Computational Design and Fabrication of Hanging StructuresBoyu Song

Peking Universityboyusong@pku.edu.cn

Jie FengPeking University

feng_jie@pku.edu.cn

Bingfeng ZhouPeking Universitycczbf@pku.edu.cn

Target shape and constraints

Configuration Optimization

Shape Simulation

Rod Modeling Rod Fabrication Assembled Result

Target

Simulation

Figure 1: An optimal configuration of the hanging structure is found through an iterative optimization, so that the structuremay resemble the target shape. Real model of the structure can also be produced using our 3D fabrication method.

ABSTRACTIn this paper, we present a method for computational design andfabrication of hanging structures. A hanging structure is composedof a series of linked rods, and forms a specific shape under gravity.Given a user-specified shape, our algorithm automatically opti-mizes the configuration of the rods, so that the resulting structurecan resemble the target shape. The optimization is achieved by atwo-phase iterative steps, including shape simulation and configu-ration optimization. Based on our structure designing algorithm,we fabricate the structure with a 3D printer. Hollow cavities andmetal materials are embedded into plastic rods to finely adjust theirweight. A set of simulation and fabrication results demonstrate theability of our method to materialize various target shapes.

CCS CONCEPTS•Computingmethodologies→ Shapemodeling; Physical sim-ulation;

KEYWORDSmodeling, computational geometry, computational design, digitalfabrication, 3D printing

ACM Reference Format:Boyu Song, Jie Feng, and Bingfeng Zhou. 2017. Computational Design andFabrication of Hanging Structures. In Proceedings of SIGGRAPH Asia 2017Technical Briefs, Bangkok, Thailand, November 27–30, 2017 (SA ’17 TechnicalBriefs), 4 pages.https://doi.org/10.1145/3145749.3149429

Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish,to post on servers or to redistribute to lists, requires prior specific permission and/or afee. Request permissions from permissions@acm.org.SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand© 2017 Association for Computing Machinery.ACM ISBN 978-1-4503-5406-6/17/11. . . $15.00https://doi.org/10.1145/3145749.3149429

1 INTRODUCTIONComputational design, along with digital fabrication, can providecustomized products to meet users’ specific requests. A commonapproach is to analyze the physical or geometrical model of thestructure, and produce it with an appropriate solution. Successfulexamples in graphics and engineering include the materializationof 2D or 3D shapes [Garg et al. 2014; Konaković et al. 2016], de-velopable surface approximation [Tang et al. 2016], and geometryprocessing for 3D fabrication [Wang et al. 2013].

In this paper we concentrate on the approximation of a kind offlexible hanging structure, which is composed of a series of linkedrods and has fixed ends. It will deform under the gravity force, untilreaching the equilibrium and forming a specific shape. The shape ofthe structure is determined by its configurations, including physicalparameters and geometry constraints.

Hanging structures are commonly applied in decorations or art-works, such as necklace, chandelier etc. Traditionally, they aredesigned manually and empirically, hence hard to form complexcustomized shapes. In order to realize the automatic design of hang-ing structures, we propose a numerical algorithm to optimize itsconfigurations according to the target shape. During the optimiza-tion, we measure the difference between the current shape andthe target shape, which is used as the criteria of convergence. Theshape corresponding to a certain configuration can be calculatedthrough a simulation algorithm.

In our implementation, we solve the configuration optimizationproblem using an improved particle-swarm optimization (PSO) algo-rithm. The difference between the shapes is introduced to improvethe converging speed of the algorithm.

We apply our method to a number of specific shapes, and fab-ricate them via 3D printing. To adjust the weight of each rod, weadopt the idea of mixed 3D fabrication [Song et al. 2016], and embedhigh-density metal material into the plastic rods. For fine adjust-ments, sphere cavities are distributed evenly inside the rods, usinga blue-noise volume sampling method [Wang et al. 2017].

