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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 composites , 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 accomplishment of the required symmetry of the composite homogenized elasticity tensor combined with the topology simplicity, are assessed through numerical simulations 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.
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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.
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: (A.E. Huespe).
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
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
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.
Accepted in Journal of the Mechanics and Physics of Solids, March, 2019
1.1 Inverse material design as a topology optimization prob-
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
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
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) = 0y2
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:
χ d
such that: kCh(χ)ˆ
Ck= 0 .
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
Accepted in Journal of the Mechanics and Physics of Solids, March, 2019
Scale of the material
Macro-structure scale
Phase M (c=0)
2 2
(domain )
Phase M (c=1)
(domain )
Micro-cell W
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
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
nfold 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
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.
Bravais lattice: Oblique
(a) (b)
Wigner-Seitz cell
Wigner-Seitz cell
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
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.
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.
Accepted in Journal of the Mechanics and Physics of Solids, March, 2019
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.
Accepted in Journal of the Mechanics and Physics of Solids, March, 2019
0.250.1 0.5 0.75
0.250.1 0.5 0.75
W=(0.5,30) W'=(0.967,75)
Rp Rp Rp
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
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
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.
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.
Oblique Rectangular Square
symmetry angle
symmetry angle
symmetry angle
symmetry line
reflection line
symmetry angle
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
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
2.2 Elasticity tensor structures according with their sym-
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.
Elasticity tensors CN
O(2) (2)
0 0 (c1c2)
0 0 c3
0 0 c3
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
and Z2for fully Anisotropic materials. From higher to lower symmetry classes, they
are: O(2) D4D2Z2. 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
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
D4 square
4 p4
4mm p4mm
O(2) hexagonal
3 p3
3m p3m1
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.
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
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
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.
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,
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.
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.
Assemblage Plane Point Elastic Sym- Ch
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).
r = 0
=V1 V2
r > 0
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
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
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-
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
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) 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.
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
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
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
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
0 0.04 0.08 0.12 0.200.16 0 0.04 0.08 0.12 0.200.16
Effective bulk modulus
(of planar elasticity) K
Effective shear
modulus G
f = 0.5
Effective bulk modulus
(of planar elasticity) K
Effective Poisson ratio
(of plane elasticity theory) v*
Walpole point
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 ˆνwith 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
0.082739 0.057930 0
0.057930 0.082739 0
0 0 0.140669
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
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
χ 1
χ df1!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
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 kChˆ
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.
9 10
9 10
5 6
9 10
0 0.015 0.02 0.025 0.03 0.035
1 2 3 45 6 7 8 9 10
Run Number
Effective bulk modulus (of planar elasticity) K
Effective shear modulus G
Cherkaev et al.
upper bound
Non-isotropy coefficient t
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
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]):
where Ciso is considered as a projection of Chonto the space of isotropic tensors
and is computed with the formula:
Ciso =
Kiso +Giso Kiso Giso 0
Kiso Giso Kiso +Giso 0
0 0 2Giso
Kiso =1
11 + 3Ch
22 + 2Ch
12 + 2Ch
33)Giso ,
Giso =1
11 +Ch
22 2Ch
12 + 2Ch
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
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.
1 2 3 4 5 6 7 8 9 10
||C -C ||
h||C ||
1 2 3 4 5 6 7 8 9 10
||C -C ||
h||C ||
1 2 3 4 5 6 7 8 9 10
||C -C ||
h||C ||
1 2 3 4 5 6 7 8 9 10
||C -C ||
h||C ||
1 2 3 4 5 6 7 8 9 10
||C -C ||
h||C ||
Figure 10: Normalized errors to satisfy the constraint kChˆ
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,
1 5
3 7
3 6
0 0.015 0.02 0.025 0.03 0.035
Instance 1
Instance 3
Instance 7
Instance 4 Hp1
Instance 9
Instance 3
Instance 6
Instance 2
Instance 5
Instance 1
Instance 8
Instance 5
Instance 2
Instance 3
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.
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.
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:
0.0392 0.0578 0.0385
0.0578 0.1050 0.0931
0.0385 0.0931 0.1100
0.0364 0.0462 0.0211
0.0462 0.0785 0.0631
0.0211 0.0631 0.0846
The same tensors in Normal coordinates, denoted ˆ
Nand ˆ
N, are expressed by:
0.2254 0.0031 0.
0.0031 0.0281 0.
0.1624 0.0043 0.
0.0043 0.0363 0.
