Computational Design of Cold Bent Glass Façades
Konstantinos Gavriil, Ruslan Guseinov, Jesús Pérez, Davide Pellis, Paul Henderson, Florian Rist, Helmut Pottmann, Bernd Bickel
CComputational Design of Cold Bent Glass Façades
KONSTANTINOS GAVRIIL ∗ , TU Wien
RUSLAN GUSEINOV ∗ , IST Austria
JESÚS PÉREZ,
URJC
DAVIDE PELLIS,
TU Wien
PAUL HENDERSON,
IST Austria
FLORIAN RIST,
KAUST
HELMUT POTTMANN,
KAUST, TU Wien
BERND BICKEL,
IST Austria
Fig. 1. Material-aware form finding of a cold bent glass façade. From left to right: initial and revised panel layouts from an interactive design session withimmediate feedback on the glass shape and maximum stress (red color indicates panel failure). The surface design is then optimized for stress reduction andsmoothness. The final façade realization using cold bent glass features doubly curved areas and smooth reflections.
Cold bent glass is a promising and cost-efficient method for realizing doublycurved glass façades. They are produced by attaching planar glass sheets tocurved frames and must keep the occurring stress within safe limits. How-ever, it is very challenging to navigate the design space of cold bent glasspanels because of the fragility of the material, which impedes the form find-ing for practically feasible and aesthetically pleasing cold bent glass façades.We propose an interactive, data-driven approach for designing cold bentglass façades that can be seamlessly integrated into a typical architecturaldesign pipeline. Our method allows non-expert users to interactively edita parametric surface while providing real-time feedback on the deformedshape and maximum stress of cold bent glass panels. The designs are au-tomatically refined to minimize several fairness criteria, while maximalstresses are kept within glass limits. We achieve interactive frame rates byusing a differentiable Mixture Density Network trained from more thana million simulations. Given a curved boundary, our regression model iscapable of handling multistable configurations and accurately predicting theequilibrium shape of the panel and its corresponding maximal stress. Weshow that the predictions are highly accurate and validate our results witha physical realization of a cold bent glass surface. ∗ ACM Transactions onGraphics , https://doi.org/10.1145/3414685.3417843.
CCS Concepts: •
Applied computing → Computer-aided design ; •
Com-puting methodologies → Shape modeling ; Mixture modeling .Additional Key Words and Phrases: mechanical simulation, cold bent glass,neural networks, computational design, inverse design
ACM Reference Format:
Konstantinos Gavriil, Ruslan Guseinov, Jesús Pérez, Davide Pellis, Paul Hen-derson, Florian Rist, Helmut Pottmann, and Bernd Bickel. 2020. Computa-tional Design of Cold Bent Glass Façades.
ACM Trans. Graph.
39, 6, Article 208(December 2020), 16 pages. https://doi.org/10.1145/3414685.3417843
Curved glass façades allow the realization of aesthetically stunninglooks for architectural masterpieces, as shown in Figure 2. Thecurved glass is usually made with hot bending, a process wherethe glass is heated and then formed into a shape using a mold orusing tailored bending machines for spherical or cylindrical shapes.While being able to unleash these stunning designs from beingrestricted to flat panels, this process is laborious and expensive and,thus, an economic obstacle for the realization of exciting conceptssuch as the NHHQ skyscraper project by Zaha Hadid Architects(Figure 10). As a cost-effective alternative, in recent years, architectshave started exploring cold bending [Beer 2015]. Here, planar glasssheets are deformed by mechanically attaching them to a curvedframe. Cold bending introduces a controlled amount of strain andassociated stress in the flat glass at ambient temperatures to createdoubly curved shapes [Datsiou 2017]. Compared with hot bent glass,it has the advantage of higher optical and geometric quality, a wide
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. a r X i v : . [ c s . G R ] S e p Fig. 2. Examples ofcurved glass façades.Left: Fondation LouisVuitton, Paris, by FrankGehry. Right: Opus,Dubai, by Zaha HadidArchitects (Photo:Danica O. Kus). range of possibilities regarding printing and layering, the usageof partly tempered or toughened safety glass, and the possibilityof accurately estimating the stresses from deformation [Belis et al.2007; Fildhuth and Knippers 2011]. Furthermore, it reduces energyconsumption and deployment time because no mold, heating of theglass, nor elaborate transportation are required.However, designing cold bent glass façades comes with a challeng-ing form-finding process. How can we identify a visually pleasingsurface that meets aesthetic requirements such as smoothness be-tween panels while ensuring that the solution is physically feasibleand manufacturable? Significant force loads can occur at the con-nection between the glass and frame, and it is essential that thedeformation of the glass stays within safe limits to prevent it frombreaking.We propose an interactive, data-driven approach for designingcold bent glass façades. Starting with an initial quadrangulation ofa surface, our system provides a supporting frame and interactivepredictions of the shape and maximum stress of the glass panels.Following a designer-in-the-loop optimization approach, our systemenables users to quickly explore and automatically optimize designsbased on the desired trade-offs between smoothness, maximal stress,and closeness to a given input surface. Our workflow allows users towork on the 3D surface and the frame only, liberating the designerfrom the need to consider or manipulate the shape of flat panels –the optimal shape of the flat rest configuration of the glass panels iscomputed automatically.At a technical level, we aim to determine the minimum energystates of glass panels conforming to the desired boundary withoutknowing their rest configuration. Based on extensive simulationsof more than a million panel configurations with boundary curvesrelevant for our application domain, we observed the existence ofseveral (in most cases up to two) stable states for many boundarycurves. Identifying both minimum energy states without knowing therest configuration and potentially multiple stable states is a non-trivialproblem and cannot be easily computed using standard simulationpackages. Furthermore, as a prerequisite for enabling interactivedesign for glass façades, we need to solve this problem for hundredsof panels within seconds.To achieve these goals, we have developed a learning-basedmethod utilizing a deep neural network architecture and Gauss-ian mixture model that accurately predicts the shape and maximumstress of a glass panel given its boundary. The training data for thenetwork is acquired from a physics-based shape optimization rou-tine. The predictions of the trained network not observed originallyare re-simulated and used for database enrichment. Our model isdifferentiable, fast enough to interactively optimize and explore the shape of glass façades consisting of hundreds of tiles, and tailoredto be easily integrated into the design workflow of architects. Asa proof of concept, we have integrated our system into Rhino. Wehave carefully validated the accuracy and performance of our modelby comparing it to a real-world example, and demonstrate its ap-plicability by designing and optimizing multiple intricate cold bentglass façades.
Interactive design and shape optimization are areas that have aconsiderable history in Engineering [Christensen and Klarbring2008], architecture [Adriaenssens et al. 2014], and computer graph-ics research [Bermano et al. 2017; Bickel et al. 2018], including toolsfor designing a wide variety of physical artifacts, such as furni-ture [Umetani et al. 2012], cloth [Wolff and Sorkine-Hornung 2019],robotics [Megaro et al. 2015], and structures for architecture [Eigen-satz et al. 2010].Motivated by the digitalization of manufacturing, there is anincreased need of computational tools that can predict and sup-port optimizing the physical performance of an artifact during thedesign process. Several approaches have been developed to guar-antee or improve the structural strength of structures [Stava et al.2012; Ulu et al. 2017]. Focusing on shell-like structures, Musialskiet al. [2015] optimized their thickness such that it minimizes a pro-vided objective function. More recently, Zhao et al. [2017] proposeda stress-constrained thickness optimization for shells, and Giluretaet al. [2019] computed a rib-like structure for reinforcing shells, thatis, adding material to the shell to increase its resilience to externalloads. Considering both aesthetic and structural goals, Schumacheret al. [2016] designed shells with an optimal distribution of artis-tic cutouts to produce a stable final result. Although we share thegeneral goal of structural soundness, in our problem setting, wecannot change the thickness or material distribution. Additionally,even just determining the feasibility of a desired bent glass shaperequires not only solving a forward simulation problem, but alsoan inverse problem because the rest shape of the glass panel is apriori unknown. Finding an optimal rest shape is often extremelyimportant. Schumacher et al. [2018] investigated sandstone as build-ing material that is weak in tension, thus requiring computing anundeformed configuration for which the overall stress is minimized.Similarly, glass panels have a low tensile strength and are subjectto very high compression loads during the assembly process, whichmotivates the need for identifying minimal energy panels.Notably, several methods have recently been proposed to designdoubly curved objects from flat configurations [Guseinov et al.2017; Konaković-Luković et al. 2018; Malomo et al. 2018; Panettaet al. 2019; Peloux et al. 2013]. However, all these methods rely onsignificantly more elastic materials and are not targeted for usewithin an interactive design pipeline. In our application, having anaccurate estimation of the stress is critical to predict panel failureand interactively guide designers towards feasible solutions. Theneed to bridge the gap between accuracy and efficiency motivatesthe use of a data-driven approach.