Overall, in this paper we propose a method that allows users toautomatically design and fabricate customized hanging structure.The main technical contributions include: 1) Through a two-phase

SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand Song, Feng, and Zhou

Figure 2: The configuration of a typical hanging structure.

iterative optimization, a set of 3D-printable, parameterized hang-ing structures can be automatically created; 2) We propose an im-proved PSO algorithm and demonstrate that it perform better inour method. It is applicable to other optimization problems as well.

2 RELATEDWORKFunctional design tools for fabrication optimize 3D models to sat-isfy a specific functional request and then produce them with realmaterials. For example, Shugrina et al. [2015] provide a tool thatallows users to customize parametric models of functional objectswhile keeping them valid for 3D-printing. Konaković et al. [2016]use auxetic cut pattern to materialize 3D surfaces. Similar tools areused to assist the design of assembled structure, e.g. [Bächer et al.2015; Sun et al. 2015]. These approaches are commonly based onmechanical engineering or geometry algorithms for constructing3D-printable shapes.

The development of 3D printers promotes the researches in 3Dfabrication, such as structure analysis, printing material saving,and 3D-printing-oriented geometry design. In order to reduce thematerial cost in 3D printing, the approach of [SHAPEWAYS 2012]uniformly hollows the 3D object by extruding the outer surface andcreating a scaled-down version on its inside. Wang et al.[2013] intro-duce a skin-frame structure to achieve the goal of saving materialwhile maintaining physical stability of the product.

Minimun potential energy principle is widely used in physics,biology and engineering [Altenbach 2008; Reissner 1946]. Mostof the works combine potential energy with stress analyzation.Based on this principle, Santos and Paulo [2011] purpose a methodto analyze hyperelastic perfectly flexible cables, in which finiteelement formulation is used for shape simulation.

3 SYSTEM OVERVIEWGiven a user-specified shape, we aim to automatically design ahanging structure to approximate the target shape. Because of theforce of gravity, the shape of the structure is estimated accordingto a certain mass distribution of rods. When the variable in the con-figurations change, the structure deforms until the total potentialof the structure minimized.

We propose a configuration optimization method for designing acustomized structure. Optimal configuration of the structure is iter-atively searched until the simulated shape optimally resembles thetarget shape within a certain error range. This two-phase designingmethod is shown in Fig.1.

On the basis of the simulated results, a real structure can befabricated according to the optimal configurations we obtained.The shape of the rods is determined by the geometry parameters,while their weights are precisely adjusted by embedding metal ballsand hollow cavities into the volume.

(a) (b)Figure 3: The errors of the approximating shapes. (a)Theproximity error in shape simulation; (b)Two simulatedshapes (red) with the same Euclidean distance to the target(green) and opposite deviation directions.

4 COMPUTATIONAL DESIGN OF HANGINGSTRUCTURE

Given a hanging structure which consists of N linked rods (Fig.2),we define vector ®M , ®G as the mass vector and the geometry vector ofthe structure, whose elementmi stands for themass of ith rod, ®дi forthe geometry parameters, including length, scale factor, curvatureetc. The shape of the hanging structure is represented by the anglesof the rods relative to the horizontal plane. Let ®Θ = (θ1,θ2, ...,θN )

be the vector of the angles, namely, the shape vector. Therefore,our goal is to obtain an appropriate configuration ( ®M , ®G), whichwould result in a shape that approximates the target shape ®Θ in itsequilibrium state.

4.1 Shape SimulationGiven a certain configuration including mass ®M and a parameter-ized geometry ®G of the rods , we first give a method to simulatethe equilibrium state of the hanging structure. The system is op-timized to satisfy: 1) The structure with the optimal shape vector®Θ should minimize the total potential energy; 2) The rods shouldbe connected while both ends of the structure are fixed; 3) Geom-etry constraints should be introduced to avoid rod collisions. Weformulate a physical model to measure the energy of the structure

E( ®Θ) =N∑i=1

miдhi + λ1E1( ®Θ) + λ2E2( ®Θ), (1)

where the first part is the total potential energy of the structure,E1 and E2 are the proximity cost and the collision cost respectively,while λ1 and λ2 are non-negative weights which are set empirically.hi =

∑ij=0 lisinθi indicate the relative height of rod i , here li denotes

the length of the rod i .

