2015
Rokko Observatory Report Computational Synthesis 2014/15
BOYANA BUYUKLIEVA
ABSTRACT: The RokkĹ? Shidare Observatory by architect Hiroshi Sambuichi and engineers ARUP and Partners Japan is a landmark of the Rokko Mountain in Kobe, Japan. This report introduces the concept behind the dome, describes the implementation of the structure, as well as provides a commentary on its construction and relates to similar methods for achieving reciprocal geometries. A detailed account of the Shift Frame Geometry (SFG) solver, the purpose-developed numerical algorithm is provided and this is compared to two other methods to make an adequate judgement of ARUP’s code. As a result it is acknowledged that the SFG solver represents a more precise and efficient alternative to the other methods discussed. The application of the code is significant because the complexity of the structure would have been immensely difficult, if at all possible, to plan without technology. Finally, it is agreed that the solver enhances the design intent of the architect by referring to the aesthetics of the surrounding landscape and capturing the rime formation prominent of the region.
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Content
1. Introduction………………………………….………………………………………4 2. Description of the Algorithm………………….………………………………….5 General Overview …………………………………………………….5 Voronoi Tessellation …………………………………………………6 Shift Frame Geometry Solver ………………………………………7 Structural Analysis ………………….……………………………….10 3. Construction …………………….. ……………….………………………………11 4. Similar Approaches ……………..……………….………………………………12 5. Conclusion ………………………..……………….………………………………13 References Appendix A. SFG Solver ………………………………………………....……15 B. Construction and Detail ..…….………………………..………17 C. Parigi et al.’s Approach ..………………………………………18 D. Song’s et al.’s Approach …………….…………………………19
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1. INTRODUCTION The sixteen meter dome of the observatory is an intricate, self-supporting rod structure that provides partial shelter against the elements. Its algorithm was created specifically for the project by a team at ARUP lead by Kazuma Goto. The concept behind the distinguishing spherical, branch-like structure is the architect’s desire to create a building that blends into the landscape of the mountain and captures the unique northern wind in Rokko with the resulting frost blanket for which the area is known. ARUP saw in this the opportunity to create a self-supporting structure, specifically a reciprocal-frame structure, which in a simpler form is native to traditional Japanese shrine and temple construction (Goto et. al, 2011, p.21). The simplest form of such a structure is the three-stick model illustrated in figure 2 below. The starting point for the design was a parametric model created in Generative Components which explored potential patterns on a multi-faceted cylinder. During this stage of the design the element were not considered as interwoven because the purpose of this model was to decide on the visual impact of the dome’s density in relation to aesthetics and estimated structural need.
Fig. 1: Initial parametric model (Kidokoro & Goto, 2011, p.3)
Based on the rough parametric model the following constant were established: the primary metal structure bars would have a fixed thickness of 50mm in diameter and 1-2m in length. The secondary, wooden structure bars would vary between 15 and 20mm (Goto et. al, 2011, p. 21). To fit in the prescribed budget, the criteria for the system was the size and weight of the constituent elements, but also simple fabrication: “without the need of special connection pieces or fabrication technologies” (Kidokoro & Goto, 2011, p.2). These factors and the architect’s initial proposal led the engineers to suggest a selfsupporting system.
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Fig. 2: Three stick model
2. DESCRIPTION OF THE ALGORITHM General Overview ARUP created the shift frame geometry (SFG) solver to create the reciprocal frame (RF) system tailored to the dome. This proprietary algorithm is a numerical approach for finding the closest matching RF-structure based on input points on a geometry. The creation of the dome was a three step process. The first step was the specification of points on the dome based on a Voronoi tessellation. The following step took inputs from the first step into the SFG algorithm to search for an optimized RF topology. The final step is a structural analysis of the SFG output.
Fig. 3: Outline of how the dome was created.
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Voronoi Tessellation The architect insisted on having an open structure reminiscent of those found in nature. For this reason instead of using an arbitrary principle, the team at ARUP proposed irregular polygon openings of the dome based on a Voronoi tessellation. To create the Voronoi tessellation, a semi-random set of points on the dome was generated. These sets of points in space were the basis for creating the tessellation boundaries which are drawn as perpendicular lines midway between adjacent points. In special circumstances three, or in special circumstances more, of these bisectors intersect to give the corners of the tessellations (Fig. 4).
