On Elastic Geodesic Grids and Their Planar to Spatial Deployment
Stefan Pillwein, Kurt Leimer, Michael Birsak, Przemyslaw Musialski
OOn Elastic Geodesic Grids and Their Planar to Spatial Deployment
STEFAN PILLWEIN,
TU Wien
KURT LEIMER,
TU Wien
MICHAEL BIRSAK,
KAUST
PRZEMYSLAW MUSIALSKI,
NJIT and TU Wien
We propose a novel type of planar–to–spatial deployable structures thatwe call elastic geodesic grids. Our approach aims at the approximation offreeform surfaces with spatial grids of bent lamellas which can be deployedfrom a planar configuration using a simple kinematic mechanism. Such elas-tic structures are easy–to–fabricate and easy–to–deploy and approximateshapes which combine physics and aesthetics. We propose a solution basedon networks of geodesic curves on target surfaces and we introduce a set ofconditions and assumptions which can be closely met in practice. Our formu-lation allows for a purely geometric approach which avoids the necessity ofnumerical shape optimization by building on top of theoretical insights fromdifferential geometry. We propose a solution for the design, computation, andphysical simulation of elastic geodesic grids, and present several fabricatedsmall-scale examples with varying complexity. Moreover, we provide anempirical proof of our method by comparing the results to laser-scans of thefabricated models. Our method is intended as a form-finding tool for elasticgridshells in architecture and other creative disciplines and should give thedesigner an easy-to-handle way for the exploration of such structures.CCS Concepts: •
Computing methodologies → Shape modeling ; Opti-mization algorithms.Additional Key Words and Phrases: geometric modeling, fabrication, elasticdeformation, physical simulation, architectural geometry, elastic gridshells,active bending
ACM Reference Format:
Stefan Pillwein, Kurt Leimer, Michael Birsak, and Przemyslaw Musialski.2020. On Elastic Geodesic Grids and Their Planar to Spatial Deployment.
ACM Trans. Graph.
39, 4, Article 125 (July 2020), 12 pages. https://doi.org/10.1145/3386569.3392490
Design and construction of structures composed of curved elasticelements has a long history in the field of architecture. Alongsidetheir aesthetical aspects imposed by nature, they have a lot of func-tional advantages: they are compact, lightweight and easy to build;nonetheless practicable, durable, and of high structural performance.They have been utilized for a long time dating back to ancientvernacular architecture for formal as well as for performance rea-sons, however, the possibilities of their form-finding in the pastwere limited [Lienhard et al. 2013].
Authors’ addresses: Stefan Pillwein, [email protected], TU Wien; KurtLeimer, [email protected], TU Wien; Michael Birsak, [email protected], KAUST; Przemyslaw Musialski, [email protected], NJIT and TU Wien.© 2020 Association for Computing Machinery.This is the author’s version of the work. It is posted here for your personal use. Not forredistribution. The definitive Version of Record was published in
ACM Transactions onGraphics , https://doi.org/10.1145/3386569.3392490. αα Fig. 1. A deployed elastic geodesic gridshell (top) and its planar lattice in therest state (bottom) fabricated of wooden lamellas. The deployment of thewhole kinematic system is based on changing angle α , such that α → α . Fortunately, the currently available computational capabilitiesand advances in computer science open up avenues for direct mod-eling of complex shapes composed of elastically bending members.This goes beyond traditional architectural design and allows to aimat many general purpose products composed of such elements. Therange of potential objects encompasses gridshells, formwork, panel-ing, various types of furniture, sun and rain protectors, pavilionsand similar small-scale buildings, home decoration and accessories,like vases, bowls, or lamps, etc., and finally, also elements of future’sfunctional digital fabrics that can be utilized in engineering as wellas in fashion.This vision leads directly to the objective of this paper: a designerprovides a target surface and a computational method finds a planargrid of flat lamellas, that—when deployed—approximates the surfacewell. Figure 1 shows a planar and a deployed grid of wooden strips,where a surface with the curved lamellas being tangential to it canbe imagined. The joints between the lamellas allow for rotation andpartially also for sliding. As the lamellas connecting opposite edgesof the planar boundary quadrilateral are not parallel to each other,the grid is rigid in the plane. Given the flexibility of wooden lamellaswith regard to bending and twisting, the grid is not rigid in space.By adjusting only one degree of freedom, for example the angle α → α at one corner, the planar kinematic configuration elasticallybends continuously into a spatial gridshell which approximates thedesired surface. The deployment process is governed by the rules ofphysics, seeing the lamellas as thin elastic minimal energy beams,allowed to bend as well as to rotate and slide at their intersections.Our goal is to find a suitable planar setup of the lamellas that canbe deformed into a spatial grid, fitting the target surface as closelyas possible. To achieve this goal, we propose a solution based on ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. a r X i v : . [ c s . G R ] J u l networks of geodesic curves on the target surface. We introducea set of conditions and assumptions which can be met closely inpractice and restrict the grids to geodesics. However, at the sametime, it allows us to develop a purely geometric solution whichbuilds on top of theoretical background from differential geometry.An advantage of our approach is to omit numerical shape opti-mization and to provide a solution which allows for easy explorationof designs of geodesic curve networks. To produce large scale grid-shells, further considerations will be needed, however, our maingoal is geometric modeling and form-finding. Our work providesinsights into that domain, also due to the fact that it uses intrinsicsurface geometry only. In summary, the contributions of this paperare the following: • We identify a specific case of the inverse design problem ofspatial elastic grids which can be formulated using geometricconsiderations only. This formulation allows us to find a gridwhich is perfectly planar and can be isometrically deformedin an elasto-kinematic manner to a desired spatial grid. • We derive a mathematical method for form-finding of suchgeodesic grids based on differential-geometric properties ofgeodesic curves. In particular, we introduce distance maps and cladding functions which allow for efficient finding ofsuitable configurations without expensive numerical shapeoptimization. • Finally, we introduce physical simulation and a simple fabri-cation method for wooden small-scale elastic geodesic grid-shells and perform empirical measurements which prove thevalidity of our approach.In the following section we review related work and in Section3 we provide a set of preliminary considerations necessary for ourformulation. In Section 4 we provide the details of our geometricderivation, and in Section 5 we propose an adapted physical simula-tion. In Section 6 we present and evaluate our results. Finally, wediscuss and conclude the work in Sections 7 and 8.
