IFS-Based Computational Morphogenesis of a Hierarchical Trussed Beam

Iasef Md Rian Department of Architecture, Xian Jiaotong-Liverpool University, China Email: [email protected]

Abstract. This paper applies IFS (Iterated Function System) as a rule-based computational modeling process for modeling a hierarchical truss beam inspired by the concept of fractal geometry. IFS is a type of recursive algorithm, which repeatedly uses the outcome as a input for an affine transformation function in generating a fractal shape, i.e., a complex shape which contains the self-similar repetitions of the overall shape in its parts. Hierarchical trusses also follow a similar geometric configuration. IFS-based computational modeling, hence, allows us to parametrically morph a parent model, thus repeat the same morphing to all its self-similar parts automatically. This IFS-based morphogenesis opens a possibility to find an optimal configuration of a hierarchical truss structurally. In this parametric modeling process, the iteration number is a unique geometric parameter. This paper uses two geometric variables (iteration number and angle) to find the most efficient design of a hierarchical truss beam through an optimization process.

Keywords: hierarchical truss, fractal geometry, IFS, computational design

1 Introduction

The feature of self-similar repetition is a common configuration in structural trusses. Generally, trusses having this type of repetitive configurations are known as the hierarchical structures. The fundamental reason for adopting the hierarchical arrangement of structural members and their sub-members is to achieve high strength with less weight. In construction, accordingly, trusses can be considered as a replacement of monolithic structural elements in order to make the structure lighter, especially for large-size structures. The Great Pyramid of Giza in Egypt consumes a huge amount of masses especially at the base to reach the stable height of nearly 146 meters, while the more than double height (nearly 324 meters) was achieved by truss- filled Eiffel tower which consumes fraction of masses than that of the Pyramid. [1] The marvel of Eiffel tower is not only its beauty of trussed lattices, but also its optimality in terms of its lightweight that touches such a height. Closer looks reveal that each member of the truss is itself a smaller truss, and each sub-member of the smaller truss is again another truss, a smaller replication of its parent truss. Thus, Gustav Eiffel maximized its lightness by repeating this process hierarchically, creating a fractal-like

CAADFutures19 -996 hierarchy in the structure. The overall lattice of the tower becomes an assemblage of huge number of triangles that make the overall structure as well as its parts highly stiff. In this context self-similar repetition of the hierarchical trusses, Benoit Mandelbrot claimed with reference to the design of the Eiffel tower that

‘… the tower that Gustave Eiffel built in Paris deliberately incorporates the idea of a fractal curve full of branch points. . . . However, the A's and the tower are not made up of solid beams, but of colossal trusses. A truss is a rigid assemblage of interconnected submembers, which one cannot deform without deforming at least one submember. Trusses can be made enormously lighter than cylindrical beams of identical strength. And Eiffel knew that trusses whose 'members' are themselves subtrusses are even lighter.’ [2]

This claim clarifies the strength lies not in mass, but in cleverly designed geometric shapes and in the case of the Eiffel tower, it lies in branching points. Thus he hints that the fractal geometry, which explains the geometry of self-similarity, can be a mathematical tool that can model such branching points in many hierarchical scales. This paper was motivated by this idea and has applied it in designing a fractal-based light-weight structural trussed beam as a benchmark application for understanding the role of fractal concept in reducing weights by securing enough strength. Some recent researches have ensured the efficient advantage of fractal geometry in developing some light-weight structures in the form of hierarchical trusses [3][4][5] and also some new shapes of roof trusses [6][7]. In the family of hierarchical trusses, a fractal-based hierarchical trussed beam is an assemblage of the self-similar beam members that are further assembled by their own self-similar submembers in a similar fashion. Obviously, a geometric rule plays an underlying role to make such repetitive configuration. A recursive algorithm as a generative process can be the best way to easily model such a repetitive configuration of the proposed trussed beam. In this paper, the Iterated Function System (IFS) has been used as a generative process to model the trussed beam followed by Barnsley's contraction mapping method. [8] A finite element analysis has been applied in finding the efficiency of the differently iterated trussed beam. The proposed geometric model of the hierarchical trussed beam is a parametric model which has two different geometric variables that are the angles between two members and the diameter of the cross-section of each individual member. Iteration number is another and unique variable of the model. These variables provide different possible versions of the model. A computational morphogenesis system uses these variables and their parameters for getting an optimal version of the hierarchical beam. The finite element analyses show that the high iterated model is structurally efficient and optimal than the solid or less iterated models, while the computational search provides the most efficient configuration of the model. This study confirms Mandelbrot's claim about the merit of fractal geometry for developing light-weight and high-strength lattice structures.

