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Maetal. Architectural Intelligence (2023) 2:22
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Architectural Intelligence
Topology optimization ofshell structures
inarchitectural design
Jiaming Ma1, Hongjia Lu1, Ting‑Uei Lee1, Yuanpeng Liu1, Ding Wen Bao1 and Yi Min Xie1*
Abstract
Free‑form architectural design has gained significant interest in modern architectural practice. Due to their visually
appealing nature and inherent structural efficiency, free‑form shells have become increasingly popular in architectural
applications. Recently, topology optimization has been extended to shell structures, aiming to generate shell designs
with ultimate structural efficiency. However, despite the huge potential of topology optimization to facilitate new
design for shells, its architectural applications remain limited due to complexity and lack of clear procedures. This
paper presents four design strategies for optimizing free‑form shells targeting architectural applications. First, we
propose a topology‑optimized ribbed shell system to generate free‑form rib layouts possessing improved structure
performance. A reusable and recyclable formwork system is developed for their effective and sustainable fabrication.
Second, we demonstrate that topology optimization can be combined with funicular form‑finding techniques
to generate a rich variety of elegant designs, offering new design possibilities. Third, we offer cost‑effective design
solutions using modular components for free‑form shells by combining surface planarization and periodic constraint.
Finally, we integrate topology optimization with user‑defined patterns on free‑form shells to facilitate aesthetic
expression, exemplified by the Voronoi pattern. The presented strategies can facilitate the usage of topology
optimization in shell designs to achieve high‑performance and innovative solutions for architectural applications.
Keywords Form‑finding, Topology optimization, Shells, Architectural design
1 Introduction
Within the architectural spectrum, shell structures are
exemplified as pinnacles of design brilliance, engineering
advancement, and visual allure. Characterized by their
ethereal and curved forms, shell structures possess
unique capabilities to span vast space while employing
a minimalist approach to material usage. As a result,
shell structures are commonly seen in the construction
of roofs, domes, and pavilions. Nevertheless, free-form
shells can introduce high shape complexity, posing
significant design and construction challenges (Choong
et al., 2019; Ma, 2022). erefore, to simultaneously
balance the design of free-form shells considering specific
loads along with aesthetic and functional needs at the
same time is no trivial task. Consequently, there exists a
palpable demand for automated design approaches that
can effectively address these challenges (Xie etal., 2023).
Topology optimization is an effective structure design
technique, performed by removing inefficient materials
or relocating them to the most critical locations. It
can generate highly efficient and structurally sound
designs that are difficult or impossible to achieve
through traditional design methods, leading to its
rising popularity in recent years (Ma etal., 2021a). e
technique typically begins with a pre-defined design
domain subjected to a set of loads and boundary
constraints. After the associated mathematical
optimization problem is solved, the material
distribution is adjusted to maximize (or minimize)
the prescribed property, such as maximizing the
*Correspondence:
Yi Min Xie
mike.xie@rmit.edu.au
1 Centre for Innovative Structures and Materials, School of Engineering,
RMIT University, Melbourne 3001, Australia
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Maetal. Architectural Intelligence (2023) 2:22
structural stiffness. Many different techniques have
been developed for topology optimization in the past
few decades, including the Solid Isotropic Material
with Penalization (SIMP) method (Sigmund, 2001), the
Evolutionary Structural Optimization (ESO) method
(Xie & Steven, 1993), the Bi-directional Evolutionary
Structural Optimization (BESO) method (Huang
& Xie, 2010), the Level-set method (LSM) (Wang
et al., 2003) and the Moving Morphable Component
(MMC) method (Guo et al., 2014). Among these, the
BESO and SIMP methods are particularly popular
in designing visually striking and highly efficient
real-world structures, owing to their simplicity and
robustness (Xie, 2022). Sasaki et al. (2007) used the
extended ESO method (EESO) in the design of the
Qatar National Convention Centre in Doha, Qatar.