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
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
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
Nand ˆ
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
Material I Material II
Homogeneous initial configuration Random initial configuration
f=0.29 f=0.26 f=0.22
f=0.51 f=0.32 f=0.26 f=0.23
Figure 12: Four examples of micro-architecture designs. Target tensors are: ˆ
Nfor material I; ˆ
CII and ˆ
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.
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
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
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.
Inital configuration:
random density
Inital configuration:
uniform density
Inital configuration:
taken from database
Composite I Composite II
= 90.deg
f =0.25 PA
= 90.deg
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-
Material I Material II
(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].
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
Pentamode materials are a special class of linear anisotropic elastic solids. They
can be characterized through elasticity tensors represented by:
C=κSS; (14)
where κis 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
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:
5.893 2.250 0.
2.250 0.8590 0.
This tensor corresponds to a pentamode material with D2symmetry. The composite
is designed to copy these elastic properties.
= 81.deg
0.30 f =0.175
PA= 90.deg
0.152 f =0.175
(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;
A.1 B.1
B.2 - Instance 1
B.2 - Instance 2
A.2 - Instance 1
A.2 - Instance 2
f = 0.151
f = 0.309
f = 0.334
f = 0.133
f = 0.279
f = 0.166
f = 0.119
Figure 16: Pentamode material designs. Topologies attained with sub-cases A1, A.2
and B. The volume fraction of each case is f1.
B.2 (five instances)
A.1 (with filter)
A.1 (without filter)
1 2 3 4 5
Relative error
Figure 17: Pentamode material design. Relative errors kChˆ
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 kChˆ
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.
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.
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-
The topology optimization problem (2) can be solved by introducing a smooth level
set-funtion, ψC0(Ωµ), satisfying
ψ(y) =
0 in the interfaces
and utilizing an augmented Lagrangian technique, see Lopes et al. [45]. In this case,
the problem is rewritten as follows
ψT(ψ, λ),(17)
T(ψ, λ) = Zµ
χ(ψ)dΩ + λ(kCh(ψ)ˆ
Ck) + α
where λis the Lagrange multiplier and αis the penalty parameter of the augmented
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 λαkChˆ
C) : DψCh
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
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+αkChˆ
The penalty parameter αis hold fixed during the full process.
A local optimality criterion of problem (17), see Amstutz [46], is given by the
DψT>0 ; yµ(23)
which can be implemented by verifying the inequality
"Rµgψ dV
kgkL2kψkL2#>(1 tolψ) ; (24)
combined with
Ck< tolC; (25)
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:
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
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(C1C2).(27)
This is a smooth transition of elastic properties with density varying in the interval
0ρe1. 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:
ρ d,
such that: kCh(ρ)ˆ
Ck= 0 .
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
as a circle in 2D with the radius R. Then, the application of a density filter leads to
the variables ρe:
i=1 we
i=1 re
where Qis the number of elements in the neighborhood of element edefined by R,
iis the distance from element ito elements e, Ωiis the volume of element i,we
a weighting factor defined as:
i=max(0, R re
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]).
η[eβ(1ρe)(1 ρe)eβ] 0 ρeη
(1 η)[1 eβ(ρeη)/(1η),
+(ρeη)eβ/(1 η)η < ρe1
where βis a parameter taken as β1. When the Heaviside filter is applied in each
iteration, the volume fraction is preserved by satisfying the following equation:
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
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
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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... All 2D lattice geometries can be classified into 10 point groups depending on their symmetry property [29,39]. There are seven symmetry operations in a 2D plane: the identity transformation (1), 2-fold rotation ( ), 3-fold rotation ( ), 4-fold rotation ( ), 6-fold rotation ( ), and mirror with respect to the 1 -axis ( ) and 2 axis ( ). ...
... In general, a lattice structure can possess more than one symmetry operation; a lattice geometry can be classified into multiple point groups based on the symmetry operations to which it belongs [39]. The micropolar elasticity tensor of a point group must remain unchanged under all the symmetry operations of this point group. ...
... . From Table 1, we can classify the micropolar elasticity tensor for every point group together with the corresponding symmetry operations, as shown in Table 4. Note that there are two types of elasticity tensors for the point groups and 3 depending on the mirror symmetry axes ( 1 and 2 ), as indicated in Table 4. Thus, we have 12(= 10 + 2) types of micropolar elasticity tensor compared with three types in the Cauchy elasticity tensor [39]. The point group 3 ensures isotropy in the Cauchy elasticity [42]. ...