Developable Surfaces.
This topic has a long tradition in computergraphics and architectural geometry [Pottmann et al. 2015]. A lot ofattention has been paid to the approximation of freeform surfaceswith developable strips [Pottmann et al. 2010; Wallner et al. 2010],which can be fabricated from 2d flat material-sheets by cutting. Bybending and combining them, complex freeform surfaces can beerected. Also paneling of surfaces with planar tiles [Eigensatz et al.2010] or with general planar polygons [Chen et al. 2013] have beenproposed. Another way is the division of shapes into principal stripswhich bend automatically if combined [Takezawa et al. 2016]. Onthe theoretical side, a novel representation of developable surfacesusing quadrilateral meshes with appropriate angle constraints [Ra-binovich et al. 2018] or a definition of developability for trianglemeshes [Stein et al. 2018] have been proposed recently. Also discretegeodesic parallel coordinates for modeling of developable surfaceswere proposed [Wang et al. 2019]. All these works aim at the designof developable surfaces, which, due to their isometric properties, canbe fabricated from planar sheets. However, they do not incorporatea planar-to-spatial elastic deployment.
Deployable Surfaces.
One more way to easily construct spatialshapes from flat sheets is by appropriately folding paper [Massarwiet al. 2007; Mitani and Suzuki 2004], which is inherently relatedto the Japanese art of Origami [Dudte et al. 2016]. Another set ofworks deals with curved folding and their efficient actuation fromflat sheets to spatial objects [Kilian et al. 2008, 2017a]. Our workis related to these approaches in terms of being deployable from aplanar initial state, however, the main difference is that our gridsare elastic and approximate doubly-curved surfaces.In fact, a lot of attention has been paid to the design of doubly-curved surfaces which can be deployed from planar configurationsdue to the ease of fabrication. One way of achieving this goal is byusing auxetic materials [Konaković et al. 2016] which can nestle todoubly-curved spatial objects, or in combination with appropriateactuation techniques, can be used to construct complex spatial ob-jects [Konaković-Luković et al. 2018]. The main difference to ourapproach is that these structures do not use elastic bending to reachthe actual spatial shape.
Elastically Deployable Surfaces.
An interesting way to deploy sur-faces is to utilize the energy stored in planar configurations in orderto approximate shapes, for instance using prestressed latex mem-branes in order to actuate precomputed planar geometric structuresinto freeform shapes [Guseinov et al. 2017], or to predefine flexi-ble micro-structures which deform to desired shapes if set undertension [Malomo et al. 2018]. A combination of flexible rods andprestressed membranes lead to Kirchhoff-Plateau surfaces that alloweasy planar fabrication and deployment [Pérez et al. 2017]. Thesemethods achieve their planar-to-spatial configuration from elastictension in the network, either due to prestressing in the planar stateor by setting appropriate boundary conditions. The latter approachis more closely related to ours, however, instead of structure opti-mization, we build on top of the differential geometric properties ofgeodesic curves on the target surfaces. Thus, our method is based onthe assumption that the elastic elements can bend and twist, but notstretch and must therefore maintain the same length in the planaras well as in the spatial configuration.
Wire Surfaces.
Our work also contributes to surface approxima-tions using grids. This is not a novel approach, and previous workshave tackled this topic. For example, approximations of surfaceswith meshes based on Chebyshev nets [Garg et al. 2014], as well aswith wires that are deformed in planar configurations and assem-bled together [Miguel et al. 2016] to abstract a spatial shape, havebeen proposed. In contrast to us, these works do not focus on elastic-planar-to-spatial deployment nor on elasticity of the networks.
Physical Surfaces.
A number of methods which aim directlyat computational design of physically valid and stable architec-tural structures have been proposed. For example, design of self-supporting masonry surfaces [Vouga et al. 2012] or the design ofunreinforced masonry surfaces [Panozzo et al. 2013]. Also the pro-cess of erection of such objects has been computationally explored[Deuss et al. 2014]. Moreover, methods for fast interactive form-finding of physically stable structures [Tang et al. 2014], for the min-imization of material usage under stability constraints [Kilian et al.2017b], or physically plausible tensegrity structure design [Pietroni
ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. n Elastic Geodesic Grids and Their Planar to Spatial Deployment • 125:3 et al. 2017] have been proposed. Our method is related in termsof the goal of achieving structurally stable shapes. In turn, thesemethods do not utilize elastic bending for deployment or stability.
Classical Geometric Surfaces.
In classic differential geometry, geo-desic nets on surfaces which can be mapped onto a geodesic net ona different surface (including a plane) have been analyzed by Voss[1907] and Lagally [1910]. Regarding to their analysis, arc-lengthpreserving mappings of continuous geodesic nets onto each otherrequire rhombic geodesic nets, i.e., need a parametrization of thesurface with the net curves as parameter curves and E = G in thefundamental form. The resulting Liouville surfaces are very limitedin shapes, and therefore not useful for our freeform design purpose. Gridshells and Active-Bending.