Lai et al. (2023) applied the multi-material BESO
on the wind bracing design of a 574-m arch bridge
in Chongqing, China. Ohmori et al. (2004) used the
EESO method for the wall design of the Akutagawa
River Side building in Takatsuki, Japan. In addition to
the civil and architectural applications, Airbus (Krog
etal., 2009) successfully applied topology optimization
on its A380, A400, and A350 plane models, including
the design of wing ribs, engine mount frames, and
floor crossbeams. Apart from the artificial structures,
Ma etal. (2021b) and Zhao etal., (2018, 2020) utilized
topology optimization to facilitate the analysis of bio-
structures in plants and animals that have evolved over
millions of years.
Recently, developments targeting architecture design
requirements have been seen in topology optimization
to address the growing demands from architects.
Rong et al. (2022) created an adaptive design domain
algorithm which can help designers to effectively explore
undefined design domains during topology optimization.
He etal. (2023) proposed a hole-filling based approach
to controlling the number of cavities and tunnels in
topology optimization. Yang etal. (2019), He etal. (2020),
and Cai etal. (2021) provided several useful strategies
to generate diverse and competitive designs within the
topology optimization framework. Furthermore, Li
and Xie (2021) extend the BESO algorithm so that the
optimized designs can have multiple materials with
different properties in tension and compression. Bi etal.
(2022) and Xiong et al. (2020) extended the topology
optimization by considering manufacturing constraints
of 3D printing. Zhuang et al. (2023) combine body-
fitted mesh with topology optimization to eliminate the
zig-zag boundaries from optimization, which reduces
the post-processing workload for designers. Moreover,
Yan et al. (2023) proposed a multi-volume constraint
approach to accurately control the local volume of a
structure from topology optimization. Bao et al. (2022)
introduced a swarm-BESO approach to combine logical
structure rules into the multi-agent algorithm to balance
architectural aesthetics and structural efficiency in the
early stage of design.
Despite the widespread interest in topology
optimization by architects and engineers, its real-world
applications on shells remain embryonic due to the
absence of comprehensive guidelines and pioneering
precedents. is paper seeks to redress this imbalance
by offering four topology optimization strategies tailored
for architectural shell applications. e strategies
are proposed based on four major factors related to
Fig. 1 Topology‑optimized ribbed slab with a square design domain: a a topology‑optimized and digital‑constructed square ribbed slab (slab
dimensions: 2.2 m × 2.2 m; column height: 2.5 m) (Ma et al., 2022); b a topology‑optimized curved ribbed shell (Ma et al., 2023)
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Maetal. Architectural Intelligence (2023) 2:22
shell design, including the rib layout, design diversity,
fabrication cost, and artistic expression. Specifically,
Section 2 demonstrates free-form ribbed shells created
by topology optimization and advanced fabrication.
Section3 combines topology optimization with funicular
form-finding techniques to generate diverse designs
with exceptional structural performance. Section 4
offers cost-effective design solutions achieved using
surface planarization and periodic constraints. Section5
presents a new algorithm that can generate free-form
shells with topology-optimized Voronoi patterns.
Section 6 concludes the paper by summarizing the key
points discussed.
2 Optimizing free‑form ribs forshells
For shells in buildings, ribs (see Fig. 1) are often
employed as stiffeners. Traditional techniques typically
produce over-designed ribs for shells, resulting in exces-
sive material usage. Recently, computer-aided methods
have been improved to facilitate the design of rib layouts;
Fig.2 Topology‑optimized ribbed slabs with a 1:1.5 aspect ratio rectangular design domain supported by: a a circular column and generated
from a small filter; b a circular column and generated from a medium filter; c a rectangular column and generated from a small filter; d a rectangular
column and generated from a medium filter
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Maetal. Architectural Intelligence (2023) 2:22
nevertheless, they still rely on the conventional experi-
ence of ribbed shell designs. To overcome the traditional
shape restrictions on the ribs and maximize their contri-
bution to stiffness, we have developed a topology opti-
mization technique capable of producing free-form and
high-performance ribbed shells (Ma et al., 2022). Fig-
ure1a shows a constructed ribbed shell designed using
the proposed optimization technique. In this example,
we optimize and compare the stiffness of our ribbed
floor with that of the Gatti Wool Factory’s ribbed floor
designed by Pier Luigi Nervi. Our result shows a 26.3%
increase in terms of structural stiffness with the same
amount of material. e developed technique is also
applicable to the design of curved ribbed shells (Ma etal.,
2023), as shown in Fig.1b.
