ALGORITHMS FOR GENERATION OF IRREGULAR SPACEFRAME STRUCTURES

Franz GRUBER, Gunter WALLNER

University of Applied Arts, Vienna

ABSTRACT:Complex space frames with respect to aesthetics and stability are an important factor in contempo-rary architecture. Obviously there are many different ways to generate spatial structures, especiallyif randomness affects the generating process. In this work we present two algorithms to generateirregular space frames inside arbitrary (including non-convex) boundary volumes with predefinedsupport areas. The resulting structures are intended as input for a genetic algorithm which optimizesthe static stability. The first algorithm uses 3D-Voronoi structures as a starting point, which makessense in terms of the frameworks load capacity. The second approach uses a repulsive force field forthe calculation of curve-skeletons of three-dimensional objects.

Keywords: space frames, structures, 3D Voronoi tessellations, skeletonization, vector field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1: A viewing platform supported by aframework of Voronoi paths.

1 INTRODUCTIONThe current methods of structural design are usu-ally based on structures with a high degree of reg-ularity (see, for example, [5] for numerous casestudies). However, there is desire for irregularcomplex space frames in contemporary architec-ture. The goal of the project Algorithmic Gen-eration of Complex Space Frames is to analyzenew and innovative approaches to develop irreg-ular and at the same time effective structures [3].Part of this project was the development of al-gorithms to generate irregular space frames in-side arbitrary (including non-convex) boundaryvolumes with predefined support areas. At thispoint we should stress that it is not our concern

Figure 2: A pavilion were the girders where laidout with the skeletonization algorithm.

to generate structures in regard to statics in thefirst instance. Instead, the structures are intendedas input for a genetic algorithm which optimizesthe static stability (see Hofmann et al. [10] for adescription of an early version of the system).Space frames were independently developedby Alexander Graham Bell, who inventedspace-frames assembled from tetrahedral framesaround 1900 and by Buckminster Fuller, who’sinvestigations five decades later led to the cre-ation of the famous geodesic dome. Nowadays,not only regular but also irregular space framesare becoming increasingly popular in architec-tural design. Notable buildings are for exam-ple the Biosphere 2 in Oracle, Arizona, the Eden

Project in the United Kingdom or the Beijing Na-tional Aquatics Center.Obviously there are many different ways to gen-erate spatial structures, especially if randomnessaffects the generating process. In this work wepresent two algorithms which, after different ex-periments, turned out to be promising. The firstmethod uses 3D Voronoi structures as a start-ing point, which are known to produce stati-cally rigid structures of space-filling tetrahedra[4]. The second method uses a repulsive forcefield for the calculation of the structure and wasinfluenced by the work of Cornea et al. [7]. Fig-ure 1 and Figure 2 show concepts renderings forbuildings where the framework was calculatedwith the two described algorithms.The paper at hand is structured as follows: Sec-tion 2 reviews related work in the area of irreg-ular space frames. In Section 3 the boundingvolume is discussed and Section 4 as well as 5present the two algorithms. The paper is con-cluded in Section 6.

2 RELATED WORKBecause contemporary landmark architecture –as pointed out by [8] – continually moves awayfrom economic considerations toward increasingnumbers of building elements that are uniqueboth to the individual project as well as withinthat particular project, research on irregularstructures has increased over the past years.For example, Kanellos [12] addressed a problemsimilar to ours, where a certain volume has to befilled with a structural space frame network lat-tice consisting of a given number of nodes. Theauthor employs a particle-spring system wherethe connectivity between the particles is not pre-determined but established dynamically. Thesystem uses only local rules of inter-particle in-teraction so that the particles are able to generatecrystal-like lattices through self-organisation.Canzarra [4] also starts with a population ofpoints in space but uses a model which is usedin bone accretion, which mechanisms are rela-tively known and simple and produce structureswith good static stability. As in our case, the au-

thor was not interested in finding optimal solu-tions but instead challenging and creative ones,as pointed out in [4]. A Delauney triangulationis used as starting structure, which is the dual ofthe Voronoi tessellation.Jaworski [11] published a method to grow astructure that supports a building by providinginitial seeds and the volumes to be supported. In-fluenced by the concept of phototropic growth,stems originating at the initial seeds grow ver-tically upwards and avoid obstacles and there-fore entwine existing volumes. During growththe stems are connected to nearby points to en-sure static stability. The resulting structures looksimilar, although denser, than our structures pro-duced by algorithm (1).Fischer [8] proposed a method to create appar-ently irregular structures from relatively smallsets of identical parts, by combining a highly reg-ular space-filling structure with a bottom-up gen-erative procedure.