CAADFutures19 -997 2 Geometric Modeling

In this chapter, the concept of IFS-based fractal-inspired computational modeling has been applied for developing a truss beam as an experiment to analyze its changing structural behavior with the changing of geometric parameters and iteration numbers. The configuration of the experimental truss beam is inspired by the truss model developed at micro scale by Robert Farr and his team [3]. Here, the beam is considered as a planar truss beam.

2.1 Mathematical Formulation and IFS Hierarchical trusses are the assemblage of the self-similar members that are further assembled by their own sub-members in a similar fashion. In the case of hierarchical truss beam, each beam member is replaced by self-similar truss whose sub-members are further replaced by its corresponding truss beams. This geometric configuration follows a rule that can be constructed by a generative process, here the Iterated Function System, i.e., IFS. Here, the Barnsley's contraction mapping method has been applied to produce such a truss beam. In the beginning, a single line has been taken as a beam B0 which is the initial shape of the geometric operation. In the first iteration, B0 is transformed into a truss beam B1 which is an assemblage of 25 self-similar copies (b1, b2, b3, : : : :, b25) of B0 (Fig. 1). There are four groups of different self-similar copies that are produced by the contractions of λ1, λ2, λ3 and λ4. In the next steps, this process of repetition is continued using the same transformation rules. Thus, it results in a fractal figure which is an attractor, i.e., an intersection set of all the identical subsets of B0, as shown in Fig. 1. The attractor B can be expressed as,

∞

B = ⋂ B푖 (1) 푖=0

where, B1 = 푏1 ∪ 푏2 ∪ 푏3 … … ∪ 푏25 (2)

B1 = 푓1(B0) ∪ 푓2(B0) ∪ 푓3(B0) … … ∪ 푓25(B0) (3) At n th B = 푓 (B ) ∪ 푓 (B ) … … ∪ 푓 (B ) (4) iteration, 푛 1 푛−1 2 푛−1 25 푛−1

i.e., B푛 = ⋃ 푓푖(B푛−1) (5) 푖=1

If B1, B2, …., Bn, …. are contraction sets of B0, that are contracted by using the contractivity factor λi and transformed by using an affine transformation function fi, such that,

CAADFutures19 -998

B0 ⊃ B1 ⊃ B2 ⊃ ⋯ ⊃ B푛−1 ⊃ B푛 ⊃ ⋯ (6) then, they form a perfect self-similar fractal set. In the case of constructing the fractal hierarchical truss beam, the following function of IFS is applied that consists of 25 different affine transformations (f1, f2, f3, …. f25) of the main triangle.

푐표푠휃 −푠푖푛휃 푥 푑푥 푓푖 = 휆푖 [ ] [ ] + [ ] ; 푖 = 1 푡표 25 (7) 푠푖푛휃 푐표푠휃 푦 푑푦

In this geometric configuration of the truss beam, there are four groups of self-similar copies based on four different contractions, which are,

