form-finding of grid-shells using the ground structure and...

Struct Multidisc Optim (2018) 57:1187–1211

RESEARCH PAPER

Form-finding of grid-shells using the ground structureand potential energy methods: a comparative studyand assessment

Yang Jiang1 ·Tomas Zegard2 ·William F. Baker3 ·Glaucio H. Paulino1

Received: 8 May 2017 / Revised: 26 August 2017 / Accepted: 31 August 2017 / Published online: 8 November 2017© Springer-Verlag GmbH Germany 2017

Abstract The structural performance of a grid-shelldepends directly on the geometry of the design. Form-finding methods, which are typically based on the search forbending-free configurations, aid in achieving structurallyefficient geometries. This manuscript proposes two form-finding methods for grid-shells: one method is the potentialenergy method, which finds the form in equilibrium byminimizing the total potential energy in the system; thesecond method is based on an augmented version of theground structure method, in which the load applicationpoints become variables of the topology optimization prob-lem. The proposed methods, together with the well-knownforce density method, are evaluated and compared usingnumerical examples. The advantages and drawbacks of themethods are reviewed, compared and highlighted.

Keywords Form-finding · Grid-shells · Potential energymethod · Ground structure method

Yang Jiang and Tomas Zegard equally contributed.

� Glaucio H. [email protected]

1 School of Civil & Environmental Engineering,Georgia Institute of Technology, 790 Atlantic Drive,Atlanta, GA 30332, USA

2 Pontificia Universidad Catolica de Chile, Vicuna Mackenna4860, Santiago, RM 7820436, Chile

3 Skidmore, Owings & Merrill LLP, 224 S Michigan Ave,Chicago, IL 60604, USA

1 Introduction

The present work introduces two form-finding methodsfor grid-shells: one method based on the minimization ofpotential energy, dubbed as the potential energy method(PEM); and a second method based on an augmented formof the ground structure method (GSM) (Dorn et al. 1964).The main advantage of the potential energy method is thereliance on all equations of mechanics (kinematics, equilib-rium, and constitutive relations), all of which have directphysical interpretation. On the other hand, the ground struc-ture method approximates the absolute optimal shape (withno dependence on additional user-supplied parameters).

From the aforementioned, the PEM can be consideredas a stiffness-type method, since it not only considers forceequilibrium, but the constitutive and compatibility equationsas well. However, it does not make direct use of the stiff-ness matrix: instead, it minimizes the potential energy of thesystem considering large deformations. It is therefore akinto the experimental approach originally used by Gaudı andIsler (Bergos and Llimargas 1999; Chilton 2010). The prob-lem of potential members in compression (when seeking atension-only solution) (Haber and Abel 1982), is solved byrelaxing the stiffness to trigger the snap-through process ofthese members.

The (augmented) ground structure method’s core ideaconsists of replacing the topology, shape and sizing problemby a sizing-only problem (Hemp 1973). This sizing-onlyproblem is done for a highly-redundant and interconnectedtruss system (i.e. the ground structure), from which the opti-mal layout is extracted and sized. The standard formulationof the method is augmented to include the load applica-tion points as variables of the problem, thus making the

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1188 Y. Jiang et al.

method suitable for the form-finding of compression-onlystructures (no need to flip the result). The proposed methodconsiders the iterative refinement of the initial ground struc-ture based on previous results, thus increasing the degreeof precision of the resulting geometry. The method canbe performed over any (discretized) footprint thanks toa graph-theory based generation algorithm (Zegard andPaulino 2014, 2015), and requires no a priori connectivityor sizing. The resulting (optimal) geometry is highly pre-cise; but the lattice-like underlying structure does requirea posteriori interpretation or rationalization to finalize thedesign.

The rest of the manuscript is organized as follows:Section 1.1 reviews the literature in the field of form-finding; Section 1.2 briefly reviews the well-known forcedensity method (for the purpose of completeness and com-parison); Section 2 describes the various types of staticloads that are considered in the form-finding process;Section 3 is dedicated to the form-finding method basedon the principle of minimum potential energy; Section 4introduces the form-finding method using the augmentedground structure method; Section 5 shows three numeri-cal examples designed with the three form-finding meth-ods previously presented; Section 6 discusses the fea-tures of the two proposed form-finding methods and theforce density method; Finally, a summary and compari-son of these methods, as well as concluding remarks aregiven in Section 7. Appendix A has details on the for-mulation of the force density method, and Appendix Bpresents the nomenclature used in the main text of thismanuscript.

1.1 Literature review

Grid-shells are a type of compression-only structure, whereefficiency is attained by minimizing (or ideally elimi-nating) the bending moment and shear forces. Optimalcompression-only designs can be analytically derived underspecific conditions of loads and geometry. However, inmost practical applications a compression-only design can-not be obtained analytically. Early works in form-findingcan be traced back to the experimental models used byAntoni Gaudı: the form for the Colonia Guell was obtainedusing a scaled experimental model consisting of hangingchains and weights, which he called a funicula (or funicular)model. This model was continuously revised for 10 years;adjusting the cord lengths and hanging weights (Bergosand Llimargas 1999; Martinell et al. 1975). The resultsfrom this experimental model were mirrored vertically, andtraced over for further graphical refinement in order toobtain a compression-only solution (Collins 1977). An anal-ogous concept was later used by Frei Otto, but using soapfilm models instead of hanging chains (Addis 2007). Heinz

Isler, used hanging cloths to overcome the difficulties ofsoap films, and used this experimental technique to designimpressive thin shell concrete structures (Chilton 2010).

With the rise of computers, new (computational)form-finding methods replaced the previous experimentalapproach using scaled models. One said method consists ofconstructing the stiffness matrix using the elastic constitu-tive equations, and solving for a configuration in equilib-rium for a given set of external loads (e.g., Siev and Eidel-man 1964; Argyris et al. 1974; Tabarrok and Qin 1992).Methods in this rationale can be viewed as simulating thephysical hanging models using the finite element method.Drawbacks include the computationally expensive construc-tion of the stiffness matrices, and the possibility to obtaincompressive members in what should be a tension-onlystructure (Haber and Abel 1982). An alternative approachwas later proposed that solves for the steady state of adynamic equilibrium problem, which is equivalent to theequilibrium state of the static problem (Barnes 1977, 1988).This dynamic approach was later used in the design of axial-force-only particle-spring systems (Kilian and Ochsendorf2006). Methods based on dynamic relaxation are typicallyfaster than those based on stiffness matrices, but requireadditional parameters which may not have physical mean-ing, and may require tuning to ensure the stability andconvergence of the solution (Nouri-Baranger 2004; Mikiet al. 2014). Circa 1970, the design of the Munich OlympicComplex required form-finding capabilities beyond thoseoffered by the available methods (Lewis 2003). The ForceDensity Method (FDM) was developed in response to thisrequirement: the form is obtained in a purely geometricmanner by ensuring equilibrium of forces, but requiringassumptions on the members’ force densities, defined asthe force divided by the length of the member (Linkwitzand Schek 1971; Schek 1974). The computations involvedin the FDM make it relatively inexpensive compared tostiffness-based methods, and hence it rose quickly in pop-ularity due to the relatively scarce computational resourcesof the time. The method has been extended and modifiednumerous times to alleviate specific problems, or expandits capabilities (e.g., Haber and Abel 1982; Bletzinger andRamm 1999; Nouri-Baranger 2004; Sanchez et al. 2007;Pauletti and Pimenta 2008). In addition to the aforemen-tioned techniques, other works make use of structural opti-mization (Bletzinger and Ramm 1993, 2001; Bletzinger etal. 2005), or graphic statics (Block and Ochsendorf 2007;Block 2009; Thrall et al. 2012; Akbarzadeh et al. 2015), toname a few. None of these methods is free of issues, andtherefore there is ongoing research to improve existing ordevelop new form-finding methods. The reader is referred toLewis (2003) and Veenendaal and Block (2012) for a morethorough comparison of the most popular form-findingmethods.

Form-finding of grid-shells using the ground structure and potential energy methods... 1189

1.2 Review of form-finding by the force density method(FDM)

The force density method (FDM) by Linkwitz and Schek(1971) (in German) and Schek (1974) is a form-findingmethod for general cable network structures. In this method,a parameter called force density is defined as the ratio ofthe (signed) member force over the member length. Giventhe user-defined force densities and member connectivity, afixed geometric stiffness matrix can be constructed, whichrelates the geometry (form), with the force equilibrium ateach node. The solution of this system yields the nodalcoordinates for which the geometry is under equilibriumfor the given external loads. In the present work, an itera-tive scheme is introduced to cope with the design-dependentload calculation described in Section 2. It should be notedhowever, that this method does not consider the constitu-tive laws nor the compatibility equations. Thus, the resultinggeometry is not dependent on the initial geometrical config-uration of the network. The equations involved in the FDMare given in Appendix A for completeness.