In addition to curved shells, it is suggested that the
aspect ratio of a slab, geometry of the column top sur-
face, and filter radius in the optimization settings can
be useful options to generate diverse optimized rib pat-
terns. Figure 2 presents a collaborative study we con-
ducted with Arup on the ribbed floor designs for a depot
building in Sydney. ese designs take into account the
aforementioned factors and adopt the load case detailed
in Ma etal. (2022). e study examines two distinct col-
umn shape configurations on defining the boundary
conditions for a 1:1.5 slab design domain. Additionally,
the outcomes, which include both fine and intermediate
feature sizes, are realized by using different filter radii in
topology optimization.
Additionally, we demonstrate that utilizing a hybrid
digital fabrication approach for construction is practi-
cal and sustainable for realizing free-form ribbed shells.
By assessing the geometry complexity of the ribbed slab
and column in Fig. 1a, CNC machining, and robotic
plastic printing are selected as the measures for fabricat-
ing the formwork components. Although 3D printing
has advantages in creating complex free-form units,
its processing speed and resulting surface evenness are
not comparable to those of subtractive manufactur-
ing methods. erefore, the flat components of the slab
formwork are CNC machined by an ART XR Router
using plywood (see Fig. 3a). To produce the free-form
components with 3-dimensional curvatures, we employ
a KUKA KR 150 6-axis robot equipped with a custom-
ized polymer extruder for 3D printing, as illustrated in
Fig.3c. It can 3D print the two-meter-high column form-
work in two sections as well as the vertical rib shells in
segments using Polyethylene terephthalate glycol (PETG)
pellets. To facilitate the assembly and demolding with
the 3D-printed rib shells, interlock grooves with a depth
of 8mm are engraved on the plywood as presented in
Fig.3b. Owing to the substantial pressure from the liquid
concrete associated with the two-meter column height,
a zig-zag infill, as illustrated in Fig. 3d, is employed to
enhance the formwork strength. is infill increased the
formwork shell thickness to 25mm without excessively
increasing the material usage, formwork weight, and
printing time, which is cost-effective for construction.
e hybrid digital fabricated formwork has proved to
be feasible for traditional concrete casting. Figure4a–d
record the assembly of the column and slab formwork.
Reinforcements are prefabricated and installed within
both formworks to enhance the concrete strength. Once
the formworks and reinforcements are in place, the con-
crete can be poured, vibrated, and leveled, similar to
using conventional formworks (see Fig.4e). Unlike rely-
ing exclusively on 3D printed formworks, the suggested
hybrid formwork offers advantages in terms of time and
cost. Additionally, it can seamlessly integrate with con-
ventional construction materials and techniques that are
well-established and readily accessible.
Fig. 3 Hybrid digital formwork fabrication: a CNC machine the planar components (Ma et al., 2022); b interlock grooves engraved on the planar
components; c robotic 3D printing of the column formwork (Ma et al., 2022); d cross section of the column formwork shell with zig‑zag infill
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Maetal. Architectural Intelligence (2023) 2:22
In contrast to sacrificial 3D printed formworks (Les-
chok & Dillenburger, 2019), the newly developed form-
works can be readily dismantled without damage. is
allows for their reuse in casting repetitive structures,
thereby minimizing material waste and shortening form-
work lead times. Figure5 shows the slab formwork and
column formwork after the casting and demolding. e
column formwork is successfully reused in a subsequent
rammed earth project for construction (Gomaa et al.,
2023). Beyond their reusability, both the 3D printed and
CNC machined formwork components are recyclable.
e used PETG components can be cleaned, melted,
and reformed into new 3D printing pellets, and the used
plywood components can be ground and recycled to pro-
duce new plywood sheets.
