Optimizing Glass Panel Geometry for Freeform Architecture: A Curvature-Based Pavilion Study

Abstract

This article proposes a methodological framework for the design of a freeform glass pavilion based on surface curvature analysis and systematic panel classification. The research methodology consists of two stages. The first stage is a historical review, presenting the development of glass-bending technologies, panelization strategies, and the significance of transparency in architecture. The analysis of selected freeform realizations aims to identify structural solutions and their limitations. The second stage involves parametric modeling in Rhino/Grasshopper, applying Gaussian and mean curvature analysis to optimize surface subdivision. Finite element method (FEM) calculations and CFD simulations complemented the process by assessing structural and environmental parameters. Based on the study, a panel classification system was developed, distinguishing flat, singly curved, and doubly curved elements. This classification enables the optimization of production costs and serves as a tool for balancing geometric, structural, and economic requirements. The presented theoretical research indicates that the relationship between geometry, structure, and economic efficiency is a key factor in the design of glass architecture. The proposed methodology supports informed decision-making in the design process.

1. Introduction

In the 21st century, the use of curvilinear geometry in architectural design has significantly increased as a result of the development of digital tools that enable the manipulation of complex forms through NURBS (Non-Uniform Rational B-Spline) modeling. This technological advancement has expanded the scope of design possibilities, introducing numerous geometrically intricate forms into the digital environment. However, the mere generation of such forms does not guarantee their feasibility in the construction process. To address this challenge, various methods of rationalization and optimization of geometric complexity have been developed, aiming to ensure the constructability of curvilinear projects.

The integration of architectural form generation with curved geometry modeling represents a crucial stage in contemporary design practice, closely tied to material properties and manufacturing methods. This process requires the direct incorporation of data concerning material behavior and production technologies at the level of geometric modeling. The intersection of these three domains has become a vital field of research, emphasizing the necessity of their consideration in the early phases of the design process [1]. Geometry determines the stiffness and deformability of a structure under load, while material properties define its strength and internal force distribution, remaining simultaneously dependent on the applied manufacturing technologies. The distribution of material and the shaping of cross-sections, in turn, are constrained by production capabilities. Consequently, every design decision in one of these areas directly affects the others, justifying the need for their simultaneous and integrated consideration in the design of freeform structures.

This article places particular emphasis on economic efficiency, understood as achieving the intended effect with minimal resource consumption or obtaining maximum results with a given input. This factor plays a key role in the design process, serving as a critical criterion for balancing geometric, structural, and technological requirements.

The design process of freeform structures has undergone significant evolution: from intuitive and manual shaping methods, through the introduction of computer-aided tools, to today’s integrated digital systems that enable the simultaneous consideration of geometry, materials, and manufacturing technologies. This transformation has not only expanded the formal possibilities of architecture but also reinforced the role of rationalization, making it a key factor in balancing innovation with feasibility and economic efficiency [2]. The main research problem addressed in this study concerns the identification of architectural and structural parameters that enable the realization of freeform glass structures. This includes an analysis of the interdependencies between geometry, structure, and economic performance. An essential aspect is also the consideration of the aesthetic quality of the proposed solutions, which forms an integral part of assessing architectural value. The study seeks to reduce the gap between digital design exploration and construction practice, supporting the choice of sustainable and rational architectural solutions.

The research hypothesis presented in the article assumes that the integration of parametric modeling with CFD and FEM analyses at an early stage of freeform glass design enables the simultaneous optimization of geometry, panel curvature, and structural layout. This approach enhances material efficiency, reduces construction costs, and improves structural performance. These benefits are achieved without compromising the aesthetic value of the design.

2. Review of Freeform Glass Architecture

2.1. Freeform Geometry

Freeform surfaces can be distinguished from other structures by their unique amorphous shapes, smooth flowing lines, and complex geometries [3]. Curt Siegel distinguishes two fundamental types of free form: structural and plastic [4]. Plastic free forms have no structural relationships. They represent solely an expression of free shaping, where the appearance of the form is prioritized over its construction. They can be compared to solutions known from Baroque architecture, in which the theatricality of stucco, marble, and gilding was the main means of expression.

Structural free forms, discussed in this article, are usually asymmetrical and irregular. They are not based on elementary geometric solids or their compositions. Instead, they are subordinated to the laws of structure. The arrangement and distribution of elements remain consistent with the principles of statics and gravity. They arise from the search for unconventional shapes based on freedom and arbitrariness, and often also on deliberate deformation [4].

The primary limitation in their modeling is the necessity to preserve geometric continuity and smoothness. Digitally modeled free forms are objects whose essence lies in surfaces with a variable radius of curvature while maintaining geometric continuity and smoothness. Freeform geometry is described as a new, flexible architectural language that introduces lightness, innovation, and a certain sense of mystery into design. It constitutes not only a formal tool but also a means of expression that requires a new approach to the design process and spatial representation [5].

Contemporary research on freeform architecture focuses primarily on rationalization and adaptation to technological realities. Pottmann et al. [6] present the mathematical foundations of freeform geometry, surface modeling techniques, and their application in building design and construction. They also highlight the importance of optimizing surface geometry for fabrication and assembly [7]. Further studies developed digital tools supporting the design of single-layer grids, analyzing their regularity, panel flatness, and structural aspects [8]. Other works address the trade-off between design intent and construction costs, proposing parametric and generative methods for classifying and simplifying freeform surfaces [9]. All of these studies emphasize the technical and economic aspects of implementing freeform architecture, complementing research that focuses on the formal potential and spatial expression of such forms.

The growing interest in structural glass is also evident in academic research. Examples include analyses of how the subdivision of curved façades affects visual perception [10,11], as well as studies focusing on the practical aspects of design and structural principles of fully glazed façades [12].