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Mechanical couplings such as axial-shear and axial-bending have great potential in the design of active mechanical metamaterials with directional control of input and output loads in sensors and actuators. However, the current ad hoc design of mechanical coupling without theoretical support of elasticity cannot provide design guidelines for mechanical coupling with lattice geometries. Moreover, the correlation between mechanical coupling effects and geometric symmetry is not yet clearly understood. In this work, we systematically search for all possible mechanical couplings in 2D lattice structures by determining the non-zero diagonal terms in the decomposed micropolar elasticity tensor. We also correlate the mechanical couplings with the point-group symmetry of 2D lattices by applying the symmetry operation to the decomposed micropolar elasticity tensor. The decoupled micropolar constitutive equation uncovers eight coupling effects for 2D lattice structures. The symmetry operation of the decoupled micropolar elasticity tensor reveals the correlation of the mechanical coupling with the point groups. Our findings can strengthen the design of mechanical metamaterials with potential applications in areas including sensors, actuators, soft robots, and active metamaterials for elastic/acoustic wave guidance and thermal management.
... Formulation of the topology optimization problem. Following the design methodology presented in Podestá et al. (2019), we select beforehand the symmetry of the material configuration to be attained after solving the problem. This symmetry also stipulates the shape of the design domain Ω µ . ...
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Purpose: This work presents a topology optimization method for designing mi-croarchitectures of phononic crystals. The objective is to get microstructures having, as a consequence of wave propagation phenomena in these media, bandgaps between two specified bands. An additional target is to enlarge the range of frequencies of these bandgaps. Design/methodology/approach: The resulting optimization problem is solved employing an augmented Lagrangian technique based on the proximal point methods. The main primal variable of the Lagrangian function is the characteristic function determining the spatial geometrical arrangement of different phases within the unit cell of the phononic crystal. This characteristic function is defined in terms of a level-set function. Descent directions of the La-grangian function are evaluated by using the topological derivatives of the eigenvalues obtained through the dispersion relation of the phononic crystal. Findings: The description of the optimization algorithm is emphasized and its intrinsic properties to attain adequate phononic crystal topologies are discussed. Particular attention is addressed to validate the analytical expressions of the topological derivative. Application examples for several cases are presented, and the numerical performance of the optimization algorithm for attaining the corresponding solutions is discussed. Originality: the original contribution results in the description and numerical assessment of a topology optimization algorithm using the joint concepts of the level-set function and topological derivative to design phononic crystals.
... Previous studies showed that the mechanical and acoustic properties of representative PMs are sensitive to the structural parameters (Cai et al., 2016) and anisotropy degree . Recently, it was shown that topology optimization based on elasticity tensor and microstructural symmetries can realize static microstructures in the rectangular lattice for a target pentamode tensor by using the prescribed initial geometry conditions (Podestá et al., 2019). Inspired by similar crystal symmetry, topology optimization of only minimizing the effective static shear modulus (Year et al., 2020) also generated three isotropic pentamode microstructures. ...
Pentamode metamaterials (PMs), a kind of metafluids composed of complex solid medium, have shown enormous potential for both elastic wave and underwater acoustic wave manipulation. However, due to the lack of thorough understanding of the formation mechanism, most reported artificial and empirical PMs share very similar topological features, thus depriving the possibility of obtaining rigorous combination of wave parameters that are required to deliver desirable and prescribed properties and functionalities. To tackle this challenge, with the assumption of C2v-, C4v- and C6v-symmetries in both square and triangle lattices, we propose a unified inverse strategy to systematically design and explore a series of novel isotropic or anisotropic PM microstructures through bottom-up topology optimization. Optimized PM microstructures are designed to provide customized effective mass density, elastic modulus, anisotropy degree and pentamode features on demand. We demonstrate that most optimized microstructures possess broadband single-mode range of exclusive longitudinal waves; some even feature record-breaking relative single-mode bandwidths exceeding 150%. Upon shielding lights on the beneficial topological features of the broadband PMs, we extract the main topological features to form simplified PM configurations, i.e., multiple symmetric solid blocks with slender rods, which can induce the multiform multiple-order rotational vibrations or the integration of the low-order rotational vibrations and anisotropic local resonances for the broadband single-mode nature. At a higher design level, we establish a dedicated inverse-design strategy, under the function-macrostructure-microstructure paradigm, to conceive a novel broadband subwavelength underwater pentamode shielding device, which enables the conversion of propagating acoustic wave to the evanescent surface wave mode within the frequency range [1000 Hz, 4000 Hz]. Our study offers new possibilities for the practical realization of broadband PMs and underwater pentamode devices with rigorously tailored effective parameters, thus bring the PM-based technology within reach for practical applications.
