Layout optimization of truss structures with modular constraints

Abstract

Truss layout optimization is a well-established technique for designing efficient structures, but the optimized structures are often complex and associated with high manufacturing costs. To address this problem, repetitive modules are often used. However, relying solely on a single type of module may negatively impact the structural efficiency. To mitigate this trade-off, this study presents a novel approach for designing truss structures that incorporates multiple types of modules. Additionally, both the module arrangement and module structures are determined by the optimization approach. To achieve this, a novel mixed-integer linear programming problem is developed, and a heuristic method is proposed to enhance computational efficiency. The proposed approach is validated through several numerical examples, which demonstrate its ability to produce truss designs with low manufacturing costs and high structural efficiency.

Full text

Structures 55 (2023) 1460–1469
2352-0124/© 2023 The Author(s). Published by Elsevier Ltd on behalf of Institution of Structural Engineers. This is an open access article under the CC BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Layout optimization of truss structures with modular constraints
Yufeng Liu
a
, Zhen Wang
b
,
*
, Hongjia Lu
c
,
*
, Jun Ye
a
,
e
,
*
, Yang Zhao
a
,
d
, Yi Min Xie
c
a
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
b
Department of Civil Engineering, Hangzhou City University, Hangzhou 310015, China
c
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne 3001, Australia
d
School of Civil Engineering, Shaoxing University, Shaoxing 312000, China
e
Center for Balance Architecture, Zhejiang University, Hang Zhou, 310014, China
ARTICLE INFO
Keywords:
Structure optimization
Modular constraints
Truss layout optimization
Mixed integer linear programming
ABSTRACT
Truss layout optimization is a well-established technique for designing efcient structures, but the optimized
structures are often complex and associated with high manufacturing costs. To address this problem, repetitive
modules are often used. However, relying solely on a single type of module may negatively impact the structural
efciency. To mitigate this trade-off, this study presents a novel approach for designing truss structures that
incorporates multiple types of modules. Additionally, both the module arrangement and module structures are
determined by the optimization approach. To achieve this, a novel mixed-integer linear programming problem is
developed, and a heuristic method is proposed to enhance computational efciency. The proposed approach is
validated through several numerical examples, which demonstrate its ability to produce truss designs with low
manufacturing costs and high structural efciency.
1. Introduction
Truss layout optimization is a technique used to design structures
that comply with certain design constraints, such as design domain,
loading, and support conditions, while also possess optimized properties
such as minimal material usage. However, the optimized structures
obtained through this method can be expensive to manufacture due to
their complex geometries. Additive manufacturing emerges as a prom-
ising cost-saving technique for creating complex designs. However, it
tends to be more suitable for small-scale applications like mechanical
components, rather than large-scale projects such as buildings and
bridges. Therefore, previous studies have considered member classi-
cations [1–3] and structural complexity constraints [4]. In this paper, an
alternative approach is proposed that involves using repetitive modules
to design truss structures. By using multiple types of modules, efcient
designs with low manufacturing costs can be achieved.
The concept of repetitive modules has attracted much interest in the
eld of continuum topology optimization. The related studies can be
classied into two categories: (1) the design of microstructural mate-
rials; (2) the design of repetitive macro structures. Category (1) aims at
customizing the microstructural architecture of identical unit cells ar-
ranged in a periodic array to create materials with exceptional
properties such as the maximum bulk modulus [5,6], negative Poisson’s
ratio [7] and negative thermal expansion factor [8]. Reviews for this
category of studies can be found in [6,9,10].
Category (2) focus on designing efcient structures comprised by
identical modules. One of the earliest studies in this area was conducted
by Huang and Xie [11], who combined the periodic constraints with the
Bi-directional Evolutionary Structural Optimization (BESO) technique
to design modular structures with maximum stiffness using a single type
of module. The method was subsequently expanded to include thermal
compliance optimization [12] and frequency optimization [13].