This study attempts to address the key limitations arising at the intersection of the design and construction of freeform glass structures. The most significant technical challenges include ensuring the geometric and structural feasibility of complex surfaces while accounting for the properties of glass and the manufacturing constraints related to the availability and cost of panel-bending molds. Economic barriers concern the high cost of producing unique doubly curved panels, which strongly impacts economic efficiency. Environmental issues are related to optimizing material efficiency. A comprehensive analysis of these aspects allows for the development of sustainable and rational solutions that combine design innovation with the realities of production and construction.

The original contribution of this article lies in the proposal of an integrated framework for parametric design. It combines curvature-based surface analysis with a systematic classification of glass panels adapted to different manufacturing techniques. The study offers a practical methodology for optimizing freeform glass surfaces. Based on experimental prototyping and case studies, it demonstrates how adaptive panelization strategies can reconcile aesthetic goals with structural and cost requirements. The paper makes a significant contribution to the discourse on digital tectonics and sustainable architectural design, providing architects and engineers with tools for realizing innovative glass structures within practical constraints.

2.2. Brief History of Modular Glass Structures

In the history of architecture, numerous examples of modular glass structures can be identified, with origins dating back to the 19th century. Their geometry was usually based on regular forms, such as geodesic domes or structures with repetitive divisions. A characteristic feature of this approach was its economy, allowing the use of a single type of panel. The modular concept of the Crystal Palace, composed of 300,000 glass panes forming a cohesive structure, can be regarded as an early precursor to contemporary methods of form rationalization [13].

The development of iron and steel production in the second half of the 19th century enabled the creation of skeletal structures that allowed for large glazed areas. The demand for larger glass panes drove advancements in production technology. In 1904, methods for producing drawn glass were developed in the United States and Belgium, which remained in use throughout much of the 20th century [14]. Rhythmic structural divisions significantly influenced the aesthetic of buildings.

The Behrens Turbine Factory (1909) demonstrated that glass and iron could serve as artistic expression if skillfully employed by the designer [15]. The Fagus Shoe Factory in Alfeld (1911–1913), designed by Gropius and Meyer, was the first significant building reflecting the principles of a new aesthetic. Its façade no longer had a load-bearing function but acted as a transparent “skin” of the building [15].

Bruno Taut’s Glass Pavilion (1914) foreshadowed modernist architecture, combining innovative material approaches with expressionist symbolism. Various types of glass were employed—colored, opalescent, prismatic, and mirrored—creating a complex interplay of light and color [16]. In this context, glass served not only an aesthetic function but also a symbolic one, introducing a spiritual dimension through lightness, transparency, and the play of light, evoking a sense of “anti-gravity” and transcendence [16]. Transparency, luminosity, and a sense of lightness were not merely formal qualities—they symbolized ideals. Glass thus assumed an almost metaphysical character, becoming a tool for dematerializing architectural form and influencing human perception. This spiritual aspiration, rooted in modernist beliefs about the transformative power of architecture, remains a reference point for contemporary projects.

In 1951, Mies van der Rohe realized the twin towers at Chicago’s Lake Shore Drive (1948–1951), whose façades were almost entirely glazed. The façade profiles were placed externally, creating a uniform rhythm. The purity of form achieved through steel and glass became a key element [14].

In the 20th century, glass technology developed rapidly. The material became stronger, safer, and more energy-efficient. A breakthrough came with Pilkington’s float glass method in 1953, which revolutionized the glass industry [17].

In the second half of the 20th century, with the development of structural glazing and digital design tools, façades with greater geometric freedom emerged. Architecture gradually moved away from rhythmic divisions in favor of glass “skins” responsive to context, light, and function. Historical modular approaches became the starting point for contemporary design strategies. In the 1970s, efforts were made to overcome monotony and allow greater creative freedom. Improvements in installation and structural glazing technology reduced limitations in designing building envelopes, and façade divisions became less visually prominent [18].

Contemporary methods of curvature optimization and panel classification build on the tradition of modularity, transforming it through parametric tools and geometric analyses. Modularity no longer implies mere repetition; it has become a tool for balancing aesthetics, efficiency, and technological feasibility—even for complex curvilinear surfaces.

Digital tectonics allows architects and engineers to adapt structures to the requirements of each project, embedding parametric data into geometric grids. The development of glass panel manufacturing technologies expands the range of available shapes and functions, opening the door to architectural innovation. Consequently, the future of modular glass structures appears nearly limitless, promising both aesthetic innovation and efficient, sustainable solutions.

Contemporary design strategies do not abandon the principles of rationalization and repetition but transform them into flexible tools that allow designs to meet both aesthetic requirements and economic efficiency criteria. The selection of appropriate geometric grids and panel types plays a key role in achieving harmony between appearance, functionality, and construction costs. Technological advances open new possibilities for modular glass structures, revealing their transformative potential in architecture and enabling the retention of complex forms while optimizing material use.

The expressive potential of glass as a building material remains strong, and the era of digital design further enhances it. Digital tools and algorithms allow surfaces to be shaped with greater precision, maintaining lightness, transparency, and poetic expression. At the same time, they provide control over structural parameters and costs, ensuring that aesthetics, functionality, and economic efficiency coexist coherently and deliberately.

2.3. Strategies for Panelizing Curved Surfaces

Panelization of freeform surfaces is a key stage in the design of glass structures with irregular shapes. It involves selecting the appropriate type of panel—flat, singly curved, or freely curved—depending on geometry, manufacturing capabilities, and construction costs [19].

In freeform architecture, the greatest challenge lies in panels with irregular shapes, particularly doubly curved panels, which require costly molded production. Each panel differs from the others, eliminating repetition and increasing costs. One solution is to use simpler panels, such as flat panels, at the expense of surface smoothness. Many studies, including those by Pottmann et al., address methods for optimizing such geometries [7,20,21].

Panel design can be treated as an optimization problem involving discrete variables (panel type, mold assignment) and continuous variables (panel parameters and placement). Eigensatz et al. proposed a method to minimize costs while maintaining tolerances with respect to the design surface, allowable gaps, and angles between panels [2,21]. Flat panels, usually obtained through triangulation, unfolding, or linear subdivisions, provide an economical solution in “blob” structures, such as the Blob in Eindhoven (Fuksas, 2010) or the New Milan Fair (Fuksas, 2005). In freeform roof structures, triangular or quadrilateral panels are mainly used, differing in the method of surface discretization [22,23].

