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). Therefore, 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 (** Journal, 27, 1749–1758." href="/article/10.1007/s44223-023-00042-z#ref-CR17" id="ref-link-section-d24448823e360">2021a). The 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 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 (**e & Steven, 1993), the Bi-directional Evolutionary Structural Optimization (BESO) method (Huang & **e, 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 (**e, 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 et al., 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 et al. (2021b) and Zhao et al., (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 et al. (2023) proposed a hole-filling based approach to controlling the number of cavities and tunnels in topology optimization. Yang et al. (2019), He et al. (2020), and Cai et al. (2021) provided several useful strategies to generate diverse and competitive designs within the topology optimization framework. Furthermore, Li and **e (2021) extend the BESO algorithm so that the optimized designs can have multiple materials with different properties in tension and compression. Bi et al. (2022) and **ong 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. This paper seeks to redress this imbalance by offering four topology optimization strategies tailored for architectural shell applications. The strategies are proposed based on four major factors related to 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. Section 3 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. Section 5 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 for shells

For shells in buildings, ribs (see Fig. 1) are often employed as stiffeners. Traditional techniques typically produce over-designed ribs for shells, resulting in excessive material usage. Recently, computer-aided methods have been improved to facilitate the design of rib layouts; nevertheless, they still rely on the conventional experience of ribbed shell designs. To overcome the traditional shape restrictions on the ribs and maximize their contribution to stiffness, we have developed a topology optimization technique capable of producing free-form and high-performance ribbed shells (Ma et al., 2022). Figure 1a 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. The developed technique is also applicable to the design of curved ribbed shells (Ma et al., 2023), as shown in Fig. 1b.

Fig. 1
figure 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)

Full size image

In addition to curved shells, it is suggested that the aspect ratio of a slab, geometry of the column top surface, and filter radius in the optimization settings can be useful options to generate diverse optimized rib patterns. Figure 2 presents a collaborative study we conducted with Arup on the ribbed floor designs for a depot building in Sydney. These designs take into account the aforementioned factors and adopt the load case detailed in Ma et al. (2022). The study examines two distinct column 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.

Fig.2
figure 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

Full size image

Additionally, we demonstrate that utilizing a hybrid digital fabrication approach for construction is practical 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 fabricating 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 manufacturing methods. Therefore, 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 customized polymer extruder for 3D printing, as illustrated in Fig. 3c. It can 3D print the two-meter-high column formwork 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 8 mm 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. This infill increased the formwork shell thickness to 25 mm without excessively increasing the material usage, formwork weight, and printing time, which is cost-effective for construction.

Fig. 3
figure 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

Full size image

The hybrid digital fabricated formwork has proved to be feasible for traditional concrete casting. Figure 4a–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 concrete can be poured, vibrated, and leveled, similar to using conventional formworks (see Fig. 4e). Unlike relying exclusively on 3D printed formworks, the suggested hybrid formwork offers advantages in terms of time and cost. Additionally, it can seamlessly integrate with conventional construction materials and techniques that are well-established and readily accessible.

Fig. 4
figure 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

Full size image

In contrast to sacrificial 3D printed formworks (Leschok & Dillenburger, 2019), the newly developed formworks can be readily dismantled without damage. This allows for their reuse in casting repetitive structures, thereby minimizing material waste and shortening formwork lead times. Figure 5 shows the slab formwork and column formwork after the casting and demolding. The 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. The 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 produce new plywood sheets.

Fig. 5
figure 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)

Full size image

3 Optimizing free-form shells in funicular 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 optimization can be combined with funicular form-finding techniques to generate diverse and efficient shell designs for a custom single-story residential house project. Figure 6a 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 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). The 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. The right-hand side of Fig. 6b–d shows the optimized topology using BESO under gravity (Yan et al., 2019). Note that funicular form-finding techniques usually generate shells with different amounts of material under different boundary conditions; thus, the volume 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 fractions of topology optimization are 30%, 63%, and 63%, respectively. This case testifies that by simply combining topology and curvature optimization techniques, diverse shell designs can be generated on a single floor plan possessing unique curved shapes and optimized topologies.

Fig. 6
figure 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

Full size image

4 Optimizing free-form shells with surface planarization and periodic constraint

The fabrication of shell structures, especially with complex curved forms, is often difficult to perform accurately 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 fabrication could be simplified from a 3D problem into a 2D one, which is simpler and less expensive. Figure 7 shows a symmetrical curved shell planarized into multiple 2D panels. Note that all the panels at the same axial position 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.

Fig. 7
figure 7

A free-form shell in its: a curved form composed of a monolithic piece; b planarized form composed of multiple 2D panels

Full size image

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 internal 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 connectivity. To fabricate the whole shell structure, nine different 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 constraints can be incorporated into topology optimization, so that panels within the same category will have the same topology (**e et al., 2012). For example, a simple periodic constraint can be achieved by averaging the sensitivities of panels in the same category during the optimization process. Figure 8b shows the optimized result after applying the period constraint. The panel category number is reduced from nine to three, which has great cost-saving benefits if the shell is to be built using casting. Meanwhile, the compliance only increases slightly by 13%. Although the combination of category-reduced planarization and periodic-constrained optimization sounds trivial, it can substantially reduce the challenges and expenses related to fabrication and logistics when creating free-form shells in practical applications.

Fig. 8
figure 8

Topology-optimized roof with surface planarization: a without periodic constraint; b with periodic constraint. C denotes normalized structure compliance

Full size image

5 Optimizing free-form shells with Voronoi 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.

The workflow of the proposed method is demonstrated through a shell example as shown in Fig. 9. By taking advantage of symmetry, topology optimization is performed on a quarter of the shell (see Fig. 9a) using the SIMP method. Figure 9b displays the boundary conditions of the optimization setting as well as the resulting 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 vertex density proportional to the optimized material distribution (see Fig. 9c). The darker regions on the optimized design have higher material density. Then, 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, mitigating 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 et al., 2023) to control the volume of the final design (see Fig. 9e). Figure 9g 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 the shell. The large openings in the topology-optimized design, which is less practical in building applications, are eliminated.

Fig. 9
figure 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

Full size image

To demonstrate the improvement of structural performance by generating Voronoi patterns based on topology-optimized results, we present and compare the Topology-optimized-Voronoi shell against its counterpart that does not undergo such optimization. Figure 10a shows the optimized design from Fig. 9, which has more materials and smaller cavities allocated to the darker 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). The 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. Therefore, the structural stiffness of the optimized shell, which is the reciprocal 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.

Fig. 10
figure 10

Shell design in Voronoi patterns: a with topology optimization; b without optimization

Full size image

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. Three 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 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. They 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.