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Article published in
Comput. Methods Appl. Mech. Engrg., Vol. 389 (2022) 114382
https://doi.org/10.1016/j.cma.2021.114382
Structural topology optimization with an adaptive design domain
Yi Rong a, Zi-Long Zhao a,b,*, Xi-Qiao Feng c, and Yi Min Xie a
a Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne
3001, Australia
b Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing
100191, China
c Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua
University, Beijing 100084, China
* To whom correspondence should be addressed. Email: zilongzhao@buaa.edu.cn (Z.L. Zhao).
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Abstract
Topology optimization has rapidly developed as a powerful tool of structural design in multiple
disciplines. Conventional topology optimization techniques usually optimize the material layout within a
predefined, fixed design domain. Here, we propose a subdomain-based method that performs topology
optimization in an adaptive design domain (ADD). A subdomain-based parallel processing strategy that can
vastly improve the computational efficiency is implemented. In the ADD method, the loading and boundary
conditions can be easily changed in concert with the evolution of the design space. Through the automatic,
flexible, and intelligent adaptation of the design space, this method is capable of generating diverse high-
performance designs with distinctly different topologies. Five representative examples are provided to
demonstrate the effectiveness of this method. The results show that, compared with conventional
approaches, the ADD method can improve the structural performance substantially by simultaneously
optimizing the layout of material and the extent of the design space. This work might help broaden the
applications of structural topology optimization.
Keywords: Topology optimization; Adaptive design domain; Subdomain-based parallel strategy; Structural
performance improvement; Computational cost reduction
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1. Introduction
Topology optimization is a powerful design tool that aims to maximize the performance of a structure
by optimizing its material layout within a prescribed space and under certain conditions. Quite a few
topology optimization techniques have been developed in the past decades, e.g., the homogenization
method [1], the solid isotropic material with penalization (SIMP) [2-4], the level set [5-7], the bi-directional
evolutionary structural optimization (BESO) [8-10], and the moving morphable components (MMC) [11,
12]. These techniques have been extensively used in, e.g., mechanical engineering, advanced
manufacturing, architectural design, and aerospace engineering [13-15]. By imposing complex constraints
in the form-finding process, advanced manufacturing techniques such as 3D-printing can be used directly
to fabricate free-form designs generated by structural topology optimization [16, 17]. Recently, a
transdisciplinary computational framework was established to reveal the developmental mechanisms of
animal and plant tissues through biomechanical morphogenesis [18, 19]. Besides, much effort has been
directed toward increasing the resolutions of the design domain [20, 21], enhancing the manufacturability
of the optimized results [22], improving the multi-material compatibility of the optimization process [23],
and controlling the structural complexity and connectivity [24, 25].
Despite the remarkable achievements in the field of structural topology optimization, there still remain
many challenging issues which require further research. For example, an interesting issue is how to make
the design domain automatically and intelligently evolve during the form-finding process. In case that the
resolution is specified, the larger the design space, the higher the computational cost. The design space
cannot be excessively large due to the limit of computing resources. In the conventional topology
optimization, the predefined design space keeps fixed. However, it is desirable to update or reshape the
design space due to the following reasons. First, in many optimization problems, the designer cannot
predefine an optimized design space that guarantees a satisfactory solution. Second, in transdisciplinary
form-finding problems such as biomechanical morphogenesis, the evolution of the design space is
necessary to mimic the real biological growth in nature. A grid-based method was proposed to adjust the
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design space layer by layer [26, 27]. However, the grid is predefined and fixed, and the expansion of the
design space features low efficiency. There is a lack of an adequate method for updating the design domain
during the form-finding process. Such a method should help broaden the application of structural topology
optimization in many disciplines.
In this study, we propose an adaptive design domain (ADD) technique for structural topology
optimization. The ADD method can be easily integrated into the existing computational frameworks. The
initial design domain is divided into a certain number of subdomains. After each optimization cycle, the
design domain is reshaped by adding new subdomains or removing the existing ones. In the ADD method,
different adaptive strategies can be adopted to update the design domain, and different termination criteria
can be introduced to meet the design requirements. In comparison to conventional approaches, the ADD
method may achieve the optimized solutions with improved performance.
The paper is outlined as follows. In Section 2, the BESO-based computational framework is first
described. In Section 3, the ADD optimization method is formulated and integrated into the computational
framework. In Section 4, five examples of compliance minimization problems are presented to demonstrate
the effectiveness of the ADD method. In Section 5, the advantages and potential applications of this method
are discussed. In Section 6, the main conclusions drawn from this study are summarized.
2. Computational framework
We develop the ADD method within the BESO-based computational framework, which has become a
widely adopted optimization technique due to its simple concept and clear structural boundaries in the
optimized results [28]. This framework is briefly described below.
In structural optimization, the material is first modeled as a solid assemblage of n elements that are
uniformly distributed in the design domain. Less efficient elements are removed iteratively from the design
domain, and the removed elements can be re-admitted in later evolutions if their efficiency turns to be
sufficiently high. Finite element analysis (FEA) and design variable update are involved in each iteration.