Some researchers have focused on optimizing for repeatable elements by adjusting mesh vertex positions to create prefabricated, reusable modules. For example, Fu et al. [24] studied quadrilateral meshes, while Singh and Schaefer [25] investigated triangular meshes.

Singly curved panels, produced either cold or hot, offer higher visual fluidity at a moderate cost increase. Examples include the Strasbourg station canopy with cylindrically bent glass [26] and the spiral staircase at the Berlin Historical Museum.

The most advanced are doubly curved panels. In the One Blackfriars project in London, over 5000 unique elements were optimized using scripting tools [27]. Although costly, they provide aesthetic continuity and integration with the building form. Parallel developments in sustainable technologies, such as the Glass Wave system, allow the shaping of multi-curved panels while reducing emissions and material usage.

The pavilion described in this article combines all three types of panels. In the initial design stage, they were optimized through curvature analysis and classification to minimally interfere with the original aesthetic concept.

Previous research on panelization has focused primarily on flat panels. This study analyzes the design challenges associated with creating freeform shapes and presents potential solutions using the example of a digital prototype glass pavilion with an organic form composed of three types of panels: flat, singly curved, and doubly curved. The discussion includes the necessary modifications for project realization and the application of modern parametric design and 3D modeling methods.

3. Materials and Methods

The article employs a variety of research methods, including historical analysis and case studies, as well as computational and experimental techniques. (Figure 1) The historical analysis allowed for an examination of the development of glass technology in architecture and the significance of transparency in design. This approach allowed the studied project to be placed within a broader context of material and formal development. Case studies of selected freeform structures enabled the identification of proven construction solutions, which then served as the basis for further modeling and optimization.

In the next stage, advanced NURBS surface modeling techniques were applied using Rhino 3D. This allowed the creation of a precise model of the pavilion with an organic shape. It should be emphasized, however, that “organic” does not imply arbitrariness, but rather a harmonious shaping of space based on geometric and curvature principles. In this study, the term refers to architecture inspired by natural forms, usually nonlinear and irregular, which nonetheless adhere to mathematical rules and can result from precise geometric calculations [28]. Parametric design using Grasshopper enabled dynamic control over the geometry through changes in input parameters. The process began with the generation of the surface, which was then classified according to geometric properties such as curvature types. Analyses of Gaussian curvature, mean curvature, and curvature radii were performed to assess the surface quality in terms of structural stability and suitability for subsequent design stages.

Next, various surface subdivision methods were applied, including tessellation with triangular and quadrilateral meshes, aiming to balance aesthetics with panel production costs. The use of UV coordinate systems and NURBS curves allowed precise surface division and assignment of panels to appropriate zones, which was critical for the actual construction implementation. After classifying the surfaces and identifying available panelization methods, a key stage involved selecting the most suitable panelization strategy.

The subsequent stage included environmental analysis combined with geometric classification, enabling the identification of areas requiring optimization. Multi-criteria curvature-based optimization was then applied, resulting in several design variants. These variants were archived and evaluated to select the optimal solution. In case of unsatisfactory results, the process allowed returning to earlier stages for further refinement.

The design process was iterative in nature. This study builds on previous research and publications on surface subdivision and the use of glass panels in architecture [29,30,31]. This review served as the starting point for geometric analysis and pavilion form optimization, with the objective of ensuring economic efficiency. The entire process was conducted within the Rhino and Grasshopper environment.

The design procedure of the pavilion was carried out according to the scheme presented in Figure 2. This diagram illustrates the process of optimizing the geometry of the initial form through an in-depth study of the panelization process based on curvature analysis. The form-finding process was supported by numerical computations, including both computational fluid dynamics (CFD) simulations and finite element analysis (FEA), which enabled the integration of aesthetic, functional, and environmental aspects into a single coherent design model.

4. Results

The design of the freeform pavilion is intended as an example illustrating the potential geometric complexity of glass panels. The concept is based on the possibility of using a sample geometry to identify potential visual and technical challenges in various architectural applications.

4.1. Concept Development—From Idea to Geometry

The conceptual process began with formal explorations inspired by natural structures. The aim was to create a form that is both expressive and functional, serving not only a utilitarian role but also a symbolic and experimental one.

In the first design phase, an initial model was created in Rhino using parametric software, with the assumption of maximum geometric freedom (Figure 3). A key design objective was to develop a form that would allow the application of various glass-bending technologies—from flat panels to doubly curved surfaces. To verify this concept, curvature analysis tools were introduced at an early stage, enabling an assessment of the feasibility of the proposed geometry. To maintain the necessary design flexibility, these dependencies were implemented in the Grasshopper parametric add-on. This made it possible to flexibly shape the form and quickly test different variants. For the description of doubly curved surfaces, a standard modeling method—a mathematical surface model—was applied (Figure 4). The surface was subdivided using a standard 12 × 12 grid, yielding 144 panels. Panel areas vary widely; the largest measures 3.09 m² and the smallest 0.30 m². The parametric approach also ensured that the pavilion’s footprint did not exceed 100 m², while its height provided sufficient usability of the interior space.

4.2. Surface Classification Based on Curvature

To effectively analyze and optimize surfaces in freeform architecture, it is essential to understand their geometric structure. A key parameter is Gaussian curvature, which enables the classification of surfaces based on their geometry [32]. In the subsequent stage, geometric analyses were conducted to identify the local curvature properties of the surface, with the aim of determining the optimal selection of panel types (Figure 5).

