that curve are plot in Figure 8-b. The point MP with coordinates ( ˆ

K, ˆ

G) =

(0.012404,0.070334) is the one satisfying (7) with the minimum Poisson’s ratio

ˆν∗=−0.7002. Then, the eﬀective elasticity tensor that is related to the MP point

results:

ˆ

CN=

0.082739 −0.057930 0

−0.057930 0.082739 0

0 0 0.140669

,(8)

which is taken as the target elasticity tensor to formulate the topology optimization

problem. Note that this tensor is isotropic, and therefore, it is given in the Normal

coordinates.

5.1.2 The topology optimization algorithm

The volume fraction constraint (f1= 0.5) is satisﬁed in the present case by redeﬁning

the objective function of the optimization problem (2) as follows

min

χ 1

|Ωµ|ZΩµ

χ dΩ−f1!2

.

Problem (2) with the above objective function is solved with a topology op-

timization technique that uses the concepts of topological derivative and level-set

23

function, similar to the procedure explained in the Amstutz’s works, see [35] and

[5]. A brief summary of this technique is given in Appendix A. Also, considering

that the target tensor ˆ

CNmay not be attainable, the convergence of the algorithm

is reached when the following conditions are satisﬁed: i) the error in the normalized

constraint kCh−ˆ

Ck/kˆ

Ckhas stabilized to a value less than tolC= 0.4, and ii) the

criterion (25) of the Appendix A has stabilized to a value greater than 1 −tolψwith

tolψ= 0.1. Stabilization in both values means that they do not substantially change

during the last 10 iterations.

To facilitate the comparative analysis in these cases, the topology optimization

algorithm forces solutions with material conﬁgurations displaying only one length

scale. Therefore, the isotropic homogeneous phases are spatially distributed in such

a way that their ﬁnest width is limited below by a size of the order of the micro-cell

size. Under this condition, typical layered sub-micro-structures are topologies not

admitted. This criterion is imposed on the topology optimization algorithm through

a conventional Helmoltz’s type spatial ﬁlter that is described by equation (26) in

Appendix A, see additional details in [36] and [37] for local length scale control. The

ﬁlter size rls is constant and identical in all computed cases.

5.1.3 Voronoi cells, symmetries and stiﬀ phase conﬁgurations for initial-

izing the topology optimization algorithm

Considering that the target composite is isotropic, and following the procedure pro-

posed in Section 4 and Table 2, the design domain of the topology optimization

problem could be taken as a regular hexagon. Also, the material distribution could

be compatible with the plane groups p3, p3m1, p31m,p6 or p6mm. These choices

guarantee the isotropy of the designed material. Particularly in this Section, we test

the Hexagonal cell with p1, p3 and p6 plane groups. We additionally test a square

cell enforcing symmetries consistent with p1 and pm plane groups.

The ﬁve tested cases are identiﬁed as follows: Hp1,Hp3and Hp6refer to hexagonal

cells with plane groups p1, p3 and p6, respectively. Sp1and Spm refer to square cells

with plane groups p1 and pm, respectively. In the cases with p1 and pm plane

groups, the isotropy of the designed material is not guaranteed.

In order to attain a representative response of the computational procedure, ten

instances of each one of these ﬁve tested cases are solved with the iterative topology

optimization algorithm. So, they are a total of 50 designs. All of them are initialized

with random distributions of the stiﬀ phase satisfying the condition f1= 0.5. Note

that the item (3) of Section 4 is not brought into play for solving these cases.

24

0.05

0.054

0.058

0.062

0.066

0.07

1

2

3

4

5

6

7

8

9 10

1

2

3

4

5

6

7

8

9

1

2

3

4

5

7

8

9 10

1

2

3

4

5 6

7

8

9 10

2

3

4

6

8

9

10

H

S

H

H

S

p1

pm

p3

p6

p1

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.015 0.02 0.025 0.03 0.035

1 2 3 45 6 7 8 9 10

Run Number

(Instance)

Eﬀective bulk modulus (of planar elasticity) K

Eﬀective shear modulus G

MP

Cherkaev et al.

upper bound

Non-isotropy coeﬃcient t

v*=-0.35

v*=-0.45

v*=-0.5

v*=-0.25

Figure 9: Micro-structure design of a composite with negative Poisson’s ratio. Three

design cases denoted Hp1,Hp3and Hp6are solved with hexagonal cells and p1, p3

and p6 plane groups. Two cases with square cells denoted Sp1and Spm are solved

with p1 and pm plane groups. Ten instance of each one of the ﬁve cases have been

solved. a) Solutions plotted in the space of the eﬀective bulk and shear moduli.