5
Here we consider the compliance minimization of structures for illustration. In this case, the optimization
problem can be expressed as
TT
1
11
min : ( ) 22
n
p
iiii
i
Cx
x
x
UKU uku
, (1)
*
1
subject to : and ( )
n
ii
i
VxvV
FKU x , (2)
where C is the total compliance of the structure, and
|1,2,,
i
x
inx are the design variables.
1
i
x or min
x
(we take min 0.001x in this study) indicates the presence (solid) or absence (void) of
element i, respectively. U, K, and F are the global displacement vector, global stiffness matrix, and
global force vector, respectively. i
u and i
k are the nodal displacement and stiffness matrix of element
i that are obtained from the FEA.
p
is the penalty exponent, which is usually set as 3 [29]. V is the
total volume of material, i
v the volume of element i , and *
V the target volume of the optimized
structure.
The design variables are updated according to the efficiency of each element, which is assessed through
the elemental sensitivities. The elemental sensitivity of element i is defined as [10]
T
1
22
p
iiii i
i
i i ii ii
x
c
C
p
vx vx vx
uku , (3)
where the elemental strain energy i
c is obtained from the FEA.
The above raw sensitivity is smoothed by using the filtering technique to avoid numerical instabilities
such as mesh dependency and checkerboard patterns [30-32]. The following filtering scheme is employed:
1
1
n
ij j
j
in
ij
j
w
w
, (4)
where
f
max 0,
ij ij
wrd is the weight function, f
r the filter radius, and ij
d the distance between the
centroids of elements i and
j
. ij
w is independent of the elemental sensitivities and can be pre-
6
determined before each optimization cycle. To achieve a stable and convergent solution, the smoothed
elemental sensitivity is averaged with its value in the previous iteration [10]:
(1) (1) ()
1
ˆˆ
+
2
kkk
iii
%, (5)
where ()
ˆk
i
and (1)
ˆk
i
denote the smoothed sensitivities of element i in the -thk and (1)-thk
iteration, respectively. The gradually decreasing material volume takes the form as
(1) ()
1
kk
VV
, (6)
where the evolutionary ratio
is here set as 0.02 . The optimization process is convergent and will be
terminated if
() *k
VV and
11 1
11
TT
kt ktT kt
tt
CC C
, (7)
where k is the current iteration number, and ()k
V and ()k
C are the material volume and the structural
compliance in the -thk iteration, respectively [9]. We here take the integer number 5T, implying that
the change in the mean compliance over the last 10 iterations is acceptably small. Typically, the allowable
convergent error
is set as 0.001 or smaller.
3. Optimization within an adaptive design domain
In the ADD method, the form-finding process contains a certain number of optimization cycles, each
cycle consisting of multiple iteration steps. The design space will evolve into a new configuration at the
beginning of each cycle (except for the first one), and the optimized structural design will be achieved at
the end of each cycle.
3.1. Division of the design domain
Now we implement the ADD optimization method in the BESO-based computational framework.
Before the form-finding process, the design domain is first divided into a certain number of
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subdomains i
( 1, 2, ,iN), which are referred to as cells. The number N of cells may be different
in each optimization cycle. Let ()l
N denote the number of cells in the -thl cycle. For simplicity, we adopt
cuboid cells and regular hexahedral elements (Fig. 1A).
Figure 1. Division of the design space and the evolution strategy. (A) The design domain consists of a certain number of cells,
and each cell consists of multiple elements. (B) New cells (brown, green, and purple) are added through target cells 1–3, while
three cells are removed from the original design space in (A).
For a cuboid initial design domain, we can divide it into cuboid cells of the same sizes. In case that the
initial design domain has an irregular shape, we can expand it to a cuboid region, and then divide it into
cuboid cells. In Section 4, we will show that cells may have different sizes. It is not allowed that only part
of an element is included in a cell, or an element is included in more than one cells. Denote the element and
node sets of i
as i
E and i
N, respectively.
Refer to a Cartesian coordinate system
,,
x
yz. Element i
E
belongs to
j
E if
j
ij
E
x
xx
,
j
ij
E
yyy
, and
j
ij
E
zzz
, (8)
where i
E
x
, i
E
y, and i
E
z are the center coordinates of element i
E
.
j
x
(
j
x
),
j
y
(
j
y
), and
j
z
(
j
z
)
are the upper (lower) boundaries of
j
in the
x
, y, and z directions, respectively. The boundaries are
8
predefined by the designer.
j
N can be readily obtained from node connectivity once
j
E has been
known.
3.2. Adaptive strategy of the design domain
New cells will be added to the design domain in a new cycle if they are adjacent to the available faces
of the existing cells. In the following, we discuss how to find the available faces. As shown in Fig. 1B, new
cells are added through the available faces of cells 1, 2, and 3.
