Design Modelling Symposium Copenhagen 2015

From 3 - point-constellations to selforganizingly folded/bent spatial configurations Günther H. Filz, Stefan Kainzwaldner, University of Innsbruck, Faculty of Architecture, Institute of Design | unit koge. Structure & Design, Innsbruck, Austria. SUMMARY. Our investigation is based on a seemingly simple phenomenon that we all know from kinking sheets of paper. In this sense it is related to selforganizing processes and forms. The controlled elastic deformation of thin, planar sheet elements into 3dimensional objects describe above mentioned phenomenon by predefining 3 points (MAB) on the surface with M fixed in space and identified shape characteristics of the short selforganizingly curved ridge-line, which is smoothly fading to an elastically bent surface. This paper presents the geometrical and kinematic simulation of controlled elastic deformation of planar sheet elements of any shape into selforganizingly folded/bent spatial configurations by predefining any 3-point-constellation (M,A,B) with A,B equidistant and symmetrical to M (Fig.1). The geometrical description of spatial configurations in all transition phases and its kinematic simulation is based on a microscopic analysis of physical models. From this we receive a line-pattern, which can be applied to the sheet-element as a basis for simulating spatial movements of surface-points. At present simulation and physical models show maximum 5% deviation only. Several tests and various 3-point-constellations verified that our simulation can be a powerful and reliable tool in exploring design freedoms as well as limits, which will be extended to more complex arrangements.

Keywords: simulation, (active) bending, selforganized crease curve, folding, kinematic, architectural geometry, form finding.

Figure 1: Visualization of six simulated, additively arranged sheet elements in fully closed state

1

Design Modelling Symposium Copenhagen 2015

CONTEXT. Our investigation is based on a seemingly simple phenomenon that we all know from kinking sheets of paper. As illustrated in Figure 2, a characteristic curved ridge-line selforganizingly arises when a slightly vaulted sheet of paper kinks in perpendicular to the vaultdirection. With regard to convertibility, minimized material use, structural performance and architectural qualities its basic geometric and structural dependencies as well as formal, kinematic and bending active phenomena can be of great interest for architecture and engineering (Lienhard 2011). The authors described the phenomenon as controlled elastic deformation of thin, planar sheet elements into 3dimensional objects by predefining 3 points (MAB) on the surface with M fixed in space and identified shape characteristics of the short selforganizingly curved ridge-line, which is smoothly fading to an elastically bent surface (Fig.3) (Filz; Kainzwaldner, 2014).

Figure 2: Simulation of selforganizingly folded/bent spatial configurations by predefining any 3-point-constellation - inclination angle αI in M in the range of 0° to maximal 152.66° for 1,0mm copolyester and angle αMAB = 90°

General geometric specifications can be summarized as follows (Filz; Kainzwaldner, 2014):  The basic set-up is described by a 3-point-constellation (M,A,B) with A,B equidistant and symmetrical to M, which is fixed in space.  Kinematic and spatial response of the elastic sheet - change of inclination angle αI in M (Fig.2, Fig.4) in non-linear dependence of transition from straight line to a sinus-wave-like, meandering and increasingly spatial curve AB – is given due to counterdirectional in-plane-movement of A and B. There is need for actuating, but not shape defining impulses, which are triggering controlled elastic deformation.  The comparison of short selforganizingly curved ridge-line shows no perfect match with any other mathematically defined curve. (Nettelbladt 2013; Peternell 2009; Pottmann 2010)  The surface curvature analysis provides a constant relation of convex (87%) and concave surface area (13%). Simultaneously the surface curvature analysis can be read as equivalent to a stress-diagram. In general, selforganizing processes and forms contain physical properties. So specific “architected” designs follow precise rules in the genesis of form - a coalesce consideration of geometrical aspects and structural performance. Finding these rules represents the common basis for tools such as simulation, which is presented with this paper and feeds back designs in meaningful and productive ways – the designer can rely on a certain assurance. Based on above specified findings this paper presents the geometrical and kinematic simulation of controlled elastic deformation of planar sheet elements of any shape into selforganizingly folded/bent spatial configurations by predefining any 3-point-constellation (M,A,B) with A,B equidistant and symmetrical to M, as only point fixed in space. Presented simulation precisely describes any surface point and its spatial movement and therefore a continuously variable transition from plane sheet element to 3dimmensional object in real time. In this context it is possible to define any shape and size of a planar element as well as the geometrical position and orientation of 3-point-constellation (M,A,B) at will. New geometrical insights regarding these 3 point constellations at all phases from plane to object show that A respectively B are part of straight lined, symmetrical legs starting in M. This simplified the evaluation of different arrangements and their digital simulation. Since M is the only spatially fixed surface point, the planar element can have any orientation in space. Thus, we are able to broadly explore freedoms and limits of these forms within the digital environment (Filz; Kainzwaldner, 2015).