Computational design of façades.
Covering general freeform sur-faces with planar quadrilateral panels is a fundamental problem in
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. omputational Design of Cold Bent Glass Façades • 208:3
Fig. 3. Doubly curved surface panelized using a planar quad mesh following the principal curvature network (left). This is the smoothest possible panelizationof this surface achievable with flat panels [Pellis et al. 2019]. The solution using cold bent glass panels designed with our method (right) shows much smootherresults. The bottom pictures show the corresponding zebra stripping for both solutions; clearly, smoother stripes are indicators of higher visual smoothness. architectural geometry and has received much attention [Glymphet al. 2004; Liu et al. 2006, 2011; Mesnil et al. 2017; Pottmann et al.2015]. The difficulties lie in the close relationship between the curva-ture behavior of the reference surface and the possible panel layouts.Problems occur especially in areas of negative curvature and if thedesign choices on the façade boundaries are not aligned with the cur-vature constraints imposed by planar quad meshes (Figure 3). Usingtriangular panels, the problems are shifted toward the high geo-metric complexity of the nodes in the support structure [Pottmannet al. 2015]. Eigensatz et al. [2010] formulated relevant aspects forarchitectural surface paneling into a minimization problem that alsoaccounts for re-using molds, thereby reducing production costs. Re-stricting the design to simple curved panels, Pottmann et al. [2008]presented an optimization framework for covering freeform sur-faces by single-curved (developable) panels arranged along surfacestrips. However, glass does not easily bend into general developableshapes, limiting the applicability of this technique for paneling withglass.A recent alternative for manufacturing doubly curved panels iscold bending of glass. A detailed classification and description of theperformance of cold bent glass can be found in [Datsiou 2017]. Evers-mann et al. [2016a] explored simulations based on a particle-springmethod and a commercially available FE analysis tool. Furthermore,they compared the resulting geometries to the measurements ofthe physical prototypes. For designing multi-panel façade layouts,Eversmann et al. [2016b] calculated the maximum Gaussian curva-ture for a few special types of doubly curved panels. This defined aminimal bending radius for exploring multi-panel façade layouts.Berk and Giles [2017] developed a method for freeform surface ap-proximation using quadrilateral cold bent glass panels. However,they limited their fabricability studies to two modes of deformation.Although conceptually simple, we found these approaches too lim-iting for general curved panels and, thus, have based our approachon a data-driven method.
Machine learning for data-driven design.
Finite element methods(FEM) are widely used in science and engineering for computingaccurate and realistic results. Unfortunately, they are often slowand, therefore, prohibitive for real-time applications, especially inthe presence of complex material behavior or detailed models.Dimensionality reduction is a powerful technique for improvingsimulation speed. Reduced space methods, for example, based onmodal analysis [Barbič and James 2005; Pentland and Williams 1989],are often used to construct linear subspaces, assuming that the de-formed shape is a linear combination of precomputed modes. Simu-lations can then be performed in the spanned subspace, which, how-ever, limits its accuracy, especially in the presence of non-linear be-havior. Non-linear techniques such as numerical coarsening [Chenet al. 2015] allow for the reduction of the models with inhomoge-neous materials, but usually require precomputing and adjustingthe material parameters or shape functions [Chen et al. 2018] of thecoarsened elements. Recently, Fulton et al. [2019] proposed employ-ing autoencoder neural networks for learning nonlinear reducedspaces representing deformation dynamics. Using a full but linearsimulation, NNWarp [Luo et al. 2018] attempts to learn a mappingfrom a linear elasticity simulation to its nonlinear counterpart. Com-mon to these methods is that they usually precompute a reducedspace or mapping for a specific rest shape but are able to performsimulations for a wide range of Neumann and Dirichlet boundaryconstraints. In our case, however, we are facing a significantly differ-ent scenario. First, we need to predict and optimize the behavior ofa whole range of rest shapes, which are defined by manufacturingfeasibility criteria (in our case, close to, but not necessarily perfect,rectangular flat panels). Second, our boundary conditions are fullyspecified by a low-dimensional boundary curve that corresponds tothe attachment frame of the glass panel. Instead, we propose directlyinfer the deformation and maximal stress from the boundary curve.Recently, data-driven methods have shown great potential forinteractive design space exploration and optimization, for example,
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. for garment design [Wang et al. 2019], or for optimized tactile ren-dering based on a data-driven skin mechanics model [Verschooret al. 2020]. An overview of graphics-related applications of deeplearning can be found in Mitra et al. [2019]. In the context of compu-tational fabrication, data-driven approaches were used, for example,for interactively interpolating the shape and performance of pa-rameterized CAD models [Schulz et al. 2017] or learning the flowfor interactive aerodynamic design [Umetani and Bickel 2018]. Al-though these methods are based on an explicit interpolation schemeof close neighbors in the database ([Schulz et al. 2017]) or Gaussianprocesses regression ([Umetani and Bickel 2018]), in our work, wedemonstrate and evaluate the potential of predicting the behaviorand solving the inverse problem of designing a cold bent glass façadeusing neural networks. This entails the additional challenge of deal-ing with multistable equilibrium configurations that, to the bestof our knowledge, has not been addressed before in a data-drivencomputational design problem.