Fig. 4: Example of Voronoi tessellation (Nicoptere, 2008)
To accommodate the environmental design of the observatory the density of the random points was manipulated. At the lower perimeter of the structure the polygons are more generous to allow for a panoramic view, but also to allow air movement, particularly in the summer. The structure is also less-dense at its upper-north corner to allow for the cold wind to promote ice crystal formation in the winter. The uppersouthern area of the dome is more densely populated with points to reduce solar radiation (Goto et. al, 2011, p.21). The Voronoi tessellation can be considered as a 2D pattern embedded in a 3D space. The impression of this is the design intent of the project because the points of the Voronoi define the beginning and end of the rods and thereby the input of the SFG solver.
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Shift Frame Geometry Solver Parameters: The purpose of the SFG solver is to create a self-supporting, RF structure that is as close as possible to the design intent, in this case a dome with irregular Voronoi polygons. By definition the reciprocal frames are based on elements that sit on top of each other whose axes do not align. Therefore, a self-supporting RF necesserily extends into the third dimension. The SFG solver looks for the solution that produces the flattest possible RF structure. This is so because the flatter the RF-frames, the closer they sit to the desired geometry. The solver provides a numerical approach for generating the dome geometry. It is simply a tool consisting of a set of equations that give the an estimated solution of an RF-frames based on some coordinate input points. To understand how the solver works it is important to understand the three factors that influence the flatness of the structure:
Fig. 5a Eccentricity: Sum of the radii of the rods (Parigi et al., 2012,p.2) As the size of the radii increases, so does the displacement in the third dimension.
Fig. 5b Engagement length: Touching point of the pipes (Parigi et al., 2012,p.2) The longer the engagement length, the more protrusion in the third dimension.
Fig. 5b Topology: Top/Bottom configuration (Parigi et al., 2012,p.2)
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As the width of the pipes is a constant, the other two parameters are considered in the algorithm. To achieve the flattest RF-system, it is important to maximize the engagement length while optimizing the topological configuration to minimise the total length of the rods in the system.
Engagement length: A distance vector was created to describe the relationship between overlapping rods (Kidokoro & Goto, 2011, p.4). This vector (d) represents the engagement length and is such that it starts at the center core of one rod at the rods’ touching points and ends at the center core tip of the adjacent rod (Fig.6). The algorithm solves for an � - a vector for which the engagement length produces optimum value for d, which is �� the smallest ι, whilst the rods are still overlapping, i.e. the point at which the top pipe is just about to fall off the bottom one. (More details in Appendix A).
Fig. 6 Vector relationship between adjacent rods
Total sum of elements: The possible top/bottom arrangements of the rods for the different engagement lengths result in a vast number of permutations, however there is only a single combination that yields the flattest structure. Numerically this combination is described as the system with the ‘shortest total element length’ (Kidokoro & Goto, 2011, p.5). Conditional Optimizations: It is important to note that rods in an RF-structure necessarily lie on top of each other. Therefore, when optimizing the total sum of elements, one must also work with constraints imposed by optimization of the engagement. A Lagrange multiplier was used to solve for the minimum element length with the constraint of having them lie on top of each other. (More details in Appendix A) Accelerating Convergence: At its core, the SFG solver is a set of equations that gives an estimate solution of a complete free form RF geometry based on some input coordinate values. Therefore, the closer the initial input value to the solution, the better the output solution. 8
By definition the RF system has no structural hierarchy because all the rods are interconnected and therefore equally important. The interdependency of the rods means that the search space gets exponentially large and therefore significantly more time-consuming to explore with each rod. Kidokoro and Goto state that initially the algorithm took hours to converge (Kidokoro & Goto, 2011, p.5). In order to find successively better approximations to the solutions and lower computational time, the Newton-Raphson Method was used. This method bases each consecutive guess on the x intercept of a function’s derivative (Fig. 7).
Fig. 7 Example of the Newton-Raphson method: X1, the zero of the tangent line at point X0, is used as the input of the y = f(x) because it is closer to the actual zero of the function. X2 is zero of the tangent line at point X1, which is yet a better estimate to the actual y = 0. (Afandi, 2012)
Because the SFG solver produces a complex non-linear equation, repetitively applying the Newton-Raphson method yields better estimations, but not an exact answer. Therefore, it is up to the design team to decide on the degree of precision desired and based on this apply this process recursively until this intent is satisfied. For the SFG solver the function closest to zero is the nearest optimal point for the given input. (Kidokoro & Goto, 2011, p.6) In this case edges of the Voronoi polygons are the input.