These calculations enabled the mapping of curvature typologies across the entire pavilion surface, forming the basis for subsequent technological classification. Preliminary simulations revealed the presence of flat, saddle-shaped, and convex regions, indicating the necessity of applying diverse glass-forming technologies. This study builds upon the work of Liu, Zhang, Sun, Gao, and Fu, in which Gaussian curvature (K) and mean curvature (H) were employed for classifying local surface geometry [34]. Principal curvatures provide detailed external information but are computationally demanding and highly sensitive to noise. In contrast, the values of K and H serve as efficient, direction-independent descriptors. The type and sign of these curvatures at a given point define the local characteristics of the surface—as illustrated in

Table 1. Relationship between surface shape and geometric parameters, authorial study based on Liu, H. et al. [34].

Gaussian Curvature (K)	Mean Curvature (H)	Point Feature	Local Surface Shape	Example/Notes
K > 0	H > 0	Oval	Concave area	Inner side of sphere
K > 0	H < 0	Oval	Convex area	Outer side of a sphere
K < 0	H > 0 or H < 0	Hyperbola	Saddle area	Hyperbolic paraboloid
K = 0	H = 0	Parabola	Flat area	Plane
K = 0	H > 0	Parabola	Concave area	Inside of a cylinder
K = 0	H < 0	Parabola	Convex area	Outside of a cylinder
Kvaries	Hvaries	Mixed	Freeform surface	Requires point-by-point classification

of the cited article.

This paper introduces an extended classification system adapted to freeform surfaces. The original research contribution includes an additional category of regions with mixed or undefined curvature signs, which require point-by-point assessment and special consideration during the design and panelization process.

Surface modeling is one of the most challenging areas related to prototyping techniques. A properly developed surface model must meet both functional and technological requirements while also possessing adequate aesthetic qualities. Geometric continuity serves as the primary criterion for assessing model quality [35].

The analysis of NURBS surface curvatures makes it possible to evaluate different aspects of geometry through visualization in the form of a color gradient. Parameters such as Gaussian curvature, mean curvature, minimum radius of curvature, and maximum radius of curvature are examined [36]. This categorization enables the practical subdivision of the surface into zones with varying degrees of feasibility. In the subsequent stages of the project, alternative scenarios for geometry rationalization are considered, including surface segmentation, with the aim of maximizing the share of flat panels, in accordance with cost-optimization principles.

Gaussian curvature is defined as the product of the two principal curvatures, k₁ and k₂, at a given point on the surface. This analysis makes it possible to distinguish between developable regions (K ≈ 0), singly curved regions (K > 0), and doubly curved or saddle-shaped regions (K < 0). To visualize the results, a color scale was applied, ranging from negative values (blue), through zero (green), to positive values (red) (Figure 6). The analysis revealed the presence of regions with positive curvature, corresponding to convex, dome-like shapes, as well as zones with negative curvature, indicating saddle geometries.

Most of the surface exhibited values close to zero, corresponding to nearly cylindrical or flat segments. This indicates a hybrid form that combines zones of single and double curvature—typical of architectural freeform surfaces. Saddle-shaped regions may be more susceptible to material stresses and require the application of more advanced forming techniques [37].

Mean curvature, calculated as the arithmetic average of the principal curvatures, makes it possible to determine the local bending characteristics of the surface (convexity, concavity, or flat segments) [38]. The results of the analysis revealed a predominance of positive values, suggesting that convex curvature is the dominant feature of the structure (Figure 7).

Zones with lower values, approaching zero, indicate areas that are flatter or only slightly curved. This information is crucial for identifying which parts of the surface can be more easily manufactured using conventional glass-bending methods, and which may require special treatment.

The analysis of the minimum radius of curvature focused on the sharpness of bending in different parts of the surface (Figure 8). Larger radii correspond to gently curved areas, which are easier to manufacture. Smaller radii, in turn, indicate zones of sharper bending, which may present production challenges [39].

The pavilion’s form consists largely of surfaces with large radii of curvature, which supports the feasibility of construction using bent glass. However, the presence of several local areas with small curvature radii should be considered during the detailed design stage and material selection. These may play a critical role in the production and assembly process.

Gaussian curvature maps revealed that a significant portion of the surface is characterized by values close to zero, corresponding to developable or cylindrical zones. These areas can be realized using flat or singly curved panels. Regions of positive Gaussian curvature, with dome-like geometry, as well as those of negative Gaussian curvature, with saddle-like geometry, indicate the need for doubly curved panels or subdivision of these fragments into smaller segments. Such solutions are technically feasible.

The analysis of mean curvature confirmed the predominance of convex forms, which is advantageous in terms of resistance to operational loads. However, it requires maintaining high precision in the bending process to avoid residual stresses. The results of the minimum curvature radius analysis confirmed that most of the surface has radii suitable for conventional glass-bending processes, while small areas with very tight radii represent critical points where the use of specialized forming techniques, geometric adjustments, or panel size reduction may be necessary.

A classification of the panels was carried out based on Gaussian curvature K (1/mm²) with a “near-flat” threshold ε = 1 × 10⁻⁸ mm⁻². Three categories were distinguished: near-flat (|K| ≤ ε), synclastic (K > ε), and anticlastic (K < −ε). The results show a predominance of synclastic panels (~62.5%), a substantial share of anticlastic panels (~31.3%), and a small fraction of near-flat panels (~6.3%) (Figure 9).

To complement the geometric and structural analysis, technological limitations for bending 28 mm laminated, chemically tempered glass were investigated.

Table 2. Summary of bending technologies for 28 mm laminated, chemically tempered glass panels. Source: sedak GmbH, Gersthofen, Germany.

Technology	Minimum Bending Radius Rmin	Maximum Panel Dimensions	Technological Limitations	Cost and Production Notes
Slumped glass (traditional bending using bespoke steel mold)	No coating: R = 150 mm With coating (e.g., low-E): R = 500 mm	Up to 12 × 3.6 m	Requires steel mold; 8 h bending + annealing cycle	More expensive than machine bending due to mold fabrication
Machine bending (no mold; fully tempered or heat strengthened)	No coating: theoretically R = 150 mm (requires special oven), readily achievable R = 700 mm With coating: R = 700 mm	Up to 18 × 3.6 m (height × arch length) or 3.6 × 8.5 m (height × arch length)	Depends on furnace capacity; coatings and interlayers limited to max width 3.2 m	Most cost-efficient option; fast production (~20 min per ply); no molds required
Lamination bending (during autoclave cycle at 140 °C; vacuum bag process)	Not specified; depends on lamination process and geometry	Up to 20 × 3.6 m	Coatings and interlayers limited to max width 3.2 m	8 h lamination cycle; glass conforms to framing/system shape

summarizes the minimum achievable bending radii, maximum panel sizes, and relevant production constraints for three bending methods: slumped glass, machine bending, and lamination bending.

