Constructing topologically optimized spatial structure using innovative mortise-and-tenon joints

Abstract

Topology optimisation methods have been implemented on many architectural projects, and their design layouts always are fabricated with additive manufacturing methods due to their complex organic forms. However, there are still some difficulties that limit the application potentials of additive manufacturing in real projects, including size limitations of the fabrication devices, construction cost and transportation probabilities. Therefore, different from the conventional fabricating methods, this research achieves topologically optimized models using a subtractive manufacturing method with timber mortise-and-tenon joints from the Chinese traditional wooden tectonic system. In this paper, an initial shell model with specific plane angles is generated for Bi-directional Evolutionary Structural Optimization (BESO) due to the carpentry assemble requirements. Then, different classical mortise-and-tenon joints are selected to connect timber parts in different force-transferring paths and fabrication limitations, including plugging joints, connecting joints, and steering joints. Finally, with the convenience of on-site augmented reality (AR) assisted techniques, the lightweight timber parts can be easily assembled only by several people in an extremely short time. This project presents an effective and innovative way to achieve material efficiency and lightweight through a combination of innovative advanced tools and traditional woodworking techniques. These timber joints can be also used in architectural design in the context of Chinese traditional tectonics. The prototype pavilion is proof of the concept for the great potential in applications of on-site modular buildings, installations, and artworks.