The center coordinate of the -thi cell i
is calculated as
,, , ,
222
iii iii
iiii
x
xy yz z
xyz
c. (9)
Then the center of a face
j
F in i
is calculated as
,, , ,
222
jjj jjj
jjjj
FFF FFF
FFFF
x
xy yz z
xyz
c, (10)
where
j
F
x
(
j
F
x
),
j
F
y
(
j
F
y
), and
j
F
z(
j
F
z) are the upper (lower) boundaries of
j
F in the
x
, y, and z
directions, respectively. A cell centered at
2ji
F
cc
that has the same size with i
is denoted as
new
. Face
j
F is available if the volume of new
is 0. A cell is available if it has available faces.
To avoid an abrupt change in the structural topology, the newly added cells are filled with void elements
at the beginning of a cycle. Different strategies can be adopted to reshape the design domain. Here, the
adaptive strategy is sensitivity-related and based on the efficiency of each cell. The efficiency of an
available cell i
is assessed by its average sensitivity. This strategy is valid even when the cells have
different sizes. Through the filtering operation (Eq. (4)), void elements in the newly added cells will be
gradually switched to solid as they are in the vicinity of solid elements with large sensitivities.
Notice that a cell may have a vast number of void elements and a small group of solid elements with
large sensitivities. Although such a cell may have a moderate or even small average sensitivity, the regions
9
occupied by its solid elements could play a critical role in determining the overall performance of the
structure. Therefore, a high pass filter is introduced to avoid underestimating the efficiency of such a cell.
Only a certain percentage
of the most efficient elements in a cell are selected to estimate its efficiency.
The set of the most efficient elements selected for estimating the available cell i
is denoted as i
P. For
illustration, Fig. 2A shows two cells, i
and
j
; the former has more solid elements (black) than the
latter. The elements in i
P and j
P are marked with yellow dots. i
P can be used to assess the
efficiency of i
properly. However, the performance of
j
will be overestimated. As the percentage
is too small, the average sensitivity of j
P, consisting of only very few elements, could be very large, but
it is unnecessary to add solid elements on the boundaries of the available cell
j
. Therefore, a certain
number of void elements need to be involved in assessing the performance of cells such as
j
.
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Figure 2. (A) Illustration of the high pass filter in the calculation of cellular sensitivity. Elements in i
P and j
P are marked
with yellow dots. The solid elements (black) without yellow dots are not used for assessing the cellular efficiency. i
P reflects
the performance of cell i
properly, while j
P overestimates the performance of cell
j
. (B) Update of the filter matrix.
The expanded regions are labelled as ˆi
and i
, respectively. i
is a newly added cell. For the original design domain,
the weight functions of elements only in the green regions need to be updated due to the addition of i
.
For simplicity, the percentage
is set as the target volume fraction of the previous optimization cycle,
which can be calculated as
*
() ( 1)
f(1)
ll
l
V
vV
, (11)
where (1)l
V is the total volume of the design space in the (1)-thl(1)l optimization cycle.
11
Alternatively, i
P could be comprised by all solid elements of i
. However, it may lead to an
overestimation of the value of cells such as j
in Fig. 2A.
Once i
P has been determined, the average sensitivity of the available cell i
is calculated by
1
ˆˆ
i
i
i
j
j
P
P, (12)
where i
P is the number of elements in i
P, and ˆj
the smoothed sensitivity of element
j
(defined
in Eq. (5)).
After ranking the existing cells according to their sensitivity numbers, we determine the number of
cells that will be added in the next optimization cycle. It should be mentioned that the adaptive strategy of
only adding one cell in each cycle features low computational efficiency and is inadequate in a symmetric
optimization problem.
Let ()l
A represent the set of the target cells. The available i
is a target cell, i.e., ()l
i
A, if
() ()
ˆˆ
max
ij
ll
j
, (13)
where 0.9
in the present study, and
()
ˆ
1, 2, , l
jN and ()
ˆl
N is the number of available cells in
the -thl cycle. In Section 3.5, we will discuss the influence of
on the form-finding process. Now, we
focus on the symmetry constraint of the optimization problem. Consider that the available cells 1
and
2
are located symmetrically, and theoretically have 12
() ()
ˆˆ
ll
. Both 1
and 2
will be selected as
target cells if
12
() () ()
ˆˆˆ
max j
ll l
j
. However, 1
()
ˆl
and 2
()
ˆl
may have a little difference due to
numerical errors. In case of
12
() () ()
ˆˆˆ
max j
lll
j
or
21
() () ()
ˆˆˆ
max j
lll
j
, the two cells will not
be simultaneously selected as the target cells, and the symmetry of the optimization problem will be broken.
The symmetry of both the design space and the structure should be guaranteed by adding new cells
symmetrically in each optimization cycle.
12
3.3. Connectivity between neighboring subdomains
The newly added cells can be connected to their neighboring cells through different approaches.