3 Optimizing free‑form shells infunicular forms
In traditional shell design processes, a curved shell can be
modified by manually adjusting its curved shape through
trial and error, which is time-consuming and could result
in low performance. Here, we show how topology opti-
mization can be combined with funicular form-finding
techniques to generate diverse and efficient shell designs
for a custom single-story residential house project. Fig-
ure6a presents the floor plan of the project proposed by
our industry collaborators, meeting the client’s spatial
quality requirements, including 11 pillars and 12 walls
Fig. 4 Assembled full‑scale formwork: a column formwork with reinforcement ready (Ma et al., 2022); b slab formwork base; c unfinished slab
formwork with side panels for casting ribs (Ma et al., 2022); d finalized slab formwork with reinforcement; e leveling for the concrete in the slab
formwork
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Maetal. Architectural Intelligence (2023) 2:22
below the roof. By configuring different support locations
on the pillars and walls, distinctly different roof shells in
funicular forms can be obtained using the commercial
software Kangaroo (Version 2.42) (Piker, 2017). e left-
hand side of Fig.6b–d shows three examples of selected
support locations highlighted in blue and the resulting
funicular roofs. Subsequently, topology optimization can
be performed on the three distinctive funicular shells to
generate diverse roof designs with exceptional structural
performance. e right-hand side of Fig. 6b–d shows
the optimized topology using BESO under gravity (Yan
etal., 2019). Note that funicular form-finding techniques
usually generate shells with different amounts of mate-
rial under different boundary conditions; thus, the vol-
ume constraint in topology optimization shall be chosen
accordingly by the designer, considering the openings for
natural lighting, spatial aesthetics, and material usage
factors. In the cases of Fig.6b–d, the target volume frac-
tions of topology optimization are 30%, 63%, and 63%,
respectively. is case testifies that by simply combining
topology and curvature optimization techniques, diverse
shell designs can be generated on a single floor plan pos-
sessing unique curved shapes and optimized topologies.
4 Optimizing free‑form shells withsurface
planarization andperiodic constraint
e fabrication of shell structures, especially with com-
plex curved forms, is often difficult to perform accu-
rately and within budget in real projects. One possible
approach to overcoming these challenges is to rationalize
a curved shell with planar panels. By planarizing the shell
surface while maintaining the overall geometry, the fab-
rication could be simplified from a 3D problem into a 2D
one, which is simpler and less expensive. Figure7 shows
a symmetrical curved shell planarized into multiple 2D
panels. Note that all the panels at the same axial posi-
tion have the same shape. In this case, the shell in Fig.7a
can be fabricated by using only three different categories
of planar panels, as shown in Fig.7b, which may reduce
the fabrication costs significantly compared to creating a
monolithic curved shell.
Since panels within the same category are located at
different positions of the shell and thus bear different
loads, those panels will have distinctively different inter-
nal geometries after topology optimization. Figure 8a
shows an optimized design of the shell in Fig.7b under
gravity using BESO. Non-design solid edges are applied
to each panel to improve structural integrity and con-
nectivity. To fabricate the whole shell structure, nine dif-
ferent panel categories are required, which are shown
in different colors alongside the structure. Hence, the
advantage of using category-reduced planarization has
disappeared. To overcome this setback, periodic con-
straints can be incorporated into topology optimization,
so that panels within the same category will have the
same topology (Xie et al., 2012). For example, a simple
periodic constraint can be achieved by averaging the sen-
sitivities of panels in the same category during the opti-
mization process. Figure8b shows the optimized result
after applying the period constraint. e panel category
number is reduced from nine to three, which has great
cost-saving benefits if the shell is to be built using cast-
ing. Meanwhile, the compliance only increases slightly by
Fig. 5 Reassemble the used: a slab formwork ready for casting
the next slab; b column formwork for a rammed earth project
(Gomma et al., 2023)
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Maetal. Architectural Intelligence (2023) 2:22
13%. Although the combination of category-reduced pla-
narization and periodic-constrained optimization sounds
trivial, it can substantially reduce the challenges and
expenses related to fabrication and logistics when creat-
ing free-form shells in practical applications.