Full text

Proceedings of the IASS Annual Symposium 2023
Integration of Design and Fabrication
10–14 July 2023, Melbourne, Australia
Y.M. Xie, J. Burry, T.U. Lee and J. Ma (eds.)
Copyright © 2023 by Xin YAN, Ding Wen BAO, Congcong REN and Yi Min XIE. Published in the Proceedings of
the IASS Annual Symposium 2023 with permission.
Constructing topologically optimized spatial structure using
innovative mortise-and-tenon joints
Xin YANa, Ding Wen BAOb,c,*, Congcong RENd, Yi Min XIEb,*
b,* Centre for Innovative Structures and Materials, School of Engineering, RMIT University,
Melbourne, 3001, Australia
nic.bao@rmit.edu.au
mike.xie@rmit.edu.au
a Tsinghua University, Beijing, 10080, China
c School of Architecture and Urban Design, RMIT University,
Melbourne, 3001, Australia
d School of Architecture and Urban Planning, Beijing University of Civil Engineering and Architecture,
Beijing, 102627, China
Abstract
Topology optimisation methods have been implemented on many architectural projects, and their design
layouts always are fabricated with additive manufacturing methods due to their complex organic forms.
However, there are still some difficulties that limit the application potentials of additive manufacturing
in real projects, including size limitations of the fabrication devices, construction cost and transportation
probabilities. Therefore, different from the conventional fabricating methods, this research achieves
topologically optimized models using a subtractive manufacturing method with timber mortise-and-
tenon joints from the Chinese traditional wooden tectonic system. In this paper, an initial shell model
with specific plane angles is generated for Bi-directional Evolutionary Structural Optimization (BESO)
due to the carpentry assemble requirements. Then, different classical mortise-and-tenon joints are
selected to connect timber parts in different force-transferring paths and fabrication limitations,
including plugging joints, connecting joints, and steering joints. Finally, with the convenience of on-site
augmented reality (AR) assisted techniques, the lightweight timber parts can be easily assembled only
by several people in an extremely short time. This project presents an effective and innovative way to
achieve material efficiency and lightweight through a combination of innovative advanced tools and
traditional woodworking techniques. These timber joints can be also used in architectural design in the
context of Chinese traditional tectonics. The prototype pavilion is proof of the concept for the great
potential in applications of on-site modular buildings, installations, and artworks.
Keywords: BESO, mortise-and-tenon joint, topology optimization, timber structures
1. Introduction
Topology optimization has gained continuously growing attention in architecture design due to its
potential for generating elegant and lightweight structures with high structural performance. During the
last two decades, there are several classical topology optimization methods have been developed widely
and deeply, e.g., Homogenization method [1, 2], the solid isotropic material with penalization (SIMP)
method [3, 4], the evolutionary structural optimization (ESO) [5, 6], Bi-directional evolutionary
structural optimization (BESO) [7, 8] and the level-set method (LSM) [9, 10]. And there are two main
tendencies of topology optimisation in architectural research fields. On one hand, more and more
contemporary architectural projects absorb topology optimisation as their concept design sources, e.g.,
Akutagawa River Side in Takatsuki City [11], tree-like pavilions [12], Qatar National Convention
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Centre & Shanghai Zendai Himalayan Art Centre [13, 14], concrete slab [15], Volu Dining Pavilion
[16], and so on. On the other hand, topology optimisation methods can also be introduced for structural
form analysis for traditional architectures, e.g., Sagrada Familia [17], Palazzeto Dellospori of Rome
[18], Chinese wooden mortise and tension joints [19], Chinese rainbow bridge [20], etc. However, there
is seldom research combining the above two directions to perform a new structural-effective
architectural design with traditional tectonic form features and constructing probabilities.
Furthermore, although most classical topology optimisation methods are effective in finding the globally
optimised solutions around the whole design domain, there are usually some objective constraints or
subjective preferences which also play important roles in architectural form-finding. Faced with these
non-structural requirements, many researchers have been developing lots of modified topology
optimisation methods for specific aims, e.g., introducing detail constraints for concept design [21, 22],
eliminating enclosed voids [23], obtaining self-supporting optimized models [24], generating diverse
truss structures[25], simulating leaf morphogenesis [26], etc. And in this project, the carpentry
manufacture techniques of mortise-and-tenon joints make special demands on the topologically
optimised layouts, which requires modified BESO algorithms as well as customised finite element
analysis models.
Therefore, an innovative structural design-fabrication working-flow based Multi-volume constraint
BESO method [27] and Chinese traditional mortise and tension joint carpentry are introduced with a
pavilion project named X-Form 3.0 in this paper. Compared with its two previous versions (X-Form 1.0
& 2.0) [12, 28], X-Form 3.0 consists of timber components connected with mortise and tension joints,
which means that the fabrication of this project is subtractive manufacturing rather than additive
manufacturing in the past projects. Therefore, this paper proposes a new structural topology optimisation
workflow oriented to subtractive manufacturing. And with the Chinese traditional mortise and tension
joints, this installation can be transferred easily with several light slender parts and assembled on-site
with a few people, which shares the great potential in applications of on-site modular buildings,
installations, and artworks.
2. Structural form-finding
2.1. Multi-volume constraint BESO method (MV-BESO)
To form a structural design with slender tree-like components, the multi-volume constraint BESO (MV-
BESO) method [27] is introduced. This is because conventional topology optimization methods are
aimed at achieving the solution which maximizes the structural performance under a certain global
volume fraction constraint and always form continuum solid parts instead of slender rods. However, the
Multi-volume BESO aims to achieve the same object with several local volume fraction constraints and
share the ability of adjusting local structural designs. The basic problem of MV-BESO is different from
the description of conventional topology optimization methods, and it can be written mathematically as
below:
1
11
( )= 22
N
T p T
i i i i
i
min C x
=
=
X U KU u k u
(1)
*
1
( ) k
N
k k i i k
i
subject to SUBV x v SUBV
=
=

X
(2)
in which
C
,
X
,
U
,
K
,
N
are the objective function (compliance), design variable vector, displacement
vector, global stiffness matrix and total element number in the global domain, respectively.
*
k
SUBV
,
k
X
and
k
N
are the local target volume, local design variable vector and local element number of k-th sub-
domain. The terms
i
v
,
i
x
,
i
k
and
i
u
are the volume, design variable, stiffness matrix and nodal
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displacement vector for i-th element. Particularly, the value of
i
x
is alternatively
min
x
(a prescribed small
positive value, e.g., 0.001) or 1.
For stiffness optimization, the sensitivity
i

for i-th element, which is the criterion for design variable
i
x
, can be calculated as the gradient of compliance with respect to the design variable [29],
1
( ) 1
2
pT
i i i i
i
Cp
x
−
=−

Xx u k u
(3)
1
1 when 1
12
when
2
T
i i i i
ipT
imin i i i i min
x
C
px xxx

−
=


= − = 
=


u k u
u k u
(4)
And when the penalty coefficient p tends to infinity, the sensitivity number
i

becomes the one in hard-
kill BESO method [29],
1 when 1
12
0 when 0
T
i i i i
i
ii min
x
C
px xx