Diferent instances of each plane group are identiﬁed with a run number. The iso-

lines of eﬀective Poisson’s ratios of plane elasticity theory ˆν∗=−0.25,−0.35,−0.45

and −0.5 are shown; b) non-isotropy coeﬃcients that correspond to the complete

set of solutions.

5.1.4 Numerical results

The eﬀective moduli of the 50 solved micro-architectures are depicted in the space

deﬁned by the eﬀective shear modulus and eﬀective bulk modulus shown in Figure

9-a. The Cherkaev et al.’s bounds and the MP point are also depicted in the plot.

In the cases Hp1,Sp1and Spm, where isotropy cannot be guaranteed, the parameters

ˆ

Kand ˆ

Gare computed using the expression (11) below. In Figure 9, ﬁve diﬀerent

symbols identify the ﬁve cases, while the numbers (from 1 to 10) beside the symbols

identify the corresponding instances of each case.

We remark that all of the 50 designs have been solved with the same ﬁlter to

avoid micro-structure solutions displaying subscales.

The isotropy of Chcan be estimated through the non-isotropy coeﬃcient τthat

measures its distance to the space of isotropic tensors (see [8]):

τ=kCh−Cisok

kChk(9)

where Ciso is considered as a projection of Chonto the space of isotropic tensors

25

and is computed with the formula:

Ciso =

Kiso +Giso Kiso −Giso 0

Kiso −Giso Kiso +Giso 0

0 0 2Giso

,(10)

where

Kiso =1

8(3Ch

11 + 3Ch

22 + 2Ch

12 + 2Ch

33)−Giso ,

Giso =1

8(Ch

11 +Ch

22 −2Ch

12 + 2Ch

33).(11)

Expressions (11) have been taken from Meille and Garboczi [38].

The so-deﬁned non-isotropy coeﬃcient τhas been computed for the complete

set of solutions. Figure 9-b plots these coeﬃcients. These results conﬁrm that the

solutions obtained with the p3 and p6 plane groups are exactly isotropic (τ= 0)9.

Figure 10 presents the errors of the solutions to satisfy the constraint condition

of the problem (2). Each sub-Figure display the normalized distance between the

target elasticity tensor ˆ

CNand the homogenized elasticity tensor Chof each micro-

architecture design case. Note that this distance is not necessarily equal to the

distance between the point MP and each solution point displayed in Figure 9-a.

For example, compare in both Figures the results of instances 2 and 8 with plane

group Spm. Figure 10-b displays that the error for instance 8 is lower than the error

for instance 2. However, the contrary eﬀect is observed in Figure 9-a. This result

comes from the fact that Figure 10 disregards the isotropy property of the obtained

homogenized elasticity tensors.

Figure 11 displays some designed micro-structures. The plot of Figure 9-a is

repeated in Figure 11 but only including the instances whose micro-structure are

shown in this Figure.

5.1.5 Discussion of results

From the analysis of the solutions obtained in these tests, it can be drawn some

observations:

•Solutions obtained by enforcing the Hp3plane group are the closer ones to the

MP-point. This conclusion is attained with the distances computed with the

9Computational homogenization of solutions obtained with p3 and p6 plane groups and using

unit cell ﬁnite element meshes that preserve the symmetry p3 or p6 provides a value τbeing exactly

zero to the machine precision.

26

Sp1

1 2 3 4 5 6 7 8 9 10

Instance

0

0.1

0.2

0.3

0.4

0.5

||C -C ||

N

h||C ||

N

Hp1

1 2 3 4 5 6 7 8 9 10

Instance

0

0.1

0.2

0.3

0.4

0.5

||C -C ||

N

h||C ||

N

(a)

(c)

Spm

1 2 3 4 5 6 7 8 9 10

Instance

0

0.1

0.2

0.3

0.4

0.5

||C -C ||

N

h||C ||

N

Hp3

1 2 3 4 5 6 7 8 9 10

Instance

0

0.1

0.2

0.3

0.4

0.5

||C -C ||

N

h||C ||

N

(b)

(d)

Hp6

1 2 3 4 5 6 7 8 9 10

Instance

0

0.1

0.2

0.3

0.4

0.5

||C -C ||

N

h||C ||

N

(e)

Figure 10: Normalized errors to satisfy the constraint kCh−ˆ

CNk. Plane groups:

a) Sp1; b) Spm; c)Hp1, d) Hp3, e) Hp6.

metric induced by (9)–(11). Solutions with Hp3plane group have intermediate

material conﬁguration symmetries, between Hp1and Hp6plane groups. Alter-

natively, the farther solutions with hexagons have been obtained by enforcing

the higher symmetry, i.e., Hp6plane group. However, this result cannot be

taken as a general conclusion. As a counterexample, we remind the case of

the composite designed to attain the Walpole point, see Sigmund [10]. In this

case, it has been shown that enforcing the Hp6mm plane group, instead of Hp3,

provides better solutions, see M´endez et al. [39].