Furthermore, in [14], each periodic structure is allowed to rotate or
mirror in order to increase the optimization search region and identify
structures with better performance. Most of these studies have used pre-
dened non-overlapping sub-domains for the modules, but Rieser [15]
proposed using overlapping sub-domains to reduce the compliance in-
crease caused by the periodic constraints. Nevertheless, all the previous
research concentrated on continuum topology optimization, which is an
approach most suitable for problems with large volume fractions (i.e.,
ratio between structural volume and design domain volume), and may
not perform well when the volume fraction is low. In addition, these
methods often require intensive post-processing to produce practical
results [16]. As an alternative, we propose using the truss layout
* Corresponding authors.
E-mail addresses: wangzhen@zucc.edu.cn (Z. Wang), hongjia.lu@rmit.edu.au (H. Lu), jun_ye@zju.edu.cn (J. Ye).
Contents lists available at ScienceDirect
Structures
journal homepage: www.elsevier.com/locate/structures
https://doi.org/10.1016/j.istruc.2023.06.071
Received 2 February 2023; Received in revised form 31 May 2023; Accepted 14 June 2023

Structures 55 (2023) 1460–1469
1461
optimization approach to design modular structures.
The eld of truss layout optimization has a long history, dating back
to Michell’s pioneering work in 1904 [17]. Dorn et al. [18] later pro-
posed a numerical approach based on linear programming to identify
optimal truss structures. Subsequently, research in this area has focused
on plastic design with multiple loading conditions [19,20]. Building on
these foundations, Gilbert et al. [21] proposed a “member adding”
approach that signicantly increases numerical efciency without
compromising solution optimality. Additionally, geometry optimization
methods have been developed to further simplify optimized structures
obtained from layout optimization [22,23]. Recently, [24] also consid-
ered layout optimization with global stability constraints. Despite these
advances, research on the use of repetitive modular design in truss
layout optimization is limited. A recent study [25] investigated a
modular tall building skeleton system assembled from tetrahedral units,
but the pattern of the basic modular units was pre-dened rather than
optimized. Further research in this area is needed to fully explore the
potential of modular design in truss layout optimization.
Beside the gradient-based truss layout optimization methods, meta-
heuristic algorithms, such as genetic algorithm (GA) [26–28], shufed
shepherd optimization algorithm (SSOA) [29], adaptive hybrid evolu-
tionary rey algorithm (AHEFA) [30] and hybrid algorithm that
combines different meta-heuristic algorithms [31,32] have also been
applied to truss structure design. For these meta-heuristic algorithms, a
key characteristic is to balance the capabilities of overall exploration
and of local search [33]. Optimizations considering dynamic excitations
[34], natural frequency [35] and other restrictions often need meta-
heuristic algorithms, for the introduction of these related factors often
make the mathematical problem complex and hard to solve with tradi-
tional methods. The expanding leads truss optimization to a more
complete and more inclusive frame [36,37]. However, as suggested by
Sigmund [38], the computational costs for meta-heuristic approaches
can be high. Although strategies like seeding initial population with
feasible solutions in some certain meta-heuristic algorithms have the
potential to improve computational efciency [39], most of the studies
in this eld have focused on size optimization rather than layout
optimization.
In summary, many previous investigations into modular structure
design were restricted to a single type of variable module or a selection
of pre-dened modules. In an effort to ll this gap, our study considers
the truss structural optimization problem that incorporates multiple
module types. The optimization process not only determines the internal
structure of each module but also their spatial arrangement. This is
achieved by formulating a new mixed integer linear programming
(MILP) problem. However, the high computational cost of solving MILP
problems can be an obstacle. To address this issue, we also present a
heuristic two-step approach to reduce the computational costs. The
paper is arranged as follows: Section 2 describes the proposed approach;
Section 3 presents numerical examples; Section 4 provides relevant
discussions, and conclusions are drawn in Section 5.