PHYSICAL EXPERIMENTS AND RESULTING CONSTRAINTS FOR GEOMETRICAL SETUP. The selforganizingly curved ridge-line is a result of mainly two parameters, firstly a result of physical properties of used material with material thickness to be the most determining factor and secondly a result of applied angle αMAB between MA and MB (Fig.3, Fig.4). 2

Design Modelling Symposium Copenhagen 2015

Above described phenomenon was tested by numerous physical experiments by means of elastically deformable materials including paper, respectively cardboard, veneer wood, aluminum and copolyester. The size of physical models range from few centimeters to a realized structure of 4 x 4 meters shown in Figure 5. This full-scale object was built in aluminum with a material thickness of 2 millimeters. For this reason investigated material thicknesses mainly range from 0.5 to 1.5 millimeters and focus on isotropic material such as copolyester with a Young`s Modulus of 2.020 MPa according to ISO 527-2/1B/1. Self-weight has to be taken into account under the aspect of active bending, which is defined as follows: “Active-bending is applied intentional … the same straight/flat elements can be used for different curvatures. Actively-bent elements are defined as elements that are transformed from the stress-free start-geometry to their end-geometry by elastic bending.” (Lienhard et.al. 2015). Within a predefined material thickness executed physical experiments show matching forms in all stages. In this sense, physical experiments show that 3-point-constellations M,A,B with a maximum angle αMAB between MA and MB of 135°and its minimum in dependence from material thickness – e.g. 63° for 0.5mm copolyester – result in a selforganizingly curved ridge-line and therefore the objects desired kinematic effect. Other values for αMAB have theoretical meaning only. The meaningful range for αMAB is illustrated in Figure 4 center and bottom.

Figure 3: Geometrical setup of predefined 3 point constellation (A,B,M) generating a selforganized, curved “ridge-line”, smoothly fading to an elastically bent surface

Figure 4: top: Angle of inclination αI center: Angle αMAB between MA and MB bottom: Range of possible angles αMAB in dependence from material thickness

In practice angles for αMAB smaller than minima shown in Figure 4 (bottom) usually result in more than one kink in M, angles bigger than 135° cause uncontrollable lateral spreading of curved ridge-line. So 72 meaningful physical experiments resulted from circular copolyester sheet elements, with a diameter of 45cm and a material thickness of 0.5, 1.0 and 1.5mm and αMAB with 135°, 90°and in dependence from material thickness minimum αMAB of 63°, 72° and 81° (Fig. 4 bottom).

Physical experiments also define following constraints for the geometrical setup of M,A,B (Filz; Kainzwaldner, 2015):    

M,A,B must be surface points of predefined planar sheet element M,A,B must not have identical position on the sheet element Minimum distance of M to A,B is mainly dependent on material thickness A,B must not be part of selforganizingly curved ridge-line

3

Design Modelling Symposium Copenhagen 2015

Figure 5: “Serles Curl”, realized structure in aluminum, 4 x 4 meters with a material thickness of 2 millimeters.

4

Design Modelling Symposium Copenhagen 2015

3D SCANS AS REFERENCES.

Figure 6: 3D scans by Kinect during the kinematic transition-phases

Above mentioned, meaningful physical models were 3D scanned by the means of Kinect (Fig.6), which originally is a hardware motion sensing input device by Microsoft video game consoles (Filz; Kainzwaldner, 2014). Each scanned surface is represented by a mesh, consisting of 224,500 faces in average. This equates to a resolution of about 30.75 dpi respectively a point-cloud with 0.8mm distance between single points. Basically Kinect builds on software technology by using an infrared projector and camera. So this 3D scanner system employs a variant of image-based 3D-reconstruction. Due to this fact steep angles between Kinect and Surface to be scanned impede precise surveying (Fig.7). Microscribe-3D-Scan provides single points. This is suitable for edge-curves and foldline but less practical for surfaces, which have to be remeshed from pointclouds. Therefore only selected surface-points as well as edge curves were scanned by means of Microscribe-3D scanner in order to define boundaries with higher precision and for control purposes.