=

= − = 
==

u k u
(5)
The above sensitivity is usually modified to solve the mesh-dependent problem [30], using a filtering
scheme:
1
1
N
ij j
j
iN
ij
j
w
w


=
=
=

(6)
max(0, )
ij min ij
w R d=−
(7)
in which
ij
d
is the distance between the center of the j-th element and the i-th element.
min
R
and
i

are
the filter radius and the original sensitivity of the j-th element.
To achieve a convergent solution, another historical average of
i

in different iterations is introduced
[30]:
( ) ( 1)
2
nn
ii
i


−
+
=
(8)
In MV-BESO method, for k-th sub-domain, the element sensitivities are ranked in each iteration to
determine a threshold with local target volumes of next iteration,
( )
n
k
SUBV
, which is defined based on
the current volume
( 1)n
k
SUBV −
and the evolutionary ratio

.
( ) ( )
1(1 )
nn
kk
SUBV SUBV

−
=−
(9)
The threshold can be used to evaluate if the element shall be changed in a such way that if one solid
element’s sensitivity is lower than the threshold, its design variable will be switch from 1 to
min
x
, and
the design variable of a void element will be changed from
min
x
to 1 as well if its sensitivity is higher
than the threshold.
The cycle of FEA and BESO evolution continues until the following convergence criterion defined in
the variation of the objective functions is satisfied [30]:
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( )
11
1
1
1
N
k i k N i
i
N
ki
i
CC
C