•The double bar mechanisms depicted in Figure 11, case Hp3instances 1 and

3, may suggest for auxetic materials a similar observation to that reported by

[10], in the sense that better structures are obtained by splitting individual

bars into “multiply laminated bars”. In fact, Sigmund proposed a very gen-

eral class of optimal stiﬀness microstructures based on this principle which

achieves the maximum energy bounds. Even when, we cannot guarantee this

conclusion in the present study, this important aspect of auxetic material de-

signs with isotropic properties deserves additional research by repeating the

design process with ﬁner grids to generate a sequence of solutions tending to

the theoretical value.

•Chiral materials ([27],[28]) appear to be the micro-architecture topologies at-

tained when enforcing the Hp6plane group. This type of architecture can be

observed in instances 6 and, though less clearly, in instances 2 and 3. Also,

27

0.05

0.054

0.058

0.062

0.066

0.07

1 5

2

3

5

8

4

9

1

3 7

2

3 6

0 0.015 0.02 0.025 0.03 0.035

K

G

MP

Hp3

Sp1

Hp3

Instance 1

Hp3

Instance 3

Hp3

Instance 7

Hp1

Instance 4 Hp1

Instance 9

Hp6

Instance 3

Hp6

Instance 6

Hp6

Instance 2

Sp1

Instance 5

Sp1

Instance 1

Hp6

Hp1

Spm

Instance 8

Spm

Instance 5

Spm

Instance 2

Spm

Instance 3

Spm

Figure 11: Several instances of the designed micro-structures of a composite with

negative Poisson’s ratio. plane groups Hp1,Hp3,Hp6,Sp1and Spm. Composites with

phases M1and M2and properties given by: K1= 5/7; K2=K1/200; G1= 5/13;

G2=G1/200 (units of moduli in GPa). The volume fraction of phase M1is f1= 0.5.

28

there can be observed a tendency for capturing chirality in the solution Sp1,

instance 1.

•The attainment of isotropy with Hp1,Sp1and Spm plane groups lies purely

on the eﬀectiveness of the topology optimization algorithm. Some solutions of

Spm plane group show micro-structures whose eﬀective responses are closely

isotropic. However, this is not the general response for these plane groups.

•Solutions obtained with Spm plane group, instances 2 and 5, display a similar

micro-architecture to the one previously reported in the literature, see Larsen

et al. [32] and Andreassen et al. [24]. As can be observed in Figure 9-b and

ignoring a small error, both particular instances (2 and 5) display isotropic

eﬀective elastic responses, agreeing, therefore, with the reported results in

Andreassen et al. [24].

•Micro-architecture conﬁgurations with higher symmetries appear to be more

easily parametrized.

•Solutions obtained by enforcing the Sp1plane group indicate a preference to at-

tain topologies with higher symmetries, compatible with the Spm plane group.

An identical conclusion can be drawn for the solutions obtained by enforcing

the Hp1plane group; these topologies have a preference to attain symmetries

compatible with the Hp3plane group. It is particularly observed that the so-

lutions of instance 2, 5 and 8, of the case Spm, present a vertical glide plane.

Therefore, these micro-structures result with a Sp2mg plane group which is a

higher symmetry than the one originally imposed.

•It is also important to remark that the number of optimal topologies in the

design problem decreases by enforcing higher symmetries. For example, non-

centered or ﬂipped topologies in the base cell can be avoided by enforcing that

the symmetry lines cross the central point of the cell.

5.2 Micro-structure designs of materials with D2symmetry

Elasticity tensors with D2symmetry and positive Poisson’s ratios are taken for

assessing the topologies that are obtained with the procedure of Section 4.

29

5.2.1 Studied cases

Two composite micro-structure designs are assessed. These composites are denoted

by I and II, and their eﬀective elasticity tensors in Cartesian coordinates are:

ˆ

CI=

0.0392 0.0578 −0.0385

0.0578 0.1050 −0.0931

−0.0385 −0.0931 0.1100

;ˆ

CII =

0.0364 0.0462 0.0211

0.0462 0.0785 0.0631

0.0211 0.0631 0.0846

.(12)

The same tensors in Normal coordinates, denoted ˆ

CI

Nand ˆ

CII

N, are expressed by:

ˆ

CI

N=

0.2254 0.0031 0.

0.0031 0.0281 0.