Figure 7: Steep angles impede precise surveying – comparison of scans coupling Kinect and Microscribe-3D-Scan

5

Design Modelling Symposium Copenhagen 2015

CURVATURE ANALYSIS. The geometrical description of spatial configurations in all transition phases and its kinematic simulation is based on a microscopic analysis of physical models (Fig.8). Configurations of e.g. copolyester plates are showing a visual change from transparent to milky in the region of the selforganizingly curved ridge-line eccentrically passing with M due to plastic deformation of material, which occurs at an angle of inclination αI of about 55°. The microscopic analysis shows formations of microcracks with clear orientation, according to local stress peaks. So formations of microcracks match with surface curvature analysis of largest local radii of curvature (Fig.9, Fig.10) and stress diagram. From this we receive a line-pattern (Fig.11), which can be applied to the sheet-element as a basis for simulating spatial movements of surface-points.

Figure 8: Formations of microcracks in the region of the selforganizing curved ridge-line in physical model made of copolyester, 0.5mm

Figure 9: Surface curvature analysis of largest local radii on 3D scan

6

Figure 10: Planarized line pattern of the 3D surface curvature analysis

Figure 11: Application of the curvature analysis resulting in a line pattern to simulate spatial movements of surface-points

Design Modelling Symposium Copenhagen 2015

SIMULATION OF GEOMETRY. Since investigated phenomenon is resulting from a selforganized process, the access to its simulation by Origami-techniques or by means of softfolding-software, which operate with predefined patterns, was not appropriate (Zhu et.al. 2013). The developed numerical simulation can be executed by means of Grasshopper (Fig.12), whereas physical models, respectively the comparison and evaluation of 3D scans during all kinematic transition-phases served as references. Surface analysis and line pattern resulting from microcrack formations can be seen as basis for found geometrical solution in order to setup numerical simulation (Fig.11). According to circular elements in physical models, a predefined number of points (here 160 points) is evenly distributed along a circle in our digital simulation. This is done in plane state. More points increase precision but simultaneously increase processing time. The characteristic curved ridge-line is implemented and tangentially extended to the boundary circle. These tangents divide points along the circle into two groups, points on the convex side (usually larger number of points) and points on the concave side of curved ridge-line. The convex respectively concave side of curved ridge-line is subdivided accordingly to the number of points of each group. From this procedure we get a line pattern, shown in Figure 11. Between boundary circle and a circle, which is bigger than the curved ridge-line, a predefined number of circles (in our case 6 circles) are concentrically arranged. This pattern is transferred to all kinematic transition-phases of the element and mapped onto scanned surfaces (Fig.12left). In the area of the smallest circle, which is about 9% of the total area content of investigated physical models, data from 3D scans was unconsidered but described pattern is interpolated by the means of geometrical constraints like real length and directions, known from plane state. There are two reasons for this, firstly due to steep angles 3D scans are quite imprecise for this part of the element (Fig.7), secondly physical models show a small area with anticlastic curvature on the concave side of the curved ridge-line driving from plastic deformation. This anticlastic area covers less than 1% from area content of examined physical models. Like in physical model, the planar sheet element can be brought into any selforganizingly folded/bent spatial configuration with an angle αI between 0° and in dependence of αMAB and material thickness up to 174.19° (Fig.2, Fig.4). Since A respectively B are part of straight lined, symmetrical legs starting in M with a predefined αMAB, they can be equidistantly shifted, which is advantageous for a later physical realization. Since described geometrical dependencies and principles are identical for numerical simulation, the definition of 3-point-constellations in terms of position and orientation as well as the geometrical outline of used sheet material can be chosen by the user and offers a broad variety of constellations and freedom in design. Constraints in design might occur by possible collision/intersection of two or more sheet elements. For these cases, our simulation tool provides several solutions, which are not further described in this paper.