1 휆 = (8) 1 7

1 1 휆 = ∙ (9) 2 7 푐표푠휃 2 휆 = ∙ 푠푖푛휃 (10) 3 7

2 2 1 2 1 휆 = √( ∙ 푡푎푛휃) + ( ) = ∙ √4 ∙ 푡푎푛2휃 + 1 (11) 4 7 7 7

Table 1: IFS variables and their values

Group IFS Contra Rotation Displacement functions -ctivity

f λ θ δx δy

f1 L/7 (L/7).tanθ f2 2L/7 (L/7).tanθ f3 3L/7 (L/7).tanθ f4 4L/7 (L/7).tanθ f5 5L/7 (L/7).tanθ o 1 0 Group 1 f6 λ L/7 - (L/7).tanθ f7 2L/7 - (L/7).tanθ f8 3L/7 - (L/7).tanθ f9 4L/7 - (L/7).tanθ f10 5L/7 - (L/7).tanθ

CAADFutures19 -999 f11 θ 0 0 f12 - θ 0 0 o Group 2 f13 λ2 90 + θ L 0 o f14 - (90 + θ) L 0

o f15 90 L/7 o f16 90 2L/7 o f17 90 3L/7 o Group 3 f18 λ3 90 4L/7 - (L/7).tanθ o f19 90 5L/7 o f20 90 6L/7

f21 2L/7 f22 3L/7 180o – arctan(2.tan θ) Group 4 f23 λ4 4L/7 - (L/7).tanθ

f24 5L/7 f25 6L/7

The above IFS function has been transformed into a scripting code for making a parametric digital model of the structure where θ, L and the iteration number n are made variables. The code results in a graphic model shown in Fig. 1.

Fig. 1. Hierarchical truss configuration after four iterations.

CAADFutures19 -1000 2.2 Fractal Dimension Calculation Fractal dimension is a measure of how dense a figure is. It is a non-integer dimension that lies in between two successive integer dimensions. Based on the Barnsley's contraction mapping theory [8], the fractal dimension, more precisely, the Hausdorff dimension of a fractal, is linked to the contractivity factors by the following relation:

25 2 ∑(휆푖) = 1 (12) 푖=1 where λi is a contractivity factor of transformation fi and k is the number of transformations [8].

Therefore, in the case of fractal-based hierarchical truss beam:

퐷 퐷 퐷 퐷 10(휆1) + 4(휆2) + 6(휆3) + 5(휆4) = 1 (13)

1 1 1 2 1 109 ∙ ( ) 퐷 + 4 ∙ ( ∙ ) 퐷 + 6 ∙ ( ∙ 푡푎푛휃) 퐷 + 5 ∙ ( ∙ √(2푡푎푛휃)2 + 1) 퐷 = 1 (14) 7 7 푐표푠휃 7 7

The above Eq. 14 shows that the Hausdorff dimension of the shape of fractal-based hierarchical truss beam is the function of angle θ. The value of Hausdorff dimension D can be obtained by the Newton-Raphson method. In the experimental model, the angle is taken 20o, and therefore the calculated Hausdorff dimension of the shape of the beam truss is approximately 1.656. The higher the value of the fractal dimension, the higher is the density of the truss beam lattice. Structurally, denser the truss, the higher is the stiffness of a truss. Later, we will check the changing of structural deformation of the truss with the changing of its fractal dimension.

3 Parametric Modeling

The mathematical formulation of the IFS for generating the hierarchical truss has been transformed into a code in python script and developed a new Grasshopper component where Grasshopper is a parametric modeling program for Rhinoceros3D. Thus, this new component is now able to generate the parametric model of the hierarchical truss beam. This model is changeable with the changing of few key geometric variables such as the iteration number and the angle. Fig. 2 and Fig. 3 show different parametric models of hierarchical truss beam. This parametric model allow us to understand the

CAADFutures19 -1001 changing structural behavior of the truss beam with the changing of its iteration number and angle. Thus it eventually enable us to perform an optimization process in order to find the most efficient design of the fractal-based hierarchical truss beam.

Fig. 2. Parametric model of the hierarchical truss when angle is fixed at 40o and the iteration number is changeable from 0 to 4.

Fig. 3. Parametric model of the hierarchical truss when the iteration number is fixed at 2 and the angle is changeable from 20o to 40o.