Consider two cell faces 1
F (newly added or already existed) and 2
F (newly added). The node sets of the
two faces are denoted as 1
F
N and 2
F
N, respectively. 1
F and 2
F are adjacent in case that there are at
least three non-collinear nodes in 12
F
F
NN
. The neighboring faces 1
F and 2
F can be directly merged
as a new one if 12
F
F
NN
. Constraints are specified to connect the neighboring faces F1 and F2 if
12
F
F
NN
, and thereby the deformations of the two faces are coupled. Consider two different nodes 1
N
and 2
N, where 11
F
NN and 22
F
NN. If the two nodes are close, their displacements are related to
each other. The difference between the displacements of 1
N and 2
N decreases with their distance. In the
past decades, different techniques have been developed for imposing such constraints and integrated into
multiple suits of FEA software. In the current study, 1
F is tied to 2
F if they are adjacent but 12
F
F
NN
.
After connecting the newly added cells to their neighbors, available cell faces of the updated design domain
are searched by using Eqs. (9) and (10).
3.4. Variable update
For a given design space, the filter matrix can be assembled before the optimization. Calculation of
the weight functions features high computational cost, especially when the number of elements in the design
domain is vast and the filter radius is large compared with the element size. A subdomain-based algorithm
is here developed to calculate the weight functions in parallel. It is found that the computational efficiency
can be significantly improved by the parallel algorithm.
As aforementioned, the design domain is divided into N cells, expressed as
|1,2,,
iiN . To calculate the weight functions of i
, two expanded cuboid regions ˆi
and
13
i
are introduced respectively as
ff
ii
x
rxx r
, ff
ii
yryyr
, ff
ii
zrzzr
, (14)
ff
22
ii
x
rxx r
, ff
22
ii
yryyr
, ff
22
ii
zrzzr
. (15)
For any element j
E
in i
, the following set is introduced:
fˆ
jkj
EkEE ki
ErEccW, (16)
where k
E
c and
j
E
c are the center coordinates of elements k
E
and j
E
, respectively. Only the elements
in
j
E
W are used to calculate the weight functions of j
E
.
The overlapped region ˆˆ
ij
is f
2r in thickness if i
is adjacent to j
, ensuring that the
weight functions of all elements near the interface of i
and j
can be calculated properly. However,
the elements in the overlapped regions will be involved in calculations repeatedly. The duplicated data are
deleted when assembling the global filter matrix. Using the expanded regions, calculations of the filter
matrix of each subdomain can be performed separately and in parallel. The above parallel strategy is able
to make the most of the computing resources, especially when the computers or workstations support large-
scale and multithreaded calculation.
Once the adaptive design space has been changed, the global filter matrix needs to be updated, but it
is unnecessary and inefficient to recalculate the weight functions of all cells. The weight functions of the
cells far from the newly added ones keep unchanged. An efficient method is introduced below to update the
filter matrices by taking advantage of the division of design space.
Consider a newly added cell i
. As illustrated in Fig. 2B, the elements in
ˆii
(highlighted in
green, e.g., 1
E
) are affected by those in both i
(red) and
ˆ
ii
(blue) in the filtering analysis,
while the elements in
ˆ
ii
or i
(e.g., 2
E
) are not affected by each other. The weight functions
of elements in
ˆii
are changed due to the addition of i
, and therefore, the elements in i
need
14
to be considered when updating the weight functions of elements in ˆi
. The above parallel strategy is used
to calculate the filter matrix of i
, where only the elements in ˆi
need to be updated.
Figure 3. Flow chat of the ADD optimization method. The loop formed by the red lines represents an optimization cycle.
The flow chart of the ADD optimization method is shown in Fig. 3, where the loop formed by the red
line represents an optimization cycle. Two criteria are used to terminate the whole optimization process. In
the first criterion, the optimization ends when the number of solid elements in the newly added cells is
sufficiently small. The second criterion, similar to Eq. (7), is defined as
() *l
VV and
''
1'1 1
11
'
TT
lt ltT lt
tt
CC C
, (17)
15
where ()l
C is the structural compliance in the -thl cycle. In the present study, we set '3T and
'0.01
or smaller.
3.5. Extension of the ADD method
The proposed ADD method can be easily applied in different areas. For example, the subdomain-based
ADD method will be directly applicable to multi-resolution topology optimization. Generally, it is
unnecessary to require that all structural members of the obtained design have the same resolution. We
usually focus on certain regions of the optimized structure, which are referred to as the regions of interest
(ROI). A sufficiently fine mesh needs to be used in conventional optimization techniques to discretize the
whole design domain to achieve the geometric details of the ROI. The subdomain-based ADD method
allows us to increase the resolution of the obtained structure locally. One may use a relatively coarse mesh
to achieve an initial optimized design. After specifying the ROI, the cells that contain the ROI can be
selected. By re-meshing the selected cells using smaller elements and performing further optimization,
slender structural members and detailed topologies of the ROI can be obtained. In Section 4.3, we have
demonstrated that the ADD method is able to realize multi-resolution topology optimization easily and
efficiently.