3 BOUNDARY VOLUMEBecause the space frames have to be generatedinside a given (not necessarily convex) polygonalboundary volume and multiple solutions whichcan be used as input for the genetic algorithmhave to be generated, an efficient representationof the boundary volume is essential. In fact, thedata structure has to allow fast intersections witha ray and tests if a point is inside or outside. Akd-tree [2] is therefore used as spatial data struc-ture to allow fast traversal of the mesh for inter-section tests. Furthermore, it has to be possible todefine certain faces as supporting areas, in otherwords areas where support points of the frame-work can be placed.

4 VORONOI PATHSThe algorithm starts by distributing points insidethe bounding box of the given boundary volumewhich are used as basis for a 3D Voronoi tessel-lation. Afterward the tessellation is cropped atthe boundary volume. Then a given number ofpaths is traced along the cropped tessellation be-tween two points from different supporting areas(Voronoi paths). Finally, the paths are smoothed

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Figure 3: Left: The Voronoi tessellation iscropped at the boundary (red) which serves as

traffic system for the path finding. Right:Boundary points (red and green dots) fromdifferent boundary areas are connected by

individual paths.

Figure 4: 3D Voronoi tessellation cropped at anon-convex boundary surface.

and yield the irregular structure. The viewingplatform in Figure 1 was constructed with thisalgorithm. For the sake of clarity let us explainthe process in two dimensions.

4.1 PreprocessingIn the first step, points are uniformly distributedinside the bounding box of the given boundaryvolume, whereas the density is an important pa-rameter for the fineness of the final framework.These points are used as generating points forthe software package qhull [1] to calculate a 3DVoronoi tessellation. The results are read backand the edges are cropped at the boundary vol-ume. The intersection points are called boundarypoints in the following. These steps are depictedin Figure 3 (left) and a three dimensional exam-ple is shown in Figure 4.

Figure 5: Left: The network from Figure 3 aftera few smoothing steps. Right: A fully

straightened network. In both cases cross points(blue) were fixed at their initial position.

4.2 Voronoi path findingObviously, the cropped tessellation does not meetthe condition that its boundary points are solelylocated at the predefined support areas and froman aesthetical point of view it does not sat-isfy the required irregularity. However, we usethis structure as a kind of traffic system to ex-tract Voronoi paths between two randomly cho-sen support points pA and pB on different supportareas. Finding a path between pA and pB is nota well-defined task, however, in the current im-plementation we try to connect these points ona preferably short way, by using the followingmethod.Starting at position pA the path follows at eachcrossing point pC that adjacent edge which hasthe smallest angle to the target direction pB −pC. This way we obtain a randomized net-work of crossing lines which sometimes run par-tially identical, bifurcate or converge (see Figure3 (right) and Figure 6 (left) for an example in3D). Since points and edges can occur multipletimes, the network is not suitable for the follow-ing force-directed smoothing algorithm whichshould smooth the network as a whole and noteach path separately. Therefore multiple pointsand edges are merged together.

4.3 SmoothingAt this point the network already has the topol-ogy of the final framework. However, the currentnetwork is characterized by zigzag lines whichdo not make much sense neither from a staticalnor from an aesthetical point of view.

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Figure 6: Left: With the help of a random 3D -Voronoi structure we define a Voronoi-Path (red)

between two support points located at the topand bottom (yellow). Right: Complex structurecomposed of many smoothed Voronoi-Paths.

For this reason the network is smoothed itera-tively by replacing edges with elastic springs.For each vertex v a disposition vdisp is storedwhich is set to zero at the beginning of each it-eration. Afterward, the algorithm loops througheach edge with incident vertices v1 and v2 andadds ε ⋅d0 to vdisp

1 , respectively subtracts it fromvdisp

2 , where d0 = (vpos2 − vpos

1 )/∥∥vpos

2 −vpos1

∥∥and ε is a small constant. At the end of eachiteration the disposition vdisp of each vertex isadded to its position vpos. For boundary pointsthe last step is omitted to keep their initial po-sitions fixed. Conventional space frame struc-tures feature completely straight girders and canbe best achieved by additionally locking the po-sition of cross points.In general, the higher the number of iterations,the more straight the network will become. Fig-ure 5 compares a slightly smoothed with a fullystraightened network. Further three dimensionalexamples are shown in Figure 6 (right) and Fig-ure 7.

5 SKELETONIZATIONThe second approach follows the work of Corneaet al. [7], who use a repulsive force fieldfor the calculation of curve-skeletons of three-dimensional objects. Although the connectionto architecture might not seem obvious at first,because their research was originally targetedto areas like virtual colonoscopy or animation,we found it appropriate for our purposes. The

Figure 7: Left: Voronoi paths from the red to thegreen support area. Right: The fully straightenedstructure after smoothing with fixed cross points.

method uses a generalized potential field [6] togenerate a discretized vector field inside an ob-ject by charging the object’s boundary.