These parameters directly affect panel curvature classification and grouping strategies, as tighter radii or larger panel sizes may require specific production methods, impacting both cost and fabrication feasibility.

4.3. Penalization and Panel Classification

After classifying the surfaces and identifying the available panelization methods, and after recognizing potential manufacturing challenges, such as excessive curvature, dimensional limits, or technological constraints, a crucial stage follows: the selection of an appropriate panelization strategy.

4.3.1. Surface Division Methods (UV, Grids, Tessellations)

The panelization procedure consists of adopting a grid of curves based on the reference surface (the design form) and arranging panels within its cells. One of the commonly used methods of panel arrangement is the subdivision of the surface according to UV coordinates, using B-spline (NURBS) curves [40]. In this context, two types of curves are distinguished: U and V. U-curves have a defined direction, while V-curves run perpendicular or almost perpendicular to them, resembling the layout of rows and columns in a table. Together, they form a grid in which a control point is located at each intersection (Figure 10). Such a coordinate system makes it possible to define and describe arbitrarily complex surfaces.

A simpler subdivision method relies directly on the definition of the NURBS grid or constructs a new grid by interpolation, using a different number of curves in the V or U direction. This tessellation method consists of breaking polynomial curves into a series of segments [41]. The method disregards issues of seriality, focusing instead on reproducing the original geometry. Panels generated through this type of subdivision typically change their dimensions smoothly and are characterized by significant variability and heterogeneity in size between smaller and larger elements. Despite its simplicity, this method is not recommended for complex projects, as numerous technical difficulties may arise during construction due to the large number of panel types and considerable geometric variability.

For complex forms, where uniform panel fitting is essential, more advanced methods such as adaptive tessellation are recommended [42]. This method ensures that the deviation of the 3D approximation from the original surface remains within a specified model space tolerance (ε), while the number of generated panels depends on the second derivatives of the surface. The algorithm provides a practical error analysis, adapting the tessellation density in parametric space to the surface curvature to achieve a consistent level of geometric accuracy [42].

Flat panels can be adjusted to surface curvature during assembly by cold bending. Cold-bent glass is produced from flat sheets that are forced into moderate curvature, limited by the process constraints [43]. Hot-bent glass, on the other hand, can take on more complex shapes, but its production is more expensive and technologically demanding [44]. The smaller the panels, the more accurate the approximation of the reference geometry—although this simultaneously increases the cost of the supporting structure [45]. For this reason, the choice of subdivision method and panel types should be balanced against the required geometric precision and the project budget.

4.3.2. Panel Analysis and Classification

Advanced geometric modeling techniques allow panels to be assigned to specific families for standardization and replication. In the first step, the surface curves were subdivided into segments.

Variant analyses of surface subdivision were carried out, testing different strategies for panel grid generation. Both regular and irregular subdivisions were considered, including rectangular, triangular, rhomboidal, and hybrid layouts combining different types of modules (Figure 11). The colors indicate the assignment of panels to individual groups. Each variant was evaluated in terms of its fit to the pavilion surface curvature, the technological feasibility of panel fabrication, and its aesthetic and visual qualities.

Visualizations revealed clear differences in the curvature distribution as well as in the proportions and orientation of panels in the original surface subdivision. In the case of rectangular layouts, a simpler panel geometry was obtained, which facilitated the optimization of their shapes with respect to manufacturing technologies. In the analyzed model, a new subdivision into quadrilateral panels arranged in a 10 × 16 grid was adopted, which allowed for more favorable panel proportions and reduced disparities in their surface area. As a result, 160 panels were obtained. The new subdivision reduced size disparities and eliminated the largest elements.

For the classification of the pavilion panels, a clustering method based on the K-Means algorithm was applied, implemented in the Grasshopper environment using the LunchBox Machine Learning plugin. Based on the digital surface model, the panelization process was carried out, dividing the geometry into a grid of quadrilateral panels (Figure 12). A common set of features was adopted for all panels to define their geometry. Feature set 1 included two parameters: the surface area of each panel and its perimeter. Feature set 2 additionally incorporated geometric properties such as Gaussian curvature, mean curvature, normal vector, and deviation from flatness, enabling a spatial analysis of panel form. The process was performed with varying numbers of clusters, analyzing the influence of the selected features on the clustering results. The results were visualized in Grasshopper by assigning colors to panels according to their cluster membership. This method enables the identification of groups of panels with similar geometric properties, which serves as a basis for further geometry rationalization and for reducing the number of unique panel types in the manufacturing process.

Table 3. Summary of panel clastering: surface area, Gaussian curvature, and technological characteristics. Threshold for near-zero curvature: ε = 1 × 10⁻⁸ mm⁻². Units: K in 1/mm².