We propose a method for the interactive design of freeform sur-faces composed of cold bent glass panels that can be seamlesslyintegrated in a typical architectural design pipeline. Figure 4 showsan overview of the design process. The user makes edits on a basequad mesh that is automatically completed by our system to a meshwith curved Bézier boundaries. Our data-driven model then inter-actively provides the deformed shape of the cold bent glass panelsin the form of Bézier patches conforming to the patch boundariesand the resulting maximal stress. This form-finding process helpsthe designer make the necessary decisions to avoid panel failure. Atany point during the design session, the user can choose to run oursimulation-based optimization method to automatically compute asuitable panelization while retaining some desirable features suchas surface smoothness and closeness to the reference design.In Section 4, we show how the base mesh controlled by the useris extended through special cubic Bézier curves to the set of patchboundaries. Each patch is delimited by planar boundary curvesof minimum strain energy. These special Bézier patch boundariesare convenient for modeling glass panels because they facilitatethe construction of supporting frames while providing a smoothapproximation to the desired design.Bézier boundaries do not convey any information on the de-formed or undeformed configuration of the panel. Our method usessimulation to compute both configurations of the panel such thatcertain conditions are met, which are derived from manufacturingconstraints. First, current panel assembly does not guarantee C continuity at the boundary between neighbor panels because it isvery hard to enforce normals along the frame in practice. Second,glass panels have a low tensile strength and are prone to breakingduring the installation process in the presence of large tangentialforces. Following these criteria, we let the panel be defined by theboundary curve of the frame and compute both the deformed andundeformed shapes of the panel such that the resulting total strainenergy is minimal. In this way, we ensure our panelization has atleast C continuity and that the assembly of the panels requiresminimal work, thus reducing the chances of breakage. In Section 5, flat panel shapessurface geometryand stress estimationDNNframe Béziercurvesquadpanelization material-aware interactivedesign and optimizationinput referencesurface shapeoptimization Fig. 4. Overview of our design tool workflow. The user makes edits ona quadrilateral base mesh and gets immediate feedback on the deformedshape and maximal stress of the glass panels. When needed, an optimizationprocedure interactively refines the surface to minimize safety and fairnesscriteria. If desired, any target reference surface may be used to initialize theprocess. we describe in detail the physical model and the computation ofminimal energy panels.Panel shape optimization provides us with a mapping between ourdesign space of Bézier boundary curves and theoretically realizablecold bent panels, in both undeformed and deformed configurations.Our material model also accurately estimates the maximum stressendured by the glass. The user is free to interactively edit the basemesh while receiving immediate feedback on the maximum stress,but this neither ensures the panels will not break, nor does it fosterthe approximation of a target reference surface. To achieve this goal,we solve a design optimization problem: Bézier boundary curves areiteratively changed to minimize closeness to an input target surface(and other surface quality criteria) while keeping the maximumstress of each panel within a non-breaking range. In Section 7, wedescribe in detail our formulation of the design optimization.However, accurately computing the minimal energy panels iscomputationally very challenging, which makes physical simula-tion infeasible for being directly used within the design optimizationloop. Furthermore, the mechanical behavior of glass panels undercompression often leads to multiple stable minimal energy config-urations depending on the initial solution. This complicates theoptimization even more: not only does the problem turn into acombinatorial one, but there is no algorithmic procedure that can ef-ficiently count and generate all existing static equilibria given someboundary curve. We address this challenge by building a data-drivenmodel of the physical simulation. First, we densely sample the spaceof the Bézier planar boundary curves and compute the correspond-ing minimal energy glass panels together with an estimation of themaximum stress. Then, we train a Mixture Density Network (MDN)to predict the resulting deformed shape and maximum stress giventhe boundary of the panel. The MDN explicitly models multistabilityand also allows us to discover alternative stable equilibria that canbe used to enrich the training set. In Section 6, we elaborate onthe characteristics of our regression network and our sampling andtraining method. The trained regression network can finally be usedto solve the inverse design optimization problem. Once the user is
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. omputational Design of Cold Bent Glass Façades • 208:5 satisfied with the design, our shape optimization procedure gener-ates the rest planar panels, which are ready to be cut and assembledinto a beautiful glass façade.
A panelization of an architectural surface is built upon a quadran-gular base mesh M = ( V , E , F ) , where the vertices V determine thepanel corner points and each quad face in F is filled by one curvedpanel. In practice, the user interacts with the design tool by makingedits to M through any parametric mesh design method, in ourcase a Catmull-Clark subdivision from a coarser mesh (see Figure 1and the supplementary video). This helps achieve fair base meshesand gives a reasonable control for edits. However, any other meshdesign scheme could be potentially used.Each edge in E is then automatically replaced by a planar cubicBézier curve defining the boundaries of the panel, and the innercontrol points are predicted using our regression model. In thissection, we describe the details for getting from M to the unionof curved panels. Moreover, we show how to express the panelswith a minimal number of parameters, which are later used for thedata-driven model. We model each glass panel as a bicubic Bézier patch S : [ , ] → R ,which is defined by 16 control points c ij , where i , j ∈ { , , , } . Thecorner points c , c , c , c are vertices in M . Each edge e of M is associated with a patchboundary curve C e . To describe its construction, we focus on asingle edge e with vertices v , v , and we denote the unit vectorsof the half-edges originating at v i by e i (see Figure 5). We optedfor planar boundary curves representing panel’s edges; thus, wefirst define the plane Π e that contains C e . We do this by prescribinga unit vector s e ∈ R that lies in Π e and is orthogonal to e . Notethat this parameterization is non-injective (vector − s e representsthe same plane Π e ), but its ambiguity can be resolved using thecompact representation in Section 4.2. The two inner control pointsof the cubic curve C e lie on the tangents at its end points. Tangentsare defined via the angles θ i they form with the edge. Hence, theunit tangent vectors are t i = e i cos θ i + s e sin θ i , In view of our aim to get panels that arise from the flat ones throughbending, we further limit the cubic boundary curves to those witha minimal (linearized) bending energy, as described in [Yong andCheng 2004]. For them, the two inner control points are given by v i + m i t i , i = ,
2, with m = ( v − v ) · [ t − ( t · t ) t ] − (cid:0) t · t (cid:1) , and m is obtained analogously by switching indices 1 and 2.The boundary of S is thus fully parameterized by the 4 cornervertices c ij , i , j ∈ { , } , the 4 edge vectors s e , and the 8 tangentangles θ (2 per edge). This parameterization of the panels is used inthe regression model and the design tool implementation describedin Sections 6 and 7, respectively. v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t v + m t s e s e s e s e s e s e s e s e s e s e s e s e s e s e s e s e s e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e Π e t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ C e C e C e C e C e C e C e C e C e C e C e C e C e C e C e C e C e eeeeeeeeeeeeeeeee Fig. 5. Parameterization of a panel boundary curve from a pair of tangentdirections t , t corresponding to dual halfedges. The final boundary curve(red) is computed by minimizing a linearized bending energy. The interior control points c ij , i , j ∈ { , } express the shape of a panel enclosed by a given boundary. Wefound that within the admissible ranges of the boundary parameters,any optimal glass panel (see a detailed description in Section 5) canbe very closely approximated by fitting the internal nodes of theBézier patch. Moreover, we need to regularize the fitting, becausefor a given Bézier patch, it is possible to slide the inner controlpoints along its surface while the resulting geometry stays nearlyunchanged.We denote the vertices of the target panel shape x i and the corre-sponding vertex normals n i . For every x i , we find the closest points y i on the Bézier surface and fix their coordinates in the parameterdomain. The fitting is then formulated as follows:min c ij (cid:13)(cid:13)(cid:0) y i ( c ij ) − x i (cid:1) · n i (cid:13)(cid:13) A i + w B (cid:213) k E k ( c ij ) , i , j ∈ { , } , where A i are Voronoi cell areas per panel vertex, E k are the lengthsof all control mesh edges incident to the internal nodes, and w B isthe regularizer weight that we set to 10 − . To achieve independenceof rigid transformations, we express the inner control points in anorthonormal coordinate frame adapted to the boundary. The framehas its origin at the barycenter of the four corner points. Using thetwo unit diagonal vectors g = c − c ∥ c − c ∥ , g = c − c ∥ c − c ∥ , the x -axis and y -axis are parallel to the diagonal bisectors, g ± g ,and the z -axis is parallel to b = g × g , which we call the facenormal. The panel boundary is used as an input to a neural network topredict the shape and stress of the minimal energy glass panel(s)conformal to that boundary. Thus, it is beneficial to reduce the inputto the essential parameters, eliminating rigid transformations of theboundary geometry.We consider d ∈ R to be the vector of the six pairwise squareddistances of vertices c ij , i , j ∈ { , } . Given d , we can recover twovalid mirror-symmetric embeddings of the 4 corner points. Assum-ing that the order of the vertices is always such thatdet ( c − c , c − c , c − c ) ≥ ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. holds, the embedding is unique up to rigid transformations. Weassume such a vertex ordering from now on. The plane Π e for eachedge is then characterized by its oriented angle γ e with the facenormal b . Finally, we define p ∈ R as the concatenation of thedistance vector d , the 4 edge plane inclinations γ e , and the 8 tangentangles θ (2 per edge). The vector p is used as an input to the neuralnetwork defined in Section 6. Our method leverages mechanical simulation to create a large datasetof minimal energy panels that conform to cubic Bézier bound-aries. Given some boundary curves, we are interested in findingdeformed glass configurations that are as developable as possible.Non-developable panels result in high tangential forces that com-plicate the installation of the panel and increase the chances ofbreakage. By finding the pair of deformed and undeformed shapesof the panel that minimize the strain energy subject to a fixed frame,we ensure the work required for its installation is minimal, helpingto reduce the tangential force exerted at the boundary. This datasetis used to train and test a model that predicts the deformed stateand maximum stress of such panels, which is suitable for rapidfailure detection and inverse design. In this section, we describe thesimulation method used for the computation of the deformed andundeformed states of a minimal energy glass panel.