5 Optimizing free‑form shells withVoronoi
patterns
Large open spaces in the optimized results may not
always satisfy functional requirements from buildings,
and the unique solutions from traditional topology
optimization often lack user-defined artistic expression.
Here, we offer a new method that can integrate topology-
optimized shells with customized design patterns such as
the Voronoi pattern, which is commonly seen in nature.
e workflow of the proposed method is demonstrated
through a shell example as shown in Fig. 9. By taking
advantage of symmetry, topology optimization is per-
formed on a quarter of the shell (see Fig.9a) using the
SIMP method. Figure9b displays the boundary condi-
tions of the optimization setting as well as the result-
ing optimized structure with a 30% volume constraint.
Based on the optimization outcome, a multi-resolution
re-meshing algorithm (Botsch & Kobbelt, 2004; Piker,
2013) is first adopted to create a triangle mesh with ver-
tex density proportional to the optimized material distri-
bution (see Fig.9c). e darker regions on the optimized
design have higher material density. en, the Voronoi
pattern can be extracted as the dual of the triangle mesh
by connecting adjacent face centroids, as indicated in
Fig.9d. In the generated pattern, the density of polygons
intensifies within the darker regions of the shell, mitigat-
ing the presence of large cavities that could compromise
structural performance. To further improve the organic
appearance of the design, randomness can be introduced
to the vertex positions. Subsequently, the width of the
structural members can be meticulously adjusted using a
single parameter (Lu etal., 2023) to control the volume
of the final design (see Fig.9e). Figure9g and h show the
final shell in a quarter and a full model. Spline curves are
implemented to smooth the Voronoi pattern as presented
in Fig.9f. In this case, the solid region from the optimized
results and the non-design domain near the edges are
reserved to improve the functionality and aesthetics of
Fig. 6 Funicular roofs with different support locations highlighted in blue (left) and their topology‑optimized designs (right): a floor plan; b a shell
supported purely by columns; c a shell supported at the two ends; d a shell supported by columns and partially selected walls
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Maetal. Architectural Intelligence (2023) 2:22
Fig. 7 A free‑form shell in its: a curved form composed of a monolithic piece; b planarized form composed of multiple 2D panels
Fig. 8 Topology‑optimized roof with surface planarization: a without periodic constraint; b with periodic constraint. C denotes normalized
structure compliance
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Maetal. Architectural Intelligence (2023) 2:22
the shell. e large openings in the topology-optimized
design, which is less practical in building applications, are
eliminated.
To demonstrate the improvement of structural per-
formance by generating Voronoi patterns based on
topology-optimized results, we present and compare the
Topology-optimized-Voronoi shell against its counter-
part that does not undergo such optimization. Figure10a
shows the optimized design from Fig.9, which has more
materials and smaller cavities allocated to the darker
Fig. 9 Shell design with topology‑optimized Voronoi patterns: a initial shell model subjected to a concentrated load; b boundary conditions
and the topology‑optimized result of the quarter shell; c triangular mesh generated by a multi‑resolution re‑meshing algorithm; d extract Voronoi
pattern as the dual of the triangle mesh; e adjust member thickness and include randomness; f interpret the Voronoi polygons using spline curves
and trim with black regions; g final design of the quarter shell; h final design of the whole shell
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Maetal. Architectural Intelligence (2023) 2:22
regions of the topology-optimized result. In comparison,
a shell with a uniformly distributed Voronoi pattern is
generated with the same material volume (see Fig.10b).