4.1.1 Proximity cost. The proximity cost is used to guaranteethat all rods are connected in order, and the ends of the structureare fixed to the target position.We assume that the rods are alwaysconnected at the junctions following the shape vector ®Θ, besidesthe left end of the chain is fixed, the other end will deviate fromits target position(as illustrated in Fig.3(a)). This is because theflexible chain can deform optionally. To force the both ends fixed,we introduce proximity cost, which describes the position distanceof the end point

E1(Θ) = (

n∑i=0

licosθi − ly )2 + (

n∑i=0

lisinθi − lx )2, (2)

where lx and ly stand for the horizontal and the vertical distancebetween the two fixed ends of the target shape, respectively.

Computational Design and Fabrication of Hanging Structures SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand

Figure 4: Without geometry constraint, two adjacent rodswould bump into each other while the angle decreases.

4.1.2 Collison cost. As illustrated in Fig.4, the rotation anglebetween two adjacent rods is in fact limited to a certain range, sincethe boundary is determined by geometry parameters ®дi . Hence, toavoid collision and correctly simulate the shape of the structure,we introduce a collision cost:

E2( ®Θ) =N−1∑i=1

C(ϕi,i+1), (3)

where ϕi,i+1 denote the angle between rod i and rod i +1.C(ϕi,i+1)is a Boolean function, whose value is 1 if collision occurs betweenthe two rods with an angle ϕi,i+1; otherwise its value is 0.

In our implementation, we provide a set of candidate parameter-ized rod models. Therefore we can feasibly modify the outer shapeof the rods by adjusting ®G. Another benefit from this constraint isthat it allows the structure to form some more complex shapes, e.g.concave curves, because the force acting between the adjacent rodsinfluences the system performance. As demonstrated in Fig.9, thisconstraint brings more innovations and possibilities to our design.

4.2 Configuration OptimizationOur designing algorithm aims at minimize the difference betweentarget shape T and simulated shape S ,denoted as D(T , S), whichis estimated by a certain configuration ( ®M, ®G). We therefore useshape simulation as a intermediate process of designing. Based onthe simulated shape, we optimize configurations to minimize thedifference D.

4.2.1 Distance function of shapes. An elementary measurementof the shape difference is the Euclidean distance between two curves

D̃(T , S) =N+1∑i=1

|pT i − pSi |. (4)

wherepT i andpSi are the vertices onT and S , respectively. However,the Euclidean distance cannot fully represent the shape difference.As exemplified in Fig.3(b), the two curves have inverse deviationdirection to the target, while their Euclidean distances are the same.

Thus we introduce another error descriptor ®D to express thedeviation direction of S relative to T . It is a weighted sum of thedeviation vector, i.e. the directed distance between the vertices:

®D(T , S) =N+1∑i=1

(pT i − pSi ) (5)

4.2.2 The improved PSO algorithm. We follow the particle swarmoptimization (PSO) algorithm [Kennedy 2006] to solve our non-linear optimization problem of minD(T , S). The algorithm main-tains a group of candidate solutions, namely particles, and itera-tively move them in a search-space towards the optimal solution.In each iteration, each particle’s movement is influenced by its localbest known position γl and global best known position γд in thesearch-space.

(a) (b)Figure 5: The deviation directions (a) and the new guidepoints for PSO algorithm (b).