We aim to define a mechanical model that is sufficiently preciseto accurately predict glass stresses under small strains, but stillsuitable for the fast simulation of a very large number of deforma-tion samples. Consequently, we make some reasonable simplifyingassumptions in a similar way to Gingold et al. [2004]. We geomet-rically represent a glass panel as a planar mid-surface extruded intwo opposite normal directions by a magnitude h /
2, where the totalthickness h is much smaller than the minimal radius of curvature ofthe reference boundary frame. We assume the lines normal to themid-surface always remain straight and do not undergo any stretch-ing or compression. Under a linearity assumption, the followingexpression for the volumetrically defined Green’s strain tensor E with offset z in the normal direction can be derived: E ( x , ¯x , z ) = ¯E ( x , ¯x ) + z ˆE ( x , ¯x ) . (1)Here, x and ¯x are, respectively, the deformed and undeformed con-figurations of the mid-surface, and ˆE is the quadratic bending strain,equivalent to the shape operator of the deformed mid-surface. Themembrane strain ¯E = . ( F T F − I ) is the in-plane Green’s straintensor defined in terms of the deformation gradient F . We refer toGingold et al. [2004] for a detailed explanation of the continuousformulation. We will focus on our discrete formulation, which hasbeen previously considered by Weischedel [2012]. We discretize glass panels using a triangulated surface mesh with N nodes and M edges. We separately consider the membrane andbending strains from Equation (1) and define two correspondingmid-surface energy densities integrated over the panel thickness. MPa > Fig. 6. A comparison between the stress distribution produced with thetypical shape operator used in, e.g., [Pfaff et al. 2014] (left), and ours, assuggested in [Grinspun et al. 2006] (right). The latter is much smoother andresults in a more reliable estimation of the maximal stress.
To discretize membrane strain,we assume piecewise constant strains over FEM elements. In thiscontext, in-plane Green’s strain is computed as follows: ¯E = A (cid:213) i s i ( t j ⊗ t k + t k ⊗ t j ) , (2)where A is the triangle area, s i = ¯ l i − l i ( i ’th edge strain), t j and t k are the two other edge vectors rotated by − π /
2. For computing thecorresponding membrane energy density integrated over the panelthickness, we adopt the Saint Venant-Kirchhoff model:¯ W = h (cid:16) λ (cid:0) Tr ¯E (cid:1) + µ Tr (cid:0) ¯E (cid:1)(cid:17) , (3)where λ and µ are, respectively, first and second Lamé parameters. The bending strain is directly de-fined as the geometric shape operator of the continuous surface.We compute a discrete approximation of the shape operator usingthe triangle-based discretization suggested in Grinspun et al. [2006],which faithfully estimates bending strain regardless of the irregu-larity of the underlying triangle mesh. In addition to mesh nodes,this metric considers additional DoFs per edge by defining the devi-ation of the mid-edge normals from the adjacent triangle-averageddirection: ˆE = (cid:213) i θ i / + ϕ i Al i ( t i ⊗ t i ) . (4)Here, θ i is a dihedral angle associated with the edge i and ϕ i isthe deviation of the mid-edge normal toward the neighbor trian-gles normals. Overall, the discrete deformed state of the glass panelis defined with a vector x ∈ R N + M . We denote the correspond-ing undeformed configuration ¯x . The bending energy density inte-grated over the panel thickness is then defined by the Koiter’s shellmodel [1966]: ˆ W = µh (cid:16) λλ + µ (cid:0) Tr ˆE (cid:1) + Tr (cid:0) ˆE (cid:1)(cid:17) . (5)Contrary to the simpler thin shell bending models commonly usedin computer graphics [Pfaff et al. 2014], the discrete shape opera-tor suggested in [Grinspun et al. 2006] more faithfully capturesprincipal strain curves and outputs smoother stress distributions(Figure 6). In the next section, we will describe how we find theminimal energy configuration corresponding to some given Bézierboundaries. ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. omputational Design of Cold Bent Glass Façades • 208:7
Given a parametric design of a façade composed of a quadrangularmesh with Bézier curves at the edges, we aim to find a suitable pan-elization using cold bent glass. Although deforming a glass panel toconform to cubic boundaries is feasible, the fragility of this materialimposes non-trivial constraints on the maximum amount of stresstolerated by the panels. Thus, we designed a method to compute theglass panel design with the lowest possible strain energy that stillwill fit our installation constraints. Note that in practice, the existingassembly methods do not preserve normals across neighboring pan-els; thus, we restrict our problem to guarantee only C continuity.By computing the fabricable panel with the lowest possible totalstrain energy, we minimize the net work required to install the paneland notably reduce the local tangential stress suffered by the glass.Overall, our pipeline takes as an input the 16 control points of theBézier boundaries and automatically computes both the deformed x and undeformed ¯x configurations of a planar glass panel thatconforms to the boundary and has minimal energy. This is done intwo steps. At first, we generate a regular mesh that uni-formly discretizes the parameter domain of the surface (a unitsquare) and lift the vertices to an initial Bézier patch defined by theboundaries. In our pipeline, such a patch can be obtained in twoways: • Generated by our prediction model, when shape optimizationis used to enrich the database or to compute the undeformedshape of the final design panels. • Initialized as a surface patch with zero twist vectors at thecorners (a quad control mesh has parallelograms as cornerfaces) when shape optimization is used to build the initialdatabase.The lifted mesh is conformally flattened with minimal distortion.We uniformly resample the boundary of this mesh targeting a totalnumber of edges M b and triangulate the interior using Delaunaytriangulation with the bounded maximal triangle area. Finally, thevertices of this mesh are mapped back to the parameter domainand lifted to the initial Bézier patch. As a result, we obtain an ini-tial configuration for a deformed glass panel conforming to Bézierboundaries and its corresponding undeformed configuration. The initial solution is not in static equilibriumand has arbitrarily high stresses. We compute the minimal energyconfiguration by minimizing the discrete strain energies defined inEquations 3 and 5 over deformed x and undeformed ¯x configurations. Fig. 7. Comparison between two alternative stable equilibria for a givenBézier boundary. The two resulting panels produce radically different Gaussmaps (right), leading to very distinguishable reflection effects.