2. Formulation for truss layout optimization with modular
constraints
2.1. Review of layout and geometry optimization
Numerical truss layout optimization methods provide a powerful
means to generate highly efcient truss-like structures with given con-
ditions (e.g., domain, loading and supporting conditions). The standard
process involves a series of steps, as shown in Fig. 1(a–c). Firstly, the
design domain, load and support conditions are specied (Fig. 1(a));
secondly, nodes are generated inside the design domain and potential
members are created by interconnecting these nodes, generating a
‘ground structure’ (Fig. 1(b)); thirdly, the optimal layout is identied
(Fig. 1(c)) by solving the following problem:
objective :min
a,qV=lTa(1a)
subject to :Bq =f(1b)
−
σ
ca≤q≤
σ
ta(1c)
where V represents the volume of structure. l and a are the length vector
and cross-sectional area vector, respectively; B is known as the equi-
librium matrix, q is the internal force vector and f is the external force
vector;
σ
c
and
σ
t
are the allowable compressive and tensile stresses of the
material. The objective function (1a) represents the optimization goal of
minimum structural volume, constraint (1b) represents the mechanical
equilibrium constraint, and (1c) represents the stress constraint. a and q
are taken as the design variables.
Problem (1) uses the rigid-plastic material assumption. According to
Rozvany [20], in single load condition, the optimal results obtained by
plastic design are consistent with elastic optimal designs. Therefore, the
layout optimization approach is frequently used during the conceptual
design stage.
In addition, Problem (1) is a linear programming (LP) problem, thus
can be tackled effectively using modern interior point solvers. However,
the optimized structures are usually complex. To address this issue, the
geometry optimization [40] can be used. In this approach, the node
coordinates are treated as additional design variables on the basis of
layout optimization. The optimization formulation is written as follow:
objective :min
a,q,x,yV=l(x,y)Ta(2a)
subject to :B(x,y)q=f(2b)
−
σ
ca≤q≤
σ
ta(2c)
where x =[x
1
, x
2
, …, x
d
]
T
and y =[y
1
, y
2
, …, y
d
]
T
are vectors of nodal x-
and y-coordinates with d denoting the number of nodes.
Problem (2) uses the optimized structures from Problem (1) as
starting points. In addition, geometrical modications such as merging
close nodes and creating crossovers are performed in-between the iter-
ations. Due to the non-convexity of Problem (2), the returned structural
Fig. 1. Workow of the traditional layout optimization approach: (a) design domain, loading and support conditions; (b) generate the ground structure; (c) the
optimized structure and (d) rationalize the structure by moving the nodes.
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Structures 55 (2023) 1460–1469
1462
volume may uctuate up or down slightly compared to the input
structure. For more details of the geometry optimization, the research
conducted by He and Gilbert [40] can be referred.
2.2. Modular constraints
In the current study, the truss structures comprised by multiple types
of modules will be designed. For clarity, the process is demonstrated
through a simple cantilever example shown in Fig. 2. In this example, we
rst divide the design domain into a number of identical sub-regions
(Fig. 2(b)), which are used as the potential slots for the structural
modules. In the next step, identical node grids and ground structures are
constructed in each slot (Fig. 2(c)). In the nal step, by solving the
optimization problem, the module arrangement (i.e., the type of module
in each slot) and the inner structure of each module are determined
(Fig. 2(d)).
In this process, the number of module types is pre-dened as p. To
apply the modular constraints, the module type variable tk,ζ is dened to
represent the existence of ζ-th type of module on the k-th slot. tk,ζ is a
binary variable (i.e., tk,ζ∈ {0,1}) and tk,ζ=1 donates that the ζ-th type
of module is assigned to the k-th slot. To ensure that each slot adopts
only one type of module, the following constraint is used:
tk,1+tk,2+⋯+tk,ζ⋯+tk,p=1,∀k∈ {1,2, ...,
η
}(3)
where
η
represents the number of module slots.
In addition, the following constraint is used to guarantee that the
slots adopting the same module type have the same structure:
Am,1+M× (tk,2+tk,3+⋯+tk,p) ≥ ai≥Am,1−M× (tk,2+tk,3+⋯+tk,p)
Am,2+M× (tk,1+tk,3+⋯+tk,p) ≥ ai≥Am,2−M× (tk,1+tk,3+⋯+tk,p)
⋮
Am,p+M× (tk,1+tk,2+⋯+tk,p−1) ≥ ai≥Am,p−M× (tk,1+tk,2+⋯+tk,p−1)
(4)
where a
i
is cross-sectional area of the i-th member in the structure, Am,ζ
donates the area of the m-th member for the ζ-th module type. Note that
index m is a local member index associated with the ground structure of
each module and i is a global member index associated with the ground
structure for the design problem. The corresponding index m for the i-th
member can be obtained in the modular ground structure construction
process (Fig. 2(c)). M is a pre-dened large number, which is commonly
used to activate and deactivate constraints in MILP problems [41].