Group	Surface Area—Range (cm2)	Gaussian Curvature—Range (1/mm2)	Curvature—Characteristics	Technological Conclusions	Panel Characteristics
0	3732–11,310	|K| ≤ ε	Nearly flat/developable	Flat glass or cold-bent (large R)	Best for standardization.
1	4446–26,113	ε < K ≤ 1 × 10−6	Synclastic, low	Hot-bent (gentle)	Typically dimensionally feasible.
2	4109–9123	1 × 10−6 < K ≤ 5 × 10−6	Synclastic, medium	Hot-bent	Consider finer subdivision.
3	4354–8026	5 × 10−6 < K ≤ 1 × 10−5	Synclastic, high	Hot-bent (tight)	Prefer smaller panels/grid re-orientation.
4	4671–12,185	K > 1 × 10−5	Synclastic, very high	Hot-bent/precise molds	Re-panelization recommended.
5	2772–14,564	−1 × 10−6 ≤ K < −ε	Anticlastic, low	Cold/hot forming per requirements	Acceptable.
6	3567–16,078	−5 × 10−6 ≤ K < −1 × 10−6	Anticlastic, medium	Anticlastic forming	Consider smaller panels.
7	2892–7029	K < −5 × 10−6	Anticlastic, high	Special molds	Prefer grid correction.

presents a comparative summary of the eight panel groups identified during the curvature-based classification process. For each group, the surface area range (in cm²) and Gaussian curvature range (in 1/mm²) are given, along with concise descriptions of their dimensional characteristics, curvature properties, and potential technological implications. The grouping was developed to support panel standardization, optimize manufacturing methods, and identify elements suitable for flat or curved glass production. The panel dimensions fall within the manufacturing standards and are feasible to produce. However, for the purpose of comparing the aesthetic effect, it would be worth considering an increased number of subdivisions.

4.4. Environmental Analysis—CFD Simulation

The final shape of the form and its subdivision should take into account not only aesthetic and technological considerations but also environmental aspects. Analyses such as CFD simulations make it possible to assess the impact of the form on user comfort and the airflow inside and around the structure. Environmental analyses serve as tools supporting the design process and the optimization of the form.

4.4.1. Design Objectives

One of the key design issues in the case of the glass pavilion was the analysis of wind effects, particularly the pressures exerted on the surface of the structure. Due to the assumed streamlined shape of the construction, there was a high probability of the occurrence of suction zones that could exert a “pulling” effect on individual glass panels, especially those located on the roof and side surfaces. The complex, irregular geometry of the pavilion excluded the possibility of applying analytical methods to determine the pressure distributions. Therefore, a numerical airflow analysis was carried out using computational fluid dynamics (CFD). The calculations were performed in the open-source simulation environment OpenFOAM, which enables the modeling of turbulent flows around objects with complex geometry (Figure 13).

As part of the analysis, simulations of the given geometry were carried out for ten different wind inflow directions, which made it possible to assess the airflow behavior around the structure under various exposure conditions (Figure 14). Control points were distributed across the surface of the structure, where the pressure values acting at each point were examined (Figure 15). For each case, the wind speed was set at 100 km/h.

4.4.2. Analysis Results

The results are presented both graphically and in the form of a table summarizing the outcomes for selected control points. Based on these, it can be observed that different wind directions are critical for different points (

Table 4. CFD analysis results for the pavilion.

Wind Velocity = 100 km/h	Pressures at Control Points [kPa]
Wind angle	0	20	40	60	−20/340	−40/320	−60/300	150	180	210
Control point ID
1	−0.01	−0.01	−0.21	−0.68	0.14	−0.37	−0.55	−0.27	−0.01	0.0
2	−0.05	−0.1	−0.27	−0.62	0.14	−0.25	−0.31	−0.27	−0.05	−0.06
3	−0.12	−0.06	−0.14	−0.24	0.26	−0.38	−0.17	−0.29	−0.07	−0.11
4	−0.55	−0.37	−0.38	−0.42	−0.70	−0.63	−0.53	−0.62	−0.44	−0.17
5	−0.02	−0.06	−0.16	−0.35	−0.09	−0.23	−0.25	−0.19	−0.04	−0.09
6	−0.07	−0.09	−0.13	−0.32	0.14	−0.26	−0.33	−0.22	−0.08	−0.13
7	−0.07	−0.13	−0.2	−0.22	0.15	−0.2	−0.3	−0.21	−0.07	−0.2
8	−0.01	−0.12	−0.14	−0.11	−0.07	−0.19	−0.21	−0.18	−0.11	−0.21
9	−0.05	−0.07	−0.13	−0.25	−0.12	−0.25	−0.27	−0.18	−0.09	−0.14
10	−0.26	−0.39	−0.08	−0.18	−0.20	−0.16	−0.22	−0.16	−0.39	−0.53

). The results indicate that wind incidence from the front (point 1) is the least critical. Due to the streamlined geometry, the occurring local overpressures are relatively small, on the order of 0.1 kPa, and appear primarily on surfaces oriented at angles close to perpendicular to the wind direction. In contrast, negative pressures reach values of approximately −0.5 kPa (with critical values up to −0.7 kPa) and occur in areas where the form exhibits strong local curvature. In the table, green colors correspond to the lowest pressure values, while red indicates the highest negative pressures.

The analysis allowed for the identification of areas exposed to the highest positive and negative pressure values, which was of key importance for determining the required glass panel thicknesses and the design of fixing solutions (Figure 16).

For the analyzed pavilion geometry, under windless conditions, the primary load acting on the structure remains the self-weight of the glass panels. In a simplified sense, this corresponds to a “positive” pressure. A suction pressure equal to the self-weight implies that the structure (theoretically and in a first approximation) is not subjected to any load. Suctions exceeding the self-weight of the glass result in the connections being loaded in the opposite direction compared to normal operating conditions.

The results extrapolated for a wind velocity of 140 km/h indicate that, for the given velocity, suction pressures are expected to significantly exceed the self-weight of the glass. This implies that the structure will be governed by higher negative loads, which then become critical in terms of structural resistance.

However, once the suction pressure surpasses the self-weight of the glass, the load-bearing behavior of the entire system changes: the fasteners and the supporting structure must transfer forces acting in the opposite direction compared to static conditions. Instead of carrying “compressive” loads, the connections must resist forces attempting to detach the glass from its supports. Such a condition may be particularly critical for fastening systems and sealing elements.

From a design perspective, the unit mass of the applied glass panels is also of key importance. For the considered configuration, it amounts to approximately 70 kg/m², corresponding to laminated and chemically tempered composite glazing with a nominal thickness of 28 mm. The glass thickness was selected “arbitrarily” due to technological considerations.