− + − − +
=
−+
=
−


(10)
where
k
is the current iteration number,

is an allowable convergence error (

= 0.001 in this paper),
which means stable compliance at least in successive 10 iterations.
2.2. Topology optimization with structural predesign
As Fig.1 shows, the pavilion in this project has a centrosymmetric free-form roof with width and length
of about 3960mm and 3900mm, respectively. The curved roof is consisted of several quadrilateral or
triangular plates (grey parts) and supported by three equal tree-like wooden structures (blue parts).
Therefore, in the following structural form-finding process, only one-sixth part is introduced with
symmetric condition settings.
Figure 1. The top view of whole X-Form 3.0 pavilion.
The timber structure components are required to be hexagonal or octagonal columns to leave the spaces
for mortise-and-tension joints carpentry works, which means that the timber components need to link
each other with the fixed plane angles of 0, 45, 90, 135, or 180 degrees. Therefore, as illustrated in Fig.2,
an initial shell model with the specific plane angles is generated as the basic design domain for the
topologically optimised form-finding. Moreover, the above initial design domain is divided into 30 sub-
domains and 6 non-design domains to get the locally optimised structure design with MV-BESO method
effectively.
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Figure 2. Initial model and sub-domain settings for MV-BESO methods.
Fig.3 illustrates some typical stages of MV-BESO stiffness optimisation process from iteration 1 to
iteration 94. This FEA model is divided into 100,000 shell elements with the thickness of 10 mm. The
material assumption is assumed as Young’s Modulus of 700 MPa and Poisson ratio of 0.45. The MV-
BESO parameters are Rmin of 20 mm and ER of 1%. All the local volume fraction constraints for sub-
design domains are 45%. With the boundary conditions and loading cases (Fig.4 a), it is obvious that
each sub-design domain removes its materials during the evolutionary loop. After 94 iterations, the final
optimised structural layout is generated and reshaped as octagonal rods according to the requirements
from carpentry work (Fig.4 b).
Figure 3. MV-BESO evolutionary history of X-Form 3.0.
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(a) (b)
Figure 4. Boundary conditions & load case (a) and modified structural layout(b).
3. Wooden mortise-and-tenon joint and on-site assembles
One of the project’s aims is to form an innovative wooden working flow which can be transferred and
assembled easily, so the above optimised structure with MV-BESO needs to be divided into several
slender timber rods with the maximal length of 1500mm and section diameters from 60mm to 20mm.
Then, several different kinds of Chinese traditional mortise-and-tenon joints are introduced to link
different structural members without any nails. The selected mortise-and-tenon joints are straight
mortise-and-tenon, dovetail mortise-and-tenon, silver ingot mortise-and-tenon, mantis head mortise-
and-tenon, and sleeve-shoulder mortise-and-tenon, etc.
Figure 5. Diagram of different mortise-and-tenon distributions in X-Form 3.0.
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3.1. Plugging joints
In this pavilion fabrication, the most commonly used plugging joint is straight mortise-and-tenon joint,
which is usually applied to insert the end of one rod into the middle of another rod (Fig. 6). With this
joint, it is really convenient to form a structural rod like a “tree branch”. Therefore, it is very suitable
for generating tree braches.
Figure 6. Diagrams of straight mortise-and-tenon joint.
3.2. Connecting joints
The word “connecting joints” here represents the mortise-and-tenon joints which are implemented for
the main purpose to link two structural parts, instead of generating new parts or new directions. And
there are two kinds of connecting joints in this pavilion, which are mantis head mortise-and-tenon joint
and sleeve-shoulder mortise-and-tenon joint, respectively (Fig. 7-8). The former one is used to combine
two parallel rods into one. It is really necessary to link the three tree supports together with the mantis
head mortise-and-tenon joint. And the function of sleeve-shoulder mortise-and-tenon joint is to fix roof
panel to the top of tree branch.
Figure 7. Diagrams of mantis head mortise-and-tenon joint.
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Figure 8. Diagrams of sleeve-shoulder mortise-and-tenon joint.
3.3. Steering joints
The last kind of mortise-and-tenon joint in this project is steering joint, and it is the main component to
change rod direction. As Fig. 9-10 shows, both dovetail mortise-and-tenon joint and silver ingot mortise-
and-tenon joint can undertake this task. One biggest difference between the two joints is that dovetail
mortise-and-tenon joint can only link light parts, while silver ingot mortise-and-tenon joint is always
used in the main structure components.
Figure 9. Diagrams of dovetail mortise-and-tenon joint.
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Figure 10. Diagrams of silver ingot mortise-and-tenon joint.
Figure 11. Rendering view of X-Form 3.0.
3.4. Augmented reality on-site assembles
Taking the advantages of mortise-and-tenon joints, X-Form 3.0 is carrying-friendly and assembling-
friendly. For example, the whole weight of wooden material is only 24kg, and the slender structural rods
can be divided into two suitcases easily. Furthermore, the mortise-and-tenon joints also share the
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advantages of on-site assembling with Fologram. Fologram is an augmented reality equipment which
can transform 3D models into full-size construction guides through augmented reality glasses such as
Hololens. This program is dedicated to overlaying digital guidance in the workspace to assist in the
construction of complex projects that require a series of measurements, verification and targeted
management, enabling step-by-step guidance for masonry work during construction. Thus, using
Fologram, anyone can easily and effectively assemble the rods together (Fig. 12).
Figure 12. Fologram augmented reality construction technology.
4. Conclusion
This paper introduces the whole working flow of X-Form 3.0 including structural form-finding, joint
fabrication and on-site assemble. The most notable contribution of X-Form 3.0 is that it presents a
successful example to integrate topology optimisation based form-finding method with Chinese
traditional wooden tectonics.
Firstly, a modified BESO method, MV-BESO method, is introduced to form the slender structural rods
for timber fabrication. To achieve this, it is necessary to pay more attention to the initial model design
according to carpentry requirements. Then, five different mortise-and-tenon joints are implemented to
appropriately link different structural members. In this phase, the traditional carpentry also plays a great
role in the fabrication because there are always some complex shapes in the mortise-and-tenon joints
which can not be fabricated by robots. Finally, taking the advantage of mortise-and-tenon joints, the
team can conveniently transfer all the structural components and assemble them on-site with Fologram.
Furthermore, it is notable that X-Form 3.0 is only a preliminary attempt to integrate topology
optimisation with traditional subtractive manufacturing, and there are still some parts which require
more in-depth studies, e.g., an effective BESO method to generate timber rods, a robotic fabrication
method to make complex wooden joints, and so on.
This innovative pavilion is a demonstration of the combination of new design and traditional
construction techniques, explaining the design and construct process of the pavilion through exploration
of emerging technologies in both digital design and subtractive manufacturing, which shares great
significance for architects, engineers, and designers.
Acknowledgements
The authors gratefully acknowledge the financial support provided by the Australian Research Council
(FL190100014) and National Natural Science Foundation of China (51908543).
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