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Bioinspired Generative Architectural Design
Form-finding and Advanced Robotic
Fabrication Based on Structural Performance
Ding Wen BAOa,b,1, Xin YANa,c,1, Roland SNOOKSb (), Yi Min XIEa ()
a Centre for Innovative Structures and Materials, School of Engineering, RMIT
University,
Melbourne, 3001, Australia
mike.xie@rmit.edu.au
b School of Architecture and Urban Design, RMIT University,
Melbourne, 3001, Australia
roland.snooks@rmit.edu.au
c Centre of Architecture Research and Design, University of Chinese Academy of
Sciences,
Beijing, 100190, China
1 These authors contributed equally to this work.
Abstract. Due to the potential to generate forms with high efficiency and elegant
geometry, topology optimization is widely used in architectural and structural de-
signs. This paper presents a working flow of form-finding and robotic fabrication
based BESO (Bi-directional Evolutionary Structure Optimization) optimization
method. In case there are some other functional requirements or condition limita-
tions, some useful modifications are also implemented in the process. With this kind
of working flow, it is convenient to foreknow or control the structural optimization
direction before the optimization process. Furthermore, some fabrication details of
the optimized model will be discussed because there are also many notable technical
points between computational optimization and robotic fabrication.
Keywords: generative design, form-finding, BESO method, robotic fabrication,
topological optimization, 3d printing, pavilion
Article published in
Architectural Intelligence, pp 147 - 170, 2020, Springer
2
1 Introduction
Throughout the history of architecture, there is always a close relationship between
the development of traditional architectural forms and the evolution of structural
morphology. From Rome to Gothic, the popular architecture prototype changes
from barrel arch to pointed arch based on the technology of flying buttress. Simi-
larly, after the Renaissance, many physical analysis methods are developed to help
architects to achieve more complex architecture forms structurally, like graphic stat-
ics and inverse lifting models by Antonio Gaudi. Furthermore, after Modernism, the
new construction techniques with glass and steel make the international style pop-
ular over the world. Recently, with the fast development of computational tech-
niques, the topic of form-finding based on structural performance has gained new
momentum [1]. Not only researchers have focused their view on structural form-
finding methods, but also many architects are attracted because it can generate
forms with a characteristic of high structural efficiency and more elegant shape po-
tentials.
Among the form-finding methods, the topology optimization method of ESO
(Evolutionary Structural Optimization), developed by YM Xie and GP Steven in
1993 [2] and its modified version, known as BESO (Bi-directional evolutionary
structural optimization), published in 2006 [3], are widely implemented in architec-
ture practices, such as Qatar National Convention Centre and Shanghai Himalayas
Center.
With the further development of Ameba, a new GH plugin based on BESO al-
gorithm by YM Xie and his team [4], more and more architects and designers will
have opportunities to use a new intelligent method to work with the computer inter-
actively, to create innovative, efficient and organic architectural forms and facilitate
the realization of mass customization in the construction industry through the intro-
duction of advanced 3D printing technologies, such as large robotic 3D printing and
some hybrid fabrication strategies developed by Roland Snook and his research
team in RMIT Architectural Robotic Lab. The concept of topological optimization
and the inspiration of Gaudi’s Sagrada Familia Basilica will be reflected through
the pavilion form-finding and its optimization. The new approach of generative ar-
chitectural design and fabrication will be introduced in this project, which explores
the architectural implications of topological optimization design through robotic 3D
fabrication.
2 Basic Theory of BESO Method
BESO algorithm aims to find the solution with the highest structural performance
under certain material limitations by removing or adding material elements step by
step. The basic problem can be described mathematically as follows:
3
ii
T
i
N
i
p
i
T
xC ukuKUUX
∑
=
==
1
2
1
2
1
)( min
(2.1)
subject to
*
1
)
(V
vxV
N
i
ii
≤= ∑
=
X
(2.2)
in which C, U, K, ∗and X are the objective function (compliance), displacement
vector, global stiffness matrix, the objective volume and the global design variable
vector, respectively. The terms , , and are the volume, design variable,
stiffness matrix and nodal displacement vector for i th element. Furthermore, there
are only two alternative values for in BESO, which are 1 for the solid element or
prescribed value for the void element.