Figure 12: (Non-radial) Distribution of line pattern between circle and curved ridge-line transferred to all kinematic transition-phases of the element and mapped onto scanned surfaces by the means of Grasshopper

7

Design Modelling Symposium Copenhagen 2015

SIMULATION TOOL. Basically our numerical simulation tool consists of 5 components, which can be executed by the means of Grasshopper (Fig. 13-18). Component 1 contains the data for the simulation. Component 2 represents the actual simulation. The other components allow analyses and further elaborations of the model.

Figure 13: Component 1 Data

Figure 14: Component 2 Simulation

Figure 15: Component 3 – Sort Geo

Figure 16: Component 4 – Mesh Intersection

Figure 17: Component 5 – Cut Mesh

In general Component 1 - Data (Fig.13) contains all refined data from scans and material data for simulation. This component is separated from other components to make general data available for simulation of multiple elements, instead of feeding this information each time, which would increase processing time of total simulation and memory capacity. Component 2 – Simulation (Fig.14) allows for smooth simulation from plane state to 3D object up to defined maximum angle of inclination αI and user definable angle αMAB as well as continuously variable adjustment of material thickness in the range of 0.5mm to 1.5mm. Possible settings for Component 2 – Simulation are Component 1 – Data, M_Motion, describing non-linear motion of point-cloud from plane state to 3D point-cloud, T_Thickness, offering continuously variable adjustment of material thickness, A_Angle, setting user definable angle αMAB, F.Pt_Force Point, defining position and orientation of A and B and their spatial path during motion, Rot_Rotation, defining orientation of curved ridge-line in plane state, New F, considering any user definable outline of sheet element, which is different from circular default setup, and Pl_Plane, defining the position of element regarding World XY-plane. Component 2 – Simulation output provides I_Info, information on selected αMAB, selected material thickness and angle of inclination αI, Geo_Geometry, geometrical information as described below, M Pt_Mesh Points, resulting point-cloud in order to generate and illustrate the resulting surface, U_U-direction, number of points in U-direction and V_V direction, number of points in V-direction (Fig.14). Component 3 – Sort Geo (Fig.15) is fed in by Geo_Geometry and provides 2D output for later fabrication. Ri_ridge displays curved ridge-line in plane state, LMA/LMB 2D_Leg MA/MB defines leg MA respectively MB and AB 2D_Points AB 2D defines position of point A respectively point B in plane state. Component 4 – Mesh Intersection (Fig.16, 17, 18) is a useful component for the simulation of more than one sheet elements and the detection of possible intersections respectively collisions in realized structures. This component considers parallel, serial and all possible intermediary motion sequences of elements regarding possible intersections/collisions.

8

Design Modelling Symposium Copenhagen 2015

Figure 18: Analysis, Feedback and visualization of mesh intersections

Together with Component 5 – Cut Mesh (Fig.17) it is not only meant to be useful in terms of collision detection but also in terms of a creative design process, where elements can be used to “tailor” their neighbor elements. So there are basically three options how to deal with intersections, firstly to search for serial, non-intersecting motion sequences of elements, secondly to trim at a user defined position, which can either be the starting or end-position of elements and thirdly to select elements, which are monitored within a range of αI,and which are trimmed according to their motion sequence. In this case all select elements (M1_Mesh 1, M2_Mesh 2) are fed into Component 5 – Cut Mesh, where all spatial positions are continuously variable simulated within selected range of αI. Before starting this process it is possible to define that the elements equally trim each other or that there is a hierarchy between elements (Trim; 0 = both elements, 1 = M1 gets trimmed, 2 = M2 gets trimmed). Component 5 – Cut Mesh provides output i_info, informing the user in case that trimmed lines or boundary line of elements are intersecting with curved ridge-line, M1_Mesh 1, trimmed geometry of mesh 1, Crv 1_curve 1, generated new boundary line of element 1 and/or M2_Mesh 2, trimmed geometry of mesh 2, Crv 2_curve 2, generated new boundary line of element 2.

9

Design Modelling Symposium Copenhagen 2015

Figure 19: Simulation of additively arranged sheet elements from planar geometry through continuously variable adjusted states to fully closed 3Dstate

Figure 20: Simulation of surface, randomly subdivided into sheet elements from planar geometry through continuously variable adjusted states to fully closed 3D-state.