We refer to the vector of all the deformed nodes at the boundaryand the internal nodes as b and i , respectively. To reduce the com-plexity of the problem and keep a high-quality triangulation of theundeformed configuration, we assume internal nodes at the restconfiguration ¯i are computed through Laplacian smoothing of theboundary vertices ¯i = L¯b . Then, the aforementioned minimizationproblem results in the following:min i , ϕ , ¯b W ( x , ϕ , ¯b ) + R ( ¯b ) , (6)where W is the sum of all strain energy terms, ϕ are the mid-edgenormal deviations, and R is a regularization term removing thenull space due to the translation and rotation of the undeformedconfiguration. In particular, it is formulated as a soft constraint: thecentroid of the boundary nodes is fixed to the origin, and one of thenodes has a fixed angle with the x-axis. Note that we only considerundeformed boundary nodes ¯b as DoFs of the optimization; aftereach solver iteration, we project the internal nodes’ coordinates ¯i through Laplacian smoothing. In addition, the boundary nodes ofthe deformed configuration remain fixed and conforming to Bézierboundaries.As can be seen in Figure 7, minimizing Equation 6 does not al-ways produce a unique solution. For a given boundary, glass panelscan potentially adopt multiple stable equilibria corresponding tolocally optimal shapes that depend on the initialization of the prob-lem. Although for some boundary curves there is a clearly preferredshape that is more energetically stable than the rest, in other cases,several stable equilibria are valid solutions that might be practi-cally used in a feasible panelization. Furthermore, the maximumstress levels differ a lot between stable configurations. Multistabilityimposes two challenges for building a data-driven model of glasspanel mechanics. First, we do not know in advance the number oflocal minima that exist for a given boundary nor how energeticallystable these configurations are in practice; second, we do not knowhow to initialize the minimization problem to obtain such solutions.Both challenges motivated the use of a MDN as a regressor for theshape and corresponding stress of the glass panels. In Section 6, wedescribe our regression model and the methodology we followed toenrich the database by discovering new stable equilibria throughan iterative process. To estimate whether the panel is going to break, we compute themaximal engineering stress across all the elements of the discretiza-tion. We estimate the stress of an element by computing the firstPiola-Kirchhoff stress tensor P = FS . Here, F is the deformationgradient of the element, and S is the corresponding second Piola-Kirchhoff stress tensor. In a similar fashion to Pfaff et al. [2014],we compute the total stress of a panel using our estimation of thecombined bending and membrane strain introduced in Equation 1: S (cid:0) E ( ¯E , ˆE , z ) (cid:1) = λ Tr ( E ) I + µ E . (7)The maximal engineering stress is then evaluated as the maxi-mum absolute singular value of P across all elements. That is, for ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. each element, we compute S (cid:0) E ( ¯E , ˆE , ± h / ) (cid:1) , where the bending con-tribution to the stress is at its maximum, and pick the highest ab-solute singular value. The global maximal stress value is generallyat most C -continuous with respect to the panel boundary curves,which makes its direct usage in a continuous optimization undesired.Instead, we compute an L p -norm of maximal principal stress perelement. In practice, we found that p =
12 suffices. We denote theresulting value σ and refer to it as the “maximal stress” for brevity.Taking our assumptions, it is important to note that neither theoverall shape nor the maximal stress value changes for a given panelunder uniform scaling. This implies that only the ratio of the thinshell dimensions and the panel thickness matters. For simplicity,we choose 1 mm as our canonical thickness for the simulationsand scale the obtained results accordingly for every other targetthickness. We require a model that can efficiently predict the shape and stressof the minimum-energy panels for a given boundary. The simulationdescribed in Section 5 calculates these quantities, but is too slowto incorporate in an interactive design tool. Our data-driven modelaims to predict the deformed shape and corresponding maximumstress of the panels more efficiently. Moreover, we use it to calculatethe derivatives with respect to the input boundary, which is requiredfor gradient-based design optimization.Therefore, we use a statistical model that maps panel boundariesto the shapes and stresses of minimal-energy conforming cold bentglass surfaces. Section 6.1 describes the model and training process.The training requires a large dataset of boundaries and the resultingpanel shapes and stresses; in Section 6.2, we describe the space ofboundaries we sample from and how the shape optimization andstress computation of Section 5 is applied to them. To improve theresults further, we augment the dataset to better cover the regions ofthe input space where the predictions do not match the training databecause of multistability of the glass panels (Section 6.3) and retrainthe model on this enriched dataset. We will release our dataset andpretrained model publicly.
Our prediction model takes as an input a vector p ∈ R repre-senting a panel boundary. As noted in Section 5.3, several differentsurfaces may conform to a given boundary, corresponding to differ-ent local minima of the strain energy. Therefore, predicting a singleoutput yields poor results, typically the average over possible shapes.Instead, we use a mixture density network (MDN)—a neural networkmodel with an explicitly multi-modal output distribution [Bishop2006]. For a given boundary, each mode of this distribution shouldcorrespond to a different conforming surface.Whereas training a neural network to minimize the mean squarederror is equivalent to maximizing the data likelihood under a Gauss-ian output distribution, an MDN instead maximizes the likelihoodunder a Gaussian mixture model (GMM) parameterized by the net-work. Therefore, it must output the means and variances of a fixednumber K of mixture components, as well as a vector ˆ π of compo-nent probabilities. In the rest of the paper, all variables with hats denselayer-norm layer-norm p residual block is repeated 5× residualdense512 dense512 ζ ˆ ξ ˆ ζ ˆ ξ ˆ π ˆ Fig. 8. Architecture of our data-driven model. The input is a panel boundary p ; the model predicts means ˆ ζ k , variances ˆ ξ k , and component weights ˆ π for a two-component Gaussian mixture over the shape and stress of theminimal-energy surface. The numbers in dense layers indicate the numberof output units. denote predictions from our data-driven model, as opposed to valuesfrom the physical simulation. In our model, each component is a (12+ 1)-dimensional Gaussian with diagonal covariance, correspondingto the four interior control points of the shape, c ij ∈ R , i , j ∈ { , } ,and stress, σ , of one possible conforming surface. We denote themean of the concatenated shape and stress of the k th mixture com-ponent by ˆ ζ k and the variance by ˆ ξ k ; both are output by the neuralnetwork and hence depend on the input boundary p and networkweights w . We use a densely connected model withsix layers of 512 exponential linear units (ELU) [Clevert et al. 2015],with residual connections [He et al. 2016] and layer-normalization [Baet al. 2016] at each hidden layer (Figure 8). We trained tens of modelsusing combinations of these hyperparameters’ values and selectedthe one that performed the best on the held-out validation set. Insimulation, we observed that a given boundary could potentiallyadmit more than two stable states. However, these cases were ex-tremely rare; therefore, we set K =
2. This suffices for capturing thevast majority of stable states observed in our dataset, resulting in alow validation error. Hence, the output layer has 54 units, with noactivation for the means ˆ ζ k , exponential activation for the variances ˆ ξ k , and a softmax taken over the mixing probabilities ˆ π . The model is trained to minimize the negativelog-likelihood of a training set T under the GMM: L (T ; w ) = − (cid:213) ( p , ζ )∈T log K (cid:213) k = ˆ π k ( p ; w ) N (cid:16) ζ (cid:12)(cid:12) ˆ ζ k ( p ; w ) , ˆ ξ k ( p ; w ) (cid:17) (8) where ζ is a true output for panel p , i.e. the concatenation of shapeand stress from one simulation run, and N represents a diago-nal Gaussian density. We also add an L2 regularization term withstrength 10 − on the weights w , to discourage over-fitting.We use the stochastic gradient method Adam [Kingma and Ba2014] to minimize the above loss function with respect to the net-work weights w . We use a batch size of 2048, learning rate of 10 − ,and early stopping on a validation set with patience of 400 epochs.We select the best model in terms of the validation loss obtainedduring the training process. A single epoch takes approximately ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. omputational Design of Cold Bent Glass Façades • 208:9
30 seconds on a single NVIDIA Titan X graphics card, and in total,training takes around 20 hours.
For brevity, in the remainder of the paper, fora given panel boundary p and for a possible state k ∈ { , } , wewrite: • ˆ S k p : [ , ] → R for the Bézier surface patch that is definedby the boundary p and the predicted interior control nodesfrom the mean of the k th component (i.e., the leading 12elements of ˆ ζ k ). • ˆ σ k p for the stress value (i.e., the last element of ˆ ζ k ). • ˆ π k p for the k th component probability ˆ π k ( p ; w ) .Furthermore, we write ˆ S p and ˆ σ p (i.e., without the k superscript)to refer to the best prediction for boundary p , which is determinedby two factors:(1) if any of the component probabilities ˆ π k p is greater than95%, we discard the alternative and define the correspondingshape/stress prediction as best, or(2) otherwise, we consider both components as valid, and thebest one is determined depending on the application, eitheras the lower stress, the smoother shape, or the closer shapeto a reference surface.We discard components with ˆ π k p < .
05 since the modes with a near-zero probability imply a low level of confidence in the correspondingprediction.