The idea of gridshells—structuresthat gain their strength and stiffness through their curvature —wereintroduced by Shukhov for the Rotunda of the Panrussian Exposition[Shukhov 1896] and further pursued by famous architects, e.g., byFrei Otto for the construction of the roof of the Multihalle at theMannheim Bundesgartenschau [Happold and Liddell 1975].The introduction of the active bending paradigm [Lienhard et al.2013] together with enhanced and easy-to-use computational meth-ods increased the interest of the scientific community in systemati-cally utilizing elastic bending to realize curved shapes. Until recentadvances in computer science they could only be form-found em-pirically [Gengnagel et al. 2013].Existing design approaches are often based on particular kinds ofsurface curves, e.g., curvature lines [Schling et al. 2018]. Emergingconcepts for the erection of elastic gridshells facilitate the construc-tion process or even eliminate the need for scaffolding [Quinn andGengnagel 2014].Architectural works which aim at the approximation of gridshellsand combine lightweight structural design with aesthetics [Soriano2017] also inspired our work. Soriano et al. [2019] also proposedmechanisms for the deployment of geodesic gridshells using an evo-lutionary solver to form-find the grids. However, the design processis rather complex and time consuming, using numerical gradient-free optimization methods. In contrast, our approach is based ongeometric considerations and omits expensive computations. Be-sides gridshells, kinetic structures, bending plate structures, andtextile hybrids form a new class of structures explored in the activebending research community [Lienhard and Gengnagel 2018].Recently [Panetta et al. 2019] introduced an interactive approachfor finding deployable grid structures. Their method requires theuser to create an initial grid design by iterating between layoutediting and grid simulation steps. Once an overall satisfying shapeis found, the layout is then optimized to reduce the internal elasticenergy of the flat assembly state and the deployed target state.In contrast, our design approach only requires the user to providea target surface patch. Based on its geometry, our algorithm pro-duces a grid layout to approximate the target surface patch whendeployed. Furthermore, our approach guarantees that the planarconfiguration is in a zero-energy state.
Fabrication and Elastic Simulation.
The computer graphics com-munity started to deal with fabrication and computational design[Bermano et al. 2017], for this reason many novel methods aim at aaaa a
Fig. 2. The principle behind our planar to spatial deployment system. Toprow: all members of a family are parallel and rigid, the kinematic linkagecan move freely in the plane. Bottom row: non parallel layout produces adeadlock when trying to change the shape, inner members are too long.Allowing members to elastically deform, they buckle out of plane. fast but physically valid simulations. Our simulation is based onthe method of discrete elastic rods [Bergou et al. 2010, 2008], whichhave been adapted and utilized for works on sparse rod networks[Malomo et al. 2018; Pérez et al. 2015; Vekhter et al. 2019]. Recentlythis method has been also used for the simulation of hemisphericalelastic gridshells [Baek and Reis 2019].
The main idea behind our planar-to-spatial deployment is based ona very simple kinematic mechanism, as depicted in Figure 2. It isa special case of a planar quadrilateral four-bar linkage with rigidmembers, rotating joints and one degree of freedom.If we change the angle at one corner and all links of a family areparallel, the system can move freely in the plane (Figure 2, top row).If we introduce stiff inner links which are not parallel, the system isdeadlocked. By introducing bending and twisting flexibility to themembers, they buckle out of plane in order to preserve their lengthand form a spatial grid (Figure 2, bottom row). To construct such amechanism, the lengths of the members must match on the surfaceas well as in the planar configuration. Mathematically, this behaviorcan be modeled by geodesic curves on a surface.A geodesic locally minimizes the arc length between two distinctpoints and maintains its length under isometric deformations of thesurface. Moreover, its principal normal falls into the surface normal,i.e., it allows normal curvature, but prohibits geodesic curvature. Asa consequence, a carefully chosen network of such curves can beused to build the elasto-kinematic deployment mechanism and atthe same time to abstract the surface’ characteristics.Additionally, gridshells of the nets should be easy to manufacture,transport, assemble, and deploy. To meet these properties in practice,we use thin straight lamellas with a cross section ratio of about 1 : 10,creating a distinct weak axis for easy bending and a strong axis thatprohibits bending. These lamellas can be wrapped on a surfaceand interpreted as tangential strips with a geodesic centerline. Alsotheir connections, which are essential for the kinematic deployment,imitate the intersections of geodesics well: the lamellas can rotatewith the axis of rotation being always parallel to both of the principalnormals of the centerlines, and their connections can slide alongthe tangents of the centerlines.
ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020.
Fig. 3. Overview of our approach and the notation. Left: the user selectsfour corners on a desired target surface. Center: the surface patch P withmembers of the д and h family. Each family is parameterized with pairs ( u , u ) and ( v , v ) respectively. Right: a corresponding planar patch P with corresponding members of the д and h family (cf. Section 3.2). Besides apparent advantages of easy production, geodesics offera lot of theory and give us a great set of tools to analyze surfacepatches and find suitable solutions.
The input to our computational system is a surface patch P whichis a convex bounding shape defined on a designer created targetsurface by four corners. They are connected by geodesic curves onthe surface which constitute the boundaries of the surface patch P as depicted in Figure 3. The output of our system is a planarquadrilateral, denoted as planar patch P , filled with interconnectedstraight lines. Its corners are the counterparts of the spatial corners.The patches consist of two families of grid members: д , h -membersare geodesics on the surface patch, and д , h -members are their corre-sponding straight lines in the planar patch with matching lengths (cf.Figure 3). The grid members are parameterized along the boundarieswith parameter-pairs ( u , u ) and ( v , v ) respectively. Using geodesics to model the grid members also poses restrictionson the representability of the target surfaces. There are two waysto compute geodesics: defining a start point and a direction vector,which has a unique solution, or defining a start and an end point,which delivers the shortest path between these two points, but doesnot necessarily have a unique solution [Polthier and Schmies 1998].To maintain the length of a curve between the boundaries, weneed to compute geodesics between two points on opposite bound-aries, so for our application we use the second case, which we willdenote as shortest geodesics from now on.A feature of shortest geodesics—namely the possibility of nonunique solutions—can have disadvantageous effects for the approxi-mation. It