A Novel Method for Revolved Surface Infrastructures
Gökhan KınayoğluBilkent University, Turkey http://www.tectonicon.com gokhan.kinayoglu@bilkent.edu.tr
Abstract. This paper presents an algorithm for the formation of single or double curved
revolved surface’s infrastructures through standardized parts. Any revolved surface can be generated with only two types of parts, interconnected by a ribbed structure technique. The proposed method differs from the accustomed orthogonal rib structures by the varying angle in-between coupling parts. The algorithm can be customized through several parameters like the number, width of parts and thickness of the material used for the infrastructure. The algorithm also offers an advantageous nesting pattern with minimum loss of material regardless of the revolved surface cross-section.
Keywords. Revolved surface; standardization; ribbed structure; contouring; nesting
pattern.
INTRODUCTION
Following the design process, the transformation of a complex shape from the digital medium of com-puter software into the physical reality through material existence requires a further computation and rationalization (Griffith et al., 2006). The ration-alization process, depending on the geometrical complexity of the intended shape, aims for the standardization of parts for possible ways of manu-facturing, while lowering the manufacturing costs at the same time. Therefore, the process of ration-alization becomes an inevitable part of computa-tional design, as long as the target shape is a non-standard surface and requires non-non-standard parts for its constructability. By non-standard surface, any surface - single or double curved - that cannot be built through conventional manufacturing and con-struction techniques, and requires a computational design process for both rationalization and manu-facturing is indicated. In the case of non-standard surfaces, the number of units may end up in a large
number of unique variations, each requiring an in-creased degree of computation and a heavy process of production. Therefore, rationalization is required to transform the non-standard parts into standard-ized ones, in terms of their geometry, variability and economy. Besides rationalizing the surface parts, the infrastructure that builds up the surface also needs a process of computation and rationalization (Figure 1). It is the latter one, the infrastructure of a surface that this study is going to focus on specifically.
This study is an attempt to devise a methodol-ogy for producing single or double curved surfaces through standardized parts. This kind of approach can be considered as the primary step for enhanc-ing the current computational processes and brenhanc-ing- bring-ing forth a novel method for the fabrication of sin-gle or double curved surfaces. As Branko Kolarevic states, the production of a surface, whether single or double curved, can be realized through “contour-ing, triangulation, use of ruled, developable surfaces
and unfolding” (Kolarevic, 2003, p. 43). The devised method can be regarded as an innovative contribu-tion to the contouring technique in which the slices are not ran parallel to each other in two orthogonal directions.
As the starting point of this study, revolved sur-faces have been chosen, for their constant curvature along the rotation axis. The constancy of curvature among the rotation axis enables the standardiza-tion of parts and apart from the cross-secstandardiza-tion of the revolved surface, the algorithm generates only two types of parts which form the final infrastructure. The method introduced in this paper focuses mainly on small scale infrastructures, which are possible to be manufactured from sheet materials, like card-board, acrylic, fiberboard or metal.
The final infrastructure can be denoted as a diagrid system, which is being widely used in
archi-tectural projects, especially in skyscraper structures. The environmental and structural factors make it suitable for the diagrid system to be used in high-rise buildings. Therefore, another objective of this paper can be considered as the investigation of possible extensions of the diagrid system through a smaller scale implementation.
RELATED WORK
Apart from the orthogonal ribbed structures, which have been studied and implemented extensively, studies made on non-orthogonal ribbed structures have been analyzed. Agnieszka Sowa’s study, at ETH Zurich, explores the possibility of separating parts in relation to each other, for going beyond the sca-lar limitations of the material and manufacturing techniques. (Sowa, 2004) Sowa’s method focuses on generating and optimizing a cubic structure as an
Figure 1
An example of the infrastruc-ture for a spherical revolved surface.
instance of many possible forms, through a number of cross-sections integrated through a ribbed struc-ture, by using an algorithm. The parts are manufac-tured from planar timber elements with a constant thickness. Similar to this study’s approach, the varia-tion of angles between the parts has not been con-sidered in Sowa’s study, but instead all parts were manufactured by a two-axis CNC milling machine, hence the perpendicular slits, apart from the angle of intersection. The main aim of the study is to de-vise an algorithm that is capable of generating the intersecting ribs, optimizing the structure through some parameters, separating each rib where nec-essary and nesting the manufacturing drawings for generating the cutting scheme.
In Kenfield Griffith, Larry Sass and Dennis Michaud’s study of generating a strategy for irration-al building design, contouring technique is adopted for irrational surfaces, generating horizontal and vertical ribs, with every part different from each oth-er (Griffith et al., 2006, p. 467). While horizontal ribs are located at differing levels with equal intervals and remain parallel to each other, the vertical ribs are generated perpendicular to the surface, end-ing up with a more complex arrangement. The final structure resembles a waffle slab system used in concrete constructions, projected onto a curvilinear irrational surface. The perpendicularity of horizontal and vertical ribs solves the problem of slit angle vari-ation, thus enabling the manufacturing of the com-ponents on a two-axes milling machine. However, it should be noted that the variance among shapes creates a highly irregular nesting pattern.