To sum up, by combing Problem (1) and constraints (3) and (4), the
truss layout optimization problem with modular constraints is:
objective :min
a,q,t,Am
V=lTa(5a)
subject to :Bq =f(5b)
−
σ
ca≤q≤
σ
ta(5c)
tk,1+tk,2+⋯+tk,p=1,∀k∈ {1,2, ..., q}(5d)
Am+MTi>=aie>=Am−MTi,∀i∈ {1,2, ..., n}(5e)
where A
m
=[A
m,1
, A
m,2
, …, A
m,p
]
T
; n is the number of members in the
structure; e= [1,1,1,⋯]is an all-one vector; Ti is dened by:
Ti=
⎡
⎢
⎢
⎣
0tki,2⋯tki,p
tki,10 ⋯ tki,p
⋮ ⋮ ⋱ ⋮
tki,1tki,2… 0
⎤
⎥
⎥
⎦
,
where ki is the slot number of the i-th member.
In Problem (5), a, q, A
m
and T
i
are considered as design variables.
Specically, a, q, Am are continuous variables while t= [tk,1,tk,3,⋯,tk,p]
are binary variables. In addition, the objection function (5a) and the
constraints (5b) to (5e) are all linear. This characteristic classies
Problem (5) a MILP problem. Therefore, we employ the reputable
commercial solver Gurobi [42] to solve Problem (5).
2.3. Geometry optimization of modular truss structure
The geometry optimization approach (i.e., Problem (2)) described in
Section 2.1 can be used on the modular structure as well. The optimi-
zation formulation is:
objective :min
a,q,x,y,t,Am
V=l(x,y)Ta(6a)
subject to :B(x,y)q=f(6b)
−
σ
ca≤q≤
σ
ta(6c)
tk,1+tk,2+⋯+tk,p=1,∀k∈ {1,2, ..., q}(6d)
Am+MTi>=aie>=Am−MTi,∀i∈ {1,2, ..., n}(6e)
Similar to Problem (2), Problem (6) is also a non-linear non-convex
problem. Thus, the structure obtained from Problem (5) is used as the
start point of Problem (6). Considering that, tk,i are all xed constants in
Problem (6), making all the variables in Problem (6) continuous. It is
worth noting that Problem (6) is only carried out on the optimized
structures which have much less members than the full ground structure
used in Problem (5). Therefore, the computational cost is negligible
compared to Problem (5).
2.4. Heuristic two-step approach for improving numerical efciency
MILP problems are usually companied by high computational costs.
Although in Problem (5), ground structures are constructed within the
module slots, resulting in much fewer potential members compared to
the traditional layout optimization problem, the computational costs
still increase rapidly as the design domain size expands. Therefore, a
heuristic approach is proposed to increase the computational efciency.
This approach is based on an assumption as following:
Assumption 1.The optimized module arrangements obtained using low-
or high-complexity modules are similar, with high-complexity modules being
those with a denser node grid in the module ground structure.
Based on this assumption, the two-step approach illustrated in Fig. 3
is proposed for the efciency improvement. In the rst step, the inter-
mediate modules (i.e., with low-complexity) are used (Fig. 3(b)) in
solving Problem (5) to obtain the module arrangements (Fig. 3(c)),
Fig. 2. Workow for truss layout optimization with repetitive modular constraints: (a) design domain, loading and support conditions; (b) divide the design domain
into potential module slots; (c) generate the ground structure for each module slot and (d) solve the problem and obtain the optimized structure.
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Structures 55 (2023) 1460–1469
1463
where Problem (5) is a MILP. In the second step (Fig. 3(d-e)), the ob-
tained module arrangement is used with the target modules (i.e., with
high-complexity) to obtain a more-efcient structure. Since the binary
variables are now constant, Problem (5) in the second solving step be-
comes a LP problem which can be solved with much less computational
costs. Considering that the MILP problem costs the majority of the
solving time, the computational costs of the second step is therefore
negligible. Therefore, compared to the direct solving approach in Sec-
tion 2.2, this heuristic approach can signicantly reduce the computa-
tional costs by reducing the numerical size of the MILP problem in the
rst step.
