Content uploaded by Yi Min Xie
Author content
All content in this area was uploaded by Yi Min Xie on Jan 15, 2023
Content may be subject to copyright.
1
Article published in
Composite Structures, Vol. 306 (2023) 116609
https://doi.org/10.1016/j.compstruct.2022.116609
Topology optimization of structures composed of more than two materials with
different tensile and compressive properties
Yu Lia,b, Philip F. Yuana, Yi Min Xieb,*
a College of Architecture and Urban Planning, Tongji University, Shanghai, 200092, China
b Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne, 3001, Australia
* Corresponding author. E-mail address: mike.xie@rmit.edu.au (Y.M. Xie)
Abstract:
A novel method is proposed to optimize the topology of composite structures made of more
than two materials with different mechanical properties in tension and compression. In this method,
the design domain of the structure is divided into tensile and compressive regions according to the
first invariant of the stress tensor. Then two groups of materials suitable for tension and
compression are arranged in the tensile and compressive regions of the structure, respectively.
Using a bridge-type beam with a concentrated force as an example, the study of the four-material
topologically optimized structures reveals the effects of the volume fractions and material
mechanical properties on the optimization results. Further, the three-material topology optimization
method, which is derived from the four-material topology optimization method, is used to design a
series of novel sandwich structures. Application examples demonstrate that the proposed method
can achieve a balance between enhancing structural stiffness and saving material costs, providing
solutions competitive in various aspects and exploiting the performance and potential of different
materials better than previous single- or dual-material topology optimization methods.
Keywords: Bi-directional evolutionary structural optimization (BESO), multi-material topology
optimization, multiple materials, sandwich structure.
1. Introduction
Since the 1980s, with the rapid advances in computer technology, topology optimization
techniques based on finite element analysis have been widely developed, providing a new strategy
2
for the automatic shape-finding of structures [1]. The commonly used topology optimization
methods mainly include the solid isotropic material with penalization (SIMP) method [2–4], the
level set method [5–7], and the evolutionary structural optimization (ESO) method [8–10]. Among
these methods, the ESO method and its advanced version, the bi-directional evolutionary structural
optimization (BESO) method, are popular among many researchers and designers because of their
simplicity and the availability of well-developed commercial software such as Ameba [11]. The
ESO method and the BESO method have been widely used in various fields such as structure,
architecture, aerospace, medicine, and biomechanics [12–14]. Besides, some new structural
optimization algorithms have been developed, such as YUKI, Jaya, and Cuckoo [15–17], which
provide new strategies and perspectives for structural optimization and are promising to be applied
to future topology optimization to enhance the efficiency of the computation. However, there are
still some modifications to be done to combine these novel algorithms with topology optimization
algorithms.
Although having been extensively studied and applied, these classical topology optimization
algorithms need to be improved for dealing with complex structures, such as composite structural
systems. Most research [18–20] in the field of topology optimization has long been based on a
single material, and how to optimize the topology of composite structures composed of multiple
materials has always been a challenging problem. For the topology optimization of dual-material
structures, previous studies [21–24] typically set high-efficiency materials for critical regions of the
structure with high stress or sensitivity number and set inefficient materials for unimportant parts
where the stress or sensitivity number is low. For example, based on the material interpolation
scheme, a new BESO method [25] with penalty parameters is developed and extended to
multi-material topology optimization. Some researchers [26,27] have presented a framework for
multi-material compliance minimization by solving a series of explicit convex (linear)
approximations to the volume-constrained compliance minimization problem. Similarly, the
compliance-based multi-material topology optimization design method [28] is proposed to prevent
the propagation of crack patterns.
The material arrangement strategies mentioned above are reasonable, but they have limitations
when applied to actual composite structures because the fact that some materials show completely
different mechanical properties in tension and compression has not been taken into consideration.
3
For example, when a structure is composed of steel and aluminum, these strategies are applicable
because each of the two metals can be considered to have an equal Young’s modulus in tension and
compression, but it is not applicable to a structure made of steel and concrete since it is obvious that
concrete has an excellent performance in compression but weak performance in tension. To solve
this problem, the design domain of the structure can be divided into tensile and compressive regions,
and a material suitable for tension and a material suitable for compression are arranged in these two
areas, respectively. Based on this strategy, in the steel-concrete composite structure, steel and
concrete can be arranged only in the tensile and compressive regions of the structure, significantly
improving the efficiency of material utilization and bringing into play the respective potentials of
multiple materials [29–31, 46].
Building upon the two strategies mentioned above, this study combines these two methods and
divides the multiple materials into two groups: materials suitable for tension and materials suitable
for compression. These two groups of materials are arranged in the tensile and compressive regions
of the structure, respectively. At the same time, in the single tensile or compressive region,
high-rigidity materials are arranged in the area where the sensitivity numbers are relatively high,
while low-rigidity materials are arranged in the area with relatively low sensitivity numbers. Since
the two criteria, including the different mechanical properties of a single material under tension and
compression and the difference in mechanical properties between different materials under the same
stress state (tension or compression), are considered simultaneously, this method can further exploit
the efficiency of materials and save the use of high-performance materials.
The paper is organized as follows: Section 2 presents the methodology and the main procedures
of the proposed method. In Section 3, based on a beam with a concentrated force, the effects of
material volume fractions and material mechanical properties on the results of four-material
topology optimization are comprehensively studied, and the three-material topology optimization
method is then developed. Subsequently, in Section 4, two practical applications based on the
proposed method are carried out to demonstrate the advantages of the method in cost control and
enhancing structural stiffness. Finally, the conclusions are drawn in Section 5.