0.0.0.0006

;ˆ

CII

N=

0.1624 0.0043 0.

0.0043 0.0363 0.

0.0.0.0008

(13)

being that the rotation angles between the Cartesian and the Normal bases are

θI=−arctan(√2) and θII = arctan(√2) for materials I and II, respectively.

The composites are constituted of a stiﬀ phase M1and void. Young’s modulus

and Poisson’s ratio of the stiﬀ phase are E1= 1.[GP a] and ν= 0.3, respectively.

The plane stress hypothesis is assumed.

5.2.2 The topology optimization algorithm.

A density-based topology optimization technique combined with the Solid Isotropic

Material with Penalization (SIMP) interpolation formulation ([3]) is adopted in this

Section for solving the problem (2). The resulting formulation is iteratively solved

with the IPOPT interior point primal-dual algorithm, see W¨achter and Biegler [40]

and the bibliography cited therein. The coupling of these procedures has been

analyzed by Rojas-Labanda et al. [41]. Therefore, in Appendix B, we sketch a brief

description of this approach remarking the ﬁlters that are used to solve these cases.

Algorithm parameters: Similar to the previous examples, we use a density ﬁlter

that forces the optimum solution to display a material topology with only one length

scale and a Heaviside projection ﬁlter to alleviate the issue related to the presence

of gray material.

The density ﬁlter described in Appendix B.2, uses a value R=αL, where Lis

the cell size and α(with 0 < α ≤1) is the adimensional parameter scaling the ﬁlter

radius. We adopt α= 0.01. The gray material presence is diminished by using the

Heaviside ﬁlter described in Appendix B.3, also discussed in Wang et al. [37]. The

parameter βhandles the Heaviside ﬁlter. An external loop of the IPOPT algorithm

increases the β-parameter according to the sequence {0,1,2,4,8, ...}. Finite element

meshes with 40000 elements are used; SIMP density exponent p= 3.5. All the

30

solutions satisfy the normalized constraint of the problem (2) with the tight tolerance

(tol = 1.e −4) deﬁned in the IPOPT algorithm.

5.2.3 Performance assessment of the design procedure

Micro-structures designed by enforcing the rules 1 and 4

Squares cells are used and the IPOPT algorithm is initialized with two diﬀerent

conﬁgurations: a stiﬀ material density randomly distributed or uniformly distributed

(with initial density ρ= 0.5). Additionally, two procedures are followed to solve

four topology optimization problems:

i) The ﬁrst procedure performs the inverse design of composites Iand II with

the target tensors ˆ

CIand ˆ

CII in Cartesian coordinates without imposing

symmetries.

ii) In the second procedure, tensors ˆ

CIand ˆ

CII are ﬁrst rotated to Normal

coordinates. Therefore, the inverse designs are performed with the target

tensors ˆ

CI

Nand ˆ

CII

Nand the Voronoi cells are aligned with their Normal basis.

Furthermore, a symmetric material distribution consistent with a p2mm plane

group is imposed on the SIMP methodology. The horizontal and vertical

central lines of the cells are symmetry lines. In summary, the micro-structure

design is performed according to the two rules 1 and 4 describe in Section 4.

The solutions of these four cases, with the two initial conﬁgurations, are com-

pared in Figure 12. We observe that the solutions, whose cells are aligned with the

Normal basis of the target tensors and with the imposed symmetry being consis-

tent with a p2mm plane group (second and fourth column of the Figure), display

simpler topologies. These solutions are similar to laminates which can be easily

parametrized. Alternatively, the micro-architectures designed with the elasticity

tensor in the original Cartesian basis display more complex topologies which result

as a consequence of using square cells and periodic boundary conditions not aligned

with the symmetry lines of the materials.

It is additionally noted that the solutions in Figure 12 are not very sensitive

to the initial conﬁguration taken to start the IPOPT algorithm. Even when this

conclusion will be partially relativized in the following numerical tests, it is in the

same line with observations reported in the literature, see Rojas-Labanda et al. [41].

Micro-structures designed by enforcing the four rules

The same composites Iand II, with target elasticity tensor ˆ

CIand ˆ

CII , are next

designed by following the full set of rules described in Section 4. The Voronoi cells

compatible with a target elasticity displaying D2symmetry corresponds to Rpor Rc

Bravais lattices. The cell slendernesses, i.e., the aspect ratio between the larger and

31

Material I Material II

Unit-cells

CICI

NCII CII

N

Homogeneous initial conﬁguration Random initial conﬁguration

f=0.31

Unit-cells

f=0.29 f=0.26 f=0.22

f=0.51 f=0.32 f=0.26 f=0.23

s

n

n

s

s

s

n

n

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

Figure 12: Four examples of micro-architecture designs. Target tensors are: ˆ

CIand

ˆ

CI

Nfor material I; ˆ

CII and ˆ

CII

Nfor material II. Pictures correspond to the assembled

micro-architectures in a direction agreeing with the Cartesian basis. Unit cells and

volume fractions are also shown.