may happen that two points on a surface patch can beconnected by more than one shortest geodesic. The existence ofsuch points is linked to the Gaussian curvature K of the surface.They result in areas of the patch P that cannot be covered withshortest geodesics connecting the boundaries. For the quality of theapproximation, it needs to be ensured that every point on patch P can be reached by a shortest geodesic of the д and h -curves family. Ifthis is not the case, surface features cannot be captured with shortestgeodesics and cannot be encoded in the planar grid.Figure 4 illustrates the problem: when drawing shortest geodesicsfrom point p to all points on the opposite boundary, the central area p ppp ir(p) Fig. 4. Shortest geodesics between point p and points on the oppositeboundary (top) and distance fields emanating from p (bottom). Left: thepeak area cannot be covered by shortest geodesics, cut locus L( p ) andinjectivity radius ir ( p ) are indicated. Right: Uncovered area sufficientlyreduced by smoothing (cf. Section 3.3). of high positive K remains uncovered and produces a gap in thecoverage. Taking a look at the distance field (Figure 4, left), we canidentify singularities as it approaches the opposite boundary. Thesesingularities form the cut locus L( p ) on P and each point ∈ L( p ) can be reached from p by two distinct geodesics of the same length.The geodesic distance d between p and its nearest point on L( p ) is called the injectivity radius ir ( p ) [do Carmo 1992] given as ir ( p ) = inf d ( p , L( p )) . Using a corollary of the Rauch comparison theorem [do Carmo1992] we obtain the following inequality: ir ( p ) ≥ π √ K max . (1)It gives us a lower bound for the injectivity radius ir ( p ) for each sur-face point p . Evaluating it at local peaks of Gaussian curvature K max serves as a quick check for the uniqueness of shortest geodesics.If the lengths of all members are smaller than the right hand sideof Expression (1), the patch can be used as it is. If this is not the case,the surface patch cannot be covered completely (unless the peak ison the boundary).Although Expression (1) indicates the existence of these areas,the size of the gaps remains unclear. Small gaps may not pose bigproblems for the quality of the approximation, while big gaps do.They indicate that there is a considerable difference in length be-tween the shortest geodesic next to the peak and the (start-direction)geodesic over the peak, thus the quality of the approximation of thesurface by the planar grid will be worse. In order to handle surfacepatches that cannot be covered with shortest geodesics completely,we propose an iterative smoothing procedure.To check for uncoverable areas around a Gaussian curvature peak p max , we first compute two distance fields: one from the peak p max and one from the boundary point p , where we choose p to be theclosest point to p max on the boundary.They provide us with distances d ( p , q ) to the points q of theopposite boundary as well as d ( p , p max ) and d ( p max , q ) . We computethe minimum of d ( p , p max ) + d ( p max , q ) − d ( p , q ) , which is reachedat a point q . If the minimum is close to zero, the peak p max is notproblematic and there is no gap. If not, the factor: η = d ( p , p max ) + d ( p max , q ) d ( p , q ) is used to measure the size of the gap. In order to remove the unreach-able gaps, we perform Laplacian smoothing of P with cotangent ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. n Elastic Geodesic Grids and Their Planar to Spatial Deployment • 125:5 du u du u Fig. 5. Distance fields on a planar patch P and a surface patch P , computedfrom a single point shown on the left. By sampling all point-pairs alongcorresponding ( u , u ) -domains, we create distance maps D u ( u , u ) and D u ( u , u , α ) . Note that the planar distance map D also depends on theshape of P and thus the angle α (cf. Section 4.2). weights iteratively [Desbrun et al. 1999], until η falls below a certainthreshold η max . In practice we choose η max = . Our goal is to find a grid of geodesics on P , which can be “planarized”to P with a certain angle α . The grid curves are allowed to reducetheir curvature and torsion but should keep their total lengths as wellas the lengths between points of intersection. At each configuration,the grid curves should be geodesics on a hypothetical surface.Inversely, the planar grid is deployed to a spatial grid as the planarangle approaches the spatial angle, i.e., α → α such that the planarcorners approach their spatial counterparts, and the planar straightlines bend to geodesic curves tangential to the target surface.In order the meet these requirements, both the planar and thespatial grids need to obey the following geometric demands:(i) Length correspondence : All straight lines д , h have the samelengths as their corresponding geodesics д , h . (ii) Boundary correspondence : On boundaries, the ( u , u ) and ( v , v ) coordinates of connections are identical for the 2dand the 3d grid.(iii) Bijectivity of correspondence : Each point on one boundaryhas one and only one corresponding point on the oppositeboundary, defining a grid member uniquely.(iv)
Convexity of boundary : the corresponding patches P and P need to be convex.Criterion (iv) is necessary, since otherwise the kinematic mechanismcan run into a deadlock. It is fulfilled if each of the four inner anglesof P is less than π , which can be argued with the triangle inequalityof the surface metric and the convexity of sufficiently small areas[do Carmo 1992].In the following, we introduce mathematical tools which allowto identify geodesic grids which fulfill all posed criteria. We explainthe process only for one family of members. Note however thatthe shape of the planar patch is chosen with respect to both fami-lies, satisfying interconnecting constraints, thus they are not foundindependently. du u eeff du u F(i, ) a u u F(i, ) a u u Fig. 6. Intersection of distance map D u ( u , u , α ) for planar patch in blueand distance map D u ( u , u ) for surface patch in orange. Left: proper in-tersection, fulfilling the constraints (cf. Sec. 4.3). Center: partial intersection,providing an invalid cladding function F u . Right: piecewise linear functions F u of both cases evaluated on a discrete grid (cf. Section 4.3). As a tool to match the distances on the surface patch P and theplanar patch P , we introduce distance maps D u and D v . To createthem, distance fields are spread from all points p ( u ) on one bound-ary to all points q ( u ) on the opposite boundary, measuring thegeodesic distances d ( p ( u ) , q ( u )) between them (cf Figure 5, left).Transforming the distances into the ( u , u , d ) -3d space creates arepresentation of the geodesic lengths of the surface patch, which isillustrated in Figure 5. While the distance maps of the surface patch D u ( u , u ) and D v ( v , v ) have a predefined angle α induced bythe choice of the surface patch and depend only on the coordinates u , u and v , v respectively, the distance maps of the planar patch D u ( u , u , α ) and D v ( v , v , α ) also depend on the angle α . Thechoice of that angle changes the shape of the planar grid and hencealso the shapes of the distance maps D u and D v .In our implementation, distance maps are represented as quadmeshes; their resolution is chosen according to the resolution of theinput surface mesh. In practice, it is around 100 ×
100 vertices.