2. Methodology
2.1 Problems statements
4
The topology optimization of a structure composed of one solid material that has different
properties in tension and compression can actually be regarded as a dual-material topology
optimization problem. The material, MT, which has better performance in tension, can be allocated
in the tension region of the structure, and the other material, MC, suitable for compression, can be
allocated in the compressive region.
Whether a material is suitable for tension or compression depends on the comparison of the
tensile and compressive mechanical properties, the price, and the nonsubstitutability of the material.
For example, for a material like concrete, the compressive properties are much better than its tensile
properties, so it can be intuitively concluded that it is suitable for compression. For a metal material
such as steel, it can be generally considered that the tensile and compressive properties are the same.
However, when forming a composite structure with concrete, which is quite cheap and suitable for
compression, steel is considered more suitable for tension than compression.
In stiffness-based topology optimization problems, the mean compliance is used as the
objective function to find the optimal structures. In the dual-material case, the optimization problem
can be stated as
Minimize: =
Subject to:
= 0
= 0
(1)
= 0
= or 1
= or 1
where C is the total strain energy or the mean compliance of the structure. K is the global stiffness
matrix, and u is the displacement vector. V* is the target volume fraction of all the solid materials,
and Vi is the volume of the i-th element. and are the total volume fractions of all the
tensioned and compressed materials, respectively. N is the total number of elements in the design
domain. For the dual-material structure, xi and yi are two variables used to describe the material
layout. xi = 1 or xmin indicates the presence or absence of the tensioned solid material in the i-th
element. Similarly, yi = 1 or ymin describes the compressed solid material in the i-th element. To
avoid singularity in the stiffness matrix, two small values, xmin and ymin, are used, which equal 0.001
and 0.002 in this study. As for the volume fraction of the tensioned or compressed region, it cannot
5
be predetermined for the final structure before the start of the optimization procedure, but it can be
calculated for the current structure according to Eq. (5) in each iteration.
When the structure is made of 2n different materials where n kinds of materials are used for
tension and the other n materials are applied for compression, the 2n non-zero solid materials can be
classified into two groups. One group is called the tensioned group, while the other is named the
compressed group. Young’s moduli of the two groups of materials are assumed Et,1 > Et,2 > … > Et,n
and Ec,1 > Ec,2 > … > Ec,n. The optimization problem can be stated as
minimize: =
Subject to:
= 0
,
,
= 0
,
,
= 0
(2)
,
=,
,
=,
= or 1 (= 1,2,...,1)
= or 1 (= 1, 2, . . . , 1)
where ,
and ,
are the volumes of the j-th tensioned and compressed materials, respectively.
xij and yij are the j-th groups of variables used to present the tensile and compressive state of the i-th
element, and their definition of the four-material topology optimization can be seen in Table 1. ,
or , is the ratio of the volume of material Et,j or Ec,j to the volume of all the tensioned or
compressed materials, which is a predetermined constant value. Like the dual-material topology
optimization problem, the total volumes of the tensioned and compressed materials cannot be
predetermined before the topology optimization, but they can be calculated according to Eq. (5) in
the optimization process. The two design variables, xij and yij, are defined as
=1 for >,, and in tension
for ,(), or in compression (3)
=1 for >,, and in compression
for ,(), or in tension (4)
2.2 Criterion of tension or compression
Judging whether an element is in tension or compression has always been a difficult problem.
In some past research [32], the first principal stress is used as the criterion, but the process is
6
complicated. Li and Xie [29,30] used the first invariant of stress tensors as the criterion for judging
an element in tension or compression and achieved excellent results. For a 3D element, the first
invariant of the i-th element , can be calculated as
,=,+,+, (= 1, 2, . . . , ) (5)
where ,, ,, , are the maximum principal stress, second principal stress, and minimum
principal stress, respectively. For 2D problems, only the maximum principal stress and the
minimum principal stress are needed.
In a physical sense, the first invariant of stress tensors represents the mean normal stress or
pressure [33]. The first invariant of stress tensors is also three times the hydrostatic stress (also
called volumetric stress) [34]. So, I1,i ≥ 0 means the i-th element is enlarged in volume, and it can be
regarded as in tension. While I1,i < 0 means the volume of the element is reduced, which can be
considered as compressed.
2.3 Material interpolation scheme and sensitivity numbers
By referring to the material interpolation scheme used in the original multi-material topology
optimization [25] and taking the compression and tension state into consideration, the material
interpolation scheme between the j-th and (j + 1)-th tensioned materials can be stated as:
=
,+1
,() (= 1, 2, ..., ) (6)
where p denotes the penalty coefficient, which is set to 3 in this study. Similarly, between the j-th
and (j + 1)-th compressed materials, the material interpolation scheme can be stated as
=
,+1
,() (= 1, 2, ..., ) (7)
Then, the two different sensitivity numbers αij and βij, which are global variables but used in tension
and compression region, can be calculated as
=
=
(,
,
) (8)
=
=
(,
,
) (9)
which can also be expressed as
7
=
[1 ,()
,],
for materials M,M,...M
(,,())
,(
),(),
for materials M(),M(),...,M,,...,M
(10)
=
[1 ,()
,],
for materials M, M, ..., M
(,,())
,(
),(),
for materials M(), M(), ..., M, M, ..., M
(11)
If the material MTj, which is supposed to be in tension according to the assumption, is
compressed during the optimization iteration, it can be regarded as the corresponding compressed
material MCj, and then its sensitivity numbers are calculated according to Eq. (11).
2.4 Volume evolution program, filter scheme, and stabilization process
Before the start of the topology optimization process, the whole design domain will be assigned
a single material MT1 or MC1. After the first iteration, the tension region and compression region of
the structure can be quickly identified according to Eq. (5). Meanwhile, the tension region will be
allocated MT1, and the compression region will be allocated MC1, respectively.