32

shorter size of the cells, are assessed through the database and the criterion deﬁned

by expression (6). The shape of the cells for material I and II determined with

this criterion are the rectangular cells, compatible with RpBravais lattices, shown

in Figure 13-a. Furthermore, an adequate plane group determining the material

distribution geometry would be one of the seven plane groups denoted in Table 2 by

pm,pg,cm,p2mm,p2mg,p2gg or c2mm. In particular, we choose the plane group

p2mm.

Figure 13-b depicts the topologies computed with this procedure. The eﬀect of

taking an initial conﬁguration given by the database (topologies shown in the third

column of these results) is evaluated by comparing the micro-architectures solved

with diﬀerent initial conﬁgurations, a stiﬀ material density randomly distributed or

uniformly distributed (with density ρ= 0.5), such as shown in the ﬁrst and second

columns.

It can be seen that the ﬁnal micro-architecture, obtained with the initial conﬁg-

uration taken from the database, is the simpler one constituted by bars of (almost)

uniform thickness, but diﬀerent length ratios. The bar lengths typically agree with

the cell slenderness.

Figure 14 show the micro-architectures of material I and II that have been ob-

tained by using SIMP with the initial conﬁguration taken from the database.

CI

N

CII

N

Inital conﬁguration:

random density

Inital conﬁguration:

uniform density

Inital conﬁguration:

taken from database

11

7.6

4.2

Composite I Composite II

(a)

(b)

= 90.deg

0.13

f =0.25 PA

= 90.deg

0.24

f =0.20 PA

Figure 13: Micro-structure designs of composites I and II. The topology optimization

problem uses the four rules of Section 4. a) Voronoi cells determined with criterion

(6); b) Unit-cell solutions of the topology optimization algorithm by adopting three

diﬀerent initial material conﬁgurations.

5.2.4 Discussion of results

The obtained solutions show that enforcing periodic boundary condition along di-

rections not aligned with the Normal basis may be the cause of attaining micro-

structures with complex topologies if compared with solutions obtained by follow-

33

Material I Material II

x

y

n

s

x

y

n

s

(a) (b)

f =0.31 f =0.25

Figure 14: Micro-structure asasamblages of the composites I and II. The topology

optimization problem uses the four rules of Section 4. Initial conﬁgurations are

taken from the database. The resulting volume fractions are f= 0.31 and f= 0.25

for composites I and II, respectively.

ing the ﬁrst rule of the procedure here proposed. Even simpler topologies can be

attained when the initial conﬁguration of the iterative algorithm are taken from

the database. Nevertheless, a simple conﬁguration may not be the global optimum

solution of the problem.

In a similar way, the sensitivity of the attained material conﬁguration with dif-

ferent design domains is evidenced in these tests. Notably, the aspect ratios of the

cells also play an important role in obtaining diﬀerent topology types.

5.3 Pentamode material design

According to Norris [42], pentamode materials are useful for constructing acoustic

cloaking devices. A Pentamode material is a class of extremal material having ﬁve

easy (compliant) modes of deformation in a three-dimensional space, and having only

one non-easy (hard) mode of deformation10. The elasticity tensor of this material

has one non-null eigenvalue and ﬁve null eigenvalues (hence the name of pentamode

given to this class of material). In 1995, Milton and Cherkaev [43] have coined the

10In the following, we preserve the name of pentamode material for plane (two-dimensional)

problems. Removing the third dimension and the out-of-plane ﬁeld components, the elasticity

tensor has only three eigenvalues. Particularizing the same concept of pentamode material, two

of these eigenvalues are related to compliant modes of deformation and only one is related to a

hard mode. Strictly, this material should be called bi-mode material. Additional discussion about

bi-mode materials can be found in [23].

34

name of pentamode materials in the context of linear elasticity. In the same year,

Sigmund [2] has independently introduced it in the context of inverse homogenization

problems.

Pentamode materials are a special class of linear anisotropic elastic solids. They

can be characterized through elasticity tensors represented by:

C∗=κ∗S⊗S; (14)

where κ∗is a pseudo-bulk modulus with the dimensions of stress and Sis an adi-

mensional symmetric second order tensor with norm not necessarily equal to one.

As usual, the symbol ⊗denotes the tensorial product.

The micro-structure design of a pentamode material proposed for constructing

an acoustic cloaking device is here studied.