For stiffness optimization problem, the sensitivity for i th element, which is
the criterion for design variable , can be calculated as the gradient of compliance
with respect to the design variable [5],
ii
T
i
p
i
i
px
x
Cuku
X
1
2
1)(
−
−=
∂
∂
(2.3)
=
=
=
∂
∂
−=
−
min
1
min
when
2
1 n whe
2
1
1
xx
x
x
x
C
p
iii
T
i
p
iii
T
i
i
i
uku
uku
α
(2.4)
And when the penalty coefficient p tends to infinity, the sensitivity number be-
comes the one in hard-kill BESO method [5], i.e.,
min
1 when 1
12
0 when 0
lim
T
i ii i
i
i
i
p
x
C
px xx
α
→∞
=
∂
=−=
∂= =
u ku
(2.5)
The above sensitivity is usually modified to solve the mesh-dependent problem
[6,7] using a filtering scheme with
∑
∑
=
=
=N
j
ij
N
j
jij
i
w
w
1
1
~
α
α
(2.6)
4
),
0
max(
ij
f
ij
d
rw −=
(2.7)
in which is the distance between the centre of the j th element and the i th ele-
ment and
, is the filter radius and the original sensitivity of the j th element.
To achieve a convergent solution, another historical average,
�, indifferent itera-
tions are introduced [8], i.e.,
2
~~
)1()( −
+
=
n
i
n
i
i
αα
α
(2.8)
In BESO method, the element sensitivities are ranked in each iteration to determine
a threshold with a target volume of next iteration, (), which is defined based on
the current volume (−1) and the evolutionary ratio δ.
)1
(
)
1(
)
(
δ
−= −
n
nV
V
(2.9)
The threshold can be used to evaluate if the element shall be changed in such a way
that if one solid element’s sensitivity is lower than the threshold, its design variable
will be the switch from 1 to , and the design variable of a void element will be
changed from to 1 as well if its sensitivity is higher than the threshold.
3 Form-finding based on BESO
In this work, a pavilion structure is introduced to make a discussion about the BESO
form-finding details for architecture. There are mainly three steps in the form-find-
ing process:
1. Model definition
2. BESO topology optimized iterations
3. Post-modification
To generate an optimized model with not only high structural performance but
also some other characteristics to meet other functional requirements or aesthetic
preferences, there are many detail modifications which we should pay much atten-
tion to during the process.
3.1 Model Definition
BESO method is an FEA (Finite Element Analysis) based iterative process. The
BESO sensitive number, which is the criterion to add or remove the elements, is
also calculated with the data of FEA result. Therefore, defining an appropriate FEA
model is a fundamental work for BESO algorithm.
5
3.1.1 Initial Geometry
Before generating the FEA mesh, the initial geometry should be made as accurately
as possible if there is a rough objective form. Theoretically, BESO object is to find
the best answer within a solution domain about the material distribution, and we can
shrink the initial material distribution possibilities by modifying the initial geometry
to fit the objective form. In this pavilion design, with the inspiration of Gaudi’s
Sagrada Familia Basilica, the concept of this pavilion is to generate a tree-like struc-
ture form (Fig. 3.1).
Fig. 3.1 Natural tree branch (left), columns in Sagrada Familia Basilica (middle)
and one structure in this pavilion (right)
If the initial geometry were designed as a solid block (Fig. 3.2-a), BESO algo-
rithm would generate the form with the highest structural stiffness in Fig. 3.2-b,
which is far from the design concept. Therefore, to produce the branch-like supports
and leave enough space for visitors to go through, the initial geometry should be
modified with some cavities and thin columns (Fig. 3.3).
(a) Initial geometry (b) BESO optimized model
Fig. 3.2 The initial geometry without modification and its BESO result
6
(a) Initial geometry (b) BESO optimized model
Fig. 3.3 The modified initial geometry and its BESO result
3.1.2 Mesh Discretization
Another way to manually predesign the BESO result is to modify the calculation
mesh. In BESO method for continuum structures, the main element types of FEA
calculation mesh are solid for block structures or shell for surface structures, respec-
tively. For this pavilion design, considering symmetric geometry and boundary con-
ditions of the structure, only a quarter is generated with solid elements in the mesh
discretization process.
In BESO method, the structure evolves based on the element addition or deletion
so that the element size can influence the structure details on some level with a
proper BESO filter radius. In other words, the smaller the element size and the filter
radius are, the more details can be generated using BESO.
Furthermore, some mesh modification can also help us to control the final model.
For example, a space near the symmetric plane is imposed to interrupt some certain
force paths (Fig. 3.5) in case the unsafe horizontal beam (Fig. 3.4) occurs.