... Recently, Wang et al. (2019) showed that near-optimal and periodic truss lattice structures could be obtained for multiple load cases by distorting simple Bravais-like lattice structures to a parallelepiped. Besides using a parallelogram in 2D or a parallelepiped in 3D, one can use many more different types of polygons to solve the homogenization equations (Barbarosie et al. 2017;Podestá et al. 2019). For example, in 2D, a hexagon can be used to describe a periodic isotropic hexagonal microstructure (Sigmund 2000). ...
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Multi-scale structures, as found in nature (e.g., bone and bamboo), hold the promise of achieving superior performance while being intrinsically lightweight, robust, and multi-functional. Recent years have seen a rapid development in topology optimization approaches for designing multi-scale structures, but the field actually dates back to the seminal paper by Bendsøe and Kikuchi from 1988 (Computer Methods in Applied Mechanics and Engineering 71(2): pp. 197–224). In this review, we intend to categorize existing approaches, explain the principles of each category, analyze their strengths and applicabilities, and discuss open research questions. The review and associated analyses will hopefully form a basis for future research and development in this exciting field.
... Originally initiated in the field of crystallography, the study of the links between spatial invariances of physical phenomena and the invariances of the underlying matter have now spread all over engineering sciences. The tools initially introduced by Curie [15] are central to contemporary materials science where materials are designed for specific applications [1,9,46,53]. These materials that can be termed composites, architectured or meta have in common the fact that their internal geometry is specifically designed to produce, or inhibit, physical couplings within the matter. ...
... Combined with the symmetry theory, the Cauchy elasticity also indicates an axial-shear coupling of lattice structures. Oblique structures having either two-fold rotational symmetry or no symmetry are known to have an axial-shear coupling [30]. The new finding of the axial-shear coupling with four-fold rotational symmetry in the tetra-chiral lattice is significant, demonstrating the micropolar elasticity's superiority in identifying coupling effects. ...
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Chiral lattices are generally considered to possess auxetic properties with negative Poisson's ratios and high compressibility. However, the effects of anisotropy and handedness (chirality) on their mechanical properties, such as directional moduli, Poisson's ratio, and other coupling effects, are not clearly understood due to the lack of analytical methods to handle both anisotropy and chirality. Herein, we construct a generalized micropolar homogenization method to characterize the elastic constants of tetra-chiral and tetra-achiral lattices having different joint types. The developed method can identify the directional moduli and Poisson's ratio and coupling effects in the anisotropic wavy lattices. The generalized micropolar homogenization of the wavy square lattices reveals i) the axial–shear coupling effect of tetra-chiral structures, ii) the weak correlation of the chirality of tetra-chiral lattices to auxeticity, and iii) the directional positive and negative Poisson's ratio of tetra-achiral lattices. This work offers a powerful platform for metamaterials' design via geometric reconfiguration of lattice structures to adjust both anisotropy and chirality.
... Originally initiated in the field of crystallography, the study of the links between spatial invariances of physical phenomena and the invariances of the underlying matter have now spread all over engineering sciences. The tools initially introduced by Curie [15] are central to contemporary materials science where materials are designed for specific applications [1,9,46,53]. These materials that can be termed composites, architectured or meta have in common the fact that their internal geometry is specifically designed to produce, or inhibit, physical couplings within the matter. ...
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The piezoelectricity law is a constitutive model that describes how mechanical and electric fields are coupled within a material. In its linear formulation this law comprises three constitutive tensors of increasing order: the second order permittivity tensor S, the third order piezoelectricity tensor P and the fourth-order elasticity tensor C. In a first part of the paper, the symmetry classes of the piezoelectricity tensor alone are investigated. Using a new approach based on the use of the so-called clips operations, we establish the 16 symmetry classes of this tensor and provide their associated normal forms. Second order orthogonal transformations (plane symmetries and-angle rotations) are then used to characterize and classify directly 11 out of the 16 symmetry classes of the piezoelectricity tensor. An additional step to distinguish the remaining classes is proposed.