Besides, the subdomain-based ADD method enables us to shrink the existing design domain while
adding new cells. Conventional optimization techniques remove less efficient materials at the element level,
while the removal of material in the ADD method can be performed at the cell level, which will significantly
improve the efficiency of the form-finding process. The cells that have no or very few solid elements can
be directly removed before starting a new optimization cycle. The simply-connected regions that are
occupied by void elements can be partly removed by reducing the sizes of relevant cells. In an extreme
case, one may have a fixed, prescribed volume for the design space. At the beginning of each optimization
cycle, the existing design domain is reduced by the same amount as that of the newly added cells. Then, the
optimal design domain and the optimal topology would be achieved simultaneously for given loading and
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boundary conditions.
Another important extension of the ADD method is the adaptive strategy of the design domain. In
Section 3.2, new cells are added in the vicinity of those with relatively large average sensitivities. Apart
from this, other strategies can be easily developed for updating the design space. For example, new cells
could be added in the neighborhood of the existing cells which have more solid elements or a larger volume
fraction of solid elements. In Eq. (13),
is set as a constant. Alternatively,
can be defined as a function
of ()
f
l
v. After a certain number of optimization cycles, the average sensitivities of the boundary cells are no
longer significantly larger than that of the interior cells. The average sensitivities of the available cells tend
to be distributed evenly over the whole design domain. ()l
, the value of
in the -thl cycle, is defined
using the following Heaviside functions:
() () ()
f
() ˆ
1max1 , , if '
1, otherwise
ll l
lNv
, (18)
where 1
and ()l
is calculated as
'1'1
1
()
'1
1
Tlt ltT
t
l
Tlt
t
CC
C
. (19) Using this
definition, multiple cells can be added in one optimization cycle. The target volume fraction ()
f
l
v decreases
with the increasing of l , and in Eq. (18),
increases with decreasing ()
f
l
v . Therefore,
increases
gradually during the form-finding process, which can slow down the expansion of the design space and
help achieve a convergent solution.
is set as 1 when ()l
approaches '
. This could help avoid
excessive cell additions in the converging stage.
It should be pointed out that, by adopting different adaptive strategies of the design space, cross-
disciplinary mechanisms can be integrated into the ADD optimization method. Certain regions could be
considered as priority to be added as a new part of the design domain. In the morphogenesis of animal and
17
plant tissues such as human bones and plant leaves, the factors that affect the evolution of the design space
could be, for examples, the concentration of nutrients and the intensity of sunshine. By penalizing the yet
to be occupied design space with different coefficients, the ADD method allows us to drive a biological
organ or tissue to grow in a certain direction or a region. By integrating biophysical factors into the ADD
optimization method, we can explore the formation, growth, and optimization mechanisms of biological
structures.
In this study, the computer code is developed using Python and linked to the commercial software
Abaqus, in a similar manner as in [33]. The Python code developed here can be used to solve general three-
dimensional topology optimization problems.
4. Examples
In this section, five typical compliance minimization problems are provided to verify the effectiveness
of the proposed ADD method. For illustration, the material properties, external forces, and geometric
parameters are normalized as dimensionless parameters. All materials are assumed to be homogenous,
isotropic, and linearly elastic. The Young’s moduli of the solid and void elements are set as solid 1E and
9
void 110E
, respectively. The Poisson’s ratio is set as 0.3 . In the first four examples, three-dimensional
models are established, and the filter radius is set as three times of the element size. In the fifth example, a
two-dimensional model is established, and the filter radius is set as 10 times of the element size. Eight-
node hexahedral elements with size 111 and four-node quadrilateral elements with size 11 are used
for the finite element discretization of the three- and two-dimensional models, respectively.
18
Figure 4. Loading and boundary conditions of the support structure.
4.1. Support structure
We first consider a simple support structure. The initial cuboid design domain and the
loading/boundary conditions are shown in Fig. 4. A concentrated force is applied on the initial design
domain at the center of its top surface. Four bottom corners of the design domain are fixed. The initial
design domain is divided into 1283 cubic cells and each cell consists of 10 10 10 elements. The
target volume of the support structure is set as 28,800 , which is 10% of the volume of the initial design
domain. New cells are not allowed to be added below the bottom of the initial design domain. The structures
optimized within the initial design domain and the automatically expanded design domain are shown in
Fig. 5A–C, respectively.
The updated design domain has more than half a million elements after 10 optimization cycles. In
each cycle, multiple cells are added in the vicinity of the existing cells with high sensitivities. It is seen
from Fig. 5C that new cells are first added near the acting point of the external force. Usage of more
materials in these high-stress regions can effectively stiffen the support structure. The structural compliance
drops rapidly at the beginning of each optimization cycle, and then decreases gradually till the end of the
cycle. The structural compliance decreases significantly in the second optimization cycle. More and more
void elements in the newly added cells evolve into solid elements, while less efficient solid elements in the
old design domain are gradually removed. The topology of the optimized support structure varies distinctly
with the update of the design domain. The compliance is normalized by the value in the initial design
19
domain. The compliance (0.604) of the final design obtained from our ADD method is significantly lower
than that (1.146) of the structure optimized within the initial design domain. The structural performance
improvement reaches as large as approximately 50% .