(a) Initial design mesh (b) BESO optimized model
Fig. 3.4 Initial mesh and BESO result without modification
7
(a) Initial design mesh (b) BESO optimized model
Fig. 3.5 Initial mesh and BESO result with modification
3.1.3 Material Property
The material property setting is another important aspect of FEA modelling. Differ-
ent material properties can also influence the BESO result indirectly. In the finite
element analysis, the material is assumed to be homogenous. For the homogenous,
isotropic, and linearly elastic materials, there are two main parameters, Young’s
modulus and Poisson's ratio. For the model with only one material, the BESO results
will change if they are assigned with different Poisson's ratios (Fig. 3.6) or Young’s
moduli (Fig. 3.7), and Poisson’s ratio has a closer correlation with BESO result than
Young’s modulus.
Fig. 3.6 BESO result with different Poisson’s ratios
(0.15 for the left, 0.30 for the middle and 0.45 for the right)
8
Fig. 3.7 BESO result with different Young’s moduli
(0.01GPa for the left, 1GPa for the middle and 100GPa for the right)
However, for the model with multi-materials, different relative material Young’s
modulus can be designed purposely to generate different forms. For example, Fig.
3.8 shows a control experiment about façade topology optimization. The initial de-
sign domain is divided into two parts, the non-design domain and the design do-
main. One bottom corner is fixed in all three displacement directions, and the non-
design domain is assigned with uniform pressure. The optimized structures vary
significantly with the different materials assigned to the non-design domain. Spe-
cifically, the following figures show the different results with different Young’s
modulus value of the non-design domain, and it can be concluded that with a de-
crease of non-design domain’s Young’s modulus, there will be an increase in the
area of the branch structure’s top to hold the soft materials.
Fig. 3.8 Initial FEA model settings
9
(a) 100GPa (b) 10GPa (c) 1GPa (d) 0.1GPa (e) 0.01GPa
Fig. 3.9 BESO result with different Young’s moduli of non-design domains
3.1.4 Load Case and Boundary Condition
Different from the above geometry part, load case and boundary condition are the
force defining part in FEA process, which can make significant influences on BESO
results.
3.1.4.1 Load Case
Load case and boundary condition are two aspects in FEA to define the force field
where the model is located. Also, some impressive points can be concluded to de-
scribe the relationship between BESO results and the force field.
In architecture design, concentrate load and distributed load are two load types
which are usually used to define load cases. For concentrate load, especially pointed
load, materials always tend to concentrate around the local area where the concen-
trated load acts and form a local structure in the load direction to support that load.
While the distributed load, especially pressure, usually acts on a surface which is
treated as a non-design domain in BESO and supported by some branches in the
final BESO results.
Fig. 3.10 Initial model with pointed load and its BESO result
10
Fig. 3.11 Initial model with pointed load and its BESO result
3.1.4.2 Boundary Condition
It is well-known that one point has six degrees of freedom in 3D space, including
three displacements and three rotations. Boundary conditions describe which direc-
tions are fixed in the model boundaries. The boundary condition should be made
based on the physical conditions around the model. However, when designing a
form, the BESO results can be different with various boundary conditions.
For example, BESO algorithm may generate some structures in certain directions
to resist the displacements or rotations of the boundary if there is not any constraint
in that direction. In the pavilion design, if the bottom corners are only fixed in z
directions, the ring beam at the bottom will be generated to resist the horizontal
displacements of the bottom. However, if the bottom points are pinned in three dis-
placement directions, the ring beam will be unnecessary and avoided by BESO.
Fig. 3.12 BESO results with different boundary conditions
11
3.2 BESO topology optimized iterations
Besides the above details in FEA model definitions, BESO algorithm also provides
users with some algorithm constraints and parameters to modify the designs. In the
past ten years, the topic of modifying the topology optimization method to solve
some specific problems, such as generating symmetric or periodic structures, print-
ing concrete and reserving functional parts, has attracted many attentions. In the
process of this pavilion form-finding, the modifications about the non-design do-
main and symmetric constraint have been introduced. And adjustments of BESO
parameters, such as filter radius (FR), evolution ratio (ER) and volume fraction (VF)
are also considered.
3.2.1 BESO Parameters
BESO main parameters are evolution ratio (ER), filter radius (FR) and volume frac-
tion (VF), describing the number of variable elements, the sample range of averag-
ing sensitivity number and the volume of the final model, respectively.