Mechanical couplings such as axial–shear and axial–bending have great potential in the design of active mechanical metamaterials with directional control of input and output loads in sensors and actuators. However, the current ad hoc design of mechanical coupling without theoretical support of elasticity cannot provide design guidelines for mechanical coupling with lattice geometries. Moreover, the correlation between mechanical coupling effects and geometric symmetry is not yet clearly understood. In this work, we systematically search for all possible mechanical couplings in 2D lattice structures by determining the non-zero diagonal terms in the decomposed micropolar elasticity tensor. We also correlate the mechanical couplings with the point-group symmetry of 2D lattices by applying the symmetry operation to the decomposed micropolar elasticity tensor. The decoupled micropolar constitutive equation uncovers eight coupling effects for 2D lattice structures. The symmetry operation of the decoupled micropolar elasticity tensor reveals the correlation of the mechanical coupling with the point groups. Our findings can strengthen the design of mechanical metamaterials with potential applications in areas including sensors, actuators, soft robots, and active metamaterials for elastic/acoustic wave guidance and thermal management.
Purpose The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept. Design/methodology/approach The optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional. Findings The optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms. Originality/value The proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.
To realize extraordinary wave phenomena, metamaterials need to attain unique effective material properties. In this work, we propose an inverse design strategy for metamaterials with specific anisotropic EMD (effective mass density). Although the conventional inverse homogenization technique has been extended to various fields, few works have been published to explore the inverse realization of an EMD tensor, each component of which is supposed to gain a given value at a target frequency. To this end, we propose a calculation scheme, in which the EMD tensor can be calculated in a much similar way to the homogenized static stiffness. Therefore, the scheme is quite convenient for sensitivity analysis. The coating layer interfacing the core and matrix is chosen as the design region because it directly determines the motion of the core. The matrix layout, which not only contributes to the stiffness of the metamaterial but also highly affects the core's local motion, is chosen carefully. The perfect transmodal Fabry–Pérot interference phenomenon is considered in this work, and through several numerical examples, the phenomenon is ideally realized. The proposed design strategy could be critically useful in designing locally resonant metamaterials with general anisotropy.
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New tools for the design of metamaterials with periodic micro-architectures are presented. Initially, a two-scale material design approach is adopted. At the structure scale, the material effective properties and their spatial distribution are obtained through a Free Material Optimization (FMO) technique. At the micro-structure scale, the material micro-architecture is designed by appealing to a Topology Optimization Problem (TOP). The TOP is based on the topological derivative and the level set function. The new proposed tools are used to facilitate the search of the optimal micro-architecture configuration. They consist of the following: i) a procedure to choose an adequate shape of the unit-cell domain where the TOP is formulated. Shapes of Voronoi-cells associated with Bravais lattices are adopted. ii) a procedure to choose an initial material distribution within the Voronoi cell being utilized as the initial configuration for the iterative topology optimization algorithm.
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By combining the two basic deformation mechanisms for auxetic open-cell metamaterials, re-entrant angle and chirality, new hybrid chiral mechanical metamaterials are designed and fabricated via a multi-material 3D printer. Results from mechanical experiments on the 3D printed prototypes and systematic Finite Element (FE) simulations show that the new designs can achieve subsequential cell-opening mechanism under a very large range of overall strains (2.91%-52.6%). Also, the effective stiffness, the Poisson's ratio and the cell-opening rate of the new designs can be tuned in a wide range by tailoring the two independent geometric parameters: the cell size ratio [Formula: see text], and re-entrant angle θ. As an example application, a sequential particle release mechanism of the new designs was also systematically explored. This mechanism has potential application in drug delivery. The present new design concepts can be used to develop new multi-functional smart composites, sensors and/or actuators which are responsive to external load and/or environmental conditions.
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Materials and structures with negative Poisson’s ratio exhibit a counter-intuitive behaviour. Under uniaxial compression (tension), these materials and structures contract (expand) transversely. The materials and structures that possess this feature are also termed as ‘auxetics’. Many desirable properties resulting from this uncommon behaviour are reported. These superior properties offer auxetics broad potential applications in the fields of smart filters, sensors, medical devices and protective equipment. However, there are still challenging problems which impede a wider application of auxetic materials. This review paper mainly focuses on the relationships among structures, materials, properties and applications of auxetic metamaterials and structures. The previous works of auxetics are extensively reviewed, including different auxetic cellular models, naturally observed auxetic behaviour, different desirable properties of auxetics, and potential applications. In particular, metallic auxetic materials and a methodology for generating 3D metallic auxetic materials are reviewed in details. Although most of the literature mentions that auxetic materials possess superior properties, very few types of auxetic materials have been fabricated and implemented for practical applications. Here, the challenges and future work on the topic of auxetics are also presented to inspire prospective research work. This review article covers the most recent progress of auxetic metamaterials and auxetic structures. More importantly, several drawbacks of auxetics are also presented to caution researchers in the future study.
