Decomposing Three-DimensionalShapes into Self-supporting,Discrete-Element Assemblies

Ursula Frick, Tom Van Mele and Philippe Block

AbstractThis study investigates a computational design approach to generatevolumetric decompositions of given, arbitrary, three-dimensional shapesinto self supporting, discrete-element assemblies. These assemblies arestructures formed by individual units that remain in equilibrium solely as aresult of compressive and frictional contact forces between the elements.This paper presents a prototypical implementation of a decomposition toolinto a CAD software, focusing on user-controlled design to generate suchassemblies. The implementation provides an interactive design environ-ment including real time visual feedback, in which the design space ofself-supporting block assemblies can be explored and expanded. Somesurprising results of such explorations are included and discussed.

Introduction

Volumetric decomposition as a means to reduceelement size in assemblies is relevant to thebuilding industry because it simplifies fabricationand transport. The connections between theindividual units needed to establish equilibriumof the assembly are often problematic, materialintensive or complicated. Especially tensile,mechanical connections often result in compli-cated detailing and can be expensive and intru-

sive. Glued connections are mainly simple, buttypically difficult to adjust or remove.

This research presents a prototypical decom-position tool as a means to design volumetricdecomposition of three-dimensional shapes intoself-supporting, discrete-element assemblies. Thegenerated structures, formed by individual, dis-joint units, rely solely on spatial compressionflows (arching), friction, balancing, and anycombination of these actions to stand in equi-librium (Fig. 1). This means they are stablewithout additional mechanical or physical joinerybetween the blocks, which keeps the connectionssimple and adjustable.

Considering the structural integrity of suchassemblies, the defining principles that must beevaluated are the assembly’s overall stability and

U. Frick (&) � T.V. Mele � P. BlockBlock Research Group, ETH Zurich Institute ofTechnology in Architecture, Zurich, Switzerlande-mail: frick@arch.ethz.ch

© Springer International Publishing Switzerland 2015M.R. Thomsen et al. (eds.), Modelling Behaviour, DOI 10.1007/978-3-319-24208-8_16

187

Fig. 1 Photographs of a discrete element assembly in equilibrium as a result of arching, friction, and balancing, andwithout mechanical connections or glue

188 U. Frick et al.

material failure at the scale of the individual unit.As stability is predominantly an issue of geom-etry and not of internal stress distributions,methods to address geometrical stability areembedded in the presented computationalsetup. Material failure of the discrete elements isnot considered at this stage of the research.

Related Work

Discretization of architectural geometries anddesign of self-supporting structures are ongoingand active areas of research. Both topics play animportant role in architectural design and asso-ciated fields and have a strong influence onmanufacturing, assembly and building cost.

In recent years, various innovative computa-tional techniques for topological andsurface-based discretization of architecturalgeometries have been developed, such as Pott-mann et al. (2008) and Eigensatz et al. (2010).However, few of these techniques consider struc-tural and/or assembly constraints e.g. Rippmannet al. (2013), Panozzo et al. (2013) and Deuss et al.(2014). Furthermore, in most cases topologicaland surface based discretization approaches arenot applicable to volumetric shapes.

Most existing volumetric discretizationapproaches, such as methods to decompose sol-ids into parts optimized for layered fabrication(Hu et al. 2014), do not consider interactions thatkeep disjoint assemblies in equilibrium. Whitinget al. (2009), Whiting (2012) presented anapproach for generating structurally soundmasonry assemblies by refining coarse volumet-ric models with known/typical structural ele-ments such as walls, arches, domes, etc.

Objectives and Outline

The presented approach focuses on interactivedecomposition of given arbitrary shapes thatwould naturally not be considered as suitable

shapes for self-supporting, discrete-elementstructures. By providing real-time visual feed-back, it allows exploring and extending thedesign space of such assemblies. From this designperspective, the proposed approach offers ameans of creating surprising equilibrium assem-blies that go beyond the scope of known struc-turally sound configurations for unreinforcedmasonry and other discrete-element structures.

“Decomposition process” gives an overviewof the decomposition process, the used structuralanalysis method and the computational imple-mentation. In “Results”, the results of explora-tions are presented with three case studies. Thestability of the generated digital models are val-idated with 3D-printed physical models andillustrated with photographs. “Conclusion” dis-cusses the presented approach and gives an out-look to future work.

Decomposition Process

This section gives an overview of a prototypicalimplementation of a decomposition tool in Rhi-noceros (2014). Grasshopper (2009) was used tobuild up an interactive design environment.Equilibrium calculations were written in Python(2015) and solved with quadratic programming.