The analysis confirmed that the suction loads are close to or even exceed the self-weight of the panels. Under such parameters and the resulting suction pressures, it is essential to implement fastening systems that ensure effective anchorage of the glass to the supporting structure, with sufficient resistance to the uplift forces induced by airflow.

The conclusion drawn from the CFD analysis can serve as an important criterion in further shaping the form and panelization strategy. The simulation results identify regions particularly exposed to strong suction or overpressure, which generate the highest loads acting on the glass surface. In such areas, a denser subdivision of panels is recommended, as it allows for a more uniform distribution of forces and reduces the risk of local damage. Conversely, in zones with a more moderate pressure distribution, the use of larger panels is feasible, which helps minimize the number of joints and improves the overall aesthetics of the form.

In practice, this means that the results of the airflow analysis may directly influence the panelization strategy—leading to differentiation in the size and shape of glass elements depending on local aerodynamic conditions. Such an approach not only enhances operational safety but also enables the optimization of fastening solutions and the rationalization of production and installation costs.

To verify the structural behavior under extreme aerodynamic loading conditions, a strength analysis was subsequently carried out using the finite element method (FEM).

4.5. Optimization of Glass Pavilion Composite Structure—MES Digital Analysis

The glass pavilion structure was analyzed using the finite element method (FEM). The objective of the calculations was to assess the strength and stiffness of the structure under different laminate configurations and to determine the optimal solution in terms of load-bearing capacity and rigidity. Carbon fiber composite was adopted as the primary structural material, and the analysis accounted for both permanent and variable loads acting on the longitudinal beams.

Table 5. Load analysis of the longitudinal beam, including contributions from glass, snow, wind, and structural elements.

Longitudinal Beam
Beam span	1.5 m	m
Support width	1.5 m	m
Glass weight	50	kg/m2	490.5
Snow weight	200	kg/m3	0.4905
snow thickness	20	mm
Linear Load
Glass weight	0.4905	kN/m2	47%
Snow weight	0.03924	kN/m2	4%
Wind load	0.5	kN/m2	48%
Web weight	0.16704	kg/m
Capping weight	0.42	kg/m
Composite beam load	0.005758862	kN
Working load	1.04	kN/m2
Linear load on beam	1.55	kN/m

presents a detailed summary of the loads considered in the analysis. The beam span and support width were assumed to be 1.5 m. Permanent loads included the self-weight of the glass panels, snow load, and additional structural elements, while the variable load was wind pressure. The results indicated that glass (47%) and wind (48%) represented the dominant contributions to the total load, whereas snow accounted for only 4%. The total linear load acting on the beam was calculated at 1.55 kN/m, which served as the basis for further numerical computations within the FEM model.

4.5.1. Computational Model

The structural model comprises two types of beams: transverse (80 × 120 mm) and longitudinal (80 × 80 mm). The FEM model incorporates the glass as a source of loading but does not account for its strength properties. Standard boundary conditions were applied: supports at the contact points with the foundation and hinged connections between the beams.

4.5.2. Structural Variants Investigated

Eight structural variants were analyzed: variants 1–7: identical laminate configurations to all beams; variant 8: optimized with differentiated laminates, locally adapted to specific stress conditions (Figure 17).

4.5.3. Results of the Analysis

As part of the structural analyses, eight design variants of load-bearing beams were evaluated. Each variant met the strength requirements, assuming a minimum safety factor of 1.14. (Figure 18) Among all the configurations analyzed, only two—variant 7 and variant 8—satisfied the additional displacement criterion, limiting maximum deflections to less than 10 mm. (Figure 19)

Variants 1–7 employed a uniform laminate type for all beams, irrespective of their role within the structure. However, the analysis revealed that the majority of these beams were oversized, leading to excessive material use without a corresponding improvement in the overall stiffness of the structure. Only a small subset of beams had a significant influence on the maximum displacement of the entire system.

Variant 8 represents a preliminary optimized solution incorporating two different laminate types—a lighter version for beams with minimal influence on displacement, and a reinforced version used selectively in key beams critical to the structure’s rigidity. This approach enabled a significant reduction in the overall weight of the structure while preserving its functional performance. Compared to variant 7, the total weight of variant 8 was reduced by 430 kg, making it the most material- and mechanically efficient option among all those analyzed (Figure 20).

4.5.4. Variant 8—Detailed Analysis

Variant 8, as a preliminary optimization, utilizes two different laminate types, with reinforcements applied only to beams that are critical in terms of maximum displacements and stress concentrations. The greatest deflection of 9.95 mm was observed in the CROSS G beam, while the lowest safety factor of 1.96 occurred in the CROSS E beam due to critical sheet stress (Figure 21).

4.5.5. Summary of the Analysis

The numerical analysis demonstrated that the uniform application of laminates resulted in oversized elements and excessive structural mass. Variant 8, based on differentiated material use, enabled a weight reduction of 430 kg while simultaneously limiting displacements to below 10 mm. This made it possible to preserve the intended geometry of the pavilion and its aesthetic coherence while achieving optimized structural performance parameters.

5. Discussion

In the 21st century, the architectural landscape has undergone a profound transformation with the emergence of buildings featuring curvilinear glass envelopes. This phenomenon has confronted designers with new technological and structural challenges. Advances in manufacturing have enabled the more economical production of curved glass, expanding both design and aesthetic possibilities. Curvature, however, is only one aspect of the complex requirements imposed on contemporary architectural glass.

The concept of freeform design, particularly in relation to transparent structures, extends beyond the purely geometric definition. It requires the integration of material solutions, glazing technologies, and structural performance analysis [44]. In the literature, authors such as H. Pottmann, J. Wallner, and M. Eigensatz emphasize that the development of parametric and algorithmic tools allows not only for the generation of complex forms but also for their optimization in terms of cost and production efficiency [2,44,46]. An increasing number of studies highlight the importance of incorporating environmental simulations—such as solar exposure, thermal comfort, daylighting, or wind effects—at the early stages of the design process [47,48]. In the present study, aerodynamic analyses (CFD) were employed, directly influencing the pavilion geometry and leading to forms optimized both environmentally and structurally.