Figure 5. Optimized topologies of the support structure and the final design domains (semi-transparent), where the acting point
of the external force (red arrow) keeps unchanged. The structures are optimized within (A) the initial design domain and (B) the
automatically deformed design domain, respectively. (C) Evolution histories of the normalized volume and compliance of the
support structure. The initial design domain, the first and the last expanded design domains are presented. The color changes in
the curves are used to distinguish different optimization cycles.
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Figure 6. Optimization under varying loading conditions. Adaptive strategy of the design domain. (A) A limited update region
is introduced to divide the optimization process into several stages, where the optimized structure (blue) is obtained in the
previous stage. (B) New elements are added only near the concentrated force before the structure raises to the top of the limited
update region. The current stage will be completed until a convergent optimized solution is achieved. Optimized topologies of
the support structure which are optimized within (C) the initial design domain and (D, E, F) the automatically deformed design
domains, respectively.
Further, we show that the ADD method can be used in an optimization problem where the loading
conditions vary with the deformation of the design space. This is of importance as the external loads applied
on a growing biological tissue/organ are usually dependent of the morphological evolution of the later [34-
36]. Here, the acting point of the concentrated force (Fig. 4) changes with the form-finding process. The
force is always applied at the top center of the optimized structure. The elements near the acting point of
the concentrated force have higher sensitivity than the other elements. New cells will be added to the
neighborhood of the concentrated force. To avoid such a problem, limited update regions are introduced to
21
divide the form-finding process into several stages, as shown in Fig. 6A. Each stage consists of a certain
number of optimization cycles. In each stage, new elements will be first added in the vicinity of the
concentrated force until the structure raises to the top of the limited update region (Fig. 6B). During this
process, the structural compliance increases rapidly due to the continuous change in the loading condition.
In each stage, the design domain will be reshaped after the loading condition is fixed, and the structural
compliance will decrease distinctly as a result of the ADD optimization. A new stage will start after a
convergent solution is achieved. Here, the height h of the limited update region is set as the length of a
cell, and the whole optimization is terminated after five stages. It is seen from Fig. 6C–E that the optimized
structures become taller and taller, and their four legs become thinner and thinner. The evolution histories
of the normalized volume and compliance of the support structure are shown in Fig. 7. As there are too
many optimization cycles, we do not distinguish them using different colors in the curves. The five sharp
increase–decrease changes in the compliance–iteration curve correspond to the five stages.
Figure 7. Evolutions of the normalized volume and compliance of the support structure, where the acting point of the external
force raises with the structure. Several updated design domains are presented.
4.2. Arch bridge
In the second example, an arch bridge with a horizontal non-design deck on the top surface is
optimized using the ADD method, where the loading and boundary conditions are shown in Fig. 8.
22
Uniformly distributed forces are applied on the top surface of the initial cuboid design domain, which is
divided into 10 2 3 cuboid cells and each cell consists of 20 20 10 elements. The target volume of
the bridge is set as 48,000 , which is 20% of that of the initial design domain. New cells are not allowed
to be added below the bottom of the initial design domain. The selected optimized structures at the end of
each optimization cycle are shown in Fig. 9. The evolution histories of the normalized volume and
compliance are shown in Fig. 10, where several updated design domains are presented.
Figure 8. Loading and boundary conditions of the bridge, where the deck is non-designable.
23
Figure 9. Optimized topologies of the bridge. The structures are optimized within (A) the initial design domain and (B, C, D)
the automatically deformed design domains, respectively.
Figure 10. Evolution histories of the normalized volume fraction and compliance of the bridge. The initial design domain and
two updated design domains are presented. The color changes in the curves are used to distinguish different optimization cycles.
24
There are 688, 000 elements in the updated design domain after 14 optimization cycles. Fig. 9A–
D shows that the bridge-type structures at the end of each optimization cycle are distinctly different in both
the overall topology and the component shape. The compliance (1.859 ) of the final design obtained from
our ADD method is significantly lower than that (2.763 ) of the structure optimized within the initial design
domain. The structural performance improvement reaches as high as approximately 13
.
All designs presented in Fig. 9 are structurally efficient and aesthetically pleasing. They can be used
under different traffic conditions. The structure optimized within the initial design domain is a deck-type
bridge, while that obtained by our ADD method is a half-through type bridge. It is of interest to notice that
the ADD method has helped us generate a bridge design, of which the type is completely different but
significantly better than the initial design. The results show that the ADD method is capable of producing
multiple innovative and appealing structural forms for an engineering project.
Figure 11. (A) Loading and boundary conditions of the bridge, where the deck is non-designable. The design domain L
is
sufficiently large, of which the boundaries are highlighted in blue. (B) Optimized topology of the bridge and the design domain
L
(semi-transparent).