The decomposition process starts with aninitial geometry, which is refined step by stepuntil a satisfactory result is obtained. After everyuser-controlled refinement, interfaces betweenblocks, and between blocks and the surroundingsare detected automatically. At every step, thediscretization of the geometry can be changed orupdated, whereafter no-tension equilibrium hasto be (re-)established. An equilibrium solutioncan be found by changing the boundary condi-tions, modifying the location and orientation ofthe interfaces, changing the material properties,or any combination of these options. Providingintuitive, visual feedback on the current state ofthe model is clearly essential during this process.

Decomposing Three-Dimensional Shapes into Self-supporting… 189

Equilibrium Calculation

The equilibrium calculations are based on themethod described in Whiting (2012) and Whitinget al. (2009, 2012).This method extends the RigidBlock Limit Equilibrium Analysis method byLivesley (1978, 1992) by including penalty forcesto allow for configurations of discrete-elementassemblies in which tension is required. Thisstructural analysis method enables computingestimates of the occurring forces in a givenstructure that satisfy the equilibrium equations,including friction constraints, with quadraticprogramming.

Here, we briefly summarize the essentialequations of the optimization problem. The staticequilibrium equations (Whiting et al. 2009) canbe set up in matrix form as follows:

Aeq � cþ b ¼ 0

The matrix Aeq contains the sub-matrices Aj,kof size 6 × 4 vk, with vk the number of vertices ofinterface k, representing the (global)xyz-components of the force and moment

interactions between block j and interface k in thelocal coordinate system ðn̂k; ûk; v̂kÞ of interfacek (Fig. 2):

Aj;k � ck þ bj ¼ 0

which expands to:

fkxfkyfkzm1j;kxm1j;kym1j;kz

fkxfkyfkzm2j;kxm2j;kym2j;kz

. . .

fkxfkyfkzmvj;kxmvj;kymvj;kz

26666664

37777775�

c1kc2k...

cvkk

26664

37775þ

FjxFjyFjzMjxMjyMjz

2666664

3777775

¼ 0

The sub-vectors fkx, fky, and fkz of Aj,k containthe xyz-components of the local coordinate sys-tem of interface k.

fkx ¼ ðn̂kÞx; �ðn̂kÞx; ðûkÞx; ðv̂kÞx� �

The sub-vectors mij;kx ; mij;ky

and mij;kz containthe xyz-components of the moment contributionsof the interface forces of vertex i of interface k,

Fig. 2 Diagram of the interaction between block j and interface k

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acting on block j. The vector rij; k defines therelative position of vertex i with respect to themass centroid of block j.

mij; kx

¼ ðrij; k � n̂kÞx; � ðrij; k � n̂kÞx;ðrij; k � ûkÞx;ðrij; k � v̂kÞxh i

The 4vk × 1 vector ck, with vk the number ofvertices of interface k, contains the unknownnormal and in-plane force coefficients (signedforce magnitudes) for all vertices of interface k,including the penalty formulation as described inWhiting et al. (2009), Whiting (2012).

cnormal ¼ cþn � c�ncfriction ¼ cu þ cv ) c

ik ¼

ciþknci�kncikucikv

2664

3775

The 6 × 1 sub-vector bj contains thexyz-components of the body forces applied to thecentroid of block j.

To improve stability of the calculation, aneight-sided friction pyramid (Livesley 1992)rather than the four-sided one used in Whiting(2012) has been used for friction constraints (µ isthe static friction coefficient of the used material).

cuj j; cvj j; 1ffiffiffi2

p cuj j þ 1ffiffiffi2

p cvj j� �

� l � ccþn

The energy function, as described in Whiting(2012), has then been used to calculate theinterface-forces.

minimizec

f cð Þ such thatAeq � c ¼ �bcnormal � 0cfriction � l � cþn

8<:

This structural analysis method enables thecalculation of infeasible self-supporting structures.Infeasible self-supporting structures are those forwhich the no-tension equilibrium is violated. Incomparison, stability simulations with physics

engines, e.g. Bullet-Physics-Library (2012) andNvidia physx library (2013), would typically resultin a Yes/No answer. Furthermore, the equilibriumequations of the used method can be solved rea-sonably prompt with quadratic programming. Bothof these points makes the approach particularlyadequate to be implemented into a computationalsetup with emphasis on interactivity.

Interface-Force Diagrams

The models are visualized using interface-forcediagrams, in which contact interfaces are repre-sented by coloured surfaces. The colours providedifferent information about the forces at theinterfaces depending on the selected feedbackmode.