In this section we derive the cladding functions which determinethe distribution of the corresponding members in P and P . Thisis done via finding a suitable angle α , such that the grid criteriadefined in Section 4.1 are fulfilled.The cladding function F u is built by first projecting the intersec-tion of the distance maps D u and D u to the u , u -plane (respec-tively, F v is built using a projection to the v , v -plane). Points onthis function represent geodesics which connect opposite bound-aries and have the same length on both the planar and the spatialpatch. Please recall that the shape of the distance map D u ( u , u , α ) also depends on the choice of the angle α , hence the shape of thecladding function does as well.Grid criteria (i) and (ii) are fulfilled by the nature of these func-tions. Our goal is now to determine the parameter α such that alsogrid criteria (iii) and (iv) are fulfilled. This implies that the claddingfunction F u must be continuous and bijective over the entire do-main, which means its first order partial derivative (cid:219)F u w.r.t. u should nowhere reach 0 nor ∞ (cf. Figure 6, right).Additionally, bounds can be set on (cid:219)F u in order to avoid too steepor too flat tangents, which would result in a strong concentration of ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. a aa a β γ β γ
Fig. 7. The influence of α on the cladding with grid members: its choice affects the distribution and coverage of the members д and h on the surface patch P .Right: the shape of the cladding function F u with indicated members (cf. Section 4.3). Please note also the angles β and γ , which are used to determineminimum distances between lamellas with a certain width (cf. Section 4.4). members on a boundary and an uneven coverage of the patches P and P as shown in Figure 7. Moreover, if criteria (iii) and (iv) are notfulfilled, triangular member connections may appear in the planargrid, destroying the kinematic deployment mechanism.With this picture in mind, we denote the cladding functions as u = F u ( u , α ) and v = F v ( v , α ) with u , u ∈ [ , ] ( v , v respectively). Refer to Figure 7 for adepiction. Please note that for the cladding functions to exist, thelength of the diagonals e , f of the surface patch P and e , f (cf.Figure 6) of the planar patch P must fulfill the following inequality: ( e − e ) · ( f − f ) < . (2)In other words, this inequality is a necessary condition for a properintersection of the distance maps. Figure 6 depicts how the diagonals e , f of the surface patch and e , f of the planar patch appear in thedistance maps.To find a feasible domain for the angle α under the condition ofbijective cladding functions F u ( u , α ) and F v ( v , α ) , we formulateit as an optimization problem using Expression (2) as a constraint.Note that at ( , ) and ( , ) distance maps always intersect, so F u is always defined there. However, the function might be not definedor not continuous over the entire domain of u ∈ [ , ] , as depictedin Figure 6, center. To deal with this case, we introduce a piecewiselinear parametric representation F u ( i , α ) = ( u ( i ) , u ( i ) , α ) givenover the entire domain and range of F u (cf. Figure 6, right).Using the slopes of the segments (cid:219) F u and (cid:219) F v simultaneously as con-straints, we cast the following optimization problem to determine afeasible domain for the angle:min α s.t. ( e − e ) · ( f − f ) < k min < (cid:219) F u ( i , α ) < k max , . . . nk min < (cid:219) F v ( i , α ) < k max , . . . n , (3)with n being the number of segments and with k min and k max beingslope bounds which we have determined empirically as k min = . k max =
10. We evaluate (cid:219) F u , (cid:219) F v using finite differencing (cid:219) F u ( i , α ) = ∆ u ( i ) ∆ u ( i ) at all segments, as shown in Figure 6, right. To tackle the case where (cid:219) F u = ∞ , we set its value to c ∆ u with c ≫ k max ; cases with (cid:219) F u = F . We solve Problem (3) using sequential quadratic programmingwith numerical gradients w.r.t. α . First we determine the minimumfeasible α min with the lower bound for α from the convexity restric-tions of grid criterion (iv). Then we find a maximum feasible α max using the same concept. Values of α between these bounds ensurethe cladding functions F u and F v to be bijective.Note, that setting bounds for α also makes it possible to introducedesigner constraints on the shape of the planar patch P . In practice,we choose α min for our examples, which results in a compact planarpatch design. After checking the validity of the surface patch (with smoothing, ifneeded) and fixing α , we choose the number and positions of the gridmembers. Patches with many curvature features (compare Figure4) obviously need a minimum number of well placed members tocapture all surface features well. For this specific example, all thebumps of the surface have to be encoded in the planar grid.Our approach for fitting grid members is a geometrically moti-vated heuristic. It reuses the information from the intersections ofthe respective distance maps D u and D u in the ( u , u , d ) space(cf. Section 4.3). Along their intersection curve, we can constructan associated function C u ( s ) of geodesic lengths d of the members.Its maxima and minima correspond to longest or shortest geodesics( д i , д i ) on the surface patch P and provide good candidates forphysical members of the elastic grid.Hence, members are first placed at the extrema of C u ( s ) and nextat the extrema of the curvature of C u ( s ) . The first pass ensures tocover major features (large peaks) since these members correspondto locally longest and shortest geodesics. The second pass ensuresto capture finer features (smaller bumps), since the correspond-ing members are also locally the longest or the shortest members,however on a smaller scale. Figure 8 depicts these steps.In order to avoid the members to be placed too close to each otheror to overlap, we compute the offsets d ( + ) ( β ( u ) , γ ( u ) , w m ) and d (−) ( β ( u ) , γ ( u ) , w m ) which give the minimum distance between a member and its pre-ceding and subsequent neighbors. The angles β ( u ) and γ ( u ) arethe enclosed angles between a member and the boundaries, and w m is the member width (cf. Figure 7).If members are too dense, we prioritize them using the absolutevalue of curvature of C u ( s ) . The assumption behind this choice isinspired by the observation that the more curved C u locally is, themore distinct surface features the corresponding geodesic captures. ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. n Elastic Geodesic Grids and Their Planar to Spatial Deployment • 125:7 C u ( s ) s sC u ( s ) d d Fig. 8. One iteration of the member placement procedure. Left: membersplaced based on geometric features. Right: additional members placed inthe gaps and distributed without affecting the initial members. Bottom rowdepicts the C -function with indicated members (cf. Section 4.4). If members are too sparse, we add new members in the gaps,which fulfill the restrictions imposed by d ( + ) and d (−) . After addingthem, we minimize the sum of the squared distances to existingmembers in order to achieve a more equal distribution.Note that the same procedure is applied to D v and D v to obtainthe function C v and the members of the ( h , h ) family. Deploying the planar grid with rotational-only connections deliversan approximation of the surface patch P , but the centerlines ofthe physical lamellas cannot become geodesics on P . The reasonis that they are held back by their fixed intersections with innermembers of the other family. This restriction is a consequence ofthe grid criteria (i) and (ii). Note that as shown by Lagally [1910],an arbitrary geodesic grid cannot be planarized in general.To address this issue, we introduce sliding notches at the connec-tions of inner members. These notches provide two translationaldegrees of freedom at each connection, enabling the respective mem-bers д i and h j to slide by the notch lengths ℓ д i , ℓ h j (cf. Figure 9).We can identify unique optimal sliding directions and notch lengthsfrom comparing the difference of the locations of the connectionsw.r.t. the arc length between the geodesic members д , h and theirplanar counterparts д , h .In other words, traversing an inner member pair ( д i ( s ) , д i ( s )) ∈( д , д ) along its arc length parameters s and s , the notch length ℓ д i at a particular connection is given by ℓ д i = s − s . The notch length ℓ h i along the ( h i ( s ) , h i ( s )) member pair is givenin an analogous way (cf. Figure 9).The corresponding sliding directions are given by the sign of thisequation. If each connection slides to the end of both its notches, thecenterlines of the lamellas move towards the geodesics on P . Dueto the extra degrees of freedom, notches enable the structure to takea lower energy state by reducing the torsion and curvature of themembers. The notches are physically realized by simply elongatingthe holes of the corresponding lamellas. When changing the angle α → α , an elastic grid buckles out ofplane into a curved configuration. While the surface patch P has afixed shape, the grid can deform to multiple spatial configurations, Fig. 9. Left: deployment without notches, where orange dots indicate opti-mal connections in the spatial state. Right: Notches ℓ д , ℓ h computed forone particular connection q (cf. Section 4.5). since an elastic grid for a specific surface patch is also suitable forall isometric surface patches. This is given by the fact that our gridsare constructed using the intrinsic metric on P , which is invariantto isometries. Isometries of a surface can be imagined by bendingthe surface without stretching it.To force the grid into the desired configuration, we introduceadditional anchors which pin connections of members to fixed pointson the target surface. We systematically introduce them on selectedconnections of inner members with boundary curves, such that theypush the elastic grid into a configuration in agreement with theshape of P .For practical reasons, we only allow anchors on the boundaries.In particular, we identify points of locally extreme curvature on theboundary geodesics and filter for small extrema. The connections ofmembers closest to these points serve as anchor locations (cf. Fig. 10). To simulate the physical behavior of the deployed grid, we use asimulation based on discrete elastic rods [Bergou et al. 2010] andbuild upon the solution of [Vekhter et al. 2019]. We refer the readerto those papers for the details. Note, that the associated materialframes of the rods do not need to be isotropic, which allows us alsoto model the exact cross sections of lamellas with a ratio of 1 : 10.A central aspect of the kinematics of elastic geodesic grids isthe ability of grid members to slide at connections, denoted in thefollowing as q . In general, they do not coincide with the verticesof the discretized grid members. To handle them, we introducebarycentric coordinates β q to describe the location of a connectionon a rod-edge. We also take the physical thickness t of the lamellasinto account, which is modeled by an offset between the members д and h at each connection. Hence, a connection q consists of twopoints q д and q h with an offset t . Apart from sliding, members areallowed to rotate around connections about an axis that is parallelto the cross product of the edges q д and q h lie on. Simulation.
Our aim is to find the equilibrium state of the givenelastic grid, which corresponds to an optimization problem of mini-mizing the energy functional E = E r + E q + E a + E n + E p , where E r is the internal energy of the rods, E q is the energy of theconnection constraints, E a is the energy of the anchor constraints, E n is the energy of the notch-limit constraints, and E p is an addi-tional notch penalty term that also serves to account for friction.We perform the simulation by minimizing the entire energy E for ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020.
Fig. 10. The influence of anchors and notches on the example
Archway .Left: Anchors at the corners are not sufficient to push the grid into theright configuration. Center: Deployed state without notches, local bucklingand irregularities in smoothness can be observed. Right: Notches relax thestructure to a more natural, lower energy shape (cf. Sections 4.5 and 4.6). the rod centerline points x using a Gauss-Newton method in a simi-lar fashion as proposed by Vekhter et al. [2019]. In Section 6.2 weperform an empirical evaluation of the accuracy of the simulationby comparing it to laser-scans of the makes.For the sake of readability, we will define the constraint energyterms only for a single constraint each. E r is the sum of stretching,bending and twisting energies of each individual rod. As a fullexplanation of the DER formulation is out of scope for this paper,we refer the reader to the work of [Bergou et al. 2010] for a detaileddescription of these terms.The connection constraint energy E q is given by E q = λ q , (cid:13)(cid:13) q д − q h + tm д (cid:13)(cid:13) + λ q , (cid:13)(cid:13) q h − q д − tm h (cid:13)(cid:13) + λ q , (cid:13)(cid:13) ∠ (cid:0) m д , m h (cid:1)(cid:13)(cid:13) , with m д and m h denoting the material vectors of д and h at q respec-tively. The term tm accounts for the thickness of the rods, while λ q , and λ q , are the constraint weights for the position and directionterms.The anchor constraint energy E a ensures that both the position q and material vector m of the given connection do not deviate fromthe position q a and material vector m a of the corresponding anchor.It is given by E a = λ a , ∥ q − q a ∥ + λ a , ∥ ∠ ( m , m a )∥ , with λ a , and λ a , as weights. This constraint applies to the gridcorners and anchors.The notch-limit constraint energy E n ensures that the connectionpoint remains within the bounds of the notch. They are specifiedby the notch length l and the sliding direction (cf. Section 4.5): E n = δ (−) (cid:18)
110 log (cid:16) β q − β (−) (cid:17)(cid:19) + δ ( + ) (cid:18)