CAADFutures19 -1002 4 Structural Modeling and Analyses

4.1 Structural Modeling and Preprocessing The above digital model is transformed into a finite element model for the finite element analysis by taking the following considerations:

. The length of the truss beam is 10 meter. . Only the finitely iterated fractals are applicable in real-world constructions. Therefore, here from the initial model to second iterated model has been taken. The high iterated model has not been taken to avoid heavy calculation by FEM (Finite Element Modeling) solver that would take a painful long time. . The overlapping lines are removed, and the remaining lines are considered as beam elements. . Each beam element is hollow steel tube having the circular cross-section of 200 cm diameter and 10 mm thickness for initial model, 25 cm diameter and 8 mm thickness for the first iterated model, and 5 cm diameter and 4 mm thickness for the second iterated model. . The overlapping points are removed, and the remaining points are considered as structural nodes. . Each node is a welded joint connecting the corresponding beams. . The overlapping of beams near the joints is ignored. . One end of the main structure is horizontally as well as vertically restrained, while the other end has been vertically retrained only. . Self-weight and the vertically applied distributed point loads of 10 KN are considered for the analyses.

4.2 Structural Analyses and Results After preprocessing and applying the loads, the FEM solver calculates the maximum displacements of the structures, shown in Fig. 4. Here, the allowable displacement has been set 20 mm. The finite element analyses have been calculated by Karamba3D, a FEM solver that works in Grasshopper for a parametric model. Karamba3D gives instant feedback with the changing of parametric model of a structure. The whole operation is operated and visualized in Rhinoceros3D. After the finite element analyses, it has been found (Fig. 4 and Table 2) that the loss of mass of the solid beam B0, when transforming into the first iterated truss beam B1, does not affect significantly to the overall strength of beam structure. This fact is the same when the first iterated truss beam B1 transforms into the second iterated beam B2. Until the allowable maximum deformation under a certain practical load, the beam can be iterated for several generations, thus reduce as much mass as possible. The analytical 8

CAADFutures19 -1003 results, hence, indicate the advantage of using fractal concept for reducing mass as much as possible, as hypothesized by Mandelbrot in the beginning [2] and studied some examples by researchers from applied sciences and engineering [3][4][5]. In this case, the number of iteration plays the main role to strategically reduce the mass from the main solid beam.

Fig. 4. Finite element analyses of differently iterated hierarchical trusses when angle is fixed at 40o.

CAADFutures19 -1004 Table 2: Changing weight and deformation of the truss with the increasing of iteration number

Weight Deformation Iterated beams (Kg) (mm) Initial beam 2441 0.5 1st Iterated beam 1894 3.0 2nd Iterated beam 712 5.6

However, apart from this variable, there is another geometric variable that is angle which can significantly influence the strength of the truss. Table 3 shows the decreasing deformation of the hierarchical truss with the increasing of its angle from 20o to 40o when it was demonstrated only on the second iterated model. However, there is another geometric variables that can further help to reduce the mass by maintaining enough strength. One of such variables is the cross-sectional size of beam elopements. In the following section, the optimal form of this hierarchical truss has been searched by changing the uniform cross-sectional size (diameter) of the beam members.

Table 3: Changing fractal dimension and deformation of the truss with the increasing of angle.

Deformation Angle Fractal Dimension (mm) 20o 1.66 51.7 25o 1.72 33.1 30o 1.80 22.6 35o 1.91 16.3 40o 2.0 12.2

4.3 Structural Optimization The fractal-based hierarchical truss, modeled here as a benchmark experiment, has been made parametric where iteration number is a key variable. However, apart from the iteration number, the cross-sectional sizes of the members and sub-members are a significant factor for obtaining an optimal design of the truss beam. From the above finite element analyses, it is observed that the second iterated model is the lightest among initial and the first iterated truss beam and yet its maximum displacement is about 6 mm which is within the allowable maximum deformation of 20 mm. This

CAADFutures19 -1005 means the model with further iterations is possible that will be lighter than the second iterated model within the allowable deformation of 20 mm. Fig. 5 illustrates the schematic strategy of finding the optimal design of the hierarchical truss by changing the angle (ranging from 20o to 40o) and the cross-sectional diameter (ranging from 1.0 to 5.0 cm) while the thickness of each beam has been kept constant (4 mm).