e non-design domains of the solid edges are preserved
in both cases. By performing finite element analysis on
both designs, the normalized structural compliances
of the optimized shell and the non-optimized shell are
obtained as 0.45 and 1, respectively. erefore, the struc-
tural stiffness of the optimized shell, which is the recip-
rocal of compliance, has increased by 122% compared
to the non-optimized shell. Besides the Voronoi pattern,
bespoke patterns crafted by architects also possess the
possibility to integrate with topology optimization. Such
amalgamation can produce shells that can accommodate
both aesthetic preference and technical requirements.
6 Concluding remarks
In this paper, we have introduced four topology
optimization strategies tailored for free-form shells in
architectural applications. In the first strategy, we show
that topology optimization can be flexibly employed to
generate structurally efficient and aesthetically pleasing
stiffened ribs for shells. Nine ribbed shell designs are
created by considering different curved forms, aspect
ratios, and optimization settings. A hybrid digital
formwork system capable of producing topology-
optimized ribbed shells is developed and demonstrated
by a 2.5-meter-high ribbed slab unit, which also exhibits
promising reusability and recyclability. In the second
strategy, we illustrate that curvature optimization
techniques such as funicular form-finding can be
combined effectively with topology optimization to
achieve diverse solutions for shells. ree distinctly
different designs generated by the proposed combined
strategy are given on a custom single-story residential
house project as a proof of concept. In the third strategy,
we demonstrate that topology optimization subject to
periodic constraints can be used with planarization
techniques to reduce the fabrication challenge for shells.
A simple example is provided to illustrate that a free-form
shell can be rationalized and optimized into only three
kinds of different planar panels rather than nine, which
can potentially save significant manufacturing costs.
Finally, we show that customized design patterns, such
as the Voronoi pattern, can be integrated with topology
optimization to incorporate aesthetic preferences into
shells with optimized structural performance. A stiffness
enhancement of 122% is achieved on a free-form shell
customized with the Voronoi pattern. In addition to
applying the four strategies individually, they can also
be flexibly integrated to serve multiple purposes. For
instance, combining strategy three with strategy four
can produce cost-effective shells with topologically
optimized customized patterns. Although most of the
Fig. 10 Shell design in Voronoi patterns: a with topology optimization; b without optimization
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Maetal. Architectural Intelligence (2023) 2:22
examples shown in this study are solved using the BESO
method, the design strategies presented in this paper
are equally applicable to other commonly used topology
optimization techniques. ey can be readily used by
designers and engineers to automatically balance the
design requirements between artistic preference and
technical performance for shell design in architecture
applications.
Acknowledgements
This project is supported by the Australian Research Council (FL190100014).
The authors are grateful to Alex Edwards and Phillip Lathourakis from Arup’s
Sydney office, and to Peter Felicetti from Felicetti Pty Ltd and Dr Zhonggao
Chen for their collaboration on the residential building projects.
Author’s contributions
Jiaming Ma: Conceptualization, Methodology, Investigation, Software,
Writing – original draft. Hongjia Lu: Methodology, Software, Writing – review &
editing. Ting‑Uei Lee: Methodology, Writing – review & editing. Yuanpeng Liu:
Methodology, Software, Writing – review & editing. Ding Wen Bao: Resources,
Writing – review & editing. Yi Min Xie: Funding, Resources, Writing – review &
editing, Supervision.
Funding
This project is supported by the Australian Research Council (FL190100014).
Availability of data and materials
The datasets generated and/or analyzed during the current study are available
from the corresponding author upon reasonable request.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
Two of the authors are Guest Editors of the special issue “Integration of Design
and Fabrication” for Architectural Intelligence but were not involved in the
journal’s review, or any decisions, related to this submission.
Received: 28 September 2023 Accepted: 30 October 2023
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