Then, the volumes of the MT1 and MC1 decrease gradually, and the total volume of all the solid
materials reduces simultaneously. At the same time, the volumes of other low-rigidity materials, like
MT2 and MC2, increase accordingly from 0 to their target volumes. To accelerate the process, the
evolutionary volume ratio of all the materials can be specially designed so that all the materials can
reach their defined volume fractions at the same time. For example, in the four-material topology
optimization in this study, the evolution rates are set as
,= 1 ,
() (12)
,= 1 ,
() (13)
where , and , are the evolutionary volume ratios of MT1 and MC1, and er is the
evolutionary volume ratio of all the materials. One thing that needs to be pointed out is the volume
evolution scheme shown in Eqs. (12) and (13) is specially designed for this study, and other slowly
changing volume evolution schemes are also acceptable.
2.5 Filter scheme and stabilization process
8
To avoid checkerboard patterns and mesh dependency, using a filter scheme is an effective
method [35–37]. The two groups of sensitivity numbers are modified as
=()
()
(14)
=()
()
(15)
where M is the total number of elements in the sub-domain , which has the radius of rim, and
() is the weight factor of the m-th element in the sub-domain . It can be calculated as
() = for <
0 for (16)
where rmin indicates the filter radius.
Although the same defined filter function and filter radius are used in one multi-material
structure, due to the differences in the mechanical properties of the materials, high-rigidity materials
tend to appear as fine components. This is because high-rigidity material components with smaller
cross-sections can provide similar contributions as low-rigidity material components with larger
cross-sections to the overall structure. This phenomenon can also be seen in previous studies of
multi-material topology optimization [21,24].
Next, to stabilize the iteration, the historical data is introduced to be averaged, and the
sensitivity numbers become
=
(,+,) (17)
=
(,+,) (18)
where k is the current iteration number.
2.6 Convergence criterion
The convergence of the evolution can be confirmed by calculating the relative change in the
compliance of the structure [38]. The iterations will be looped until the following convergence
criterion is satisfied and the volume fraction of each material is reached.
()|
(19)
Where k is the current iteration number, and S is the number of historical data involved, which is set
to 5 in this paper. τ is the allowable convergence error, which is set to 1 × 10-6 in this study.
2.7 Procedure
Take four-material topology optimization as an example. As shown in Fig. 1, the evolutionary
9
procedure of multi-material topology optimization based on the BESO method can be described as
follows:
Step 1: Discrete the design domain with finite element mesh, including the non-design domain.
Define relevant settings and parameters, such as loads, boundary conditions, and design variables.
Step 2: Define the volume fraction of each material and design the volume change program.
Step 3: Carry out the finite element analysis (FEA) and calculate sensitivity numbers,
compliance, displacements, and the first invariant of stress tensors I1.
Step 4: Filter the sensitivity numbers and average the sensitivity number with the
corresponding history data.
Step 5: Calculate the volumes of the materials for the next design.
Step 6: Update the design variables xi and yi to get a new design.
Step 7: Repeat steps 3-6 until the volume constraints are reached and the compliance
converges.
Fig. 1. Evolutionary procedure of the multi-material BESO method.
From the above flow and Fig. 1, it can be seen that in terms of computational cost, compared to
Start
Discrete the design domain.
Define design domian, loads,
boundary conditions and FE
mesh. Initialize x
i
and y
i
Carry out FEA and calculate
sensitivity number, compliance,
displacement, and I
1
Calculate the volumes of the
materials v
k
, v
t1,k
, v
t2,k
, v
c1,k
, v
c2,k
for the next design
Define V
*
, r
t,1
, r
c,1
, and design
the volume change schedule
Update the design variables x
i
and y
i
and get a new design
End
Filter the sensitivity number
and average the sensitivity
number with the corresponding
history data
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Is the compliance
converged?
Are the volume
constraints satisfied?
Yes
No
Yes
No
10
single-material topology optimization, only two main steps are added in the proposed approach,
including the determination of the tensile or compressive state for each element and the assignment
of different materials, while the time-costing FEA step is the same. In addition, due to the utilization
of the efficient algorithm [39], the computation time for most of the examples in this paper is within
about 30 minutes.
3. Numerical examples
3.1 Bridge-type beam made of four materials
Firstly, four-materials topology optimization is researched. Four solid and one void material,
MT1, MT2, MC2, MC1 plus void material, are used in the example of a 2D bridge-type beam, as shown
in Fig. 2. In this structure, the concentrated force is applied at the bottom of the mid-span, and the
left and right ends are supported by a hinge and a roller, respectively. Among the solid materials
used, MT1 and MT2 are suitable for tension, while MC2 and MC1 have better properties in
compression. Young’s moduli of the four materials are assumed as Et,1 > Et,2 > Emin and Ec,1 > Ec,2 >
Emin. In this example, they are set as Et,1: Et,2: Ec,1: Ec,2 = 4: 2: 3: 1. The volume fractions are set as
V* = 0.4, rt,1 = 0.3, rt,2 = 0.7, rc,1 = 0.6, and rc,2 = 0.4, which means the total volume fraction of all
the four solid materials to the design domain is 0.4, the volume ratio of MT1 to all the tensile
materials (MT1 and MT2) is 0.3, and the volume ratio of MC1 to all the compressed materials (MC1
and MC2) is 0.6.
Because of symmetry, only half of the structure is discrete with 600 × 200 four-node elements.