Fig. 5. Schematic diagram of the computational searching.

Fig. 6. Optimized model of the hierarchical truss.

For avoiding the high calculation and for the limited memory and speed of the personal computer, the trial with higher iterated models have not been attempted. The load condition for the optimization process remains the same. Karamba, calculates the preprocessed finite element model of the structure and gives instant feedback with the changing of the value of diameter. It works interactively in the Grasshopper's parametric environment and produces the digital model in the Rhinoceros3D. This set up has been connected to Galapagos which is a search algorithm component that helps

CAADFutures19 -1006 to find the optimal shape of the structure. In this case, the allowable maximum deformation of 20 mm has been defined as the fitness function. The optimization process has been stopped at 100th generation, and the optimal truss beam design has been found shown in Fig. 6 whose end angle is 39.2o. The optimized cross-sectional diameter of the truss beam has been found 2.1 cm. The weight of this optimal beam truss is 225 kg and the maximum deformation touches to the allowable deformation of 20 mm. The optimization process that has been adopted here is a shape and size optimization method based on evolutionary algorithm which searches the optimal shape out of a population of varied fractal-based configurations. Topological optimization is another popular structural optimization method which works around limiting pure shape optimization which, in some cases, results in forms that are loosely resemble with fractal-like figures, and does not offer many shape variations with fine configurations. Besides, in topological optimization, the degree of hierarchy (the parameter of iteration) is not there and controllable, thus restricting the hierarchical outcome to a certain limit. On the contrary, fractal method followed by evolutionary optimization method provides hierarchical level or iteration as a unique geometric parameter that results in finer outcomes with high level hierarchies.

5 Conclusion

The computational morphogenesis and optimization of the fractal-inspired truss beam presented in this paper as a benchmark application shows that a trussed structure having high degree of hierarchical sub-trusses within it is structurally efficient and optimally lightweight. This experiment confirms Mandelbrot's claim about the merit of fractal geometry for developing light-weight and high-strength lattice structures. It also ensures and gives a confidence to explore further applicability of fractal concept in designing efficient and innovative structural shapes. Our next research will emphasize the applicability of fractal method for novel and inventive design possibilities, such as the geometric variations of the macro form of various trusses, in the fields of architecture, civil engineering and creative designs. Thanks to the advancement of computers, the computational modeling system has made it possible to adopt the mathematical concept of fractal geometry for modeling a large range of hierarchical scaling from micro scale of truss configuration to macro scale of space structure, thus giving theoretically an infinite range of scale hierarchies (iteration number can be theoretically zero to towards-infinity). Practically, this opportunity, given by computational modeling system, offers at least a feasible range where a huge-scale trussed structure can be seen as an assemblage of trusses at micro-scale, even possibly at nano-scale too. The recent highly advanced fabrication technologies can make it possible to fabricate such an ‘extra-light’ massive structure. Future research could be dedicated to this possibility of designing and fabricating ‘extra-light’ massive and varied forms of lattice structures.

CAADFutures19 -1007 In this paper, the optimization process considers the cross-sectional sizes of all the members are uniform. However, the ideal process of optimization to find the optimal size of the cross-section of each member is based on its minimum capacity to carry internal distributing force within the allowable limit. In this case, the process will result in different sizes of all the members and they will not be uniform anymore. Therefore, in the future research, some other detailed structural and practical constructional aspects will be considered in order to find the best efficiency of fractal-based structural designs. For the evaluation, this paper has focused on assessing the load capacity behavior of the global model of the fractal-based beam truss. However, for the future research, the load capacity behavior will need to be assessed on member by member in order to understand more detailed picture of the structural behavior of the trusses and their sub- members developed by fractal method. Besides, the future study will also focus on the structural behavior based on other important parameters such as the varying angles throughout the truss.

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