To train our prediction model, we require a dataset of boundariesthat is representative of our target application. These are then pairedwith the shapes and stresses of the conforming surfaces with min-imal energy. Recall from Section 4 that a panel boundary may beparameterized invariantly to rigid transformations by corner pair-wise squared-distances d , edge-plane inclinations γ , and halfedgetangent directions θ . We generate boundaries by sampling theseparameters from the ranges and distributions described in Appen-dix A. Note that the physical model for the deformed shape andstress is invariant under the scaling of all geometric magnitudes;we choose our sampling ranges so that it would be possible to scalethe results to panel length-to-thickness ratios commonly used incold bent glass façades (e.g., 150–600). By applying the shape op-timization described in Section 5 to these boundaries, we obtainfine discrete meshes representing the deformed cold bent panels.We obtain a Bézier representation of such panels by keeping thesampled boundaries and fitting interior control points to matchthe simulated surface using the method described in Section 4.1.2.Plus, in our representation, any non-flat panel geometry can beequivalently represented in four alternative ways, depending onvertex indexing, and can be mirrored. Therefore, we transform eachsimulated panel into eight samples by permuting the vertex indexesand adding their mirror-symmetric representations.In total, we simulated approximately 1.5 million panels, whichcorresponds to 12 million samples after vertex permutations andadding mirror-symmetric panels. We reserved 10% of these samplesas a validation set for tuning the optimization hyperparameters and network architecture. To acquire such large amounts of datarequiring massive computations, we employed cloud computing. When a given boundary has multiple conforming panels, the physi-cal simulation returns only one of these determined by the twist-freeBézier patch initialization. Conversely, our data-driven model al-ways predicts K = π k , indicating it is unlikely to be a valid optimalpanel. We observed that after training, the model often predictsshapes for boundaries in its training set that differ from those re-turned by the simulation—however, re-simulating these boundarieswith a different initialization recovers a solution close to that pre-dicted by the data-driven model. This observation suggests a methodto extend the dataset with new samples to improve prediction error.Specifically, we use the prediction from the model as an initializa-tion for the simulator, which is then likely to converge to a stablesurface that was not reached from the default initialization. Theresulting surface can be added to the training set, so after retrain-ing, the model will give an even more accurate prediction in thesame region of parameter space. We apply the data-driven modelto every panel in the training set, and collect the predicted shapesˆ S k p , k ∈ { , } , where π k > ∼
15% of the original training set) for which this deviation isthe largest. For each such boundary, we re-run the simulation, usingthe predicted shape ˆ S k p as the initialization. Finally, we select allpanels which have at least 2 mm difference along any dimension ofany internal control node compared to the panel obtained originallyand add these to the training set. The resulting, enriched trainingset is used to retrain the model. In this section, we show how we arrive at a practical interactivedesign tool for freeform surface panelization using cold bent glasspanels. We aim to produce a tool compatible with the standarddesign workflow of an architectural designer. At every momentduring the editing process, the user gets immediate feedback onthe physical properties of the panelization (i.e., shape and stresspredictions for the panels). Upon request, an automated processrunning at interactive rates uses an optimization to “guide” thedesign. Figures 1 and 9 show two different doubly curved glasssurfaces that have been interactively designed from scratch usingour tool. Although it is generally desired to create designs free frombreaking panels, in a real project, one might like to assume thecost of hot-bending a small proportion of the panels. Therefore,there is a practical trade-off between the smoothness and aestheticsof a design and its manufacturability. We consider this option byexplicitly weighting various design criteria in the formulation ofour inverse design problem.
Depending on the specific application domain, the desired propertiesmight vary. This translates into the minimization of a composite
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. MPa > ba Fig. 9. An initial design includes panels exceeding stress limits (left, a). It is optimized for stress-reduction (left, b) and rendered (center). Right: a differentfaçade designed with our tool. target functional E : E = w σ E σ + w s E s + w f E f + w p E p + w c E c . (9)Overall, the total energy E depends on the vertex positions v ∈ V ,the edge plane vectors s e defining the plane Π e associated with e ,the tangent angles θ i , and some auxiliary variables associated withinequality constraints. Each weighted contribution to E representsa desired property of the final design, which we discuss in detail inthe following sections. E σ . The most important property is the manu-facturability of the final design. Failure in a specific type of glassis modeled by estimating the maximal stress present in the glasspanel and comparing it to the maximum allowed stress value.The MDN from Section 6 acts as a stress estimator. We constrainthe predicted stress value ˆ σ p for a given boundary p to be less thana stress bound σ max . We assign σ max to a value lower than the stressvalue at which failure occurs, taking into account a safety factor andthe estimator error. The inequality constraints ˆ σ p ≤ σ max are con-verted to equality constraints by introducing an auxiliary variable u p ∈ R per panel boundary p , and formulating the manufacturabilityenergy as E σ = (cid:213) p ( ˆ σ p − σ max + u p ) . (10)Figure 13 shows the effect of limiting the maximum stress of thedesign for a section of the façade of the Lilium Tower. E s . Here, we collect some terms in the final objec-tive function that aim in various ways to obtain as smooth as possiblepanelizations. As shown in Figure 11, this is essential for achiev-ing the stunning look of curved glass façades because it greatlyaffects the reflection pattern. The smoothness term is the sum oftwo individual functionals, i.e. E s = E + E . Kink angle smoothing E . It is generally not possible to get smoothconnections along the common boundary curves of panels, but wecan try to minimize the kink angle. For each pair of faces f i , f j sharing a common edge e , we consider their respective predictedpanels ˆ S i , ˆ S j and minimize the angle between their surface normals n i , n j evaluated at the parameter t = . E = . (cid:213) e ∈ E I ( − n i · n j ) , (11) MPa > a b Fig. 10. Optimization of the NHHQ skyscraper design (Zaha Hadid Archi-tects). We first optimize for smoothness of the overall design, and thenoptimize selected high stress areas for stress reduction. (a) Stress on panelscomputed on the original shape and panel layout. Red panels exceed thethreshold of MPa. (b) Stress on panels after optimization. The inset showsan area with clearly visible shape change. We decrease the number of panelsexceeding MPa from to . where E I is the set of interior edges of M , and 0 . S i , ˆ S j are shape predictions for the respective boundary curves of the twofaces f i , f j . Thus, optimization involves computing the Jacobian ofthe MDN output w.r.t. the input boundaries. Figure 12 shows theeffect of including the kink smoothing term in the design of theNHHQ façade. Fig. 11. Effect of optimization on visual smoothness. On the left, a selectionof cold bent panels computed on a given layout. On the right, the samepanels after optimization of the layout for the kink angle and bending stressreduction.