It is worth noting that Assumption 1 may lead to sub-optimal solu-
tions. However, it is found that the gap between the results obtained by
the heuristic approach and the global optimal results (i.e., obtained with
the approach in Section 2.2) is small, which will be demonstrated with
the examples in Section 3.
3. Numerical examples
In this section, several numerical examples are presented to
demonstrate the effectiveness of the proposed approaches. In these
classic examples which employ commonly used boundary conditions,
the self-weight of the structure is not considered. The allowable tensile
stress and compressive stress are both taken as
σ
t
=
σ
c
=1 MPa. The
calculation is conducted in a workstation with an Intel i7-12700 k
(3.61Ghz) CPU and 32 GB running memory. The parameter “complexity
of modules”, which is expressed as n
x
×n
y
, represents that each module
has n
x
nodes in horizontal direction and n
y
nodes in vertical direction.
For the termination criteria of Gurobi [43], 1% gap of objective function
is used for all the examples. As Gurobi utilizes a gradient-based solving
Fig. 3. Workow for the heuristic two-step approach: (a) divide the design domain into modules, (b) generate ground structure with intermediate module
complexity, (c) solve the intermediate optimization problem to identify the module arrangement, (d) generate ground structure with target module complexity, (e)
solve the target problem and obtain the nal optimized structure.
Fig. 4. Simply supported example: (a) case
description; (b) solution obtained with 2 ×
2 complexity modules, V =0.544 V
0
; (c)
solution obtained using the module
arrangement of (b) but switching to mod-
ules with 4 ×4 complexity, V =0.537 V
0
;
(d) geometry optimized result of (c), V =
0.536 V
0
; (e) solution obtained with direct
solving approach using 4 ×4 complexity
modules, V =0.539 V
0
; V is the structural
volume, the results are normalized against
the structural volume in Fig. 5(a); members
with the same color share the same module
type.
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Structures 55 (2023) 1460–1469
1464
algorithm, the solving process is deterministic, and the results are global
optimal. This characteristic effectively eliminates any potential inu-
ence from the solving algorithm.
3.1. A simply supported domain subjected to a point load
Firstly, we use a simple example to demonstrate the proposed
approach. The case description is shown in Fig. 4(a). In this example we
consider the module type number p=2 and adopt the heuristic two-step
approach in Section 2.4. In the rst step, the design domain is discretized
into 32 module slots as shown in Fig. 4(a) and the module arrangement
is identied by solving Problem (5) using 2 ×2 intermediate complexity
modules (i.e., Fig. 4(b)). In the second step, the module arrangement is
xed and 4 ×4 target module complexity is considered, and the opti-
mized structure is shown in Fig. 4(c). The geometry optimization
approach described in Section 2.3 is then used to further rationalize the
structure (Fig. 4(d)). For comparison, the result obtained by directly
using 4 ×4 modules with Problem (5) is shown in Fig. 4(e). Compare
Fig. 4(d) to Fig. 4(e), the volume difference is 1.47%, which is consistent
with Assumption 1. In terms of computational costs, Fig. 4(d) takes 72 s
of CPU time while Fig. 4(e) takes 275580 s. By using the two-step
approach, the solving time is reduced by 99.97% but the obtained so-
lutions are similar, highlighting the effectiveness of the proposed
approach.
To analyze the inuence of module type number p, we further
consider structures with p=1and 4. The optimized structures are
shown in Fig. 5(a–b). Note that Fig. 5(a) is a result from the traditional
layout optimization algorithm that does not contains integer variables.
Since only one type of module is used in Fig. 5(a), much of the structural
material is non-fully stressed, causing an 86.68% increase in volume
compared to Fig. 4(d). When p is increased to 4, an extremely simple
truss structure is obtained (Fig. 5(b)), in which two modules consist of
only one inclined member, one module consists of only one horizontal
member, and one module is empty. Note that Fig. 5 (b) is obtained using
the heuristic two-step approach, with a CPU cost of 1325 s. From Figs. 4
and 5, it is clear that increasing the number of module types signicantly
reduces the structural volume, highlighting the benets of using multi-
ple types of modules.