The filter radius is set to 0.04 m for all elements in the design domain, including solid elements and
void elements. The evolutionary volume ratio er is set to 2 %, which indicates 2 % of the solid
material will be deleted in every iteration. To represent the topology of the four-material structure,
four variables, xi1, xi2, yi1, and yi2, are introduced according to Eq. (5), as shown in Table 1.
Table. 1. Variables of the four solid and one void materials.
Material MT1 MT2 Vo i d MC2 MC1
Young’s modulus Et,1 Et,2 Emin Ec,2 Ec,1
xi1 1 xmin xmin xmin xmin
xi2 1 1 xmin xmin xmin
yi1 ymin ymin ymin ymin 1
yi2 ymin ymin ymin 1 1
As can be seen from the optimized structure shown in Fig. 3(a), the proposed multi-material
topology optimization is quite different from previous single- or dual-material topology
optimization [25]. In the optimized beam, the tensile materials MT1 and MT2 are allocated in the
11
lower part of the structure. While the two compressed materials are allocated in the upper region,
and material MC1 forms an arch. The elemental strain energy distribution shown in Fig. 3(b)
indicates that, overall, although the mechanical properties of these materials are different, the
elemental strain energy is relatively close except for a few extreme values at the concentrated force
point and the support points. This confirms that the strain energy-based material update criteria used
in this study can lead to a very close contribution of each solid element to the compliance reduction
of the whole structure.
Fig. 2. Design domain, loading and supporting conditions of the beam.
Fig. 3. Result of four-material topology optimization of the beam: (a) Optimal topology; (b)
Distribution of the normalized elemental strain energy, where Max1 and Max2 are the maximum
elemental strain energy of tensile and compressive elements, respectively.
Fig. 4. Evolutionary history of materials’ volume fractions and mean compliance of the beam.
0.8m
4.8m F = 200 kN
12
The evolutionary history of the volume fractions and the mean compliance is shown in Fig. 4. A
total of 300 iterations are shown in Fig. 4. Actually, the convergence criterion (Eq. 19) is satisfied
after the 102nd iteration and remains satisfied since then, so the topology optimization of the
structure can be considered to have converged thereafter. It is clear that due to the specially
designed volume evolution program, the volumes of these four materials reach their target values
almost at the same time. Then, the compliance of the structure reaches a stable value at the end.
3.2 Variation of volume fractions
From the result above, it is obvious that the layout of the optimized structure is strongly related
to the components of the materials. When the materials used are given, different materials fractions
will lead to completely different results. To illustrate this point, based on the same beam structure
shown in Fig. 2, a series of cases, I(a), I(b), I(c), I(d), I(e), I(f), I(g), and I(h), with the same
materials but different material volume fractions, are set up. Similarly, due to the symmetry, only
half of the design domain is discrete with 600 × 200 elements. Young’s moduli of the four materials
are set as Et,1: Et,2: Ec,1: Ec,2 = 4: 2: 3: 1. The target volume fraction is set as V* = 0.4, the filter radius
is set to 0.04 m for all these cases, and the evolutionary volume ratio er is set to 2 %.
Fig. 5 shows the optimized designs. When the ratio of MT1 to all the tensile materials rt,1 = 0.1,
the material MT1 is only allocated in the lower middle area of the beam, where the tension force is
concentrated. From case I(a) to I(d), with the increase of the MT1’s volume, the material MT1
expands along the lower chord. If the volume of the material MC1 increases, the layout of the
structure changes accordingly. From case I(e) to I(h), the material MC1 expands from the upper
middle area to the whole upper chord, forming an arch.
As can be seen from the elemental strain energy distribution shown in Fig. 5, although the
material volume fractions of these cases differ significantly, the values of the elemental strain
energy are relatively close in each case, except for the stress concentration regions (the areas near
the mid-span point and the support points at the two ends). The results indicate that although the
mechanical properties of these materials are quite different, in general, they have similar
contributions to the overall compliance control of the structure. However, it can also be noticed that
it is not always the high-rigidity materials that contribute more to the structure than the low-rigidity
ones. Near boundaries between two different materials, the elemental strain energy of the
low-rigidity material is greater than that of the high-rigidity material.
13
Fig. 5. Optimal topology and the corresponding distribution of normalized elemental strain energy
when the volume fractions of MT1 and MC1 are varied.
Fig. 6 shows the deflection of the beams at the point of concentrated forces. Because of the
different material volume fractions adopted in these cases, the mean compliance cannot be
compared directly, but the mid-span deflection can clearly reflect the stiffness of the beam. The
comparison in Fig. 6 reveals a very simple fact: the rigidity of the optimized structure increases
14
with the volume fractions of the high-rigidity sub-materials.
Fig. 6. Comparison of deflection of the beams with different material volume fractions.
3.3 Variation of Young’s modulus
In addition to the volume fraction of each material, the material properties also strongly
influence the results of the multi-material topology optimization. To demonstrate this point, based
on the same beam shown in Fig. 2, a series of cases, Ⅱ(a), Ⅱ(b), Ⅱ(c), Ⅱ(d), Ⅱ(e), Ⅱ(f), Ⅱ(g), and
Ⅱ(h), are set up, as shown in Fig. 7. In these cases, all the parameters remain the same, except for
Young’s moduli of the materials. Half of the design domain is discrete with 600 × 200 elements.
The volume fractions are set as V* = 0.4, rt,1 = 0.3, rt,2 = 0.7, rc,1 = 0.6, and rc,2 = 0.4. The filter
radius is set to 0.04 m for all the cases, and the evolutionary volume ratio er equals 2 %.
