Awarded : Architectural Institute of Korea(AIK) Spring Conference

Outstanding Presentation Paper Award

Abstract

This study explores the possibility of integrating topological optimization with diamond vault architecture from Karl Bötticher's tectonic perspective. According to Bötticher's concept of tectonics, topological optimization constitutes a kernform but not an a kunstform. To address this insufficiency, this paper proposes the possibility of combining the post-Gothic style of diamond vaults with topologically optimized forms. It is observed through a historical and conceptual overview and geometric analysis that both share the resultant effect of volumetric efficiency and the geometric principle of being interpreted and created based on points. Furthermore, in the procedural aspect, topologically optimized forms undergo the rationalization process to be transformed into industrially producible forms, where this research suggests that the reinterpretation of forms through diamond vaults becomes possible. Broadly interpreted, this study attempts to reconsider and restore the degree of architects' involvement in form, which has been gradually diminishing in the computational process, from Karl Bötticher's tectonic viewpoint.

1. Introduction

1.1 Background and Objective

In architecture, the concept of tectonics deals with the core and controversial issue of how the physical and mental realms can integrate causally. This aporia, as Kenneth Frampton points out, is "neither new nor unique, but rather has been continuously posed to modern consciousness since the end of the 18th century." Despite this, the question remains relevant in contemporary architecture, as it has either been left unresolved or has become even more acute. Karl Bötticher, a 19th-century German architectural historian who sparked the tectonic discourse, distinguished between 'kernform' (core form) and 'kunstform' (art form) within the tectonic concept, emphasizing their complementarity. Kernform is associated with realism, emphasizing construction, structure, and materials, while kunstform is related to the artistic symbolism represented by Greek architecture. Bötticher's tectonic theory posits that "the kunstform form is necessary because of the a historicity of the kernform. That is, the kunstform, by representing historical form, connects the structural beauty's historical concept up to the present day." His equivalence of art to historicity was influenced by the historical context of the 19th century, but in modern times, this could be sufficiently interpreted without reducing it to historicity. The advent of topological optimization in the 1980s, which also influenced architecture, is a technique that optimizes structure by interpreting shapes through the Finite Element Method (FEM). The forms created through topological optimization belong to the realm of kernform, as they are essentially about force and its transmission. However, in contemporary architecture, topological optimization has been uncritically accepted due to the trend and necessity of generative design, without critical reflection on the kunstform. This process appears to increasingly limit architects' ability to intervene in form. This paper explores the possibilities for architects to intervene in topologically optimized forms through procedural and resultant balance between kernform and kunstform. Procedural balance means that both core and kunstforms are equally considered during the design process, while resultant balance refers to the structural advantages of the kernform and the aesthetic beauty of the kunstform being equally represented in the design outcome.

1.2 Scope and Methodology

This study aims to explore the reinterpretation possibilities of topologically optimized slab forms through the architectural style of the diamond vault, faithful to Bötticher's tectonic theory. For this purpose, the study will primarily refer to <Diamond Vaults: Innovation and Geometry in Medieval Architecture> by Zoë Opačić, Professor at the University of London, for content on diamond vaults, and <3D-Printed Stay-in-Place Formwork for Topologically Optimized Concrete Slabs> by Dr. Jipa et al. from ETH Zurich, 2016, for content on topological optimization. The research will first outline the architectural value of diamond vaults and the application of topological optimization methodology in architecture through theoretical examination. Next, it will analyze the geometric characteristics of each and examine the reinterpretation possibilities of topologically optimized forms through diamond vaults based on these characteristics. Finally, the paper will reflect on the results, discussing the architectural significance and possibilities for expansion of the research, as well as future research directions.

2. Theoretical Consideration

2.1. The History of Diamond Vaults

Figure1. Diamond Vault in Trebsen Castle(Early 16 Century)

The diamond vault emerged in the late 15th century as part of the Late Gothic style in Meissen, eastern Germany, and became fashionable for about a century, primarily in Central Europe, including the Czech Republic and eastern Germany. As seen in Figure 1, a primary characteristic of this style is the ceiling surface divided into geometric patterns (tessellated) typical of the Gothic style. Although diamond vaults, a development within the Gothic vaults that lack ribs, have been evaluated as modern in design, they have not been widely known. Zoë Opačić, in her mentioned book, meticulously documents the buildings featuring the diamond vault style remaining in the Czech Republic. The characteristics of diamond vaults in this region are as follows: Firstly, high geometric complexity and perfection. Secondly, applicability to spaces with various purposes. Thirdly, an active response to both natural and artificial lighting. These features, especially the secularization of the diamond vault's application in the Czech Republic, are presumed reasons for their unique attributes. Therefore, this paper designates the diamond vault as the main subject of study as follows. The diamond vaults found in the Czech Republic are distinguished by space within the same building because numerous instances were discovered where different types of vaults were used depending on the space.

2.2. Theoretical Exploration of Topological Optimization

Topological optimization, which originated from the research by Bendsøe & Kikuchi in 19888), has evolved through various methods such as SIMP (Solid Isotropic Material with Penalization), ESO (Evolutionary Structural Optimization), and BESO (Bi-directional Evolutionary Structural Optimization). At the heart of these methodologies lies Finite Element Analysis (FEA). Finite Element Analysis, or the Finite Element Method, is a technique that replaces a structure with a finite number of elements to conduct structural analysis on each element. The ultimate goal of topological optimization is to optimize, i.e., make more efficient, the distribution of materials in a structure based on given conditions and objectives. This optimization typically results in the minimization of material use. Such characteristics are being applied across various scales in architecture. In 2011, Arata Isozaki's Qatar National Convention Centre implemented this method extensively in its architecture, and in 2016, Dr. Jipa and others developed a case of topologically optimized slabs.