3.2. Inuence of intermediate module complexity
The second example is a cantilevered structure, as illustrated in
Fig. 6. The design domain has dimensions L =6 m and H =3 m and is
subject to a vertical unit load F =1 N applied at the top right corner and
two pin supports at the two corners on the left boundary.
In this example, we consider the module type number p=4. The
intermediate and target module complexities are set to 2 ×2 and 4 ×4.
By using the heuristic two-step approach, the optimized structure is
shown in Fig. 7(a). In addition, the optimized structure obtained directly
using 4 ×4 modules are shown in Fig. 7(b). Compare Fig. 7(a) to (b), the
volume of Fig. 7(a) is 6.1% higher than that of Fig. 7(b), while the
associated CPU time is only 1/800 of the latter. However, Fig. 7(a) and
(b) display distinct geometries. While Fig. 7(b) contains a few empty
regions, all the slots are lled with structures in Fig. 7(a). A possible
reason for this is that the complexities of the intermediate modules are
too low to provide a close estimate of the optimal module arrangement,
causing the differences in both inner structures of modules and module
type arrangements in Fig. 7(a) and (b).
To verify the assumption, the problem is solved again with the in-
termediate module complexity switched to 3 ×3. With the modication,
the obtained result become identical to Fig. 7(b), and the CPU time is 3%
of Fig. 7(b). This suggests that intermediate modules with too low
complexity may lead to sub-optimal solutions. However, in this case, the
effect is minor since the volume difference between Fig. 7(a) and (b) is
only 6.1%.
Interestingly, two modules in Fig. 7(b) form a super module, a mini-
cantilever that duplicates three times, leading to only two types of
modules for manufacturing (i.e., a horizontal bar and the mini-
cantilever). To investigate the potential material saving from using a
more complex structure for the mini-cantilever, we consider 6 ×6 and 8
×8 modules and the optimized results are shown in Fig. 8. It can be seen
that using more complex mini-cantilever structures resulted in minor
volume decreases (i.e., less than 1%). This suggests that once the module
arrangement is determined, the inuence of the module complexity on
the optimized structural volume is minor. It is worth noting that Fig. 8(c)
has slightly higher volume than Fig. 8(a). This is because the nodes of
the 6 ×6 grid is not a subset of 8 ×8 grid, which further makes the
search region of the optimization problem in Fig. 8(a) not a subset of
Fig. 8(c).
3.3. The beam example
In this section, we consider a beam structure with aspect ratio equal
to 1:5 (height: length). The case description is shown in Fig. 9(a), in
which a unit load F =1 N is applied at the center of the bottom edge. The
number of the module types p=4 and the target module complexity is 4
×4. The problem is solved using the heuristic two-step approach (i.e.,
using 3 ×3 intermediate module) and the direct approach. These two
approaches lead to the same optimized structure as shown in Fig. 10.
However, the heuristic two-step approach (2362 s) takes only 1.96% the
Fig. 5. Simply supported example with different number of module types: (a) with one type of modules, V =V
0
; (b) with four types of modules, V =0.363 V
0
, where
V represents the structural volume. The results are normalized against the structural volume in Fig. 5(a).
Fig. 6. Schematic diagram of the cantilever case.
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Structures 55 (2023) 1460–1469
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CPU time of the direct approach (120498 s), demonstrating the nu-
merical efciency of the proposed approach. Despite this, here the
structural efciency can be further improved. Firstly, in Fig. 10, modules
with mirror topologies are considered as two types; secondly, the
modules located at the top corners (i.e., marked with red dashed lines)
are zero-stressed, but the limited number of module types forces the two
slots to be lled with structures (i.e., an additional module type is
required for using empty modules). Therefore, to address these issues, a
problem that take advantage of the symmetry is considered, as shown in
Fig. 9(b).