As Kolarevic and Oxman argue, the processes of form generation and performance optimization should proceed in parallel and iteratively [42,49]. Contemporary parametric and simulation tools make it possible to test multiple variants of shape, panel orientation, and loading conditions, supporting more informed design decisions. However, the diversity of glass panels directly impacts construction costs. Compared to standard flat panels, the production of curved glass entails significantly higher material and technological expenditures [50]. A greater number of unique elements increases logistical complexity and elevates the costs of prefabrication and installation [51,52]. The complexity of freeform geometries necessitates precise adjustment and an individual approach to each panel, which can extend construction schedules and raise the risk of fabrication or installation errors. The literature underscores the need to balance formal expression with façade standardization. As Eigensatz notes, modularity and panel repetition are key to reducing costs and optimizing construction processes [44]. Panelization strategies that combine standard elements with a limited number of unique panels allow for design flexibility while lowering production and assembly costs.

Beyond technological and economic considerations, an equally important question concerns the impact of freeform architecture on perception and user experience. As Pallasmaa observes, architecture engages not only visually, but also tactilely, acoustically, and atmospherically [53]. Research in neuroaesthetics indicates that curvilinear forms are more often perceived as visually appealing and comfortable than sharply angular ones [54]. At the same time, Hildebrand highlights the importance of spatial legibility and safety, often associated with modularity and repetition [55]. This opens a field of reflection on the relationship between the degree of curvature and spatial clarity—from monumental, enclosed interiors with minimal curvature to expressive, open geometries that encourage fluid user circulation.

In summary, modern design tools provide vast opportunities for generating complex forms, yet they call for further research into algorithmic design, particularly regarding freeform surfaces and panelization strategies. The pursuit of formal expression must be accompanied by cost optimization and economic efficiency. Integrating cost, environmental performance, and spatial perception analyses from the early stages of design enables the creation of solutions that are not only innovative and aesthetically refined but also economically and functionally sustainable.

Logical Inversion of Architectural Form Generation

Contemporary methods of form generation in architecture increasingly depart from the traditional design sequence, in which the process begins with a spatial intention—understood as the designer’s compositional and functional vision—followed by the selection of geometry corresponding to that vision, and concludes with adjustments to meet structural and technological requirements. This approach, deeply rooted in modernist and postmodernist architectural practice, emphasizes creative freedom, allowing the full expression of aesthetic and spatial intentions. Its drawback, however, lies in the risk of producing solutions that are difficult or even impossible to realize without significant modifications in the later stages of design. In the case of complex freeform surfaces, this may result in costly changes to geometry or materials.

In the study presented in this article, an inverse design logic was applied based on the following sequence: curved surface → curvature analysis → feasibility assessment. In this approach, the design is not shaped solely according to aesthetic intentions but, from the outset, considers technological parameters—such as the minimum glass-bending radius, maximum panel dimensions, and constraints arising from manufacturing and transportation capabilities. Early-stage curvature analysis allows for determining which parts of the surface can be realized as flat panels, singly curved panels, or doubly curved panels. As a result, the final form largely emerges as a compromise between the desired visual effect and technological feasibility.

6. Conclusions

Since the early modernist period, when architects such as Bruno Taut and Mies van der Rohe recognized the spatial and emotional potential of glass, the design of glass structures has combined aesthetic exploration with technological constraints. The design of freeform buildings requires an integrated approach that encompasses geometric, environmental, aesthetic, and structural aspects. The analysis of panel geometry and curvature plays a key role in this process—panels with large curvatures demand advanced bending technologies and customized fastening systems, which generate considerably higher financial costs compared to flat or slightly curved panels.

In the analyzed pavilion, the initial panelization scheme consisted of 144 panels. After modifying the grid layout, the number of panels increased to 160, which resulted in reduced variation in both panel surface areas and curvature values. The curvature analysis enabled the unambiguous classification of panels in relation to the technological limits for bending laminated glass with a thickness of 28 mm. Most panels fall within the range of curvatures achievable through machine bending, which allows for cost-efficient production. Panels exceeding the adopted technological limit require either geometric modifications or the use of alternative technologies, such as bespoke mold bending or panel segmentation. The selection of the minimum bending radius threshold has a significant impact on the classification outcome—adopting a more conservative limit may increase the number of panels requiring special solutions.

The integration of parametric modeling with CFD and FEM analyses enables the early testing of form variants, panel arrangements, and loading conditions. This increases material efficiency, reduces construction costs, and improves structural performance without compromising aesthetic value. Structural analysis of the beam variants demonstrated that the use of differentiated laminates in critical elements allows for a mass reduction of 430 kg compared to the uniform laminate solution while maintaining full functionality. These results highlight the importance of material and structural optimization as an integral part of the design process.

Based on the conducted analyses, it is now possible to perform detailed cost and material assessments that can evaluate the impact of panel diversity on overall construction costs. At the same time, this opens a field of research into how curvature and panelization of freeform surfaces affect interior perception, acoustics, light dispersion, and user experience.

With regard to production technology, it should be emphasized that physical prototyping of panels, as well as testing of cold and hot bending methods and the verification of joining techniques for panels produced by different methods, has not yet been carried out. Such studies will be crucial for the practical validation of digital models and the optimization of the manufacturing process.

Furthermore, future work must also consider issues of maintenance and durability, which are essential for the long-term use of freeform glass structures. Research into innovative fastening systems capable of adapting to variable wind and thermal loads may contribute to enhanced durability, reduced weight, and lower material consumption of the supporting system.

Ultimately, the development of comprehensive simulation tools integrating CFD and FEM enables an even more holistic approach to the design of freeform glass structures. Such a process allows designers to make informed and sustainable decisions at all stages of realization—from concept, through production, to operation—ensuring that aesthetics, functionality, safety, and economic efficiency remain integral components of the project.