In compression-tension mode, blue indicatescompression, and red tension. Colour gradientsindicate variations in the distribution of forcesover the interface and interfaces without com-pression are grey. Note, that the colour gradientsrepresent the force distribution normalized perface and do not reflect the force magnitudes.Relative contact-force magnitudes are visualizedby switching to vector mode.

In friction mode, interfaces without frictionare also grey. Interfaces with friction have solidcolours between yellow and red. All frictionforces below a user-defined threshold are illus-trated in yellow. Red indicates that the frictionforce exceeds the allowed maximum. Occurringfriction forces between those bounds are illus-trated with orange shades. For example, darkorange indicates that the resulting friction force isclose to the allowed maximum.

For easier understanding of the assembly’sequilibrium, mass-center locations are displayedand additional information of contact-forcemagnitudes in text form are optional. Figures 3,4 and 5 illustrate the gradient and vector visual-izations modes and the influence ofuser-controlled modifications.

Decomposing Three-Dimensional Shapes into Self-supporting… 191

Feedback and EquilibriumModification

Figure 3 depicts different configurations of asolid object in the shape of the letters SC. Theinterfaces are shown in compression-tensionmode. In Fig. 3a, the notension constraint isviolated at the support interfaces. The left sup-port is entirely in tension. At the right supportinterface, both tension and compression forcesoccur simultaneously. This equilibrium is theresult of the location of the center of mass of theshape in relation to the supports. No-tensionequilibrium can be established by adding anadditional support interface (Fig. 3b), or bychanging the overall geometry (Fig. 3c).

Figures 4 and 5 depict different configurationsof a discretization of the letter “M”. Figure 4shows the interfaces in compression-tensionmode. In configuration a, no-tension equilib-rium is violated due to the need for tension forcesat the internal interface (between the top andbottom element). In configurations b and c, theviolation is resolved by changing the location ofthe cut. The colour gradient in Fig. 4b indicatesthat, at this location, the compression forces areunevenly distributed over the interface. Movingthe cut further down results in a more even

distribution and thus a more robust equilibrium(Fig. 4c). Note that in all cases compressionforces at the support interface are evenly dis-tributed, as these forces depend only on globalequilibrium, which remains unaltered in the threecases. Figure 5 shows the interfaces of a slightlydifferent discretization of the M-shaped object infriction mode. In configuration a, the orientationof the cut is such that maximum friction isexceeded. By rotating the cut in configurations band c, friction is reduced to allowable levels.Friction is lowest in configuration c. Note that innone of the configurations friction occurs at thesupport interface, since the applied loads arevertical and there is no arch action.

Results

This section presents the results of explorationsof the no-tension discrete-element assemblydesign space with the decomposition tool dis-cussed in “Decomposition process”. The equi-librium of the generated digital models has beenverified with physical models. The physicalmodels were 3D printed with a ZCORP ZPrinter650, using a composite of zp150 powder and

Fig. 3 Different configurations of a solid object in the shape of the letters SC. a Violation of no-tension equilibrium atthe support. No-tension equilibrium established by (b) adding a support interface, or c changing the geometry

192 U. Frick et al.

Fig. 4 Tension forces at the internal interfaces (a) can be removed by changing the location of the interface (b). Morerobust solutions are recognized by more uniform colouring of the interfaces (c)

Fig. 5 Violation of friction limitations at an internalinterface (a) can be resolved by, for example, rotation ofthat interface (b). The amount of friction is indicated by a

colour ranging from yellow to red. Red indicates thatmaximum friction is exceeded (a). Yellow indicates thatfriction is below a user-defined threshold (c)

Decomposing Three-Dimensional Shapes into Self-supporting… 193

zb61 clear binder and impregnated with Z-Bond101. The density of the composite material hasbeen approximated with 0.60 g/cm3, and thefrictional angle of 40° (Van Mele et al. 2012). Allinterfaces have been modeled flat and planar inthe computational and physical models, to notchange the frictional behaviour between them.This means no male-female interlocking mecha-nisms have been used, for example to assistduring the assembly process.

Case Study One

Figure 6 shows two different decompositions ofthe DMSC acronym of the symposium. In bothcases, we started from one solid geometry, sepa-rated the letters where possible, and then furtherdiscretized the letters into smaller pieces. All cutswere positioned and adjusted manually, based onthe visual feedback of the equilibrium calcula-tions. Figure 6b, c depict the tension-compressionand frictional contact forces at all interfaces. Bothassemblies require compressive and frictionalforces for equilibrium. Friction does not occur atthe support interfaces.

A physical test with a 3D-printed modeldemonstrates the designed assembly is indeedstable by itself without any mechanical connec-tions or glue (Fig. 1).