The goal of this assessment is to show that diﬀerent design domains Ωµprovide

markedly dissimilar solutions of the topology optimization algorithm and that the

solution obtained with the systematic procedure of Section 4 display the simpler

topologies.

5.3.1 Studied cases

Norris [42] has reported that an acoustic cloaking device can be realized by design-

ing a layered graded pentamode material whose eﬀective elastic properties can be

determined with the analytical results presented by Gokhale et al. [44].

The design and realization of the idealized layer have been addressed in a number

of works. In particular, the reported solution by M´endez et al. [39] divides the layer

into 20 sub-layers whose micro-structures are then determined by means of an inverse

homogenization design. The layered composite is constituted of aluminum, with a

bulk modulus κAl = 70.GP a and shear modulus GAl = 25.5GP a, and a ﬂexible

polymer foam characterized by an isotropic material whose elastic modulus has a

contrast factor γ= 0.00001 times the elastic modulus of the aluminum.

Sub-layer number 15 of the acoustic cloaking device described in M´endez et al.

is chosen to perform this study. Here, we only consider a partial aspect of the total

layer design for the mentioned device, i.e., the micro-architecture design is only

addressed without enforcing the required density constraint.

For this sub-layer in particular, the target elastic properties are characterized by

the elasticity tensor described in Normal coordinates as follows:

ˆ

CN=

5.893 2.250 0.

2.250 0.8590 0.

0.0.0.

.(15)

This tensor corresponds to a pentamode material with D2symmetry. The composite

is designed to copy these elastic properties.

35

= 81.deg

0.30 f =0.175

PA= 90.deg

0.152 f =0.175

PA

(a) (b)

Figure 15: Pentamode material design. a) Hexagonal cell taken from the database

using the criteria (6). b) Rectangular cell being compatible with the Bravais lattice

whose Voronoi cell is the hexagonal cell in picture a).

5.3.2 Topology optimization algorithm

The level-set algorithm presented in the previous sub-Section is used in this case to

solve problem (2). Similar to the technique adopted in the previous examples, we

use a spatial ﬁlter to avoid topology designs with multiple length scales.

5.3.3 Obtained results

First, we remark that it has not been possible to attain acceptable results with

square cells and the adopted mesh resolution. Therefore, the analysis is restricted

to rectangular and hexagonal cells (Rpor RcBravais lattices)which are compatible

with the D2symmetry of the target tensor. Furthermore, the solved topologies are

enforced to satisfy the p2mm plane group.

Case A: The hexagonal cell shown in Figure 15-a has been determined with the

criterion (6). The resulting parameters are: ω= 0.3, ς= 81.deg, pattern PA

and f= 0.175. Taking a domain of analysis deﬁned by this hexagonal cell, we

test two sub-cases, A1 and A.2, which are deﬁned with diﬀerent initial stiﬀ phase

conﬁgurations, as follows.

A.1 The initial stiﬀ phase conﬁguration is the one given by the solution of problem

(6), such as shown in Figure 15-a.

A.2 The initial conﬁguration consists of a random stiﬀ phase distribution. Five

diﬀerent instances of this sub-case are solved.

Case B: the rectangular cell shown in Figure 15-b is taken as the design domain.

This cell with parameters ω= 0.1517, ς= 90.deg can reproduce a similar micro-

structure as that given by the hexagonal cell of the case A. The three sub-cases are

solved with the following initial conditions:

B.1 stiﬀ bars are disposed in the boundary of the cell;

36

A.1 B.1

B.2 - Instance 1

B.2 - Instance 2

A.2 - Instance 1

A.2 - Instance 2

B.3

f = 0.151

f = 0.309

f = 0.334

f = 0.133

f = 0.279

f = 0.166

f = 0.119

1

1

11

1

1

1

Figure 16: Pentamode material designs. Topologies attained with sub-cases A1, A.2

and B. The volume fraction of each case is f1.

37

B.1

B.2 (ﬁve instances)

A.2

A.1 (with ﬁlter)

A.1 (without ﬁlter)

0

0.1

0.2

0.3

0.4

1 2 3 4 5

Relative error

Instance

B.3

Figure 17: Pentamode material design. Relative errors kCh−ˆ

Ck/kˆ

Ckcomputed

for the solutions obtained with the Level-set algorithm.

B.2 a random distribution of stiﬀ phase. Five instances are solved with initial

random conﬁgurations.

B.3 stiﬀ bars are disposed in a hexagonal array. This hexagonal array is the same

as that of the initial conﬁguration of Case A.1, but here, it is projected onto

the rectangular cell such as shown in Figure 15-b.