3.2.1.1 Evolution Ratio (ER)
With different ER values, the topology optimization process will be completed in a
different time, and the results can also be different significantly. It is because, with
a large evolution ratio, the number of variable elements in each iteration will in-
crease too much to get the global optimized structure. In traditional topology opti-
mization theory, to achieve getting effective structures, ER value is suggested
smaller than 5% and as small as possible. However, for designers, the global opti-
mized structure is not necessary sometimes and changing ER value comes to be a
simple way to generate diversity local optimized results with similar structural per-
formances, although a little lower than the global best one.
Fig. 3.13 The BESO results with different ER values
(from left to right 4% 2% 1%)
12
Table 3.1 Iterations and compliances of BESO processes with different ER values
ER = 4%
ER = 2%
ER = 1%
Iteration
83
167
307
Compliance
0.622
0.616
0.603
3.2.1.2 Filter Radius (FR)
Filter radius (FR) is vital in predesigning the BESO result. In topology optimization
theory, the filter radius is introduced to solve the checkboard problem. However,
from the appearance of the final results, the filter radius can be used to predesign
the minimum size of whole structure details. As what the following figures show,
the BESO results of the same model can be different with different filter radiuses.
Fig. 3.14 The BESO results with different FR values
(from left to right 16mm, 24mm, 32mm and 40mm)
Table 3.2 Iterations and compliances of BESO processes with different FR values
FR=16mm
FR=24mm
FR=32mm
FR=40mm
Iteration
82
83
96
106
Compliance
0.591
0.622
0.654
0.797
3.2.1.3 Volume Fraction (VF)
Volume fraction (VF) is the parameter to define the remaining part’s number, and
it is comprehensible that volume fraction has an obvious influence on BESO results.
However, there are also two points should be treated carefully. The first one is that
for some model, VF value cannot be too small in case the whole structure collapses
due to lack of materials. The other one is that the shell element model is easier to
get transparent holes than solid element model with the same VF value, while solid
element model can represent which parts should be thicker than other parts.
13
Fig. 3.15 The BESO result of the top surface with solid elements
Fig. 3.16 The BESO result of the top surface with shell elements
3.2.2 Algorithm Constraint
The three above parameters are the normal parameters in traditional BESO algo-
rithm, and there are also many types of research about the algorithm modifications
in topology optimization. For the pavilion design in this paper, two main algorithm
constraint parts are as follows.
3.2.2.1 Non-design Domain
For some functional requirements, there are always some local parts which should
be reserved during the topology optimization process. Therefore, BESO method
permits the users to set the non-design domain in the initial model, which will take
part in the FEA calculation but will be reserved in the following optimization
14
iterations. To generate the tree-like structures in the pavilion, the initial domains are
set as Fig.3.17.
Fig. 3.17 The initial domain's settings of the pavilion
3.2.2.2 Symmetric Constraint
Because of the symmetric characteristic, this pavilion needs to be kept symmetrical
during the iterations. However, numerical calculation errors or odd void element
numbers may cause asymmetries to the whole model. In BESO, there is also a con-
straint function to keep the model symmetrical all the time.
Fig. 3.18 The BESO result without symmetric constraint
15
Fig. 3.19 The BESO result with symmetric constraint
3.3 Post-modification
It is easy to see that the rough mesh model optimized by BESO method cannot
satisfy the atheistic and fabrication requirements for architecture. As a result, the
optimized mesh model should be modified carefully after the optimization for the
following fabricating works.
3.3.1 Mesh Smooth
For finite element analysis, the calculation mesh is composed of some fundamental
elements, such as triangles or quadrangles for shell and cubes or tetrahedrons for
solid. As a result, the BESO model is always a mesh with a coarse, irregular surface.
Fortunately, the GH plugin, Ameba, has a really strong mesh optimization functions
to deal with that problem. With the help of Ameba mesh tools, it is easy to get a
smooth mesh model for the following fabrication works [9].
Fig. 3.20 The smooth mesh workflow in Ameba
16
Fig. 3.21 The pavilion generation process
3.3.2 Over-hanging
In this pavilion work, large 3D printing techniques are implemented. The current
technique has some printing limitation by the issue of large overhang angles without
any supporting material, so the model should be modified to avoid large draft angles
in the model. The maximal overhang angles are 32 degrees.
3.3.3 Fine-tuning based on the feedback of FEA analysis
Once the form of pavilion is finalised, it has been imported into Abaqus for finite
element analysis to get the more accurate structural performance feedback which
helps to re-test and fine-tune the form to fix some structural defects and ensure the
pavilion has a better structural performance based on keeping the basic generated
geometry (Fig. 3.22).