Note that the process of assembling the modelwas difficult, because interim stability duringassembly was not considered in the design of thedecomposition. In fact, equilibrium could not beachieved for the combination of “D” and “M”alone, without the additional weight of the letters“S” and “C”.

Case Study Two

This case study demonstrates the potential of theproposed interactive procedure to explore avariety of equilibrium solutions for the samegiven initial shape (Fig. 7). As can be seen inFig. 7a, the initial shape for this case study is athreedimensional, kinked loop positioned on a

horizontal plane. The initial geometry is inequilibrium, with evenly distributed compressionforces at the support.

In Fig. 7b, the object has been partitioned intotwo L-shaped and two cuboid geometries by fourhorizontal cuts. The correspondinginterface-force diagrams illustrate that the therebygenerated assembly is stable. All interface forcesare compressive and vertical. No friction isrequired to establish equilibrium.

Even after further decomposition with verticalcuts through the L-shaped elements, the assem-bly remains self-supporting (Fig. 7c). All inter-face forces are still compressive and vertical.Therefore, as before, no friction is required.Furthermore, there is no force interaction on thetwo vertical interfaces. The assembly could thusbe separated into two independent parts andremain stable.

A completely different decomposition isachieved with inclined cuts, as seen in Fig. 7d. Interms of force transfer, this configuration is moreinteresting. It requires both compressive andfriction forces to be in equilibrium, because archaction is activated by the orientation of the cuts.

The physical models in Fig. 8 demonstratethat the designed assemblies of this case studyare indeed stable by themselves.

Case Study Three

Figure 9 shows the outcome of a completelydifferent decomposition strategy. Starting with asimple box with an open bottom, a decomposi-tion pattern has been applied on the outer boxsurface. From that cutting pattern (Fig. 9a),interface geometries with the potential toself-interlock in an assembly have been gener-ated. This has been achieved through extrusionof the cutting pattern to a single point. Therefore,the resulting interfaces are planar and conicaldirected to that point. The contact-force diagramsin Fig. 9a illustrate that the assembly isself-supporting. All contact forces are eithercompressive or frictional and the maximumfriction is not exceeded.

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Fig. 6 Two differentdecompositions of theDMSC acronym (left), withthe correspondinginterface-force diagrams incompression-tension mode(middle) and friction mode(right)

Decomposing Three-Dimensional Shapes into Self-supporting… 195

Furthermore, even after removing severalparts of the assembly, as seen in Fig. 9b, theresulting configuration remains in self-supporting

equilibrium. Again, the physical models inFig. 10 demonstrate the self-supporting equilib-rium of the designed decompositions.

Fig. 7 Variousequilibrium solutions forthe same initial geometry(left), with thecorrespondinginterface-force diagrams incompression-tension mode(middle) and friction mode(right)

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Fig. 8 Photographs of two different decompositions from the same initial geometry

Decomposing Three-Dimensional Shapes into Self-supporting… 197

Fig. 9 Resulting decomposition of case study three before (top a) and after removing several parts of the assembly (topb), with the corresponding interface-force diagrams in compression-tension mode (middle) and friction mode (bottom)

198 U. Frick et al.

Fig. 10 Photographs of case study three before (top) and after removing parts from the assembly (bottom)

Decomposing Three-Dimensional Shapes into Self-supporting… 199

Conclusion

In practice, volumetric decomposition intoself-supporting assemblies has applications inareas ranging from the construction of large-scale,cut-stone masonry structures (Howeler et al.2014) and prefabricated building assemblies, tosmaller-scale prototyping. When applied to thedesign and production of prefabricated elementsin the building industry, optimization of joints canbe achieved as a means of limiting joinery topredominantly no-tension force transfer.

The prototypical decomposition tool pre-sented in this paper is based on realtime, clear,visual feedback. The advantage of such aninteractive process has been demonstratedthrough the surprising results shown in “Results”.As can be seen in those case studies, thestep-by-step, interactive discretization processhas successfully been used to explore the designspace of discrete-element assemblies and exper-iment with previously unseen forms.

Despite the promising results shown in“Results”, the interactive decomposition proce-dure can be time consuming and the handling oflarge models can be problematic. Furtherresearch will be necessary to develop general,(semi-)automated decomposition strategies.A digital self-supporting equilibrium solutiondoes not automatically guarantee a physicalself-supporting application. As explained inprevious research (Whiting 2012), an alternativeequilibrium state could exits where friction con-straints are violated. Furthermore, the imple-mented algorithms do not consider imperfectionsof element’s geometry or assembly.