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Fig. 8 Voronoi tessellation after the SFG is applied (Kidokoro & Goto, 2011, p.7)
Structural Analysis The SFG solver produces an accurate geometric model, however this model is still merely a mathematical object. As Kazuma points out, the physical dome will be subject to various conditions including snow and typhoon loads. Therefore, to test the viability of the design, a structural analysis of the RF-dome was done based on the eccentricity of the tubes (Goto et. al, 2011, p. 24). Most likely, a finite element analysis with a starting point of the ground-supported rods was used to calculate the stress on the structure. If any weak areas are identified, the number of points used to create the Voronoi in that area are adjusted and the SFG-solver is re-applied. This is, again, an iterative process which can be repeated until a solution is produced that fufils the structural and aesthetic goals.
Fig. 9 Structural analysis results displaying possible node displacement under simulated loads (Kidokoro & Goto, 2011, p.8)
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Fig. 10 Final mathematical model from the SFG (Kidokoro & Goto, 2011, p.6)
Fig. 11a Data list and specification used in construction phase (Kidokoro & Goto, 2011, p.8) Fig. 11b Mock-up of the dome (Kidokoro & Goto, 2011, p.9)
3. CONSTRUCTION Although the primary rods were connected simply by welding (Goto et. al, 2011, p. 24), the interdependency of elements made the use of conventional 2D construction plans insufficient. Additional specifications needed to be provided, including rod position, dimension and engagement point (Fig 9a/b). Also a close collaboration with the manufacturer had to be established from early on in the construction phase. This collaboration included the construction of full-scale mock-ups of certain areas before assembly on site (Kidokoro & Goto, 2011, p.8). For the secondary structure, thermal analysis and on-site experiments were conducted to establish that rough, thin timber with low thermal capacity promoted the best rime formation (Goto et. al, 2011, p.25). 11
4. SIMILAR APPROACHES The appendices contain two alternative computational approaches to creating RF structures which are compared to the SFG solver here. Despite their differences, all three algorithms take a numerical approach, because the emergent property of reciprocity is too complex to model analytically. Parigi et al. approach the generation of an RF geometry differently from ARUP because they adjust individual nodes iteratively, instead of solving for all the elements ‘simultaneously’ (Kidokoro & Goto, 2011, p.5). In the SFG-solver the whole structure is shifted at once, whilst in Parigi et al.’s method every element is adjusted “one node at a time one after the other” and then optimized multiple times to account for the error caused from one node to the other (Parigi et al., 2012, p.6). One could argue that the two algorithms are similar in output, but differ in terms of computational expenses, because Parigi et al.’s method relies on more iterative moments: once for topology, once for geometry and finally for regulating adjacent nodes to each other. Song et al. propose a more simplified and interactive computational tool for designing RF-structures. The main contrast between this and the SFG solver is that Song et al. start with an RF-pattern and map it to a surface. By contrast, ARUP uses a desired surface and, based on its requirements, an RF - frame is generated. Song et al.’s tool allows for more interactive control over the aesthetic of the RF-structure at the cost of precision. In their approach there is a ‘two-level hierarchy’ (Song et al., 2013, p.5), in comparison, the structures of the SFG are completely interdependent, thereby closer to Parigi et al.’s definition of reciprocity (Parigi et al., 2012, p.1). Because of the lack of hierarchy, ARUP’s model is able to produce more geometric detail with less RF frames. In this sense ARUP and Parigi et al. provide the better RF-solution (A more detailed comparison is available in Appendix D). The SVG solver provides a more tailored and computationally efficient self-supporting structure in comparison to the other methods presented. It can work on any desired surface pattern and produce detail with less frames in comparison to Song et. al.’s approach. Also the SFG-solver relies on less iterations in comparison to Parigi et al.’s algorithm and can thereby be considered a more elegant approach.