In Fig. 7, from case Ⅱ(a) to Ⅱ(d), the topology of the structure varies considerably as Young’s
modulus of the high-rigidity material MT1 is increased, and the lower chord of the beam becomes
progressively thinner. Similarly, from case Ⅱ(e) to Ⅱ(h), the compressive members in the structure
become gradually thinner as Young’s modulus of the material MC1 is increased, and the shape of the
upper arch also varies. As can be seen from the distribution of the normalized elemental strain
energy in Fig. 7, from case Ⅱ(a) to Ⅱ(d) or from case Ⅱ(e) to Ⅱ(h), the contribution of the
high-rigidity sub-materials in the structure becomes more and more significant as they are
enhanced.
Then, from the comparison of the deflection of these beams in Fig. 8, it can be seen that the
overall stiffness of the optimized structure increases with the maximum rigidity of the
sub-materials.
18.93
17.47
16.39
15.57
20.54
18.40
16.39
16.09
15
16
17
18
19
20
21
Case I(a) Case I(b) Case I(c) Case I(d) Case I(e) Case I (f) Case I(g) Case I(h)
Deflection (mm)
15
Fig. 7. Optimal topology and the corresponding distribution of normalized elemental strain energy
when Young’s moduli of MT1 and MC1 are varied.
16
Fig. 8. Comparison of the deflection of four four-material beams composed of materials with
different Young’s moduli.
3.4 Topology optimization with three materials
After the four-material topology optimization has been developed, the topology optimization
with three materials can be derived quite easily. Assuming that two among the four materials have
the same mechanical properties, they can be regarded as one material. For example, in Fig. 9(a), Ec,1
= Ec,2, the two compressed materials, MC1 and MC2, are regarded as one single material MC.
Similarly, in Fig. 9(b), Et,1 = Et,2, the two tensioned materials can be treated as one material MT.
Then, in Fig. 9(c), the two low-rigidity materials (MT2 and MC2) have the same properties in tension
and compression. The sandwich structure got in Fig. 9(c) becomes the prototype of the application
example in Fig 14. Similarly, if the two relatively low-rigidity materials have the same properties in
tension and compression, they can be treated as one material, and the optimal topology is shown in
Fig. 9(d).
Fig. 9. Three-material topology optimization for the beam.
16.39
15.25
13.58
13.19
16.39
13.60
12.26
11.69
11
12
13
14
15
16
17
Case Ⅱ(a) Case Ⅱ(b) Case Ⅱ(c) Case Ⅱ(d) Case Ⅱ(e) Case Ⅱ(f) Case Ⅱ(g) Case Ⅱ( h)
Deflection (mm)
17
4. Applications
4.1 Cost analysis of different material schemes
Because the proposed multi-material topology optimization is applicable to the topology
optimization of composite structures composed of different materials, cheaper and inefficient
materials can be used to fabricate the relatively unimportant areas of the structure where the stress
or sensitivity number is low. Thus, the use of expensive or high-performance materials can be saved,
which can significantly reduce the cost of the structure.
As shown in Fig. 10, to show the advantages of the structures designed via multi-material
topology optimization in cost-saving and stiffness-enhancing, four different material arrangement
schemes are designed for the same beam based on the beam shown in Fig. 2. The first beam, design
1, is made of steel only. The second beam, design 2, is made of two materials: steel and
ultra-high-performance concrete (UHPC), while the beam of design 3 is made of iron and C30
concrete. In designs 2 and 3, the volume fraction of each sub-material is determined by the stress
state in the optimization process according to Eq. 5. While, in the last design, four materials, steel,
iron, UHPC, and C30 concrete, are used in design 4. In design 4, the volume fractions satisfy: rt,1 =
0.3, rt,2 = 0.5, rc,1 = 0.6, and rc,2 = 0.4. Young’s moduli and the price of the four materials can be
seen in Table 2. The price of the four materials is referenced in [40], which is an e-commerce
platform for building materials. Half of the design domain is subdivided using a mesh of 600 × 200
elements in each case, and the target volume fractions are equally set to 0.4. Other parameters are:
rmin = 0.04 m and er = 2 %.
After the four beams are topologically optimized, the deflection, cost and mass of the optimized
structures are compared in Fig. 11 and Table 3. It should be noted that here, in order to simplify the
comparison, only the total material cost of each design is considered. However, the utilization of
multiple materials may incur additional costs, such as those due to increased nodes and increased
complexity of construction. These additional costs are not considered in the present study.
The data show that as an optimal structure composed of a single material, design1 has the best
rigidity but also the highest material cost. If two materials are used to make this beam, such as
design 2 and design 3, the cost can be reduced, and the rigidity will be decreased accordingly.
Design 4, with four materials used, falls between designs 2 and 3 in terms of both stiffness and
18
material cost. While from the weight of the structures, Design 1 is the heaviest, and the other three
are relatively close.
As for which design is optimal, it will depend on the evaluation criteria. For example, design 1
has the best rigidity when the material cost is not constrained. While design 3 is the best solution
with the lowest material cost and total weight when the deflection of all four designs is satisfied.
When the mid-span deflection is limited to 6 mm, design 4 can achieve the lowest material cost and
lightest weight while meeting the deformation constraint. And when the material cost is limited to
2,500 CNY, design 4 has the best rigidity with the cost requirement satisfied.
It should be noted that based on the four materials shown in Table 2, there are more material
combination options than the four designs displayed in Fig. 10. In summary, the four-material
composite beam achieves a balance between structural rigidity and material cost compared to the
single- or dual-material solutions, providing designers with an option competitive in various aspects.
In the actual design of composite structures, the solution may be finalized according to the design
requirements with additional costs considered.
Fig. 10. Four different designs of the beam via different materials’ arrangements.
Table 2. Young’s moduli and the price of four materials.