25
Further, we optimize the bridge within a sufficiently large, predefined, fixed, cuboid design domain
L
(Fig. 11A). L
is divided into 1.92 million elements, and its volume is 7 times larger than that
of the original design domain . The target volume of the bridge is set as 48,000 . The optimized
topology of the bridge is shown in Fig. 11B. L
is large enough as in the final design there are no solid
elements near the initial boundaries of L
(except for the bottom). The structure (Fig. 11A) optimized in
L
is almost the same as that (Fig. 9D) obtained by our ADD method, where the difference in structural
compliance is less than 0.3% . It should be noted that the bridge optimization using our ADD method is
completed by an ordinary PC with 16 GB of memory. However, this computer is out of memory when
running the example in Fig. 11. As a result, a high-performance workstation is needed to obtain the
optimized bridge design from the sufficiently large design domain L
. It is seen that the ADD method can
substantially reduce the computational cost.
Figure 12. Loading and boundary conditions of the table, where the top cover is non-designable.
26
4.3. Table
This example will show that cells with different sizes and resolutions can be used in the ADD method.
We optimize a table with a non-design deck, supported by a non-design single square column (Fig. 12). The
column and the table are made up of the same material. The bottom surface of the column is fixed, and its
top surface is tied to the bottom of the table. Consider that the deck is subjected to uniform pressure on its
top surface. The initial design domain is divided into 192 non-uniform cells which contain 268, 000
elements in total. There are 1, 000 elements in a small cell and 2, 000 elements in a large cell. The target
volume of the table is set as 67, 000 , which is 25% of the volume of the initial design domain. New cells
are not allowed to be added above the top surface of the table or below the bottom surface of the column.
Cells that are newly added below the bottom surface of the initial design domain have a smaller size than
those in other regions. Elements with different sizes are used here. Interfacial constraints between the new
cells and the column are imposed. The structures optimized within the initial design domain and the final
design domain are shown in Fig. 13A and B, respectively. The evolution histories of the normalized volume
and compliance are shown in Fig. 13C, where several updated design domains are presented.
Figure 13. Optimized topologies of the table. The structures are optimized within (A) the initial design domain and (B) the
automatically deformed design domain, respectively. Elements covering the column have smaller sizes. (C) Evolution histories
of the normalized volume and compliance of the table, where the initial design domain and two updated design domains are
27
presented. The color changes in the curves are used to distinguish different optimization cycles.
There are more than half a million elements in the updated design domain after eight optimization
cycles. It is seen that new cells are added to the column along the vertical direction. The thickened column
renders higher stiffness and stability of the whole structure. The internal pattern of the table changes
substantially in each optimization cycle, which is adapted to the evolution of the column. The elements
surrounded by the column have smaller sizes. It is interesting to notice that the straight branches that support
the square deck in Fig. 13A evolve into curved ones in Fig. 13B. The compliance (1.430 ) of the final design
obtained from our ADD method is much lower than that (1.635 ) of the structure optimized within the initial
design domain. The structural performance improvement is approximately 12.3% . This example shows
that the sizes of both cells and elements could be adjusted flexibly in our ADD method.
4.4. Hinge arm
In this example, we consider a hinge arm [37]. The initial design domain is shown in Fig. 14. An
upward surface traction is applied at the handle (green), while the bolt holes (blue) are fixed. The inner
surfaces of the bolt holes (red) are pinned, and the region highlighted in yellow is fixed in the vertical
direction. The initial design domain is divided into 148 uniform cells which contain 296, 000 elements
in total. There are 2, 000 elements in each cell. The target volume of the hinge arm is 74,000 , which is
25% of the initial design domain. The side and top views of the structures optimized within the initial
design domain are shown in Fig. 15A and B, respectively. The structures optimized within the automatically
expanded design domain are shown in Fig. 15C–F, where 0.9
in Case 1 (Fig. 15C and D) and
0.99
in Case 2 (Fig. 15E and F). The evolution histories of the normalized volume and compliance are
shown in Fig. 16, where the initial and final design domains are presented. As there are too many
optimization cycles, we do not distinguish them using different colors in the curves.
28
Figure 14. Loading and boundary conditions of the hinge arm.
In Case 1, there are approximately half a million elements in the updated design domain after 15
optimization cycles (Fig. 15C and D). The design domain expands continuously in both the vertical and
horizontal directions. The compliance (1.716 ) of the final design obtained from our ADD method is
significantly lower than that ( 2.384 ) of the structure optimized within the initial design domain. The
structural performance improvement is approximately 28.1% . It is found that the fixed region (highlighted
in blue in Fig. 14) is enlarged with the update of the design domain. It suggests that our ADD method is
able to automatically change the boundary conditions of a topology optimization problem during the form-
finding process. In Case 2, we adopt another adaptive strategy ( 0.99
) for the design domain, which
leads to a final design with a distinctly different topology (Fig. 15E and F). There are 26 optimization
cycles in the whole optimization. The compliance of the final design in Case 2 is 1.718 , which is almost
the same as that in Case 1. The bolt holes are outside the sidewalls of the optimized design in Case 1, while
the holes are inside the sidewalls in Case 2. These two designs, both highly efficient, can be used under
different working conditions.
29
Figure 15. Optimized topologies of the hinge arm. (A) Side and (B) top views of the structures optimized within the initial design
domain, respectively. (C) Side and (D) top views of the structures optimized within the automatically deformed design domain,
respectively. (E) Side and (F) top views of the structures optimized within the automatically deformed design domain,
respectively. We take 0.9
in (C) and (D), and 0.99
in (E) and (F).