2 Load derivation

In the following section, the loads in the {x, y, z} directionsfor a node i are defined as fi . These nodal loads are builtfrom three types of sources: loads applied directly at thenodes f•i , line loads on the edges f�i , and area loads from thepolygonal panels f�i . The total load is therefore:

fi = f•i + f�i + f�i ∀ i = 1, 2 . . . Nn (1)

where Nn is the number of nodes in the network.In the form-finding process, the nodal loads are depen-

dent on the displacements: the load contributions comingfrom f�i and f�i are updated based on the (new) nodal loca-tions. Thus, the algorithm iterates until some convergencecriterion is met. In the case of the PEM and FDM methods,we use a displacement-based criterion:

∣∣unew − uold

∣∣

Nf

< ε (2)

where unew and uold are the updated and previous nodal dis-placements, respectively, Nf is the number of degrees offreedom, and ε is a user-defined tolerance. Whereas in theGSM, there is a user-defined number of iterations. Nonethe-less, for this last method, an automatic convergence criterioncan be implemented based on variations of the interpolatedsurface.

2.1 Panel loads

The panel loads are resolved to their respective nodes usinga tributary area distribution. The methodology can not onlyhandle triangles and quadrangles, but any convex polygon1.The polygonal panels are triangulated from the center ofmass (CM) as shown in Fig. 1a; which is calculated usinga temporary triangulation from the average of the panel’snodal coordinates. It should be noted that this procedure willintroduce error if the panels are significantly warped out-of-plane. Thus, it is assumed that the panel partition has arelatively small out-of-plane warping.

The area of the polygon’s triangles are calculated sequen-tially: half of each triangle’s area is assigned to the corre-sponding exterior node, as shown in Fig. 1a. This process isrepeated for all load panels, and thus the tributary area asso-ciated with a node may receive contributions from multipleload panels (see Fig. 1b).

The panel load can itself be of three types: self-weightby unit area q, projected load w in the {x, y, z} directionsrespectively, and an internal pressure p normal to eachpanel. A two-dimensional simplified illustration of these isshown in Fig. 2.

Thus, for a node i in the network, the loads in the {x, y, z}direction due to panel loads are:

f�i =∑

�0.5

(

δij + δik

) (

a�jk

) (

q e3 + w : n�jk + pn�

jk

)

(3)

where a�jk is the area of a single triangle in the polygon’s

triangulation associated with the exterior nodes j and k (seeFig. 1a), δij is the Kronecker delta (ensuring loads are onlyadded to the exterior nodes in the triangle), q is the self-weight of the panel by unit area applied in the positive z

direction (i.e. q is usually negative), e3 is the unit vector inthe positive z direction, w is the projected distributed loadin the {x, y, z} directions respectively, n�

kj is the unit nor-mal vector of a single triangle associated with the exteriornodes j and k, and p is the internal pressure load by unitarea (positive outwards). For each triangle in the polygon’striangulation, we derive its area aij and unit normal vectornij by

n�ij = vi × vj

a�ij = 0.5

∣∣∣n�

ij

∣∣∣

n�ij = n�

ij /

∣∣∣n�

ij

∣∣∣ (4)

where vi is the vector from the polygon’s center of mass tonode i (see Fig. 1a).

1Minor concavities can be handled, but are generally not advised sincethey may introduce structural instabilities in the node–bar network.


0.5aΔ51

aΔ12

aΔ23

aΔ34

aΔ45

0.5aΔ51

v4

v3

v2

v1

v5

1

2

3

4

5

(a)

(b)

Fig. 1 Tributary area derivation: a A polygonal panel is triangulatedfrom the center of mass. For each triangle, the area is calculated andthen evenly assigned to the two adjacent nodes. Arrows indicate thenodes that the tributary area is assigned to; b Tributary area associatedwith a single node in the network

2.2 Edge loads

The edge loads are calculated based on the length �jk andcross-sectional area ajk of the bar between nodes j and k.Half of the bar’s weight is assigned to each node. The imple-mentation of projected loads in linear members is relativelysimple; however, these loads are rare in practice, and aretherefore not considered in this work. Thus, for a node i inthe network, the nodal loads in the {x, y, z} direction due tothe edges are:

f�i =∑

�0.5

(

δij + δik

) (

�jkajk

) (

γ e3)

(5)

where γ is the specific weight of the bars.

ρ

(a)

wz

wx

(b)

p

(c)

Fig. 2 Two-dimensional illustration for the 3 types of panel load: aSelf-weight; b Projected load; c Internal pressure

3 Form-finding by the potential energy method(PEM)

The potential energy method (PEM) finds the form in equi-librium by minimizing the total potential energy of thesystem, as its name suggests. This approach considers largedeformations in the expression for the potential energy,where the designer can implicitly control the resulting formby manipulating the material properties and the initial nodalpositions. The design is in an up-side-down configura-tion, which mimics the 3D hanging chain models used byAntoni Gaudı (Bergos and Llimargas 1999; Collins 1977;Martinell et al. 1975). The reciprocal relationship betweenthe up-side-down tension-only structure and the standingcompression-only structure under the same loading case isillustrated in Fig. 3, which is inspired by Zalewski and Allen(1997). To prevent members in undesired compression, thesnap-through process is triggered by relaxing the stiffness of


initial (unloaded) configuration

deformed configuration

(a) (b)

(c) (d)

Fig. 3 A funicular in pure tension can be inverted vertically toobtain a compression-only funicular and vice-versa: a Tensile funic-ular form obtained by suspending three weights from springs orcables; b Compression-only funicular obtained from inverting the

three-weight tensile form; c Deformed configuration for a cable in ten-sion under a uniformly distributed load; d Compression-only quadraticarch obtained from inverting a cable under a uniformly distributed load

these members in compression (Jiang 2015). The followingsubsections elaborate on the details of the PEM.

3.1 Initial network generation

The first step for grid-shell form-finding with the PEM isto generate an initial network. This initial network is com-prised of hinge nodes, bar members, boundary conditions,and (optional) load panels. Figure 4 illustrates how an ini-tial network is generated for a small example. The nodes aredefined by their 3-dimensional coordinates in space, withsupport boundary conditions added where appropriate. Barmembers are defined by a pair of nodes with their orderinghaving no effect in the network definition. Load panels aredefined by the numbering of their vertices: the panel has apositive normal outwards based on counter-clockwise order-ing. This can define panel meshes such as the one shown inFig. 4a. Load panels can have any number of sides and donot contribute to the structural behavior. Bar members canbe automatically generated from the load panels if required,with differentiation of the inner and boundary members ifnecessary2. If the nodes at both ends of a bar are fixed,the bar is not considered in the form-finding process and isremoved from the network. Using the automatic generation

2Each inner member forms the edge of 2 load panels while eachboundary members forms the edge of 1 load panel.

from load panels, a total of 12 bar members are defined asillustrated in Fig. 4b.

The method’s goal is to find a suitable form in equilib-rium under a single load case, where the members sufferno bending moment or shear force (other than those causedby self-weight between nodes). All bar members will besubject to axial loads only with the following assumptions:

1. All bar members are hinge-connected;2. Loads are only applied at the nodes.

3.2 Formulation of the potential energy method

The total potential energy PE is defined as the sum of mem-ber strain energy minus the work done by the external loads.Thus, the minimization statement is:

minu

PE =Nb∑

i=1

1

2ki (��i)

2 − {f(u)}T u (6)

where ki is a reduced and relatively low axial stiffnessassociated with the i-th member for the form-finding pro-cess only, ��i the i-th member’s change in length, f is thenodal load vector (dependent on the displacements) calcu-lated as per Section 2 of the present manuscript, and u is thenodal displacement vector. The unconstrained minimizationproblem is solved by a quasi-Newton algorithm with the


Fig. 4 Initial networkgeneration for a simple example:a Eight load panels enclosed bydashed lines; b Twelve active barmembers between hinged nodes

12

34

56

78

(a)

12

3

4

5

6

78

9

10

11

12

(b)

BFGS Hessian updating scheme. Note that f is dependenton the displacements, which introduces a nonlinearity intothe problem. Thus, we iterate until the form is in equilib-rium for the loads under the updated configuration (within asmall error tolerance).

The final grid-shell form is determined by deformationof the initial configuration, which means that the resultingform is dependent on the initial domain geometry. Whilethis can be seen as a drawback of the current method, itcan be used as an additional degree-of-freedom to tunethe resulting shape (in addition to the reduced memberstiffnesses ki).