In terms of our problem, the guide points γl and γд are deter-mined by D̃. To speed up the convergence, we utilize the devi-ation direction ®D as an additional reference. Since each particleγi (i = 1, ...,m) stands for a certain configuration, it has a corre-sponding shape Γ , and a deviation direction ®D(T , Γ ). As shownin Fig.5(a), if the angle between ®D(T , Γ1) and ®D(T , Γ0) is approxi-mately zero while | ®D(T , Γ1)| > | ®D(T , Γ0)|, we consider γ1 as a worsesolution (shown in red) than the current particle γ0. On the contrary,γ2 is a better solution (in green) than γc , for it has a smaller length.When the direction of ®D(T , Γ3) and ®D(T , Γ0) are opposite, γ3 is alsoa better solution, since the length of ®D(T , Γ0) would decrease ifγ0 moves toward γ3. Hence the particles are encouraged to movetowards a better solution while away from a worse solution. Finally,one better solution γb and one worse solution γw will be selectedfrom all the known particle positions. Then, the moving directionof current particle in the next iteration is the weighted average offour guide directions {−−−→γ0γд , −−→γ0γl ,

−−−→γ0γb ,−−−→γwγ0}. (Fig.5(b)).

5 FABRICATION OF HANGING STRUCTUREAfter generating the optimal configuration ( ®M, ®G) for rods of thehanging structure, we develop a 3D fabrication method to producereal model of the hanging structures.

3D printer can easily fabricate the outer shell of rods accordingto the geometry parameters ®G . As 3D printing always concentrateson reconstructing models rather than mass management, the mostchallenging thing here is to accurately adjust the weights of therods to coincide with the mass configuration ®M .

Since the 3D printer we adopted uses low-density homogeneousmaterials (ABS or PLA plastic), the weight is in proportion to theamount of the material used. Therefore, when producing a light-weight rod, sphere cavities are set inside the rod to reduce its weight.To avoid the offset of its barycenter, the cavities should be dis-tributed evenly in the volume of the rod. Here, we utilize a blue-noise volume sampling method [Wang et al. 2017] to guaranteethe cavity distribution. The radius of the cavities can be furtheradjusted to fine tune the weight. For some heavy-weight rods whichare impossible to realize using only plastic materials, we follow theidea of mixed 3D fabrication [Song et al. 2016], and embed high-density metal material inside the plastic rods. Thus, the variationrange of the rod weight can be expanded.

Combining both methods of embedding hollow cavities andmetal materials, we are able to finely adjust the rod weight in awide range. Fig.6 demonstrates an example of fabricated rod, inwhich metal balls are used to increase the total weight and thecavities are for fine tune.

SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand Song, Feng, and Zhou

Figure 6: An example of mixed-material rods. Metal balls(red) are embedded to increase its total weight, and hollowcavities (green) are used for fine tuning.

(a) Simulated results, colors indicate the weight of the rods

(b) 3D-printed and assembled structuresFigure 7: The simulation and fabrication results of hangingstructure with different target shapes

Figure 8: Simulation and rendering result of a chandeliermodel, composed of multiple hanging structures.

6 EXPERIMENTAL RESULTSIn this section, we show some experiments to demonstrate theeffectiveness of our computational design and fabrication approach.

Fig.7 shows a group of simulation and fabrication examples.Given a set of target shapes (from left to right: a semi-circle, atriangle and a polygon), the simulation of the structures with theseconfigurations are shown in Fig.7(a), where colors changing fromblue to red correspond to weights from light to heavy. Then, usingour fabrication method, 3D-printed rods are assembled and resultin the real hanging structure in Fig.7(b). In this experiment, wemodify the rod weights while keeping their outer shell unchanged.

Fig.8 shows a 3D space application of ourmethod, wheremultiplehanging structures compose a more complex model of chandelier.In Fig.9 we design a heart-shape structure which is benefitted fromthe collision constraint. To form such a special shape, the centralrod (highlighted) is printed as a connected rigid body. Moreover,by replacing part of the rods, the lower part of the structure willdeform accordingly.

7 CONCLUSION AND FUTUREWORKIn this paper we propose an automatic method to design and fab-rication hanging structures with a specific shape. The weight andgeometry configuration of the structure is calculated through aniterative optimization.The resulting structure can be fabricated us-ing metal and plastic materials. Our method is proven effective

Figure 9: Aheart-shaped structurewith differentweight con-figuration.

by experimental results, and may provide an automatic tool forindustrial or artwork designing.