In Fig. 9(b), half of the problem is considered by adding roller sup-
ports at the right boundary. The heuristic two-step approach is used with
2 ×2 intermediate modules and 4 ×4 target modules. The module type
number p=3,4and 5 are considered and the optimized results are
shown in Fig. 11(the structure is supplemented according to symmetry).
Since in Fig. 11 the modules with mirror topologies are regarded as one
type, an additional empty module can be used, leading to the more
simplied structures and lower material usages (i.e., Fig. 11(b) has
Fig. 7. Cantilever case: (a) result obtained using heuristic two-step method with 2 ×2 intermediate modules, V =V
0
; (b) result obtained via directly using 4 ×4
modules, V =0.942 V
0
, where V represents the structural volume, the results are normalized against the structural volume in Fig. 7(a).
Fig. 8. Results of the cantilever case with more complicated modules (a) optimized structure with 6 ×6 target modules, V =0.939 V
0
; (b) geometry optimized result
of (a), V =0.937 V
0
; (c) optimized structure with 8 ×8 target modules, V =0.940 V
0
; (d) geometry optimized result of (c), V =0.936 V
0
, where V represents the
structural volume, the results are normalized against the structural volume in Fig. 7(a).
Fig. 9. Simply supported beam: (a) the full case; (b) the half case with symmetrical boundary conditions.
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7.03% lower volume compared to Fig. 10). It is noteworthy that Fig. 11
(b) still features four modules with horizontal zero-stressed members
due to the restriction on the number of module types, but this issue is
resolved when p is increased to 5, as shown in Fig. 11(c). In addition, as p
increases from 3 to 5, the volume of the optimized structure and the
layout complexity reduces, which is consistent with the conclusion ob-
tained from Fig. 4 and Fig. 5.
3.4. Inuence of the number of modules
To explore the impact of varying module slot discretization, we
conducted another cantilever example in this section. The problem
description is shown in Fig. 12(a). By taking advantage of the symmetry,
half of the problem is investigated. This example engages an interme-
diate module complexity of 2 ×2 and a target module complexity of 4 ×
4. We utilized four module types (p=4) while varying the module
numbers across cases.
Firstly, the result involving 4 ×2 modules, as shown in Fig. 12(b),
closely mirrors the result obtained from classical layout optimization
without modular constraints. This is understandable as all four modules
in the lower half exhibit different patterns, suggesting that the modular
constraints are not active in this case. Upon increasing the module
number to 8 ×4 (Fig. 12(c)) and 12 ×6 (Fig. 12(d)), the structural
volume rises by 33.7% and 47.0% respectively. This volume increase
might seem counterintuitive since, in most optimization problems, a
more rened node grid often yields improved results. Nonetheless, it’s
important to note that the number of module types does not increase
with the number of modules. As a result, the modular constraints
become more restrictive with the increase in module numbers. In this
scenario, the adverse effect of modular constraint restrictions surpasses
the benets offered by the rened node grid, leading to an increase in
structural volume.
3.5. Modules with 1:2 aspect ratio
In this section, we consider a building bracing design problem, which
targets at identifying the optimized structure subjected to horizontal
wind loads as shown in Fig. 13(a). The module type number p=4 and
the aspect ratio of each module slot is 2:1. In the heuristic two-step
approach, 2 ×2 intermediate modules and 6 ×6 target modules are
used and the optimized result is shown in Fig. 13(b).
For comparison, the result adopting only one type of modules is
Fig. 10. Optimized solution of the simply supported beam example, V =V
0
, where V represents the structural volume.
Fig. 11. Simply supported beam example with symmetrical boundary condition: (a) solution with 3 types of modules, V =1.120 V
0
; (b) solution with 4 types of
modules and, V =0.930 V
0
; (c) solution with 5 types of modules, V =0.701 V
0
, where V represents the structural volume, the results are normalized against the
structural volume in Fig. 10.
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Structures 55 (2023) 1460–1469
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shown in Fig. 13(c). It can be found that Fig. 13(b) have a volume of only
27.54% of Fig. 13(c). The volume reduction obtained via using more
modular types is larger than the previous examples in Fig. 5 and Fig. 11.