3.3 Snap-through triggering

A network of pin-connected bars may have more than oneequilibrium configuration, with the resulting shape deter-mined by the loading history. Under certain conditions,the PEM may converge to a solution with one or more

Fig. 5 An example of a possible equilibrium configuration with mem-bers in compression. These members comprise a kink, which may beavoided if a snap-through action takes place. Arrows on nodes denotenodal loads

members in compression. An illustrative example in Fig. 5shows an equilibrium configuration with an undesirablekink composed of members in compression. The configu-ration indicates a local minimum of the potential energy.These members in compression are likely to have anothervalid configuration in tension, which can be attained afterundergoing a snap-through. Figure 6 illustrates the conceptof these two configurations separated by a snap-through.Consider a 2-bar truss system loaded vertically downwardsat the center node by a force p, as shown in Fig. 6a.If the load is large enough (force pA) to load beyondthe snap-through multi-stable region, the only equilibrium

p

(a)

uA

pA

(b)

pB

uB

(c)

pB

*uB

(d)

Fig. 6 A two-bar truss system loaded at the center with a snap-throughbehavior: a Unloaded configuration; b Case A—equilibrium config-uration for a large load; c Case B—equilibrium configuration for asmall load, where bars are in compression; d Case B—equilibriumconfiguration for a small load, where bars are in tension


configuration is in tension (Case A) and is shown in Fig. 6b.However, if the load is small (force pB ), there may be twopossible equilibrium configurations that are stable (Case B).Based on the initial configuration and assuming a relativelyslow and monotonic loading history, the equilibrium config-uration is reached after a small displacement uB with bothmembers in compression, as shown in Fig. 6c. The secondstable configuration, shown in Fig. 6d, has both membersin tension: the node has snapped-through downwards andthe displacement u�

B is large compared to uB . In the contextof form-finding, the configuration in Fig. 6d is preferred,since this will result in a compression-only grid-shell oncethe result is inverted (or flipped) up-side-down: a featureusually sought in grid-shell structures.

The snap-through load-displacement behavior for thistwo-bar problem is illustrated in Fig. 7. Because theload history and initial configuration determine the finaldeformed shape of the loaded structure (Leon et al. 2011),the computational algorithm by which the structure isloaded and solved will have impact on the solution. Start-ing from the configuration shown in Fig. 6a: for Case Athe solution is unique (pA, uA), although depending on thealgorithm used, the load history or path might not be cor-rect; in Case B however, the stable solution

(

pB, u�B

)

isreached before the preferred configuration at (pB, uB). Tocope with the situation, a snap-through relaxation proce-dure is implemented. If a member is found in a state ofcompression, its stiffness will be reduced to a relativelysmall value (e.g. 1% of the original value), such that thesnap-through condition is artificially triggered. Later in thesolution iterations, if said member returns to a state oftension, its original stiffness is restored. Thus, in the con-text of the two-bar problem shown in Fig. 6, for Case B

u

p

0

pA

*uB

pB

uB uA

Solution with

Theoretical load-disp curvesnap-through relaxation

Fig. 7 The load-displacement plot for the two-bar system. The trueload-displacement curve and the one using snap-through relaxation,coincide in the tension-only region but not on the compression regime

the equilibrium configuration will likely be the preferredtension-only (pB, uB), as illustrated in Fig. 7.

3.4 Member sizing

The member cross-sectional areas are designed such that themembers are all fully-stressed. Coelho et al. (2014) docu-ment several experiences of grid-shell form-finding usingcommercial software and introduce an additional membersizing step after the final form is found (Coelho et al. 2014).The member sizing approach in the present work consid-ers large displacements in the analysis, which is performedusing the minimum potential energy approach. Based onthe analysis result, the members are sized using the stressratio method (Gallagher and Zienkiewicz 1973; Barneset al. 1977). The stress ratio method can consider differ-ent limits in tension and compression, however due to thesnap-through relaxation described in Section 3.3, only thetension limit is used by the algorithm (compression once theresult is flipped up-side-down). The stress ratio method isan iterative procedure which can include considerations forbuckling or other criteria. Common steel section familieshave their moment of inertias proportional to their cross-sectional areas (Baker 1992). Thus, the inclusion of elasticbuckling in the stress limit is straightforward provided thata section family is chosen. However, for the sake of sim-plicity and generality of the present work, this limit value isassumed constant (i.e. no buckling criterion is used).

Given a set of member cross-sectional areas, the equilib-rium state is solved by the PEM:

min.u

PE =Nb∑

i=1

1

2ki (��i)

2 − fT u (7)

where the stiffness of the i-th member is defined as

ki = Eiai

�i

(8)

in which Ei is the material Young’s modulus, ai is the cross-sectional area and �i is the member length.

Note that ki is the (real) member stiffness (i.e. ki �ki), and, assuming small displacements, the load does notdepend on the displacements. The form-finding and mem-ber sizing formulations presented ((6) and (7) respectively),bear many similarities with the key differences being:

1. In the member sizing, the member stiffness ki is calcu-lated by (8); whereas ki is user-defined and compara-tively low.

2. In the member sizing, the nodal load vector f is assumedto be constant; whereas in the form-finding (6), f isdependent on the form. In the member sizing stage, we


assume that the cross-section change introduces a rel-atively small change in the overall form, resulting innegligible changes in the loads3.

The cross-sectional member areas are then updated bythe stress ratio method (Gallagher and Zienkiewicz 1973;Barnes et al. 1977), where σadm,i is the limit admissiblestress4 for the i-th member:

ak+1i = ak

i

(

σki

σadm,i

)η

(9)

where aki and ak+1

i are the cross-sectional areas of the i-thmember in the current and next iteration respectively. Thestress ratio is defined as the ratio of the current memberstress σk

i over the member’s admissible stress σadm,i , and isused as a factor to update the cross-sections. The parameterη is a numerical damper 0 < η ≤ 1 that improves the con-vergence and stability of the method. Based on experience,good results are expected for η in the range of [1/2, 1/3],with lower values recommended in the event that buck-ling limits are considered (buckling introduces an additionaldesign nonlinearity).

During some design trials, we observe that the membersizing may result in a design that is almost fully-stressed butwith some grid members with a stress ratio less than 1. Asiterations proceed, these non-fully-stressed grid membersshould vanish with negligible cross-section areas. It shouldbe noted, however, that the stresses of these vanishing mem-bers can be non-zero and still be determined without across-section: stresses can be calculated with displacementsand constitutive relations only (case of a singular topology,refer to Rozvany (2001)).

3.5 Group member sizing

For cost and constructibility purposes, the design may berestricted to a reduced number of distinct cross-sectionalareas. This can be achieved by a group member sizing,where bar members in the same user-defined group willhave identical cross-sectional areas. All members within agroup g will have the same cross-section ag throughout thedesign process (including the initial guess). The stress ratioupdating scheme is modified to be

ak+1g = ak

g

[

max.j∈Gg

(

σkj

σadm,g

)]η

(10)

where Gg is a user-defined group of members with equalcross-sectional area ag , and admissible tension limit σadm,g ,

3The load could depend on the displacements or the cross-sections, butbecause the displacements are small, they are assumed constant for thesake of simplicity.4Considering the material’s yield limit, and optionally including con-siderations for buckling or other criteria.

for all g = {1, 2, 3 . . .} number of groups. This updatingscheme will not result in a fully-stressed design; insteadeach group should have at least one fully-stressed member(assuming the solution is not degenerate (Rozvany 2001)).

3.6 Summary of form-finding by the PEM

The processes of form-finding by PEM and member sizingare summarized with flowcharts in Fig. 8. Note that as theform-finding and member sizing are separate processes, theform and load input for member sizing is not restricted tothe PEM results.

4 Form-finding by the ground structure method(GSM)

The ground structure method (GSM) is a topology opti-mization technique developed for obtaining the optimalgeometry, connectivity, and cross-sections of pin-jointedframeworks or trusses (Dorn et al. 1964). This method hasseen extensive use as an aid in the derivation of closed-formanalytical solutions for optimal structures (Lewinski et al.1994; Rozvany et al. 1997; Pichugin et al. 2012; Lewinskiet al. 2013; Sokoł and Rozvany 2013; Rozvany and Sokoł2013 Sokoł 2011, 2014). The power of the method liesin its ability to formulate the optimization statement as alinear programming5 problem (Hemp 1973). The linear pro-gramming problem can be solved very efficiently usingthe interior-point method family of linear program solvers(Karmarkar 1984; Wright 2004). The method requires aninitial highly-redundant and interconnected truss network,the ground structure, from which the best possible solutionis extracted, with this being an approximation of the theoret-ical optimum. Since the solution will only contain membersin the original ground structure, this ground structure canbe tailored (or specially generated) to meet the designrequirements or geometric restrictions. The method is onlylimited by the user’s ability to generate this ground struc-ture. Recently, a methodology was introduced to generateground structures in any two-dimensional domain, includingconcavities and holes, based on graph theory and collisiondetection (Zegard and Paulino 2014). This methodology waslater extended to three-dimensional space (Zegard 2014;Zegard and Paulino 2015), and thus serves as the foundationfor this section.