This research is part of a larger research pro-ject, which aims to develop better understandingand novel techniques for the equilibrium designof masonry and other discrete-element assem-blies. Near-future goals related to the researchpresented are the implementation of additionalmethods for the equilibrium design, such asmanipulation of weight distribution within thevolume (Bacher et al. 2014), and the develop-ment of general, automatic decomposition strat-egies. Furthermore, the research will focus on the

structural integrity and robustness by consideringpossible material failure at unit scale and interimstability during assembly.

Acknowledgments This research was supported by theNCCR Digital Fabrication, funded by the Swiss NationalScience Foundation (NCCR Digital Fabrication Agree-ment # 51NF40-141853).

References

Bächer M, Whiting E, Bickel B, Sorkine-Hornung O(2014) Spin-it: optimizing moment of inertia forspinnable objects. ACM Trans Graph 33(4):96:1–96:10

Bullet-Physics-Libary (Copyright (c) 2012) Bullet colli-sion detection and physics library. http://bulletphysics.org

Deuss M, Panozzo D, Whiting E, Liu Y, Block P,Hornung-Sorkine O, Pauly M (2014) Assemblingself-supporting structures. ACM Trans Graph 33(6):214:1–214:10

EigensatzM, KilianM, Schiftner A,Mitra NJ, PottmannH,Pauly M (2010) Paneling architectural freeform sur-faces. ACM Trans Graph 29(4):45:1–45:10

Grasshopper (Copyright 2009) Grasshopper—algorithmicmodeling for rhino version 27 August 2014. http://www.grasshopper3d.com

Höweler E, Yoon JM, Ochsendorf J, Block P, DeJong M(2014) Material computation—the collier memorialdesign using analog and digital tools. In: Gerber DJ,Ibaez M (eds) Paradigms in computing—making,machines and models for design agency in architec-ture, chapter 0. eVolo Press, Los Angeles

Hu R, Li H, Zhang H, Cohen-Or D (2014) Approximatepyramidal shape decomposition. ACM Trans Graph 33(6):213:1–213:12

Livesley RK (1978) Limit analysis of structures formedfrom rigid blocks. Int J Numer Meth Eng 12:1853–1871

Livesley RK (1992) A computational model for the limitanalysis of threedimensional masonry structures.Meccanica, 27(3):161–172. Nvidia-PhysX (Copyright2013)

Nvidia physx library (2013). http://www.nvidia.com/object/physx-9.12.0213-driver.html

Panozzo D, Block P, Sorkine-Hornung O (2013) Design-ing unreinforced masonry models. ACM TransGraph 32(4):91:1–91:12

Pottmann H, Schiftner A, Bo P, Schmiedhofer H,Wang W, Baldassini N, Wallner J (2008) Freeformsurfaces from single curved panels. ACM TransGraph 27(3):76:1–76:10

Python (Copyright 2001–2015) Python programminglanguage. https://www.python.org

200 U. Frick et al.

http://bulletphysics.orghttp://bulletphysics.orghttp://www.grasshopper3d.comhttp://www.grasshopper3d.comhttp://www.nvidia.com/object/physx-9.12.0213-driver.htmlhttp://www.nvidia.com/object/physx-9.12.0213-driver.htmlhttps://www.python.org

Rhinoceros (Copyright 1993–2014) Rhinoceros modelingtools for designers, version 5. https://www.rhino3d.com

Rippmann M, Curry J, Escobedo D, Block P (2013)Optimising stonecutting strategies for freeformmasonry vaults. In: Proceedings of the internationalassociation for shell and spatial structures (IASS)symposium 2013. Wroclaw, Poland

Van Mele T, McInerney J, DeJong M, Block P (2012)Physical and computational discrete modeling ofmasonry vault collapse. In: Proceedings of the 8thinternational conference on structural analysis ofhistorical constructions. Wroclaw, Poland

Whiting E, Ochsendorf J, Durand F (2009) Proceduralmodeling of structurally-sound masonry buildings.ACM Trans Graph 28(5):112

Whiting E, Shin H, Wang R, Ochsendorf J, Durand F(2012) Structural optimization of 3d masonry build-ings. ACM Trans Graph 31(6):159:1–159:11

Whiting EJW (2012) Design of structurally-soundmasonry buildings using 3d static analysis. PhDthesis, Department of Architecture, MassachusettsInstitute of Technology

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https://www.rhino3d.com

16 Decomposing Three-Dimensional Shapes into Self-supporting, Discrete-Element AssembliesAbstractIntroductionRelated WorkObjectives and Outline

Decomposition ProcessEquilibrium CalculationInterface-Force DiagramsFeedback and Equilibrium Modification

ResultsCase Study OneCase Study TwoCase Study Three

ConclusionAcknowledgments