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. omputational Design of Cold Bent Glass Façades • 208:11 ◦ kink > ◦ a b Fig. 12. Comparison of the kink angle between panels in the NHHQ model,before (a) and after (b) running our design optimization algorithm. As aresult, the mean kink angle is lowered from . ◦ to . ◦ , while the maximumis reduced from . ◦ to . ◦ . Curve network smoothing E . Each edge in the dominant mesh poly-lines of M determines a cubic patch boundary curve, and the se-quence of these curves should also be as smooth as possible. At eachconnection of two edges, the corresponding tangents should agree,and thus, the inwards directed unit tangent vectors satisfy t i = − t j ,or equivalently t i · t j + =
0. This tangent continuity constraintexplains the first part in the smoothness term E = (cid:213) ( t i · t j + ) + (cid:213) (cid:2) s e · ( n i + n i + ) (cid:3) . (12)The second part concerns the planes Π e . We consider an edge e withendpoints v i , v i + . The discrete osculating plane at v i is spanned by ( v i − , v i , v i + ) and has a unit normal n i . Likewise, ( v i , v i + , v i + ) defines a discrete osculating plane with normal n i + at v i + . Wewant Π e to be the bisecting plane between these two, i.e. s e · ( n i + n i + ) =
0. Of course, the sums are taken over all occurrences of thedescribed situations.Finally, in practice, a few other parts are added to the smoothnessterm E s , which concern special cases. At combinatorially singularvertices of M , we constrain the tangent vectors to lie in a tangentplane. Plus, there are various symmetry considerations that are usedat the boundary, but those could easily be replaced by other termswith a similar effect. E f . So far, we have dealt with the smoothness ofthe panelization to a given mesh M . Because we also allow the mesh M to change during the design, we need to care about its fairness.This is done in the standard way using second-order differences ofconsecutive vertices along dominant mesh polylines, E f = (cid:213) ( v i − − v i + v i + ) . (13) E p . When designing a paneliza-tion for a given reference geometry, it is not sufficient to have themesh M . One will usually have a finer mesh M ref describing thereference geometry (Figure 4). To let M change but stay close tothe reference surface, we need a term that allows for the gliding of M along M ref . This is done in a familiar way: to let a vertex v i stayclose to M ref , we consider its closest point v ∗ i on M ref and the unitsurface normal n ∗ i at v ∗ i . In the next iteration, v i shall stay close to MPa > a b c Fig. 13. Optimization of the Lilium Tower (model by Zaha Hadid Architects)for different target properties. (a) Stress values for the initial panel layout.(b) Optimizing the design only for stress reduction and proximity to theoriginal design leads to more panels within the stress threshold, but alsoto a non-smooth curve network. (c) Allowing the design to deviate fromthe input and including fairness, produces a smoother result with reducedstress. Number of panels exceeding MPa is, respectively, , , and . the tangent plane at v ∗ i , which is expressed via E p = (cid:213) v i ∈ V (cid:2) ( v i − v ∗ i ) · n ∗ i (cid:3) . (14) E c . Since we want the neural net-work to produce reliable estimates, we need to ensure the panelboundary curves remain within the range used for training (Sec-tion 6.2). This is achieved as the sum of two constraint functionals E c = E + E . First, we constrain the tangent angles to | θ i | = ∠ ( t i , e i ) ≤ . ◦ for all angles θ i of halfedges e i with tangent vec-tors t i . We again convert the inequality constraints to equalityconstraints by introducing auxiliary variables u i , E = (cid:213) ( θ i − ( . ◦ ) + u i ) . (15)Second, we are working under the assumption that the vectors s e are unitary and orthogonal to their respective edges e , which resultsin E = (cid:213) e [( s e · e ) + ( s e − ) ] . (16) The minimization of E results in a nonlinear least-squares problemthat we solve using a standard Gauss-Newton method. The deriva-tives are computed analytically, and, since each distinct term of E has local support, the linear system to be solved at each iteration issparse. We employ Levenberg-Marquardt regularization and sparseCholesky factorization using the TAUCS library [Toledo 2003]. The edge plane vector s e of an edge e is ini-tialized so that Π e is the bisecting plane of two discrete osculatingplanes, as in the explanation for Equation (12). The angles θ i areinitialized so that they are at most 5 ◦ and so that the tangents lie asclose as possible to the estimated tangent planes of the referencegeometry. After initializing all other variables and computing anestimated stress value per face panel, the auxiliary variables areinitialized such that they add up to the inequality constraint boundor zero otherwise (i.e., the inequality constraint is not satisfied). Theshape S p of each panel is initialized with the MDN prediction usingthe initial boundary parameters p . In case there are two possible ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. shapes, we use the one that provides the best solution consideringapplication-dependent criteria (e.g., stress reduction). When lookingfor the smoothest fit, we pick the one minimizing (cid:205) ( − n i · n e ) ,i.e., a measure of angle deviation between each edge normal (sumof two adjacent face normals orthonormalized to e ) and the surfacenormal n i evaluated at the parameter t = . The weights associated with the targetfunctional E act as handles for the designer to guide the outputof the optimization toward the desired result. We do not opt for afixed weight configuration because the ideal balance is not uniquelydefined, but is instead governed by project-dependent factors suchas budget and design ambition.In all our experiments, we found it sufficient to assign the weightseither to zero or to the values 10 {− , − , } . Figure 13 shows one ex-ample of the different effects possible when changing the propertyimportance. In practice, and as a rule of thumb for a standard opti-mization where we prioritize stress reduction and smooth panels(in that order), we use weight values w σ = w s = w c = − , and w p = w f = − . We also reduce the fairness importance at the i -thoptimization iteration by scaling its weight by 0 . i . We experimentally validated our simulation results and design work-flow. For practical reasons, the experiments were done at a smallscale using borosilicate thin glass of about 180 ×
130 mm and0.35 mm thickness.For the validation of the simulation results (see Figure 15), highprecision frames were machined from cast aluminum. The glasspanel is pressed down on a 2 mm wide smooth support frame by adense array of stainless steel finger springs which are cushioned by0.5 mm polytetrafluoroethylene (PTFE). The support frame matchesa thin boundary strip of the simulated glass panel. To test the ac-curacy of the predicted shape, we selected and 3D-scanned a panelwith a high estimated maximal stress (98 MPa), which is beyond oursafety limit but still manufacturable (see Figure 15). The obtained Fig. 14. Realization of a doubly curved surface using 3x3 cold bent panels. surface was registered to the output of our shape optimization rou-tine, and we observed a worst-case deviation of 0.12 mm. Note thatwe registered an offset surface from the optimal mid-surface toaccount for the glass thickness.The frames for the design model created with our tool are illus-trated in Figure 14 and were built from laser cut and welded 1.2 mmthick stainless steel sheet metal. The glass, which is cushioned bytape, is pressed down on to the frame by L-shaped stainless steelfixtures spot welded to the frame. The presented design model isnegatively curved and consists of nine individual panels, each about200 ×
170 mm in size. The expected stress levels range from 20 to62 MPa. As predicted, all panels are fabricable and intact.During bending, our panels usually do not need to go throughmore extreme deformations than the final one, meaning that wedo not expect higher stresses while bending. Because normallypanels have a dominant bending direction, we observed that itis possible to “roll” the glass onto the frame accordingly duringassembly. Clamping the glass to the frame fixes the normals at theboundary, which makes one of the alternative shapes preferable. Our data-driven model (Section 6) must reproduce the output of thephysical simulation model efficiently and accurately. To evaluate itsaccuracy, we generate a test set of 10K panel boundaries and use thedata-driven model to predict the conforming surfaces. We consideronly admissible surfaces, i.e. those with a predicted probability of atleast 5%. The surface predictions are used to initialize our physicalshape optimization routine to obtain the true shape and stress valuesfor a comparison. In the resulting test set, the mean maximal stressvalue is 83 MPa, and the standard deviation is 52 MPa with 57% ofpanels having a maximal stress value above the threshold.We evaluate the shape prediction on panels with maximal stressbelow 65 MPa, which results in a mean average error (MAE) of ∼ ∼ L p -norm) exceedsthe 65 MPa threshold. From our test set, we obtained below 1% offalse negatives (when the model incorrectly predicts the panel isfeasible) and 15% of false positives. From a manufacturing point of view, the simplest solution to cladarchitectural surfaces is the use of planar panels. However, thissimplification sacrifices the visual smoothness of the surface. More-over, planar panels impose a restriction on the panels layout, andin negatively curved areas, there is often no other choice than tofollow the principal curvature directions of the surface. On the otherhand, a panelization with doubly curved panels is often prohibitivebecause of the high production cost of custom molds. Cold bentglass can be then a suitable solution. In Figure 3, we compare thevisual appearance of the smoothest possible panelization achievable
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. omputational Design of Cold Bent Glass Façades • 208:13 mm . Fig. 15. A doubly curved panel of a thickness of 0.35 mm with off-plane corner deviation of 6.9 mm. A white coating has been applied to it for 3D-scanning.Right: deviation from the simulation by at most 0.12 mm. with planar panels with a cold bent one, while in Figure 16, we showa panelization layout that is mostly feasible with cold bent glass,but not with planar panels. In the following, we illustrate how userscan employ our workflow for architectural panelization and design.