Qatar National Convention Centre(2011)

Topological Optimized Slab Prototype A(2016)

3. Geometric Analysis and Review of Reinterpretation Possibility

3.1. Geometric Characteristics of Diamond Vaults

The geometric principle of diamond vaults is based on 'tessellation,' which typically involves dividing a surface with lines drawn between points (or more) on each plane and their vertices. The method of determining these points varies in each case, either by using points on the surface excluding edges, such as centroids, for division (as shown in Figure 3 Collin-Luther House-Room Step3, Architecture No. 45 Step2), or by using midpoints that bisect the edges for division (as shown in Figure 3 Collin-Luther House-Aisle Step2, Monastery and Church-Aisle Step2). In Figure 3, a 'bay' refers to a single span from wall to wall, wall to pillar, or pillar to pillar. The parts folding upwards in each bay are indicated with dashed lines, while those folding downwards are marked with solid lines.

Figure3. Bay Tessellation Process

Based on the analysis, the geometric characteristics of diamond vaults can be summarized as follows:

ⓐ In the initial division, the direction of folding is consistently maintained.

ⓑ The last divided surface is always folded upwards.

ⓒ The final division typically results in tri-partition or quad-partition of the previously divided surface.

ⓓ A surface folded upwards is not folded upwards again.

These characteristics suggest that diamond vaults could achieve complex yet refined geometric forms. Additionally, characteristics ⓐ and ⓓ contribute to structural stability, while ⓑ and ⓒ allow for a higher perceived ceiling height and maximize the beauty of the space through the use of light and shadow. Although it may not have been a considered aspect during the era when diamond vaults first appeared, a significant observation in contemporary times is that the application of diamond vaults reduces the overall volume of the vault, i.e., the slab, compared to when traditional vaults are applied.

3.2. Geometric Features of Topological Optimization

Topological optimization, which requires computational interpretation through FEA, results in data for each element being output in a specific format. In 3D modeling software, the most fundamental data form of topological optimization results is the mesh. A mesh contains data for each point, including coordinates and color values, and these points are connected using specific algorithms to form surfaces within the 3D modeling software. Figure 4 demonstrates the implementation of a topologically optimized slab by Dr. Jipa and others on a computer using McNeel's 3D modeling software Rhinoceros 7 and its add-on Millipede. The color values held by each point of the mesh indicate areas receiving more force in white and areas receiving less force in black.

These values were then used to apply z-vector values to create a three-dimensional surface with varying heights. The process of transforming the generated mesh data into a form that can be physically produced is known in digital fabrication as Rationalization. Dr. Jipa and others mentioned using the Catmull-Clark and Loop Subdivision Algorithms during this process11). Through Rationalization, surfaces that appeared rough are refined, and areas that were unproducible due to technical constraints are transformed into producible forms. Based on this analysis, the geometric characteristics of topological optimization can be summarized as follows:

ⓐ The process of topological optimization yields a mesh that includes color values for each point.

ⓑ The color values contained in each point can be converted into shape-related data, such as height

ⓒ Data processed through topological optimization must undergo rationalization to be producible, a stage where aesthetic judgments can be integrated.

3.3. Considering the Integration of Topological Optimization and Diamond Vaults

According to the analyses, topological optimization and diamond vaults share a common result—volume efficiency. While the primary goal of topological optimization is volume efficiency, the efficiency achieved by diamond vaults is conjectured to be an aesthetic intention rather than the original aim. Both topological optimization and diamond vaults share the commonality of being interpreted and created based on ‘points’. Considering that data from topological optimization can be transformed into shape-related data such as height, it suggests the possibility of reinterpreting the outcomes of topological optimization through the form logic of diamond vaults.

Particularly, as mentioned earlier, the outputs of topological optimization must undergo an appropriate Rationalization process to be transformed into producible forms (Figure 5). This stage potentially allows for aesthetic judgments that were not possible in other phases. However, the existing views predominantly regard the Rationalization process as a ‘problem-solving step’ for production, relating to physical realm discussions, i.e., the kernform, but not to the spiritual realm, i.e., the kunstform.

4. Conclusion and Further Research

This study explores the integration possibilities between topological optimization, which emerged in the 1980s and is still actively researched, and diamond vaults that appeared in the late 15th century as part of the Late Gothic style. This work attempts to reconsider and restore architects' involvement in form from Karl Bötticher's tectonic perspective in design fields that have actively incorporated CAD (Computer Aided Design) such as topological optimization. The analysis of the geometric characteristics of both topological optimization and diamond vaults revealed a shared basis on ‘points’. Additionally, although not initially intended, diamond vaults result in volume efficiency, aligning with the goal of minimizing material usage in topological optimization. Form intervention can occur during the Rationalization process, where topologically optimized forms are transformed into producible shapes. This is because, in the entire computational process, the Rationalization stage uniquely allows for architects' intervention in form. This research can broadly be interpreted as an exploration of how to embed architects' intentions in forms derived from computer calculations, like topological optimization. What is proposed here follows Bötticher's tectonic perspective faithfully, suggesting that interpreting his theory's art from a contemporary viewpoint (rather than historical context) could still yield significant discussions. Future research should not stop at evaluating possibilities but should also visualize and materialize the study by presenting prototypes, as done by Dr. Jipa and others. Such follow-up research could provide architects with the opportunity to apply the aims of this study to their practical work.