This is because in this example, the design problem is dominated by the
bending effect caused by the horizontal loads, with the maximum
bending moment occurring at the bottom edge and minimum bending
moment at the top. Consequently, using a single type of module for both
top and bottom results in a signicant material waste. However,
allowing for multiple types of modules, as seen in Fig. 13(b), allows for
the strategic placement of heavier modules at the bottom and lighter
modules at the top, resulting in a signicant reduction in material usage.
4. Discussions
From the results shown in Section 3, it can be seen that signicant
volume reductions can be achieved by using multiple types of modules
(compared to one type of module), suggesting the superiority of the
proposed approach. Meanwhile, the complexity of modules has a rela-
tively minor inuence on the optimized structural volume (as shown in
Figs. 7 and 8). In addition, it can be observed that the proposed heuristic
two-step approach can achieve similar results as the direct approach but
with much less computational costs. However, it should be pointed out
that even with the heuristic approach, the computational costs increase
Fig. 12. The cantilever case: (a) the case illustration (b) case with 4 ×2 modules, V =V
0;
(c) case with 8 ×4 modules, V =1.337 V
0
; (d) case with 12 ×6 modules, V
=1.470 V
0
; where V represents the structural volume, the results are normalized against the structural volume in Fig. 12(b).
Fig. 13. The horizontal wind load case (a) design domain, loading and supporting conditions; (b) solution with 4 types of modules, V =0.275 V
0
. (c) Solution with
only one type of modules, V =V
0
, where V represents the structural volume, results are normalized against structural volume in Fig. 13(c).
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Structures 55 (2023) 1460–1469
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rapidly when more module slots are used, or 3D problem is considered.
Therefore, the future studies can focus on further improving the
computational efciency. This problem may be solved or eased by
abandoning the global optimality and turning to non-linear mixed
integer programming solver as suggested in [44].
In this study, we focused primarily on integrating only modular
constraints into the layout optimization problem for obtaining results
close to the global optimum. Given that our formulation is rooted in
truss optimization, it holds the potential to be combined with other truss
optimization approaches that take into consideration other factors like
stability, natural frequency, and dynamic loads. Notably, in some ex-
amples, the internal structures of modules can still be complex, sug-
gesting that further structural simplication techniques may be
benecial. An effective method has been proposed by Fairclough [45],
which holds promise for integration with our current algorithm.
5. Conclusions
This study explores the design of efcient modular truss structures by
optimizing both module arrangements and module structures concur-
rently. This approach presents a distinct advantage over previous
methods that restrict designs to a single type of module or a selection of
pre-dened modules. To facilitate this, we propose a novel mixed-
integer linear programming (MILP) formulation based on the tradi-
tional layout optimization approach. To mitigate the computational
costs associated with MILP problems, we employ a two-step heuristic
approach. In the rst step, we use a sparse node grid to determine
module arrangements. The second step then further renes the struc-
tural efciency of the solution from the rst step by utilizing a denser
node grid. The efcacy of this novel approach is underscored by several
illustrative examples, which yield the following conclusions:
(1) By increasing the number of module types, signicant material
savings can be achieved. In one example, a 72.46% reduction in
material usage is achieved by switching from one type of module
to four.
(2) The proposed heuristic two-step approach can lead to similar
optimized results as directly solving the MILP problem. However,
the computational cost of the former is only ~ 1% of the latter,
highlighting the efciency of the heuristic approach.
(3) In the cantilever example, it is observed that when the module
type number remains constant and the number of modules is
relatively low (e.g., ≤72), an increase in the number of modules
tends to have a counterproductive effect. This occurs as the re-
strictions imposed by modular constraints surpass the benets
gained from a rened node grid, resulting in an overall increase in
structural volume.
(4) Geometry optimization can be applied to further rationalize the
structure from the MILP problem to reduce structure complexity,
especially for these cases which have complicated inner struc-
tures in the modules.
Declaration of Competing Interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Acknowledgement
The authors would like to acknowledge the project supported by the
National Natural Science Foundation of China (No. 52078452,
52208215), the Basic Public Research Project of Zhejiang Province (No.
LGG22E080005, LQ22E080008) and the Australian Research Council
(No. FL190100014).
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