The topologies obtained in all these cases are shown in Figure 16.

Figure 17 plots the relative errors kCh−ˆ

Ck/kˆ

Ckfor all the tested cases solved

with the Level-set algorithm. The solution with the lower error is computed under

the following conditions: the hexagonal cell and the initial conﬁguration are taken

from the database, and the algorithm is run without the ﬁlter. In the remaining

cases, the ﬁlter penalizes the solutions to attain eﬀective elasticity tensors closer

to the target one. In any case, it is notable that the hexagonal cell provides more

accurate solutions if compared with those obtained, in comparable circumstances,

with the rectangular cell.

6 Conclusions

The associations presented in Tables 2, 3, 4 and Figure 6 constitute the fundamen-

tal ingredients introduced in this paper to connect the symmetries characterizing

the material conﬁgurations and the physical properties. They also constitute the

basic components to identify an adequate micro-architecture design procedure of

composites whose eﬀective properties fulﬁll a given elastic response.

38

Based on these notions, four rules have been proposed in this work. These rules

deﬁne a systematic procedure that can be followed to facilitate the micro-structure

design carried out by means of an inverse homogenization technique.

Through a number of numerical assessments, it has been shown that the con-

ventional square cells, adopted as the design domain for the inverse homogenization

techniques formulated as a topology optimization problem, are not adequate to ob-

tain simple micro-architecture conﬁgurations in general situations. Instead, appro-

priate cells and plane groups, identifying the material conﬁguration symmetry within

the cell, can alternatively be chosen to guarantee the attainment of composites with

simple topologies and eﬀective elastic responses displaying identical symmetries to

the target ones. These conclusions are independent of which topology optimization

algorithm is taken to solve the inverse homogenization problem.

An observation here remarked is that the proposed rules could be extended

to inverse homogenization techniques involving 3D elastic material designs. The

generalization of the concepts involving crystal physics and the related symmetries

in 3D could provide some hints to conceive base cells for more general domain designs

in 3D problems.

Acknowledgments

The authors acknowledge the ﬁnancial support from CONICET and ANPCyT (grants

PIP 2013-2015 631 and PICT 2014-3372 and 2016-2673).

A APPENDIX: Solving the topology optimiza-

tion problem with Topological Derivate Algo-

rithm

The topology optimization problem (2) can be solved by introducing a smooth level

set-funtion, ψ∈C0(Ωµ), satisfying

ψ(y) =

<0∀y∈Ω2

µ

>0∀y∈Ω1

µ

0 in the interfaces

,(16)

and utilizing an augmented Lagrangian technique, see Lopes et al. [45]. In this case,

the problem is rewritten as follows

max

λmin

ψT(ψ, λ),(17)

39

with:

T(ψ, λ) = ZΩµ

χ(ψ)dΩ + λ(kCh(ψ)−ˆ

Ck) + α

2(kCh(ψ)−ˆ

Ck)2(18)

where λis the Lagrange multiplier and αis the penalty parameter of the augmented

term.

The algorithm for solving the problem (17) utilizes two nested loops. In an

internal loop, the objective function Tis minimized by holding ﬁxed λand α. This

loop, with index denoted k, consists of a level-set function-based iteration. While

an external loop, with index denoted l, modiﬁes iteratively λ.

The minimum of Tin the internal loop is searched with a descent direction

algorithm. For problem (17), the topological derivative used to estimate the descent

direction is given by

DψT(ψ, λ) = 1 − λ−αkCh−ˆ

Ck(Ch−ˆ

C) : DψCh

kCh−ˆ

Ck!(19)

where DψChis the topological derivative of the homogenized elasticity tensor, see

[5] for an additional description of this term. Then, we deﬁne the function :

g(y) =

−(DψT)if :ψ < 0

+(DψT)if :ψ > 0

,(20)

The updating formula for ψ, at the (k+ 1)-th internal loop, is deﬁned by

ψk+1 =ψk+τg, (21)

with the scaling factor τbeing determined by means of a line search technique.

In the (l+ 1)-th external loop, the Lagrange multiplier λis updated using the

Uzawa algorithm

λl+1 = max(0, λl+αkCh−ˆ

Ck).(22)

The penalty parameter αis hold ﬁxed during the full process.

A local optimality criterion of problem (17), see Amstutz [46], is given by the

condition

DψT>0 ; ∀y∈Ωµ(23)

which can be implemented by verifying the inequality

"RΩµgψ dV

kgkL2kψkL2#>(1 −tolψ) ; (24)

combined with

kCh−ˆ

Ck< tolC; (25)

40

Additionally, a Helmoltz-type ﬁlter taken from [36] is implemented. The smooth

level set function ˜

ψin each iteration (k+ 1)-iteration is computed by solving the

ﬁeld equation:

r2

ls∇2˜

ψk+1 +˜

ψk+1 =ψk+1 (26)

with homogeneous boundary conditions d(˜

ψk+1)/dn= 0 on the boundary of Ωµ.