An ongoing study by Yuliy Schwartzburg and Mark Puly, at Ecole Polytechnique Federale Laus-anne, Switzerland, explores the possibility of con-structing any shape through intersecting planar shapes (2013). A devised algorithm searches for solutions taking into consideration the cases of in-tersection and optimizing the inin-tersection angles and positioning of every rib through assembly limi-tations, slit constraints and material qualities. The study also takes into account the varying angle in-between the parts and comes up with a solution of
a predefined maximum amount of variation in the intersection angles. Therefore the firmness of the structure is attained when the assembly of the parts is completed.
ALGORITHM
The algorithm was devised through a script writ-ten in Autodesk Maya’s Maya Embedded Language (MEL) and later re-implemented in Rhinoceros and Grasshopper plug-in. It produces the production drawings of the parts of the infrastructure and they can be manufactured by a two-axis router, laser-cut-ter or walaser-cut-ter jet, for the planar quality of the parts at the current stage of the study.
The algorithm starts with a cross-section curve on the XY plane for the revolved surface, where Y-axis is the axis of revolution; hence farther the curve to Y-axis, larger the revolved surface will be. The cross-section curve should not intersect the Y-axis for the preservation of tubular quality of the revolved surface. Additionally, because of the limitations of the infrastructure, the cross-section curve should pass the horizontal line test, i.e. a line in X direction should intersect the cross-section only once at any point, however a line in Y direction can intersect the cross-section any number of times. By revolving the cross-section around the Y-axis for 180˚, a revolved surface is formed.
The surface is then intersected by a plane ly-ing in the YZ plane and inclined through Z-axis. The degree of inclination is the first parameter of the infrastructure. To guarantee the infrastructure cover-age of the whole revolved surface, the intersection points of the plane and the revolved surface at the top and bottom points should be checked, and the angle of inclination may be lessened in order to fully intersect the revolved surface.
The resulting curve of the intersection is cop-ied towards the Y-axis with a certain distance, the parameter for the width of all infrastructure parts, and two intersecting curves form the main construc-tive element by lofting to generate a surface. The resulting surface is arrayed radially N times around Y-axis with a total angle of 360˚, where N is the third
parameter of the infrastructure. By mirroring the re-sulting surfaces across XY plane, the infrastructure is formed by planar elements with a total number of elements of 2*N.
Beyond the computation of the slit dimensions, thickness of the material is also useful for visualiza-tion purposes (Figure 2). By extruding the parts with thickness of the material, the infrastructure is basi-cally formed, except for the slits.
Each part consists of a number of intersections depending on the formal characteristics and size of the cross-section, degree of the inclination angle and the number of elements. The inclined nature of each part results varying intersection degrees, but they are limited to the number of intersecting parts, i.e. if each part has 7 intersections, the model will have a maximum 7 varying angles at total. To determine the position and size of each slit, angle between the parts and thickness of material are used. The diagram shows the mathematical relation between the intersection angle and the slit width (Figure 2). With the intersection angle decreasing
in-between the parts, slit width increases and a 90˚ intersection produces slit width equal to the mate-rial thickness.
ADVANTAGES, LIMITATIONS AND
FU-TURE WORK
When the same revolved surface is produced through orthogonal ribbed structures, in which par-allel horizontal and axial vertical sections are used to generate the infrastructure, the method introduced displays some advantages. First of all, the devised al-gorithm guarantees that there will be only two types of parts to generate the infrastructure. However in orthogonal ribbed structures, while the vertical sec-tions will be identical due to the revolving quality of the surface, the horizontal sections differentiate according to the cross-section, all being circular. Ad-ditionally, for all of the slits are parallel to each other on every part, stability of the infrastructure depends on the frictional forces or the infrastructure requires additional elements to fixate the parts; whereas the introduced method has the advantage of
interlock-Figure 2
Three examples of the infra-structure.
ing itself due to the varying directions of each slit. Moreover, the standardization of parts also allows the perfect nesting in the cutting scheme regardless of the shape of the final part or the cross-section of the surface (Figure 3).
For the structural quality of the infrastructure is not crucial in small-scale implementations and ma-terial efficiency has a higher priority over structure, in the algorithm the intersection curve is not offset-ed to attain a constant width among the curve but instead copied towards the Y axis. As a direct out-come, the parts have varying width throughout, but on the contrary they can be nested in the produc-tion drawings regardless of the initial cross-secproduc-tion of the revolved surface. Additionally, as the nesting pattern also allows a lossless configuration of the parts, routing time is also decreased by the shared edges in between the parts. As long as the manu-facturing techniques allow, instead of cutting each part separately and producing left-over material in-between the parts, the parts are arranged perfectly without any loss of material. However, it should also be noted that this kind of approach may bring forth structural inadequacies, for the uncontrolled vari-ance in the material widths. Nevertheless, the nest-ing algorithm may be updated dependnest-ing on the structural necessities for each case, still protecting the advantages of the lossless nesting pattern.