5There are two common versions of the GSM: one based on the plasticformulation, and one based on the elastic formulation. Only the plasticformulation can be formulated as a linear programming problem. Theelastic formulation (Bendsøe et al. 1994; Christensen and Klarbring2009; Ramos and Paulino 2015), despite being computationally moreexpensive, considers compatibility and can include material and geo-metric non-linearities to name a few features. The present work usesthe plastic formulation in the form-finding process.


Generate the initial domain and

Convergence?

Finish

Start

Y

N

-ration by minimization of

potential energy (large displacements)

Recalculate loads with the new

Load the form and loads; input actual material properties

Convergence?

Finish

Start

Y

N

-ration by minimization of

potential energy (small displacements)

Calculate member stresses

Member sizing

Store member sizing results

Update member cross-sectional areas with stress ratios

(a) (b)

Fig. 8 Process of form-finding by the PEM and member sizing: a Flowchart for form-finding by the PEM; b Flowchart for member sizing

The optimization formulation for the (plastic version ofthe) GSM is:

mins+,s−

{

�T κ �T} {

s+s−

}

= σT V

s.t.[

BT −BT]{

s+s−

}

= f

s+, s− ≥ 0 (11)

where s+i and s−

i are (positive) slack variables associatedwith the tension and compression force of the i-th memberrespectively, κ = σT /σC is the ratio of the tension to com-pression stress limits, � is the vector of member lengths, andV is the volume of the resulting structure. The matrix BT

is the nodal equilibrium matrix built from the directionalcosines of the members, and f is the nodal force vector, bothexcluding the components associated with supports. Thecross-section of the i-th member is ai = s+

i /σT + s−i /σC ,

and the respective force is ni = s+i − s−

i . The solutionto this problem assumes the design is fully stressed, whichis valid for problems with a single load case. For addi-tional details on the assumptions and derivations leading to(11) the reader is referred to Hemp (1973) and Zegard andPaulino (2014).

4.1 Formulation with transmissible loads

The GSM finds the optimal structure (within the groundstructure) given the nodal loads and supports. This posesa difficulty for grid-shell form-finding: since the grid-shellgeometry is not known a priori, the points of application

of the loads are unknown. Rozvany and Wang (1983) andWang and Rozvany (1983) use the concept of transmis-sible loads: the load is applied to fictitious nodes con-nected by virtual members to the real structure. Thesemembers are essentially free and allow the load to posi-tion itself optimally. It may be incorrectly assumed thatthis is equivalent to making the vertical members in theground structure free, which in the context of (11) trans-lates into artificially assigning a zero member length toverticals: the (ideally) unique solution will be repeated ver-tically multiple times because the optimal lies in an edgeor facet of the feasible domain (infinite solutions). As aconsequence, the numerical result will vary with the lin-ear program solver used: for the benchmark problem inFig. 9a, the solution using the modified version of LIP-SOL in MATLAB® (Zhang 1996a, b) is shown in Fig. 9b,where the repetition artifact of this flawed approach isclear.

The approach of transmissible loads using virtual barshas also been applied to continuum optimization (Fuchsand Moses 2000; Chiandussi et al. 2009; Yang et al. 2005).Instead of virtual vertical members, the transmissible loadscan be analogously implemented by extending the prob-lem in (11) to include the vertical loads as variables, andadding equations to maintain equilibrium (Gilbert et al.2005; Darwich et al. 2010). The present work considersa generalization of this extended formulation, allowing forany number, direction and magnitude of the transmissibleloads. This approach is combined with an adaptive andunstructured discretization of the domain used to gener-ate the ground structure (Zegard and Paulino 2015). Thesolution in Fig. 9c is the result of this proposed method,


q=1 / length

L =3X

L =

2Z

(a)

(b)

(c)

Fig. 9 Regular and orthogonal simple mesh to illustrate the trans-missible loads in the GSM: a Base mesh of 60 × 40 elements usedin the ground structure generation; b Assigning zero length to thevertical members leads to an (incorrect) degenerate solution that isrepeated vertically; c Solution obtained using the augmented GSMwith transmissible loads, with equal limits in tension and compression(i.e. κ = σT /σC = 1)

where the spokes radiating or branching from the supportsare indeed part of the optimal solution for the case of equallimits in tension and compression (Darwich et al. 2010; Tyaset al. 2010). Conversely, the solution for a compression-onlyarch under a constant distributed load is a parabolic arch(Rozvany and Wang 1983), which will be used to verify theproposed method.

The unknown vertical loads are integrated as variables inthe linear program by augmenting the system in (11). This isdone by splitting the equations associated with the vertical

Fig. 10 Simple domain discretization with 9 transfer groups. Transfergroups can have different numbers of nodes

loads (z direction, as in Fig. 10) and horizontal in-planeloads (x and y direction):

(12)

where fxy are the nodal loads associated with the x and y

coordinates, and fz are the vertical nodal loads. The sub-scripts xy denote an aggregation of terms associated withthe x and y coordinates, i.e.:

BTxy =

{BT

x

BTy

}

, fxy ={

fxfy

}

(13)

The vertical loads fz are allowed to transfer (or move) ver-tically to optimally position themselves within a transfergroup: a transfer group Ti is defined by all nodes sharingthe same x and y coordinates (within some small toler-ance), as shown in Fig. 10. It should be noted that there isno order or structured requirement; i.e. the transfer groupscan have different size and are automatically generated. Thetotal load (fv)i associated with the i-th transfer group Ti

will be distributed among the nodes of the group, thus:

(fv)i =∑

j∈Ti

(fz)j (14)

with (fv)i = 0 if any vertical (z direction) degree-of-freedom in group Ti is supported. If the (vertical) load canposition itself anywhere in the transfer group, the existenceof a support will draw all the loads to itself.

The nodal vertical loads fz are split into a (positive)magnitude f�z and direction β, therefore:

fz = β : f�z (15)


This splitting is convenient since standard linear programsrequire all variables to be positive, and the goal is to makefz a variable in the optimization statement. Thus, the nodalequilibrium in the z direction becomes:

BTz

(

s+ − s−) − diag (β) f�z = 0 , (16)

with the load direction vector β defined as:

βi ={

1 if i ∈ Tj and fj ≥ 0−1 if i ∈ Tj and fj < 0

(17)

A rectangular matrix A indicates the correspondence of anode to a transfer group:

Apq ={

1 if q ∈ Tp

0 otherwise(18)

with an explicit relationship between the correspondencematrix A and the load direction vector: β = AT sgn� (fv),with sgn� (·) being the sign operator reserving thatsgn� (0) = 1. It is assumed that every node belongs to a sin-gle transfer group Ti , and that all nodes belong to a transfergroup (even if the group contains a single node). By namingfv the vector of vertical transfer loads associated with eachtransfer group, the augmented ground structure formulationis:

(19)

where the relation Af�z = |fv| is enforcing the vertical loadsto add to the specified external vertical loads (14). Equation(19) is the main optimization formulation adopted in thiswork.

4.2 Surface interpolation and refinement

The location of the grid-shell surface is interpolated basedon the point of application of the transmissible load. Foreach transfer group Tj , the vertical location of the grid-shellsurface zj is the weighted average of the node vertical posi-tion zi , weighted by the fraction of transmissible load in it(

f �z

)

i:

zj =∑

i∈Tj(zi)

(

f �z

)

i∑

i∈Tj

(

f �z

)

i

=∑

i∈Tj(zi)

(

f �z

)

i

|fv|j(20)

with |fv|j being the magnitude of load applied in the j -th transfer group. In the event that no vertical loads areassigned to a transfer group, an equivalent procedure can bedone using the relative cross-sections of members connect-ing to each node within a group.

It has been shown that the tension region spokes in thearch solution in Fig. 9c result in a better (lighter or less-compliant) structure than a compression-only arch (Darwichet al. 2010; Tyas et al. 2010). In the present work how-ever, the interest is in compression-only solutions; which arealso known as Prager structures (Rozvany and Prager 1979;Rozvany et al. 1982). The spokes radiating or branchingfrom the supports in Fig. 9c constitute a compression-tension orthogonal field that will bias the surface interpola-tion towards a shallow configuration. The compression-onlysolutions can be filtered by reducing the tension to com-pression limit ratio κ in (19). It has been proven that fora two-dimensional arch, such as the benchmark problem inFig. 9, a tension to compression limit ratio κ = σT /σC <

1/3 is sufficient to yield a compression-only solution (Dar-wich et al. 2010). In the three-dimensional case, and to covera wide range of geometries and load configurations, ide-ally κ → 0 to obtain compression-only solutions. However,a value of κ = 0.01 > 0 is used for numerical stabilitywhile still rendering close-to compression-only solutions:problems with overhanging awnings or cantilevers requirea tension region, thus κ > 0 allows the algorithm to finda solution in those cases (though severely punishing thetension region).