Materials volume fractions
Steel Iron UHPC C30 concrete
Young’s modulus (GPa) 200 100 60 30
Poisson’s ratio 0.2 0.2 0.2 0.2
Price (CNY/t) 39,500 23,700 8,000 400
Density (kg/m3) 7,850 6,600 2,300 2,360
19
Fig. 11. Comparison between the four beams in terms of deflection, material cost, and total mass.
Table 3. Data of the four materials and the four beams.
Materials volume fractions Deflection
(mm)
Cost
(CNY)
Mass
(kg)
Steel Iron UHPC C30 concrete
Price (CNY/t) 39,500 23,700 8,000 400
Young’s modulus
(GPa)
200 100 60 30
Design 1 Steel 0.40 - - - 2.19 6,067.20 6,028.8
Design 2 Steel + UHPC 0.14 - 0.26 - 4.55 2,922.24 3,258.24
Design 3 Iron + C30 concrete - 0.14 - 0.26 9.10 1,314.05 2,952.19
Design 4 Steel + Iron + UHPC +
C30 concrete
0.076
0.076
0.15
0.10
5.43 2,161.21 3,211.46
4.2 Design of sandwich structures
A sandwich structure is a kind of high-performance structure, generally consisting of two thin
but stiff upper and lower layers and a less stiff and lighter middle core [41,42]. In a sandwich
structure, multiple materials work together to take full advantage of all materials, providing
sufficient bending stiffness and reducing overall weight at the same time. Because of these
advantages, sandwich structures are widely used in aviation, aerospace and other fields [43].
Since most previous topology optimization techniques can only deal with a single material, it is
difficult to apply these algorithms directly to the design of composite structures. Thus, most
previous topology optimization has been performed only for the middle core [44,45].
For a sandwich structure with unidirectional bending, as shown in Fig. 12, it is clear from a
2.19
4.55
9.10
5.43
6,067.20
2,922.24
1,314.05
2,161.21
6028.8
3258.24
2952.19
3211.46
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
0
1
2
3
4
5
6
7
8
9
10
Steel Steel+ UHPC Iron+ C30 Concrete Steel+ UHPC+ Iron+
C30 Concrete
Cost (CNY) or Mass (kg)
Deflection (mm)
Deflection Cost Mass
20
simple analysis that the upper layer is generally subjected to compression and the lower layer
generally to tension. If these two layers are made of materials with excellent compressive and
tensile properties, respectively, this solution can give full play to the potential of the materials
compared to the design where the upper and lower layers are made of the same material.
Thus, three materials are planned to be used for the sandwich structure, including a
high-rigidity material (MT1) suitable for tension, a high-rigidity material (MC1) suitable for
compression, and a low-rigidity material (MTC2) that can be subjected to both tension and
compression. Young’s moduli of the three materials are Et,1: Ec,1: Etc,2 = 5: 4: 1. The upper and lower
edges with a thickness of 2 mm are set as the non-design domains, and the upper layer is subjected
to a uniform load q = 100 N/m. The total volume fraction of all the materials is set as 0.4. The
volumes of MT1 and MC1 occupy 50 % and 60 % of the tensile and compressive zones, respectively.
Other parameters are: er = 2 % and rmin = 0.05 m.
When the calculation is performed, half of the structure is sufficient for the calculation due to
the symmetry, which is discrete using 480 × 80 four-node elements. The periodic topology
optimization method is used with four duplicated cells in each half of the structure. By setting
different symmetry constraints, different topology optimization results can be obtained, as shown in
Fig. 13. One significant difference between these structures and other sandwich structures is that the
materials of the upper and the lower layer of each sandwich structure in Fig. 13 are different. So,
two high-performance materials, like carbon fiber and ceramic tile, can be used to fabricate the two
layers. One soft but light material, like plastic, can be used as the core. The different layers can be
joined together with glue.
The optimized result in Fig. 13(d) is selected as the prototype for the post-design of the
sandwich structure. The cell, shown in Fig. 14(a), can be evolved into two 3D forms, one is suitable
for bending in one direction (Fig. 14b), and the other is suitable for bending in two directions (Fig.
14c). The sandwich structures designed based on these two cells are shown in Figs. 14(d) and 14(e).
Because these sandwich structures made of three materials can further exploit the potential of these
materials while retaining the advantages of conventional sandwich structures, it has the potential
and prospect for application in some special fields, such as spacecraft.
21
Fig. 12. Design domain, load and support conditions of the sandwich structure.
Fig. 13. Sandwich structures designed by using three-material topology optimization.
Fig. 14. Post design and application of the three-material sandwich structure: (a) 2D three-material
22
sandwich structure designed by the proposed method; (b) 3D sandwich structure cell suitable for
bending in one dimension; (c) 3D sandwich structures cell suitable for bending in two dimensions;
(d) Sandwich structure panel suitable for bending in one dimension; (e) Sandwich structure panel
suitable for bending in two dimensions; (f) Application of the designed sandwich structure panel in
aviation.
5. Conclusion
This paper has developed a method that optimizes the topology of a composite structure
composed of more than two materials with different mechanical properties in tension and
compression. The first invariant of the stress tensor of each mesh element is used as the criterion to
determine whether the element is tensioned or compressed. The materials in the structure are
divided into two groups: materials suitable for tension and materials suitable for compression,
respectively. Then, the two groups of materials are arranged in the tension and compression regions.
For the individual tension or compression region, high- and low-rigidity materials are assigned
according to the magnitude of the sensitivity numbers.