(a) Displacement (b) Mies Stress (c) Strain Energy Density
Fig. 3.22 FEA analysis
17
4 Advanced Robotics Fabrication
Fig. 4.1 The digital model of the innovative pavilion for fabrication
The digital pavilion structure (Fig. 4.1 left) is finalized based on generative method
topological optimization. To fabricate it, it has been further designed for fabrication
and construction that includes main three parts: top transparent 12mm thickness
acrylic panel, 3d printing main structural bodies and timber base (Fig 4.1 right).
4.1 Application of KUKA Robotics
Fig. 4.2 RMIT Architectural Robotic Lab
18
The Architectural Robotic Lab (Fig.4.2) sits within the RMIT University School of
Architecture, and Urban Design directed by Associate Professor Roland Snooks
leads the school’s development of architectural robotic research and advises on the
development of its infrastructure. The lab consists of nine industrial robots ranging
in scales from a large Kuka KR 150 mounted on a five-meter track, down to small
UR10 robots. Currently, the main robotic research is primarily focused on large 3D
printing of polymers. Roland Snooks and his team have developed a series of inno-
vative 3D printing technologies to build up several pilot project and large-scale pro-
totypes, such as Monash SensiLab (2017) and NGV Floe pavilion (2018) [10].
4.1.1 Advantages of Robotic 3D printing
Fig. 4.3 The eight pieces of components of the pavilion structure
The innovative technology that combines KUKA KR 150 6 axis robot with a 3D
polymer printing extruder is applied on printing large scale prefabricated building
components. The folding, corrugated, translucent printed polymer components can
refract intricate patterns of light and create varying transparencies. The X-Form 1.0
pavilion was printed in 8 pieces with non-screw joint connections (Fig.4.3). This
updated “start-stop” 3D polymer printing approach is a development from previous
one-curve continuous printing path. It achieves the aim of printing fractal-like
19
geometries. The total printing time is 64 hours, including four upper part tree col-
umns and four lower part base columns (Fig. 4.4).
4.2 Modified Printing Path Code for fractal-like geometries
Fig. 4.4 The digital model of fractal-like structure & grasshopper simulation
4.2.1 Start-Stop script development
Fig. 4.5 The updated start-stop script based on grasshopper KUKA PRC
Due to the tree branches system of the pavilion columns, the new start-stop script
can achieve the aim of printing fractal-like forms. The script is originally written in
C# code by Roland Snooks’s research team (Fig.4.5).
20
4.2.2 Import the printing file into KUKA machine & run printing
Firstly, the geometry needs to be imported into rhino/grasshopper. Later, the code
convert geometry into the mesh, slice it in the Z direction and generates the printing
file “prc_kuka.src” based on the printing parameters (speed, layer height, tempera-
ture. etc.). Import the generated .src file into KUKA and run the machine. (Fig 4.6)
Fig. 4.6 The process of robotic large 3D printing using updated code
4.3 Printing Parameters
Fig. 4.7 Testing results of polymer materials with various parameters
21
Materials behaviour in the printing process is an important factor that impacts the
quality of the printing result. Due to some uncontrollable factors, it is hard to avoid
uncertainty, such as the interior humidity, temperature, old/new plastic. However,
it can decrease the risk of uncertainty through repeated experiments and data record.
The main factors include printing speed, layer height, bead size, extrude tempera-
ture and purging step. From table (Fig.4.7), stability of printing will be significantly
influenced by printing speed once it is more than 200 mm/s; the Z height impacts
the stability and speed of printing, and bead size will cause the thickness of extru-
sion; one of the most influent parameters is extruded temperature, it significantly
affects the transparency of printing result. Thus, the most successful result is with
the 60 mm/s, 2.8mm Z-height, 4.2 bead size and 210 degrees extrude temperature
[10]. The purging step is important that can clean the nozzle and ensure the fused
polymer extruded from nozzle equally and smoothly. (Fig. 4.8)
Fig. 4.8 Printing examples with various qualities (from left to right: bad to good)
4.4 Joint Design & Assembly Methodologies
Fig. 4.9 Plug-in joint design for connecting lower part and upper part structures
22
There are two types of joints design applied to this pavilion. One is the plug-in joint
design without any screws. The plug-in joint provides convenience for connecting
the lower and upper part of structures through printing an internal offset layer and
inserting the lower part tubes into the upper part tubes. (Fig. 4.9). The other joint
design is applied to the connection between the top panel and structure branches.
Instead of screws, nails and glues, the white reusable cable ties are used to tie up
through the reserved holes on 3d printing structures, and laser-cut acrylic top panel
(Fig. 4.10).