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Structural topology optimization problems are commonly defined using continuous design variables combined with material interpolation schemes. One of the challenges for density based topology optimization observed in the review article (Sigmund and Maute Struct Multidiscip Optim 48(6):1031–1055 2013) is the slow convergence that is often encountered in practice, when an almost solid-and-void design is found. The purpose of this forum article is to present some preliminary observations on how designs evolves during the optimization process for different choices of optimization methods. Additionally, the authors want to open a discussion on how to properly define and identify the boundary translation that is often observed in practice. The authors hope that these preliminary observations can open for fruitful discussions and stimulate further investigations concerning slowly moving boundaries. Although the discussion is centered on density based methods it may be equally relevant to level-set and phase-field approaches.
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A topology optimization technique based on the topological derivative and the level set function is utilized to design/synthesize the micro-structure of a pentamode material for an acoustic cloaking device. The technique provides a micro-structure consisting of a honeycomb lattice composed of needle-like and joint members. The resulting metamaterial shows a highly anisotropic elastic response with effective properties displaying a ratio between bulk and shear moduli of almost 3 orders of magnitude. Furthermore, in accordance with previous works in the literature, it can be asserted that this kind of micro-structure can be realistically fabricated. The adoption of a topology optimization technique as a tool for the inverse design of metamaterials with applications to acoustic cloaking problems is one contribution of this paper. However, the most important achievement refers to the analysis and discussion revealing the key role of the external shape of the prescribed domain where the optimization problem is posed. The efficiency of the designed micro-structure is measured by comparing the scattering wave fields generated by acoustic plane waves impinging on bare and cloaked bodies.
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The surge of interest in so-called “designer materials” during the last few years together with recent advances in additive manufacturing (3D printing) techniques that enable fabrication of materials with arbitrarily complex nano/micro-architecture have attracted increasing attention to the concept of mechanical metamaterials. Owing to their rationally designed nano/micro-architecture, mechanical metamaterials exhibit unusual properties at the macro-scale. These unusual mechanical properties could be exploited for the development of materials with advanced functionalities, with applications in soft robotics, biomedicine, soft electronics, acoustic cloaking, etc. Auxetic mechanical metamaterials are identified by a negative Poisson's ratio and are perhaps the most widely studied type of mechanical metamaterials. Similar to other types of mechanical metamaterials, the negative Poisson's ratio of auxetics is generally a direct consequence of the topology of their nano/micro-architecture. This paper therefore focuses on the topology–property relationship in three main classes of auxetic metamaterials, namely re-entrant, chiral, and rotating (semi-) rigid structures. While the deformation mechanisms in the above-mentioned types of structures and their relationship with the large-scale mechanical properties receive most attention, the emerging concepts in design of auxetics such as the use of instability in soft matter and origami-based structures are discussed as well. Furthermore, the data available in the literature regarding the elastic properties of auxetic mechanical metamaterials are systematically analyzed to identify the spread of Young's modulus–Poisson's ratio duos achieved in the auxetic materials developed to date.
Additive manufacturing (AM) enables fabrication of multiscale cellular structures as a whole part, whose features can span several dimensional scales. Both the configurations and layout pattern of the cellular lattices have great impact on the overall performance of the lattice structure. In this paper, we propose a novel design method to optimize cellular lattice structures to be fabricated by AM. The method enables an optimized load-bearing solution through optimization of geometries of global structures and downscale mesostructures, as well as global distributions of spatially-varying graded mesostructures. A shape metamorphosis technology is incorporated to construct the graded mesostructures with essential interconnections. Experimental testing is undertaken to verify the superior stiffness properties of the optimized graded lattice structure compared to the baseline design with uniform mesostructures.
Materials that become thicker when stretched and thinner when compressed are the subject of this review. The theory behind the counterintuitive behavior of these so-called auxetic materials is discussed, and examples and applications are examined. For example, blood vessels made from an auxetic material will tend to increase in wall thickness (rather than decrease) in response to a pulse of blood, thus preventing rupture of the vessel (see Figure).
The present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian , i.e. orthotropic, square unit cell and nonabelian , i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a symmetric hexagonal unit cell.