The solution with κ = 0.01 for the benchmark prob-lem in Fig. 9 is shown in Fig. 11a; with its interpolated

(a)

(b)

Fig. 11 Ground structure solution for the compression-only singlearch: a Solution obtained using κ = 0.01 for the problem in Fig. 9a; bSurface interpolation obtained from (20)


L =

0.5

Z

(a)

L =

0.12

5Z

(b)

Fig. 12 Refinement of the domain for the problem in Fig. 9a: a Firstrefinement. Surface is extruded in both directions for a total height of0.5; b Second refinement. Surface is extruded in both directions for atotal height of 0.125

surface shown in Fig. 11b. This interpolated surface isreasonably smooth, despite the coarseness of the domainpartition and having a large portion of the ground structureoutside the scope of the optimal arch (hence a large compu-tational overhead). This interpolated surface is used to refinethe discretization used in the ground structure generationby shrinking the domain, thus achieving better solutions.Figure 12a shows the result after shrinking the domain to1/4 of the initial vertical span, based on the interpolated sur-face in 11(b): the supports are centered on the mesh, and thecollinear tolerance6 is reduced by 5% (initially ColT ol0 =0.999999, thus ColT ol1 = 1 − 0.05 (1 − ColT ol0) =0.99999995). The result shown in Fig. 12b is obtained afterrepeating the process once more (again reducing the verticalspan by 1/4 and the collinear tolerance by 5%).

The quality of the results and the efficacy of the refine-ment can be evaluated with help of the analytical solution:the summary of the results is displayed in Table 1. The GSMmakes use of discrete nodal locations, and thus the surfaceinterpolation can over- or under-estimate the height of thearch depending on the partition. However, the optimal vol-ume will converge to the optimal volume from above as thesolution improves.

In the three-dimensional space, the geometry (form-finding) of the grid-shell is determined by the surfaceinterpolation procedure. The ground structure solution asso-ciated with it provides information on the optimal load-path

6Refer to Zegard and Paulino (2014) for details on the effect of thecollinear tolerance on the ground structure generation process.

Table 1 Convergence and error of the arch height and volume withrefinement for the problem in Fig. 9a

Case Arch height Volume V

Analytical√

3Lx

4 = 1.2990 4qL2x

4√

3σC= 5.1962

Orthogonal 1.3209 5.2200

grid error = 1.68% error = 0.46%

1st 1.2962 5.2010

refinement error = −0.22% error = 0.09%

2nd 1.2964 5.1962

refinement error = −0.20% error = 0.01%

of the forces in the optimized grid-shell geometry, whichcan be used to guide the optimal layout of the mem-bers in the grid-shell. In addition, previous work has alsofocused on the automatic layout of the members for a givengrid-shell geometry (Richardson et al. 2013).

The method of transmissible loads using the groundstructure generation framework from Zegard and Paulino(2015) was here exemplified with a two-dimensional exam-ple. While this section dealt with a known two-dimensionalproblem and a design-independent load, the power of theproposed method lies in its extension to three-dimensionalspace, and the capability of addressing (almost) any domainshape. The inclusion of self-weight, and design-dependentloads can be readily applied, and is described in Section 2.This is a remarkable property of the present GSM becausein the case of a continuum density-based method, specialprocedures are needed to handle design-dependent loads(Bendsøe and Sigmund 2003).

5 Numerical examples

The three form-finding methods mentioned above areimplemented and tested with three numerical examples. Theexamples include a quadratic synclastic dome, a polygonallytessellated pavilion and a doubly-curved bowtie surface.

– Quadratic synclastic dome: the simplicity of this prob-lem allows for easy comparison and visual evaluationof the various results obtained with each method.

– Polygonally tessellated pavilion: this problem high-lights the ability of the methods to address grid-shellswith unstructured polygonal load panels with the possi-bility of holes and concavities.

– Doubly-curved bowtie surface: the capability to handlenon-flat curved boundaries and to yield compression-only results are showcased with this example.

The units used are kilonewton (kN) for forces and meters(m) for length unless otherwise specified.


Fig. 13 Quadratic synclastic dome: rendering of the design competi-tion used as inspiration for the design (image courtesy of SOM LLP)

5.1 Quadratic synclastic dome

This example addresses the form-finding of a synclasticdome with quadratic edges in the XY plane, which is illus-trated in Figs. 13, 14, 15, 16, 17, 18, 19, 20 and 21. Thisproblem is inspired by a competition design of Skidmore,Owings & Merrill LLP (SOM LLP) (see Fig. 13). The orig-inal design was obtained using a heavy soap film-inspiredapproach based on isogeometric continuum shell optimiza-tion. This approach considers a different treatment of thesupports, with their geometry being part of the optimizationprocess. The reader is referred to Mitchell (2013) for detailson the procedure used to obtain the result in Fig. 13.

The grid-shell is meant to provide cover for a 2808 m2

footprint. The edges are both quadratic in plane, with the

63m, 7 Bays

4.5m

36m

,7 B

ays

9m

a = 4aunita = 2aunita = 1aunit

Fig. 14 Quadratic synclastic dome: footprint, panel mesh, boundaryconditions, and the prescribed cross-sectional areas used in the PEM.The value of aunit is 5, 000 mm2 in the PEM-based form-finding stage

(a)

z

x-30 -20 -10 00

5

10

10 20 30

(b)

z

y-20 -10 00

5

10

10 20

5.34

m11

.33m

(c)

Fig. 15 Quadratic synclastic dome: form-finding result by the PEM.a Isometric view; b Front view; c Side view

longer edge convex outwards, and the shorter edges con-cave inwards as shown in Fig. 14. The longer edges arefixed to the ground level, while the opposite edges are open.Requirements of the design include:

– The opening should be no less than 5 m high;– The apex height should be approximately 12 m.

a/ amax Plot, amax=5665 mm2

0

0.2

0.4

0.6

0.8

1

Fig. 16 Quadratic synclastic dome: area-ratio plot showing the ratioof the members’ cross-sectional area over the maximum value in thestructure. Both the line width and color-scale in the plot are corre-lated with the area-ratio.The maximum cross-sectional area amax =5, 665 mm2


a1=4238 mm2

a2=5670 mm2

a3=1053 mm2

(a)

Resulting stress ratios are in the range [0.298, 1.000]

0

0.2

0.4

0.6

0.8

1

(b)

Fig. 17 Quadratic synclastic dome: group member sizing results. aResulting cross-sectional areas for the three groups; b Stress ratioplot shows that each group has at least one fully-stressed member, asindicated by an arrow

5.1.1 PEM solution

The design domain is discretized into the structured networkwith 64 nodes, 49 load panels, and 98 bars (edges) as shownin Fig. 14. The system is loaded by its panels, considering aself-weight load of 4.5 kN/m2 and an upward wind suctionload of 0.5 kN/m2. The member stiffnesses are determinedby ki = Eiai/�i , in which E is assumed to be 1 GPa for allmembers, and �i is the original length of the i-th memberin the initial configuration (i.e. z = 0, ∀x = {x, y, z}).The cross-sectional areas are shown in Fig. 14, where aunit

equals 5, 000 mm2. These fictitious cross-sections are tunedto yield the desired geometric requirements (opening height> 5 m, and apex height of ≈ 12 m).

The isometric, front, and side view of the form-findingresults by PEM are shown in Fig. 15a, b and c, respec-tively. It can be observed from Fig. 15c that the openingsare 5.34 m in height, with an apex height of 11.33 m, whichconforms with the design requirements.

Member sizing is conducted after the form-finding stageusing the stress ratio method with a numerical damping

of η = 0.5. The design material is assumed to be A500Gr. B carbon steel, with a Young’s modulus of 200 GPa,and an admissible stress of 315 MPa. The admissible stressreduction factor is 0.9 against yielding. The critical loadcombination of 1.2 times the dead load and 1.0 times thewind load is considered, with these loads inherited from theform-finding stage. Starting from a uniform initial guess of5, 000 mm2, the ai values converge to those shown in Fig. 16after 28 iterations. The maximum cross-sectional area is5, 665 mm2, with all members reaching a stress ratio equalto 1.0 (within a tolerance of 0.0005).

Group member sizing is conducted for comparison. Themembers are divided into 3 groups as shown in Fig. 17a.Starting with an initial guess of ag = 5, 000 mm2 forall groups, the cross-sectional areas converge to a1 =4, 238 mm2, a2 = 5, 670 mm2, and a3 = 1, 053 mm2, after16 iterations (see Fig. 17a). Figure 17b confirms that eachgroup has at least one member with stress ratio equal to 1.0.