The ﬁlter measure rls deﬁnes the minimum length scale in the topology optimization

problem.

B APPENDIX: Solving the topology optimiza-

tion problem with SIMP

The SIMP technique reported in Bendsoe and Sigmund [3] is here adopted. In this

technique, the design domain Ωµis subdivided using a ﬁnite element mesh, and each

element eis assigned a density ρe. This density represents the presence of soft or

stiﬀ materials (zero density for soft phase, unit density for stiﬀ phase) by deﬁning a

non-linear interpolation of the elasticity tensors, C2for soft and C1for stiﬀ phases,

as follows:

Ce(ρe) = C2+ (ρe)p(C1−C2).(27)

This is a smooth transition of elastic properties with density varying in the interval

0≤ρe≤1. The intermediate densities (0 < ρe<1) are penalized by deﬁning the

power law of ρwith exponent p > 1.

Given the distribution of densities in Ωµand the interpolated elasticity tensors

(27), the homogenized elasticity tensor, Chcan be computed through standard

procedures. This notation: Ch(ρ), remarks the dependence of Chwith the spatial

distribution of the so-deﬁned densities.

B.1 Topology optimization problem

The inverse homogenization technique can be formulated as follows:

min

ρZΩµ

ρ dΩ,

such that: kCh(ρ)−ˆ

Ck= 0 .

(28)

B.2 Density ﬁlter

The density ﬁlter taken from [36] is also implemented in the context of the SIMP

methodology. The density of an element ρeis taken as being the weighted average

of the design variables of the neighbor ﬁnite elements. The neighborhood is deﬁned

41

as a circle in 2D with the radius R. Then, the application of a density ﬁlter leads to

the variables ρe:

ρe=PQ

i=1 we

iΩiρi

PQ

i=1 re

iρi

(29)

where Qis the number of elements in the neighborhood of element edeﬁned by R,

re

iis the distance from element ito elements e, Ωiis the volume of element i,we

iis

a weighting factor deﬁned as:

we

i=max(0, R −re

i)

B.3 Volume preserving Heaviside ﬁlter

The ﬁltered variables ρeare further transformed into element densities ˆ

ρeby means of

an additional Heaviside ﬁlter: values smaller than a threshold value g are projected

to 0; values larger than ηare projected to 1. In this work it is used a Volume

preserving Heaviside ﬁlter ([47]).

ˆ

ρe=

η[e−β(1−ρe/η)−(1 −ρe/η)e−β] 0 ≤ρe≤η

(1 −η)[1 −e−β(ρe−η)/(1−η),

+(ρe−η)e−β/(1 −η)η < ρe≤1

(30)

where βis a parameter taken as β≥1. When the Heaviside ﬁlter is applied in each

iteration, the volume fraction is preserved by satisfying the following equation:

N

X

i=1

ρiΩi=

N

X

i=1

ˆ

ρi(η) Ωi.(31)

where, the left-hand side is the volume before the ﬁltering, and the right-hand side

is the volume after applying the ﬁlter. This equation is satisﬁed by adjusting the

parameter ηwith a bisection iterative method. The number of elements is N.

Finally, the objective function and the homogenized elasticity tensor Chin prob-

lem (28) are computed with the ﬁltered density ˆ

ρe.

B.4 Procedure for imposing the plane group symmetry on

the micro-architecture topology

The design variable are updated at the end of each iteration of the topology op-

timization algorithm to satisfy the symmetry requirement speciﬁed by the plane

group. For the level set methodology, the design variable update is computed ac-

cording to the following sequence of operations. Initially, the set of points of the

42

spatial domain related through the space group symmetry operations are found. One

node of this set is chosen as the master one, and the remaining are the slave ones.

At each iteration: 1) calculate element sensitivities and project to nodes; 2) com-

pute the averaging of sensitivities corresponding to nodes that are linked through

the symmetry operations, 3) using equation (20), update the level-set variable only

for the master nodes and 4) copy the value of the master node level-set variable

to all the slave nodes. For the SIMP methodology, the updating procedure is the

following: 1) calculate sensitivities for all elements; 2) identifying “master elements”

or design variables, 3) compute the averaging of sensitivities that are linked through

the symmetry operations, 4) solve the optimization update only for the reduced set

of master elements, 5) copy the master design variable value to the linked elements.

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