However, the hanging structures under study is of the mostgeneral type, and there are many directions for future study. Here,we mainly concentrate on realizing shapes by 2D chains. To designmore complicated 3D structures, joints with more than two adjacentrods must be taken into account. In another aspect, our methodcurrently considers only the influence of the gravity field. A morecomplex stress field, e.g. a mixture of gravity and wind field wouldprovide more possibilities for shape designing and simulation.

ACKNOWLEDGMENTSThis work is partially supported by National Natural Science Foun-dation of China grants No.61370112 and No.61602012, and the KeyLaboratory of Machine Perception (Ministry of Education), PekingUniversity.

REFERENCESHolm Altenbach. 2008. Book Review: J. N. Reddy, Theory and Analysis of Elastic Plates

and Shells. (2008), 827-827 pages.Moritz Bächer, Stelian Coros, and Bernhard Thomaszewski. 2015. Linkedit: Interactive

linkage editing using symbolic kinematics. Vol. 34. ACM. 99 pages.Akash Garg, Andrew O Sageman-Furnas, Bailin Deng, Yonghao Yue, Eitan Grinspun,

Mark Pauly, and Max Wardetzky. 2014. Wire mesh design. ACM Trans. Graph. 33,4 (2014), 66–1.

James Kennedy. 2006. Swarm intelligence. In Handbook of nature-inspired and innova-tive computing. Springer, 187–219.

Mina Konaković, Keenan Crane, Bailin Deng, Sofien Bouaziz, Daniel Piker, and MarkPauly. 2016. Beyond developable: computational design and fabrication with auxeticmaterials. ACM Transactions on Graphics (TOG) 35, 4 (2016), 89.

Eric Reissner. 1946. Analysis of shear lag in box beams by the principle of minimumpotential energy. Quart. Appl. Math. 4, 3 (1946), 268–278.

HAFA Santos and CI Almeida Paulo. 2011. On a pure complementary energy prin-ciple and a force-based finite element formulation for non-linear elastic cables.International Journal of Non-Linear Mechanics 46, 2 (2011), 395–406.

SHAPEWAYS. 2012. Tutorial: How to hollow objects for 3Dprinting. http://www.shapeways.com/tutorials/reating-hollow-objects.html. (2012).

Maria Shugrina, Ariel Shamir, and Wojciech Matusik. 2015. Fab forms: Customizableobjects for fabrication with validity and geometry caching. ACM Transactions onGraphics (TOG) 34, 4 (2015), 100.

Peng Song, Bailin Deng, Ziqi Wang, Zhichao Dong, Wei Li, Chi-Wing Fu, and LigangLiu. 2016. CofiFab: coarse-to-fine fabrication of large 3D objects. ACM Transactionson Graphics (TOG) 35, 4 (2016), 45.

Timothy Sun, Changxi Zheng, Y Zhang, C Yin, C Zheng, K Zhou, Y Yue, B Smith, CBatty, Z Xu, et al. 2015. Computational design of twisty joints and puzzles. ACMTrans. Graph. 34, 4 (2015), 101–1.

Chengcheng Tang, Pengbo Bo, Johannes Wallner, and Helmut Pottmann. 2016. Inter-active design of developable surfaces. ACM Transactions on Graphics (TOG) 35, 2(2016), 12.

Ke Wang, Jiaojiao Zhao, Jie Feng, and Bingfeng Zhou. 2017. A Three-dimensionalError-diffusion Algorithm for Importance Sampling with Blue-noise Property.. InVISIGRAPP (1: GRAPP). 70–81.

Weiming Wang, Tuanfeng Y Wang, Zhouwang Yang, Ligang Liu, Xin Tong, WeihuaTong, Jiansong Deng, Falai Chen, and Xiuping Liu. 2013. Cost-effective printing of3D objects with skin-frame structures. ACM Transactions on Graphics (TOG) 32, 6(2013), 177.