110 log (cid:16) β ( + ) − β q (cid:17)(cid:19) , with β (−) and β ( + ) denoting the barycentric coordinates of the notchbounds on their corresponding edges. The term is only active whenthe connection lies on the same rod-edge as one of the notch bounds,so δ (−) = δ ( + ) = E p controls the movement ofa connection q between two adjacent edges. If q switches edges, itneeds to be reprojected to the neighboring edge at the next iterationof the simulation. Within an iteration, E p prevents q from moving Fig. 11. The effect of the weighting parameter µ in E p (from left to right):surface shaded with K and geodesics; µ = . , rods slide onto geodesics; µ = . , sliding in high K areas reduced (our setting); µ = , sliding isheavily reduced. Refer to Section 7.3 for a further discussion on µ . too far beyond the end of the current edge: E p = (cid:0) µ log (cid:0) ϵ + β q (cid:1)(cid:1) + (cid:0) µ log (cid:0) ϵ + − β q (cid:1)(cid:1) , with ϵ denoting how far q is allowed to move past the end of theedge and µ acting as a weighting parameter (we choose ϵ = . µ = . E p is not 0 even inside the edge, it penalizes very smallsliding movements that would otherwise accumulate over many it-erations. In other words, E p creates a pseudo-frictional effect, whichis controlled by µ . In a physical grid, friction creates a force actingagainst the sliding movement of a connection. If the driving forceof the movement and the frictional force counterbalance, the move-ment stops. This situation has an analogy in our grids. A connectionstops moving inside a notch if ∂ E q ∂ β q + ∂ E p ∂ β q = µ . Using our method, we have approximated a number of surfaceswhich are depicted in Figures 13 and 14. We used input surfaceswith positive and negative Gaussian curvature regions, as well aspurely elliptic and hyperbolic surfaces.The fabricated models we present in Figure 14 are made of limewood lamellas and placed on 3d-printed supports after assembly.To position the notches precisely, lamellas are laser-cut from thinlime wood plates. Members are connected by simply using screwsand nuts. The support structures fix the shape of the boundarymembers to anchors as described in Section 4.6 and also providecorrect orientation for the lamellas by inclined contact areas.
Quantitative Results.
In Table 1 we summarize quantitative resultsof our method for seven models (Figure 13 and 14). The presentedvalues RMS and RMS denote the root mean square distance be-tween grid vertices and the mesh representing P without and withnotches respectively. As can be seen, notches allow for closer prox-imity between the rods and P . Please note that the model width,depth and height listed in Table 1 are dimensionless and that wescale the model by a global factor for fabrication.The computation time for the geometric grid generation (c.f. Sec-tion 4) mainly depends on the mesh resolution of P , which also ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. n Elastic Geodesic Grids and Their Planar to Spatial Deployment • 125:9
Fig. 12. Comparison of the simulation result (Section 5) to a laser scan ofthe example
Double Vault . The figure shows the point cloud with simulationresults overlayed. The notches are indicated in red. The lamellas have crosssection of . . cm. The color indicates the L distances of the points tothe lamellas. The total RMS error of the comparison is . cm. determines the number of distance fields that are computed. Smooth-ing additionally requires the computation of several distance fieldsin every iteration. Simulation time of the deployed state of the gridwith and without notches mainly depends on the number of gridvertices. Evaluation of Simulation.
To evaluate the agreement of the sim-ulated results with the fabricated wooden makes, we used a state-of-the-art laser-scanning device (Metris MCA 36M7) to capture thedeployed gridshell. To enable precise agreement of the cartesiananchor coordinates q a and the point cloud, we registered them usingthe ICP algorithm.The material properties of the wood were not determined bytesting, but estimated using reference values for deciduous woods.Figure 12 shows the results of the comparison. Note that the rootmean square error between the point cloud and the simulated modelis 0.06 cm, which is only about half the thickness of a lamella. Table 1. Quantitative results of our method. We measure the root meansquare error (RMS) between the member centerlines and the target mesh:RMS refers to grids without notches and RMS to grids with notches.Timings are in seconds, t grid refers to the computation times of generatingthe geometric elastic grid, t refers to the simulation without notches and t to the simulation with notches. | M V | expresses the number of mesh verticesand | G V | the number of grid vertices. Captions refer to examples TorusWide, Waves Bump (Fig. 13), and Sphere, Double Vault, Waves, Archway, andTriple Vault (Fig. 14) respectively. Measured on an Intel Xeon E5-2687W v4. T.W. W.B. Sph. D.V. W. A.w. T.V.width 100 . . . . . . . . . . . . . . . . . . . . . | M V | | G V |
767 388 414 300 328 625 494 t smooth − − − − t grid t t Our grid design algorithm is implemented in Matlab, utilizing itssequential quadratic programming solver for solving the optimiza-tion Problem (3) using numerical gradients w.r.t. α . We furthermoreimplemented the DER-simulation in C ++ , building upon the frame-work of [Vekhter et al. 2019]. To compute the distance fields onthe surface patch P we use the VTP algorithm by [Qin et al. 2016].For the computation of the geodesic paths we use the algorithm forexact geodesics between two points by [Surazhsky et al. 2005]. In order to design general grids, the paths of the surface curvesneed to be flexible. In our method, we focus on geodesic curves dueto their properties, in particular allowing only the normal curva-ture on surfaces (cf. Section 3). The directions of the curves on thesurface can only be controlled by changing the angle α becauseof the restrictions induced by the cladding functions. Creating anelastic geodesic grid that approximates an arbitrary curve networkis therefore not possible.As a consequence of our design choice, cross sections of fabricatedmembers need to be rectangular with a high width to thickness ra-tio. While this ensures easy fabrication, at the same time it poses alimitation on the design space. As shown by Panetta et al. [2019], theshape-space of similar grid structures can be controlled by changingthe profile of cross sections. However, when using more complicatedcross sections, parts of them may buckle during deployment. Thiscauses nonlinearities in stiffness parameters requiring to account forbuckled cross sections. We avoid this necessary nontrivial update ofthe stiffness parameters, as the choice of our cross section minimizesthese geometric second order effects.Note that in our models, the size of the cross sections is uniform.Allowing different dimensions for every rod or even every segmentwould allow for an even better approximation of the surface patch. Elastic geodesic grids can only approximate surfaces, that are “clad-dable” by unique shortest geodesics. If this is not the case, oursmoothing algorithm ensures cladding, but surface details could belost. Also the number and the density of members influences therepresentable shapes. If the shape is of very high frequency geomet-ric details, it might not be representable by a too sparse networkof physical members. In turn, in order to ensure fabricability, onlya limited number of members is possible. This relationship is aninteresting issue for future work.To approximate the extrinsic shape of P , we introduce anchorson the boundaries of an elastic grid. They act as constraints onthe shape of the grid and are supposed to reduce the number ofpossible configurations to a single one. However, in some cases ourdefinition of anchors is not sufficient. Imagine a high-frequencysurface: fixed boundaries may not suffice to uniquely determinethe direction of inner bumps. Although we did not encounter thisproblem in our examples, there certainly exist surface patches thatrequire additional anchors inside the grid to pin down its shapeuniquely. ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020.