Fig. 4.10 white cable plastic ties are applied on connecting top panel and structure
Due to the efficient fabrication process by large-scale robotic 3D printing in the lab,
the construction process only took one hour to assemble the whole pavilion by five
students supervised by authors. Five students elevated the upper parts of structures
up to 1 meter, then authors moved the lower part columns to the corresponding
location, and let lower columns are plugged into the columns of the upper part (Fig.
4.11).
Fig. 4.11 on site assembly process
23
5 Conclusion and Future works
Fig. 5.1 The final built pavilion (2m x 2m x 2.5m)
The paper explores the integration of emerging technologies in both digital design
and advanced manufacturing, respectively topological optimization-based genera-
tive architectural forming finding and advanced robotic large 3D printing fabrica-
tion. In this work, a pavilion is introduced to demonstrate the combination of new
design & construction techniques and explain the design & construction process
(Fig. 5.1).
Pavilion X-Form 1.0 is an experimental prototype which tested how important role
the optimized structure play in architectural form-finding (Fig 5.2). The Bi-direc-
tional Evolutionary Structural Optimization (BESO) method provides not only an
efficient structure but also elegant architectural form. The integrated technologies
have the potential to serve the building industry due to its capability of producing
large-scale free form architectural components with high structural performance
and efficient materials.
In a further study, major barriers to the implementation of these technologies in the
building industry will be resolved to apply this new technology widely to the mass
customized design and manufacturing in the building industry. Also, the further re-
search project X-Form 2.0 will be investigated and focus on more complex topo-
logical optimization form-finding, curved top panel, and more advanced 3D printing
technique.
24
Fig. 5.2 The X-Form 1.0 pavilion in the Digital FUTURES 2019 exhibition
(From left to right: Feng ‘Philip’ Yuan, Wen Jun Zhi, Dingwen ‘Nic’ Bao, Mark
Burry, Yi Min ‘Mike’ Xie, Xin Yan, Tong Yu Sun)
Acknowledgements
The authors would like to thank several colleagues whose support helped fulfil the
research project described in this paper:
• Professor Feng ‘Philip’ Yuan (Archi Union, Fab Union, DigitalFU-
TURES, Tongji University)
• Professor Yi Min ‘Mike’ Xie (Centre for Innovative Structures & Materi-
als, RMIT University)
• Associate Professor Roland Snooks (School of Architecture and Urban
Design, RMIT University)
• Dr Jiawei Yao, Dr Xiang Wang, Miss Reina Zhewen Chen (Tongji Uni-
versity)
25
6 References
1. Huang, X., & Xie, Y.M. (2010). Evolutionary topology optimization of continuum structures:
methods and applications. John Wiley & Sons.
2. Xie, Y. M., & Steven, G. P. (1993). A simple evolutionary procedure for structural optimi-
zation. Computers & structures, 49(5), 885-896.
3. Huang, X., Xie, Y. M., & Burry, M. C. (2006). A new algorithm for bi-directional evolution-
ary structural optimization. International Journal Series C Mechanical Systems, Machine El-
ements and Manufacturing, 49(4), 1091-1099.
4. Zhou, Q., Shen, W., Wang, J., Zhou, Y. Y., & Xie, Y. M. (2018). Ameba: A new topology
optimization tool for architectural design. In Proceedings of IASS Annual Symposia (Vol.
2018, No. 19, pp. 1-8). International Association for Shell and Spatial Structures (IASS).
5. Huang, X., & Xie, Y. M. (2010). A further review of ESO type methods for topology opti-
mization. Structural and Multidisciplinary Optimization, 41(5), 671-683.
6. Sigmund, O., & Peterson, J. (1998). Numerical instabilities in topology optimization: a sur-
vey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struc-
tural optimization, 16(1), 68-75.
7. Sigmund, O. (2007). Morphology-based black and white filters for topology optimization.
Structural and Multidisciplinary Optimization, 33(4-5), 401-424.
8. Huang, X., & Xie, Y. M. (2007). Convergent and mesh-independent solutions for the bi-
directional evolutionary structural optimization method. Finite Elements in Analysis and De-
sign, 43(14), 1039-1049.
9. http://ameba.xieym.com
10. Bao, D. W., Yan, X., Snooks, Roland., & Xie, Y.M. (2019). Design and construction of an
innovative pavilion using topological optimization and robotic fabrication. In Proceedings of
IASS Annual Symposia (Vol.2019, pp. 474-481). International Association for Shell and
Spatial Structures (IASS).