Different optimization results can be obtained for the same beam by varying the volume
fractions or Young’s moduli of the materials. The comparison of these results shows that the
stiffness of the optimized structure using the proposed method increases with the volume fraction or
rigidity of the high-rigidity sub-materials. In addition, by comparing the four designs of the beam
based on different material selections, it can be seen that the four-material solution can achieve a
better balance between enhancing stiffness and saving material cost compared to other single- or
dual-material solutions. Then, the four-material topology optimization method can be very easily
derived into a three-material topology optimization method. Finally, the sandwich structure
designed based on the three-material topology optimization algorithm demonstrates the innovation
and potential of the proposed method in lightweight design, which has practical application
prospects in the aviation field.
However, in the method proposed in this study, the volume fraction of each material in the
tensile or compressive material group is predetermined, which limits the further optimization of
multi-material structures. Besides, in the comparative study in this paper, the additional cost
increase due to the increased nodes and fabrication difficulties are not considered in the present
23
study, and they will be explored in our future research.
CRediT authorship contribution statement
Yu Li: Investigation, Methodology, Software, Validation, Visualization, Writing ‐ original draft.
Philip F. Yuan: Supervision, Conceptualization. Yi Min Xie: Supervision, Conceptualization,
Methodology, Writing ‐ review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.
Data Availability Statement
The data that support the findings of this study are available from the authors upon reasonable
request.
Acknowledgment
This project was supported by the Australian Research Council (FL190100014).
References
[1] Wu J, Sigmund O, Groen JP. Topology optimization of multi-scale structures: a review. Struct Multidiscip
Optim 2021;63(3):1455-1480. https://doi.org/10.1007/s00158-021-02881-8.
[2] Rietz A. Sufficiency of a finite exponent in SIMP (power law) methods. Struct Multidiscip Optim
2001;21(2):159-163. https://doi.org/10.1007/s001580050180.
[3] Bendsøe MP, Sigmund O. Material interpolation schemes in topology optimization. Arch Appl Mech
1999;69(9-10):635–654. https://doi.org/10.1007/s004190050248.
[4] Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O. Efficient topology optimization in
MATLAB using 88 lines of code. Struct Multidiscip Optim 2011;43(1):1-16.
https://doi.org/10.1007/s00158-010-0594-7.
[5] Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Comput Methods Appl
Mech Eng 2003;192(1-2):227–246. https://doi.org/10.1016/S0045-7825(02)00559-5.
[6] Wei P, Yang Y, Chen S, Wang MY. A study on basis functions of the parameterized level set method for
topology optimization of continuums. J Mech Des Trans ASME 2021;143:041701.
https://doi.org/10.1115/1.4047900.
24
[7] Luo Z, Wang MY, Wang S, Wei P. A level set-based parameterization method for structural shape and
topology optimization. Int J Numer Methods Eng 2008;76(1):1-26. https://doi.org/10.1002/nme.2092.
[8] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct
1993;49(5):885-896. https://doi.org/10.1016/0045-7949(93)90035-C.
[9] Querin OM, Steven GP, Xie YM. Evolutionary structural optimisation (ESO) using a bidirectional algorithm.
Eng Comput (Swansea, Wales) 1998;15(8):1031-1048. https://doi.org/10.1108/02644409810244129.
[10] Xie YM, Steven GP. Evolutionary structural optimization for dynamic problems. Comput Struct
1996;58(6):1067-1073. https://doi.org/10.1016/0045-7949(95)00235-9.
[11] Zhou Q, Shen W, Wang J, Zhou YY, Xie YM. Ameba: A new topology optimization tool for architectural
design. Proc. IASS Annu. Symp., 2018.
[12] Seifi H, Rezaee Javan A, Xu S, Zhao Y, Xie YM. Design optimization and additive manufacturing of nodes
in gridshell structures. Eng Struct 2018;160:161-170. https://doi.org/10.1016/j.engstruct.2018.01.036.
[13] Lencus A, Querin OM, Steven GP, Xie YM. Aircraft wing design automation with ESO and GESO. Int J
Veh Des 2002;28(4):356-369. https://doi.org/10.1504/IJVD.2002.001995.
[14] Zhao ZL, Zhou S, Feng XQ, Xie YM. Morphological optimization of scorpion telson. J Mech Phys Solids
2020;135:103773. https://doi.org/10.1016/j.jmps.2019.103773.
[15] Benaissa B, Hocine NA, Khatir S, Riahi MK, Mirjalili S. YUKI Algorithm and POD-RBF for Elastostatic
and dynamic crack identification. J Comput Sci 2021;55: 101451.
https://doi.org/10.1016/j.jocs.2021.101451.
[16] Zitar RA, Al-Betar MA, Awadallah MA, Doush IA, Assaleh K. An Intensive and Comprehensive Overview
of JAYA Algorithm, its Versions and Applications. Arch Comput Methods Eng 2022;29:763–792.
https://doi.org/10.1007/s11831-021-09585-8.
[17] Cuong-Le T, Minh HL, Khatir S, Wahab MA, Tran MT, Mirjalili S. A novel version of Cuckoo search
algorithm for solving optimization problems. Expert Syst Appl 2021;186: 115669.
https://doi.org/10.1016/j.eswa.2021.115669.
[18] Eschenauer HA, Olhoff N. Topology optimization of continuum structures: A review. Appl Mech Rev
2001;54(4):331-390. https://doi.org/10.1115/1.1388075.
[19] Sigmund O, Maute K. Topology optimization approaches: A comparative review. Struct Multidiscip Optim
2013;48(6):1031-1055. https://doi.org/10.1007/s00158-013-0978-6.
[20] Rozvany GIN. A critical review of established methods of structural topology optimization. Struct
Multidiscip Optim 2009;37(3):217-237. https://doi.org/10.1007/s00158-007-0217-0.