5.1.2 GSM solution

The initial domain extrusion assumes a flat configurationof the surface (i.e. z = 0 everywhere), and is extruded toa height of 26 using 12 elements in thickness as shown inFig. 18a. The structure is loaded by the self-weight of thepanels, which is taken to be q = 1 unit/m2, and calcu-lated based on the current surface configuration. After theoptimal (pin-jointed) solution is found, a new surface is cal-culated (20) and the loads are updated. The domain is againextruded around the calculated surface, solved and updateda total of 6 times. With the exception of the very firstdomain, the surface is extruded equally in both directions.The extrusion thickness is continuously decreased through-out the iterations: the second domain is extruded to a totalthickness of 13 and is shown in Fig. 18b; the sixth and finaldomain is extruded to a total thickness of 2.6 and can beseen in Fig. 18c. The interpolated surface obtained fromthe last iteration in the process is shown in Fig. 18d. Theground structure solution (Fig. 19a) provides information onthe optimal load-path of the forces in the optimized grid-shell geometry, with the relative magnitude of these forcesillustrated by the member thickness: the load-path hints ata mostly one-way structural system, with smaller membersproviding lateral support and resolving the perpendicularcomponents. The resulting (interpolated) height shown inFig. 19b does not depend on user-defined parameters, and itclosely approximates the (tessellated) optimal shape.

5.1.3 FDM solution

The initial domain is discretized in the same manner as inthe PEM solution (see Fig. 14). The force density is assignedas shown in Fig. 20, with runit = 50 kN/m.


Fig. 18 Quadratic synclasticdome: domain discretizationsused in the ground structuregeneration algorithm (GSM). aInitial discretization; b Domaindiscretization after the 1st

refinement; c Domaindiscretization after the 5th

refinement; d Final interpolatedsurface geometry

(a) (b)

(c) (d)

The form converges after 5 iterations, and the result isshown in Fig. 21. The geometric requirements are met withan apex height of 12.19 m and an opening height of 6.15 m.After this form-finding process, the members can be sizedin the same manner as in the PEM method (if needed).

(a)

0

5

10

15

20

22.23

(b)

Fig. 19 Quadratic synclastic dome: optimized ground structureobtained after 5 domain refinements (6 analysis processes). a Memberlayout; b Optimized height

5.2 Polygonally tessellated pavilion

The second example is a polygonally tessellated pavilionwith internal supports and multiple openings, which is illus-trated by Figs. 22, 23, 24, 25, 26 and 27. The domain isdiscretized into polygonal panels in the XY plane using theeducational software PolyMesher (Talischi et al. 2012). Thisexample demonstrates that the proposed methods work withunstructured polygonal load panels, holes, and concavities.

5.2.1 PEM solution

Figure 22 shows the domain partition, consisting of 270nodes, 138 load panels, and 369 bars (edges). The systemis loaded by its panels, considering a self-weight load of

r =4runitr =2runitr =1runit

Fig. 20 Quadratic synclastic dome: user-defined force densities


(a)

z

x-30 -20 -10 00

5

10

10 20 30

(b)

z

y-20 -10 00

5

10

10 20

6.15

m12

.19m

(c)

Fig. 21 Quadratic synclastic dome: form-finding result by FDM. aIsometric view; b Front view; c Side view

4.5 kN/m2 and an upward wind suction load of 0.5 kN/m2.The bars have an identical Young’s modulus of 1 GPa andcross-sectional area of 120 mm2. The isometric and frontview of the resulting form after 5 iterations are shown inFig. 23a and b, respectively. We observe that the apex occursnear the center of the domain with a height of 4.10 m.

Member sizing is done after the form-finding stageusing the stress ratio method with a numerical dampingof η = 0.5. The adopted material properties, admissi-ble stress reduction and load combination are the sameas used in Section 5.1.1. Starting with an initial guess of5, 000 mm2 for all cross-sections, the values converge to the

22.21 m

14.6

3 m

Fig. 22 Polygonally tessellated pavilion: footprint with load panelpartition and support boundary conditions

(a)

-5 0 5 10 150

2

4z

x(b)

Fig. 23 Polygonally tessellated pavilion: form-finding result by thePEM. a Isometric view; b Front view

fully-stressed design shown in Fig. 24 after 115 itera-tions. The maximum cross-sectional area in the structure is218 mm2.

5.2.2 GSM solution

The initial domain extrusion assumes a flat configurationof the surface (i.e. z = 0 everywhere), and is extrudedto a height of 6 using 8 elements in thickness as shownin Fig. 25a. The structure is loaded by the self-weight ofthe panels, which is taken to be q = 1 unit/m2, and cal-culated based on the current surface configuration. Afterthe optimal (pin-jointed) solution is found, a new surfaceis calculated (20) and the loads are updated. The domainis again extruded around the calculated surface, solved andupdated a total of 6 times. With the exception of the veryfirst domain, the surface is extruded equally in both direc-tions. The extrusion thickness is continuously decreasedthroughout the iterations: the second domain is extruded to

a/ amax Plot, amax=218 mm2

0

0.2

0.4

0.6

0.8

1

Fig. 24 Polygonally tessellated pavilion: area ratio plot based on thePEM solution


Fig. 25 Polygonally tessellatedpavilion: domain discretizationsused in the ground structuregeneration algorithm (GSM). aInitial discretization; b Domaindiscretization after the 1st


refinement; d Final interpolatedsurface geometry. (a) (b)

(c) (d)

a total thickness of 3 and is shown in Fig. 25b; the sixthand final domain is extruded to a total thickness of 0.6 andcan be seen in Fig. 25c. The interpolated surface obtainedfrom the last iteration in the process is shown in Fig. 25d.The ground structure solution (Fig. 26a) provides informa-tion on the optimal load-path of the forces in the optimizedgrid-shell geometry, with the relative magnitude of theseforces illustrated by the member thickness. The resulting(interpolated) height shown in Fig. 26b does not depend

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.30

(b)

Fig. 26 Polygonally tessellated pavilion: optimized ground structureobtained after 5 domain refinements (6 analysis processes). a Memberlayout; b Optimized height

on user-defined parameters, and it closely approximates the(tessellated) optimal shape.

5.2.3 FDM solution

The FDM solution uses the same domain discretization asthe PEM solution (see Fig. 22). A uniform force density of15 kN/m is applied to all members. With this setup, the formconverges after 6 iterations. Figure 27a and b show the iso-metric and front view of the resulting form, respectively. Weobserve that the apex occurs near the center of the domainand has a height of approximately 4.25 m.

5.3 Doubly-curved bowtie grid-shell

This example covers a doubly-curved bowtie-shaped wavysurface with a targeting apex height of 25 m approximately,which is illustrated by Figs. 28, 29, 30, 31, 32, 33, 34 and

(a)

-5 0 5 10 150

2

4z

x(b)

Fig. 27 Polygonally tessellated pavilion: form-finding result by theFDM. a Isometric view; b Front view


153m, 11 Bays4.5m

54m

,6 B

ays

9m

a= 4aunita= 2aunita= 1aunit

Fig. 28 Doubly-curved bowtie grid-shell: footprint, panel mesh,boundary conditions, and the prescribed cross-sectional areas used inthe PEM. The prescribed cross-sectional areas used in the PEM adoptan aunit value of 6, 000 mm2

35. The grid-shell lies in a quasi-rectangular domain in theXY plane, with doubly curved longer edges and concaveshort edges. The edge geometry uses spline curves in itsgeometric definition and is shown to scale in Fig. 28.

5.3.1 PEM solution

The grid mesh is structured with 11 bays in the x-directionand 6 bays in the y-direction. The discretized domain shownin Fig. 28 is comprised of 84 nodes, 124 bars and 66 loadpanels. The constrained nodes are not flat on the ground(i.e. z �= 0 m), but can be 9 m away from the ground level.The system is loaded by its panels, considering a self-weightload of 4.5 kN/m2 and an upward wind suction load of0.5 kN/m2. The initial interior nodes’ height (z direction) isinterpolated from the exterior boundary in a smooth manner:a user-defined initial condition to achieve a desired form.The varying nodal elevations in the initial configurationcause the structure to have compressive local equilibriumconditions: i.e. the snap-through relaxation is necessary toachieve a tension-only form. Due to symmetry, the resultingform is expected to have a doubly-curved surface with twoapexes aligned along the x axis. In addition to the domainspecification, Fig. 28 also shows the cross-sectional areasof the members in the form-finding process, with aunit =6, 000 mm2. All bars share the identical Young’s modulusof 1 GPa.

The resulting form is obtained after 5 iterations, and isshown in Fig. 29. To better illustrate the variable height sup-ports, the final configuration is shown with the columns thatprop the grid-shell in said position. We observe that the formis indeed doubly-curved as desired, with a symmetric apexheight of 24.99 m, and an opening height of 13.90 m, bothmeasured from the zero level.