Torus Wide Waves Bump
Fig. 13. Computed and simulated results without make, renderings of the simulation and the planar grid. The orange lines follow our simulation with notches.The dark lines follow the shortest geodesics on P . Besides this geometric view on multiple deployed configurations,they can also be looked at from an equilibrium point of view. Ifdeployed and anchored correctly, a structure in equilibrium willmaintain its shape. Further conclusions about the nature of theequilibrium would require a sensitivity analysis which could give in-teresting insights to the properties of elastic grids like the pronenessto pop into a different configuration in a loading scenario.Notches allow the grid to relax into a lower energy state andincrease the accuracy of the approximation. If a grid without notchesis deployed, it cannot approximate the surface patch P , becausedistances between connections do not agree with the metric of P . The effects can be observed in local buckling of members andgeneral deviations from P (cf. Figure 10).Finally, the current definition of distance maps is not compatiblewith holes in the surface, so the surface patch needs to maintain asingle boundary. In our simulation, the energy term E p is not physical, nonetheless, itacts as a source of pseudo-friction. We incorporated it to speed up theconvergence of sliding movements and to make the simulation morerealistic. As E p causes connections to not fully utilize the notches,it interferes with the quality of the approximation (cf. Figure 10).However, in our simulated models we registered that successivelyincreasing µ first penalizes notches that belong to members withgeodesics in areas of high K . Here geodesics are sensitive to impre-cisions (e.g., from discretization of P or our numeric algorithm) andcan exhibit deviations from the desired optimal path. This results innotches that are overly long.The effects of E p penalize sliding in high K regions first, whichhelps to trim such locally overly long notches (c.f. Figure 13, WavesBump and Figure 14, Archway). Using the suggested settings, thereis no significant negative effect of E p on the quality of approxi-mation as Table 1 and the Figures 13 and 14 show. It would beinteresting to investigate a notch-penalty term that goes beyondimitating friction, but controlling the quality of the approximationvia systematically reducing notch-lengths. A further investigationinto similar concepts of handling notches is an attractive topic forfuture work.The used simulation is based on the DER formulation and there-fore uses the concept of linear material elasticity. It does not accountfor non-linear elastic effects like plasticity or the failure of members.Since we prescribe deformations in the deployment scenario, the resulting stresses have to be kept within an acceptable range. Thesearising stresses are higly influenced by crosssectional sizing. The deployment of an elastic grid is achieved by changing the angle α and applying additional bending to guide it to the desired extrinsicshape. While our treatment of the deployment process is limited tothe start and end configurations, without investigating intermediatestates, we expect the process to be feasible if the end configurationis physically sound. All our experiments performed in accordancewith this expectation, although a proof remains future work.While deploying our physical models, we encountered that thestatic friction of wood can hinder connections from sliding freely. Itthereby prevents the system from moving into a configuration oflower elastic energy. This can be countered by introducing someextra energy into the system that helps to overcome friction. Alsofinding fabrication methods that minimize friction between mem-bers are interesting problems to explore in the future.Our approach is intended as a form-finding tool for 2d-3d elasti-cally deployable gridshell structures. Although we only validatedour approach with small scale models, [Panetta et al. 2019] exam-ined the deployment of structures that use a similar deploymentmechanism, but are bigger in size. Investigating how our approachcan be adapted to the challenges of large scale architecture is aninteresting engineering problem and a potential topic for futurework. We presented a novel approach for computational design of elasticgridshell structures that approximate smooth freeform surfaces byplacing grid elements close to geodesic curves on the surface. Ourmethod is inspired by architecture and design, and aims at simple fab-rication, assembly, and most importantly at easy planar–to–spatialdeployment. Moreover, it should provide an easy to handle toolfor designers to create physically sound and aesthetically pleasingspatial grid structures based on the active bending paradigm.Our solution is based on theoretical considerations and combinesgeometrical background with physical simulation. We have pro-posed a concept for the computation and simulation of such elasticgrids. Additionally, we compared the results of the simulation toreal fabricated grids and show that they match very well. Finally,we presented a set of examples with varying Gaussian curvatureand fabricated a subset of them as wooden small-scale gridshells asa proof of our concept.
ACM Trans. Graph., Vol. 39, No. 4, Article 125. Publication date: July 2020. n Elastic Geodesic Grids and Their Planar to Spatial Deployment • 125:11
SphereDouble VaultWavesArchwayTriple Vault
Fig. 14. Computed, simulated, and fabricated results of our method. Left: computed planar grids and renderings of the simulation. The orange strips followour simulation with notches, the dark lines follow the shortest geodesics on P . Right: photographs of our makes. Best seen in the electronic version in closeup. ACKNOWLEDGMENTS
This research was mainly funded by the Vienna Science and Tech-nology Fund (WWTF ICT15-082) and partially also by the AustrianScience Fund (FWF P27972-N31). The authors thank Florian Rist,Christian Müller, and Helmut Pottmann for inspiring discussions,as well as Etienne Vouga and Josh Vekhter for sharing code.
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