Figure 16. Evolution histories of the normalized volume and compliance of the hinge arm. The initial and the final design
domains for both adaptive strategies are presented.
30
4.5. MBB beam
It is of interest to find that the MBB beam optimized with an extended design domain has an wavy
bottom edge [38, 39]. Motivated by this phenomenon, we optimize a two-dimensional MBB beam using
the ADD method. Boundary and loading conditions of the MBB beam example are shown in Fig. 17A. The
design domain is divided into 24 4 square cells and each cell consists of 25 25 elements. New cells
are not allowed to be added above the acting point of the concentrated force. The target volume of the MBB
beam is set as 30, 000 , which is 50% of the volume of the initial design domain. The beams optimized
in the initial and the final design domains are shown in Fig. 17B and C, respectively. The evolution histories
of the normalized volume and compliance are shown in Fig. 17D, where several updated design domains
are presented. As there are too many optimization cycles, we do not distinguish them using different colors
in the curves.
Figure 17. Optimization of the MBB beam. (A) Loading and boundary conditions of the MBB beam. Structures (black)
optimized within (B) the initial design domain (grey) and (C) the final design domain (grey). (D) Evolution histories of the
normalized volume and compliance of the MBB beam, where the initial design domain and two updated design domains are
presented.
The structures in Fig. 17B and C have distinctly different topologies. The compliance (0.458) of the
31
final design obtained from our ADD method is significantly lower than that of the structure optimized
within the initial design domain which is 1.420 . The structural performance is improved by approximately
67.7% . The abrupt changes in the compliance–iteration curve (Fig. 17D) is induced by the breakage of
structural members. It is seen that the design domain mainly evolves downward, and finally the MBB beam
evolves into a truss-like structure.
5. Discussions
Following the above examples, the advantages and potential applications of the developed ADD
optimization method are summarized below. Firstly, this method has been successfully integrated into the
BESO-based computational framework. Owing to its simplicity of concept, it can be easily integrated into
other computational frameworks, e.g., the SIMP technique. Secondly, the filtering scheme (Eq. (16))
enables us to conveniently generate the sensitivities of the emerging elements around the boundaries of the
newly added cells. Thirdly, the numerical results show that the presented method can automatically, flexibly,
and intelligently optimize the design domain during the form-finding process. Both loading and boundary
conditions can be changed automatically during the form-finding process when using this method. Fourthly,
compared with the conventional topology optimization, which is performed within a fixed design space,
the ADD method is able to provide optimized design solutions with significantly higher structural
performance. Fifthly, in the ADD method, the initial design domain is divided into a certain number of
subdomains (cells). Reshaping the design space at the cell level instead of the element level makes the
algorithm more efficient. On the one hand, we only need to analyze the newly added cells and their
neighboring cells when updating the weight functions, as the elemental and nodal information of the rest
subdomains keep unchanged. On the other hand, calculations for different subdomains can be performed
separately and in parallel.
It is worth mentioning that the existing optimization techniques usually aim to produce a single optimal
solution for a given set of loading and boundary conditions. In practice, it is highly desirable to obtain
32
multiple design options which not only possess high structural performance but also have distinctly
different shapes and forms [40, 41]. By imposing different constraints on the evolution of the design space
(e.g., setting a non-design region), or using different termination criteria for the form-finding process, the
ADD method is capable of providing the designer with diverse and competitive structural designs.
Finally, it is noticed that in an optimization problem, there are two types of design variables, i.e., the
design domain and the structural topology. In this study, we handle the two types of design variables
separately. They are not updated simultaneously in the same step of iteration but alternately in different
steps of iteration. If the two types of design variables are simultaneously updated, the potential space
surrounding the current design domain would need to be involved in the finite element analysis and the
sensitivity analysis, which would require significant additional computational cost.
6. Conclusions
In this paper, we have developed a novel and efficient structural topology optimization method using
an adaptive design domain (ADD), which enables the design space to evolve automatically, flexibly, and
intelligently during the form-finding process. It has been successfully integrated into the BESO-based
computational framework. The effectiveness of the proposed methodology is verified by a series of
representative examples, including the compliance minimization of a support structure, an arch bridge, a
table, a hinge arm, and a MBB beam. Compared with the structural designs optimized within a fixed design
domain, the solutions obtained by the ADD method exhibit significantly higher structural performance for
the same amount of material. Besides, the ADD method can be readily integrated into other computational
frameworks. It can be easily extended to solve various optimization problems by, for examples, adopting
different adaptive strategies for different regions of the design space, and using different termination criteria
for the form-finding process. The developed method can not only create more efficient structural designs,
but also can be used in transdisciplinary computational problems, for example, the morphogenesis of
growing biological tissues. Through making the design space adaptive, this work might help broaden the
33
potential applications of topology optimization.
Acknowledgments
This research was supported by the Australian Research Council (DE200100887 and FL190100014)
and the National Natural Science Foundation of China (11921002).
34
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