Fig. 29 Doubly-curved bowtie grid-shell: isometric view of the PEMform. Note that the grid-shell support columns are plotted only to showheight fluctuation of boundary nodes, but they are not part of analysisnor optimization

The member sizing is done after the form-finding stageusing the stress ratio method with a numerical dampingof η = 0.5. The adopted material properties, admissi-ble stress reduction and load combination are the same asused in Section 5.1.1. The member sizing results are plot-ted in Fig. 30, with a maximum cross-sectional area of13, 515 mm2.

5.3.2 GSM solution

The initial domain extrusion assumes a flat configurationof the surface (i.e. z = 0 everywhere), and is extrudedto a height of 24 using 10 elements in thickness as shownin Fig. 31a. The structure is loaded by the self-weight ofthe panels equal to 4.5 kN/m2 and an upward wind suctionof 0.5 kN/m2, both calculated based on the current surfaceconfiguration.

After the optimal (pin-jointed) solution is found, a newsurface is calculated (20) and the loads are updated. Thedomain is again extruded around the calculated surface,solved and updated a total of 6 times. With the exceptionof the very first domain, the surface is extruded equallyin both directions. The extrusion thickness is continuouslydecreased throughout the iterations: the second domain isextruded to a total thickness of 12 and is shown in Fig. 31b;the sixth and final domain is extruded to total thicknessof 2.4 and can be seen in Fig. 31c. The interpolated sur-face obtained from the last iteration in the process is shown

0

0.5

1a/ amax Plot, amax= 13515 mm2

Fig. 30 Doubly-curved bowtie grid-shell: area ratios based on thePEM form


Fig. 31 Doubly-curved bowtiegrid-shell: domaindiscretizations used in theground structure generationalgorithm (GSM). a Initialdiscretization; b Domaindiscretization after the 1st


refinement; d Final interpolatedsurface geometry.

(a) (b)

(c) (d)

in Fig. 31d. The ground structure solution (Fig. 32a) pro-vides information on the optimal load-path of the forces inthe optimized grid-shell geometry, with the relative magni-tude of these forces illustrated by the member thickness: thesurface works as two independent synclastic domes inter-connected by an anticlastic region in the middle; i.e. thereis (relatively) no force exchange between the two synclas-tic regions. The resulting (interpolated) height shown inFig. 32b does not depend on user-defined parameters, and itclosely approximates the (tessellated) optimal shape.

Symmetry of the interpolated surface is enforced in thisproblem. The numerical nature of the interior-point methodsolver, which stops (within some tolerance) before reachingan absolute vertex of the feasible domain, together with the

(a)

0

5

10

15

20

24.65

-1.43

(b)

Fig. 32 Doubly-curved bowtie grid-shell: optimized ground structureobtained after 5 domain refinements (6 analysis processes). a Memberlayout; b Optimized height

relative flatness of the objective near the optimum, causethe symmetry of this problem to be lost. This loss of sym-metry is unnoticeable after a single step. However, after theseries of 6 domain refinements, load updates, optimizationand interpolation steps, this asymmetry is accentuated (notethe minor differences in Fig. 32a). Thus, for a symmetricproblem (like the one addressed here), it is recommended toenforce symmetry of the interpolated surface.

5.3.3 FDM solution

The FDM process uses the same domain discretization as inthe PEM section (see Fig. 28). The force density of all barsis defined as 106 kN/m. The resulting form converges after7 iterations and is shown in Fig. 33. As done for the PEMform, the final configuration is shown with the columns thatprop the grid-shell in said position to better illustrate thevariable height supports. We observe that the form is indeeddoubly-curved as desired, with an symmetric apex height of24.98 m, and an opening height of 19.60 m, both measuredfrom the zero level.

5.3.4 Convergence history

In the form-finding process, the nodal loads are design-dependent. Thus, iterations are needed to obtain a convergedform. The convergence history for the bowtie exampleis shown in Fig. 34. For the PEM and FDM, the result

Fig. 33 Doubly-curved bowtie grid-shell: isometric view of FDMform. Note that the grid-shell support columns are plotted only to showheight fluctuation of boundary nodes, but they are not part of analysisnor optimization


Fig. 34 Doubly-curved bowtie grid-shell: convergence history for theform-finding methods (PEM, GSM, and FDM)

converges after 5 and 7 iterations, respectively. Convergenceis achieved once the residual (2) is less than a specified tol-erance. For the bowtie example, Nf = 180 and ε = 10−3.For the GSM, a specific number of iterations (6) is specified,and the residual is evaluated.

5.3.5 Layout investigation

Figure 35 shows the PEM, GSM and FDM form of thedoubly-curved bowtie grid-shell considering XY (planeview) and XZ (front view) projections. Note that the GSM

form is obtained using the interpolated surface with a struc-tured mesh similar to the PEM and FDM form. In the leftcolumn of Fig. 35, the red dashed line denotes the horizon-tal location (along Y dirction) of the top of the left edge archin the FDM form diagram. We observe that the FDM formdiagram has a smaller footprint as the edge arch curves sig-nificantly towards the center. In the right column of Fig. 35,the dark dashed lines denote the apex heights. We observethat the PEM form diagram is the least vaulted in the sensethat in the front view, most of the top boundary is lowerthan the apex height. On the other hand, the FDM form dia-gram is the most vaulted, and the GSM form diagram is inbetween the other two diagrams.

5.3.6 Tonnage evaluation and comparison among themethods

From a design point of view, the effectiveness of a layoutcan be evaluated by the material tonnage. The closer to theoptimal layout the design is, the lower the tonnage. Here weuse preliminary member sizing results to estimate (a lowerbound of) the material tonnage. Table 2 shows the estimatedtonnages and the vertical load sums for the three formsobtained by PEM, GSM and FDM for the doubly-curvedbowtie grid-shell.

From Table 2, we observe that the GSM form requires theleast material tonnage among the three forms. In addition,the load sums are different as load forces are dependent onthe geometry. The load sums of the PEM form and the GSM

Fig. 35 Doubly-curved bowtiegrid-shell: comparison of thePEM, GSM and FDM formconsidering XY (plane view)and XZ (front view) projections


Table 2 Doubly-curved bowtie grid-shell: tonnage and load sumcomparison of the designs obtained by the three form-finding methods

Method PEM GSM FDM

Tonnage [ton] 71.1 63.2 71.9

Normalized Tonnage 1.12 1.00 1.14

Load sum [kN] 44629 45287 36515

The tonnage is calculated by multiplying the total material volume bythe material density. Assume that the construction material is A500 Gr.B carbon steel with a density of 7.85 ton/m3. The normalized tonnageis calculated by dividing the tonnage by the GSM tonnage

form are comparable. The vertical load summation of theFDM form is lower than the other two, because the FDMform has a smaller footprint. As a result, the total tributaryarea for load calculation is smaller. Comparing both thetonnage and the vertical load sum, we observe that the GSMform supports the largest vertical load with the least amountof material.

To understand why the GSM design is more efficient,we investigate several parameters related with the load path.One parameter of interest is the Michell number (Michell1904), which is defined as

∑ |P |L, where P is the mem-ber force (positive for tension)and L is the member length.Another important quantity, Maxwell number (Maxwell1890), is defined as

∑P ·L. In our cases, the grid-shell is or

is close to compression-only and thus the Michell numberand the Maxwell number are of similar magnitudes. Accord-ing to Maxwell’s theorem (Maxwell 1890), the Maxwellnumber equals the sum of the force-distance product fromall forces. Thus∑

P · L =∑

FT D (21)

where F is the external force vector and D is the distancevector pointing from an arbitrary origin to the point wherethe force F is applied. With the above definition, the con-cerned parameters are calculated and tabulated in Table 3.Specifically, we calculate the total force-distance producttogether with the components from the load forces andreaction forces, respectively.

First of all, we verify that the calculation is correctby noticing that the normalized tonnages are equal to thenormalized Michell numbers. For fully-stressed truss struc-tures, the Michell number is linearly correlated with thematerial volume (Baker et al. 2013), as∑

|P | L =∑

σadmAL = σadm

∑

AL = σadmV (22)

where A denotes the member cross sectional area and V

is the total material volume. Similarly, the tonnage is alsolinearly correlated with V , as

Tonnage = ρV (23)

Table 3 Doubly-curved bowtie grid-shell: Michell number, Normal-ized Michell number, Maxwell number and force-distance product ofthe designs obtained by the three form-finding methods

Method PEM GSM FDM

Michell Number [kN · m] 2.57 × 106 2.29 × 106 2.60 × 106

Normalized Michell number 1.12 1.00 1.14

Maxwell number [kN · m] −2.57 × 106 −2.28 × 106 −2.60 × 106

Force-distance product[kN · m]

−2.57 × 106 −2.28 × 106 −2.60 × 106

Force-distance prod-uct from load forces[kN · m]

−6.83 × 105 −8.32 × 105 −7.11 × 105

Force-distance product fromreaction forces [kN · m]

−1.89 × 106 −1.44 × 106 −1.89 × 106

The normalized Michell number is calculated by dividing the Michellnumber by the GSM Michell number

where ρ is the material density. Thus the normalized ton-nage should be equal to the normalized Michell number ifthe calculation is correct.