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5. CONCLUSION The greatest strength of ARUP’s solver and the project in general is that a high-tech approach is applied to create an efficient low-tech solution both in the environmental design and the dome geometry, which is the focus of this report. Computation is employed to organize the complexity that arises at a macro level from having interdependency at a micro level, a task that would have taken an unthinkable amount of time to do by hand. However, to truly judge the dome and draw conclusions it is important to return to the design intent. The main inspiration behind the Rokko dome was the shifting processes in nature and more precisely the transition of Kobe’s subtropical green of the mountain into a pristine white landscape in the winter. The frame of the dome succeeds in capturing the latter because of the strategic density distribution of the dome and the low surface temperature of the frames that attract frost. The main aesthetic tribute to the nature of the Rokko Mountain is given by the Voronoi polygons that form the RF-structure. The irregularity of the structure recreates the sensation of being protected under the shade of branches. However, a more deep metaphor for the interwoven and complex relationships in nature can be seen in the interdependency of simple elements and emergent reciprocal frame structure, so iconic of the Rokko Observatory. In this sense the collaboration between ARUP and Sambuichi succeeds in creating an extension of the local environment which captures correlation between the elements and the surrounding landscape.
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REFERENCES AFANDI, M. (2012) Metode Newton-Raphson. [Online] 27 September 2012. Available from - http://blog.ub.ac.id/derryafandi/category/uncategorized/ [Accessed: 05.03.15] GOTO K., KIDOKORO R., MATSUO T. (2011) Rokko mountain observatory. The Arup Journal. [Online] 46. (2). p 20-16. Available from publications.arup.com/Publications/T/The_Arup_Journal/2011/The_Arup_Journal_20 11_Issue_2.aspx. [Accessed: 03.03.15] HYBEL, J. (2013) RokkĹ? Shidare Observatory. [Online] 03 April 2013. Available from - ww.arcspace.com/features/sambuichi-architects/rokko-shidare-observatory/. [Accessed: 06.03.15] KIDOKORO R., GOTO K. (2011) "Rokko Observatory" - Application of Geometric Engineering. In ALGODE TOKYO 2011. Tokyo, March 14-16, 2011. NICOPTERE (2008) Delaunay triangulation and Voronoi diagram. [Online] 9 September 2008. Available from - http://en.nicoptere.net/?p=10. [Accessed: 12.03.15] PARIGI D., KIRKEGAARD P., SASSONE M. (2012) Hybrid optimization in the design of reciprocal structures. In Proceedings of the IASS Symposium 2012: from spatial structures to spaces structures. Seoul, 21-24 May, 2012. SONG P., GOSWAMI C., Zheng J., MITRA N., COHEN-OR, D. (2013) Reciprocal frame structures made easy. ACM Transactions on Graphics (TOG). [Online] 32.(4). Available from - http://dl.acm.org/citation.cfm?id=2461915. [Accessed: 10.03.15]
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Appendix A: SFG Solver
Fig. 1: SFG solver as described by ARUP
a. Distance Vector − đ?’™đ?’™đ??´đ??´ ďż˝ − đ?’™đ?’™đ??ľđ??ľâ€˛ ) đ?’…đ?’… = −(đ?’™đ?’™đ??´đ??´ + đ?‘ đ?‘ ďż˝đ?’™đ?’™đ??´đ??´â€˛ďż˝
This amendment was made to match the formula to the vector in the diagram provided by ARUP. A vector d was created that describes the distance from the bottom rod at the point of overlap to the end of the top rod (in both cases we are considering points on the rod’s axes). In essence, the vector is a means of measuring the degree to which the rods are out of plane. b. Boundary Condition ďż˝ = 0 or otherwise stated đ?’…đ?’… = đ?’…đ?’… ďż˝ đ?’…đ?’… − đ?’…đ?’…
Because there are many ways to make the order of arrangements (top/bottom) and place the location of the crossing positions the number of permutations is a vast one. However, not all placements will yield the desired flatness and an overlapping structure. Therefore to achieve both, a boundary condition was introduced. This requires the d vector to be equal to the vector perpendicular to the axis of both rods and with a length equal to the sum of their radii. In other words, this requires two rods to be perched at the farthest possible point before falling off one another. c. Conditional Optimization
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For the conditional optimization, the Lagrange multiplier was used to simultaneously find the total minimum length of all rods in the system with regards to the constraint given by the boundary condition. Lagrange multipliers are a way of finding an optimum value subject to constraint (Fig.2). This method involves establishing an expression that combines both the equation and its constraint. In this case the equation that is being optimized is the sum of the lengths of all the rods in the system. This sum is to be as small as possible, because the smallest possible sum is also the flattest possible solution. Because we are dealing with many rods we consider the Lagrange multiplier for every intersection of the rod, therefore our constraint is (d-đ???đ???Ě‚) for every pair of rods and the function to be optimized is ∑i li where li are the lengths of the rods.