The perfect nesting quality of the parts results in high efficiency in terms of material use when the number of parts being manufactured increase. Therefore, an infinite number of parts manufactured from a roll of steel leads to a 100% material effi-ciency, which is a distinctively advantageous feature when mass fabrication is considered.
The assembly process follows a relatively sim-ple system of formation. There are only two types of parts, inner and outer. Parts are interconnected to each other through corresponding slits, i.e. first slit to first, second to second and so on. For the varying direction of slits, assembly process is a bit problem-atic at the beginning when only a few number of parts are assembled. With the increasing number of parts, the infrastructure becomes more interlocked
and around the implementation of 70% of the parts, the interlocking becomes completed. It should be noted that the elasticity of the material used plays an important role for the different directions of the slits. While assembling the infrastructure, the parts need to be bended to a certain degree for joining them. This property also ensures the interlocking of the infrastructure. Bendable elastic materials like acrylic, medium-dense fiberboard (MDF), spring steel or cardboard suit well to the infrastructure.
If two-axis manufacturing techniques and thick materials are used for the infrastructure, the surface quality will be highly coarse. This can be better visu-alized by increasing the thickness of the material in the algorithm (Figure 4). Additionally, ribbed struc-ture technique has a problematic slit connection in cases other than the parts are intersected perpen-dicularly. Both problems may be overcome through the use of thin materials and further manufacturing techniques. The algorithm is capable of generat-ing the exact three dimensional model for precise interconnection between parts, resulting from the varying degrees of intersection. Therefore, when the parts require a higher degree of precision, a five-axis milling machine becomes more adept. Instead of planar pieces, five-axis milling machine will be able to incorporate the surface curvature of the revolved surface into the parts through thicker materials, be-sides the angle variations in the slit connections.
Another possible manufacturing method, and maybe the most suitable and optimized one for the infrastructure is the use of injection molding technique. As the number of parts required for the implementation of the infrastructure is limited only to two, in cases of mass-production of a specific cross-section, the manufacturing of two molds will be enough. Together with the decreased manufac-turing times and costs, the precision of the final in-frastructure is achieved through the use of molding. Since in its current formation the model does not offer any structural properties, there should be further studies for the optimization of parameters in terms of structural criteria. Number of elements, width and thickness of parts, degree of inclination
Figure 3
Visual representation, param-eters and nesting patterns of the previous three examples.
should be tested and analyzed structurally to find out the structural advantages and deficiencies of the method. Alterations to the algorithm may be in-troduced according to the findings about the struc-tural behavior of the infrastructure. Additionally, the setbacks of the varying width may be studied as an outcome of the structural findings, together with the nesting possibilities apart from the algo-rithm’s current potentials. Consequently, material limitations may be overcome and larger prototypes and implementations may be further achieved. For attaining larger scale infrastructures, Sowa’s tech-niques may be adopted (2004). By dividing each part into sub-parts and connecting them with additional elements, dimensional restraints may be extended.
CONCLUSION
Apart from the formal qualities of the form, stand-ard or non-standstand-ard, through designed methods and techniques, the part can be rationalized. The proposed algorithm shows an example of rationali-zation of a non-standard surface through computa-tional processes and attaining standardized parts. In a larger scope of context, this study proposes an
inquiry into the relation between individual part and overall form, together with their integral rela-tion in between, which cannot be separated in any phase of the design process. This attempt reflects an understanding of an approach, which prioritizes the potentials of rationalization from the initial steps of the design process, while also taking material con-siderations into account.
REFERENCES
Griffith, K., Sass, L., Michaud, D 2006 ‘A strategy for complex-curved building design: Design structure with Bi-lat-eral contouring as integrally connected ribs‘, SIGraDi
2006 [Proceedings of the 10th Iberoamerican Congress of Digital Graphics] Santiago de Chile, Chile pp. 465-469.
Kolarevic, B (ed) 2003, Architecture in the Digital Age: Design
and Manufacturing, Spon Press, New York.
Schwartzburg, Y. and Puly, M. 2013 ‘Fabrication-aware de-sign with intersecting planar pieces‘, Eurographics, 32 (2), pp.317-326.
Sowa, A. 2004 Generation and Optimization of Complex and Irregular Construction/surface: On the Example of NDS2004 Final Project. Postgraduate studies final the-sis in ETH Zurich. ETH Press: ETH Zurich.
Figure 4