5.1 OutlineThe algorithm starts by distributing point chargeson the surfaces of the boundary volume. Af-terward a discretized vector field is calculatedwithin the surface’s bounding box. Althoughthis would not be strictly necessary, it acceleratesthe numerical integration later in the process andeases the computation of critical points. Once thevector field is established, the particle trajecto-ries, starting from the supporting areas toward acritical point, are calculated with numerical inte-gration. Because these paths are running partiallyparallel they are merged together to avoid unaes-thetic clutter. Finally, cross-links depending onan angle-threshold are inserted into the existingspace frame structure. Figure 8 illustrates thesesteps.

5.2 PreprocessingAfter the boundary volume has been defined npoints p1...pn with charges q1...qn are placed ina small distance orthogonal to the surfaces, asopposed to Cornea et al. [7] who place themat the center of cells which are intersected bythe volume. This is necessary in our case be-cause the particles start at the boundary surfacesand should not move outside the boundary vol-ume right away. The point charges are placed atthe vertices of the the bounding volume and onthe center between the barycenter vbc and eachtriangle vertex v1,v2,v3. The placement is re-

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Figure 8: Left: After point charges (circles with arrows) have been uniformly distributed on theobject’s boundary, a discretized vector field (shown schematically as gray grid) is derived which at

least contains one critical point (red circles). Right: The trajectories of particles (green) whichoriginate on the object’s boundary are the core of the final framework. Separate components are

joined by straight lines (blue, dot-dashed) which connect the two closest crossings (blue circles) toensure that the entire structure is a coherent whole. Cross-links (orange, dashed) are inserted to

ensure better stability.

peated recursively for each triangle (vbc,v1,v2),(vbc,v2,v3) and (vbc,v3,v1) until a given subdivi-sion level is reached.Based on the bounding box of the volume adiscretized vector field is constructed where thenumber of division in each direction depends onthe associated side length. At each cell center avector

v = c ⋅n

∑i=0

(qi

∥di∥m

)⋅di (1)

is calculated, where di is the vector pointing fromthe cell center to the position of point charge i. mis a quantity for the descent of the vector fieldand c a small constant.In a discretized vector field a critical point mayonly occur in cells where all three components ofv pass through 0, which in case of trilinear in-terpolation can be found with a simple heuristicas described in [9]. If for each component of theforce vector at each cell vertex both negative andpositive values exist then the components mustchange sign somewhere inside the cell and thecell is a potential candidate for containing a criti-cal point. If the condition is fulfilled then the cellis recursively subdivided and the test is repeatedfor each sub-cell until either the test fails or amaximum number of subdivisions is reached. Aspointed out by Globus et al. [9] this is only anecessary condition and the cell must not con-

tain a critical point. They therefore use Newton’smethod to better estimate the location of the crit-ical point after a fixed number of subdivisions.However, in our case the exact location of thecritical point is not necessary and incorrect clas-sifications do not interfere with the correct work-ing of the algorithm. Locating the critical pointsis the most time consuming step in the prepro-cessing step. However, the pre-process must onlybe performed once and does not need to be re-peated to generate different space frames for agiven boundary volume. Figure 8 (left) showsthe status of the system after the pre-process isfinished.

5.3 Particle TrajectoriesOnce the pre-process has finished, paths throughthe vector field are traced which build the foun-dation of the final framework. This process startsby placing particles randomly on the supportingareas. For each particle the trajectory is calcu-lated by explicit Euler integration1. Because allparticle trajectories have to end at a critical point(or to be more specific at an attracting node),and there must be at least one critical point inthe closed boundary volume, the integration is

1More precise integration schemes, like Runge Kutta 4thorder integration can also be used but the additional ef-fort may not be necessary because the accurate path isnot required for the matter in hand.