In this case, the input is a quadrilateralmesh that encodes both the design shape and the panel layout.Once the edges of the input mesh are smoothed via cubic Béziercurves, we can predict the panels’ shapes and stresses. Those panelsexceeding the failure criterion shall be realized with custom molds.At this point, the user can optimize the shape for the reduction ofstress and kinks between panels, and tune the weights describedin Section 7.2 for choosing an appropriate compromise betweenfidelity with the original shape, number of custom molds needed,and visual smoothness (see Figures 10, 12, and 13). To show cold bentglass’ capabilities in façade panelization, we tested this workflowon the challenging NHHQ and Lilium Tower models by Zaha HadidArchitects, which were never realized. The initial quadrangulationsfor these two designs are planar quad meshes which were createdby the architect. The results are shown in Figure 17.
Besides the panelization of a given shape, ourworkflow is very well suited as an interactive design tool. In this case,the user can interactively modify the quad mesh that representsthe panel layout and gets immediate feedback on which panelscan be produced with cold bent glass, while exploring different MPa > a b Fig. 16. Bent glass capabilities. (a) A quadrilateral mesh where the red facesexceed a deviation of a planarity of . (measured as the distance betweenthe diagonals divided by average edge length) and, therefore, not suitablefor a flat glass panelization. (b) A cold bent panelization with correspondingface stresses. The stress values for the six central panels have been computedvia simulation because they were outside the MDN input domain. Accordingto a stress limit of MPa, most of the panels optimized are feasible. Theresulting cold bent panelization is shown in Figure 17. designs. Initial values are computed as in 7.2.1 and we use the meshvertex normals for the estimated tangent planes at the vertices.The estimation times are compatible with an interactive designsession. Once the user is satisfied with a first approximate result, thepanelization can be further optimized, as described in Section 8.3.1,to improve the smoothness and reduce panel stresses. In this step,we can further reduce the number of panels that are not feasible forcold bending. Figures 1, 9, and 16 show some sample architecturesdesigned with this procedure. Furthermore, the accompanying videodemonstrates the interactive feedback capabilities of our systemfor efficient form finding and exploration of the constrained designspace while keeping the designer in the loop.All interactive design sessions were performed on an Intel®Core™ i7-6700HQ CPU at 2 .
60 GHz and NVIDIA GeForce GTX 960M.The MDN is implemented in TensorFlow 2.1 and is run on the GPU.For 1K panels, the prediction time is 0 . ∼ elements. Note that this routine is not fullyoptimized for speed because it is not required during the interactivephase but mainly used for acquiring training data. We have introduced an interactive, data-driven approach for material-aware form finding of cold bent glass façades. It can be seamlesslyintegrated into a typical architectural design pipeline, allows non-expert users to interactively edit a parametric surface while pro-viding real-time feedback on the deformed shape and maximumstress of cold bent glass panels, and it can automatically optimizefaçades for fairness criteria while maximal stresses are kept withinglass limits. Our method is based on a deep neural network archi-tecture and multi-modal regression model. By coupling geometricdesign and fabrication-aware design, we believe our system willprovide a novel and practical workflow, allowing to efficiently find acompromise between economic, aesthetic, and engineering aspects.Identifying such a compromise usually involves multiple compet-ing design goals. Although we have demonstrated the applicabilityof our system for several design criteria, it would be interestingto extend the design workflow by adding capabilities, for example,
ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020.
Fig. 17. Dominant cold bent glass realizations of the NHHQ model (left). The Lilium Tower (center) after optimization for smoothness and stress reduction.The surface from Figure 16 as an architectural design (right). Panels exceeding the maximum stress (check Figures 10, 13, 16) are realized with hot bending. for strictly local edits, marking some panels as a priori hot bentor specifying kink edges. Because of our differentiable network ar-chitecture, in theory, it should be trivial to incorporate additionalcriteria into our optimization target functional or even employ adifferent numerical optimization algorithm if desired.Similar to all data-driven techniques, we should only expect ac-curate predictions from our network if similar training data wasavailable. Surprisingly, we noted that we were able to discover stablestates that we initially did not find with the traditional optimizationapproach, and used these to enrich our database. Each boundaryin the training set is associated with a single stable surface outputby the simulator. Nevertheless, the network may correctly predictthe existence of a second stable state for that boundary, because itspredictions implicitly incorporate information from similar panelsin the training set, where the second state is seen. However, we can-not guarantee that our database contains all relevant stable statesand that all of them will be predicted. Identifying all stable statesand optimally sampling the database using this information wouldbe an interesting avenue for future work. For fabrication, in ourexperiments, reproducing the desired particular state was trivialand emerged when intuitively attaching the glass to the frame.In the presence of more than one potential state, our system cur-rently selects in each iteration per panel the state that best fits ourapplication-dependent criteria. An alternative would be to computea global, combinatorial optimal solution among all potential states.However, because of the combinatorial complexity, this would resultin a much harder and probably computationally intractable opti-mization problem. We also considered solving the combinatorialproblem by using a continuous relaxation but ultimately did not findevidence in our experiments that would indicate the need for suchan approach as we observed stable convergence to satisfactory re-sults. However, identifying the global minimum would neverthelessbe an interesting research challenge.By design, our workflow is not limited to a particular shell model,and in theory, more advanced and extensively experimentally vali-dated engineering models could be used if needed. We believe thebenefit of our learning-based approach would even be more evidentwith more complex mechanical models, because they are computa-tionally significantly more expensive. Our workflow could serve asan inspiration for many other material-aware design problems. Forfuture work, it would be exciting to explore extensions to different materials, for instance metal, wood, or programmable matter thatcan respond to external stimuli, such as shape memory polymers orthermo-reactive materials.
ACKNOWLEDGMENTS
We thank IST Austria’s Scientific Computing team for their support,Corinna Datsiou and Sophie Pennetier for their expert input on thepractical applications of cold bent glass, and Zaha Hadid Architectsand Waagner Biro for providing the architectural datasets. Photo ofFondation Louis Vuitton by Francisco Anzola / CC BY 2.0 / cropped.Photo of Opus by Danica O. Kus.This project has received funding from the European Union’sHorizon 2020 research and innovation program under grant agree-ment No 675789 - Algebraic Representations in Computer-Aided De-sign for complEx Shapes (ARCADES), from the European ResearchCouncil (ERC) under grant agreement No 715767 - MATERIALIZ-ABLE: Intelligent fabrication-oriented Computational Design andModeling, and SFB-Transregio “Discretization in Geometry and Dy-namics” through grant I 2978 of the Austrian Science Fund (FWF). F.Rist and K. Gavriil have been partially supported by KAUST baselinefunding.
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A SAMPLING PANEL BOUNDARIES
We briefly describe how the panel boundaries forming the trainingset for our data-driven model are sampled (Section 6.2). We param-eterize panel boundaries invariantly to rigid transformations, bycorner pairwise squared distances d , edge-plane inclinations γ , andhalfedge tangent directions θ (Section 4). In order to sample d such ACM Trans. Graph., Vol. 39, No. 6, Article 208. Publication date: December 2020. that it represents a valid quad, we start with two adjacent edgelengths l , l , an angle α between them, and a displacement a of theremaining vertex from the point that would form a parallelogram.We sample each of these parameters as follows: • l , l ∼ Uniform [ . , . ] ; this corresponds to 15–60 cm fora 1 mm thick panel, • α ∼ Uniform [ ◦ , ◦ ] , • a is given by sampling a point on the unit sphere, then scalingit by a factor drawn from Uniform [ , min { l , l }/ ] , • γ i ∼ Uniform [− ◦ , ◦ ] , • θ i is given by arccos of a value sampled from Uniform [ cos 5 ◦ , ] ,negated with probability 1 /
2, so θ i ∈ [− ◦ , ◦ ] .Note that our model for the deformed shape and stress is invariantunder scaling of all geometric magnitudes. Our sampling rangesare chosen to allow scaling the results to thickness/curvature ratioscommonly used in cold bent glass façades..Note that our model for the deformed shape and stress is invariantunder scaling of all geometric magnitudes. Our sampling rangesare chosen to allow scaling the results to thickness/curvature ratioscommonly used in cold bent glass façades.