CONCLUSION AND PERSPECTIVE. At present simulation and physical models show a maximum deviation of 5%. This can mainly be explained by imprecision like asymmetry caused by the fabrication process of the physical model and difficult determination of boundary lines from 3D scans. Within the digital environment, our simulation delivers a maximum deviation of less than 3% in surface area and less than 2% in metrical length of boundary lines between plane state and fully closed 3D state. Like in physical model, the planar sheet element can be brought into any selforganizingly folded/bent spatial configuration with an angle αI between 0° and in dependence of αMAB and material thickness up to 174.19°. So the geometrical precision of our simulation tool can be rated as sufficient and reliable. 10

Design Modelling Symposium Copenhagen 2015

At present this is proved for arrangements, respectively naked edges of sheet elements forming a common pattern in plane and arrangements with naked edges of sheet elements forming a common 3dimensional pattern. In conclusion several tests and various 3point-constellations in terms of position and orientation on planar sheet elements of various shapes and their transition into selforganizingly folded/bent spatial configurations verified that our simulation can be a powerful and reliable tool in exploring design freedoms as well as limits. Ongoing investigations focus on the wealth of possible forms that can be simulated and accordingly fabricated (Fig.19, 20). Therefore, we are looking into constellations and patterns of sheet elements regarding their intersections. Thus, we will develop strategies in order to avoid or using these intersections in the creative design process. Arranging more than one 3-point-constellation on one sheet element is another option to increase formal variety (Filz; Kainzwaldner, 2015). In this context our investigations seem to have close relation to principles of rigid Origami folding (Demaine 2011, pp. 39-50; Jackson 2011). These investigations will be accompanied by architectural design studies and applications to highlight functional, spatial and structural qualities of generated forms and configurations.

REFERENCES. Demaine E, Demaine M, Koschitz D (2011) Reconstructing David Huffman’s Legacy in Curved-Crease Folding. In: Yim, M. 2011: Origami 5, Fifth International Meeting of Origami Science, Mathematics, and Education. A K Peters/CRC Press Filz G-H, Kainzwaldner S (2014): Between Bending and Folding - buckling from plane to object by 3 predefined points. Paper presented at the VI. Latin-American Symposium on Tension Structures (IASS-SLTE). Shells, Membranes and Spatial Structures: 'Footprints', Brasilia, 2014 Filz G-H, Kainzwaldner S (2015) Freedoms & limits in the arrangement of 3 point constellation on planar sheet elements, generating spatial, selforganized folded/bent forms. Paper presented at the International Association for Shell and Spatial Structures (IASS), Amsterdam, 2015 (Abstract accepted, Full Paper in review) Fuchs D, Tabachnikov S (2007) Mathematical Omnibus. American Mathematical Society, USA, pp.185-197, 207-215 Jackson P (2011) Folding techniques for designers – form sheet to form. Laurence King Publishing Ltd, London, pp. 176-187 Kilian M, Flöry S, Chen Z, Mitra J-N, Sheffer A, Pottmann H (2008) Curved folding. Proc. ACM Transactions on Graphics Leopoldseder S, Pottmann H (2003) Approximation of Developable Surfaces with Cone Spline Surfaces. Institut für Geometrie, Wien Lienhard J, Schleicher S, Knippers J (2011) Bending-active structures – research pavilion ICD/ITKE. Paper presented at the International Symposium of the IABSE-IASS Symposium, Taller Longer Lighter, London, 2011 Lienhard J, Alpermann H, Gengnagel Ch; Knippers J (2012) Active Bending, a review on Structures where Bending is used as a Selfformation Process. In: Kim, Seung Deog: From Spatial Structures to Space Structures, IASS-APCS 2012. Abstract Book. Seoul: Korean Association for Spatial Structures Nettelbladt M (2013) The geometry of bending. Mårten Nettelbladt, Stockholm, pp. 9-11,17 Peternell M (2009) Unterteilungskurven und – flächen. Proc. Fortbildungstagung für Geometrie, Wien, 2009 Pottmann H, Asperl A, Hofer M, Kilian A (2010) Architekturgeometrie. Wien, Springer-Verlag, pp. 221, 222, 247-257 Tirapegui E, Martinez J, Tiemann R (2000) Insatbilities and Nonequilibrium Structures VI. Dordrecht: Kluwer Academic Publishers, pp. 269-274 Zhu L, Igarashi T, Mitani J (2013) Soft folding; Computer Graphics Forum, Volume 32, Number 7, Proc. of The 21st Pacific Conference on Computer Graphics and Applications. Singapore: Pacific Graphics

11