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Figure 9: Top: Calculating each particletrajectory independently from each other results

in clutter, since paths frequently run partiallyparallel to each other. Middle: The sameexample after merging the paths during

integration which circumvents the cluttering.Bottom: The result after the components have

been connected and the paths have beensmoothed. The red circles show two areas where

unaesthetic spikes have been removed by thesmoothing algorithm.

aborted if the last position is in proximity of sucha critical point. This point is added as the lastpoint to the particle trail. These paths are shownin green in Figure 8 (right).Because the time step ∆t for numerical integra-tion is usually relatively small it is not practicalto add each position to the final path. Thereforea minimum distance εd > ∆t between two pointsmust be fulfilled.As shown in Figure 9 (top) these paths run fre-

quently in parallel which makes the result lookcluttered and not suitable for a space frame struc-ture. To circumvent this problem the integra-tion stops if the current location is in proxim-ity of an already existing position, which even-tually becomes the final point of the current path.This is implemented with a simple space parti-tioning scheme which divides the bounding vol-ume into small cuboids. Each time a new po-sition p is added to the path it is inserted intothe appropriate cuboid c by mapping its coordi-nates to indices. Then the distances between pand all other positions of different paths in c arecalculated and p will be connected with the po-sition which is closest to it. Positions on whichmultiple paths converge will be called crossingshenceforth and critical points are automaticallyconsidered as crossings regardless of the numberof incident edges. The merged paths are shownin Figure 9 (middle).

5.4 PostprocessingDepending on the vector field, the emergingstructure may consist of multiple non-connectedcomponents. These are linked to each otherby connecting the two closest crossings betweenthem. Once a single component exists thesmoothing algorithm as described in Section 4.3is applied. Figure 9 (bottom) shows the result af-ter the components are connected and the pathshad been smoothed. Afterward cross-links –which are shown as orange slashed lines in Fig-ure 8 (right) – are added to the structure as fol-lows.First, for each crossing c and every pair of ad-jacent branches with vertices (u0 = c,u1, ...,um)and (v0 = c,v1, ...,vn) a cross-link is added be-tween ui and vi provided that none of the follow-ing conditions is fulfilled:

1. the angle α=∠(−−−→ui−1ui,−−−→vi−1vi) is larger than

a definable threshold αmax2. either ui or vi is a crossing or a boundary

point3. the distance between ui and vi is larger than

a definable maximal length lmax4. a cross-link has already been added between

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ui and vi5. m > n or n > m

The process is then repeated recursively betweenui+1 and vi+1

2 until one of the above mentionedconditions is violated.Although the basic appearance depends on thevector field which in turn depends mostly on thegeometry of the bounding volume the results canbe altered to a certain degree by changing thenumber and starting position of the particles aswell as the parameters εd , αmax and lmax and thesize of the cuboids (used for merging). Figure 2shows a pavilion which was constructed with thisalgorithm.

6 CONCLUSIONSAmong the infinite number of possibilities ofgenerating spatial structures inside a given vol-ume, we described two methods: one is basedon Voronoi tessellation and the other on repulsiveforce fields.Regarding the former we currently use a uni-formly distributed point cloud to generate theVoronoi tessellation. For future work, the den-sity of the point cloud could depend on geomet-ric properties of the boundary volume. For ex-ample, the density could be higher in critical ar-eas inside the volume, like constrictions or bot-tlenecks. Furthermore, different topologies canbe generated by replacing the Voronoi tessella-tion with alternative patterns like a regular grid.The main disadvantage of the latter method isthat the resulting structures – despite randomlychoosing the support points – look quite similarbecause the underlying vector field is defined bythe bounding volume. If the vector field leads tostatically unfeasible space frame structures thenaltering the support points will not have much ef-fect and one cannot expect that the genetic algo-rithm will create a good solution. For example, asimple box only has one critical point and all par-ticle trails will converge to this point. Therefore,

2ui and vi can each only have one successor because oth-erwise they would either be a crossing (two or more)or a boundary point (one) which would terminate theprocess.

the vector field should be disturbed by placingpoint charges randomly inside the volume whichwill influence the number and location of the crit-ical points.

ACKNOWLEDGMENTSThis work was supported by grant L358 of theAustrian Science Foundation (FWF).

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ABOUT THE AUTHORS1. Franz Gruber studied Technical Mathematics

at the Johannes Kepler University in Linz.

He is currently working as Research Scien-tist at the University of Applied Arts in Vi-enna, where he did his dissertation about theApplication of classical geometric methodsin architecture and computational aesthetics.His main interests are the numerical devel-opment of algorithms in geometric applica-tions and surface design. He can be reachedby e-mail: franz.gruber@uni-ak.ac.at.His web site is: www1.uni-ak.ac.at/geom/staff_fg.php

2. Gunter Wallner received his Diploma degreein Computer Science from the Technical Uni-versity Vienna in 2005. He is currently a Re-search Assistant at the Department of Geom-etry (Institute for Art and Technology) at theUniversity of Applied Arts Vienna where hefinished his doctoral thesis about GPU En-hanced Algorithms for Radiosity and ShadowVolume Rendering in 2009. His research in-terests include global illumination, photore-alistic rendering, GPU programming as wellas procedural methods in computer graph-ics. He can be reached by e-mail: wallner.guenter@uni-ak.ac.at

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