Fig. 2: In the diagram above the circle is considered as a constraint for the given function. The maximum and minimum values that satisfy the constraint for are a and b respectively.
d. Optimal Point Derived Using Variation Method The Lagrange method involves partial differentiation to be solved. The expression provided by ARUP in d. displays the differentiation of the function and its constraint. Differentiation is used to find the zeros of the expression, which is an approximation to the minimum value of the total sum of rods. This minimum is the flattest RF frame from the coordinates of the end of each rod provided in the beginning. e. Tangent Matrix using the Newton-Raphson Method From the method of Lagrange multipliers follows a very large system of equations that equals zero. These systems of equations, which are stored in a tangent stiffness matrix, can be solved to find the values for the variables in the system for the lowest value of the total sum of rods.
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Appendix B: Construction and Details
The dome during construction and afterwards. The primary carrying structure consists of metal rods and the second one is thin timber that promotes rime formation.
Images courtesy of ARUP and Partners Japan
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Appendix C: Parigi et al.’s Approach Parigi et al. propose a method for the generation of RF-structures by means of a hybrid optimization consisting of a genetic algorithm (GA) for establishment of topology and a gradient based descent to accelerate convergence. This method is based on a node, consisting of a three stick model that can be repeated to create a free form structure by means of multiple assembly. The optimization is run “one node at the time one after the other … multiple times, assuming that the errors induced from one node to the other will be every time smaller” until convergence is reached (Parigi et al., 2012, p.6).
Fig. 1: Bar 1 is an element with points P1 and P2 that belongs to two nodes, n1 and n2. The algorithm is run once to adjust bar 1 to the n1, and then the same is done for n2. However the second adjustment distorts the first and therefore the optimization must be run multiple times.
The fitness of the GA is equivalent to the boundary condition considered by ARUP. It accounts for whether each rod sits on an adjacent one, but also how it sit in relation to its neighbors. This difference arises from the fact that in a three stick model there is more favorable top/bottom configuration, therefore a penalty factor was added to account for those cases (Fig. 2a/b).
Fig. 2a: Penalty factors imposed for top/bottom configurations Fig. 2b: Fitness landscape of the GA, which takes into consideration the penalty factors
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Appendix D: Song’s et al.’s Approach
Fig. 1 Outline of Song et al.’s interactive RF tool
RF units are defined and connected based on simple grammar rules to form a 2D grid. The two-level hierarchy of the structures comes from the fact that the small RF units are independently self-supporting before being attached to form the larger pattern. Thereafter this pattern is lifted into the third dimension by a conformal mapping that introduces some distortion (Song et al., 2013, p. 2). The distortions are dealt with by a relaxation which is bound by three constraints. The first constraint is ‘contact constraint’, which is similar to the boundary condition of the SFG solver (Song et al., 2013, p. 7). This constraints is concerned with making the joints necessarily come in contact. The second constraint is ‘surface constraint’. This is similar to the SFG solver because its goal is to make the RF as flat as possible. However to achieve this, Song et al.’s method does not calculate the total length of all elements, but rather requires each of their centroids to align as close as possible to the defined surface. The final constraint is a conformality constraint. This constraint is concerned with preserving the appearance of the single RF-units in the system, by regulating the angles that define the junction of the rods. This condition does not have an equivalent in the SFG-solver because the connection points given as an input to the solver already have a 3D relation to each other.
Fig 2. Contact constraint
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It is important to note that both approaches produce a complex structure because of the interdependencies of elements at a micro level has a direct impact on the geometry at a macro level. Peng et al.’s approach alleviates this problem by using regular self-supporting ‘bricks’ to build the structure. This could be considered as a geometric approximation, which accounts for the model’s scale dependency for detail, which is less the case in ARUP’s algorithm. However, it is worth mentioning that Song’s et al. intention is to create an interactive tool for a broader audience that focuses on the visual aspect of the RF-structure. From this point of view it is clear that some precision had to be sacrificed in favor of aesthetic control.
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