[21] Li D, Kim IY. Multi-material topology optimization for practical lightweight design. Struct Multidiscip
Optim 2018;58(3):1081-1094. https://doi.org/10.1007/s00158-018-1953-z.
[22] Gangl P. A multi-material topology optimization algorithm based on the topological derivative. Comput
Methods Appl Mech Eng 2020;366:113090. https://doi.org/10.1016/j.cma.2020.113090.
[23] Park J, Sutradhar A. A multi-resolution method for 3D multi-material topology optimization. Comput
Methods Appl Mech Eng 2015;285:571-586. https://doi.org/10.1016/j.cma.2014.10.011.
[24] Liu P, Shi L, Kang Z. Multi-material structural topology optimization considering material interfacial stress
constraints. Comput Methods Appl Mech Eng 2020;363:112887.
https://doi.org/10.1016/j.cma.2020.112887.
[25] Huang X, Xie YM. Bi-directional evolutionary topology optimization of continuum structures with one or
multiple materials. Comput Mech 2009;43(3):393-401. https://doi.org/10.1007/s00466-008-0312-0.
[26] Sanders ED, Aguiló MA, Paulino GH. Multi-material continuum topology optimization with arbitrary
volume and mass constraints. Comput Methods Appl Mech Eng 2018;340:798-823.
https://doi.org/10.1016/j.cma.2018.01.032.
25
[27] Sanders ED, Pereira A, Paulino GH. Optimal and continuous multilattice embedding. Sci Adv 2021;7(16).
https://doi.org/10.1126/sciadv.abf4838.
[28] Banh TT, Lee D. Multi-material topology optimization design for continuum structures with crack patterns.
Compos Struct 2018;186:193-209. https://doi.org/10.1016/j.compstruct.2017.11.088.
[29] Li Y, Xie YM. Evolutionary topology optimization of spatial steel-concrete structures. J Int Assoc Shell
Spat Struct 2021;62(2):102-112. https://doi.org/10.20898/j.iass.2021.015.
[30] Li Y, Xie YM. Evolutionary topology optimization for structures made of multiple materials with different
properties in tension and compression. Compos Struct 2021;259:113497.
https://doi.org/10.1016/j.compstruct.2020.113497.
[31] Li Y, Lai Y, Lu G, Yan F, Wei P, Xie YM. Innovative design of long-span steel–concrete composite bridge
using multi-material topology optimization. Eng Struct 2022;269: 114838.
https://doi.org/10.1016/j.engstruct.2022.114838.
[32] Liu S, Qiao H. Topology optimization of continuum structures with different tensile and compressive
properties in bridge layout design. Struct Multidiscip Optim 2011;43(3):369-380.
https://doi.org/10.1007/s00158-010-0567-x.
[33] Principal Strains. https://www.continuummechanics.org/principalstrain.html (accessed on November 5,
2022).
[34] Megson THG. Structural and Stress Analysis, Second Edition. Academic Commons, 2011.
[35] Sigmund O, Petersson J. Numerical instabilities in topology optimization: A survey on procedures dealing
with checkerboards, mesh-dependencies and local minima. Struct Optim 1998;16(1):68-75.
https://doi.org/10.1007/BF01214002.
[36] Li Q, Steven GP, Xie YM. A simple checkerboard suppression algorithm for evolutionary structural
optimization. Struct Multidiscip Optim 2001;22(3):230-239. https://doi.org/10.1007/s001580100140.
[37] Huang X, Xie YM. Convergent and mesh-independent solutions for the bi-directional evolutionary
structural optimization method. Finite Elem Anal Des 2007;43(14):1039-1049.
https://doi.org/10.1016/j.finel.2007.06.006.
[38] Huang X, Xie YM. Evolutionary topology optimization of continuum structures. Wiley, 2010.
https://doi.org/10.1002/9780470689486.
[39] Ferrari F, Sigmund O. A new generation 99 line Matlab code for compliance topology optimization and its
extension to 3D. Structural and Multidisciplinary Optimization 2020;62(4):2211-2228.
https://doi.org/10.1007/S00158-020-02629-W/TAB LES/ 4.
[40] MySteel Net. https://m.mysteel.com/ (accessed on November 5, 2022).
[41] Hunt CJ, Morabito F, Grace C, Zhao Y, Woods BKS. A review of composite lattice structures. Compos
Struct 2022;284:115120. https://doi.org/10.1016/j.compstruct.2021.115120.
[42] Feng Y, Qiu H, Gao Y, Zheng H, Tan J. Creative design for sandwich structures: A review. Int J Adv Robot
Syst 2020;17(3):1-24. https://doi.org/10.1177/1729881420921327.
[43] Castanie B, Bouvet C, Ginot M. Review of composite sandwich structure in aeronautic applications.
Compos Part C Open Access 2020;1. https://doi.org/10.1016/j.jcomc.2020.100004.
[44] Zhu JH, Liu T, Zhang WH, Wang YL, Wang JT. Concurrent optimization of sandwich structures lattice core
and viscoelastic layers for suppressing resonance response. Struct Multidiscip Optim 2021;64(4).
https://doi.org/10.1007/s00158-021-02943-x.
[45] Galletti GG, Vinquist C, Es-Said OS. Theoretical design and analysis of a honeycomb panel sandwich
structure loaded in pure bending. Eng Fail Anal 2008;15(5).
https://doi.org/10.1016/j.engfailanal.2007.04.004.
[46] Lai Y, Li Y, Huang M, Zhao L, Chen J, Xie Y.M. Conceptual design of long span steel-UHPC composite
26
network arch bridge.