From Table 3, we observe that the GSM form providesthe least magnitude of the force-distance product (equiva-lently, the Michell number or the magnitude of the Maxwellnumber) among the three forms. Due to the large loadsum and relatively vaulted form, the GSM form has thelargest magnitude of the force-distance product from loadforces. However, the GSM form results in smaller reactionforces in the XY plane than the other two forms, and itsforce-distance product from reaction forces is also smaller.Overall, the total force-distance product of the GSM form isstill smaller than that of the other forms. Thus, we concludethat the structural efficiency of the GSM form comes fromthe more benign horizontal reaction forces.

6 Discussion

Several aspects can influence the choice of the form-finding method. One aspect of concern is the computational

Table 4 Computational cost of the form-finding methods used in thenumerical examples section. The computational costs are measured inreal time in seconds a

Method PEM GSM FDM

Dome 2 1168 0.1

Pavilion 17 19635 0.3

Bowtie 4 773 0.2

aThe computations were conducted using MATLAB R2015b on adesktop computer with an Intel® Xeon® CPU E5-1660 v3 @ 3.00GHzprocessor and an installed memory (RAM) of 64 GB


Table 5 Qualitativecomparison of the form-findingmethods presented

Method PEM GSM FDM

Computational complexity moderate heavy mild

Physics equilibrium, mechanics, and compatibility equilibrium equilibrium

Initial configuration dependency yes no no

Pre-defined connectivity yes no yes

complexity, which can be preliminarily evaluated by meansof the computational runtime as shown in Table 4. Thisaspect, as well as other metrics of interest are qualitativelycompared in Table 5.

In summary, the PEM yields solutions with full physi-cal consideration of the standard constitutive relationships.Thus, all the parameters required have direct physical mean-ings, and the method can be easily understood from anintuitive point of view. A key drawback of the method is thedependence of the resulting form on the initial configura-tion. However, it may also be argued that this dependencecan be exploited to drive the solution towards some desir-able form.

The GSM with transmissible loads requires no a prioriassumption on the geometry nor the member connectivity,thus making it the most agnostic of the three methods. How-ever, it ranks high in computational complexity and requiresinterpretation (or rationalization) of the results. The result-ing configuration does not depend on additional parameterssuch as member stiffness or force density; it only requiresthe footprint geometry. In other words, for a given problem,the method converges to a unique grid-shell configuration.

The FDM is the most computationally inexpensive of thethree form-finding methods. A major drawback of FDM isthat the force density, a parameter that the method heav-ily relies on, may not have a clear physical interpretation.Therefore, the outcome from the FDM is relatively hard topredict and control, relying on the user’s experience or somedegree of trial-and-error. While the FDM does not dependon the initial configuration, connectivity information is stillrequired (see Appendix A for further information).

7 Concluding remarks

In this paper, two form-finding methods for grid-shelldesigns are proposed: a method based on minimum potentialenergy, and a method based on an augmented form of theground structure method, including domain refinement. Theload derivation scheme used in the present work allows forvarious types of static loads, which can be applied to loadpanels, bar members, and/or directly at the nodes. Theseloads can be design-dependent, and thus the form-findingprocess is iterative. The results from the two proposed form-finding methods, together with the widely-known force

density method, were compared with three design examples.These examples demonstrate the capability of the form-finding methods to handle various load types, unstructureddomains, and polygonal discretizations. In particular, weinvestigate the layout and tonnage for the doubly-curvedbowtie grid-shell to compare the structural efficiency of theforms obtained with different methods.

Last but not least, we note that the three form-findingmethods (two proposed and one reviewed) do not considerbuckling in their form-finding nor sizing stages. Considera-tion for elastic buckling can be integrated into the proposedmethods provided that a section family is defined (referto Section 3.4). However, the results should be consideredas an initial approach towards a complete design, whichcomplies with local building codes (e.g. International CodeCouncil 2015; ACI Committee 318, American ConcreteInstitute 2014; American Institute of Steel Construction2011).

Acknowledgments The authors acknowledge the financial supportfrom the US National Science Foundation (NSF) under projects#1559594 (formerly #1335160), which is a GOALI (Grant Opportuni-ties for Academic Liaison with Industry) project with SOM (Skidmore,Owings & Merrill LLP), and project #1321661. Ms Haley Simms pro-vided valuable comments that contributed to improve the manuscript.We are also grateful for the endowment provided by the RaymondAllen Jones Chair at the Georgia Institute of Technology. The informa-tion provided in this paper is the sole opinion of the authors and doesnot necessarily reflect the views of the sponsoring agencies.

Appendix A: Force density method formulation

The formulation of the force density method presented herefollows the procedure in Schek (1974), and is presented herefor the sake of completeness.

The nodes are numbered from 1 to Nn and bar members(or branches) from 1 to Nb. For any bar member j , there are2 corresponding nodes with number cj and dj . The Nb ×Nn

branch-node matrix Cs is constructed as follows:

Cs (j, i) =⎧

⎨

⎩

+1 for cj = i

−1 for dj = i

0 otherwise(24)

The branch-node matrix Cs can be subdivided into C andCf , which corresponds to the columns associated with free


and fixed nodes, respectively. The unknown nodal coordi-nates are defined as x, y, and z; and the coordinates of thefixed nodes as xf , yf and zf . The vectors denoting the coor-dinate difference between 2 connected nodes are u, v, andw (in the x, y and z coordinates respectively). Thus, therelations between the nodal coordinates and the coordinatedifferences are:

u = Cx + Cf xf

v = Cy + Cf yf

w = Cz + Cf zf (25)

For each element j in the network its length is calculated

as �j =√

u2j + v2

j + w2j , and its force density is defined

as qj = nj/�j , where ni is the member internal force.The member lengths are grouped in a vector �, the mem-ber forces in n, and the user-defined force densities for allmembers in the network in q.

We denote the diagonal matrices of vectors u, v, w, � andq as U, V, W, L and Q. The nodal force equilibrium canthen be expressed as:

CT UL−1n = fxCT VL−1n = fyCT WL−1n = fz (26)

where fx , fy , and fz, are the nodal force vectors in the x, y

and z coordinates, respectively. Using the definition of forcedensity in (26) we obtain:

CT Uq = fxCT Vq = fyCT Wq = fz (27)

where a simple transformation gives:

Uq = Qu

Vq = Qv

Wq = Qw (28)

Using the definition of coordinate difference (25) and thedefinitions in (27) and (28), we obtain:

CT QCx + CT QCf xf = fxCT QCy + CT QCf yf = fyCT QCz + CT QCf zf = fz (29)

Finally, defining the matrix D as D = CT QC, and Df asDf = CT QCf , a linear system for the free (unknown)nodal coordinates that ensures force equilibrium at everynode is obtained:

Dx = fx − Df xf

Dy = fy − Df yf

Dz = fz − Df zf (30)

Appendix B: Nomenclature

A Correspondence matrix of nodes belonging to atransfer group

a Vector of member cross-sectional areasa

�ij Area of one single triangle associated with the

edge nodes i and j

aki Cross-sectional area of the i-th bar in the k-th

iterationakg Cross-sectional area of bars in the g-th group in the

k-th iterationBT Nodal equilibrium matrixD Distance vectorF External force vectorf Nodal load vectorfxy Nodal loads associated with the x and y coordinatesfz Nodal loads associated with the z coordinatefv Vector of loads associated with each transfer groupf•i Load vector applied directly at the i-th nodef�i Line loads associated with the edgesf�i Area loads associated with the load panelski Reduced bar stiffness for the i-th member used in

the PEM form-finding processki Real bar stiffness for the i-th memberL Member length� Vector of member lengths��i Length change for the i-th memberNf Number of degrees of freedomn Vector of member forcesn�

ij Unit normal vector of the triangle associated withthe edge nodes i and j

P Member forcePE Potential Energyq Self weight by unit areas+ Slack variables associated with tensions− Slack variables associated with compressionu Displacement vectorunew Displacement vector in the current iterationuold Displacement vector in the previous iterationvi Vector from a polygon’s center of mass to the i-th

nodeV Material volume of the resulting structurew Projected area load acting on the load panelsβ Transfer group’s load direction vectorγ Material specific weight of the bar membersη Numerical damping for the stress-ratio methodκ Ratio of tension to compression stress limit, i.e.

σT /σC

ρ Material density of the bar membersσadm,i Admissible stress associated with the i-th barσi Stress in the i-th barσC Compression stress limitσT Tension stress limit


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